morphology, dynamics and forecasting power of …

WARSAW SCHOOL OF ECONOMICSCOLLEGIUM OF ECONOMIC ANALYSIS

MORPHOLOGY, DYNAMICS AND

FORECASTING POWER OF THE POLISH

GOVERNMENT YIELD CURVE

Marcin Dec

A doctoral thesis submitted for the degree of Doctor of PhilosophySupervisor: dr hab. Marcin Kolasa

Warsaw, 28th of December, 2020

SZKO LA G LOWNA HANDLOWA W WARSZAWIEKOLEGIUM ANALIZ EKONOMICZNYCH

MORFOLOGIA, DYNAMIKA I SI LA

PREDYKCYJNA POLSKIEJ RZADOWEJ

KRZYWEJ DOCHODOWOSCI

Marcin Dec

Praca doktorska napisana pod kierunkiem naukowym:dr hab. Marcina Kolasy

Warszawa, 28 grudnia 2020

ABSTRACT

The main objective of this empirical and quantitative study is to understand what expectationsof the future interest rates evolution are confined in the government yield curves of less liquid mar-kets as in the case of Poland and to explore methods to improve their forecasting power. We offernovel insights on how the risk-free interest rates term structure can be obtained from the govern-ment bonds yields as well as we offer solutions to the following issues: (i) generally shorter recordedhistory of less liquid markets compared to advanced economies, (ii) extreme sensitivity of risk pre-mium inference due to the issues with estimation of the short end of the yield curve, (iii) insufficientdiversity of maturities of available bonds, (iv) insufficiently precise price quotes for many of theoff-the-run securities.

In the yield curve estimations we use classic Nelson-Siegel-Svensson parsimonious yield curvefunctional form. What is unique in our approach is a newly invented, stylised facts based, systemof weights. Having tested different systems and having ranked them in the space of goodness-of-fitand smoothness we confirmed that there exists a class of weights that systematically gives betterresults than the classic approach of all equal weights. It proves that inferring from the liquidity inparticular maturities raises the information content and quality of yield curve estimation, which isone of the hypothesis of this enquiry.

On the basis of these unique, high quality, estimated time series of the yield curves, we havefurther decomposed them into the risk-free layer and the term premia structure using the methodof three stepped linear regressions proposed by Adrian, Crump and Moench. Contrary to theseauthors’ findings from the US market, in estimations for Poland the factor responsible for level riskwas not priced on a statistically significant level. The main reason for that is non-stationarity ofthis factor due to much shorter available history for Polish bonds than for the US-Treasuries. Polishterm premia proved to be countercyclical and positively correlated with market uncertainty mea-sures, also to be, on average, upward sloping, and significantly different from zero for medium andlonger term bonds (staring from 3-year tenor). The level of risk premia is sensitive to choices ofthe beginning and end dates of the sample, but we offered an ameliorating algorithm to this problem.

In this thesis we verify number of hypotheses. The Pure Expectations Hypothesis (PEH) doesnot hold universally for Polish government bonds yield curve and contrary to the research conductedon the US markets where PEH is almost always rejected, we have found that for Poland there is,although limited, domain (in a space spanned by different investment horizons and selected bondtenors) where PEH cannot be ruled out. Moreover, we showed that the Expectations Hypothesis(EH) holds in Poland for almost all choices of long and short term investments, and that existenceof term premia structure explains the differences between compounded rates of returns from shorterinvestments and corresponding maturity longer term zero-coupon bond’s yield. Finally, we havefound evidence that, unlike in the case of liquid markets, professional forecasters expectations donot help to increase the informational content of the yield curves in the case of Poland.

STRESZCZENIE

Contents

Introduction 1

1 Chapter 1. Focused literature review 61.1 Building blocks: interest rates and yield curves . . . . . . . . . . . . . . . . . . . . . 61.2 Yield curve estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Risk premia estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Bond market liquidity measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5 Pure Expectations Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . 271.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Chapter 2. Polish government yield curve fitting 312.1 Characteristics of Polish government bonds market . . . . . . . . . . . . . . . . . . . 312.2 Data selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Stylised facts on Polish bonds (and their impact on filtering) . . . . . . . . . . . . . 362.4 Filtering rules and weight system framework . . . . . . . . . . . . . . . . . . . . . . . 442.5 Polish yield curve estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.6 Estimation results for the best weight system . . . . . . . . . . . . . . . . . . . . . . 532.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Chapter 3. Expectations Hypothesis domain in Poland 593.1 Excess return on term premium regressions . . . . . . . . . . . . . . . . . . . . . . . 603.2 Realized change in the spot rate on term premium regressions . . . . . . . . . . . . . 613.3 One year excess return on average one year forward rates regressions . . . . . . . . . 633.4 Rolling realised returns on term premia regressions . . . . . . . . . . . . . . . . . . . 663.5 Realised spread on pro rata temporis current spread regressions . . . . . . . . . . . . 683.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Chapter 4. Extracting Term Premia from Polish yield curves 734.1 Establishing state space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Term premia in the Polish yield curves . . . . . . . . . . . . . . . . . . . . . . . . . . 774.3 Polish term premia robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4 Term premia regressions to test the Expectations Hypothesis . . . . . . . . . . . . . 914.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Chapter 5. Forecasting power of the Polish government yield curve 975.1 Forecasting strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Ex-post level forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.3 Ex-post slope forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.4 Ex-post short term interest rates forecasts . . . . . . . . . . . . . . . . . . . . . . . . 1175.5 Professional forecasters short interest rates time series . . . . . . . . . . . . . . . . . 1195.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Summary and conclusions 124

Literature 127

Appendix. Auxiliary figures. Detailed tables 141

You can’t depend on your eyes when yourimagination is out of focus

Mark Twain (1835-1910)

Introduction

The general objective of this thesis is to understand what expectations of the futureinterest rates evolution are confined in the government yield curves of less liquid mar-kets (henceforth, LLM) such as Polish one and to explore methods to improve theirforecasting power. The notion of less liquid markets is not precise in the literature, it is definedby comparison to liquid bond markets such as the major advanced economies1.

The risk-free interest rates’ term structure obtained from the yields of government bonds is in-dispensable for valuation of assets and liabilities, macroeconomic modelling, financial planning atfirms and financial institutions. However, for many practical reasons, these standardized estimatesare unavailable for emerging and less liquid markets, due to a number of conceptual challenges. Theobjective of this thesis is to offer novel insights on how the risk-free interest rates term structure canbe obtained for government bonds in less liquid markets, using the case of Poland.

There are well defined challenges in modelling yield curve in such markets as Polish one, and themajor novelty proposed here concerns the ways how these challenges are tackled. We offer quanti-tative solutions to the following issues: (i) generally shorter recorded history of less liquid marketscompared to liquid markets, (ii) extreme sensitivity of risk premium inference due to the issues withestimation of the short end of the yield curve, (iii) insufficient diversity of maturities within the bondtypes, (iv) insufficiently precise price quotes for many of the off-the-run maturities. These issuesimply that the estimates of the yield curve run a high risk of being spurious, due to over-fitting.These issues are characteristic for many LLMs. For the estimation purposes, we will use data fromPoland as an example of such markets. The true innovation of our this thesis rests upon proposinga comprehensive set of solutions to all these issues in an internally coherent, model-consistent way.Ultimately, we provide insights about the information confined in the yield curves and about theexpectations to be extracted for policy purposes from them.

We develop the methodology on how to overcome all of the (i)-(iv) problems inherent in lessliquid market modelling and produce plausible yield curve decompositions in such markets that areusually not covered by the mainstream research. We acknowledge that it would be hard if not impos-sible to overcome in full the challenge of short recorded history (i) and the fact the recent 15 yearscoincides in its majority with new normal monetary and fiscal policies developed and implementedafter the Financial Crisis all over the world. Nevertheless we will show way to mitigate some of theproblems arising from this situation and we posit here that the methods presented may be carriedover into the future as the times series are appended everyday with new data on the yield curvesand the new normal status of the policies prevail, which in turn is more and more probable dueto unfolding of the COVID-19 stress in the global economy and financial systems. We believe thatin the scarcity or sometimes absence of longer time series of professional forecasters surveys thatone may use for calibration, the vertical integration of the whole process of yield curve and termstructure estimation is necessary. We explore the possible trade-offs in the modelling environment

1For the sake of this thesis we define less liquid government bond markets indirectly via measure of marketsize as the ones with at least 80 bln USD equivalent outstanding amounts of general government debt securitiesin local currency, but excluding sovereign issuers from the United States, the United Kingdom, Japan, theEuro Area, Switzerland, Canada and Australia. Using Bank for International Statistics data as of the endof December 2019 the following countries would fall into this group: Brazil, Chile, Czechia, Hungary, India,Indonesia, Israel, Malaysia, Mexico, Korea, Poland, Singapore, South Africa, Thailand and Turkey

Morphology, Dynamics and Forecasting Power of the Polish Government Yield Curve. Marcin Dec, SGH 2020 1

and conduct an extensive robustness checks in order to generate dataset(s) of daily time series ofyields and risk premia for a particular market (Polish) made available for further studies and to helpto verify the following main hypotheses that in less liquid markets:

1. Hypothesis 1: There exists a class of weighting schemes which improves fit rela-tive to conventionally used methods. We operationalise fit as mean absolute error andsmoothness of estimated forward curves. We propose to combine the outstanding amountsand market liquidity data with market level information on prices of government bonds inthe form of weighting schemes. Prior literature (i.e. Dziwok (2004, 2013)) for LLMs andGurkaynak et al. (2011) for developed markets), offered approaches that lack a reference toliquidity measures such as: equal weights of yield differences or weights derived from modifiedduration on price differences. We postulate that inferring from the liquidity in particularmaturities raises the information content of the estimation and, unlike earlier approaches, ismodel-consistent.

2. Hypothesis 2: Pure Expectations Hypothesis (PEH) does not hold universally.Empirical research demonstrated that in liquid markets (US) Pure Expectations Hypothesis,claiming that that future rate is an unbiased predictor of the future spot rate, holds for shorthorizons and durations, but not universally (Fama & Bliss (1987), Campbell & Shiller (1991),Cochrane & Piazzesi (2005)). We will obtain verification of PEH across horizons and durationsanalogous to the literature for the US and compare the results between the less liquid (Poland)markets and liquid market (US).Note that Hypothesis 1 and Hypothesis 2 have straight forward implications for the generalpractice in the financial less liquid markets to utilize IBOR2 type rates to construct the shortend of the yield curves. The use of IBOR type rates is motivated by availability, but they arenot strictly speaking risk free instruments. Consequently, incorporating them in the processof obtaining the risk premia is internally inconsistent. Our proposed method replaces IBORtype rates with internally consistent method of blending transactional data of governmentbonds and some implied money market instruments without turning to indicative prices. Thisis particularly relevant, because in many markets, as Polish one, low liquidity in governmentbonds is accompanied by structural over-liquidity of the banking sector.

3. Hypothesis 3: Expectations hypothesis (EH) holds. In the light of a perception thatterm premia structure has influence on the yield curve’s behaviour and level we will test if theestimated term premia are indeed explaining the differences between compounded rate of re-turn on rolled investments in short term bonds versus long term bond yield (of a correspondingmaturity with the end date of the strategy applied).

4. Hypothesis 4: Unlike in the case of liquid markets, professional forecasters expec-tations do not help to increase the informational content of the yield curves inthe case of LLMs. For the purposes of yield curve decomposition into the term structureand risk premium, necessary assumptions can be pinned down by readily available expecta-tions of the professional forecasters (Crump et al. (2016)). We operationalise this hypothesisby studying if in the case of less liquid markets, surveys of professional forecasters raise thepredictive power of yield curve. Technically, professional forecasters surveys report effectivelyconcurrent implied forward rates rather than the actual expected future short interest.

In order to properly verify our hypotheses we put forward the following detailed researchquestions, which we will use as a guide in the focused literature review (Chapter 1) as well asduring the whole enquiry.

2IBOR stands for Interbank Offered Rate. This class includes such examples as: WIBOR, LIBOR, PRI-BOR, EURIBOR. The class is currently being reformed in line with new regulations (BMR) in order to betransaction-based with scope to, so called expert judgement, limited to minimum.


1. How liquidity and volume traded of certain series of bonds changes in their lifetime andgrowing outstanding amounts?

2. What is the information quality of professional forecasters expectations on interest rate futurepaths in the less liquid markets with a special focus on Poland?

3. What are the benefits and potential drawbacks of such a more complex weighting system inthe environment of available bonds’ scarcity as compared to develop markets?

4. Is there a tested and robust weighting system used in yield curve estimation which has beenoriginated from liquidity data or outstanding amounts of bonds’ data on a certain market(especially the less liquid markets, including Polish)?

5. What data are available for Polish government bond markets including expectations on thefuture paths of interest rates (quality, timespan, depth, sources) and how this compares withliquid markets in terms of depth and breadth?

6. What are the classic regressions use to test Pure Expectations Hypothesis?

7. How robust are the results on the pure expectations hypothesis to weight system changes?

8. What are the bond tenors and investment horizon’s time domains in which the pure expecta-tions hypothesis may not be rejected for Poland?

9. How sensitive the estimations of term premia are to changes of sample’s exact dates? Whatare the methods available to make these estimates more stable?

The method proposed in this thesis is innovative along four main lines. First, our de-cision to jointly study yield curve fit and term structure estimation is a unique proposal,to our best knowledge, not found in the literature covering less liquid government bond markets.This vertical integration in the modelling of term structure of interest rates is crucial for precisionand meaningful inference mainly due to the challenges of blending information from relevant moneymarket sub-segment into the curve on one hand and the sensitivity of term premia extraction tothe behaviour of short-end of the yield curve, on the other. This modelling complexity is even morepronounced in the light of interest rate benchmarks reform (henceforth: BMR), because after theFinancial Crisis of 2007-8 (henceforth: FC) and LIBOR manipulation scandal, the regulatory pres-sure towards transaction-based rather than survey-based indices has increased greatly. In principle,in our modelling and estimations we hold to this rule indeed that the raw data we use shall besolely based on transactional information (from trading platforms and central banks) but not theindicative prices from the general financial information distributors. Our approach, in contrast tothe earlier literature, exploits to the maximum the available micro-structural data of bond market:turnover (monthly and daily, where available), outstanding amounts, bid-ask spreads, number oftransactions traded daily, bid and ask yields.

For less liquid markets, so far separate (disintegrated) research threads were conducted: theoryof parsimonious fitting, yield curve estimation, expectations hypothesis testing and the extractionof risk premia structure. Studies on risk premia in less liquid markets are rare as the literature isdominated by research conducted on developed markets. The very exceptions (i.e. Jab lecki et al.(2016), Kucera et al. (2017)) are limited to raw term premia presentation and comparisons basedon unaltered methodology used for US markets which in turn is supplied with zero coupon yieldsgenerically prepared by third party providers (Bloomberg or Refinitiv). There are examples of ex-tensive usage of estimated term premia in calibration and design of macroeconomic models (as inKolasa & Weso lowski (2020)), but yet again these calculations of term premia are based on US mar-kets originated algorithm and rely on zero-coupon bond data taken as given from external sources.We acknowledge that there are numerous studies of less liquid markets as Czech and Polish ones,


but they are concentrated on yield curve estimation challenges fitting and smoothing - and al-most never extend to term premia extraction ( Hladıkova & Radova (2012), Kladıvko (2010), Slavık(2001), Swieton (2002), Cieciwa (2003), Marciniak (2006), Kliber (2009), Dziwok (2004, 2013)). Tothe best of our knowledge no in-depth study of vertical robustness of yield curve and term-premiaestimation exists for markets other than in the US, the UK, Australia, Canada and the Euro Area.Our study draws heavily on the idiosyncrasy of a particular micro-structure of a given bond marketand hence fills that gap, at least in part.

Second, we intend to deliver more meaningful, realistic and interpretable time-seriesof yield curves than the ones produced by methods overly concentrated on their smoothness andperfect fit to all observed data without market practise view on weights of different bond series. Webelieve, that in the existent literature focuses too much on the graphic side of the curve withoutproper attention to the economical sense of the results. Yield curve is an imaginary object that spansfrom a discrete set of points; hence the major problem is how we define the continuous function ofinterest rates in the first place, rather than running an exercise in smoothing. This is why, out of arelatively diversified catalogue of potential tools of choice we would focus on merits of parsimoniousmodelling. The yield curve used in further decomposition should be a representation of marketexpectations of the future interest rates behaviour and being such should allow for some series ofbonds to be dear or cheap to the curve. The methodology developed should also identify structuralmispricing due to tax reasons and supply related information policies of a particular sovereign is-suer. These desired characteristics are absent in the results of implementations of such methods asB-spline models stabilized with a variable roughness penalty seen in Marciniak (2006).

Third, we intend to improve the forecasting procedure under spanning hypothesis forless liquid markets. As it has been shown in the literature, generally, implied forward rates are poorpredictors of future realisations of interest rate paths or in other words, that the pure expectationshypothesis may be rejected. Though splitting the yield curves into two structures: short term rateexpectations and risk premia gives rise to very promising interpretations and better forecasts ofinterest rates. We will conduct such a precise scission and relate the results to real world data onforecasts performed by professional forecasters.

Fourth, we will provide the academic community with well documented methodol-ogy of the Polish government yield curve decompositions to risk-free and term premiaparts. It is reckless to infer on the level and dynamics of risk premia in, for that matter, Polishgovernment bond yields relying on generic, not market-specific methodology used by commercialproviders of yield curve data. To the best of our knowledge, we will prepare first vertically inte-grated sets of these type of data for less liquid markets, using the example of Poland. What is more,we would like to design and produce such datasets only using publicly available data without anycommercial royalties involved. Transparent, robust algorithms may guarantee unrestricted, wideusage of these solutions amongst both members of academia, the industry (fund managers, insurersand banks) or even the central bank and the Ministry of Finance. The dataset and methodologywe develop would give a common and robust departure point for many potential new research inthe field of central bank’s and governmental stimulus packages for the economies in recession afterCOVID-19, including but not limited to studies on quantitative easing programmes, long term yieldtargeting, international spill-over effects.

The organization of the thesis is as follows. In Chapter 1 we conduct a focused literature reviewin order to establish state of art as well as introduce methods and nomenclature used in subsequentparts. We cover some necessary building blocks of interest rates and yield curve modelling, literatureon yield curve estimation, on risk premia extraction and expectations theory testing, on yield curveforecasting and on bond market liquidity measures. Chapter 2 is devoted to uncovering stylisedfacts about Polish government bonds, developing filtering rules and weight systems framework, toconclude with parsimonious yield curve fitting and ranking for the best weight system in goodness-


of-fit vs smoothness space. In Chapter 3 we explore the domain of the Pure Expectations Hypothesis(PEH) in Poland. To this end we run and check for robustness all classic regressions from Fama& Bliss (1987) and Thornton (2006) to Cochrane & Piazzesi (2005). In Chapter 4 we establishthe state space for Polish yield curve and estimate time-varying term premia. We carefully checkfor robustness, identify optimal number of factors and propose a method to produce consistentin-sample and out-of-sample time-series of term premia. We explore the forecasting power of thesepremia in explaining excess returns in Poland. On the basis of the estimated curves and term premiawe will conduct a verification of the Expectations Hypothesis (EH) in Poland. The final Chapter5 investigates if a forecasting model based on previously estimated term premia structure mayproduce yield curve shapes and short interest rate forecasts with better out-of-sample accuracy thannaive benchmarks in state space simulations. Reconciliation of forecasts with so called professionalforecasters projections is performed there as well.


1 Chapter 1. Focused literature review

The goal of this thesis as well as the hypotheses we put forward establish a framework within whichwe will conduct more focused literature review. To this end, we would further distinguish five areasfor this review to cover.

First, we have to define properly all the basic objects we will use extensively later in this thesis,such as, but not limited to: discount rate, implied forward rate, types of yield curves or term struc-ture of interest rates. We would be interested in a brief recap of the commonly used nomenclature.

Next two parts build on a broad economic and financial literature about term structure of interestrates developed after the seminal paper of Vasicek (1977). Within this branch which summarises thestate of the art of analysis of the term structure of interest rates, we distinguish two major strands:(1) yield curve and term structure estimation and (2) risk premia estimation, which areintertwined to some extend but the first builds primarily on the theory of financial mathematics andthe second - on the econometrics.

The forth part is devoted to bond markets liquidity measures, since we will be conduct-ing integrated estimation of yield curve and inference on risk premia on a given less liquid market(Poland) and we require a toolbox with which we can analyse and compare markets by liquidity.This, in consequence, should allow us to be more careful when implementing commonly acceptedtools of a yield curve decomposition in developed (liquid) markets to our LLM case.

The last section covers the pure expectations hypotheses testing, which will be conducted inChapter 3 after the Polish government bond yield curves would be estimated in Chapter 2.

In all of the above parts we will also pay attention to less liquid markets specificity and if suchresearch exists we would comment on it, when relevant.

1.1 Building blocks: interest rates and yield curves

Before a precise exposure to the definitions and literature directly related to yield curves (especiallybased on government bonds), which is needed for the clarity of the argument, it is worth to commenton current developments in so far promising other markets (from a general class of Interest RateDerivatives) that might be used to infer on term structure of interest rates. In what follows, weoffer a rationale as to why other markets are not a good choice for a macro-economically orientedmodeller, central banker or monetary policy maker.

The Financial Crisis (henceforth: FC) started in 2007 and lasting for at least two years was theturning point in many aspects of financial instruments valuation methodology and risk managementin banks and investment funds. In the interest rate products domain, it has triggered a revolutioncaused by the ceasing the no-arbitrage assumption as a fundamental basis for these products.

During the FC we observed dramatic changes in levels, volatilities and liquidity of interbankmoney market products (uncollateralised deposits, short-term repo, fx swaps, OIS3) and interestrate derivatives (FRA, IRS, basis swaps, caps, floors, swaptions) which may be summarised asfollows (cf. Ametrano & Bianchetti (2009), Bianchetti (2008), Henrard (2014), Kienitz & Caspers(2017), Dec (2019))

3OIS stands for Overnight Index Swap in which counterparts exchange fixed rate for a compound rollingovernight xONIA rate in an agreed period of time


1. dramatic outburst of spreads of money market deposit rates (xIBOR refrence rate based4)and the ones implied in OIS derivatives (xONIA based5) rates. The spreads increased from along-term historical plateau of several basis points to more than 200 bps at some point duringthe crises, still hovering around several tens of basis points now,

2. divergence of forward rate agreements rates and the corresponding rates implied by relevantunsecured deposits, and in consequence, virtually impossible arbitrage possibilities mainly dueto liquidity and capital constraints,

3. re-emergence of tenor (basis) swap spreads risks defined as the differences between interestrates of various tenors reference indices, yet another argument for rejecting the assumption ofno-arbitrage valuation,

4. explosion of credit spreads especially these of financial institution’s, including banks whichparticipated as panelists in reference rates fixings (i.e. xIBORs) - from single digit basispoints levels to 80-300 bps,

5. evaporation of credibility of so far the biggest and the oldest banks in the world, which leadto dramatic squeeze of liquidity in many market segments in almost all instruments with rareexemptions for the ones traded with central banks or being highly collateralised.

As a result of a widespread diffusion of collateral agreements reducing credit risk between coun-terparts of the over-the-counter derivatives, these instruments’ prices in the market may now beregarded as risk-free. Nowadays almost all exposures in derivative portfolios around the world haveto be collateralised now, the market cannot use the same discount curve to calculate net presentvalues and to forecast forward rates. Hence the market moved to the best available proxy of risk-freediscounting, dubbed: OIS discounting. Another important consequence of FC is that every tenorof a reference rate (i.e 3M or 6M) may now be treated as a separate underlying asset which leadsto multi-curve environment’s challenges. Some authors (i.e. Bianchetti & Morini (2013), Henrard(2014)) suggest that a good approach is to see different curves as if they were different currencies andrefrain from trying to model why the curves differ but rather describe how to incorporate multi-curvereality into one model. This proves to be a difficult task when carefully formulating no-arbitrageconditions and checking consistency in risk neutral measures used in pricing.

On the top of this interest rate modelling break-up we are currently going through the interestrates’ benchmarks reform (dubbed: BMR6). Despite few years into the reform and the wide inter-national discussion on the future of these rates, their shape, definitions and the true underlyingtransactions, nothing seems to be agreed. One of the key arguments of the BMR reform was thatthe final user of interest rate benchmarks (i.e. retail clients of banks: mortgage borrowers) shouldnot be exposed to expert judgment - based rates, which are extremely prone to manipulation as aseries of scandals reviled (cf. McConnell (2013, 2014), Hou & Skeie (2014), Vasudev & Guerrero(2014), Walker (2013)). In turn, the authors of the reform call for an extensive use of so calledVWAP (volume weighted average price) as a main mechanism behind an interest rate benchmark.Be it very wise, it seems to be equally impractical as the main problem of many underlying interestrates’ markets fo xIBOR style rates is the uncomfortable fact that, there are almost no real trans-actions to report (on unsecured interbank deposit market with longer that one week tenor, let alone

4as found in Dec (2019): xIBOR is a trimmed average reference rate for OTC money market unsecureddeposits in a currency , usually calculated at 11:00 AM local time or different maturities (from O/N to 12months, depending on the currency) on the basis of a questionnaire amongst the highest credit rated marketparticipants (panellists). It is not based on real transactions

5as found in Dec (2019): xONIA is a weighted average overnight rate in currency x usually calculated bythe relevant for that currency central bank from real O/N deposits settled between banks

6Regulation (EU) 2016/1011 of the European Parliament and of the Council of 8 June 2016 on indicesused as benchmarks in financial instruments and financial contracts or to measure the performance of in-vestment funds and amending Directives 2008/48/EC and 2014/17/EU and Regulation (EU) No 596/2014,http://data.europa.eu/eli/reg/2016/1011/oj


typical 3-month or 6-month horizons). Hence the reform is somehow flawed as it targets currentlyimpossible solutions.

This is why, even more importantly than in the previous decades, the yields of governmentbonds observed in the secondary market (which may be treated as risk-free in a given system’sjurisdiction) are undoubtedly the only unquestionable source of information on how themarket perceives the evolution of a short term interest rate in the future. Therefore, ourresearch is focused on that fixed income market segment (government bonds), instead of interestrate derivatives, for which longer term inference is somehow flawed at the moment.

One of the universally used quick measures of rate of return from an investment in a couponbearing bond j is the yield-to-maturity (ytm), which is subject to many localisations (to marketpractise i.e. calendar and day count conventions) but in its essence it is an internal rate of returnknown from investment projects’ appraisals (two versions: annual compounding and continuouscompounding will be given when relevant)7:

Pj =

N∑i=1

CFi

(1 + ytmj)ti

or Pj =

N∑i=1

CFie−ytmjti (1)

where, CFi is an i-th promised cash flow (coupons, principal or its parts), ti is a corresponding yearfraction, N is a total number of cash flows. Finding a particular j bond’s yield-to-maturity givencash flow schedule (dates and payments) and market price Pj is not trivial for N > 1. Usually somebasic numerical method is involved (i.e. Newton-Raphson).

The concept of yield-to-maturity is not a good departure point for term structure of interestrates, as every observed ytm of a bond blends information from potentially different schedules andytm may be treated (with some level of generalisation) as an average interest rate over a periodof this particular bond’s lifespan till maturity and redemption. This is why, when talking aboutterm structure or expectations of interest rates in the future or yield curve as such, researchers andpractitioners rely on concepts of spot rates and forward rates. The spot rate y(t) may be definedimplicitly as the interest rate which may be used for discounting a particular cash flow at time t:

P (t) =CFt

(1 + y(t))tor P (t) = CFte

−y(t)t (2)

and it means that a coupon bearing bond’s price is a sum of all discounted (potentially using uniquediscount rates for each cash flow) payments from that security:

Pj =

N∑i=1

P (ti) (3)

It is easy to notice that spot rate is explicitly:

y(t) =

(CFt

P (t)

)1/t

− 1 or y(t) = −1

tln

P (t)

CFt=

1

tln

CFt

P (t)(4)

In particular, if we normalize the future cash flow to 1 the P (t) objects are called discount factorsand the function P (·) a discounting function8:

P (t) =1

(1 + y(t))tor P (t) = e−y(t)t (5)

7for textbook exposure on the yield curve’s building blocks cf. Seppala & Viertio (1996)8function P : R+ → (0, 1∗] which maps tenor t into the price of a discount bond for that period (expressed

per 1 unit). The asterisk as a superscript of 1 indicates that in the possible negative interest rates environmentsit is feasible to have technically P (t) > 1, whereas before the FC the assumption that in normal timesP : R+ → (0, 1] as perfectly fine. After the FC it seems that mathematically and economically we can haveP : R+ → (0, 2], as it is currently unimaginable to have more negative rates than −100%.


Using these concepts, we may formulate first, general version of term structure of interest ratesproblem: Given a series of Pj prices of K bonds from an active market, to infer a seriesof spot rates y(t) for a non-empty subset of dates t for which a cash flow from any bondis promised. Naturally, Tmin ≤ t ≤ Tmax, where Tmin and Tmax are the nearest and the furthestcoupon or principal payment of all K bonds set. Obviously, this problem is not trivial to solveand requires advanced numerical techniques on the top of some additional assumptions or modellingdecisions are needed, amongst which we typically find in the literature:

1. discrete or continuous domain of t

2. interest rate model’s compounding

3. set of market conventions we follow when calculating year fractions ∀j∈K∀i∈Nj ti.

4. form of y(t) - functional form, in particular

5. interpolation schemes, where applicable

6. filtering of the data used (out of all K bonds set)

It is important to underline here, that in this introductory section we are interested in deterministic(static) framework of interest rates and the stochastic one will be covered in the proceeding sections,in which we elaborate on term structure of interest rates and risk premia. No arbitrage principlein this - deterministic - set-up means (Filipovic (2009)) that having today’s information only thereshould not be a difference in terms of expected rate of return between an investment in bond madefor a duration of t and proceeding investment made again at t for the period of T − t and aninvestment made today for the full period of T years. This is why we may define explicitly a forwardrate f(t, T ) assumed at time 0, which is relevant for discounting to date t < T a cash flow thatwould appear at time T by:

f(t, T ) ≡ −ln(P (T ) − ln(P (t))

T − t=

y(T )T − y(t)t

T − t(6)

An instantaneous forward rate is a theoretical concept (not observed directly in the market) in whichT − t → 0, hence using d’Hospital and chain rules we obtain:

f(t) ≡ limT→t

f(t, T ) =−∂ln(P (t))

∂t= y(t) + t

∂y(t)

∂t(7)

In consequence, the price of a zero coupon bond issued hypothetically at time t and maturing at T ,given today’s information only, is:

P (t) = e−∫ t0 f(s)ds (8)

where P (t) is a zero-coupon bond price at time 0 with t maturity.

There are applications in which we would need to assume some behaviour of the modelled spotrates between discretely estimated points. To this end we would need an interpolation scheme. Thechoice id the proper schema is out of scope of this thesis, nevertheless in this regard we will followthe recommendations of Hagan & West (2006), who found out, that based on the following criteria(cf. Dec (2019)):

• continuity and positivity of forward rates

• minimisation of little spill-over effect (locality of interpolation)

• stability of forwards (bumping does not change much in the shape of the curve)

• locality of hedges (delta risk of hedge concentrates near the underlying with no filtering toother areas of the curve)


the best results, although for distinct purposes, were achieved using a linear interpolation on thelogarithms of discount factors9 and a monotone convex interpolation on the logarithms of discountfactors (which is much more demanding numerically, hence usually implemented directly in a softwarepackage). The raw linear method of interpolation between two points Ti and Ti+1 at T may besummarized as follows:

P (t) = P (ti+1)t−ti

ti+1−ti P (ti)ti+1−t

ti+1−ti (9)

This method results in piecewise constant instantaneous forward rates, which in turn is not adesirable feature. On the other hand, monotone cubic splines class has a very tempting charac-teristics of smoothness and produces visually round forward curves, however one should be verycareful to use it for hedging and sensitivity analysis as the method suffers from such problems as (cf.Ametrano & Bianchetti (2009)): spurious inflection points, excessive convexity and lack of locality(when curve bumping). These drawbacks are mitigated in tension splines method as proposed byHagan & West (2006).

Having the interest rates defined we would now turn to their sets or collections. We define ayield curve as a mapping: Cyc : t → y(t), a forward curve (when not indicating the tenor of aforward rate we would assume instantaneous forwards) - mapping Cf : t → f(t) and a discountcurve: Cd : t → P (t). All of the curve objects may have discrete: t ∈ [Tmin, Tmax] or continuoust ∈ {T1, T2, ..., TN} domain of year fractions t. This nomenclature is typical when we consider asingle day curve, without referring to time series. In the latter case, we usually use x instead of t asa tenor variable, allowing t to be a subscript governing the time stamp of the observation (a day),which would be the case in the next section.

In the centre of this enquiry lays a notion of risk or term premium, which we define for a giventenor as the difference between current spot rate for this period and the expected average value offuture short rates. Adrian et al. (2012) underline countercyclical features of term premia and atendency to move with measures of uncertainty and disagreement about the future level of interestrates. Despite very simple definition risk premium calculations are expectations-dependent in thesense, that we have to assume and test how the expectations are formed in a particular market.Only then we can infer on the level, dynamics and structure of risk premia (if any) potentiallyimplied in the yield curve data. There exists a strand of the interest rate term-structure literaturethat focuses on hypotheses related to the forecasting power of yield curve and inquiries into thenature of expectations which may be grouped into bigger sets: (1) expectations hypothesis (ofdifferent varieties), originating mainly from the works of Fama & MacBeth (1973); Fama & Bliss(1987) and (2) spanning hypothesis (also in various combinations), which is a younger conceptused by Duffee (2002).

To help plausibly verify these hypotheses in time one needs modelling environments which in-corporate market prices of risk (of shocks), hence we introduce here dynamics of interest rates andexpectations under a risk-neutral probability measure. These will allow us to decompose the yieldcurve and analyse risk premia and their structure. In the dynamic (stochastic) environment thediscount factor (also dubbed as term structure of interest rates, or pricing kernel) is usually definedas (cf. Duffie (2001))

Λ(t) ≡ EQ0[

exp

(∫ t

0−f(s)ds

)](10)

and a zero-coupon yield of a bond maturing at t is:

Y(t) =ln(Λ(t))

t(11)

9which Hagan & West (2006) call originally: raw


Fundamental difference between the above cited definitions of a discount factor and a spot ratecompared with the ones we used in the previous parts of this text (for P (t) and y(t), respectively)is the expectations operator under a risk-neutral measure10. In the stochastic environment theno arbitrage assumption (together with some technical assumptions on the interest rate stochasticprocess) require the existence of an equivalent martingale measure (henceforth: EMM). Such ameasure is usually called Q as opposed to physical probability measure P. Under EMM the discountfactor of a form in 10 allows to discount any cash flow expected in the future to appear. Obviously,in order to use 10 we should specify the stochastic dynamics (usually via stochastic differentialequation (diffusion equation), henceforth: SDE) of the interest rate process (f(·) or r(·)) or assumedirectly a from of a pricing kernel. This specification is called interest rate modelling and is a verywide topic as such (cf. a bible on that matter Brigo & Mercurio (2007)). There are many classesof interest models that potentially are useful in our considerations of term structure of interestrates inferred directly from government bonds market. But it seems that the most prominent areaffine term structure models (henceforth: ATSM, Vasicek (1977), Duffie & Kan (1996), Dai &Singleton (2003), Duffee (2002)) which constitutes very rich and tractable class assuming both driftand diffusion parts of SDE to be linear in x and as a consequence a price at time t of a zero-couponbond maturing at x has an exponential-affine structure:

Pt(x) = eA(t)+B(t)x (12)

where A and B are some functions of time estimated using econometric methods. As Joslin etal. (2011) and Hamilton & Wu (2012) underline the most popular versions of ATSM are GaussianATMS, which specification centres around a joint distribution of yields and factors and the assump-tion that this distribution is multivariate Normal with constant conditional variances. They usequite expensive and involving maximum likelihood (MLE) and minimum-Chi-square (MCSE) esti-mation procedures.

1.2 Yield curve estimation

In the second strand of related literature on yield curve fitting two major approaches have beendeveloped: spline based and parametric. Please note that we focus in this subsection on static situ-ation on the bond market, without referring to dynamic (through the time) processes of short terminterest rates and the term structure of interest rates. The latter we will cover in the next section.There is no consensus on which method is best as the choice depends on the purpose that the yieldcurve is intended to serve. The comparative studies in the literature are variants of research ontrade-off between over-fitting and smoothness of the modelled curve. Market users (i.e. traders, in-vestment funds) tend to choose methods which result in saw tooth patterns but almost perfect fit atevery data point whereas economists prefer smoother, easily interpretable shapes of the yield curves.

We introduce some nomenclature needed for a proper exposition and discussion of the develop-ments in yield curve fitting. We roughly follow the one proposed in Bank of Canada working paperscf. D. Bolder & Streliski (1999), D. J. Bolder et al. (2004), D. Bolder & Gusba (2011) with somealterations when necessary.

We are usually given a vector x = {x0, x1, . . . , xN} of N + 1 year fractions x of the bonds forwhich we observe market prices and in consequence, yields. We may think of this as if we observevalues of unobservable function S(x) as follows: f = {f0, f1, . . . , fN}. The theoretical price of a par-ticular bond would be named Pi(S) and the observed prices’ vector: p = {p0, p1, . . . , pN}. This inentire up to imagination, rationalisation and interpretation of a modeller what form of this function

10Due to obvious limitations of this work a deeper exposure to stochastic calculus is not feasible here, forsuch a course consult: Musiela & Rutkowski (2005), Wilmott et al. (1995), Wilmott (2006), Filipovic (2009),Cochrane (2005), Deutsch (2009).


S to propose. Formally, we what S to have a domain over [x0, xN ] and we typically assume that Sis continuous on that set, namely S ∈ C[x0, xN ]. We do not adjudge morphology of this functionyet, as we let S to be a sum of many piecewise constant sub functions, a one-piece function or evena convolution of some functions.

For clear exposition of the optimisation problem and differences in the models proposed in theliterature we introduce two more concepts:

1. a weighting function Wi, which allows us to consider some observations (x, f) indexed byi more important than others (the approach to this function for LLMs will constitute one ofthe practical key deliverables of this thesis)

2. a penalty function λ(x, t), which governs the importance of smoothness in the character-istics of a final optimised function S(x). Note that the function λ has two arguments: bond’smaturity in years x and time of the observations set t (in a panel time series).

When modelling term structure of interest rates we usually are interested (during the constructionof an objective function) in two characteristic groups of measures in such set a problem:

1. goodness-of-fit, usually defined as a weighted sum of a function of distances between theo-retical and empirical values ℓ(S) or their squares ℓ2(S) in the yield space,

ℓ(S) ≡N∑i=0

Wi|S(xi) − fi|, ℓ2(S) ≡N∑i=0

Wi(S(xi) − fi)2 (13)

In the price space we would be interested in the distances between clean prices: theoreticalones that are implied by a fitted S(·) and the empirical (observed) ones, i.e.:

ℓ(S) ≡N∑i=0

Wi|Pi(S) − pi|, ℓ2(S) ≡N∑i=0

Wi(Pi(S) − pi)2 (14)

In many practical applications ℓ(S) or ℓ2(S) are measured in price space, as it is numericallycheaper. It may be odd at first sight, because calculations in price space require finding Nprices implied by a current guess of S. In real market situations it is very rare that f consistsof spot rates only. Usually it is a mixture of zero-coupon rates and coupon bearing bonds’ytm-s, hence it is not feasible for fitting S directly. We should stick to the observed prices pthen and based on a guess of S calculate a vector of theoretical implied prices, which is muchcheaper and straight forward (as it is a sum of discounted values) then the reverse operationof finding ytm of a series of bonds from their theoretical prices.There are some additional measures of goodness-of-fit as hit ratio, cheap ratio or rich ratio(D. J. Bolder et al. (2004)), but they are never seen in the objective function formulation.Hence we cite them together with econometric (ex post) measures of fit.

2. smoothness of S or its derivatives (later refered to as G(S) 11)

G(S) ≡∫ xN

0

(∂2S(xi)

∂x2

)2

dx (15)

In some particular applications the modeller may be interested not in the smoothness of theyield curve as such (represented by the function S) but the smoothness of the resultant (im-plied) instantaneous forward rates f(S(xi)). Proper adjustment is easy to handle numerically.

11we may also require smoothness of instantaneous forwards curve resultant from the yield curve S ratherthan the yield curve as such


By incorporation of these two groups of measures we may formulate the minimisation problemin the yield space we are facing in a general form as:

minS(·)

O ≡ minS(·)

{ N∑i=0

Wi(S(xi) − fi)2 +

∫ xN

0λ(x, t)

(∂2S(xi)

∂x2

)2

dx

}(16)

or in the price space:

minS(·)

O ≡ minS(·)

{ N∑i=0

Wi(Pi(S) − pi)2 +

∫ xN

0λ(x, t)

(∂2S(xi)

∂x2

)2

dx

}(17)

It is important to underline significant capacity of this formulation to incorporate almost all ofthe approaches to yield curve construction. Precise forms of S(), Wi, λ(x, t) and accompanied bythe choice of bonds used in such a construction12 would define the method in full.

Econometric measures of goodness-of-fit, which are capable to evaluate after the optimisationprocess is finished:

1. (W)MAE (Weighted Mean Absolute Error)13

MAEprx =

∑Ni=1Wi|Pi − Pi|∑N

i=1Wi

MAEyld =

∑Ni=1Wi|yi − yi|∑N

i=1Wi

(18)

2. (W)RMSE (Weighted Root Mean Square Error) - puts more weight on extreme observations

RMSEprx =

√∑Ni=1Wi(Pi − Pi)2∑N

i=1Wi

RMSEyld =

√∑Ni=1Wi(yi − yi)2∑N

i=1Wi

(19)

3. hit ratio (when we observe not only mid prices or yield but bid and ask as well)14

hit ratio =card

({Pi(S) ∈ N : pbidi ≤ Pi(S) ≤ paski

})N

(20)

where card() is cardinality operator of a set and N is a full set of all N available bonds

4. cheap ratio and rich ratio (in addition to the remark above we are interested in a directionalbias of our estimated curve)15

cheap ratio =card

({Pi(S) ∈ N : Pi(S) ≥ paski

})N

(21)

rich ratio =card

({Pi(S) ∈ N : Pi(S) ≤ pbidi

})N

(22)

It is worth mentioning that during filtering or out-of-sample robustness tests we might limit thenumber of bonds taken into consideration when estimating yield curve. But we should not leavethese observations out of goodness-of-fit calculations, especially when we would evaluate models with

12the latter is called in the yield curve literature: filtering13the abbreviations of these measures are not unique, for example Dahiquist & Svensson (1996) refer to

them as: MAPE, MAYE, RMSPE, RMSYE respectively, whereas D. Bolder & Streliski (1999) calls ourMAEprx the average absolute value of yield errors AABSEyields and effectively it is the same measure.

14after D. J. Bolder et al. (2004)15after D. J. Bolder et al. (2004)


different filtering rules affecting number of bonds taken into consideration N < N .

First branch of methods we discuss here are, spline-based methods, lucidly presented in sem-inal works of McCulloch (1971, 1975), Fisher et al. (1995), (henceforth: FNZ) or Waggoner (1997)(henceforth: VRP). The idea is based on smoothed B-splines with a control for the smoothness inthe form of additional element in the typical objective function O called roughness penalty, whichaddresses the trade-off between goodness-of-fit vs. smoothness (opposite of roughness or over-fitting)and is in the form of the second element of 16.

These methods base on an assumption that the function S(·) is a piecewise cubic polynomial,with potentially different parameters in every segment we may consider on a curve. For the sakeof clarity we introduce the notion of knots k = {k0, k1, . . . , kN} which are the set of all uniquebeginning and ending points on x for every segment we decide to split the domain into. Sometimesit is convenient to assume k = x. Then we may define S(·) as:

∀i∈[1,N ] Si(x) = a0,i + a1,i(x− ki−1) + a2,i(x− ki−1)2 + a3,i(x− ki−1)

3, x ∈ [ki−1, ki] (23)

For the system of equations be determinable (or in other words: theoretically non-singular) one hasto add as many conditions as the sum of numbers of parameters for each segment . Conditionsresulting from an obvious assumption of equalisation of the first and second derivatives for eachinner knot (twice differentiability) and equalisation of theoretical and empirical values at all knots(continuity) are not enough and we have to turn to some auxiliary assumption on outer knots,which is not obvious and in practical applications and in research there are some widely acceptedbut different ideas implemented to handle these missing pair of conditions:

1. natural - second derivatives at the outer knots (beginning and end of the curve we are fitting)are equal to zero

2. clamped - first derivatives at the outer knots are equal to the slope of a segment connectingthese points

3. financial - second derivative at the staring outer knot is equal to the zero and at first derivativeat the ending knot is equal to zero16.

When the number of knots is equal to number of observations (bonds on the curve) we reach perfectfit where goodness-of-fit is actually zero with the cost of second derivatives to change signs frequentlyalong the curve (which increases the values of G(S) measures. However, as D. J. Bolder et al. (2004)suggests, as the number of knot points increases, this approach is awkward to implement and nu-merically unstable.

Method called B-splines (B as in basis) came with a stable solution here. It is based on theidea of a cubic spline that takes positive values over only four adjacent subintervals in the overall xpartition. Everywhere else it takes the value of zero. As a consequence, when one defines a sequenceof B-splines on its own (four) adjacent intervals, there are precisely four non-zero splines on anygiven subinterval of x, which is a key characteristics allowing for the numerical procedure to bestable. S(·) in B-spline is defined (precise exposition of B-splines may be found in Lancaster &Salkauskas (1986), D. J. Bolder et al. (2004)) for i = 1, ..., N − 1 indirectly as:

S(x) =

N∑i=0

{S (ki) Φi(x) +

∂S(ki)

∂xΨi(x)

}(24)

16as a result we have a horizontal right asymptote as x → +∞


where Φi(x) and Ψi(x) are further defined as:

Φi(x) =

0 x < ki−1

− 2(ki−ki−1)3

(x− ki−1)2 (x− ki − 1

2(ki − ki−1))

ki−1 ≤ x < ki2

(ki−ki−1)3

(x− ki + 1

2(ki − ki−1))

(x− ki+1)2 ki ≤ x < ki+1

0 x ≥ ki+1

(25)

Ψi(x) =

0 x < ki−1

1(ki−ki−1)2

(x− ki−1)2 (x− ki) ki−1 ≤ x < ki

1(ki−ki−1)2

(x− ki) (x− ki+1)2 ki ≤ x < ki+1

0 x ≥ ki+1

(26)

As a intermediate result of calculations we receive sequences of S(ki) and ∂S(ki)∂x for all i.

By adding smoothing measures to the objective function in our estimation we obtain fully-fledgedsmoothing cubic spline class with two major already mentioned instances: FNZ and VRP. Maindifference between FNZ and VRP is that the first one chooses the penalty to be the same acrossmaturities and variable daily whereas the latter insists on the penalty to be dependent on thematurity of a particular bond and constant from day to day. Generally, these methods allow forbetter MAE fit and richer curve shapes at the cost of more variation in forward rate curve as a result(over-fitting). Mathematically it means that the objective function in Fisher et al. (1995) is:

minS(·)

O ≡ minS(·)

{ N∑i=0

Wi(Pi(S) − pi)2 + λ(t)

∫ xN

0

(∂2f(S(xi))

∂x2

)2

dx

}(27)

where as in Waggoner (1997):

minS(·)

O ≡ minS(·)

{ N∑i=0

Wi(Pi(S) − pi)2 +

∫ xN

0λ(x)

(∂2f(S(xi))

∂x2

)2

dx

}(28)

where ln(λ(x)) = L− (L− S) exp(−xµ ) and these smoothing parameters L, S, µ are to be estimated

(cf. B. N. Anderson & Sleath (1999) or Marciniak (2006)).

Despite giving a researcher / modeller a flexibility of a trade-off between goodness-of-fit andsmoothness there are some drawbacks of spline methods usually listed in the literature (cf. Seppala& Viertio (1996), Dahiquist & Svensson (1996)):

1. reputation of a practitioners approach

2. lack of underlying financial or economic theory

3. poor asymptotic behaviour of the long term rates (which is inevitable as the curve is a sumof polynomials which are not finite as t → +∞)

4. fluctuations (oscillations) of the estimated forward rates as they may become unstable (whichis particularly worrying in the longer end)

5. the estimations depend greatly on the location of the knot points between different segmentsof the curve, arbitrarily chosen in the procedure

Second strand is a the parametric functional approach in which the most important and widelyused forms are the exponential-polynomial family, in particular parsimonious functional formproposed by Nelson & Siegel (1987) and its Svensson (1993, 1994) extension (henceforth: NSS),


dominated the practise of the yield curve estimation and in the last couple of decades both in cen-tral banks and academia.

There is a handful of other functional forms used in the history, but are somehow forgotten andnot used in practise. It is worth noting that in the 1960s the so called Super Bell17 model waspopular because of its OLS regression based ease of use:

ytm(x, c) = β0 + β1(x) + β2(x2)

+ β3(x3)

+ β4(x0.5

)+ β5(log x) + β6(c) + β7(c · x) + ε (29)

where ytm is a function of a year fraction x and a coupon rate of a particular bond. The process ofextracting zero-coupon bonds is not very complicated, but as a result of the procedure we receiveonly zero-coupon rates for a discrete set of year fractions, no forward curve and some additionalassumptions on the interpolation is needed (which as such maybe a daunting task with may degreesof freedom, cf. Hagan & West (2006) for a lucid review of the methods and their advantages anddrawbacks. What especially disappoints, was the forward curve which is not stable and may havevery strange shapes. Therefore Super Bell was a good try, but now it is of curiosity rather than anelement of a catalogue of models to use.

General form of the exponential-polynomial family to which the NS and NSS models belong(cf. Filipovic (2009)) is:

ϕ(x; Θ) =

K∑k=0

(Pk(x) exp(−αkx)

)(30)

where Pk(x) is a polynomial of k-th degree and set of parameters Θ includes the coefficients of thepolynomials as well as the parameters in the exponential functions α. This family joins the bestqualities of polynomials with fading features of exponential function with negative exponents.

NS and NSS models are well anchored in the theory and stylised facts of yield curve behaviourand their spines are the forward rates per se. The two models are similar in construction with NSSsometimes being dubbed extended NS model as it incorporates yet another hump as compared withthe original NS. In NS (four parameters ΘNS = {β0, β1, β2, τ1}) we define the instantaneous forwardas:

f(x) = β0 +

(β1 + β2

x

τ1

)e

(− x

τ1

)(31)

Whereas the forward rate in NSS model is of a form (six parameters ΘNSS = {β0, β1, β2, β3, τ1, τ2}):

f(x) = β0 +

(β1 + β2

x

τ1

)e

(− x

τ1

)+ β3

x

τ2e

(− x

τ2

)(32)

The economical and practical meaning of the parameters may be summarized as follows:

1. β0 - the level of ultra long instantaneous forwards and hence the ultra long spot rates’ asymp-tote for x → +∞. Vast majority of literature (i.e. D. Bolder & Streliski (1999)) suggest thatβ0 > 0, but in the current environments of negative interest rates and yields in many countriesit seems that this constraint may be relaxed. Technically and mathematically it is not a issueto have β0 < 0.

2. β1 - the slope of the curve calculated as a difference between shortest interest rate (verticalintercept) and asymptotic ultra long rates β0. Positive slope (the curve sloping upward) wouldthen be represented by β1 < 0

17as it was widely used in Bell Inc.


3. β2 - drives the magnitude and direction of the first hump (in NSS, and the only one hump inNS)

4. β3 - acts as β2 but for the second hump (in NSS)

5. τ1 - determines a position of the first hump, we usually want τ1 > 0

6. τ2 - covers a position of the second hump, we assume τ2 > 0 and τ2 > τ1

From the above one may extract the constraints that we would have to impose on our minimisationproblem (limiting the available and feasible methods - which we discuss in the section that follow).Spot rates are simply a result of integration of instantaneous forward rates over the whole periodfrom the day of valuation till maturity we are interested in (x). For example for NSS we would have:

y(x) =1

x

∫ x

0f(s)ds = β0 + (β1 + β2)

1 − e

(− x

τ1

)t/τ1

− β2e

(− x

τ1

)+ β3

1 − e

(− x

τ2

)t/τ2

− e

(− x

τ2

) (33)

As it was said above, it is almost impossible to find smoothness of NS(S) reported in the literaturewhen compared with, for example spline method. In our thesis we would close that gap, by findingexplicit formula for G(f(·)), which reads18:

G(f(·)) =

∫ T

0(f ′′(s))2ds

=1

4

(2β2

2 − 6β2β3 + 5β23

τ31+

e−(2T )/τ1(−2β2

1τ21 + 2β1β2τ1(3τ1 − 2T ) + β2

2

(−5τ21 + 6τ1T − 2T 2

))τ51

−

8β3(β1(2τ22 + 3τ2τ1 + τ21

)− 2β2

(τ22 + 3τ2τ1 + τ21

))τ2τ1(τ2 + τ1)3

+1

τ22 τ21 (τ2 + τ1)3

+

8β3eT (−1/τ2+1/τ1))

(β1τ1(τ2 + τ1)

(2τ22 + τ2(τ1 − T ) − τ1T

)−

β2(2τ32 (τ1 − T ) + τ22

(6τ21 − 4τ1T + T 2

)+ 2τ2τ1(τ1 − T )2 + τ21T (T − 2τ1)

)+

5β23

τ32−

β23e

−(2T )/τ2(5τ22 − 6τ2T + 2T 2

)τ52

)(36)

where T is the maximum significant tenor x in the domain of the curve we want to use Cf .

Main implementation challenge of parsimonious (NSS) yield curve fitting techniques is to findsix parameters Θt = (β0t, β1t, β2t, β3t, τ1t, τ2t) for every date t in a dataset:

yt(x) = β0t + (β1t + β2t)1 − e

(− x

τ1t

)x/τ1t

− β2te

(− x

τ1t

)+ β3t

1 − e

(− x

τ2t

)τ/τ2t

− e

(− x

τ2t

) (37)

Being flexible enough to fit a wide range of term structure shapes, frugal in terms of number offactors used and regular in terms of smooth forward curves generation, the NSS form is somewhatchallenging on the computational side. In its core, it is a multidimensional minimisation problem

18where we use the facts:

f ′(x) =1

τ21e

(− x

τ1

)(−β1τ1 + β2τ1 − β2x) +

β3

τ22e

(− x

τ2

)(τ2 − x) (34)

f ′′(x) = − 1

τ31e

(− x

τ1

)(−β1τ1 + 2β2τ1 − β2x) − β3

τ32e

(− x

τ2

)(2τ2 − x) (35)


with constraints with the objective function being highly irregular admitting many local minima.Hence the choice of particular instruments used to build the curve, the initial vector of parameters,the weights used in the objective function as well as many market specific rules (i.e. how two repre-sent the short end of the curve) influence the final result. In developed, highly liquid markets withlong enough history (i.e. US) years of research resulted in almost commonly accepted methodologiesand standards in the yield curve estimation, with the prominent example being a daily updated andmaintained by Fed19 set of NSS parameters and corresponding estimated yields (1-30Y) with thehistory started in 1961. The estimation is done in line with Gurkaynak et al. (2011) and commonlyreferred in the literature to as GSW set. To our best knowledge, there is scarcity of alike setsprepared for less liquid government bonds markets, and for some markets as Polish one there is nosuch a set available.

Attractive properties of parsimonious yield curve fitting (NS, NSS) (cf. D. Bolder & Streliski(1999), Ahi et al. (2018)):

1. the greatest flexibility at the short end of the curve (empirically short end of the curve exhibitsthe biggest variation of interest rates)

2. asymptotically flat for x → +∞ (this is how market participants think about ultra-longrates20.

3. ability to capture occasional (local / piecewise) lack of monotonicity of the term structure(spot and forward rates) thank to hump (NS) or two humps (NSS) (which is desirable toreflect cyclical elements of monetary policy and their market expectations)

4. easy decomposition of parsimonious curve to level, slope and curvature elements (in line withempirical findings and identification that these three factors explain majority of yield curvevariability)

The typical most frequently listed drawbacks of NS(S) models are directly and indirectly linkedto the non-linearity of the objective function:

1. uneven distribution of bond maturities as well as the resulting cash flows (interim couponpayments) is leading to problems with multiple local minima and may lead to false convergence(cf. Gilli et al. (2010))

2. each market may have its own heuristics with regard to possible shapes on the yield curve

3. the constrained optimisation methods may become particularly slow

4. different combinations of starting parameters may produce an equally good fit to observeddata (cf. Gimeno & Nave (2009), Kladıvko (2010))

5. there is usually a set of starting values of ΘNS(S) needed (i.e. some 100 were used by Bankof Canada)) based on history and expert judgement to ensure the algorithm finds globalminimum, or at least substantially limiting the risk of not hitting it.

As Dahiquist & Svensson (1996) suggest, in yield curve estimation for monetary policy purposesless precision is required when compared with trading (arbitrage) and valuation needs. Also it isreasonable to postulate that zero-coupon curve is rather smooth function, as it may be viewed by(macro)economists as a collection of inter-temporal marginal rates of substitution. Last but notleast, it is worth restricting the models functional form in order to stabilise long term instantaneousforward rates at some constant level.

19https://www.federalreserve.gov/data/nominal-yield-curve.htm20cf. Technical documentation of the methodology to derive EIOPA’s risk-free interest rate term structures

Changes (2018)


Bliss (1997a) states that too flexible yield curve estimations will be incorporating unwanted mea-surement error or idiosyncratic (bond-specific) factors of no relevance in the other parts of the yieldcurve. On the other hand as D. J. Bolder et al. (2004) underlines, an overly smooth specificationof the yield curve fitting may disguise some important issue-characteristic or term structure relatedeconomic information in government bonds’ prices (i.e. tax effects, supply effects). Bliss (1997b)gives more arguments as to why smoothness is more important that spurious precision. First, thatthere is a range prices within bid-ask spread that are equally likely to be true. Second, that thequoted prices (bid, ask, mid) are all observed with error (random noise).

Parsimonious yield curve fitting is predominantly a non-linear constrained optimisation prob-lem and therefore we will leverage our research indirectly on the fundamental works in that areaof R. H. Byrd et al. (1999); R. Byrd et al. (2000), Coleman & Li (1994, 1996), Powell (1978) andWaltz et al. (2006). What is even more important than the algorithm as such is the objective func-tion numerical construction which should be as computationally light as possible. We will proposeto innovate relative to the available computational methods by turning to matrices of bonds cashflows and corresponding vectors year fractions. Our calculations so far reveal that this innovationon computational optimization can deliver gains in speed in the order of several hundred magni-tudes relative to the algorithms used currently. We will fully explore this approach and verify itsconsistency during the estimation preparatory works. Especially important here are the techniquesinvolved in finding heuristically the objective functions minimum with set of different starting pa-rameters to reflect variability in possible yield curve shapes. Without fast algorithm a single run ofan experiment with a particular set of weights would take weeks to produce 15 years daily history.

In this section it is worth to mention yet another approach to yield curve modelling, which in factcombines static and dynamic information. Diebold & Li (2006) represent neither pure no arbitragenor equilibrium approaches. The authors took out-of sample forecasting perspective - hence it is notonly important how the yield curve fits on a single day, but how it may be fitted through time -hence they look for a almost-NS form with the following parameters Θt = (β0t, β1t, β2t, λt) for everydate t of a form.

yt(x) = β0t + β1t

(1 − e−λtx

λtx

)+ β2t

(1 − e−λtx

λtx− e−λtx

)(38)

In practise the authors suggest to make a econometrically sound guess on λt ≡ λ that is a parameterresponsible for a placement of a maximum of loading of curvature (third principal component) toease on the numerical intensity. The three β-s are interpreted as PCs: a level, a slope and curvatureand may be treated as state variables. Then Diebold & Li (2006) propose to forecast these orthogo-nal state processes i.e. using AR(1) or VAR(1). Knowing the forecasted paths of state variables weeasily recover the forecasted yield curve for each day of the forecast.

A number of comparison studies using parametric and spline-based methods were conductedfor the US market with rather mixed picture: Bliss (1997b) found that parametric methods per-formed favourably relative to more flexible methods, but Sack (2000) studied the US zero-couponyield curve based on STRIPS21 using one parametric method and FNZ and found the latter moreuseful in capturing the observable STRIPS yields in the longer segment of the curve, though heacknowledges that the STRIPS themselves may be subject to mispricing on the market given notobvious tax status. B. N. Anderson & Sleath (1999) did comparative studies of these methods forthe UK government bond curve using as the criteria: smoothness, flexibility and stability and itappeared that NS and FNZ are inferior to NSS and VRP in the space of roughness and averagein-sample error. They also find out that VRP performs better than NSS for UKs market whengeneral collateral repo data is included on the short-end of the curve. D. Bolder & Streliski (1999),

21STRIPS securities made available to the market as a result of a process of detaching the interest paymentsfrom the bond (coupon stripping)


D. J. Bolder et al. (2004) , D. Bolder & Gusba (2011) did extensive studies of Canadian yield curveand experimented with different settings of spline-based and function forms (including Fourier se-ries and negative exponential basis functions) and found that the latter were superior to splines forCanadian market in 2000-2001.

For the less liquid markets the literature is much more scarce and troubled by important method-ological limitations. Some research has delved into the curve fitting methods. Marciniak (2006)proposes an extension to the VRP framework with the penalty used by Eilers & Marx (1996) andcompares NSS with his VRP method on Polish data finding out that the latter produces lowerRMSE/MAE with the same parameter stability as NSS. Unfortunately, the time series used in thisstudy are rather short (approximately 2 years). However, no further studies were conducted forlonger samples and the choices the author makes in the data selection on the curve seemed to bearbitrary and without potential for generalisation. Another limitation of the literature on the lessliquid markets is that it typically excludes bonds with duration shorter than one year as not liquidand potentially mispriced Marciniak (2006). This assumption is not actually well documented, andit is not consistently followed, because arbitrarily some shorter duration treasury bonds are includedin the dataset along with eight short run WIBOR rates as the proxy for the short end of the curve.In the light of future BMR reform this assumption has particularly worrisome bearing on inferenceabout the term structure and risk-free interest rates. Finally, the literature typically abstracts fromweighting the bonds of various duration in term structure and risk-free interest inference. One no-table exception is a Hladıkova & Radova (2012), who account for liquidity of different bonds, byincluding data on bid ask spreads (not: the volume traded) for almost 10 years of daily data. Thestudy was inconclusive with regards to whether adding liquidity data helps in overall fitting exercise.This study concerns Czech bond structure and uses objective function and weights for NS and NSS(i.e. without comparing it to spline-methods).

Against this strand of the literature, our thesis offers several important innovations. First, wewill explicitly address the issue of the short end of the term structure, rather than take arbitraryassumptions on the validity of the prices in short durations. Second, we will follow internally consis-tent approach inferring the term structure from actual bond prices, rather than take the conceptualshort-cut of using IBOR-type rates. This is particularly relevant for the inference of risk-free interestrate, because IBOR-type rates are indeed risky. Third, we will systematically approach the issue ofweighting the prices by liquidity-related information, using actual volumes of trades rather than ashort-hand of the bid-ask spreads. We will rigorously show the bearing of the short-cuts taken inthe earlier literature for the inference on term structure and risk-free interest rate.

1.3 Risk premia estimation

Only relatively recently Adrian et al. (2013) (henceforth: ACM) proposed computationally effectiveestimation approach to the term premia structure which is much faster than the classic approach ofMLE and MCSE. They used three-stepped linear regressions (ordinary least squares) for differentnumber of stochastic factors and some unspanned variables. The ACM calibration and modellingchoices depended on what risk premia structures were produced by a particular model as comparedwith available information on interest rate expectations from other sources (i.e. surveys of profes-sional forecasters, which for US market are in abundance). As with the Gurkaynak et al. (2011)dataset, the methodology Adrian et al. (2013) used is now disseminated, used and maintained by FedNew York and the world-wide academic community may build their inference on robustly estimatedterm premia.

Assumptions of the ACM method are:

1. state variables X evolve according to VAR, where innovations conditionally follow a Gaus-sian multi-normal distribution with zero mean and variance matrix Σ or in other words:


vt+1| {Xs}ts=0 ∼ N(0,Σ)

2. no arbitrage and in consequence - the existence of a pricing kernel M allowing for the followingvaluation equation:

P(n)t = Et

[Mt+1P

(n−1)t+1

](39)

where n is a particular tenor (in months) of the yield curve.

3. market prices of risk (λ), that are used in risk-neutral pricing in the expectation operatorabove are of an affine form themselves (affine in factors, multiplied by covariance matrix oferrors from VAR modelled in the first step):

λt = Σ−1/2 (λ0 + λ1Xt) (40)

Note that λ0 is a column vector and λ1 is a square matrix, all with maximum dimension ofthe number of factors in a model’s specification. We introduce augmented matrix of Λ to be[λ0, λ1]

4. hence, the kernel M is exponentially affine, such that:

Mt+1 = exp

(−rt −

1

2λ′tλt − λ′

tΣ−1/2vt+1

)(41)

where rt is a short(est) risk free interest rate, λt is a vector of prices of risk (of factors).

5. log excess holding return of a bond maturing in n-periods is defined:

rx(n−1)t+1 = lnP

(n−1)t+1 − lnP

(n)t − rt (42)

where rt is an ultra short interest rate time series.

6. log excess holding returns (rx(n−1)t+1 ) and vt+1 are jointly normally distributed

In consequence of the above mentioned assumptions Adrian et al. (2013) decomposed the unex-pected excess return into a component that is correlated with vt+1 and another component that isconditionally orthogonal:

rx(n−1)t+1 − Et

[rx

(n−1)t+1

]= γ

(n−1)′t vt+1 + e

(n−1)t+1 (43)

where e(n−1)t+1 are i.i.d. with variance of σ2. From the assumption, that log excess holding returns

rx(n−1)t+1 and innovations vt+1 are jointly normally distributed and an assignment that:

β(n−1)′

t ≡ Cov[rx(n−1)t+1 , v

′t+1]Σ

−1 (44)

they inferred that technically γ(n−1)′t from Equation 43 in the same as β

(n−1)t in Equation 44. Finally,

these assumptions led to a decomposition of one-month excess returns (rx(n−1)t+1 ) of investments in

n-month government bonds:

rx(n−1)t+1 = β(n−1)′ (λ0 + λ1Xt)︸︷︷︸

expected return

− 1

2

(β(n−1)′Σβ(n−1) + σ2

)︸︷︷︸

convexity adjustment

+ β(n−1)′vt+1︸︷︷︸return innovation

+ e(n−1)t+1︸︷︷︸

return pricing error

(45)

The above equation matrix notation is:

rx = β′ (λ0ι′T + λ1X−

)︸︷︷︸expected return

− 1

2

(B⋆ vec(Σ) + σ2ιN

)ι′T︸︷︷︸

convexity adjustment

+ β′V︸︷︷︸return innovation

+ E︸︷︷︸return pricing error

(46)


where: vec(Σ) is a vectorisation of a Σ matrix, β = [β(1)β(2)...β(N)] is a vector of factor risk ex-posures, B∗ = [vec(β(1)β(1)′)...vec(β(N)β(N)′)]′, ι′N and ι′T are column vector of ones, V - stackedvector of innovations and finally N - the number of tenors we use for calibration and T - the numberof observations in time series. Let underline here only the affinity of market prices of risk in theform of: λ0 + λ1Xt and the necessity of splitting the returns in to expected and innovation parts.This representation forms a departure point for consecutive three stepped regressions.

In its essence the ACM procedure of extracting term premia structure is as follows:

1. decide on the number of factors K and the data source (i.e. factors from PCA procedurecarried on the previously prepared and estimated spot rate time series via NSS)

2. split state factors into predictable part and innovations in a simple one period lag vectorautoregression (VAR(1)) of the form:

XKt+1 = µ + ΦXK

t + vt+1 (47)

where, XK· indicate a collection of time series of N factors (demeaned results of PCA procedure

on monthly observations, for example), µ is a vector of means (in our case it would be equalto zero vector since the factor data are already standardised), Φ is K × K matrix of VARcoefficients and v· is a vector of innovations with mean 0 and variance matrix Σ, such that:vt+1|

{XN

s

}ts=0

∼ N(0,Σ),

3. decide on the number of tenor of the yield curve the model would be calibrated to. Originally,Adrian et al. (2013) used some 12 different tenors n to infer the term premia structure.

4. choose rt - the short term interest rate

5. regress excess returns (rx(n−1)t+1 ) on a constant (column vector of ones: ι′T ), one period lagged

factors (X−) and contemporaneous factor innovations (stacked vector of innovations: V ):

rx = aι′T + β′V + cX− + E (48)

where c = β′λ1 and a = β′λ0 − 12(B∗vec(Σ) + σ2ιN )

6. estimate market prices of risk using cross-sectional regression and find explicitly that:λ0 =(ββ′)−1

β(a + 1

2

(B⋆ vec(Σ) + σ2ιN

))λ1 =

(ββ′)−1

βc(49)

where a, c and β are estimates from the previous step.

7. find pricing matrices A and B using

lnP(n)t = An + B′

nXt (50)

and the definition of log excess holding return and then state-by-state equating it to the

decomposed rx(n−1)t+1 from Equation 4522

8. knowing matrices A, B and the Λ matrix, recursively obtain back our yield curves via logprices.

9. switch off the price of risk of factors by zeroing them out and recalculating the matrices Arfn

and Brfn and obtain risk free (without term premia) yields by:

lnP(n),rft = Arf

n + Brf ′n Xt (51)

22Details of this recursive procedure are clearly exposed in Adrian et al. (2013) Section 2.4.


10. the difference in yields obtained in the last two points give the structure of term premia impliedby history of factors and current state of the yield curve.

Historical time series analysis of the results will be conducted and the short term rate expec-tations will be compared with professional forecasters in search for confirmation that the proposedextraction of risk premia is meaningful, and not just a econometric exercise.

There is an ongoing dispute as to whether use so called bias correction in the term premia cal-culations. The existence of bias in VAR model on factors due to persistence (slow-mean reversion),autoregressive process and sample size of available data (especially, when there is a limited numbercycles of interest rates covered) is somehow unquestionable. Bauer et al. (2012) suggest generally tocorrect the whole VAR, whereas Wright (2011) posit that bias correction applied to US data leads toless plausible term premium and risk-neutral yield estimations23 Usual technique applied to correctfor the bias is the one proposed by Killian (1998)24.

Finally, there are several degrees of freedom with regard to selection of bonds, weighting rulesand curve blending rules, depending on the market and purpose with which a certain curve is esti-mated in the first place. For example, Duffee (2002) and Longstaff (2000) argue to exclude Treasurybills from US curve building, Gurkaynak et al. (2011) suggest using off-the-run bonds issues ratherthan on-the-run25 and McCulloch (1971) McCulloch (1975) discusses the necessary tax adjustments.Since our ultimate goal is inference on risk term premia, in our study we will explore these choices(among others) and their impact on our results.

In the case of less liquid markets, no systematic effort was taken to evaluate if the assump-tions well founded in the US data and embedded in the algorithms behind combined approach ofGurkaynak et al. (2011) and Adrian et al. (2013) for example, are at all suitable for the market.To the best of our knowledge, no literature systematically evaluates the assumptions behind thisapproach against the data properties and trading facts for the government bonds. Indeed, typicaluse of GSW/ACM concerns professional forecasters (and professional forecaster surveys) as well asthe central banks. Robust, commonly accepted term premia estimations are lacking. For example,in the case of Poland there is only one survey, since 3Q2011, and there is no dataset of term premiapublicly available.

Against this strand of the literature, our approach offers several important innovations. First,we will formally address the pure expectations hypothesis, which to the best of our knowledge, hasnot been tested with reference to majority of less liquid markets, on a comprehensive set of data.The relevance of this hypothesis is paramount for the policy purposes (both monetary and fiscal au-thorities) and for evaluating the efficiency of the government bond markets. Lacking this evidence,decision-makers actually rely on inaccurate information when making borrowing and lending deci-sions. Second, we will formally evaluate the validity of GSW/ACM assumptions in the context ofPoland, as an example of less liquid market. The modelling choices behind GSW/ACM are suitablefor the US bonds market, our objective is to provide analogous set of alternatives for emerging mar-kets. We will root the departures between the two sets of modelling assumptions into finance theory.Above we elaborate a variety of theoretical reasons for which these two set of modelling assumptionsshould not be identical between less liquid markets and fully liquid markets. In fact, our approachwill base on actual pricing of assets (as is consistent with theory) rather than utilize shortcut (risky)xIBOR rates. Under the conditions of banking sector over-liquidity (as is a characteristic feature ofmany less liquid markets), the reference rate by the central bank is paramount for setting the term

23in the US there is abundance of time series of forecasts of interest rates and therefore it is much easier tocheck if bias correction is necessary or not. It is usually not a case for LLMs like Poland.

24implemented, for example, to Australian bonds by Jennison (2017)25Despite the fact that Sack (2000) showed that weekly off-the-run bonds turnover stood at 22% of their

outstanding amount whereas for on-the-run bonds that measure read 1400%.


structure and risk-free interest rate. Third, and related, the choice of the data to be used in termstructure and risk-free interest rates inference has obvious and important bearing on estimates, inways that can cause inference to be misguided. In principle, we will include all available assets, andfully exploit liquidity-related measures. This way we discipline our approach to minimize the scopeof arbitrary choices. Comparing our estimates with methods employing arbitrary choice of datawill help guide the decision-makers towards more accurate decisions. Our study thus contributessimilarly as Malik & Meldrum (2014) for UK government bond market and McCoy (2019) for EuroArea. However, we also extend extending the domain to levels of freedom during NSS estimationincluding different ways to blend in money market data of the short end of the curve. We discussthe details of these methodological innovations in section Research methodology below.

Based on previously fitted historical yield curve we will extract market expectations of the fu-ture interest rates via calculation of implied forward rate structure. We will use all of the data onrealised and implied rates to conduct tests of different versions of expectations hypotheses. We willuse regressions in style of Fama & Bliss (1987) with correction of the standard errors to adjust forheteroscedasticity and autocorrelation in the spirit of Newey & West (1987).

We will uncover the risk adjusted expectations about future interest rates paths that are impliedby yield curves in which we will follow the three stepped regressions in line with Adrian et al.(2013) and test different number of factors in our model to extract market prices of risk. Theseauthors’ approach in its core is based on relatively simple and well-established tools of principalcomponent analysis and a vector autoregressive model of order 1 (VAR(1)) we will obviously usein our calculations. We will use Wald (1943) or T. W. Anderson (1951) test statistics to infer onthe optimal number of stochastic latent factors to be taken in our final model as well to evaluatestatistical significance of particular coefficients estimates interpreted as the prices of risk.

1.4 Bond market liquidity measures

It this subsection we are interested in the current state of art with regard to an asset market’sliquidity measures with a particular focus on a bond market. Despite the fact that vast majorityof papers in this field explicitly deal with either shares or corporate bonds, it is possible to use theproposed measures (even without any alterations) to infer on government bond markets. We woulduse these tools and concepts to develop stylised facts about a particular LLM bond market prior toyield curve and term structure estimation later in this thesis. The liquidity characteristics accompa-nied by filtering rules would constitute a basis for unique weights system to be used in our estimation.

Literature on general asset market liquidity measures is relatively rich and may be traced backto the seminal paper by Niederhoffer & Osborne (1966) who introduced negative serial covariation(correlation) of assets prices observed in a market as a measure of illiquidity. The authors positedthat the lower the covariance (more negative) the more probable is that the market we are observingis illiquid judging by other measures (turnover, number of market’s participants, bid-ask spread).This idea was then used by many authors from different angles.

We should start with the most prominent implementation of the idea by Roll (1984) who intro-duced a measure of an effective spread observed on a market based on the first-order serial covarianceof price changes. Ross made two assumptions: (1) the asset is traded in an informationally efficientmarket and (2) that the probability distribution of observed price changes is stationary. He claimedthat when transactions are costly to effectuate (be it due to real transactional costs or price riskassociated with warehousing the asset until sufficient demand arises) a market maker must be com-pensated somehow. The market maker does that via quoted bid-ask spread. During transitoryprice movements it would be wary to infer from quoted bid-ask spread, even if we assume that highfrequency data for a particular market of our interest exist. Roll observed that when a new tradearrives on a market it is either done at bid or ask price, hence we potentially have all the informa-


tion needed to assess the effective bid-ask spread confined in the time series of prices. Hence, Roll’sfamous measure of liquidity (effective spread) reads in its stylised version:

Roll.spread =

{2√−cov if cov < 0

0 otherwise(52)

where cov stands for a first-order serial covariance of price changes (cov = Cov(∆pt,∆pt+1))26.

On generally the same assumptions, Grossman & Miller (1987) designed their famous marketliquidity model around a concept that market makers supply immediacy by their continuous pres-ence on the market and willingness to bear risk during the time period between the arrival of finalbuyers and sellers for the particular asset. They found out that the lower the autocorrelation inrates of return (closer to zero), the higher is the necessary level of liquidity.

Even works directly dedicated to transaction costs in their core base on effective spread concept.Edwards et al. (2007) examined the transaction costs on US corporate bond market and their de-pendency on transparency, and posited that the costs are lower for the bonds with transparent tradeprices. Transparency was defined by the presence of a particular issue in TRACE trading system,where orders to buy and sell with certain price limits were placed. What is more they establisheda negative dependency between mean estimated transaction costs (in basis points) and the tradesize (in USD). To estimate these costs the authors used time-series regressions separately for morethan twenty thousand bonds, and in its essence they used effective half-spreads, again idea origi-nated from Niederhoffer & Osborne (1966). Goldstein & Hotchkiss (2007) conducted a controlledexperiment in the very field and found that switching transparency on (i.e. by placing a bond inabove mentioned TRACE system) for a particular corporate bond at least in neutral for liquidityand potentially leads to lower effective spreads, however the event is not usually associated withhigher trading volume nor causes spreads for very rarely traded bonds to fall.

Corwin & Schultz (2012) proposed a way to estimate effective bid-ask spreads from additionalinformation on traded prices during a day, namely high and low prices. They infer this spread fromdifferences in high-low ratios and stock’s price variance in one-day and two-day intervals. Theyclaim that their measure outperforms other classic liquidity measures because it better captures thecross-sectional distribution of spreads.

Recently, researchers (i.e. Bao et al. (2011), De Renzis et al. (2018)) tend to treat the negativecovariance as the measure itself and skip the square root and scalar of 2 from Roll’s original proposal.For example Bao et al. (2011) used γ = −Cov(∆pt,∆pt+1). As long as we use the Roll’s spreador BPW’s γ for ordinal comparisons both within a market - for different instruments, or acrossmarkets it shouldn’t matter which one we use. However, should one wish to have cardinal measureand interpretable value it is obvious to choose Roll’s effective spread.

The described above idea of measuring liquidity of an asset is fairly general gauge of a particularmarket’s tightness (in addition to standard quoted bid-ask spread), although we should state hereclearly that it does not address the following:

1. other potential dimensions or aspects of liquidity27: depth and resilience. Depth of a marketis inevitably linked with the data on volume traded at different prices, whereas resiliencemeasures are based on such variables as: zero trading days (henceforth: ZTD), volatility(an its local spikes in particular), changes in the number of market makers (or any otheridentifiable participants). Nowadays, the depth of a market for growing number of different

26in case of bonds it is much easier to implement changes of yields: cov = Cov(∆ytmt,∆ytmt+1), whichleads to qualitatively same results as covariance of changes in prices

27cf. De Renzis et al. (2018) for broader classification


assets may be examined more closely due to trading processes automation (high frequencytrading, henceforth: HFT) and every day augmenting databases on so called limit order books(henceforth: LOB), where all of the orders (wills to trade a certain asset at a certain priceand for a certain volume with very precise timestamps) of all participants are collected forfurther studies28. However it is rare to encounter any data of such a granulation for bondmarkets, for which low frequency data is still a prevailing type. The major reason for such asplit is that LOB data are mainly extracted from stock exchange systems or over the counterelectronic platforms facilitating global foreign exchange (FX) trading, rather than purely overthe counter markets consist of a net of bilateral contacts between banks, brokers, investmentfunds without all of the trades or orders ever recorded, as is the case for majority of bondsmarkets.

2. bond market specificity: i.e. liquidity tends to cluster around so called on-the-run issuesand fades away with bond’s age, there are new issues and the old bonds are maturing orare subject to switch auctions, hence one need an aggregate measure of a constant maturitymarket segment (range of tenors: i.e. bonds with current years to maturity between 1.5 and2.5 years) rather than a measure of liquidity of a particular bond (with is subject to typicallifetime cycle).

Classic example of a depth measure29, of a market (or asset, to be precise) in low frequency setupsis the illiquidity coefficient proposed by Amihud (2002). The main idea of calculating the absolutevalue of asset return per unit traded is captured by the following formulae:

Amihud.coeff =J∑

j=1

|rj |Vj

or Amihud.coeff =1

J

J∑j=1

|rj |Vj

(53)

where J is the number of days with nonzero volume in a given period of our interest, rj stands fora daily return (in %) observed on j-th day as compared with previous trading day and Vj is a dollarvolume realised on a j-th day. The higher the ratio the less liquid an asset is considered to be, hencethe claim in the literature that this coefficient is actually a measure of illiquidity. The differencebetween these two versions of this formula is that the one on the right hand side is a daily average ofprice impact calculated on the left hand side and the average is calculated for non-zero volume day.It may be more appropriate to use this average version when analysing Amihud measure in time,when J may vary by periods. Note that the Amnihud measure is calculated for a given time-frame(i.e. month, quarter) and may be calculated for a series of such periods, resulting in a time series ofilliquidity measures, which changes and trends may be analysed in time dimension as well. It may bethe cause why some authors dubbed this measure as realised scaled volatility30. Amihud has greatercontribution in that field, in Amihud & Mendelson (1991) he found that on the US corporate bondmarket illiquidity influences yield to maturity. This finding is of our particular interest, because indesigned weight systems we will treat less liquid bonds as less informative as far as term structurein concerned. This approach will allow us to establish which bonds are rich or cheap to the curveand potentially identify the underlying reason.

Huang & Wang (2008) and Hasbrouck (2009) when studying trading costs, market crashes andthe phenomena of liquidity evaporation during such events, established that illiquidity in the marketleads to high expected returns, negative and asymmetric return serial correlation and that thereis a strong evidence that effective spread is positively related to stock returns. Vayanos & Wang(2012) in their theoretical work on asset returns under asymmetric information additionally estab-lished that imperfect competition can lower expected returns. These findings may be particularly

28cf. https://lobsterdata.com as an example of such a database open for researchers29note that some of the authors classify this measure as price impact measure rather than depth, cf. Schestag

et al. (2016)30cf. Guidelines on the management of interest rate risk arising from non-trading book activities (2018)


worrying for term structure estimations, because it means that using illiquid bonds prices especiallyduring distressed market periods may lead to over estimated forward interest rates structure (higherexpected returns). Yet again we will try to manage this problem by the weight scheme.

For the sake of completeness, we should also evoke a list of much more intuitive, if not primitivemeasures of bond market depth and resilience, which we will be using in inference on Polish data:

1. volume traded in a certain period, market segment or market type

2. turnover ratios (i.e. ratio of a sum of dollar volume traded in a period to an average outstand-ing amount31 of a certain bond or group of bonds)

3. average trade size

4. zero volume trading days (henceforth: ZTD)

5. share of bonds for which it was not possible to calculate fixing price

6. number of primary dealers obliged to fulfil market making duties

It is worth noticing that all these measures but the last two, may be calculated both on individualand aggregate (market) level. Since in this thesis we are interested in bonds’ lifetime and maturityrelated effects on liquidity, to further design appropriate weightening scheme for different bondswhen constructing a yield curve, we will incorporate all of the above measures for Polish (subjectto data availability) market across time, maturities and age of bonds).

Recent developments of HFT in almost every asset class in the financial market are not ofequal pace and depth. It seems that bond markets, presumably because of their original marketmicrostructure and set-up (OTC) are lagging behind. Therefore, and especially for less liquid govern-ment markets, the liquidity measures designed to reflect detailed price impact and price dispersionare not feasible to implement due to lack of order data (LOB). These HFT related measures are32:

1. transaction cost measure (as seen in Edwards et al. (2007))

2. difference of mean bid and ask prices (as seen in Hong & Wang (2000) and Chakravarty &Sarkar (2003)).

3. so called round-trip transaction costs33

4. Roll’s intraday measure

5. interquantile range (as seen inHan & Zhou (2008)

6. slope of the price function (as seen in Hasbrouck (2009))

1.5 Pure Expectations Hypothesis testing

One of the renowned versions of pure expectations hypothesis (henceforth: PEH) is that forward rateis an unbiased predictor of the future spot rate34. Lighter version of PEH is expectations hypothesis(henceforth: EH), which states: the long term rates are determined by the market’s expectation for

31outstanding amount is a sum of nominal values all issued bonds of a particular series adjusted for anycall backs, switch auctions or any early redemptions and calculated for a given date

32cf. Schestag et al. (2016) for complete study33these are of vital interest for intraday traders on a particular market. The lower the sum of costs of

entering into and liquidating a trading position the more intraday traders may be interested in activelytrading in that particular market, supplying immediacy services for all market’s participants and potentiallyincreasing the liquidity

34cf. Bams & Wolff (2003)


short term interest rates over the holding period of the long term asset augmented by a constant riskpremium35. Whereas the spanning hypothesis (henceforth: SH) in its simplest form posits that, theyield curve spans all information relevant for forecasting future yields and returns and no variables(including macroeconomic data) other that the current yield curve itself are needed36.

First, focus on the expectations hypothesis. Based on previously fitted historical yield curve, forexample in NSS form, we may extract market expectations in non-stochastic set-up of the future in-terest rates via calculation of implied forward rate structure and then ask typical research questionsbeing put forward when testing PEH: (1) do current forward rates forecast future interest rates? (b)do current forward rates have information about the structure of current expected returns on bondswith different maturities? These questions have a direct equivalents in the regression equationsproposed by Fama & Bliss (1987), which become canonical since the publication of their paper. It isimportant to understand specific nomenclature use in this literature strand, that allows for precisedifferentiation of time of observation of rates or formulation of expectations, from the time periodsfor which a certain type of object refers to.

Since interest rates are considered to be root or near-root processes (cf. Thornton (2006)) usuallythe regressions used to test (P)EH use spreads to some contemporary observed rate instead of levels.Nevertheless, not all time series for all of the markets in the world are unit root. For example, Sarnoet al. (2007), when testing for unit roots in U.S. bonds with different maturities from1 to 120 monthsin period: 1952:01 to 2003:12 use two statistics: (1) KPSS ( Kwiatkowski et al. (1992)) with H0

of stationarity and (2) PT (Elliott et al. (1996)) with null hypothesis of a unit root, in which theyfailed to reject the null in the first and rejected the null in the second. Therefore they assumed theirdata are not unit root processes.

1. Type 1 regression: a modo Fama & Bliss (1987) -Term premium regressions

rx(n)t+h = α + β

(f t+h,t+nt − y

(h)t

)+ ϵ

(n)t+h (54)

where the subscript in all the objects is a moment in time when we observe the market rates,know the history prior to this moment or formulate expectations about future relative thattime and the upper-script is the duration for which the object refers to, n is the time to

maturity of a bond, h is the investment horizon, rx(n)t+h is an excess return realised in the

future at time t + h of a bond with original maturity of n years over a yield of a bond with

maturity of h-years observed originally at t (y(h)t ). The null hypothesis (that PEH holds) is

H0 : β = 0, hence nothing forecasts returns. Additionally, we have the following definition ofexcess return:

rx(n)t+h =

((1 + y

(n)t )n

(1 + y(n−h)t+h )h

) 1n−h

− 1 − y(h)t (55)

The forward rate observed (implied) at t and good for discounting to time t+ h a cash-flowoccurring at t + n is given by the obvious:

f t+h,t+nt =

((1 + y

(n)t )n

(1 + y(h)t )h

) 1n−h

− 1 (56)

Usually tests are carried over for different combinations of bond’s original maturity n anda hypothetical investment horizon h such that these selected pairs (h, n) should satisfy theconstraint: ∀(h,n)h < n, as we are interested in excess returns in investments in bonds forshorter periods than their maturity. Otherwise, we would not be talking about excess returnbut simply yield do maturity.

35cf. Thornton (2006)36cf. Bauer & Rudebusch (2017)


2. Type 2 regression: a modo Fama & Bliss (1987) - Forecasts of the change in thespot rate

y(n)t+h − y

(h)t = α + β

(f t+h,t+nt − y

(h)t

)+ ϵ

(n)t+h (57)

The null hypothesis here is that PEH holds and therefore: H0 : β = 1. The pure expectationshypothesis that the long yields are the average of the future expected short yields (or in otherwords: that the excess returns should not be predictable) has been primarily rejected in manystudies for US market (Fama & Bliss (1987), Campbell & Shiller (1991)). Only for the shortinvestment horizons PEH could not be rejected. Cochrane & Piazzesi (2005) provided evenstronger evidence against PEH, but in the same time pushed the boundaries and proposedhigher order factors and their combinations as the regressors used in predicting bond returns,leading us to the third popular regression pattern:

3. Type 3 regression: a modo Cochrane & Piazzesi (2005) - tent shapeRegression on one year excess return of 1, 2, 3, 4 and 5 year bonds (separately), where regressorsare the one year forward rates. For n = 1, 2, ...5 we have:

rx(n)t+1 = β

(n)0 + β

(n)1 y1t + β

(n)2 f t+1,t+2

t + β(n)3 f t+2,t+3

t + β(n)4 f t+3,t+4

t + β(n)5 f t+4,t+5

t + ϵ(n)t+1 (58)

This regression is estimated in two steps. First, we run a regression of the average (acrossmaturities) excess return on all forward rates:∑5

n=2 rx(n)t+1

4= γ0 + γ1y

1t + γ2f

t+1,t+2t + γ3f

t+2,t+3t + γ4f

t+3,t+4t + γ5f

t+4,t+5t + ϵt+1 (59)

And then we run 4 regressions for n = 2, 3, 4, 5, to find coefficients: bn

rx(n)t+1 = bn

(γ0 + γ1y

1t + γ2f

t+1,t+2t + γ3f

t+2,t+3t + γ4f

t+3,t+4t + γ5f

t+4,t+5t

)+ ϵ

(n)t+1 (60)

4. Type 4 regression: a modo Thornton (2006) - conventionalRegression is done for different combinations of n and h, where k ≡ n

h ∈ Z, where we tryto explain variability of an average realised h period rates in n tenor with the observed andimplied by the market spread between spot rates feasible for discounting in a long n and shorth tenors. Therefore, we have:

1

k

k−1∑i=0

y(h)t+i×h − y

(h)t = α + β

(y(n)t − y

(h)t

)+ ϵt (61)

The null hypothesis (PEH holds) is H0 : β = 1.

5. Type 5 regression: a modo Thornton (2006) - contrarianHere we explain the realized spread on holding n long bond for h period over the original rate

y(n)t by the observed spread between the two spot rates scaled pro rata temporis by h/(n−h):

y(n−h)t+h − y

(n)t = α + β

(h

n− h

)(y(n)t − y

(h)t

)+ ϵt (62)

The null hypothesis is again H0 : β = 1.

Since there is no technical possibility in all of these relatively simple regressions to tellwhat causes the rejection of PEH: existence of risk premia or unexpected excess yield (we haveonly a mixture of these two contained in β0 coefficient estimators in regression of type 1 and 2 or γ0in regression of type 3) these findings gave rise for further research and formulation of alternative


hypotheses i.e. existence of a constant risk premia structure, existence of a time-varying risk premiastructure, existence of a liquidity premia structure a la Hicks.

Second, when considering the spanning hypothesis it is important use two groups of state vari-ables to test predictive power of the yield curve. As Bauer & Rudebusch (2017) state it, we typicallytry to find evidence against the spanning hypothesis in the following form:Generic spanning regression

yt+h = β′1x1t + β

′2x2t + ϵt+h (63)

where β′1 and β

′2 are column vectors of coefficients, x1t is a vector of predictors solely related to the

yield curve and x2t contains all other possible predictors we would like to test. The null hypothesisis that β2 = 0, hence the proposed set of non-yield-curve related variables have no predictive power.

First group of predictors are purely related to the information contained within the yield curve.It is customary to use the principal component analysis (henceforth: PCA) to uncover yield curvestates to be used as x1t, but it is not the only choice used in the literature. For example -Ang &Piazzesi (2003) or Hamilton & Wu (2012) use certain three representative yields (i.e. 1-, 12- and60-months). There are number of studies (R. Litterman & Scheinkman (1991) Dai & Singleton(2003), Duffee (2002), Kim & Orphanides (2012)) showing that first three principal components areexplaining the variability of the yield curve in more than 90% most of the time. It is also verystraight forward and notice that the shapes of loadings of these first three components offer inter-pretation of being: the level, the slope and the curvature of interest rate term structure. But thereare some studies (Cochrane & Piazzesi (2011), Adrian et al. (2017)) suggesting that greater numberof components from PCA is needed, despite they are not easily interpretable.

Second group x2t may contain any other variables, including: measures of economic growth(Joslin et al. (2011))37, inflation (Cieslak & Povala (2016), Kim & Orphanides (2012))38, bondsdemand/supply dynamics (Greenwood & Vayanos (2014)) or output gap Cooper & Priestley (2008).

1.6 Summary

In this chapter we have recalled fundamental and necessary research which are instrumental in ourenquiry. This forms a base for original yield curve design, calculations and inference that will follow.We have identified a gap in the literature on estimation of yield curve, term premia structure andinference on spanning and expectation hypotheses for less liquid markets, including Polish one. Evenfor developed, liquid markets (American or British) the research rarely concentrates robustness andvertical integration of the whole estimation process from filtering to professional forecasters’ perfor-mance evaluation. We intent to fill some of this gap in the subsequent chapters.

37i.e.: three month moving average of Chicago Fed National activity Index38i.e.: one-year CPI expectations from Blue Chip Financial Forecasts


2 Chapter 2. Polish government yield curve fitting

2.1 Characteristics of Polish government bonds market

Debt securities are instrumental in financing current budget deficit and in refinancing the existing(rolling) debt, amounting to approximately 86.1% of 1.088,19 bln PLN public debt as of the end ofJune 202039. There are retail, wholesale domestic and wholesale foreign (denominated in other cur-rencies than PLN and placed in the international markets) government bonds available for investingand trading in Poland. Despite playing a prominent role in socially beneficial saving behaviour pro-motion, retail segment is relatively small (3.3% of public debt) and usually linked to CPI, whereasthe internationally placed bonds (17.2% of public debt) are denominated in foreign currencies, hencetheir information potential is connected directly with foreign term structure and expectations withregards to its evolution as well as an evaluation of Poland’s default risk for external investor, butnot much with Polish interest rates term structure.

Therefore and of because the nature of this enquiry we narrow the scope of bond types of ourinterest to wholesale domestic ones with fixed coupon, without any additional features like call/putoptions (51.3% of public debt). These kind of bonds are, primarily, 2-year zero coupon bonds, 5-,10- and 20-year fixed coupon bearing bonds without special features, denominated in PLN (4.7%,18.1%, 20.3% and 6.6% of public debt, respectively). Polish government securities (T-bonds andT-bills, henceforth: TS) recent history (after the communist regime fallen in 1989) is relativelyyoung as first auction of T-bills took place in May 1991 (in material form during that time) andfirst actions of fixed coupon bonds, namely: 2-year - OS0696 and 5-year - OS0699 were held on17th February 1994. Since then the development of the secondary market was strongly correlatedwith the State Budget borrowing requirements but the quality of the market in terms of diversifiednumbers of investors in different segments of the curve (i.e. banks, investment funds, pension funds,insurance companies - including the ones offering life insurance, and foreign investors as a separate,and itself heterogeneous, group) has been growing with the pension funds reform, the developmentof interbank OTC market and Polish capital market in a broader sense.

There are two other segments in wholesale domestic group, that we leave for now as out of scopefor estimation purposes: CPI-linkers and floaters (0.4% and 16.9% of public debt, respectively), asany inference of future interest paths that may be derived and extracted from the price history ofbonds of these types is dependent on the proper estimation of the base yield curve and term premiain the first place. We will come back to these bonds in the last chapter of this thesis.

In Poland the system of Treasury Securities Dealers (henceforth: TSD, authorised PrimaryDealers and Bank Gospodarstwa Krajowego (state owned bank)) was implemented in 2002, in orderto regulate the primary and secondary bond market in consultation with buy-side participants withthe ultimate goal of better transparency and lower public debt financing costs. It involved definingtypes of participants in auctions, conditions and obligations of both: investors and the Ministryof Finance. Treasury BondSpot Poland (henceforth: BondSpot or BS)40 - as electronic wholesaletrading platform and one of the key elements in the TSD system, the successor of the ElectronicTreasury Securities Market (2002 - 2004) plays a key role in both creating cheap, transparent tradingenvironment and in price discovery and market information dissemination to the public and for debtmanagement purposes. Since the beginning the platform offered at least one - so called fixing - atime window in a day in which Primary Dealers are obliged to place two-way quotes (both bid and

39cf. Public Debt 06/2020. Monthly newsletter. Ministry of Finance. Republic of Poland40On the official BondSpot’s page we read: ”Established on November 25, 2004, BondSpot Poland is an effect

of cooperation with an Italian company MTS S.p.A., the first European electronic market for governmentbonds and the founding member of MTS Group. Today, the group of MTS companies - MTS Galaxy isthe leading market in Europe for the trading of fixed income securities. It has over a thousand participantsthroughout Europe, with average transaction volumes exceeding 85 bln EUR a day (single-counted)”.


ask prices) for the majority of existing securities with certain maximum (regulated) bid-ask-spread(BAS) depending on the current maturity of a particular series. On the other hand, Primary Dealershave been granted an exclusive right to purchase T-bills and bonds on the primary market with anintention of distributing them among others via BondSpot. The platform distinguishes betweenthree types of participants41: Market Makers (Primary Dealers) who commit themselves to con-stantly providing two-way quotes to the general market, Market Takers - who basically may placean order in the system against Market Maker offer and Institutional Investors with an access toa request for quote feature used to communicate directly with a Market Maker of their choice inorder to trade. In the moment of writing this Chapter a minimum amounts of 5 mln PLN and theirmultiples thereof can be traded. Fixing prices of securities being reference prices for the domesticdebt market are set twice a day during two fixing sessions each trading day.

There are three types of primary market operations in Poland: sale auction, switch auctionand buy-back auction. Since 2012 the sale auctions are carried out in uniform price formula whichmeans that the successful bidding investors buy bonds at minimum accepted price (cut-off ) fromthe order book42. Once the price is known, additional non-competitive bids may be placed and TSDbuy bonds at the minimum accepted price (up to 15% of total sales on a given action). Usually thereare 2-4 auctions a month with some 1-4 securities on offer. The bonds are tapped - which meansthat for several tenders the same series is auctioned - leading to increase of outstanding amount andin consequence the liquidity on the secondary market and lower servicing costs for the future issues.Less common but still very important as a tool in debt management are switch auctions in whichthe Minister of Finance buys back bonds with near redemption dates before their originally plannedmaturity date, while selling longer term bonds in exchange without any cash flows. The least com-mon auctions are the buy-backs in which the Ministry targets a few series of bonds (usually withvery short maturity) and buys them back for cash in multiple-price styled auction. The latter twotypes of operations reduce refinancing risk, help to build large outstanding amounts in benchmarkissues and reduce number of illiquid series.

2.2 Data selection

As focused literature review in the previous chapter showed, to verify our hypotheses with regardto Polish market we would need the following data on government bonds market and their charac-teristics and some auxiliary variables:

1. on government bonds secondary market (dynamic): daily43 data by bonds series ofpreferably firm44 prices or yields to maturity (ideally: bid, ask, mid), daily data on volumetraded by series, number of trades, volume weighted trading prices at least for some widelyused electronic platform and monthly data for the whole secondary market

2. on government bonds primary market (static): databases with characteristics of allbonds with regard to: (1) coupon values, payment and ex-dividend dates, (2) primary auctionresults (regular and switch auctions) and other important operations on securities which mayinfluence the proper calculation of outstanding amounts on particular settlement dates

3. ultra short risk free interest rates i.e NBP official interest rates, short term money marketrate POLONIA time series and volume of NBP bills auctioned regularly week by week as wellas, less frequent, so called tuning operations.

41In September 2020 there were, 28 participants, including 13 banks holding the status of Primary Dealer42However, in the years of 1994-2011 sale auctions were held in the multiple-price auction system - meaning

that bidders bought bonds at a price submitted in their order (potentially different price for each investor43end-of-day or other fixed time that is uniform across the whole time series44as opposed to indicative, which are prone to contribution mistakes without any economic motivation to

correct it. Firm prices are those on which it is possible to trade


4. professional forecasters expectations time series with regard to NBP reference rates, aswell as historical data on different tenors of WIBOR and POLONIA benchmark rates

We intend to use the publicly available data to the maximum and draw from privately broadcasteddata only for comparison reasons to avoid any derived data claims when making our estimationresults public. Moreover we would like everybody to be able to reproduce the input data set as wellas replicate the results of our estimation without any financial cost incurred.

Our research showed that there are the following possible choices of sources of necessary data:

1. Polish Ministry of Finance offers a general access information on public debt on its web-site45 where it provides extensive historical and current data on T-bonds and T-bills including,but not limited to: (1) schedules of coupon payments and interest periods dates for every is-sued bond since 1994, (2) securities operations (primary auctions and their results, switchoperations, early redemptions, special operations etc.) since 1994, (3) monthly turnover onthe secondary market by particular series of bonds since April 201446, (4) legal supportingmaterials (Auction procedures, Issuance procedures, Letters of issue, Rules and RegulationsGoverning the Activities of the Treasury Securities Dealer, The Public Finance Sector DebtManagement Strategy)

2. BondSpot via its webpage 47 makes available all fixings information since 26th November2004, by bond series bid/ask/mid prices and ytm-s are available in one or two fixings a dayregime. On the top of it BS publishes data about trades conducted on the platform per bondseries i.e number of trades, volume, min and max prices and best bid and ask prices. All of theprices published by BondSpot are firm in that a market taker could have always used themto trade, which makes them very valuable for our inference and estimation of term structureof Polish interest rates.

3. Refinitiv 48 collects and privately (access is subject to monthly fee and long term contractsfor majority of interested parties49) makes available probably broader information than BondSpot in some areas, but in the case of Polish government bonds no volume data on the over-the-counter markets nor electronic platforms is available and what is more important for ourinference is that the bid ask prices and ytm-s provided for each series even tick-by-tick areindicative only and no one is guaranteed to trade on these. Prices on the secondary market arecollected by Reuters since 1997, but the quality of data, judging by the frequency of changeis rather poor until 2002-2004 where the prices seem to be updated regularly.

4. Bloomberg offers Polish bond data for a slightly shorter period but of very similar depthand breadth and character as Refinitiv. The company produces their own estimations of zerocoupon curves derived from government bonds price data50. These curves may be potentiallyused to compare with our estimations.

5. National Bank of Poland provides all the history of the monetary policy decisions, timeseries of benchmark overnight interest rate POLONIA (since January 24th, 2005), historical

45https://www.gov.pl/web/finance/public-debt46In May 2019 we were granted a permission to use additional data sent by the Secretariat of Public Debt

Department in Polish Ministry of Finance for the period staring from July 200447https://www.bondspot.pl, although it is worth noting an elaborate and at some points cumbersome process

of collecting all of the data using web scrapping techniques and aggregating information from more than 8.000html tables in one Matlab database covering the period of 2005:01-2020:06

48formerly Thomson Reuters, and Reuters49The author has been granted an access to data and the right to use it in academic publications, including

dissertation by Thomson Reuters on 11 January 2018 - with side letter dated June 20th, 201850via a service called BVAL GSAC (Government, supranational, agency, corporates issuer & sector curves).

Calculation details are not publicly available and the algorithm is patent pending


and current information on open market operations including amounts tendered. Unfortu-nately it does not publish any time series of yield curve estimated parameters, in style of Fedor ECB.

6. Warsaw Stock Exchange is an important source of firm prices and volume data of all typesof Polish government bonds, but this platform is dedicated to serve retail investors and usualturnover and the size of bid and ask orders are very tiny (a few pieces of bonds in some series)and the total turnover in TS on WSE is minuscule as compared even with BS, let alone thetotal secondary market. Therefore, BS and MinFin data are more valuable given our aim.

Figure 1: Average yields in segments daily in the period of 2005:01-2020:06

2006 2008 2010 2012 2014 2016 2018 20200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

ave

rag

e y

tm in

se

gm

en

ts

1 = 0.0326

2 = 0.0357

3 = 0.0393

4 = 0.0432

[0, 1.5]

(1.5, 3.5]

(3.5, 6.0]

(6.0, 12.0]

Notes: (1) the segment of (12, 30] is not represented through out the entire history (2) the red dashed lines represent alltimespan means of yields for four segments separately, where µ1 refers to segment [0, 1.5] and µ4 to (6.0, 12.0]

Reconciling the list of data we need with what is available we have decided on the optimalmaximum lifespan of homogeneously good quality data (both on prices/ytm-s and trading activi-ty/volumes) to be the period between 1st January 2005 and 30th June 2020 (daily and monthly).Moreover our main sources would be: Polish Ministry of Finance, BondSpot and National Bank ofPoland, whereas Bloomberg and Refinitiv would be used in robustness and comparison studies. Aswe will show later in the Figure 11, amongst other characteristics, the number of bonds under fixingon BondSpot (second tile) and the cover ratio as a proportion of this number and the total fixedcoupon outstanding bonds (last tile). The number of bonds subject to fixing decreased after the FCto 13 from 15-17 in the beginning of 2005 but then it steadily rose to 18-21 in 3 years to stay on thatlevel till the time of writing this thesis and giving an average of 17.2 bonds being fixed everyday in2005:01-2020:06 period. Not all of the fixed coupon bonds are subject to fixing procedures on BS.There is a certain number of very short bonds (under 6-8 months to maturity) or currently beingissued for the first time51. Average cover ratio in the period was 84%, which means that on average3.2 bonds which had non zero outstanding amounts were not quoted on the fixing, which we assumeis a sufficient cover of the bonds traded in the market.

51OTC secondary markets are trading bonds even before the actual auction’s result - this is conditional onthe bond being issued and is called WI when-issued trading


Table 1: Dates with less then 10 eligible bonds on BondSpot fixing

Date Number of eligible bonds14-Mar-2005 419-Jul-2005 810-Oct-2008 615-Oct-2008 223-Oct-2008 7

Notes: eligible means here: with no less than 0.85 years to maturity.

There are only 5 days of data from BS that we have decided not to include for estimationpurposes because of too few bonds under fixing (we took 10 as a minimum number of bonds), whichis depicted in the Table 1. The main reason for Primary Dealers’ temporal decrease in scrutiny andcare in quoting fixing prices on the platform is a very turbulent time in the Financial Crisis mayhem.Finally, such defined a collection of data we will call a dataset (DS).


2.3 Stylised facts on Polish bonds (and their impact on filtering)

In this chapter we introduce a division of the yield curve into segments (buckets) by current yearsto maturity into [0, 1.5], (1.5, 3.5], (3.5, 6], (6, 12] and (12, 30] years. This proposal is based on thehistorical analysis depicted in the Figure 3. It will serve as an additional dimension in our enquiry.

Figure 3: Proposed segmentation of bonds on BondSpot in the period 2005:01-2020:06

2006 2008 2010 2012 2014 2016 2018 20200

5

10

15

20

25

ye

ars

to

ma

turity

12y

6y

3.5y

1.5y

Notes: (1) every light blue line represents one series of bond subject to fixing on BS (2) Red dashed lines are the proposedsegments’ division

Thorough analysis of the dataset has led to the following stylised facts:

1. Monthly secondary TS market’s turnover is of the same magnitude as monthlysum of NBP bills auctioned at reference rate. In the Figure 4 we can observe thatthe monthly average TS turnover (outright) in the whole period stood at 196.2 bln PLN withhighs around 400-450 bln PLN and lows around 100 bln PLN, where as the average monthlysum of NBP bills auctioned is 309 bln PLN with greater volatility between 50 and 650 blnPLN until the beginning of COVID19 Pandemic and around 850 bln PLN in 2020:06.

Figure 4: Turnover of fixed coupon bonds versus NBP bills monthly sales

2006 2008 2010 2012 2014 2016 2018 20200

1

2

3

4

5

6

7

8

9

vo

lum

e in

PL

N

1011

= 309 bln PLN

= 196.2 bln PLN

Fixed coupon government bonds secondary market turnover

NBP bills volume auctioned monthly

Notes: (1) secondary market turnover in outright transactions on government bonds (no repo, BSB) (2) NBP bills volume is asum of all auctioned securities at reference rate (normal and tuning operations)


Fig

ure

2:S

elec

ted

char

acte

rist

ics

ofP

olis

hfi

xed

cou

pon

gove

rnm

ent

bon

ds

trad

edon

Bon

dS

pot

in20

05:0

1-20

20:0

6bysegments

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020H1

OA([0,1

.5])

48590

44230

51631

52948

51408

66328

99858

106604

81082

72560

91508

66385

79065

76322

67685

86692

OA((1.5,3

.5])

59992

78764

87634

84715

89947

119549

106790

98955

112911

108352

96154

104826

109970

132429

144443

122702

OA((3.5,6

])60872

58962

52706

59775

62865

78647

72567

83818

91178

78282

84041

113466

126009

104692

98118

102449

OA((6,1

2])

29235

44288

54152

46564

61643

65496

96719

93326

108537

83944

86701

98821

115127

118313

134309

138281

OA((12,3

0])

4134

8262

15247

21261

26470

23287

6573

7774

10353

8713

8529

8529

2593

00

0OA([0,3

0])

202824

234506

261371

265264

292333

353307

382507

390476

404061

351851

366932

392027

432763

431756

444556

450125

NB([0,1

.5])

5.12

4.26

3.89

3.29

2.86

2.96

3.89

5.04

3.92

3.64

4.59

3.76

3.76

3.76

3.86

4.72

NB((1.5,3

.5])

4.35

4.90

4.94

4.10

4.07

5.12

4.77

4.04

5.00

5.51

5.00

4.82

5.14

6.00

6.26

5.11

NB((3.5,6

])3.80

2.86

2.47

3.02

3.02

3.21

3.15

3.81

3.93

3.82

4.28

5.25

5.48

4.24

3.82

3.73

NB((6,1

2])

2.00

2.17

2.82

2.16

2.81

2.97

4.18

4.13

4.82

4.29

3.97

3.80

4.51

4.87

5.37

5.07

NB((12,3

0])

0.76

1.00

1.00

1.00

1.91

1.72

1.00

1.00

0.99

1.00

1.00

1.00

0.30

0.00

0.00

0.00

NB([0,3

0])

16.03

15.20

15.12

13.57

14.68

15.98

16.99

18.02

18.67

18.26

18.83

18.65

19.19

18.87

19.31

18.63

V([0,1

.5])

66.37

69.13

59.35

29.89

45.91

77.56

129.03

184.25

91.88

116.78

103.86

68.64

54.34

58.54

38.84

34.92

V((1.5,3

.5])

136.31

137.09

72.14

46.72

74.92

209.60

210.38

270.84

240.61

240.06

119.52

103.98

123.42

134.79

60.20

21.48

V((3.5,6

])327.95

362.75

246.63

125.65

79.22

260.58

194.31

409.21

313.33

331.63

259.87

349.33

231.67

115.87

71.32

36.21

V((6,1

2])

84.85

138.63

170.94

59.22

65.85

271.42

288.48

494.13

448.58

379.60

333.14

283.47

198.57

150.43

113.67

63.28

V((12,3

0])

1.49

4.91

27.64

28.40

17.17

41.94

5.09

40.05

21.23

15.66

8.34

6.33

1.04

0.00

0.00

0.00

V([0,3

0])

616.98

712.50

576.71

289.87

283.07

861.11

827.28

1398.48

1115.63

1083.74

824.74

811.76

609.03

459.63

284.03

155.90

YTM

([0,1

.5])

4.96

4.23

4.80

6.29

4.53

4.02

4.55

4.27

2.76

2.28

1.58

1.44

1.46

1.25

1.24

0.71

YTM

((1.5,1

.5])

5.01

4.64

5.11

6.27

5.13

4.70

4.84

4.32

3.07

2.51

1.78

1.73

2.00

1.73

1.58

0.95

YTM

((3.5,6

])5.17

4.99

5.34

6.18

5.57

5.24

5.34

4.52

3.40

2.89

2.16

2.25

2.64

2.34

1.90

1.22

YTM

((6,1

2])

5.21

5.20

5.46

6.10

5.96

5.68

5.89

4.94

3.87

3.36

2.56

2.84

3.27

3.09

2.28

1.63

YTM

((12,3

0])

5.11

5.38

5.53

6.06

6.16

5.92

6.14

5.23

4.27

3.80

2.87

3.26

3.70

--

-YTM

([0,3

0])

5.07

4.72

5.16

6.22

5.40

5.00

5.21

4.54

3.35

2.82

2.04

2.13

2.40

2.12

1.77

1.13

YRFRA([0,1

.5])

1.00

0.96

0.98

1.01

0.98

1.00

1.01

0.87

0.76

0.98

0.83

0.87

0.87

0.87

0.88

0.98

YRFRA((1.5,3

.5])

2.14

2.33

2.35

2.30

2.28

2.36

2.20

2.36

2.46

2.34

2.36

2.43

2.41

2.50

2.38

2.38

YRFRA((3.5,6

])4.61

4.26

4.38

4.77

4.47

4.64

4.47

4.51

4.46

4.51

4.68

4.67

4.49

4.45

4.60

4.49

YRFRA((6,1

2])

9.30

8.48

8.50

8.48

8.49

8.99

9.03

8.86

8.38

8.19

8.18

8.38

8.98

8.93

8.48

8.09

YRFRA((12,3

0])

17.10

16.22

15.23

14.23

16.23

16.43

17.81

16.81

15.81

14.81

13.81

12.80

12.16

--

-YRFRA([0,3

0])

3.96

4.11

4.33

4.40

5.49

5.30

4.95

4.68

4.76

4.58

4.35

4.52

4.40

4.27

4.22

4.01

BAS([0,1

.5])

20.04

20.48

20.44

45.49

40.03

31.29

17.74

13.54

19.50

10.40

12.39

15.19

13.30

12.76

17.61

25.02

BAS((1.5,3

.5])

10.37

10.51

11.64

25.53

20.46

16.31

9.86

6.47

7.42

5.72

6.28

6.40

5.57

5.67

7.24

10.28

BAS((3.5,6

])8.93

7.14

5.48

14.81

12.96

9.81

6.72

4.76

5.87

4.69

4.64

4.58

4.21

4.47

4.62

7.51

BAS((6,1

2])

6.58

6.22

5.02

12.55

11.63

9.28

6.18

3.66

5.08

4.07

4.20

4.26

4.34

4.27

4.31

8.10

BAS((12,3

0])

10.16

9.37

5.40

10.44

22.49

19.21

16.50

3.98

14.67

7.59

18.82

18.00

14.46

--

-BAS([0,3

0])

12.65

12.04

11.25

24.66

21.38

16.75

10.53

7.30

9.48

6.19

7.63

7.85

6.58

6.48

7.99

12.93

ZTD([0,1

.5])

0.83

0.81

0.83

0.85

0.88

0.73

0.62

0.70

0.79

0.68

0.77

0.84

0.87

0.88

0.90

0.92

ZTD((1.5,3

.5])

0.64

0.68

0.75

0.81

0.80

0.64

0.51

0.48

0.52

0.54

0.64

0.73

0.77

0.78

0.83

0.91

ZTD((3.5,6

])0.52

0.35

0.41

0.56

0.71

0.42

0.31

0.21

0.28

0.27

0.42

0.46

0.56

0.67

0.63

0.78

ZTD((6,1

2])

0.47

0.42

0.48

0.55

0.64

0.40

0.26

0.14

0.19

0.25

0.34

0.35

0.47

0.52

0.60

0.76

ZTD((12,3

0])

0.88

0.80

0.57

0.52

0.81

0.62

0.66

0.25

0.49

0.52

0.77

0.72

0.83

--

-ZTD([0,3

0])

0.66

0.63

0.65

0.70

0.77

0.57

0.45

0.39

0.44

0.44

0.57

0.60

0.67

0.71

0.74

0.84

Note

:OA

-outstandingamountin

mln

PLN

,NB

-number

ofbondsunder

fixing,V

-volumetraded

onBondSpotin

mln

PLN,YTM

-yield

tomatu

rity

inpct

points,YRFRA

-timeto

matu

rity

inyea

rs,BAS-bid-ask

spreadin

bps,

ZTD

-zero

tradingdayssh

are

intotalnumber

oftradingdays,

(x,y

]mea

nsasegmen

tofbondswithcu

rren

ttimeto

matu

rity

betweenxandy

yea

rs(excludingxandincludingy).


2. The volume traded (both on BS platform and on the market as a whole), as wellas daily number of transactions follow common pattern during the lifetime of acertain bond type. In the Figure 5 (and Figures 62, 63, 64 in Appendix), especially for 2Yand 5Y bond we observe that the volume on average rises steadily to peak after approximately1/6th of bond’s lifetime (for 2Y: after 4 months, 5Y: after 8 months , 10Y: after 15 months).The maximum values are on average 2-4 times higher than the corresponding mean. Soonafter the peak volume and number of deals pull to the mean for more or less the same duration- 1/6th of bond’s lifetime. This pyramid shape is clearly recognizable on all of these figures.During the remaining 2/3rds of bond’s lifetime these statistics are eroding slowly with valuesslightly lower than the corresponding mean. This process starts approximately when theoutstanding amount stops to climb (because there were no further auction).These patters are observed not only on BondSpot volume data but also in MinFin data forthe total secondary market turnover, which is depicted at the Figure 6.

Figure 5: Selected averaged liquidity measures of 2Y, 5Y and 10Y fixed coupon governmentbonds traded on BondSpot platform

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Month

ly a

vera

ge o

f

daily

nr

of deals 10Y

5Y

2Y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

Bid

-ask s

pre

ad

in b

ps

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

Month

ly a

vera

ge o

f

daily

volu

me o

n

Bond S

pot in

mln

PLN

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.2

0.4

0.6

0.8

1

Month

ly p

ct of

zero

tra

din

g d

ays

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time fraction of a bond particular lifetime

0

1

2

3

Outs

tandin

g a

mount

104

2, 5 and 10Y Bonds

Notes: (1) Two year bonds included: OK0113, OK0114, OK0116, OK0406, OK0407, OK0408, OK0419, OK0709, OK0710,OK0711, OK0712, OK0713, OK0714, OK0715, OK0716, OK0717, OK0720, OK0806, OK0807, OK0808, OK1012, OK1018,OK1206, OK1207, OK1208, OK0419 (2) Five year bonds included: PS0310, PS0412, PS0413, PS0414, PS0415, PS0416,PS0418, PS0420, PS0511, PS0718, PS0719, PS1016 (3) Ten year bonds included: DS1013, DS1015, DS1017, DS1019


3. The bid-ask spread (on BS) rises approximately two- or threefold above bond’slifetime mean in the last year. Again the pattern is pronounced in the Figure 5 (andFigures 62, 63, and a bit less so in Figure 64 in the Appendix). The rise is accompanied byfall in the outstanding amount due to switch operations in the last year of the bonds lifespan.

Figure 6: Averaged turnover of fixed coupon 2, 5, 10Y government bonds during their lifespanin months

0 20 40 60 80 100 120 140

Months since first auction

0

1

2

3

4

5

6

ma

rke

t tu

rno

ve

r in

PL

N

1010 10Y bonds

Max = 33.86 bln PLN

Mon

th:

17

= 10.69 bln PLN

0 10 20 30 40 50 60 70


0

2

4

6

8

10

ma

rke

t tu

rno

ve

r in

PL

N

1010 5Y bonds

Max = 39.2 bln PLN

Mo

nth

: 9

= 13.6 bln PLN

0 5 10 15 20 25 30 35


0

2

4

6

8

10

12

ma

rke

t tu

rno

ve

r in

PL

N

1010 2Y bonds

Max = 14.67 bln PLN

Mo

nth

: 5

= 8.41 bln PLN

Notes: (1) thinner red lines indicate one standard deviation up and down from the mean on a certain month, (2) (2) Onlybonds with full life history contained in the period 2005:Q1 - 2020:Q3 were taken into consideration (2) Two year bonds

included: OK0113, OK0114, OK0116, OK0406, OK0407, OK0408, OK0419, OK0709, OK0710, OK0711, OK0712, OK0713,OK0714, OK0715, OK0716, OK0717, OK0720, OK0806, OK0807, OK0808, OK1012, OK1018, OK1206, OK1207, OK1208,

OK0419 (2) Five year bonds included: PS0310, PS0412, PS0413, PS0414, PS0415, PS0416, PS0418, PS0420, PS0511, PS0718,PS0719, PS1016 (3) Ten year bonds included: DS1013, DS1015, DS1017, DS1019

4. Zero trading days patterns resemble mirror reflection of the ones observed forvolume and number of trades. The minimum values of ZTD share in total trading daysin a given month are detected in the first 1/6th of bond’s lifetime (for 2Y: after 2 months, 5Y:after 10 months , 10Y: after 3 months). On average ZTD shares are lower for longer bonds,i.e. 10Y - 0.5, 5Y - 0.6, 2Y - 0.7, while the 2Y bonds experience big swing in values from 0 toabove 0.8 and 10y bonds ZTD shares mildly vary between 0.3 and 0.7.


Figure 7: Maximum time to maturity of bonds eligible to switch

2002 2004 2006 2008 2010 2012 2014 2016 2018 20200

0.2

0.4

0.6

0.8

1

1.2

year

fraction

eligible for switch-out

proposed envelope

Notes: (1) the red line represents an envelope based on moving averages (2) datascope: 2002- 2020:H1

5. Historically the lowest bid-ask spreads were observed in the segment of (6, 12]years to maturity and the highest in the short-end of the curve [0, 1.5] years.Table 2 reports the yearly average BAS for each segment for the whole DS (more than fifteenyears).

6. Ultra long end of Polish yield curve (12, 30] is very erratically inhabited with onlyone or two series quoted on fixing, and no representation since 2018 till now. TheBAS observed in this segment is comparable with less liquid segment of (1.5, 3.5] years.

7. Switch auctions influence prices by increasing BAS due to very limited motivationon both sides: potential buyers’ side who have alternative strategy of rolling NBPbills and potential sellers’ who maybe better off using these bonds to buy longerand more liquid ones. In the Figure 7 we show the history of maximum years to maturityof the bonds an investor wishing to buy longer bonds may switch from. It is clear that atthe end of 2017 qualitative change occurred and this maximum level increased from 0.6-0.8 to1-1.1. In the light of other above mentioned stylised facts it seems wise to exclude bonds withless than 0.85 years until mid 2017 and less than 1.20 from that point of time onwards.

Table 2: Statistics of liquidity measures of Polish fixed coupon government bonds traded onBondSpot in 2005:01-2020:06 - by segments

Amihud Roll GammaSegment mean std mean std mean std

[0, 1.5] 0.0446 0.0323 -184.20 115.29 0.0058 0.0491(1.5, 3.5] 0.0382 0.0305 -124.94 128.41 -0.0047 0.0358(3.5, 6] 0.0327 0.0290 -135.29 146.39 -0.0126 0.0468(6, 12] 0.0289 0.0242 -145.05 161.00 -0.0102 0.0483(12, 30] 0.0447 0.0326 -197.76 258.92 0.0068 0.1079[0, 30] 0.0375 0.0280 -138.02 110.34 -0.0047 0.0323

Notes: Amihud’s illiquidity measure (yield change as a proxy of return, volume taken from BS, when ZTD: volume of 0.1 wasimputed to avoid division by zero) , Roll’s effective spread measure (the averages reported in this table for Roll’s measure arepremultiplied by 1e6 solely for the clearer presentation), γ a measure proposed by Bao et al. (2011) (the averages reported inthis table for G measure are premultiplied by 1e6 as well), Roll’s and Gamma measures use 22 days window for covariance

Cov(∆ytmt,∆ytmt+1), where ytmt is a ytm at the fixing at t date of a certain bond we measure the liquidity.

8. Historically the least liquid segments were [0, 1.5] and (12,30] years to maturity.Table 2 presents the mean and standard deviation of three main liquidity measures we char-acterised in the previous chapter, namely: Amihud’s, Roll’s and BPW’s γ. It is importantto note that the higher Amihud’s and BPW’s and the lower Roll’s measures the less liquidsegment appears. Hence, data give a mixed picture as to which segment is the most liquid.


Since Amihud’s measure combines two types of information: total return and the volume thatfollows, we are inclined to say that the most liquid segment is (6, 12] with 0.0289 averageand standard deviation of only 0.0242 for the whole 15-year period. Table 10 shows yearlyaverages of these measures by segment, which offers yet another argument for that claim, asin 8 out of 15 years the segment of (6, 12] has the lowest Amihud’s read.

9. Switch operations make short bonds (eligible to switch from) richer than theinterpolated interest rate between NBP bills and [1.0, 1.5] segment of bonds. Keyconcept we use here is switch spread as a difference between linearly interpolated rate betweenNBP rate and the average ytm in the segment [1.0, 1.5] and the ytm of a particular shorterthan 1.25 years bonds. The rationale is based on the observation that NBP rate is a strongalternative for short term investors in risk-free instruments and it is rare (or not recorded)that the market expects the whole monetary policy easing or tightening cycle to last only ayear. Table 32 presents descriptive statistics by subsegment of [0.2, 1.0] and Figure 8 ilustratesall the pairs: years to maturity and switch spread for the whole sample period. Clearly, bondsup to 0.5 years to maturity were on average overvalued by 22.5 − 23.9 bps, the bonds withcurrent tenors in the range of (0.5, 0.7] by 10.1−10.3 bps and the ones falling into time bracketof (0.7, 1.0] by some 4 bps.

Figure 8: Switch spread of Polish fixed coupon government bonds in 2005:01-2020:06

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

years to maturity

-150

-100

-50

0

50

100

150

sw

itch

sp

read

in b

ps

22.3 23.9 21.510.1 10.3 6.4 4.7 4.1

Notes: (1) Switch spread is a difference between linearly interpolated rate between NBP rate and the average ytm in thesegment [1.0, 1.5] and the ytm of a particular shorter than 1.25 years bonds (2) light blue crosses represent single observations(3) red solid lines present averages in a given segment and dotted red lines indicate one standard deviation above and below

this mean

10. All segment-wise average yield time series are trend stationary when correctedfor long term variance a mode de Newley-West for lags of at least 18-months.Table 3 presents the results of different tests (Kwiatkowski et al. (1992)) for trend stationarity(stationarity around a deterministic trend) for various monthly lags used in long term variancecalculations. The times series of the averages can be found at Figure 1. 52

52We decided to use KPSS approach as the other frameworks like Augmented Dickey-Fuller work withdifferent null hypothesis (that the times series has unit root) and the way in which classical hypothesis testingis carried out ensures that the null hypothesis is accepted unless there is strong evidence against it. It hasbeen shown that these tests have low power against stable autoregressive alternatives with roots near unityand against fractionally integrated alternatives


Table 3: KPSS tests with different lags in 2005:01-2020:06 - by segments

[0, 1.5] (1.5, 3.5] (3.5, 6.0] (6.0, 12.0]

H(6m) 1.0000 1.0000 1.0000 1.0000H(8m) 1.0000 1.0000 1.0000 1.0000H(10m) 1.0000 1.0000 1.0000 1.0000H(12m) 0 1.0000 1.0000 1.0000H(14m) 0 1.0000 1.0000 1.0000H(16m) 0 0 1.0000 1.0000H(18m) 0 0 0 0pV (6m) 0.0100 0.0100 0.0100 0.0100pV (8m) 0.0194 0.0100 0.0100 0.0100pV (10m) 0.0381 0.0178 0.0118 0.0100pV (12m) 0.0596 0.0303 0.0223 0.0192pV (14m) 0.0836 0.0452 0.0356 0.0292pV (16m) 0.1000 0.0633 0.0479 0.0420pV (18m) 0.1000 0.0814 0.0665 0.0541stat(6m) 0.2438 0.3012 0.3255 0.3392stat(8m) 0.1909 0.2346 0.2538 0.2647stat(10m) 0.1603 0.1952 0.2112 0.2205stat(12m) 0.1408 0.1696 0.1831 0.1914stat(14m) 0.1279 0.1518 0.1633 0.1709stat(16m) 0.1189 0.1388 0.1485 0.1556stat(18m) 0.1127 0.1291 0.1371 0.1438

Notes: (1) Null hypothesis: H0 average yields in a given segment are trend stationary (2) H(lag) is a result of KPSS testevaluation and: if it reads 0 it means that the test fails to reject the null hypothesis that the times series is trend stationary,

and when it is 1 - the test rejects this null hypothesis. All tests are done witg 5% nominal significance level (3) pV (lag) indicatep-value of test statistics, (4) stat(lag) reports the KPSS statistics values (5) lags mean autocovariance lags to include in the

Newey-West estimator of the long-run variance

11. Share of BondSpot in total secondary market turnover is erratic and the list ofbonds traded is periodically shallow. Figure 9 shows the history of this ratio over the last15 years. Since 2012 the share is falling steadily from 0.15 − 0.18 to below 0.01 recently. Theprocess transpires also from Figure 11 (6th tile) with average zero trading days share risingfrom 0.35 in 2012 to above 0.80 in June 2020. With such a low representation the volume datafrom BondSpot cannot be reliably taken as a basis for weighting system in our yield curveestimations.

Figure 9: Share of BondSpot in total market turnover 2005:01-2020:06

2006 2008 2010 2012 2014 2016 2018 2020

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

sh

are

of

BS

in

to

tal m

ark

et

vo

lum

e

= 0.072

Notes: (1) outright transactions were considered only, as relevant in our enquiry.


Fig

ure

10:

Sel

ecte

dli

qu

idit

ym

easu

res

ofP

olis

hfi

xed

cou

pon

gove

rnm

ent

bon

ds

trad

edon

Bon

dS

pot

in20

05:0

1-20

20:0

6bysegments

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020H1

A([0,1

.5])

0.0589

0.0463

0.0429

0.0967

0.0787

0.0386

0.0269

0.0381

0.0484

0.0272

0.0298

0.0330

0.0275

0.0252

0.0367

0.0712

A((1.5,3

.5])

0.0538

0.0429

0.0346

0.0934

0.0700

0.0357

0.0218

0.0210

0.0299

0.0280

0.0334

0.0279

0.0228

0.0219

0.0239

0.0611

A((3.5,6

])0.0662

0.0290

0.0184

0.0720

0.0564

0.0237

0.0145

0.0110

0.0215

0.0194

0.0329

0.0269

0.0225

0.0270

0.0320

0.0679

A((6,1

2])

0.0377

0.0294

0.0197

0.0567

0.0532

0.0226

0.0119

0.0069

0.0189

0.0197

0.0221

0.0190

0.0230

0.0279

0.0359

0.0854

A((12,3

0])

--

--

0.0495

0.0314

0.0302

0.0114

0.0488

0.0398

0.0791

0.0576

0.0175

--

-A([0,3

0])

0.0559

0.0392

0.0310

0.0819

0.0651

0.0309

0.0199

0.0207

0.0313

0.0252

0.0335

0.0288

0.0245

0.0251

0.0320

0.0721

R([0,1

.5])

-268.16

-211.36

-215.77

-216.35

-251.18

-172.26

-154.04

-178.98

-211.32

-108.97

-183.37

-151.48

-136.34

-110.72

-167.67

-234.28

R((1.5,3

.5])

-244.65

-140.63

-86.70

-228.39

-112.68

-99.08

-101.95

-112.52

-139.26

-158.20

-99.25

-94.31

-51.42

-76.14

-81.01

-75.05

R((3.5,6

])-309.35

-157.19

-74.20

-184.09

-77.58

-63.08

-107.23

-114.38

-141.94

-199.43

-86.44

-79.76

-68.17

-110.41

-86.06

-46.86

R((6,1

2])

-238.83

-154.57

-52.30

-246.89

-105.82

-56.27

-120.19

-84.41

-187.66

-213.81

-63.23

-111.50

-91.36

-77.55

-48.90

-137.38

R((12,3

0])

--

--

-166.01

-133.05

-68.10

-51.79

-213.25

-165.35

-296.15

-145.29

-13.51

--

-R([0,3

0])

-266.75

-173.15

-115.19

-212.22

-137.28

-98.03

-112.62

-124.11

-170.50

-167.15

-125.35

-108.55

-81.72

-90.01

-91.16

-130.96

G([0,1

.5])

0.0233

0.0072

0.0182

-0.0314

-0.0154

0.0102

0.0093

0.0091

0.0090

0.0013

0.0138

0.0071

0.0073

0.0029

0.0087

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G((1.5,3

.5])

0.0108

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-0.0029

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-0.0385

-0.0065

-0.0022

0.0008

-0.0057

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-0.0004

-0.0032

0.0009

0.0006

-0.0339

G((3.5,6

])0.0212

-0.0082

-0.0077

-0.0248

-0.0412

-0.0112

-0.0027

-0.0041

-0.0277

-0.0015

-0.0315

-0.0108

-0.0063

0.0023

-0.0046

-0.0730

G((6,1

2])

-0.0011

-0.0065

-0.0120

0.0327

-0.0323

-0.0132

0.0002

-0.0146

-0.0161

0.0123

-0.0433

-0.0164

-0.0079

-0.0040

-0.0160

-0.0394

G((12,3

0])

--

--

0.0106

0.0031

-0.0029

-0.0147

-0.0083

0.0075

0.0723

-0.0048

-0.0071

--

-G((0,3

0])

0.0166

-0.0014

0.0007

-0.0108

-0.0306

-0.0050

0.0002

-0.0016

-0.0099

0.0030

-0.0080

-0.0045

-0.0032

0.0003

-0.0033

-0.0295

Note

s:A

-Amihud’s

illiquiditymea

sure

(yield

changeasaproxyofretu

rn,volumetaken

from

BS,when

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wasim

putedto

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effectivesp

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mea

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istable

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are

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ultiplied

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eclea

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al.(2011)(theaverages

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istable

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ereare

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use

22dayswindow

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Cov(∆

ytm

t,∆

ytm

t+1),

whereytm

tis

aytm

atth

efixingattdate

ofacertain

bondwemea

sure

theliquidity.


Figure 11: Selected characteristics of Polish fixed coupon government bonds traded onBondSpot in 2005:01-2020:06 - all segments

2006 2008 2010 2012 2014 2016 2018 2020

2

3

4o

uts

tan

din

g

am

ou

nt

in P

LN

105

= 350408

2006 2008 2010 2012 2014 2016 2018 202010

15

20

nr

of

bo

nd

s = 17.2

2006 2008 2010 2012 2014 2016 2018 20200

2000

4000

Vo

l in

mln

PL

N

= 699

2006 2008 2010 2012 2014 2016 2018 20203

4

5

6

yr

fra

c = 4.54

2006 2008 2010 2012 2014 2016 2018 20200

100

200

bid

ask s

pre

ad

= 11.31

2006 2008 2010 2012 2014 2016 2018 20200

0.5

1

ZT

D

= 0.61

2006 2008 2010 2012 2014 2016 2018 2020

0.6

0.8

1

fixin

g c

ove

r

in p

ct

= 0.84

2.4 Filtering rules and weight system framework

The yield curve estimation design in the less liquid markets (as Polish one) has to overcome somequalitative imperfections as compared with, for example US or UK markets. When deciding onfiltering out some bonds and then augmenting the information pool with available non-price data,we had the following principles in mind:

• Principle 1. Minimise the share of arbitrary decisions.

• Principle 2. Do not exclude bonds from the sample entirely, but diminish their weightaccordingly, unless the pricing is systemically distorted.


• Principle 3. Include as much reliable and useful information (static and dynamic data) aspossible to reflect importance of a certain bond in the yield curve formation.

• Principle 4. Include only risk free rates.

To that end, and bearing in mind the data availability and the stylised facts drafted in the previoussubsection we propose the following filtering (out) rules:

• Rule 1. from the broad set of Polish government bonds we exclude CPI-linkers, floaters, for-eign denominated and retail bonds. This is a usual choice in yield curve estimation literature.

• Rule 2. we take all pricing information of fixed and zero coupon bonds which are subject tofixing on BondSpot, with an exception in the next bullet:

• Rule 3. we exclude bonds with less than 0.85 or 1.20 years to maturity (till mid 2017 andafter that period) because their prices and therefore ytm are distorted significantly by theswitch operations of Polish MinFin (as it was clearly shown).

It is worth underlying firstly, that we do not exclude bonds with current maturity greater than12 years despite proven characteristics of this ultra long segment being under-represented (erraticpresence and only 1-2 bonds in the segment if any) and the most illiquid of all the segments dis-tinguished. We will let auxiliary data to speak for themselves (turnover and outstanding amounts)via the weight system. Secondly, we also do not exclude any bonds that may be described asoff-the-run or on-the-run as it is common in the literature of developed, liquid markets (Gurkaynaket al. (2011)), mainly because of bonds scaristy common to less liquid markets (even if we take allof the bonds they will constitute only a 1/4 or 1/5 of the number of available bonds in US or UKmarkets, let alone excluding more off-the-run bonds). Yet again here, we will differentiate the bondsimportance and quality of data through proper weights.

In order to reach our goal of using maximum depth and widest scope of information in theestimation of yield curves we propose the approach summarised below:

1. we include NBP bills rates (rebased to 365 days)53 as a good ultra-short interest rate whichinfluences expectations of market participants and offer a plausible alternative for any short-horizon investor and stable anchor for the beginning of yield curve. Other typical choices ofthe interest rate of short-end of the curve include: T-bills rates or xIBOR rates from moneymarket. In Poland T-bills were absent from the financial market’s history for a substantialperiod of time and their current share in both outstanding amounts of public debt as well asin total turnover is minuscule. The WIBOR rates are, of course available, but they are undergoing a methodology change. What is more, they intertwine risk free rates with some degreeof default risk, which is against our approach to yield curve modelling here - we stick to thepure default risk free instruments. Additionally, the underlying market of, say WIBOR1M,WIBOR3M or WIBOR6M is very shallow and there are almost no transactions in the realmarket. Hence the informational reliability and data quality is questionable.

2. the weight of the ultra-short point on a curve would have to be in line with its significance forfixed income investors in Poland, hence: of the magnitude of the sum of weights of all otherpoints.

3. we would use both outstanding amount and turnover based information to construct theweights, which constitutes a unique way of dealing with market data quality. Combiningthese two sources of dynamic information allows to flexibly treat on- and off-the-run issues aswell as reflect bigger relative importance in yield curve estimations of huge issues with veryhigh share in turnover and vice versa.

53recall that the bonds’ ytm in Poland is measured as if a year has 365 days, whereas NBP’s reference rateis in fact a yield to maturity in simple interest rate model calculated per year of 360 days, hence it should berebased to 365 days for comparability with the others


Turning now to the possible framework for weights system we will test in our yield curveestimation, we distinguish the subsequent degrees of freedom:

1. using or not the Rule 3 (filtering out the eligible for switch bonds) - 2 variants

2. using the full domain of tenors of leaving the longest segment (12, 30] out of estimation - 2variants

3. for each t use weights for outstanding amounts (W oat,i ) or for turnover share (W vol

t,i ) or bothfor all bonds i under fixing - 3 variants

4. use a weight for the shortest point of the curve (NBP rate) of 1 or 2 times bigger than thesum of all other weights used for bonds - 2 variants

On the top of these 24 sets54 we investigate the naive classic system of all equal weights (points3 and 4 above) in 4 settings as a result of choices made for point 1 and 2, ending up in total 28systems under investigation.

2.5 Polish yield curve estimation

We have decided to choose, estimate and experiment with a parsimonious yield curve environment,namely, in the popular Nelson-Siegel-Svensson form. The reasons for such choice are:

1. we are interested in morphology, dynamics and forecasting power of Polish yield curve with isa much closer goal to the one of monetary and fiscal authorities and further away from tradingand valuation of government securities domain in which tightness of the estimated curves iskey. In our enquiry, we recognise the following attractive characteristics of NSS fitting (recallfrom the literature review):

(a) the greatest flexibility at the short end of the curve, where it is needed the most also forthe term premia structure estimation.

(b) estimated curve is asymptotically flat for ultra long maturities by construction and def-inition.

(c) ability to capture local lack of monotonicity of the term structure (spot and forwardrates) thanks to two humps

(d) easy decomposition of parsimonious curve to level, slope and curvature elements

(e) the resultant curves are usually very smooth if compared with spline, which is a verydesirable feature as a yield curve may be viewed by (macro)economists as a collectionof intertemporal marginal rates of substitution. With such an interpretation in mind itwould be unreasonable to expect yield curve to be rough or zig-zacking.

2. in what follows, we propose to overcome almost all the numerical challenges (drawback of theNSS approach) listed in the previous chapter

(a) each market may have its own heuristics with regard to possible shapes on the yield curve- we have acknowledged it by carefully examining Polish market’s data and have ranmultiple tests and visual data inspection55 in order to create a list of different startingvectors of parameters Θ implying various shapes of yield curves encountered.

(b) the constrained optimisation methods may become particularly slow - we have tasted andused Matlab’s routine of internal-point whereas key step in achieving efficiency is thepreparation of bonds cash flow matrix and time to maturity vectors prior to the iterationpre se. The details will be given in this subsection.

54as a simple multiplication of variants listed above 2 × 2 × 3 × 255including but not limited to Matlab movie like: https://www.youtube.com/watch?v=dQYqCHBB1ow


(c) different combinations of starting parameters may produce an equally good fit to observeddata - we have not experienced the precise same goodness-of-fit measures for differentparameters, but we accept that such situation is probable, and therefore we have devel-oped more composite measure of goodness including wider scope of statistics. Rules ofsuch a ranking will be presented in subsequent subsections.

(d) there is usually a set of starting values of Θ needed [...] - as explained above, such a setwas prepared with more than 30 staring vectors, based on historical contexts.

(e) overly smooth specification of the yield curve fitting may disguise some important issue-characteristic or term structure related economic information in government bonds’ prices(i.e. tax effects, supply effects) - we have already proposed in filtering rules a treatmentof such specific bonds in Poland either by excluding them (switch bonds) or limiting theirinformational value in line with the importance for the market, which is measured bythe outstanding amounts and turnover on the secondary market.

3. we would avoid the drawbacks of spline methods, which are particularly undesired in the taskahead):

(a) lack of underlying financial or economic theory

(b) poor asymptotic behaviour of the long term rates

(c) oscillations of the estimated forward rates

(d) the estimations depend greatly on the location of the knot points between differentsegments of the curve, arbitrarily chosen in the procedure

The NSS parsimony accompanied by different weighting systems testing environment provide uswith enough degrees of freedom in our pursue of better than traditional setups. We have obtainedaverage errors of magnitude less than one basis point, whereas the bid ask spreads observed in themarket in different segments ranged from 5-6 to 30-40 basis (cf. Table 2) in the cases of ultra shortswitch bonds. Therefore we believe that there are other qualities of the estimated yield curves weshould concentrate (i.e. smoothness) rather than spurious precision.In this subsection we implement and test 28 particular filtering and weight systems to estimate fulltime series of parameters of NSS yield curve Θ = (β0, β1, β2, β3, τ1, τ2) for every date t in a dataset(DS):

y(x) = β0 + (β1 + β2)1 − e

(− x

τ1

)x/τ1

− β2e

(− x

τ1

)+ β3

1 − e

(− x

τ2

)τ/τ2

− e

(− x

τ2

) (64)

At the first glance, the form of parsimonious curve may seem assuming continuous compoundingof interest rate, but in reality the exponential function used has just a shape-forming role and wecan easily estimate yield curve for spot rates that are continuously compounded (Ccont

yc ) and withannual capitalisation (Cann

yc ) separately. Obviously, they will have slightly different parameters andshould be used to produce discount factors using different formulae. The yields are used to discountcash-flows from bonds to produce their estimated dirty prices. In the final presentation of the chosensystem that would be checked for robustness we would provide NSS parameters for both versions aswell as a refitted NSS curve into ytm* curve that is implied but previously fitted zero coupon yields.

The general problem we are facing at every date t in our dataset is finding the vector of parametersΘ which solves with assumed, sufficient accuracy:

minΘ

O = minΘ

{ N∑i=1

Wi(Pi(Θ) − pi)2 + Wnbp(R(Θ) − rnbp)2

}s.t. C(Θ) (65)


where Wi is a combined weight of modified duration Wmdi and either W oa

i or W voli or their sum:

Wi =

Wmd

i W oai , outstanding amounts weights only

Wmdi W vol

i , turnover weights only

Wmdi

(W oa

i + W voli

), combined weights

(66)

and the weight for the ultra-short end of the curve is defined as follows (with γ being a scalarmultiplier (1 or 2, in our simulations)):

Wnbp =

γ∑N

i=1Woai , outstanding amounts weights only

γ∑N

i=1Wvoli , turnover weights only

γ∑N

i=1

(W oa

i + W voli

), combined weights

(67)

Additionally, in the objective function we have R(Θ) which stands for estimated short term interestrate using set of parameters Θ and rnbp is the prevailing on that day NBP rate (rebased to 365days). It is worth underlying that the weights W oa

i , W voli ,Wmd

i are calculated for every day t in thedataset, based on static data.

Input: ∀t prices pi,t, weights Wi,t, rates rnbpt and a list starting values ΘOutput: time series of optimal Θ∗ for each tfor t ∈ DS do

retrieve from database for date t: pi and chosen set of weights Wi for every i-bondand W nbp, rnbp

create Bcf and vector t using static data at tfor s ∈ Θ do

while tolerance conditions not met docalculate P (Θk) = Bcf × df(Θk, t), R(Θk)calculate objective function value in k-th iteration

endreturn Θ∗

s

endchoose Θ∗

s with the lowest objective function’s value (O)store Θ∗ ≡ mins O(Θ∗

s) for the date tend

Algorithm 1: Calculating time series of the optimal Θ∗ (faster version)

We believe that it is reasonable in the prevailing negative interest rate environments that thetypical constraint for NSS on the short term zero-coupon rates to be greater than zero (β0 +β1 > 0)should be modified and set at −2%. On one hand, this will give sufficient space for the curve to befitted in Polish case of mid 2020, and on the other will still act as a non-slack constraint of parameterspace. The negative boundary for short term rates at this level is more and more common in yieldcurve estimation in the Switzerland or the Eurozone and we believe that is allows for some, althoughminimal these days, possibility of some negative rates, improving the fit in this segment of the curve.We reckon that the assumption that long term interest rates are positive is still valid (β0 > 0). Hencethe modified constraints are:

C(Θ) :

β0 > 0

β0 + β1 > −2%

τ1 > 0

τ2 > τ1

(68)


There are some technical niceties which are crucial for the algorithm to converge fast enough for thewhole dates in DS scope. Firstly, we implement some version of trusted region for our parametersΘ. We checked experimentally broad enough boundaries for parameters Θ when minimising: lowerbound Θ = [−0.10,−0.15,−2,−2, 0.15, 0] and upper bound Θ = [0.15, 0.15, 2, 2, 10, 305]. Secondly,in the minimisation problem 65 we need swift calculation of the objective function’s value as we areto call it roughly 1e6 times56 in the algorithm. Hence if our basic call to objective function lastsjust one second, the total calculation time for a DS would end up in the region of 2 weeks. Testingtime of all 28 weighting sets of parameters would be prohibitively long. The solution to this issueis to create an aggregated matrix of cash flows for each bond Bcf (the same for each iteration) andmultiply it by a vector of discount factors dfk = df(Θk, t) different for each iteration k in Θ space,instead of summing of weighted squared differences Pi(Θk)− pi bond by bond (for i ∈ [1, Nt], whereNt is a particular number of bonds taken to estimation at date t).In particular, we define Bcf for a given date (with dimensions: [Nt ×

∑Nti=1Mi], where Mi is the

number of future cash flows from i-th bond):

Bcf =

CF1,1 ... CF1,j ... CF1,M1 0 0 0 0 0 ... ... ... ... ... ... 00 0 0 0 0 CF2,1 ... CF2,j ... CF2,M2

0 ... ... ... ... ... 0...

......

......

......

......

.... . .

. . . 0 0 0 0 00 0 0 0 0 0 0 0 0 0 ... 0 CFNt,1 ... CFNt,j ... CFNt,M2

(69)

and the vector t consists of column-wise common year fractions of cashflows in Bcf matrix:

t = [t1,1 ... t1,i ... t1,M1 t2,1 ... t2,i ... t2,M2 ... tNt,1 ... tNt,i ... tNt,M2 ]′ (70)

Finally we allow function df() to return a vector of discount factors (annually or continuouslycompounded - depending on a set-up) taking arguments of the above-defined t and the current setof parameters in k-th iteration Θk, which in turn generate zero-coupon rates prevailing for thoseparameters. In such defined framework we have the column vector of modelled prices:

P (Θk) = Bcf × df(Θk, t) (71)

The speed advantage of this approach (Algorithm 1) as compared with calling cash flow schedule foreach bond in each objective function call (Algorithm 2 in the Appendix on page 153 ) is preciselydue to creation of re-usable big cash flow matrix of all bonds on a certain date and changing onlythe discount factor vector in every iteration and multiplication of a matrix by a vector is relativelycheap operation.

56we have approximately 4000 days and the experientially checked number of calls per yield curve estimationfor a given day is in the range of 100-300


Table 4: List of starting vectors of parameters Θ

β0 β1 β2 β3 τ1 τ2 β0 β1 β2 β3 τ1 τ2

ytmN−1 ytm1 − ytmN−1 -0.5 0.1 1 15 0.05 0 0.5 -1 2 15ytmN−1 ytm1 − ytmN−1 -0.5 -1.5 1 15 ytmN−1 0 -1 1.1 2 22ytmN−1 ytm1 − ytmN−1 0.5 2 1 15 ytmN−1 0 -1 -2 2 15ytmN−1 ytm1 − ytmN−1 1 3 1 15 ytmN−1 0 0.5 -1 2 15ytmN−1 ytm1 − ytmN−1 -0.5 0.1 1 19 0.06 -0.020 -0.01 -0.03 0.35 30ytmN−1 ytm1 − ytmN−1 -1 0.1 2 15 0.04 -0.015 -0.01 -0.03 0.35 30ytmN−1 ytm1 − ytmN−1 -2 0.1 2 15 0.02 -0.005 -0.01 -0.03 0.35 30ytmN−1 ytm1 − ytmN−1 1 -0.5 1 15 0.06 -0.020 -0.01 -0.03 0.55 20ytmN−1 ytm1 − ytmN−1 2 0.2 2 19 0.04 -0.015 -0.01 -0.03 0.55 20ytmN−1 ytm1 − ytmN−1 -0.1 0.5 1 8 0.02 -0.005 -0.01 -0.03 0.55 20ytmN−1 ytm1 − ytmN−1 -1 0.1 2 8 0.04 0.020 -0.01 0.03 0.35 30

ytmN−1 × 1.5 ytm1 − ytmN−1 -1 0.1 2 19 0.02 0.015 -0.01 0.03 0.35 30ytmN−1 × 1.5 ytm1 − ytmN−1 -1 0.1 2 22 0.01 0.005 -0.01 0.03 0.35 30ytmN−1 × 0.5 ytm1 − ytmN−1 -1 0.1 2 19 0.04 0.020 -0.01 0.03 0.55 20ytmN−1 × 0.5 ytm1 − ytmN−1 -1 0.1 2 22 0.02 0.015 -0.01 0.03 0.55 20

0.05 0 -1 1.1 2 22 0.01 0.005 -0.01 0.03 0.55 200.05 0 -1 1.1 1 10

Notes: (1) additionally we use last fitted parameters (from t− 1 date) (2) ytmN is the ytm of a bond that is the longest in afixing table for a given date t, analogously: ytmN−1 is the ytm of a bond that is second to the longest and ytm1 is the ytm of

the shortest bond

For a single optimisation run we used fmincon function in Matlab with its default set-up based onthe algorithm of interior point described and developed by R. H. Byrd et al. (1999) and Waltz etal. (2006).In what follows, we have analysed the results of Polish government yield curve estimations in 28different weight/filtering systems in the ensuing groups of measures:

1. statistics of estimated parameters Θt i.e. mean, median, standard deviation, interquartilerange, max-min range

2. statistics of goodness-of-fit MAE, WMAE, maximum absolute difference, hit ratio, cheap-/rich ratios,

3. auxiliary characteristics of estimated interest rates: smoothness (henceforth: SMO, asin Equation 36), short rates fit57 , volatility and level of synthetic interest rates in segments

4. optimisation algorithm - related: exit flags, number of iterations, number of calls toobjective function, execution time (in seconds)

With the aim to verify our Hypothesis 1 that there exist a class of weighting schemes whichimproves fit relative to conventionally used methods, we propose two angles of analysis. First weselect two dimensions: MAE for the goodness-of-fit and SME for the smoothness/roughness inwhich we plot the results for all the systems (Figure 12). Recall from the literature review, that thesmoothness measure proposed is close to zero when the yield curve is very smooth and higher inthe situations in which the yield curve is rough in the sense that the second derivative changes signquite often. Having the results for all 28 sets we made the following observations:

1. conventional weighting (all equal weights in the yield space) results in the roughest and notwell fitted yield curves

2. systems based on combined outstanding amounts and turnover give less rough yield curvesthen conventional with only slightly worse mean error

3. systems based on solely on outstanding amounts produce even smoother curves but with atrade-off with MAE

4. systems based on solely on turnover give the smoothest curves and yet again worse MAE

57a difference between fitted y(1/52) and rebased for 365 days NBP reference rate - in basis points


5. excluding eligible-for-switch bonds improves smoothness (lowers the roughness) and error (low-ers MAE) significantly (roughness by the factor of 5 and MAE by approx. 30%).

Figure 12: Goodness-of-fit and smoothness of all 28 tested weight systems

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

MAE

0

1

2

3

4

5

6

7

8

sm

oo

thn

ess

2

46

8

10 1214 16

1820

22 24

25

26

27

28

0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7

MAE

0.2

0.25

0.3

0.35

0.4

0.45

0.5

sm

ooth

ness

1

3

5

7

9

11

13

15

17

19

21

23

Notes: (1) Lower panel zooms in a part of the upper panel for MAE in the range of [0.5, 0.7] and smoothness in the range of[0.2, 0.5] (2) the labels are in line with number of weight systems in Table ??. (3) green dots - are for system with W = 4 (equalweights), cyan dots - W = 3 (combined outstanding amount and turnover), blue dots - W = 2 (turnover), black dots - W = 1

(outstanding amounts) (4) ◦ indicates a system where eligible for switch bonds are excluded from the estimation.

From the set of pairs: (MAE, SMO) we may extract the ones which form a trade-off frontier(henceforth: TOF) - the subset of pairs for which there is no better MAE without worsening SMOand vice-versa. This subset consists of the weight systems with numbers: 1, 5, 13, 17 and 25.Common features of these systems are the decisions: to exclude eligible-for-switch bonds and notto truncate the domain to 12 years. With the exception of system labelled 25 (equal weights), theTOF’s systems are based on the idea of heavy weight of the short end of the curve (with γ = {1, 2}.Solely this findings are enough to support the Hypothesis 1 of existence of a class of weights thatimproves fit relative to conventional methods of equal weights.

As a second angle we propose here a composite measure based on rankings of the full set of28 systems in the following categories: average standard deviation of yields in selected tenors of thecurve, average max-min range, average interquartile range, average MAE, average MAD (mean ab-solute difference) all over the time axis, short rate fit (an absolute difference between fitted y(1/52)and rebased for 365 days NBP reference rate) and the smoothness/roughness measure introduced inthe previous chapter. Therefore we have 3 volatility, 3 error and 1 smoothness measures fro whichthe less is better - less volatility, smaller errors and less rough curve. Upper panel of Table 5 presentsreadings of these measures whereas the lower panel consists of their rankings as well as the averagerank across the seven categories. The best 5 systems as far as this composite measure is concernedare: 1, 17, 5, 21 and ex equo: 9, 19 and 23. Yet again these systems have similar characteristics tothe ones in TOF and have a common part of systems with numbers: 1, 5 and 17.


Table 5: Volatility, error and smoothness statistics for all 28 tested weight systems

Weight systemexc. sw < 12y W γ STD MMR IQR MAE MAD SRF SMO

1. Y N 3 2 0.01549 0.06695 0.02887 0.53973 8.94 0.06911 0.334642. N N 3 2 0.01559 0.06843 0.02879 0.73507 13.30 0.18038 4.437413. Y Y 3 2 0.01551 0.06739 0.02888 0.59731 8.44 0.05495 0.431754. N Y 3 2 0.01561 0.06906 0.02880 0.82167 12.64 0.16288 4.978725. Y N 3 1 0.01549 0.06683 0.02887 0.53903 8.87 0.13172 0.362536. N N 3 1 0.01560 0.06874 0.02880 0.73691 13.26 0.35590 4.862597. Y Y 3 1 0.01551 0.06734 0.02888 0.59597 8.39 0.10495 0.380658. N Y 3 1 0.01561 0.06908 0.02881 0.81605 12.60 0.31642 5.091499. Y N 2 2 0.01549 0.06689 0.02889 0.61122 11.06 0.07343 0.2248310. N N 2 2 0.01558 0.06865 0.02882 0.82359 15.79 0.16588 2.8331311. Y Y 2 2 0.01551 0.06734 0.02891 0.65956 9.86 0.06069 0.2377912. N Y 2 2 0.01560 0.06897 0.02882 0.90533 14.80 0.15185 2.8689213. Y N 2 1 0.01549 0.06704 0.02889 0.60710 11.02 0.13718 0.2179414. N N 2 1 0.01558 0.06854 0.02883 0.82237 15.84 0.33084 2.9377915. Y Y 2 1 0.01551 0.06746 0.02891 0.64944 9.74 0.10915 0.2888416. N Y 2 1 0.01559 0.06889 0.02883 0.89948 14.74 0.28565 3.0251817. Y N 1 2 0.01548 0.06713 0.02886 0.55797 9.26 0.07709 0.2759318. N N 1 2 0.01558 0.06865 0.02878 0.75527 13.74 0.19543 3.6392719. Y Y 1 2 0.01550 0.06718 0.02888 0.61509 8.60 0.06311 0.2996720. N Y 1 2 0.01560 0.06896 0.02881 0.84746 13.21 0.18602 3.9038621. Y N 1 1 0.01548 0.06711 0.02886 0.55496 9.17 0.14036 0.3218922. N N 1 1 0.01558 0.06859 0.02879 0.75983 13.79 0.38757 3.6570423. Y Y 1 1 0.01550 0.06714 0.02887 0.61194 8.57 0.11622 0.2965424. N Y 1 1 0.01560 0.06895 0.02881 0.84522 13.24 0.35922 3.7985025. Y N 4 - 0.01551 0.06818 0.02887 0.51024 7.47 1.23639 1.2079126. N N 4 - 0.01564 0.06877 0.02877 0.70530 10.26 4.31371 6.2469127. Y Y 4 - 0.01553 0.06825 0.02888 0.57409 7.10 1.09017 1.6203528. N Y 4 - 0.01566 0.06906 0.02881 0.79804 9.86 3.93117 7.32579

Weight system ranks in the categoryexc. sw < 12y W γ STD MMR IQR MAE MAD SRF SMO Average mark

1. Y N 3 2 4.00 3.00 17.00 3.00 8.00 4.00 9.00 6.862. N N 3 2 19.00 15.00 4.00 16.00 22.00 16.00 23.00 16.433. Y Y 3 2 13.00 11.00 22.00 8.00 4.00 1.00 12.00 10.144. N Y 3 2 25.00 26.00 6.00 22.00 18.00 14.00 25.00 19.435. Y N 3 1 3.00 1.00 18.00 2.00 7.00 10.00 10.00 7.296. N N 3 1 22.00 20.00 5.00 17.00 21.00 22.00 24.00 18.717. Y Y 3 1 11.00 9.00 24.00 7.00 3.00 7.00 11.00 10.298. N Y 3 1 26.00 28.00 9.00 21.00 17.00 20.00 26.00 21.009. Y N 2 2 5.00 2.00 25.00 10.00 16.00 5.00 2.00 9.2910. N N 2 2 15.00 18.00 12.00 24.00 27.00 15.00 15.00 18.0011. Y Y 2 2 9.00 10.00 27.00 14.00 13.00 2.00 3.00 11.1412. N Y 2 2 21.00 25.00 11.00 28.00 26.00 13.00 16.00 20.0013. Y N 2 1 6.00 4.00 26.00 9.00 15.00 11.00 1.00 10.2914. N N 2 1 17.00 16.00 13.00 23.00 28.00 21.00 17.00 19.2915. Y Y 2 1 12.00 12.00 28.00 13.00 11.00 8.00 5.00 12.7116. N Y 2 1 20.00 22.00 14.00 27.00 25.00 19.00 18.00 20.7117. Y N 1 2 2.00 6.00 16.00 5.00 10.00 6.00 4.00 7.0018. N N 1 2 16.00 19.00 2.00 18.00 23.00 18.00 19.00 16.4319. Y Y 1 2 8.00 8.00 21.00 12.00 6.00 3.00 7.00 9.2920. N Y 1 2 24.00 24.00 8.00 26.00 19.00 17.00 22.00 20.0021. Y N 1 1 1.00 5.00 15.00 4.00 9.00 12.00 8.00 7.7122. N N 1 1 18.00 17.00 3.00 19.00 24.00 24.00 20.00 17.8623. Y Y 1 1 7.00 7.00 20.00 11.00 5.00 9.00 6.00 9.2924. N Y 1 1 23.00 23.00 7.00 25.00 20.00 23.00 21.00 20.2925. Y N 4 - 10.00 13.00 19.00 1.00 2.00 26.00 13.00 12.0026. N N 4 - 27.00 21.00 1.00 15.00 14.00 28.00 27.00 19.0027. Y Y 4 - 14.00 14.00 23.00 6.00 1.00 25.00 14.00 13.8628. N Y 4 - 28.00 27.00 10.00 20.00 12.00 27.00 28.00 21.71

Notes: (1) list of tenors for the calculation of statistics [1/52, 1/12, 0.25, 0.5, 1, 2, 3, 4, 5, 7, 10, 12] (2) STD stands forstandard deviation, MMR for max min range, IQR for interquartile range, MAE - mean average error, MAD - maximum

absolute difference (in basis points), SRF - short rate fit (an absolute difference between fitted y(1/52) and rebased for 365 daysNBP reference rate - in basis points) (2) first five columns indicate: identification number of a weight set, decision to exclude

eligible for switch bonds, decision to limit the time domain to 12 years to maturity, weight base: 1 - W oa, 2- W vol, 3 -W oa +W vol and 4 - all equal weights (3) short rate fit is (4) smoothness is calculated as proposed in Equation 36 (5) the bold

marks indicate that a particular system is in top 5.


Table 6: Descriptive statistics of zero coupon rates for different tenors and systems

1/52 1/12 1/4 1/2 1 2 3 4 5 7 10 12

mean(1) 0.0321 0.0321 0.0321 0.0323 0.0330 0.0348 0.0368 0.0386 0.0401 0.0425 0.0446 0.0454std(1) 0.0154 0.0155 0.0157 0.0161 0.0165 0.0167 0.0163 0.0158 0.0154 0.0146 0.0140 0.0139max(1) 0.0659 0.0657 0.0666 0.0705 0.0742 0.0752 0.0742 0.0731 0.0723 0.0722 0.0721 0.0717min(1) 0.0010 0.0008 0.0005 0.0001 -0.0002 0.0007 0.0026 0.0047 0.0066 0.0098 0.0118 0.0117Q1(1) 0.0152 0.0152 0.0151 0.0151 0.0156 0.0171 0.0198 0.0226 0.0252 0.0292 0.0323 0.0331median(1) 0.0355 0.0359 0.0369 0.0382 0.0395 0.0399 0.0405 0.0417 0.0430 0.0454 0.0469 0.0480Q3(1) 0.0456 0.0453 0.0447 0.0445 0.0454 0.0478 0.0504 0.0525 0.0542 0.0562 0.0574 0.0578mean(28) 0.0318 0.0316 0.0314 0.0316 0.0326 0.0348 0.0368 0.0386 0.0401 0.0425 0.0445 0.0453std(28) 0.0155 0.0158 0.0163 0.0166 0.0168 0.0167 0.0163 0.0158 0.0154 0.0147 0.0141 0.0140max(28) 0.0660 0.0686 0.0769 0.0779 0.0752 0.0744 0.0738 0.0731 0.0729 0.0730 0.0731 0.0730min(28) 0.0008 0.0007 0.0005 0.0002 -0.0001 0.0007 0.0026 0.0046 0.0067 0.0098 0.0116 0.0110Q1(28) 0.0152 0.0148 0.0147 0.0148 0.0153 0.0170 0.0198 0.0227 0.0253 0.0292 0.0320 0.0327median(28) 0.0344 0.0348 0.0356 0.0369 0.0389 0.0400 0.0409 0.0419 0.0430 0.0452 0.0469 0.0480Q3(28) 0.0454 0.0447 0.0439 0.0439 0.0450 0.0478 0.0504 0.0525 0.0542 0.0562 0.0573 0.0577

Notes: (1) tenors (in years) are in columns (2) statistics with label 1 are calculated for the curves obtained in the highestranked system and with label 28 - the lowest ranked.

To summarise, we have found heuristically a class of weighting system for Polish governmentbonds yield curve to be used in NSS estimation which significantly improves the fit and smoothnessas compared to the traditional approach of all equal weight. This class has three core characteristics:

• at least the same weight for the short end of the curve as a sum for all other tenors of bonds

• exclusion of eligible-for-switch bonds from the estimation

• bonds’ weights based on at least outstanding amounts (in the best systems we had either W oa

or W oa + W vol)

2.6 Estimation results for the best weight system

In this subsection, we present and discuss the results of Nelson-Siegel-Svenssson estimation of Polishgovernment yield curves for the period of 2005:01-2020:06 using the highest ranked weighting systemin Table 5, namely the one labelled 1. Table 6 provides detailed statistics of time series of zero couponrates for selected 12 tenors in two systems: the highest (label 1 ) and the lowest (label 28 ) rankedin that table and Figure 66 in Appendix provides time series of MAE errors in these two.


Figure 13: Full history of estimated zero coupon yield curves in Poland

Notes: (1) the highest ranked weighting system in Table 5 was used to produce these estimations (2) data timespan:2005:01-2020:06 (3) the fit was performed with the assumption of annual compounding of interest rates (Cann

yc )

It is worth noting that the mean (and median) curve in system 1 is lightly higher in the tenorsup to and inlcuding 1Y than in system 28, which is not surprising in the light of previous discussionon switch bonds exclusion due to existence of a non-negative valued switch option. On the otherhand in the same - short - segments standard deviation is lower in the high ranked system. Thehighest standard deviation readings in both systems sit in the 1-2Y region, whereas the lowest arefor the long end (12Y), which is desirable and was observed in the reality.

Figure 14: An example of estimated NSS family of zero coupon forward curves


Figure 15: An example of estimated par curve with rich/cheap analysis

Notes: (1) grey coloured dots represent switch bonds and are excluded from estimation due to the fact of their prices beingheavily distorted by switch operations conducted by MinFin

Visual inspection of Figure 13 confirms the advantages of the chosen weighing system. The yieldcurves are smooth, start at or very close to NBP reference rates, most of them have flat asymptoticlong end and cross time single tenor volatility looks tamed. The same qualitative results have beenachieved when fitting NSS curve with the assumption of the zero coupon rates being continuouslycompounded. Figure 14 is an example of a family of zero coupon (both annually and continuouslycompounded) and forward (1-month, 1-, 2-, 3-, 4-, 5-year) curves calculated for a trading date of10th of March 2017, which may be treated as an augmented information slice of the Figure 13.

Forward and spot rates are not directly observable on the market, hence in order to presentgoodness-of-fit of one particular curve estimated for a single date we have to either (1) calculateimplied (predicted) ytm of a series of bonds that were subject to fixing on that date using theestimated spot NSS curve or (2) calculate hypothetical curve called par curve, by introduction ofsynthetic securities which are priced at par (100% of nominal value) using the spot NSS by changingthe coupon rate. These coupon rates are in fact par yields. In the Figure 15 we combined this twoapproaches by plotting a par curve and colouring the dots that represent single bonds in line withthe difference between the predicted and observed ytm. It is clearly visible that the curve fits insidebid ask spread, except for short term bonds (switch bonds), which are excluded from estimation inthis system of weights (grey coloured). The designed features of the weight system and filtering rulesallow for very realistic construction of the curve with benchmark on-the-run issues (i.e. DS0725)being priced almost exactly on the curve and small, illiquid bonds are trading dearer to the curve, asthey sit in long horizon portfolios already and there is no active ongoing price discovery mechanism(i.e. WS0429).


Table 7: NSS parameters yearly averages

2005 2006 2007 2008 2009 2010 2011 2012

β0 0.0833 0.0772 0.0673 0.0600 0.0544 0.0650 0.0788 0.0756β1 -0.0291 -0.0360 -0.0228 -0.0022 -0.0163 -0.0297 -0.0361 -0.0287β2 -0.0364 0.0086 0.0347 0.0158 0.0280 0.0312 -0.0193 -0.0428β3 -0.1016 -0.0836 -0.0411 0.0016 0.0157 -0.0003 -0.0978 -0.1367τ1 1.37 2.95 3.94 1.17 2.56 3.72 2.74 2.48τ2 24.40 34.30 24.89 35.33 28.67 37.67 50.40 93.40

2013 2014 2015 2016 2017 2018 2019 2020 H1

β0 0.0671 0.0719 0.0707 0.0771 0.0751 0.0833 0.0767 0.0798β1 -0.0371 -0.0476 -0.0546 -0.0618 -0.0599 -0.0680 -0.0614 -0.0703β2 -0.0344 -0.0437 -0.0422 -0.0537 -0.0559 -0.0735 -0.0523 -0.0629β3 -0.0986 -0.0734 -0.1010 -0.0916 -0.1723 -0.3900 -0.1599 -0.3687τ1 2.46 2.00 2.29 2.10 1.70 1.83 2.24 2.76τ2 63.42 17.76 18.51 12.97 35.01 77.97 23.72 72.50

Figure 16: Level, slope and curvature of NSS estimated Polish zero coupon yield curve

2006 2008 2010 2012 2014 2016 2018 2020-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

level 10Y

slope 10Y-0.5Y

curvature 10Y-2*3Y+0.5Y

Notes: (1) the highest ranked weight system labelled 1 was used (2) the particular tenors to calculate level, slope and curvatureare arbitrarily chosen: 0.5Y, 3Y and 10Y.

Figure 16 contains time series for the long term spot rates, slope of the curve and its curvature.The exact choice for the tenors from which to calculate these measures is arbitrary and we proposehere to take 0.5Y, 3Y and 10Y, as representation of short, medium and long term rates. Level oflong interest rates is persistent and in line with the trends in the monetary policy, whereas the slopeis more cyclical, predominantly being positive except of short period after Lehman Brothers in 2008and curvature (measured as 10Y −2×3Y + 0.5Y ) also with the same exception period after FC wasnegative and slightly less volatile than slope.

Table 7 and Figure 67 in the Appendix inspect the NSS parameters estimated for Polish curve.The parameters β0 and β1 are volatile in separation but as a sum behave almost in line with NBPreference rates, because the system of weights here put double stress on the short term rate fit. Theparameter β2 responsible for the convexity of the first hump seems to have experienced two periodsin the 15-year history we deal with. First one ends approximately in 2011 during which we observehigher readings and higher volatility of β2. In the second part of the time series β2 is less volatileand the readings are on average lower. The parameter τ1 which determines the position of this firsthump hovers between 0.3 and 10, but yearly average - between 1.37 and 3.94. The parameters whichgovern the second hump: β3 and τ2 are very often taking extreme values, especially in the periods ofrapid descent of long interest rates. Very high readings of τ2 in some dates indicate that the secondhump is not crucial in estimating the curve with parsimonious form.


Figure 17 present empirical distributions of one month excess returns in various maturity bondsover short term rate (one month zero coupon rate) in Poland. All of the means of these excessreturns are positive and increasing in tenors, again suggesting some time structure of risk premia.All distributions up to 5 -years are leptokurtic. The ones from 2-year to 10-year are skewed to theright and the shorter distributions are skewed to the left. We will draw from these characteristicsand time series in the term premia estimation in the next chapter.

Figure 17: Distributions of one month excess returns for Polish bonds

-2 0 2 4 6

10-3

0.0

002

-0.01 -0.005 0 0.005 0.01 0.015

0.0

005

-0.02 -0.01 0 0.01 0.02

0.0

011

-0.04 -0.02 0 0.02 0.04

0.0

014

-0.04 -0.02 0 0.02 0.04

0.0

017

-0.05 0 0.05

0.0

022

-0.05 0 0.05

0.0

024

-0.05 0 0.05

normality not ruled out

0.0

026

-0.1 -0.05 0 0.05 0.1

normality not ruled out

0.0

037

Notes: (1) NSS yield estimations with the highest ranked weight system labelled 1. (2) sigma, S, K in the textboxes of everytile stand for standard deviation, skewness and kurtosis respectively. (3) black lines represent hypothetical normal distributions

with the same means and standard deviations as the original time series. (4) red vertically rotated text indicate mean of acertain time series. (5) n indicates number of months to maturity of a certain hypothetical bond. (6) normality not ruled out -

means that Anderson-Darling test with 0.05 significance level could not rule out that a particular time series in normal.

2.7 Summary

In this Chapter we have collected more than ten stylised facts on Polish government bonds market,which were consequently used in filtering and weight system design. These facts revealed importanceof liquidity in bonds lifespan, heterogeneity of liquidity in different yield curve segments, bid-askspreads behaviour as well as price distortion in the very short end of the curve due to switch auc-tions. We found that in our data maximum timespan all segment-wise average yield time series aretrend stationary when corrected for long term variance, which will have important implications inthe next chapter on expectations hypothesis testing.

Moreover, we argued that as for the ultra short end (1/52 years) rate we should use NBP bill


rebased rate (which importance in the default free instruments market in PLN cannot be over-looked). We developed here very efficient algorithm which allowed for relatively quick estimationsof 28 systems with approximately 4 thousand days each. We have tested 28 different weight systemsand rank them in the space of goodness-of-fit and smoothness and confirmed that there is a class ofweights that systematically gives better results than the classic approach of all equal weights. Thehighest ranked systems have at least the same weight for the short end of the curve as a sum for allother tenors of bonds, eligible-for-switch bonds were excluded from the estimation and weights werebe based on at least outstanding amounts.


3 Chapter 3. Expectations Hypothesis domain in Poland

The carefully estimated time series of the Polish government yield curves will now be used to extractmarket expectations of the future interest rates via calculation of implied forward rate structurewhich, in turn, is instrumental in pure expectations hypothesis testing. This chapter provides theresults of extensive tests of pure expectations hypothesis and will help to provide evidence for ourHypothesis 2 which claims that Pure Expectations Hypothesis does not hold universally. In general,we would like find sufficient evidence to reject the null hypothesis that forward rates are unbiasedpredictors of the future spot rates in Poland, which in turn will give rise to hypothesising on riskpremia existence and their structure estimation in the proceeding chapters.

Our staring point is the search for non-zero term premia structure’s existence is a short recall ofour yield curves’ static properties. Table 8 shows selected descriptive statistics of the zero couponrates in Poland in the whole dataset calculated for 3 and 6 months and from 1 to 10 years for eachyear. Additionally we report these statistics for level, slope and curvature of the yield curve, thepotential candidates for the state variables in the forecasting models. Couple of patterns transpirethere: (1) the most volatile interest rates are those in segment of 2-3Y, (2) volatility decreases in thesector 3-10Y to the lowest levels on the curve, (3) persistence (of different lags) follow the patternobserved for the volatility. But what is striking is that the mean yields are strictly increasing, with2Y − 0.25Y spread of 26 basis points, 5Y − 0.25Y - 89 basis points and the long term bonds spread10Y − 0.25Y - 123 basis points. It is hard to explain, in absence of any hypothetical term or riskpremia, because during the considered period monetary policy was in easing or natural bias andthe interest rates were, generally, in a strong downward trend. Our intuition calls for two possiblestrands of justification. First, the markets are very poor in forming expectations of future pathsof interest rates. Second, the market expectations are formed correctly with a consideration ofmonotonically increasing risk premia. In this chapter we will deal with the first intuition, whereasin the next we will check if hypothesising on risk term premia is plausible for Polish governmentbonds.

Table 8: Descriptive statistics of NSS fitted yields in Poland, 2005:01-2020:06

mean std max min ρ(1) ρ(12) ρ(24) ρ(36)3 months 0.0322 0.0158 0.0651 0.0007 0.9732 0.7016 0.5291 0.46316 months 0.0324 0.0161 0.0652 0.0005 0.9743 0.7297 0.5617 0.47541 year 0.0330 0.0166 0.0674 0.0005 0.9752 0.7644 0.6026 0.48602 years 0.0348 0.0167 0.0683 0.0017 0.9752 0.7917 0.6367 0.49023 years 0.0368 0.0164 0.0675 0.0037 0.9748 0.7954 0.6423 0.48554 years 0.0386 0.0159 0.0667 0.0059 0.9744 0.7888 0.6356 0.47505 years 0.0401 0.0154 0.0659 0.0081 0.9740 0.7775 0.6239 0.46136 years 0.0414 0.0150 0.0654 0.0100 0.9738 0.7645 0.6113 0.44647 years 0.0425 0.0147 0.0649 0.0116 0.9737 0.7518 0.6002 0.43208 years 0.0433 0.0144 0.0646 0.0128 0.9738 0.7405 0.5914 0.41929 years 0.0440 0.0142 0.0644 0.0137 0.9739 0.7310 0.5852 0.408410 years 0.0446 0.0141 0.0644 0.0143 0.9741 0.7235 0.5813 0.3998Level 0.0454 0.0139 0.0654 0.0148 0.9745 0.7137 0.5788 0.3885Slope 0.0122 0.0073 0.0244 -0.0052 0.9507 0.0473 -0.1275 -0.0181Curvature 0.0034 0.0049 0.0125 -0.0089 0.9270 0.5950 0.4466 0.2665

Notes (1) NSS fit using weight system labelled 1.

Prior to proceeding directly into testing, two remarks are technically important. First, we haveshown evidence that Polish interest rates are unit root processes (at least in the period under consid-eration) hence usually the regressions used to test expectation hypotheses would incorporate spreadsto some contemporary observed rate, i.e. 1Y rate, instead of nominal levels. This is in line withmajority of research conducted for liquid markets, despite a few authors positing that the US ratesare stationary (cf. Sarno et al. (2007)). Second, we will use beginning of month yield curves insteadof daily data set, which again is in line with the approach in vast majority of studies. Inference indaily data of the processes that are highly persistent may prove to be arduous and weary, without


any improvement on the statistical significance of the results.

3.1 Excess return on term premium regressions

The so called in the literature term premium regressions as performed by Fama & Bliss (1987) arethe first of the two classic models to consider when testing of pure expectations hypothesis (PEH).Recall from the literature review that the regressions are of a form:

rx(n)t+h = α + β

(f t+h,t+nt − y

(h)t

)+ ϵ

(n)t+h (72)

where rx(n)t+h is an excess return (over h-year yield observed at time t) of the investment in n-year

bond at time t with disinvestment after h-years (investment horizon), f t+h,t+nt is a forward rate

observed (implied) at t and good for discounting to time t + h a cash-flow occurring at t + n. Notethat the choice of the maturities of bonds and the length of investment horizons is somehow limitedby the length of our data series. Having 15.5 year history (186 months) we could have tried to testPEH in combinations of horizon and length of a bond like h = 6 months and n = 180 months butthen we would be left with just six observations to infer from, which - obviously - is not plausible.We have chosen vectors of tested horizons and bond maturities so that to have the analysed timeseries length of variables least of approximately 2/3-rds of the maximum data scope. Hence themaximum bond we consider is of 6-year maturity. Selected pairs (h, n) should satisfy ∀(h,n)h < nas explained in Chapter 1, translating here to the maximum horizon in a vector h being less then6-years, as we are interested in excess returns in investments in bonds for shorter periods than theirmaturity. In consequence of these constraints we have chosen the following vectors: of horizons -h = [6, 12, 24, 36, 48, 60] and of bond maturities n = [24, 30, 36, 42, 48, 54, 60, 72] in months.

Figure 18: Pure expectations hypothesis: regression of type 1 and 2 (Fama & Bliss) for Poland

24 30 36 42 48 54 60 72

n - bond time to maturity in months

6

12

24

36

48

60

h -

horizon in m

onth

s

0

0

-1

-1

-1

-1

0

0

0

-1

-1

-1

0

0

0

-1

-1

-1

0

0

0

-1

-1

0

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-1

-1

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-1

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1 1

1

1

1

1

1

1

1

1

1

1

1

Notes: (1) 0 indicates that there is no sufficient evidence to reject null hypothesis of PEH, (2) 1 - sufficient evidence (with at0.05 significance) to reject PEH, (3) -1 - not applicable (i.e. a horizon is longer or equal bond’s maturity) (4) NSS fit using

weight system labelled 1.

Table 19 reports all of the details of all the regressions (32) performed for different selected pairs(h, n) such that ∀(h,n)h < n. Our null hypothesis is that the coefficient β = 0, because PEH claimsthat nothing should forecasts returns including observed at t spread of forward rates over some


shorter bond. As in all our hypothesis testing in this thesis we assume significance level of 0.05. Dueto major overlapping of periods in our time series, before inference, we correct standard errors forheteroscedasticity (heterogeneity of variances) and autocorrelation (henceforth: HAC) proposed byNewey & West (1987) using Bartlett kernels, in which we assumed 3-year lag period58. Coefficientsβ are increasing within the same horizon regression along the tenors of bonds, so in general arealso the standard errors of these estimators. Bold estimates of β with their corresponding p-valuesimplied by these HAC corrected standard errors call for rejection of null hypothesis in the followingcases:

1. 6-year bond for all horizons starting and including 12 months ([12, 24, 36, 48, 60])

2. 5-year bond for all horizons starting and including 24 months ([24, 36, 48, 60])

3. 5.5-year bond for all horizons starting and including 24 months ([24, 36, 48, 60])

4. 4-year and 3.5-year bonds for 36-month horizon

Obviously, F-statistic test calls for rejection of null hypothesis that all the coefficients except inter-cept in a given regression are zero for the same pairs (h, n), since we have only one regressor andthe same null hypothesis when testing for significance of β estimator. The Figure 18 shows theseon a heatmap, from which we originate an assumption that the longer - not tested here - maturitieswith even short investment horizons would also call for the rejection of null hypothesis. Generally,we posit here that PEH holds for bonds up to 4 years and investment horizons up to 24 months, inany other combinations evidence for rejecting PEH is stronger the longer the bond and the longerthe horizon are.

For the pairs of (h, n) where we have collected sufficient evidence to reject the PEH we are notsure what is the root cause of this rejection, as R2 values are in the range of 4.7 − 45.5% and thisshare of explained variance increases with longer tenors of bonds within the same considered horizon.It means that on average only a third of the variance of excess return is explained by the volatility ofthe term premium defined as in this particular regression as a spread between corresponding forwardrate and a short rate in line in maturity with the investment horizon assumed. Hence it leaves uswith two thirds of variance unexplained and hints that potentially variable or fixed term premia areone of the variables to consider in increasing the R2 readings.

We have also performed robustness checks of the above-mentioned results using previously pre-pared 28 systems of weights (the details are reported in Figure 68 in the Appendix). Coefficients,HAC standard errors and, in consequence, p-values differ slightly between the systems, but the finaldecisions regarding null hypothesis (at 0.05 significance level) are almost the same or all 27 as forthe first system labelled 1 with one exception of for pair (24, 54), which coefficient β has in somesystems p-value greater than 0.05.

In summary, having analysed the results of regressions of excess returns over correspondingforward-to-spot term premia we found first argument that the PEH indeed does not hold universallyacross maturities and investment horizons. PEH holds with 0.95 confidence level only for short termbonds and short horizons, and this result is robust for changes of fitted yields from 28 differentweight systems.

3.2 Realized change in the spot rate on term premium regressions

We consider here yet another classic regression from Fama & Bliss (1987) sometimes referred to asforecasts of the change in the spot rate.

58as during 36 months autocorrelations of all tenors of fitted Polish yield curve fall below 0.5


Fig

ure

19:

Typ

e1

regr

essi

on:

am

od

oF

ama

&B

liss

(198

7)-T

erm

pre

miu

mre

gres

sion

s,P

olan

d20

05:0

1-20

20:0

6

hn

αSE(α

)p-val

βSE(β

)p-val

R2

R2adj.

obs

RMSE

F-stat

Fp-val

ρ(6)

ρ(12)

ρ(24)

ρ(36)

R1

6.0000

24.0000

0.0098

0.0029

0.0004

0.1443

0.5059

0.3878

0.0020

-0.0050

162.0000

0.0159

0.2700

0.6030

0.2200

-0.1300

-0.1300

0.0200

R2

6.0000

30.0000

0.0131

0.0044

0.0013

0.1785

0.5579

0.3745

0.0020

-0.0050

156.0000

0.0218

0.2600

0.6096

0.1800

-0.1400

-0.0900

0.0700

R3

6.0000

36.0000

0.0154

0.0061

0.0055

0.2408

0.6225

0.3494

0.0020

-0.0050

150.0000

0.0280

0.3300

0.5688

0.1500

-0.1600

-0.0500

0.1100

R4

6.0000

42.0000

0.0173

0.0078

0.0130

0.2296

0.7101

0.3732

0.0010

-0.0060

144.0000

0.0344

0.2100

0.6479

0.1500

-0.1600

-0.0100

0.1000

R5

6.0000

48.0000

0.0203

0.0093

0.0145

0.3175

0.7506

0.3361

0.0020

-0.0050

138.0000

0.0404

0.3100

0.5777

0.1300

-0.1800

0.0300

0.1000

R6

6.0000

54.0000

0.0225

0.0112

0.0223

0.4821

0.8143

0.2769

0.0040

-0.0030

132.0000

0.0470

0.5700

0.4529

0.1300

-0.2000

0.1000

0.0700

R7

6.0000

60.0000

0.0230

0.0135

0.0436

0.6604

0.8998

0.2315

0.0070

-0.0010

126.0000

0.0545

0.8400

0.3615

0.1400

-0.2200

0.1300

0.0600

R8

6.0000

72.0000

0.0261

0.0192

0.0878

0.8458

1.0426

0.2086

0.0080

0.0000

114.0000

0.0680

0.9400

0.3334

0.1400

-0.1700

0.0900

0.0900

R9

12.0000

24.0000

0.0069

0.0017

0.0000

-0.0895

0.2845

0.6234

0.0020

-0.0040

162.0000

0.0078

0.2900

0.5900

0.4900

-0.0600

-0.2300

0.0300

R10

12.0000

30.0000

0.0102

0.0026

0.0000

-0.0295

0.3538

0.5333

0.0000

-0.0060

156.0000

0.0118

0.0200

0.8956

0.4800

-0.0400

-0.1700

0.0600

R11

12.0000

36.0000

0.0124

0.0037

0.0004

0.1056

0.4138

0.3993

0.0010

-0.0060

150.0000

0.0159

0.1400

0.7088

0.4800

-0.0200

-0.1000

0.1000

R12

12.0000

42.0000

0.0140

0.0046

0.0011

0.1941

0.4797

0.3428

0.0020

-0.0050

144.0000

0.0200

0.3200

0.5721

0.4800

0.0100

-0.0400

0.0900

R13

12.0000

48.0000

0.0149

0.0056

0.0041

0.4316

0.5529

0.2175

0.0090

0.0010

138.0000

0.0243

1.1700

0.2817

0.4800

0.0100

-0.0100

0.1100

R14

12.0000

54.0000

0.0151

0.0069

0.0139

0.8270

0.6751

0.1103

0.0260

0.0180

132.0000

0.0283

3.4500

0.0654

0.4800

0.0000

0.0500

0.0900

R15

12.0000

60.0000

0.0143

0.0082

0.0403

1.2199

0.8233

0.0692

0.0460

0.0390

126.0000

0.0328

6.0400

0.0154

0.4800

-0.0200

0.0900

0.0800

R16

12.0000

72.0000

0.0119

0.0111

0.1403

1.9337

1.0601

0.0341

0.0820

0.0740

114.0000

0.0427

10.0100

0.0020

0.4800

0.0000

0.0900

0.1300

R17

24.0000

30.0000

0.0036

0.0007

0.0000

-0.0891

0.1020

0.8086

0.0150

0.0090

156.0000

0.0025

2.3500

0.1270

0.7200

0.2900

-0.1400

-0.0100

R18

24.0000

36.0000

0.0065

0.0014

0.0000

-0.0196

0.1792

0.5435

0.0000

-0.0070

150.0000

0.0052

0.0300

0.8622

0.7600

0.3600

-0.0900

0.0400

R19

24.0000

42.0000

0.0086

0.0020

0.0000

0.1135

0.2492

0.3244

0.0030

-0.0040

144.0000

0.0081

0.4600

0.4978

0.7700

0.4100

-0.0200

0.0500

R20

24.0000

48.0000

0.0098

0.0028

0.0002

0.3390

0.3089

0.1362

0.0180

0.0110

138.0000

0.0110

2.4900

0.1169

0.7800

0.4300

-0.0100

0.0500

R21

24.0000

54.0000

0.0101

0.0036

0.0029

0.6585

0.3954

0.0479

0.0470

0.0400

132.0000

0.0140

6.4300

0.0124

0.7700

0.4200

-0.0100

0.0500

R22

24.0000

60.0000

0.0098

0.0045

0.0151

1.0033

0.4706

0.0165

0.0830

0.0760

126.0000

0.0169

11.2900

0.0010

0.7500

0.3900

-0.0100

0.0700

R23

24.0000

72.0000

0.0099

0.0071

0.0831

1.6001

0.5068

0.0008

0.1510

0.1430

114.0000

0.0220

19.8500

0.0000

0.7500

0.4300

0.0200

0.1300

R24

36.0000

42.0000

0.0020

0.0007

0.0029

0.1809

0.0646

0.0026

0.1090

0.1030

144.0000

0.0019

17.3500

0.0001

0.7800

0.4000

0.0500

-0.1500

R25

36.0000

48.0000

0.0034

0.0016

0.0167

0.4418

0.1319

0.0004

0.1690

0.1630

138.0000

0.0039

27.6200

0.0000

0.7800

0.4500

0.0700

-0.1600

R26

36.0000

54.0000

0.0040

0.0026

0.0585

0.7852

0.2216

0.0002

0.2430

0.2370

132.0000

0.0059

41.6900

0.0000

0.7800

0.4500

0.0400

-0.1900

R27

36.0000

60.0000

0.0043

0.0035

0.1120

1.1260

0.2979

0.0001

0.2980

0.2920

126.0000

0.0080

52.6200

0.0000

0.7800

0.4500

0.0200

-0.1900

R28

36.0000

72.0000

0.0042

0.0058

0.2321

1.7795

0.4698

0.0001

0.3860

0.3800

114.0000

0.0119

70.4000

0.0000

0.7600

0.4300

-0.0100

-0.1100

R29

48.0000

54.0000

0.0016

0.0002

0.0000

0.2553

0.0307

0.0000

0.4000

0.3950

132.0000

0.0012

86.6000

0.0000

0.7800

0.5200

0.1100

-0.1800

R30

48.0000

60.0000

0.0030

0.0006

0.0000

0.5323

0.0592

0.0000

0.4320

0.4280

126.0000

0.0026

94.3500

0.0000

0.7500

0.4500

0.0500

-0.1800

R31

48.0000

72.0000

0.0054

0.0025

0.0149

1.0644

0.2007

0.0000

0.4550

0.4500

114.0000

0.0057

93.5800

0.0000

0.7400

0.4000

-0.0500

-0.0900

R32

60.0000

72.0000

0.0039

0.0009

0.0000

0.3871

0.0658

0.0000

0.3310

0.3250

114.0000

0.0025

55.4400

0.0000

0.8000

0.4800

-0.0300

-0.2300

Reg

ression:rx(n

)t+

h=

α+

β( f

t+h,t+n

t−

y(h

)t

) +ϵ(

n)

t+h

NullHypoth

esis:β=

0(P

EH

claim

sth

atnoth

ingsh

ould

foreca

stsretu

rns)

Standard

errors

are

calculatedwithheterosced

asticityandauroco

rrelationco

rrectionala

New

ley-W

est(B

artlett

kernel)

Bold

estimatesofβ

havelower

p-valueth

an0.05(ourassumed

significa

nce

level)andca

llforrejectionofnullhypoth

esis

R·indicatesth

atF-testfoundev

iden

ceto

reject

nullhypoth

esis

thatth

emodel

withnoindep

enden

tvariablesfits

thedata

aswellasyourmodel.


y(n)t+h − y

(h)t = α + β

(f t+h,t+nt − y

(h)t

)+ ϵ

(n)t+h (73)

where all the symbols have the same meaning as in the previous section and y(n)t+h is a zero coupon

rate of n-year bond (tenor is calculated at time t) observed at time t + h, hence in the future frompoint t in time. We have the same regressor as before but the regressand this time is a realisedspread of spot rates. For the PEH to hold we obviously would like that this spread is explained fullyby term premium, hence the null hypothesis is: H0 : β = 1.

The results of these regressions run for the same pairs of horizon and bond maturity (h, n) asfor the term premium regressions are shown in Table 20. Yet again the standard errors are of HACtype, but we have a fundamentally different null hypothesis which is reflected in p-value calculations.The domain where null hypothesis meets strong evidence against it is of the same exact form asfor the term premium regressions. What is more, the results are robust for changes in system ofweights, again with one exception of a pair (24, 54), for which in some systems we read p-value of βhigher than 0.05 (see Figure 69 in Appendix). Share of explained variance in total variance for theregressions with p-value of β estimates lower than 0.05 is very weak - in the range of 0.1− 19.7%59,suggesting other than term premia variables influencing our regressand.

In summary, the realised change in the spot rate on term premium regressions gave qualitativelysimilar results though the R2 values are approximately half the ones estimated for the first type ofFama and Bliss regressions. The domain where PEH holds (or at least there is no sufficient evidencethat it does not) is broader than in Fama & Bliss (1987) or Campbell & Shiller (1991), but still formedium and long term bonds PEH is rejected.

3.3 One year excess return on average one year forward rates regressions

In this subsection we present the results for Poland of almost classic ten shape regressions proposedby Cochrane & Piazzesi (2005) to strengthen evidence against expectations hypothesis, by finding asingle return forecasting factor - a combination of a series of one year forward rates and one year spotrate that explains the expected return of all bonds. In this pursue we follow the authors’ proposalclosely.

The key model here is a series of regressions on one year excess return of 2, 3, 4 and 5 year bonds(separately), where regressors are four consequitive one year forward rates:

rx(n)t+1 = β

(n)0 + β

(n)1 y1t + β

(n)2 f t+1,t+2

t + β(n)3 f t+2,t+3

t + β(n)4 f t+3,t+4

t + β(n)5 f t+4,t+5

t + ϵ(n)t+1 (74)

These models are not estimated directly but in restricted form in two-stepped regression. First, werun a regression of the average (across maturities) excess return on all forward rates and the oneyear spot:∑5

n=2 rx(n)t+1

4= γ0 + γ1y

1t + γ2f

t+1,t+2t + γ3f

t+2,t+3t + γ4f

t+3,t+4t + γ5f

t+4,t+5t + ϵt+1 (75)

to extract vector of coefficients γ. The regressand there is an arithmetic average of four excessreturns of investments into 2, 3, 4, 5-year bonds over one year spot rate observed at the beginning ofa hypothetical investments. The results of the first step are summarized in the upper panel of Table

59if we exclude the regressions where a simple model with intercept only was equally good - judging byp-value of F-statistics


Fig

ure

20:

Typ

e2

regr

essi

on:

am

od

oF

ama

&B

liss

(198

7)-

For

ecas

tsof

the

chan

gein

the

spot

rate

,P

olan

d20

05:0

1-20

20:0

6)

hn

αSE(α

)p-val

βSE(β

)p-val

R2

R2adj.

obs

RMSE

F-stat

Fp-val

ρ(6)

ρ(12)

ρ(24)

ρ(36)

R1

6.0000

24.0000

-0.0032

0.0010

0.0000

0.9466

0.1666

0.3742

0.4020

0.3990

162.0000

0.0052

107.6900

0.0000

0.2200

-0.1300

-0.1300

0.0200

R2

6.0000

30.0000

-0.0032

0.0011

0.0000

0.9487

0.1372

0.3544

0.4420

0.4380

156.0000

0.0054

121.7600

0.0000

0.1900

-0.1400

-0.1000

0.0700

R3

6.0000

36.0000

-0.0029

0.0012

0.0000

0.9443

0.1221

0.3242

0.4670

0.4640

150.0000

0.0055

129.7800

0.0000

0.1600

-0.1500

-0.0600

0.1100

R4

6.0000

42.0000

-0.0027

0.0013

0.0000

0.9534

0.1160

0.3440

0.4880

0.4850

144.0000

0.0056

135.4500

0.0000

0.1500

-0.1500

-0.0100

0.1000

R5

6.0000

48.0000

-0.0027

0.0013

0.0000

0.9459

0.1044

0.3022

0.5110

0.5070

138.0000

0.0056

142.1700

0.0000

0.1300

-0.1700

0.0300

0.1100

R6

6.0000

54.0000

-0.0026

0.0014

0.0000

0.9306

0.0986

0.2409

0.5240

0.5200

132.0000

0.0057

142.9300

0.0000

0.1300

-0.1900

0.1000

0.0800

R7

6.0000

60.0000

-0.0023

0.0015

0.0000

0.9174

0.0965

0.1960

0.5300

0.5260

126.0000

0.0059

139.6000

0.0000

0.1400

-0.2100

0.1200

0.0600

R8

6.0000

72.0000

-0.0021

0.0017

0.0000

0.9134

0.0906

0.1697

0.5620

0.5580

114.0000

0.0060

143.7600

0.0000

0.1400

-0.1700

0.0800

0.1000

R9

12.0000

24.0000

-0.0068

0.0016

0.0000

1.0784

0.2818

0.6096

0.2120

0.2070

162.0000

0.0078

42.9200

0.0000

0.4900

-0.0500

-0.2300

0.0300

R10

12.0000

30.0000

-0.0066

0.0017

0.0000

1.0092

0.2328

0.5158

0.2300

0.2250

156.0000

0.0078

46.0800

0.0000

0.4900

-0.0400

-0.1700

0.0700

R11

12.0000

36.0000

-0.0061

0.0018

0.0000

0.9385

0.2035

0.3812

0.2330

0.2280

150.0000

0.0079

45.0300

0.0000

0.4800

-0.0200

-0.1100

0.1000

R12

12.0000

42.0000

-0.0054

0.0018

0.0000

0.9155

0.1878

0.3264

0.2430

0.2380

144.0000

0.0079

45.6500

0.0000

0.4900

0.0100

-0.0400

0.0900

R13

12.0000

48.0000

-0.0048

0.0018

0.0000

0.8520

0.1791

0.2043

0.2370

0.2310

138.0000

0.0080

42.2400

0.0000

0.4800

0.0200

-0.0100

0.1100

R14

12.0000

54.0000

-0.0041

0.0019

0.0000

0.7626

0.1860

0.1009

0.2230

0.2170

132.0000

0.0079

37.4000

0.0000

0.4800

0.0100

0.0500

0.1000

R15

12.0000

60.0000

-0.0034

0.0020

0.0000

0.6970

0.1972

0.0622

0.2110

0.2040

126.0000

0.0080

33.0800

0.0000

0.4900

-0.0100

0.0900

0.0900

R16

12.0000

72.0000

-0.0022

0.0021

0.0000

0.6201

0.2008

0.0292

0.1970

0.1900

114.0000

0.0083

27.4500

0.0000

0.4800

0.0000

0.0800

0.1400

R17

24.0000

30.0000

-0.0142

0.0029

0.0000

1.3358

0.4031

0.7976

0.1790

0.1740

156.0000

0.0098

33.5500

0.0000

0.7300

0.2900

-0.1400

-0.0100

R18

24.0000

36.0000

-0.0127

0.0028

0.0000

1.0243

0.3550

0.5273

0.1240

0.1180

150.0000

0.0103

20.9800

0.0000

0.7600

0.3600

-0.0900

0.0400

R19

24.0000

42.0000

-0.0113

0.0027

0.0000

0.8394

0.3294

0.3129

0.0930

0.0860

144.0000

0.0107

14.4900

0.0002

0.7700

0.4100

-0.0300

0.0500

R20

24.0000

48.0000

-0.0096

0.0028

0.0000

0.6573

0.3059

0.1312

0.0660

0.0590

138.0000

0.0109

9.5700

0.0024

0.7800

0.4300

-0.0100

0.0500

R21

24.0000

54.0000

-0.0079

0.0029

0.0000

0.4756

0.3125

0.0467

0.0400

0.0320

132.0000

0.0111

5.3900

0.0219

0.7700

0.4200

-0.0100

0.0500

R22

24.0000

60.0000

-0.0064

0.0030

0.0000

0.3385

0.3093

0.0162

0.0230

0.0160

126.0000

0.0111

2.9800

0.0868

0.7500

0.3900

-0.0100

0.0700

R23

24.0000

72.0000

-0.0048

0.0035

0.0000

0.2143

0.2474

0.0007

0.0130

0.0040

114.0000

0.0108

1.4800

0.2266

0.7600

0.4400

0.0200

0.1300

R24

36.0000

42.0000

-0.0120

0.0044

0.0000

-0.0890

0.3862

0.0024

0.0010

-0.0060

144.0000

0.0112

0.1200

0.7302

0.7800

0.4000

0.0500

-0.1500

R25

36.0000

48.0000

-0.0100

0.0048

0.0000

-0.3217

0.3944

0.0004

0.0120

0.0050

138.0000

0.0116

1.6700

0.1985

0.7800

0.4400

0.0700

-0.1600

R26

36.0000

54.0000

-0.0079

0.0051

0.0000

-0.5579

0.4424

0.0002

0.0400

0.0330

132.0000

0.0117

5.4200

0.0215

0.7800

0.4500

0.0400

-0.1900

R27

36.0000

60.0000

-0.0063

0.0052

0.0000

-0.6697

0.4459

0.0001

0.0640

0.0570

126.0000

0.0118

8.5500

0.0041

0.7700

0.4500

0.0200

-0.1900

R28

36.0000

72.0000

-0.0041

0.0058

0.0000

-0.7503

0.4655

0.0001

0.1040

0.0960

114.0000

0.0116

13.0100

0.0005

0.7600

0.4300

0.0000

-0.1100

R29

48.0000

54.0000

-0.0129

0.0017

0.0000

-1.0174

0.2389

0.0000

0.1480

0.1410

132.0000

0.0096

22.5100

0.0000

0.7800

0.5200

0.1100

-0.1800

R30

48.0000

60.0000

-0.0121

0.0025

0.0000

-1.0980

0.2304

0.0000

0.1750

0.1690

126.0000

0.0101

26.3900

0.0000

0.7500

0.4500

0.0500

-0.1800

R31

48.0000

72.0000

-0.0107

0.0049

0.0000

-1.0895

0.3929

0.0000

0.1880

0.1810

114.0000

0.0110

25.9300

0.0000

0.7400

0.4000

-0.0400

-0.0900

R32

60.0000

72.0000

-0.0192

0.0046

0.0000

-0.9071

0.3212

0.0000

0.1030

0.0950

114.0000

0.0121

12.8400

0.0005

0.8000

0.4800

-0.0300

-0.2300

Reg

ression:y(n

)t+

h−

y(h

)t

=α+

β( f

t+h,t+n

t−

y(h

)t

) +ϵ(

n)

t+h,

NullHypoth

esis:β=

1(P

EH

claim

sth

atnoth

ingsh

ould

foreca

stsretu

rns)

Standard

errors

are


asticityandauroco

rrelationco

rrectionala

New

ley-W

est(B

artlett

kernel)

Bold

estimatesofβ

havelower

p-valueth

an0.05(ourassumed

significa

nce

level)andca


esis

R·indicatesth

atF-testfoundev

iden

ceto

reject

nullhypoth

esis

thatth

emodel

withnoindep

enden

tvariablesfits

thedata

aswellasyourmodel.


Table 9: One year excess return on average one year forward rates regressions, a modoCochrane (2005) , Poland 2005:01-2020:06

First step regressionγ0 γ1 γ2 γ3 γ4 γ5

coeff -0.0460 0.4069 1.3191 4.4494 -22.5294 17.3856SE 0.0180 1.8330 11.9305 31.0120 34.8548 14.1698p-val 0.4816 0.6580 0.9064 1.0000 0.0000 1.0000

Second step regressionn bn SE(hac) p-val R2

2 0.8144 0.1088 0.0000 0.33103 1.2088 0.1563 0.0000 0.34904 1.5780 0.2281 0.0000 0.34605 1.8485 0.2737 0.0000 0.4660

Note: Standard errors (of the second step regression) are calculated with heteroscedasticity and aurocorrelation correction a laNewley-West

Table 10: Multiregressions of excess one year return on return-forecasting error and Fama &Bliss term premium, Poland 2005:01-2020:06

n bn SE(hac) p-val cn SE(hac) p-val R2

n = 2 0.8211 0.1058 0.0000 -0.0433 0.4745 0.5363 0.3350n = 3 1.1991 0.1305 0.0000 0.0333 0.4795 0.4723 0.3460n = 4 1.5064 0.1948 0.0000 0.1752 0.5990 0.3849 0.3340n = 5 1.0961 0.5984 0.0335 1.5006 0.8250 0.0345 0.4350

Note: Standard errors are calculated with heteroscedasticity and aurocorrelation correction a la Newley-West

9. R2 is 0.35, which is precisely what the authors received for US data in their original article.In the second step we run 4 regressions for n = 2, 3, 4, 5, to find coefficients: bn

rx(n)t+1 = bn

(γ0 + γ1y

1t + γ2f

t+1,t+2t + γ3f

t+2,t+3t + γ4f

t+3,t+4t + γ5f

t+4,t+5t

)+ ϵ

(n)t+1 (76)

or rx(n)t+1 = bn

(γT ft

)+ ϵ

(n)t+1 (77)

where ft = [1, f t+1,t+2t , f t+2,t+3

t , f t+3,t+4t , f t+4,t+5

t ]. Obviously the coefficients γn and bn are not sep-arately identified, hence Cochrane & Piazzesi (2005) propose to normalise bn to have

∑5n=2 bn = 1.

This step is not necessary since at the end we are interested in the pattern (supposedly a tent) thatemerges from the cross product of both vectors of parameters, and we skip it for our inference fromPolish data.

The vector γ consists of coefficients with greater absolute value than the corresponding valuesfound by Cochrane & Piazzesi (2005) for n = {4, 5}. The shape that slowly transpires for Polishgovernment bonds is not resembling a tent, rather a seagull. The shape may be characteristic tothe country data and, in particular, the trends and cycles of monetary policy the timespan of datacovers. Cochrane & Piazzesi (2005) covered 40 years of data from 1963 to 2003, during which periodrates hit double digit percentage points in the seventies and into eighties, whereas our data set isplaced in absolute terms in much benign interest rates environment and has no impact of a serioustightening of monetary policy. Moreover not the shape as such is important, but the fact that theselinear combinations exist with statistical significance and the R2 in the range of 33.1−46.6%, whichis roughly double the reads we have reported for the Fama and Bliss regressions type 2 (Table 20).The robustness check of regression coefficients and the shapes they form are documented in Figure70 in the Appendix, where we clearly see that even with relatively small diversification in yieldnominal values in particular 28 systems the shape of regressions coefficients is still the same for allbond maturities under consideration but the shapes have sometimes more pronounced left wing ofthe seagull at 3-year maturity.


Table 11: Regression coefficients of one-year excess returns on forward rates, Poland 2005:01-2020:06

1 1.5 2 2.5 3 3.5 4 4.5 5-50

-40

-30

-20

-10

0

10

20

30

40

regre

ssio

n c

oeffic

ients

n=2

n=3

n=4

n=5

1 2 3 4 5n = 2 0.3314 1.0744 3.6238 -18.3490 14.1596n = 3 0.4919 1.5946 5.3786 -27.2341 21.0161n = 4 0.6421 2.0816 7.0212 -35.5511 27.4342n = 5 0.7522 2.4384 8.2246 -41.6449 32.1366

Notes: (1) Standard errors are calculated with heteroscedasticity and autocorrelation correction a la Newley-West (2)Restricted (2 stepped) model

In order to document even stronger arguments against PEH we run also a multiregression on both

the so called return-forecasting error bn(γT ft

)and the Fama Bliss term premium

(f t+1,t+nt − y1t

)for h = 12 months as regressors:

rx(n)t+1 = bn

(γT ft

)+ cn

(f t+1,t+nt − y1t

)+ ϵ

(n)t+1 (78)

Table 10 reports these regressions’ results for each bond maturity n = {2, 3, 4, 5}. Strikingly, co-efficients cn in front of the Fama & Bliss term premium are not statistically significant except forthe case of 5-year bond. Reported R2 are generally sightly lower than in single factor (restricted)model, with minor improvement only occurring for 2-year bond (from 33.1% to 33.5%). In FamaBliss term premium regression the coefficients where in absolute terms slightly higher except againfor n = 5. Having this picture in mind, we may conclude that the tests proposed by Cochrane &Piazzesi (2005) are extending the area of PEH rejection further in to shorter horizons and bondmaturities.

3.4 Rolling realised returns on term premia regressions

In what follows, we present the outcomes of regression proposed by Thornton (2006) and dubbedby the author as: conventional. The idea is very simple, and test the wide spread assumption ofeffective bond markets, that there should not exist a quantitatively significant difference betweenthe results of a strategy of rolling short term bonds and another one of investing in a long termbond for the same period as the cumulated length of the first strategy.

Regressions are performed for different combinations of n and h, where we try to explain vari-ability of an average realised h period rates in n tenor with the observed and implied by the marketspread between spot rates feasible for discounting in a long n and short h tenors. We obviously need


that k ≡ nh ∈ Z because the two strategies have to have the same horizons, without any gaps or

lags. Since the interest rates in Poland are near-root processes we also incorporate spreads to short

term yields y(h)t in both: regressand and regressor. The model has the following form:

1

k

k−1∑i=0

y(h)t+i×h − y

(h)t = α + β

(y(n)t − y

(h)t

)+ ϵt (79)

Notice that the rolling return is averaged in h-frequencies. For the null hypothesis that PEH holdswe require: H0 : β = 1. We have decided to choose short term investment length vector to beh = {1, 3, 6, 12, 24, 36} and bonds’ maturities vector: n = {6, 12, 24, 36, 48, 60, 72}. Obviously notall of the pairs (h,n) are feasible, because we have to comply with two rules: (1) h < n and (2)above-mentioned: n

h ∈ Z.

The results of these regressions for various pairs of (h, n) and the decisions on PEH hypothesiswith this regard are revealed in Table 23 and in Figure 21. We have found enough evidence to rejectPEH for medium to longer bonds (4-6 years or more) across all potential horizons we considered.In these cases the R2 are significantly different from zero which suggest that the slope of the yieldcurve has predictive power for the short short term rates. This predictive power, however diminishesas horizon is increased of the bond has longer maturity. The shorter the bond the shorter horizonstaring from which PEH is rejected with 0.95 confidence level. β coefficients within the same shortterm horizon h decrease in value along with bond maturities, except for the first three entries forh = {1, 3, 6} months. The H0 decisions were checked for robustness to weight system changes andthe results are revealed in Figure 71 in the Appendix. The regressions of strategies of rolling 3, 6or 12 months investments for 4 years in many weighting systems would call for rejection of PEHin approximately half of the systems and in the system labelled 1 rolling investments of 6 and 12months for 4 years are on the verge of rejection of null hypothesis (0.0552 and 0.0713 p-values).

Figure 21: Pure expectations hypothesis testing: conventional regression a modo Thornton(2006) for Poland

6 12 24 36 48 60 72


1

3

6

12

24

36

h -

horizon in m

onth

s

0

0

-1

-1

-1

-1

0

0

0

-1

-1

-1

0

0

0

0

-1

-1

0

0

0

0

-1

-1

0

0

0

-1

-1

-1

1

1

1

1

1

1

1

1

1

1

1

1



Figure 22 shows β coefficients for various horizons (1, 3, 6 and 12 months) with their confidencebands at 0.95. The shapes observed for US data (Thornton (2006), Campbell & Shiller (1991)) aresimilar but the ones for Poland seem to be horizontal reflections of their American counterparts andthe level is generally much closer to 1. Sufficient evidence to reject PEH (β = 1) at 0.05 − 0.06significance starts with 4-year bonds for all terms of short term rolled investments. In consequence,


our results qualitatively differ from the original paper in that, EH in Poland works relatively goodin the short-to-middle segments but not for longer bonds.60

Overall, the regressions presented in this subsection further confirm the inference from previousfour, that PEH works fine in Poland for horizons up to 12 months and for bonds with maturities upto 36 months (included).

Figure 22: Smile and smirk in coefficients of conventional regressions a modo Thornton (2006)for Poland

0 10 20 30 40 50 60 70 80

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 800

0.5

1

0 10 20 30 40 50 60 70 80

0.5

1

1.5

0 10 20 30 40 50 60 70 80

maturity of bonds in months

0

0.5

1

1.5

Notes: (1) NSS fit using weight system labelled 1.

3.5 Realised spread on pro rata temporis current spread regressions

The last regression we will run is the one proposed again by Thornton (2006) and called: contrarian.

Here we explain the realised spread on holding n long bond for h period over the original rate y(n)t

by the observed spread between the two spot rates scaled pro rata temporis by h/(n− h):

y(n−h)t+h − y

(n)t = α + β

(h

n− h

)(y(n)t − y

(h)t

)+ ϵt (80)

The null hypothesis is again H0 : β = 1. As the name suggests these regressions are instrumental inshowing and documenting the so-called Campbell-Schiller paradox (cf. Campbell & Shiller (1991)).The paradox in its original form says that the slope of the term structure (our regressor in themodel) almost always: (1) gives a forecasts in the wrong direction for the short-term changesin the long-term bond yields and (2) gives a forecast in the right direction for the long-termchanges in the short-term rates.

Figure 24 reports β coefficients for various horizons (3, 6, 12, 24 and 36 months) with their confi-dence bands at 0.95. Again the results for Poland differ from the ones obtained for American bondsby Thornton (2006) or Campbell & Shiller (1991). The wrong signs (as the authors called them)of coefficients are only visible and statistically different from 1 for horizons starting with h = 36

60whereas Thornton (2006) posit that EH works better at the short and long ends of the maturity spectrumand less in the middle


Fig

ure

23:

Rol

lin

gre

alis

edre

turn

son

term

pre

mia

regr

essi

ons,

Pol

and

2005

:01-

2020

:06

hn

αSE(α

)p-val

βSE(β

)p-val

R2

R2adj.

obs

RMSE

F-stat

Fp-val

ρ(6)

ρ(12)

ρ(24)

ρ(36)

R1

1.0000

6.0000

-0.0009

0.0005

0.0000

0.6461

0.2486

0.0773

0.2340

0.2300

179.0000

0.0025

54.1900

0.0000

0.4400

-0.0100

-0.2100

-0.1200

R2

1.0000

12.0000

-0.0021

0.0009

0.0000

0.6967

0.2532

0.1155

0.2550

0.2510

173.0000

0.0044

58.5700

0.0000

0.5800

0.0000

-0.3200

-0.1100

R3

1.0000

24.0000

-0.0053

0.0014

0.0000

0.8444

0.1756

0.1877

0.3580

0.3540

161.0000

0.0061

88.6800

0.0000

0.6800

0.1500

-0.4500

-0.0800

R4

1.0000

36.0000

-0.0083

0.0021

0.0000

0.8621

0.1156

0.1165

0.4300

0.4270

149.0000

0.0064

111.1200

0.0000

0.7000

0.2700

-0.3100

-0.1300

R5

1.0000

48.0000

-0.0106

0.0028

0.0000

0.7497

0.1383

0.0352

0.4150

0.4110

137.0000

0.0064

95.9200

0.0000

0.7400

0.3900

-0.1000

-0.1400

R6

1.0000

60.0000

-0.0130

0.0029

0.0000

0.6525

0.1061

0.0005

0.3650

0.3600

125.0000

0.0068

70.6600

0.0000

0.7500

0.4100

0.0200

-0.1100

R7

1.0000

72.0000

-0.0155

0.0037

0.0000

0.6175

0.0954

0.0000

0.3230

0.3170

113.0000

0.0076

52.9600

0.0000

0.8200

0.5300

0.0600

-0.0200

R8

3.0000

6.0000

-0.0006

0.0003

0.0000

0.6539

0.2721

0.1017

0.1870

0.1820

179.0000

0.0016

40.7300

0.0000

0.3900

0.0000

-0.1700

-0.1000

R9

3.0000

12.0000

-0.0018

0.0007

0.0000

0.7323

0.2659

0.1571

0.2290

0.2250

173.0000

0.0037

50.8400

0.0000

0.5300

-0.0100

-0.2900

-0.1100

R10

3.0000

24.0000

-0.0051

0.0013

0.0000

0.9030

0.1769

0.2918

0.3360

0.3320

161.0000

0.0057

80.5400

0.0000

0.6500

0.1300

-0.4100

-0.0600

R11

3.0000

36.0000

-0.0082

0.0021

0.0000

0.9076

0.1302

0.2389

0.3940

0.3900

149.0000

0.0063

95.7400

0.0000

0.6900

0.2800

-0.2600

-0.1000

R12

3.0000

48.0000

-0.0102

0.0028

0.0000

0.7535

0.1480

0.0479

0.3480

0.3430

137.0000

0.0065

72.0500

0.0000

0.7500

0.4200

-0.0400

-0.1100

R13

3.0000

60.0000

-0.0123

0.0029

0.0000

0.6210

0.0995

0.0001

0.2800

0.2740

125.0000

0.0071

47.9100

0.0000

0.7700

0.4400

0.0500

-0.1000

R14

3.0000

72.0000

-0.0148

0.0038

0.0000

0.5756

0.0953

0.0000

0.2420

0.2350

113.0000

0.0078

35.3600

0.0000

0.8300

0.5400

0.0700

-0.0200

R15

6.0000

12.0000

-0.0013

0.0005

0.0000

0.7951

0.2995

0.2470

0.1920

0.1870

173.0000

0.0026

40.6700

0.0000

0.4000

-0.0600

-0.2200

-0.1000

R16

6.0000

24.0000

-0.0047

0.0011

0.0000

0.9892

0.1993

0.4783

0.3050

0.3000

161.0000

0.0051

69.7200

0.0000

0.5900

0.1000

-0.3500

-0.0400

R17

6.0000

36.0000

-0.0078

0.0020

0.0000

0.9721

0.1692

0.4345

0.3470

0.3430

149.0000

0.0061

78.2200

0.0000

0.6800

0.2900

-0.1900

-0.0700

R18

6.0000

48.0000

-0.0096

0.0027

0.0000

0.7605

0.1500

0.0552

0.2710

0.2660

137.0000

0.0067

50.1900

0.0000

0.7500

0.4400

0.0200

-0.0900

R19

6.0000

60.0000

-0.0114

0.0030

0.0000

0.5705

0.1074

0.0000

0.1860

0.1800

125.0000

0.0073

28.1900

0.0000

0.7800

0.4700

0.0800

-0.0800

R20

6.0000

72.0000

-0.0137

0.0039

0.0000

0.5082

0.0997

0.0000

0.1530

0.1450

113.0000

0.0081

20.0500

0.0000

0.8300

0.5400

0.0900

-0.0200

R21

12.0000

24.0000

-0.0034

0.0008

0.0000

1.0810

0.2820

0.6130

0.2120

0.2070

161.0000

0.0039

42.7700

0.0000

0.4900

-0.0600

-0.2300

0.0300

R22

12.0000

36.0000

-0.0066

0.0015

0.0000

1.0651

0.2395

0.6072

0.2650

0.2600

149.0000

0.0055

53.1100

0.0000

0.6700

0.2700

-0.1300

0.0100

R23

12.0000

48.0000

-0.0081

0.0023

0.0000

0.7610

0.1630

0.0713

0.1720

0.1660

137.0000

0.0066

28.0200

0.0000

0.7700

0.4700

0.0900

-0.0600

R24

12.0000

60.0000

-0.0093

0.0028

0.0000

0.4434

0.1548

0.0002

0.0750

0.0680

125.0000

0.0074

10.0100

0.0020

0.7900

0.4800

0.1000

-0.0700

R25

12.0000

72.0000

-0.0114

0.0036

0.0000

0.3511

0.1374

0.0000

0.0520

0.0430

113.0000

0.0083

6.0900

0.0151

0.8300

0.5300

0.0900

-0.0200

R26

24.0000

48.0000

-0.0048

0.0014

0.0000

0.6578

0.3105

0.1353

0.0650

0.0590

137.0000

0.0055

9.4500

0.0026

0.7800

0.4300

-0.0100

0.0500

R27

24.0000

72.0000

-0.0069

0.0028

0.0000

-0.0357

0.2376

0.0000

0.0000

-0.0090

113.0000

0.0074

0.0400

0.8445

0.8200

0.5000

-0.0200

0.0300

R28

36.0000

72.0000

-0.0021

0.0029

0.0000

-0.7587

0.4756

0.0001

0.1060

0.0970

113.0000

0.0058

13.1000

0.0004

0.7600

0.4300

0.0000

-0.1100

Reg

ression:

1 k

∑ k−1

i=0y(h

)t+

i×h−

y(h

)t

=α+

β( y

(n)

t−

y(h

)t

) +ϵ t

Nullhypoth

esis

(PEH

holds):H

0:β=

1.

Standard

errors

are


asticityandauroco

rrelationco

rrectionala

New

ley-W

est

Bold

estimatesofβ

havelower

p-valueth

an0.05(ourassumed

significa

nce

level)andca


esis


months and they indeed are decreasing with maturity of analysed bond, not like in Thornton (2006)for almost all combinations of (h, n). For the Campbell-Schiller paradox to be present in Polish datawe would have needed negative β coefficients for the pairs of (h, n) where EH does not hold andthis is certainly not the case. Though, we have to admit that our samples are relatively small tothe ones available for US (approximately 1/3rd to 1/4th in length), and the future assessment ofCampbell-Schiller paradox existance in Polish bond data may change as time series grow.

Figure 24: Smirks in coefficients of contrarian regressions a modo Thornton (2006) for Poland

0 10 20 30 40 50 60 70 800

10

20

0 10 20 30 40 50 60 70 800

5

10

0 10 20 30 40 50 60 70 80


0

5

0 10 20 30 40 50 60 70 80


0

1

2

0 10 20 30 40 50 60 70 80


-3-2-101

Notes: (1) NSS fit using weight system labelled 1.

Figure 25: Pure expectations hypothesis testing: contrarian regression a modo Thornton(2006) for Poland

6 12 24 36 48 60 72


3

6

12

24

36

h -

horizon in m

onth

s

0

-1

-1

-1

-1

0

0

-1

-1

-1

0

0

0

-1

-1

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

1 1 1




Fig

ure

26:

Rea

lise

dsp

read

onp

rora

tate

mp

oris

curr

ent

spre

adre

gres

sion

s,P

olan

d20

05:0

1-20

20:0

6

hn

αSE(α

)p-val

βSE(β

)p-val

R2

R2adj.

obs

RMSE

F-stat

Fp-val

ρ(6)

ρ(12)

ρ(24)

ρ(36)

R1

3.0000

6.0000

-0.0011

0.0006

0.0000

1.3078

0.5443

0.7141

0.1870

0.1820

179.0000

0.0031

40.7300

0.0000

0.3900

0.0000

-0.1700

-0.1000

R2

3.0000

12.0000

-0.0013

0.0005

0.0000

3.4576

0.6499

0.9999

0.4860

0.4830

173.0000

0.0033

161.7800

0.0000

0.2800

-0.0400

-0.1200

-0.0700

R3

3.0000

24.0000

-0.0016

0.0006

0.0000

7.7072

0.8059

1.0000

0.6500

0.6480

161.0000

0.0036

295.3900

0.0000

0.1300

-0.0800

-0.0300

0.0000

R4

3.0000

36.0000

-0.0016

0.0007

0.0000

11.8230

0.9522

1.0000

0.7040

0.7020

149.0000

0.0039

349.4500

0.0000

0.0800

-0.1000

0.0100

0.0500

R5

3.0000

48.0000

-0.0016

0.0008

0.0000

15.9722

1.0931

1.0000

0.7410

0.7390

137.0000

0.0040

385.8200

0.0000

0.0600

-0.1200

0.0600

0.0400

R6

3.0000

60.0000

-0.0015

0.0009

0.0000

19.9699

1.2577

1.0000

0.7600

0.7580

125.0000

0.0042

390.2700

0.0000

0.0700

-0.1700

0.1000

0.0000

R7

3.0000

72.0000

-0.0013

0.0010

0.0000

23.9073

1.5550

1.0000

0.7880

0.7860

113.0000

0.0041

412.9000

0.0000

0.0300

-0.1300

0.0000

0.0300

R8

6.0000

12.0000

-0.0026

0.0009

0.0000

1.5902

0.5990

0.8378

0.1920

0.1870

173.0000

0.0052

40.6700

0.0000

0.4000

-0.0600

-0.2200

-0.1000

R9

6.0000

24.0000

-0.0032

0.0010

0.0000

3.7914

0.6661

1.0000

0.4030

0.3990

161.0000

0.0053

107.1400

0.0000

0.2200

-0.1300

-0.1300

0.0200

R10

6.0000

36.0000

-0.0029

0.0012

0.0000

5.6738

0.7323

1.0000

0.4670

0.4640

149.0000

0.0055

128.9900

0.0000

0.1600

-0.1500

-0.0600

0.1100

R11

6.0000

48.0000

-0.0027

0.0013

0.0000

7.5696

0.8309

1.0000

0.5130

0.5090

137.0000

0.0056

142.0400

0.0000

0.1300

-0.1700

0.0400

0.1000

R12

6.0000

60.0000

-0.0023

0.0015

0.0000

9.1994

0.9651

1.0000

0.5300

0.5260

125.0000

0.0059

138.6800

0.0000

0.1400

-0.2200

0.1200

0.0600

R13

6.0000

72.0000

-0.0020

0.0017

0.0000

10.9614

1.0944

1.0000

0.5660

0.5620

113.0000

0.0059

144.6500

0.0000

0.1200

-0.1600

0.0600

0.1000

R14

12.0000

24.0000

-0.0068

0.0016

0.0000

2.1619

0.5641

0.9803

0.2120

0.2070

161.0000

0.0078

42.7700

0.0000

0.4900

-0.0600

-0.2300

0.0300

R15

12.0000

36.0000

-0.0061

0.0018

0.0000

2.8259

0.6131

0.9986

0.2340

0.2290

149.0000

0.0079

44.8600

0.0000

0.4800

-0.0200

-0.1100

0.1000

R16

12.0000

48.0000

-0.0048

0.0018

0.0000

3.4089

0.7187

0.9996

0.2370

0.2310

137.0000

0.0080

41.8900

0.0000

0.4800

0.0200

-0.0100

0.1100

R17

12.0000

60.0000

-0.0034

0.0020

0.0000

3.4952

0.9971

0.9938

0.2100

0.2040

125.0000

0.0080

32.7100

0.0000

0.4900

-0.0100

0.0900

0.0900

R18

12.0000

72.0000

-0.0022

0.0021

0.0000

3.7274

1.2100

0.9879

0.1970

0.1900

113.0000

0.0083

27.2400

0.0000

0.4800

0.0000

0.0800

0.1400

R19

24.0000

36.0000

-0.0128

0.0028

0.0000

1.5429

0.5404

0.8424

0.1240

0.1180

149.0000

0.0104

20.8100

0.0000

0.7600

0.3600

-0.0900

0.0400

R20

24.0000

48.0000

-0.0096

0.0028

0.0000

1.3157

0.6211

0.6944

0.0650

0.0590

137.0000

0.0109

9.4500

0.0026

0.7800

0.4300

-0.0100

0.0500

R21

24.0000

60.0000

-0.0064

0.0030

0.0000

0.8309

0.8058

0.4169

0.0220

0.0140

125.0000

0.0111

2.8100

0.0962

0.7400

0.3900

-0.0100

0.0700

R22

24.0000

72.0000

-0.0048

0.0035

0.0000

0.6490

0.7442

0.3186

0.0130

0.0040

113.0000

0.0109

1.4800

0.2256

0.7500

0.4400

0.0200

0.1300

R23

36.0000

48.0000

-0.0100

0.0047

0.0000

-0.4458

0.4962

0.0018

0.0130

0.0060

137.0000

0.0116

1.7800

0.1847

0.7800

0.4400

0.0600

-0.1700

R24

36.0000

60.0000

-0.0062

0.0054

0.0000

-1.1470

0.7974

0.0035

0.0670

0.0590

125.0000

0.0118

8.8300

0.0036

0.7700

0.4400

0.0100

-0.1900

R25

36.0000

72.0000

-0.0041

0.0058

0.0000

-1.5174

0.9512

0.0041

0.1060

0.0970

113.0000

0.0117

13.1000

0.0004

0.7600

0.4300

0.0000

-0.1100

Reg

ression:y(n

−h)

t+h

−y(n

)t

=α+

β( h

n−h

)( y(n

)t

−y(h

)t

) +ϵ t

Nullhypoth

esis

(PEH

holds):H

0:β=

1.

Standard

errors

are


asticityandauroco

rrelationco

rrectionala

New

ley-W

est

Bold

estimatesofβ

havelower

p-valueth

an0.05(ourassumed

significa

nce

level)andca


esis


3.6 Summary

The evidence presented in this chapter suggest that indeed PEH does not hold universally for Polishgovernment bonds yield curve (our Hypothesis 2). Contrary to the US data and research wherePEH is almost always rejected, we have found that for Poland there is a limited domain wherePEH cannot be ruled out. The scope where pure expectations hypothesis probably holds in Polandis bounded by (1) the investment horizon of approximately 12 months and (joined condition) and(2) by maturity of the bond of circa 36 months. It is still unclear (as in the bewildering variety ofresearch) what causes the rejection of PEH for all other combinations of horizon length and maturity:existence of some kind of risk premia61 or unexpected excess yield (we have only a mixture of thesetwo contained in β coefficient estimators). This leads us to further yield structure decompositionand possible hypothesis that the yield curve spans all information relevant for forecasting futureyields and returns and no variables other than the current curve are needed, which we analyse inthe next chapter.

61i.e. EH with constant risk premium, EH with time-varying risk premium


4 Chapter 4. Extracting Term Premia from Polish yield curves

In this chapter we aim to uncover the risk adjusted expectations about future interest rates pathsthat are implied by the yield curves. Our motivation and intuition directly arise from the resultsof the previous chapter where we have found out that PEH operates in a very limited domain inPoland (although broader than in the US or in the UK). We aim here to verify Hypothesis 3: Ex-pectations hypothesis holds in less liquid markets once we estimate and check for robustnessthe term premia structure of the Polish government bonds yield curve and then use these variablesto find to what extend term premia help to explain some excess returns due to different horizons ofinvestment strategies.

We will closely follow already cited works of Adrian et al. (2012, 2013) and start with a statespace design based on principal component analysis (henceforth: PCA) of Polish yield curve andperform robustness checks for weight system change. Identification and interpretation of differentPCA resultant factors is provided. In the robustness evaluation we have been heavily inspired bythe work of Malik & Meldrum (2014) and slightly less so of McCoy (2019), who conducted extensivetests for the yield curves and term premium structures from affine term structure models for theUS and the Eurozone. We will come back to some of the tests proposed by these authors in thelast chapter where we investigate the role of professional forecasters in the yield curve forecastingpower’s assessment.

PCA as a technique originates back from the works of Pearson (1901) and Hotelling (1933).In the yield curve modelling PCA is performed to reduce dimensionality of a dataset by changingtypically large number of correlated tenors to only a few new uncorrelated variables - principal com-ponents (henceforth: PCs), while maximising variability. Putting it more formally, PCA is a solutionto the eigenvalues / eigenvectors problem or, alternatively, to the singular value decomposition ofthe centred data matrix62.

On the basis of the extracted components we will conduct the term premia estimations in three-stepped ordinary least squares regressions utilising the full scope data and propose number of factorsto be used in the optimal set-up. First, we decompose the pricing factors into predictable compo-nents and innovations. Second, we regress excess returns on lagged factors and contemporaneousinnovations. Finally, we run a cross-sectional regression to extract a matrix of market prices ofrisk for different factors and then recursively calculate risk free and risky yield curves. Almost as aside-effect we obtain the sought-for term structure of risk premia implied in the data.

4.1 Establishing state space

In the pursuit of state space variables for the term premia model (henceforth: ACM) we comply withthe pervasive approach in interest rate research of PCA application to zero coupon rates. Usually,first three factors are interpreted as level, slope and curvature and this specification dominates inthe literature but one should admit that, it is not rare to came across specifications with 4, 5 oreven 7 factors in the robustness check efforts of more advanced research.

For Poland as an example of less liquid market, we use our preferred weight system’s NSS pa-rameters (labelled 1 ) to produce time series of zero-coupon interest rates for selected vector of tenorsspread monthly between 0 and 10 years for the trading dates as close to end-of-month as possible63.

62we believe PCA is currently such a standard technique that further exposition here in this thesis isunnecessary. An interesting reader may be referred to a very good textbook by Jackson (2005) or relativelyrecent review by Jolliffe & Cadima (2016)

63use of EOM rather than daily curves for PCA is consulted and recommended by one of the authors ofACM, Richard K. Crump in our correspondence in April 2020


Table 12: PCA explained variance (in pct points) by factors for the Polish yield curve

factor daily & dense daily & sparse monthly & dense monthly & sparsePC1 97.7096 96.3887 96.2156 97.5962PC2 2.0736 3.2762 3.4562 2.1912PC3 0.1829 0.2887 0.2835 0.1801PC4 0.0308 0.0415 0.0392 0.0292PC5 0.0028 0.0044 0.0049 0.0028PC6 0.0003 0.0005 0.0006 0.0004PC7 0.0000 0.0000 0.0001 0.0000

Notes: (1) NSS fit using weight system labelled 1. (2) dense - means that tenors are equally spaced in monthly intervals from 1month to 120 months, whereas for sparse - we use only 11 tenors: [1, 3, 6, 12, 24, 36, 48, 60, 72, 96, 120] months (3) sample: EOM

and daily 2005:01-2020:06

To be precise, we use continuous compounding in our main application, because the term premiastructure is derived with exponential-affine interest rates assumption. Just for PCA, of course, itwould not matter much which compounding model we use, but we would like to hold to model’sconsistency at every possible step, hence our choice of a continuous time version. Per constructionof NSS parsimonious yield curve all of the zero coupon rates we have calculated for each tradingdate are subtly interconnected and correlated (cf. Figure 27). But for the predictive models we needsome dimensionality reduction and orthogonal variables, this is where PCA is handy as explainedin the introduction to this chapter.

Figure 27: Correlation structure of the Polish zero coupon rates

1 3 6 12 24 36 48 60 72 96 120

tenor in months

1

3

6

12

24

36

48

60

72

96

120

tenor

in m

onth

s

0.9309

0.9142

0.8989

0.8849

0.8612

0.8443

0.9321

0.9174

0.9037

0.8808

0.8643

0.9369

0.924

0.9021

0.8863

0.9293

0.9146 0.9458

0.9309 0.9142

0.9321

0.8989

0.9174

0.9369

0.8849

0.9037

0.924

0.8612

0.8808

0.9021

0.9293

0.8443

0.8643

0.8863

0.9146

0.9458

1

0.9982

0.9919

0.9764

0.9504

0.9982

1

0.9977

0.9871

0.9655

0.9478

0.9919

0.9977

1

0.9956

0.9801

0.965

0.9507

0.9764

0.9871

0.9956

1

0.9939

0.9837

0.9722

0.9605

0.9491

0.9504

0.9655

0.9801

0.9939

1

0.9972

0.991

0.983

0.9746

0.9585

0.9478

0.965

0.9837

0.9972

1

0.9981

0.9937

0.9879

0.9755

0.9646

0.9507

0.9722

0.991

0.9981

1

0.9987

0.9954

0.9864

0.9774

0.9605

0.983

0.9937

0.9987

1

0.999

0.9934

0.9863

0.9491

0.9746

0.9879

0.9954

0.999

1

0.9974

0.9923

0.9585

0.9755

0.9864

0.9934

0.9974

1

0.9985

0.9646

0.9774

0.9863

0.9923

0.9985

1

1 3 6 12 24 36 48 60 72 96 120

tenor in months

1

3

6

12

24

36

48

60

72

96

120

tenor

in m

onth

s

0

0.00178

0.002659

0.0002138

-0.003528

-0.004076

-0.003416

-0.002779

-0.002605

-0.003528

-0.004764

0.00178

0

2.903e-06

-0.001448

-0.004764

-0.006466

-0.007272

-0.007951

-0.008795

-0.01082

-0.01212

0.002659

2.903e-06

0

-0.0005889

-0.002977

-0.00467

-0.00571

-0.006591

-0.007538

-0.009506

-0.01056

0.0002138

-0.001448

-0.0005889

0

-0.0008103

-0.001546

-0.001827

-0.001994

-0.002269

-0.003167

-0.003673

-0.003528

-0.004764

-0.002977

-0.0008103

0

3.41e-05

0.0003768

0.0008484

0.001208

0.001351

0.001416

-0.004076

-0.006466

-0.00467

-0.001546

3.41e-05

0

0.0001605

0.0005296

0.0008848

0.001221

0.001633

-0.003416

-0.007272

-0.00571

-0.001827

0.0003768

0.0001605

0

0.0001123

0.0003145

0.0006223

0.0013

-0.002779

-0.007951

-0.006591

-0.001994

0.0008484

0.0005296

0.0001123

0

5.376e-05

0.0002839

0.001128

-0.002605

-0.008795

-0.007538

-0.002269

0.001208

0.0008848

0.0003145

5.376e-05

0

0.0001348

0.001011

-0.003528

-0.01082

-0.009506

-0.003167

0.001351

0.001221

0.0006223

0.0002839

0.0001348

0

0.0004818

-0.004764

-0.01212

-0.01056

-0.003673

0.001416

0.001633

0.0013

0.001128

0.001011

0.0004818

0

Notes: (1) left panel: NSS fit using weight system labelled 1. (2) right panel is an element-wise difference in correlationmatrices of two weight systems labelled 1 and 28 i.e. if the number is greater than zero - correlation in system labelled 1 is

higher. (3) sample: EOM 2005:01-2020:06

Table 12 reports share of variance of particular factor in the total variance for different combina-tions of daily and end-of-month data as well as for two densities of tenors (called: dense and sparse).The first factor (level) explains approximately 96.2 − 97.7% of variance, the second one (slope) -2.1 − 3.5%, the third (curvature) - 0.2 − 0.3% and all of the other factors contribute to less than0.03% of variance. It seems that the factors 6-th and higher are not necessary in our inference onterm premia, but we will test separately our ACM model for number of factors later in this chapter.We have checked all other weight systems (28 in total) and the qualitative results are similar.

Figure 28 depicts loadings of seven factors in PCA decomposition for Poland performed on ouroptimal weight system for the entire dataset time span with respect to time to maturity on a yieldcurve. The factor responsible for level is not precisely horizontal in that space, but it is very com-mon in the literature that these first factors are not strict flat lines. Nevertheless the first factor isinterpreted as a level because it does not change its sign along the curve and the magnitude of itsloadings is at least one hundred times greater than for the other factors. The slope factor crosses


zero approximately at 4-year mark, which again is not surprising and suggest the middle of the yieldcurve habitat and an economic activity state or monetary policy independent point. The magnitudeof its loadings are one hundred times less then the first one. The third factor further decompose thecurve into three segments: a short end up to approximately 1-year, a middle curve with instrumentsof 1- to 6-year maturity and a long end beyond the 6-year mark. This split may be interpreted asa systemic reflection of government securities supply policy: relatively large supply of 2 and 5-yearbonds, switch operations in the short end and moderate supply in the longest segment. The size ofthis factor is already small - one tenth of the one for slope. The subsequent factors of the greaterorder than 3 are of a tenth of a magnitude of the previous factor and usually one more zero than theprevious one. The shapes of these factors are not commonly interpreted in the literature and becauseof their minuscule importance in PCA of the Polish yield curve we also skip this interpretation untilpossibly we identify more than three factors model of risk premia.

Figure 28: Loadings in 7 factor PCA of the Polish yield curve

0 20 40 60 80 100 120

2.2

2.4

10-3 factor 1

0 20 40 60 80 100 120

-5

0

5

10

10-5 factor 2

0 20 40 60 80 100 120

0

10

2010

-6 factor 3

0 20 40 60 80 100 120

-1

0

1

2

10-6 factor 4

0 20 40 60 80 100 120

-1

0

1

2

10-7 factor 5

0 20 40 60 80 100 120

-1

0

1

2

3

10-8 factor 6

0 20 40 60 80 100 120

-1

0

1

2

10-9 factor 7

Notes: (1) NSS fit using weight system labelled 1. (2) factors are standardised and monthly sparse tenor grid was used (3)sample: EOM 2005:01-2020:06

Potentially more informative in our future inference in ACM model than the shapes of factorloadings is to check how the time series of the standardised factors look like and whether they resem-ble some demeaned economic or financial variables. Should the latent factors already mirror someeconomic information we may be closer to posit spanning hypothesis which claims that all neces-sary information to forecast yield curve in the future is contained in the current market data and


additional macroeconomic variables are not required. Figure 29 contains evolutions of standardisedlatent PCA factors for Polish curve. First factor’s history resembles the CPI and PPI paths exceptfor the last year or more and it is strongly interconnected with the monetary policy. The correla-tions of the time series of different factors with selected economic activity and market indicatorsare documented in Table 13, where for the first factor we observe, in addition to inflation variables,strong and significant negative correlation with manufacturing and retail sales, important and pos-itive correlation with unemployment, some correlation with VIX index (which measures compositeimplied volatility in options on US stock market indices and often is interpreted as fear factor) andreal effective exchange rate. Second factor is more cyclical and stationary, hence its usual collocationwith the industrial production and rate of unemployment in the economy. As Table 13 reports it isalso negatively and significantly correlated with stock market’s growth. The factors of higher ordersthan 2 are usually harder do interpret but their profiles with respect to correlation with economicvariables differ. For example, the third factor is similar in that respect to the first factor but themagnitude of correlation is 2-4 times lower and there is no significant correlation of this factor toCPI, manufacturing and industrial production nor the fear factor. Factor three tends to grow whenPLN depreciates, confidence falls and unemployment rises.

Figure 29: Time series of factors from PCA of the Polish yield curve

2006 2008 2010 2012 2014 2016 2018 2020

-2

-1

0

1

factor 1

2006 2008 2010 2012 2014 2016 2018 2020

-1

0

1

2

factor 2

2006 2008 2010 2012 2014 2016 2018 2020

-2

0

2

factor 3

2006 2008 2010 2012 2014 2016 2018 2020

-2

0

2

factor 4

2006 2008 2010 2012 2014 2016 2018 2020

-5

0

5factor 5

2006 2008 2010 2012 2014 2016 2018 2020

-5

0

5

factor 6

2006 2008 2010 2012 2014 2016 2018 2020

0

5

10factor 7

Notes: (1) NSS fit using weight system labelled 1. (2) factors are standardised and monthly sparse tenor grid was used (3)sample: EOM 2005:01-2020:06

Higher order factors are significantly negatively correlated with CPI except for PCA6 which is


orthogonal to almost all economic variables we have selected with one exception for positive corre-lation with stock market’s growth.

Three factors parametrisations are pervasive in yield curve related research, but there are (espe-cially for developed and liquid markets) examples of higher order factorisations. We postpone thedecision to choose the number of factors in our representation of state space until first estimationsof term premia structure in the next subsection to check if and how the fit of a model changes.

It is important to underline that adding factors in maximum likelihood procedure of estimatingterm premia as in Joslin et al. (2011) or Hamilton & Wu (2012) is computationally demanding andtime consuming. Here, in three-stepped-regressions of ACM model we do not have this particularconstraint.

Table 13: Correlation of factors from PCA with selected economic indicators

Indicator PCA1 PCA2 PCA3 PCA4 PCA5 PCA6 PCA7

CPI 0.33579 - - -0.20052 -0.20932 - -0.20580PPI 0.34633 -0.21878 0.17947 - -0.23811 - -

Manufacturing (survey) - -0.18814 - 0.25935 - - 0.21557Confidence (survey) 0.49060 - -0.16513 0.23700 - - 0.20819Leading indicators - -0.28773 - 0.18299 0.18963 - 0.21885

Industrial production - -0.34107 - 0.28532 - - 0.17143Manufacturing production -0.84093 -0.17255 - - -0.25139 - -

Retail sales -0.80446 -0.16305 - - -0.29370 - -Unfilled vacancies -0.64572 - -0.17788 - -0.36186 - 0.14516

Unemployment rate 0.61339 - 0.19809 0.17250 0.43580 - -Harmonised un. rate 0.59078 0.15011 0.18661 0.27909 0.42461 - -

Harmonised un. rate (SA) 0.59525 0.15213 0.18630 0.26829 0.42049 - -USDPLN -0.83821 - - - - - -

REER 0.71192 0.25833 -0.15927 - -0.17117 - -Stock market growth - -0.19245 - - 0.25920 0.17687 -

VIX (fear factor) 0.30628 0.14520 - -0.31227 -0.19340 - -

Notes: (1) only significant (with confidence of 95%) coefficients are shown, (2) source of time series of variables: OECD, MainEconomic Indicators - complete database, Main Economic Indicators (database), http://dx.doi.org/10.1787/data-00052-en

(Accessed on 17th October 2020) (3) data codes used: CPGRLE01PLM659N, POLPPDMMINMEI, BSPRTE02PLM460S,BVCICP02PLM460S, POLLOLITONOSTSAM, POLPROINDMISMEI, POLPROMANMISMEI, POLSARTMISMEI, LMJVTTUVPLM647S,LMUNRRTTPLM156S, LRHUTTTTPLM156N, LRHUTTTTPLM156S, CCUSMA02PLM618N, RBPLBIS, SPASTT01PLM657N, VIXCLS

4.2 Term premia in the Polish yield curves

Here again we follow the idea presented in details in chapter of this thesis on focused literaturereview. Let the K standardised factors coming from PCA decomposition from a time series aredenoted as: XK

· . Since they are already standardised we set µ = 0 in first step of ACM regressionsmethod explained in Chapter 1. Estimated matrix Φ from the VAR model of factors for K = 7 isof a form:

Φ7 =

0.9964 −0.0069 −0.0111 0.0004 0.0107 −0.0020 0.01700.0119 0.9235 0.0152 0.0160 −0.0136 0.0269 0.01650.0502 −0.0075 0.8412 0.0123 0.0194 0.0478 0.0180−0.0038 0.0303 0.0206 0.7457 −0.0185 0.1138 0.03180.0405 −0.0040 0.1476 −0.0503 0.5484 −0.1126 −0.03850.0044 0.0088 0.1092 0.1091 −0.1217 0.1273 −0.1323−0.0068 −0.0308 0.0501 −0.0358 −0.0379 0.0365 0.0887

(81)

It is worth noting that ΦK for K = {3, 4, 5, 6} resemble appropriate cut-outs of Φ7 matrix, but thereare some minor differences in particular coefficients64. Also the coefficient signs may vary for some

64for example for K = 3 we have:

Φ3 =

[0.9959 −0.0068 −0.01140.0119 0.9235 0.01530.0502 −0.0075 0.8412

](82)


factors as they depend on the method used in PCA calculation (i.e. singular value decomposition,eigenvalue decomposition of the covariance matrix or alternating least squares algorithm), but thisfact does not influence the final results of risk premia estimation.

The excess returns65 necessary for the next step of ACM procedure are calculated for one period(a month) for a chosen, arbitrary maturities of bonds in months tM = {6, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120}.The choice of maturities is line with original work of Adrian et al. (2013). However, a crucial choicewith regard to excess one month return is the one over short term rate. The ACM authors ex-perimented with FedFunds rates with additional smoothing to proxy the one month rate in thesecalculations. In preparation to solving this problem we have estimated the yield curve and previ-ously filtered the available data in order to have consistent methodology and the functional formfor the whole domain of maturities including the short ones. Therefore, in our case we use a zerocoupon rate that is simply implied by the functional form estimated from the yield curve and we donot have to smooth the data or do robustness checks of our results to changes in the short rate’s type.

In the core step of the procedure we estimate a matrix of market prices of risk - Λ, results ofwhich for K = 7 are reported in Table 14 accompanied by t-statistics for each coefficient and Wald’sstatistics values and p-values testing null hypotheses that the whole rows of Λ and λ1,· are zero.

During our robustness checks for the number of factors in ACM model for Poland it turned out,that coefficients’ significance evaluations differ sightly between these set-ups. In appendix Table33 we have documented such characteristics. Using Wald’s tests we infer that the level risk factoris not priced for any K = {3, 4, 5, 6, 7}, which stays in contrast with estimations carried over forUS data by Adrian et al. (2012) where in all tested models level risk is priced. Investors in Polishbonds do not require a positive expected excess return for holding the level portfolio as the Wald’stest in Table 14 shows. All the coefficients on PC1: λ0 and λ1,(1...7) are statistically not significant.This situation may be due to the fact that for Poland we have only 15.5-year history with yields’max-min range of approximately 6 pct points, where as for the US we have almost 4 times longerhistory66 and more than two times wider range of yields (including spikes of risk during oil crisesin 70’, wars with the US participation, banking crises in 80’, house prices bubble, dotcom bubble,Financial Crisis after 2007), which obviously lead to more correlation between risk perception andthe level of yields in different curve’s segments. For Poland, the variability of the times series ismore benign and it seems that is harder to ascribe high level of yields with riskier environments inperception of market participants. All the other factors are priced, except for factor of curvaturerisk in three factor model. No λ0 (constant component) in any setup and for any factor except,again, three factor model and the third factor, is statistically significant. Estimations for the USdata again show a bit different story. Some of the factors of higher order are not priced at significantlevel: i.e. second and forth in five factors set-up.

65log excess holding returns66cf. data set based on Gurkaynak et al. (2011)


Table 14: Market prices of risk for seven factor ACM model of the Polish interest rates

λ0 λ1,1 λ1,2 λ1,3 λ1,4 λ1,5 λ1,6 λ1,7 WΛ Wλ1

PC1 -0.016 -0.006 0.004 -0.005 0.004 0.012 -0.001 0.017 6.564 4.484t-stat1 -1.442 -0.540 0.407 -0.485 0.329 1.094 -0.109 1.573 0.584 0.723PC2 0.002 -0.032 -0.050 0.064 0.039 -0.002 0.033 0.019 15.689 15.680t-stat2 0.094 -1.221 -1.917 2.472 1.497 -0.077 1.271 0.737 0.047 0.028PC3 0.066 -0.041 -0.011 -0.084 0.101 0.054 0.069 0.026 19.825 17.110t-stat3 1.647 -1.018 -0.275 -2.106 2.518 1.344 1.730 0.649 0.011 0.017PC4 -0.002 -0.018 0.035 0.027 -0.205 0.049 0.144 0.046 30.584 30.582t-stat4 -0.047 -0.372 0.727 0.562 -4.307 1.029 3.014 0.952 0.000 0.000PC5 -0.006 0.036 -0.029 0.128 -0.042 -0.373 -0.003 -0.002 44.069 44.060t-stat5 -0.095 0.591 -0.492 2.117 -0.698 -6.205 -0.045 -0.035 0.000 0.000PC6 0.001 -0.030 -0.021 0.081 0.114 -0.020 -0.658 0.033 86.231 86.231t-stat6 0.007 -0.413 -0.293 1.127 1.585 -0.274 -9.051 0.449 0.000 0.000PC7 0.020 -0.011 -0.051 0.012 -0.064 -0.027 0.127 -0.744 97.870 97.799t-stat7 0.266 -0.149 -0.676 0.158 -0.850 -0.358 1.663 -9.679 0.000 0.000

Notes: (1) this table reports λ coefficient estimates paired with their t-statistic value based on standard error computed as inACM. (2) Last two columns provide Wald statistics to test if the whole row of Λk is statistically different from 0. (3) Certain

factor is not priced if Wald test shows we cannot reject this null hypothesis. (4) Bold text is used to annotate significantcoefficients (at 5%). (5) λ1,x for PC y is a regression coefficient of factor x in the price of factor y risk.

In particular, for K = 7 set-up we obtained the following results of market prices of risk matrix.The market price of level risk factor PC1 has no significant components. The price of different riskfactors have maximum of two significant components: i.e. for PC2 (slope risk) - on 2nd, 3rd factors,for PC3 (curvature risk) - on 3rd and 4th, for PC4 - on 4th and 5th, for PC5 - on 3rd and 5th,for PC6 and PC7- on self factors (6th and 7th, respectively). For N = {4, 5, 6} the significanceassessments and the level of coefficients are in line with the one for N = 7 with small exception infive factor model where the one for PC2 risk on factor number 2 is not significant. Three factormodel seems to be a poor choice as only one factor risk (slope) is properly priced.

Table 15: Response of Polish government yield curve to shocks to factors

factor n=1 n=3 n=6 n=12 n=24 n=36 n=48 n=60 n=84 n=120PC1 11.47389 11.88454 12.36317 12.95493 13.28235 13.08135 12.69989 12.28922 11.59293 10.96472PC2 4.32283 4.09493 3.73355 3.02496 1.79673 0.81593 0.03874 -0.57305 -1.41926 -2.06141PC3 2.06942 1.66086 1.12436 0.34269 -0.39441 -0.55722 -0.47447 -0.29563 0.10884 0.58225PC4 0.97566 0.62732 0.24080 -0.16532 -0.24083 -0.04014 0.13289 0.20866 0.10523 -0.33450PC5 0.43219 0.19770 0.00872 -0.09035 -0.00328 0.05813 0.04526 -0.00108 -0.05857 0.11464PC6 0.26146 0.07456 -0.02405 -0.02117 0.02073 0.00488 -0.01277 -0.01312 0.01160 -0.03012PC7 0.11891 0.01967 -0.01301 0.00495 0.00148 -0.00666 -0.00040 0.00528 -0.00148 0.01103

Notes: (1) Sample period 2005:01-2020:06. (2) Table reports implied yield loadings bn = − 1nBn for selected tenors in months.

(3) system of weights labelled 1. (4) number of factors K = 7. (5) in basis points.

Tables 15 and 16 present model implied responses of yield curve and of one-month excess returnsto some contemporaneous shocks to PCA factors (of one standard deviation magnitude). Whereasthe Figure 30 shows the variance decomposition of expected 10-year bond excess returns in sevenfactor model. Contribution of risk factors vary over time but it is clearly visible that the role oflevel risk (first component) is not as dominant as for US data. During and after Financial Crisis(2007-2010) we have even observed that slope is more pronounced as the source of time variabilityof excess return variance. The slope risk, in turn, is driven by movements in factors PC2 and PC3(cf. Table 14): the slope itself and the curvature.

Figures 73 and 74 in Appendix reveal the excess return decompositions for different other ma-turities: 1, 2, 3, 5, 8 - years. It turned out that for mid-curve tenors of 3 and 5-year tenors the levelrisk dominates the expected excess returns contribution, where as for both: shorter (1, 2-years) andlonger bonds (8, 10-years) the slope factor does.


Figure 30: Decomposition of expected 10y excess returns in ACM model for Poland

2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1

2

3

4

5

6

7

Notes: (1) Sample period 2005:01-2020:06. (2) system of weights labelled 1. (3) number of factors K = 7

Table 16: Response of one-month excess returns on Polish government bonds to shocks tofactors

factor n=1 n=3 n=6 n=12 n=24 n=36 n=48 n=60 n=84 n=120PC1 0.30288 0.83004 1.45476 2.25357 2.80728 2.82883 2.78354 2.78607 2.75297 1.53361PC2 0.17620 0.46424 0.79726 1.18392 0.82052 -0.76091 -3.04450 -5.61811 -10.57892 -15.56053PC3 -0.14522 -0.32182 -0.50378 -0.93950 -1.73957 -1.30545 0.80879 4.36726 13.90931 28.35265PC4 -0.22106 -0.71628 -1.50252 -2.79616 -3.10499 -1.05767 1.48994 3.13022 1.67004 -9.34076PC5 -0.11833 -0.53565 -1.26347 -2.32307 -2.96173 -3.65908 -5.59265 -8.59648 -14.65196 -12.95442PC6 -0.25425 -0.83660 -1.40922 -1.18686 0.77125 1.34632 0.93532 0.88687 3.83229 8.14427PC7 -0.29988 -1.03283 -2.01276 -3.36111 -5.83811 -8.32465 -10.29861 -11.98039 -15.38648 -16.94513

Notes: (1) Sample period 2005:01-2020:06. (2) Table reports implied one-month excess returns B′nλ1 for selected tenors in

months. (3) system of weights labelled 1. (4) number of factors K = 7. (5) in basis points.

Because ACM model does not impose an assumption that the principal components must bepriced perfectly, thus the potential discrepancies between the model-implied and the actual PCsfrom the original decomposition. Likewise, in our implementation, as Figure 31 shows for five factorsetup, the PC yield loadings implied by the model with five factors are in line with the ones estimatedvia PCA at the beginning of the procedure as far as the shape and signs are concerned. Notice thatseven and five factor models result in similar yield and expected excess return loadings for the firstfive risk factors. It is crucial to distinguish between negligible importance of higher order factors(greater than 3) for yield loadings and - in a stark contrast - their significant role in the expectedexcess return loadings. This is also yet another good reason for considering higher order models forPoland i.e K ≥ 5.


Figure 31: Model implied loadings in ACM five factor model for Poland in five factor model

0 1 2 3 4 5 6 7 8 9 10

years to maturity

-4

-2

0

2

4

6

8

10

12

14

1610

-3 Yield loadings

1 factor

2 factor

3 factor

4 factor

5 factor

6 factor

7 factor

0 1 2 3 4 5 6 7 8 9 10

years to maturity

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

310

-3 Expected excess return loadings

1 factor

2 factor

3 factor

4 factor

5 factor

6 factor

7 factor


Figure 32 provides a comparison of regression coefficients β(n) (on lagged factors and contem-poraneous innovations for selected maturities were calibrating our model to) with recursive pricingparameters Bn used to generate the fitted yields in the above cited equation. Visual inspection isenough to infer that these two are almost identical in pairs up to the forth factor and differ slightlyfor higher order factors, but the distances are economically very small and negligible. This kind offit characteristics is maintained for all K ≥ 5, therefore, we posit that five factor ACM model ofPolish term premia is replicating the real dynamics of the yield curve and excess returns.


Figure 32: Regression coefficients compared with recursive pricing parameters in ACM sevenfactor model for Poland

0 20 40 60 80 100 120

nr months

-0.15

-0.1

-0.05

0 for PC1

vector

interp.

B matrix row

0 20 40 60 80 100 120

nr months

-0.01

0

0.01

0.02

0.03 for PC2

vector

interp.

B matrix row

0 20 40 60 80 100 120

nr months

-6

-4

-2

0

210

-3 for PC3

vector

interp.B matrix row

0 20 40 60 80 100 120

nr months

-2

0

2

410

-3 for PC4

vector

interp.B matrix row

0 20 40 60 80 100 120

nr months

-15

-10

-5

0

510

-4 for PC5

vector

interp.B matrix row

0 20 40 60 80 100 120

nr months

-2

0

2

410

-4 for PC6

vector

interp.B matrix row

0 20 40 60 80 100 120

nr months

-1.5

-1

-0.5

0

0.5

110

-4 for PC7

vector

interp.

B matrix row


The difference in yields obtained in these two steps give the structure of term premia implied byhistory of factors and current state of the yield curve, which was our main goal since the beginning.Table 17 contains some fit measures and descriptive statistics both for yield and return pricing errorsfor seven factor ACM model’s specification. We have cross-analysed the curve for selected maturitiesas previously. In absolute terms, both errors are very small, much smaller than the error we madeduring the estimation of the yield curves per se. Yield pricing errors for maturities of at least 36months have close to normal distributions, similar results as obtained by the authors of ACM forUS data.


Table 17: Fit diagnostics for seven factor ACM model of the Polish yield curve

Panel A. Summary statistics: Yield pricing errorsn= 6 n= 12 n= 24 n= 36 n= 48 n= 60 n= 84 n=120

mean -0.000001 0.000000 0.000000 0.000000 0.000000 -0.000001 0.000000 0.000000std 0.000005 0.000003 0.000003 0.000002 0.000002 0.000001 0.000002 0.000003skewness -2.778 0.474 -1.854 0.113 0.101 -0.060 -0.189 -0.890kurtosis 15.848 4.066 9.642 2.576 3.518 3.502 3.161 4.622rho(1) 0.318 0.396 0.459 0.677 0.543 0.713 0.539 0.190rho(6) 0.044 0.083 0.122 0.382 0.267 0.408 0.241 0.007rho(12) -0.028 0.010 0.032 0.229 0.174 0.329 0.093 -0.075

Panel B. Summary statistics: Return pricing errorsn= 6 n= 12 n= 24 n= 36 n= 48 n= 60 n= 84 n=120

mean 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000std 0.000004 0.000001 0.000006 0.000003 0.000006 0.000003 0.000010 0.000029skewness -2.369 -1.494 -1.960 -1.228 -0.896 -1.503 -0.575 -0.694kurtosis 13.634 7.879 12.371 6.787 6.093 8.186 5.348 5.627rho(1) -0.010 0.510 -0.286 -0.236 -0.405 0.196 -0.417 -0.400rho(6) -0.040 -0.071 -0.008 0.025 0.051 -0.082 0.046 0.041rho(12) -0.144 -0.237 -0.098 -0.094 -0.094 -0.164 -0.066 -0.072

Notes: (1) Sample period 2005:01-2020:06. (2) Table reports pricing errors statistics for both: yields and returns. (3) Differenttenors in months are given in columns, rho(6) denotes autocorelation coefficients of errors of order 6. (4) Mean and standard

deviation are such that 0.0001 stands for 1 basis point.

Average yield pricing errors and their standard deviations for different number of factors speci-fications are shown in Table 18, where we can see clearly that starting from K = 5 all mean errorsare less than 1 basis point already, hence adding factors may not be necessary for relatively goodmodel fit.

Table 18: Yield pricing errors for different ACM model specifications

in basis pointsn=6 n=12 n=24 n=36 n=48 n=60 n=84 n=120

mean (K=3) 2.241 4.217 5.448 4.107 1.708 -0.618 -2.958 0.098std (K=3) 4.606 5.322 4.385 1.490 1.971 2.782 1.522 4.524mean (K=4) 1.911 2.807 1.844 -0.075 -1.176 -1.027 1.231 0.845std (K=4) 2.070 2.395 0.887 1.263 1.187 1.046 1.153 1.720mean (K=5) 0.460 0.614 0.246 0.099 0.365 0.670 0.621 0.457std (K=5) 1.437 0.825 0.465 0.327 0.498 0.442 0.239 0.539mean (K=6) 0.218 0.206 0.141 0.292 0.413 0.410 0.345 0.553std (K=6) 0.595 0.183 0.127 0.307 0.281 0.201 0.131 0.265mean (K=7) -0.005 0.001 -0.003 0.003 -0.001 -0.005 0.001 -0.004std (K=7) 0.048 0.031 0.032 0.021 0.022 0.010 0.019 0.032

Notes: Sample period 2005:01-2020:06. Table reports pricing errors statistics of yields. Different tenors in months are given incolumns. Mean and standard deviation are reported in basis points.

We tested our specification for number of factors using two statistics used in Adrian et al. (2013),namely: Anderson’s rank deficiency of β′ matrix (stacked factor loadings from the second stage ofthe procedure) and Wald’s statistics for presence of unspanned or useless factors. We conductedsuch tests for the Polish yield curve using the yield set produced by the weight system labelled 1and the Table 19 demonstrates the results. All the tests reject null hypotheses with p-values below0.001, hence we may conclude here, that the more factors the better for the model fit (i.e. maxi-mum K we have tested was 7). The drawbacks of such an approach may reveal themselves at thelater stages when we use the model of a particular specification to forecasting, where we naturallyassume that too tight fit would increase out-of-sample errors. We will come back to this in the nextchapter. For now, we take K = 5 as our benchmark number of factors, as in original works of ACM67.

67NB.: Adrian et al. (2013) tested only for K = {3, 4, 5} and concluded that the statistics they used broadlysupport five factor model, but if they test seven factor model they would have obtained the same results asours for Poland


Table 19: Identification tests for various numbers of factors K in ACM model for Poland

number of factors rkk−1 Wβ

3 625.8 4317.44 497.9 10214.25 384.1 8140.56 362.4 6374.07 327.1 20687.6

Notes: table reports rank deficiency test (Anderson) if rank of β is less than K and Wald test if row of β for k = K is zero

Figure 33 illustrates average term premia we have estimated for Poland in the period from 1stJanuary 2005 to 30th June 2020 accompanied by one standard deviation bands (for K = 5). Theterm premia band lays in non-zero area for maturities longer than 2-3 years, which seems to beconsistent with the findings on pure expectation hypothesis testing from the previous chapter. Termpremia are single digit (in basis points) for tenors up to and including one year, then for 2-year zero-coupon bond they were approximately 25 bps, for 5-year - approximately 80 bps and for 10-yeartenor the mean risk premia read 125 bps.

Figure 33: The average term premia with one standard deviation bands by tenors for thePolish yield curve

0 20 40 60 80 100 120

maturity in months

-2

0

2

4

6

8

10

12

14

16

18

term

pre

mia

10-3

mean

mean+std

mean-std

Notes: (1) Sample period 2005:01-2020:06. (2) system of weights labelled 1. (3) K = 7 factors.

Figure 34 provides an overview of historical estimated term premia for four selected maturities(1, 2, 5, and 10-years) over the full data set on daily basis68. This history reveals sometimes violentchanges in the level of 10-year premia (in one year, in 2014 this reading fell by some 120 bps from210 bps in the end of 2013) and also some periods in which the shortest risk premia were negative.

During and after the Financial Crisis, which was the most pronounced in the Polish financialmarket in the period of 2008:09-2011:06 the mid and longer end term premia (5-year and longer)stayed in the upward trend, and the shorter term premia were also lifted but the period was shorterand began slightly earlier: 2008:06-2009:09. The long bond risk premium is prone to be higher dur-ing spikes in general market uncertainty for example amongst several chapters of so called greek debtcrisis in the fourth quarter of 2009, mid 2010 and mid 2011, or quantitative easing announcementsin US and EU or other events as on 15th January 2015 when Swiss National Bank forwent to defendthe floor on a currency pair EUR/CHF69. This behaviour of term premia time series is desirable

68an adjustment to the procedure of estimated VAR and then collecting all the information back in therecursive pricing was inevitable

69which caused mayor disruptions in Polish banking sector ever since, due to significant exposure of retailclients to CHF denominated mortgages


and makes sense economically.

Figure 34: Evolution of term premia for selected tenors of Polish yield curve

2006 2008 2010 2012 2014 2016 2018 2020-0.005

0

0.005

0.01

0.015

0.02

0.025

12 months term premium





Table 20 documents significant correlation coefficients between pairs of various term premia andeconomic indicators for Poland in the full scope of our dataset. Additional to obvious remark thathigher term premia usually coexist with higher inflation (CPI, PPI), it is worth to notice that thecorrelation between term premia and CPI starts close to 0.5 for short tenors and fades away to 0.2levels for long tenors, a complete reverse of term premia relation to PPI for which we observe onstatistically significant correlation for short tenors and relatively strong correlation for long bonds.Naturally, as Malik & Meldrum (2014) explain, we would like our estimated term premia to becountercyclical and somehow positively related to uncertainty about future realisations of the keymacroeconomic or market-related variables. The countercyclicality of the term premia is lucidlypresented in the Table 20 as well, where we find that all the selected tenor’s premia are stronglyand statistically significantly negatively correlated with manufacturing production growth, retailsales growth or unfilled vacancies and moderately and statistically significantly positively correlatedwith other unemployment measures. Surprisingly though, we have observed also moderate positivecorrelation between confidence survey’s results and the term premia as well, but we would haveexpected that the confidence tops during expansion and at he peak of economic activity in the cycleaccompanied by low term premia and bottoms near the through of a business cycle, during whichtimes we should observe relatively high term premia. One of the explanations for such a phenomenonis that the confidence survey may has leading indicator characteristics. On the second, desired char-acteristics of term premia claimed by the authors - positive relation to uncertainty, we have selectedvery common indicator of fear in the global financial markets - namely, VIX index. As the sametable reports, we observed statistically significant positive correlation of this fear factor index andour term premia. Somehow counter-intuitive is that this relation decreases in power as tenors arelonger. Short to medium segments are in a tighter link with VIX than the long end, potentially doto the fact that VIX is by construction a measure of short term implied volatility of equity indicesand theoretically at least one might expect that in the case of a down trending equities one wouldobserve a run on longer term bonds (as a cyclical portfolio rebalancing). Such, explained abovecorrelation structure and qualitative inference have stood for different number of factors tested inthe ACM model (K = {5, 6, 7}).


Table 20: Correlation of term premia with selected economic indicators in Poland

Indicator n = 12 n = 24 n = 36 n = 60 n = 96 n = 120

CPI 0.42837 0.41132 0.38038 0.27002 0.18504 0.18220PPI - 0.14636 0.24035 0.42383 0.45528 0.42910

Manufacturing (survey) - - - 0.17408 0.14519 -Confidence (survey) 0.31259 0.39916 0.46996 0.46336 0.29285 0.24542Leading indicators -0.14505 - - 0.18304 0.21529 0.20988

Industrial production -0.15253 - - 0.23489 0.24023 0.19348Manufacturing production -0.58135 -0.68945 -0.74866 -0.63958 -0.43513 -0.43967

Retail sales -0.48481 -0.59333 -0.66098 -0.59717 -0.44119 -0.45589Unfilled vacancies -0.39794 -0.47403 -0.51379 -0.45363 -0.37376 -0.41165

Unemployment rate 0.22930 0.34400 0.42606 0.43412 0.35511 0.37928Harmonised un. rate 0.18555 0.31164 0.40534 0.42168 0.32257 0.33234

Harmonised un. rate (SA) 0.19309 0.31857 0.41048 0.42364 0.32452 0.33508USDPLN -0.65771 -0.73035 -0.77438 -0.69086 -0.51390 -0.50835

REER 0.68496 0.72527 0.72700 0.54207 0.29357 0.27030VIX (fear factor) 0.46103 0.41555 0.35628 0.20890 - 0.15129

Notes: (1) only significant (with confidence of 95%) coefficients are shown, (2) source of time series of variables: OECD, MainEconomic Indicators - complete database, Main Economic Indicators (database), http://dx.doi.org/10.1787/data-00052-en

(Accessed on 17th October 2020) (3) data codes used: CPGRLE01PLM659N, POLPPDMMINMEI, BSPRTE02PLM460S,

BVCICP02PLM460S, POLLOLITONOSTSAM, POLPROINDMISMEI, POLPROMANMISMEI, POLSARTMISMEI, LMJVTTUVPLM647S,

LMUNRRTTPLM156S, LRHUTTTTPLM156N, LRHUTTTTPLM156S, CCUSMA02PLM618N, RBPLBIS, SPASTT01PLM657N, VIXCLS (4)term premia are in columns, own estimations using ACM model, number of factors K = 5, NSS yields generated via system of

labelled 1, (5) sample 2005:01-2020:06, monthly data.


Table 21: Mean and standard deviation of term premia datasets for different systems ofweights for the Polish yield curve

Panel A. Mean term premia by tenors (in basis points)label n=1 n=3 n=6 n=12 n=24 n=36 n=48 n=60 n=84 n=1201 0.0 0.4 1.9 7.9 25.8 45.2 62.9 78.3 102.6 125.72 0.0 -0.6 0.1 6.1 26.5 47.6 65.3 80.0 103.6 127.03 0.0 0.2 1.8 7.9 26.2 45.6 63.3 78.7 102.9 125.64 0.0 -0.6 0.2 6.1 26.5 47.7 65.3 79.9 103.6 126.95 0.0 0.4 2.0 8.0 25.9 45.4 63.2 78.5 102.6 125.96 0.0 -0.4 0.7 7.0 27.4 48.4 66.1 80.8 104.3 128.17 0.0 0.3 1.9 8.0 25.9 45.3 63.1 78.6 102.7 124.98 0.0 -0.4 0.6 6.9 27.3 48.2 65.9 80.8 104.4 127.79 0.0 0.4 2.0 8.2 26.2 45.7 63.4 78.5 102.2 125.010 0.0 -0.4 0.5 6.5 27.0 48.1 65.8 80.5 104.4 127.811 0.0 0.4 2.0 8.3 26.5 45.9 63.7 79.0 103.0 125.312 0.0 -0.3 0.7 6.7 26.7 47.6 65.4 80.3 104.2 126.913 0.0 0.5 2.3 8.6 26.8 46.3 64.0 79.3 103.3 126.314 0.0 -0.3 0.8 6.9 26.9 47.6 65.3 80.2 104.0 126.915 0.0 0.6 2.4 8.7 26.8 46.3 64.1 79.6 103.6 125.716 0.0 -0.4 0.5 6.7 26.8 47.4 65.1 80.1 104.1 126.817 0.0 0.4 2.1 8.1 25.7 45.0 62.7 78.2 102.4 125.418 0.0 -0.4 0.5 6.5 26.6 47.5 65.3 80.2 104.1 127.519 0.0 0.2 1.6 7.6 25.7 45.2 63.0 78.4 102.5 124.920 0.0 -0.4 0.6 6.7 26.8 47.5 65.3 80.3 104.0 127.221 0.0 0.5 2.2 8.3 26.0 45.4 63.2 78.7 103.1 126.322 0.0 -0.4 0.5 6.6 26.6 47.5 65.3 80.2 104.0 128.023 0.0 0.3 1.8 7.8 25.8 45.1 62.8 78.2 102.6 125.424 0.0 -0.5 0.4 6.7 27.1 48.0 65.9 80.9 104.5 127.925 0.0 0.1 1.3 7.1 25.2 44.7 62.5 77.9 102.1 124.626 0.0 -0.3 1.1 8.5 30.6 51.7 69.1 83.7 107.7 131.227 0.0 0.2 1.8 7.9 26.0 45.5 63.4 78.9 102.9 124.928 0.0 -0.1 1.6 9.2 30.6 51.5 69.2 84.1 107.6 130.8

Panel B. Standard deviation of term premia by tenors (in basis points)label n=1 n=3 n=6 n=12 n=24 n=36 n=48 n=60 n=84 n=1201 0.0 4.2 9.6 17.1 21.7 20.6 19.7 20.6 25.6 34.32 0.0 4.2 9.7 17.3 21.8 20.6 20.5 22.8 29.1 37.73 0.0 4.8 10.5 18.0 22.2 20.8 19.4 19.5 22.8 30.04 0.0 4.1 9.6 17.4 22.4 21.4 21.0 22.6 27.4 34.95 0.0 4.1 9.4 16.8 21.3 19.9 18.7 19.5 24.1 31.76 0.0 4.4 9.9 17.2 21.6 20.5 20.0 21.5 26.8 34.37 0.0 4.9 10.6 18.1 23.1 23.0 22.9 24.2 29.7 39.68 0.0 4.3 9.9 17.5 22.0 20.5 19.6 20.8 25.5 33.39 0.0 4.6 10.1 17.2 21.5 20.4 19.8 21.3 27.6 37.510 0.0 4.0 9.6 17.7 22.5 20.3 19.1 20.8 27.3 36.511 0.0 4.8 10.7 18.3 22.7 21.9 21.6 23.2 29.5 40.512 0.0 4.0 9.6 18.1 23.3 21.5 20.7 22.3 27.6 36.413 0.0 4.6 10.4 17.8 21.5 19.3 17.5 18.0 22.9 31.014 0.0 4.9 10.4 17.7 21.8 19.9 19.1 21.0 27.4 36.615 0.0 5.3 11.4 19.3 24.0 22.7 21.2 21.4 25.1 33.316 0.0 4.5 10.1 17.7 22.1 20.1 18.8 19.7 24.1 31.817 0.0 4.6 10.3 17.8 22.0 20.9 20.1 21.4 27.2 36.818 0.0 4.2 9.8 17.6 21.8 19.8 18.9 20.6 26.2 34.419 0.0 4.4 10.1 17.9 22.6 21.7 21.4 23.1 29.0 39.320 0.0 4.3 10.0 17.7 21.9 20.3 19.3 20.3 24.4 31.721 0.0 4.7 10.5 17.8 21.7 20.0 18.6 19.2 24.2 32.722 0.0 3.9 9.2 16.7 21.3 19.7 18.5 19.8 25.2 33.123 0.0 4.5 10.2 17.8 21.6 19.5 17.9 18.3 21.8 27.924 0.0 4.0 9.5 17.3 22.1 20.8 19.7 20.5 24.2 30.825 0.0 3.8 8.9 16.3 22.4 23.1 23.2 24.3 28.5 36.326 0.0 3.5 8.4 15.6 20.4 19.4 18.4 19.3 23.5 30.227 0.0 4.5 10.2 18.2 24.0 24.1 23.6 24.1 28.2 36.828 0.0 4.0 9.3 16.7 21.4 20.7 19.8 20.4 24.0 30.7

Notes: (1) Sample period 2005:01-2020:06. (2) Table reports pricing errors statistics of yields. Different tenors in months aregiven in columns. (3) Mean and standard deviation are reported in basis points. (4) for K = 5 factors


4.3 Polish term premia robustness

The estimated market prices of risk and therefore term premia structure are robust for changes inthe weight systems (28 sets), which is revealed in Table 21. Although, it is worth to note that theparticular levels of term premia are very sensitive to changes in parameters of sampling and othermacro-level choices in the model, namely, they:

1. depend on the number of factors (cf. Tables: statistical summary in 22 in the main text or 34and 35 in Appendix) in that the less factors the more diversity of premia estimations for shortand medium term bonds (up to 3-4 years), with differences between systems reaching some6-7 bps. Moreover, on average, adding factors in the model increases term premia in tenorsup to 5-years and substantially decreases standard deviation for tenors longer than 3-years.

Table 22: Mean and standard deviation of term premia in models with different number offactors for the Polish yield curve

Panel A. Mean term premia by tenors (in basis points)K n= 1 n= 3 n= 6 n= 12 n= 24 n= 36 n= 48 n= 60 n= 84 n=1203 0.0 -0.4 0.1 4.3 20.6 41.2 61.6 79.6 106.1 125.74 0.0 -0.3 0.5 5.7 24.2 45.4 64.5 80.0 101.9 124.95 0.0 0.4 1.9 7.9 25.8 45.2 62.9 78.3 102.6 125.76 0.0 0.4 2.2 8.3 25.9 45.0 62.9 78.5 102.9 125.77 0.0 0.6 2.4 8.5 26.1 45.3 63.3 79.0 103.4 126.6

Panel B. Standard deviation of term premia by tenors (in basis points)K n= 1 n= 3 n= 6 n= 12 n= 24 n= 36 n= 48 n= 60 n= 84 n=1203 0.0 3.1 7.0 12.9 19.8 23.1 25.2 27.5 33.1 42.54 0.0 4.1 9.4 17.1 23.5 24.2 24.2 25.9 33.0 43.45 0.0 4.2 9.6 17.1 21.7 20.6 19.7 20.6 25.6 34.36 0.0 5.4 11.5 18.4 21.7 20.6 20.0 21.0 25.8 34.77 0.0 6.3 12.2 18.7 21.6 20.1 18.8 19.2 23.6 32.1

Notes: (1) Sample period 2005:01-2020:06. (2) all calculations performed using system of weights labelled 1 (3) columnsrepresent tenors in months.

2. may vary substantially between systems, if we additionally change the short rate from theone implied by the parsimonious form of a curve to some externally given rate (say, NBPbills rate or WIBOR1M).These differences are as high as 50 bps (cf. Figure 35) and provideadditional argument for vertical integration of estimation of a yield curve (blended) and ofthe term premia structure.

Figure 35: Term premia (10Y) for different short interest rates in ACM model for Poland

2006 2008 2010 2012 2014 2016 2018 2020

0

0.005

0.01

0.015

0.02

0.025

rf NSS

WIBOR1W

WIBOR1M



3. depend heavily on the beginning and end dates of a sample and differences could be as highas 100-150 bps (cf. Figures 36 and 37). Although, the variability in the overlapping periodsis similar for various sample exact start and end dates and the correlation structure showstrong positive cohabitation. Substantially different are the levels (i.e. for 10-year tenor andthe sample that starts in January 2007 we read at the beginning of 2008 some 60-70 bpsestimated term premium and for the sample starting in January 2008 for the same month weread approximately 150 bps).

Figure 36: Term premia (10Y) for different end dates of a sample in ACM model for Poland

2006 2008 2010 2012 2014 2016 2018 2020

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

end: 2020-04-30

end: 2019-04-30

end: 2018-04-30

end: 2017-04-30


Figure 37: Term premia (10Y) for different start dates of a sample in ACM model for Poland

2006 2008 2010 2012 2014 2016 2018 20200

0.005

0.01

0.015

0.02

0.025

beg: 2005-01-02

beg: 2006-01-02

beg: 2007-01-02

beg: 2008-01-02


4. change in time, though the process are quite persistent. The range of maximum to minimumfor 10 year premia in our sample is some 200 bps (cf. Figure 34).

The phenomenon indicated in the third bullet above is heuristically the most important sourceof difference between the estimated term premia levels. We propose the following solution to thisinstability and non-robustness (henceforth called: blended time series):


1. start at first working day of January 2005, since when - as previously carefully explained - wehave reliable (and deep enough) data for Polish bond market.

2. curve-out the minimum length of the dataset feasible for ACM model since the start (basesample). This length for one-monthly excess return calibration is precisely equal to the max-imum tenor of term premium we would like to estimate. As we have shown, the Polish yieldcurve liquidity is somehow confined in the tenors up to ten years, and the longer bonds areerratic in supply and turnover. Taking all these into consideration we propose to take first 10years of history as a base sample: 2005:01-2014:12

3. estimate ACM model for the base period and save all the daily history of term premia for2005:01-2014:12 in a term premia database (henceforth: TPDB)

4. iterate forward in every step adding one more day of the history to the base sample andre-estimate term premia, saving only the last row (most recent) to the TPDB.

5. continue iterating until the most current entry of the last working day of June 2020.

Such estimated time series of term premia are robust to some extent to the sample’s end datecrawling forward and allows for filtration-alike comparisons and out-of-sample performance calcula-tions, because at any point of time in the out-of-base period we use only the information availableto that point of time, without artificial insight (knowledge) of the future, which obviously would bemisleading in our inference. Moreover, the once calculated term premia are not changed with timepassing by and new information arrivals.

Figure 38: Term premia in 10Y tenor - blended time series for Poland

2006 2008 2010 2012 2014 2016 2018 2020

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

As of 30th Jun 2020

Blended series

As of 2nd Jan 2015

Notes: (1) Sample period 2005:01-2020:06. (2) system of weights labelled 1. (3) number of factors K = 5, (4) base period2005:01-2014:12.

Visual results of this approach’es implementation for the Polish yield curve term structure areshown in the Figures: 38 (main text) and 75, 76, 77 and 78 (Appendix) for 10-year, 1-, 2-, 3- and5-year tenors respectively. Due to the fact that the last couple of quarters in the base period weuse concides with the period of falling interest rates (both: official and market yields) the estimatedterm premia are generally lower than the ones calculated 5.5 year later for the same period. This isthe consequence of the ACM model’s construction, especially results directly from the VAR modelon factors estimation at the beginning of the procedure. The shorter the tenor the smaller thedifference between out-of-base period blended term premia and the ones calculated using the whole


history. Descriptive statistics of such blended term premia time series are detailed in Table 23 andreveal differences up to 44 bps in mean for 10-year tenor as well as up to two times higher standarddeviations for the blended premia than for the whole history based ones. The results for K = {5, 6, 7}do not differ much qualitatively and quantitatively form each other.

Despite higher volatility and, on average lower levels of term premia calculated using our blendedapproach, we believe it is wise to at least consider the two approaches when inferring on yield curve’sforecasting power in the next chapter. Having these two sources we could perform proper out-of-sample comparisons without a risk of being accused of drawing from the future information whenusing term premia time series estimated on a full history sample and without re-estimating the wholetime series since the beginning of the sample every day, which also might lead to false conclusions.

Table 23: Descriptive statistics of term premia by tenors - comparison of blended and normalapproaches for Poland

Panel A. blended approachmean std max min ρ(1) ρ(12) ρ(24) ρ(36)

3 months 0.0000 0.0005 0.0020 -0.0011 0.8728 0.5890 0.3420 0.14786 months 0.0001 0.0011 0.0036 -0.0016 0.9209 0.6432 0.4010 0.19121 year 0.0006 0.0018 0.0059 -0.0021 0.9457 0.7058 0.4943 0.28522 years 0.0018 0.0027 0.0082 -0.0021 0.9644 0.7752 0.6071 0.40323 years 0.0032 0.0033 0.0096 -0.0033 0.9706 0.7656 0.6025 0.38824 years 0.0044 0.0039 0.0108 -0.0046 0.9700 0.7179 0.5472 0.31215 years 0.0055 0.0045 0.0129 -0.0057 0.9680 0.6673 0.4939 0.23636 years 0.0063 0.0050 0.0148 -0.0068 0.9666 0.6270 0.4583 0.18077 years 0.0069 0.0054 0.0163 -0.0077 0.9663 0.5989 0.4395 0.14668 years 0.0074 0.0058 0.0175 -0.0084 0.9668 0.5813 0.4332 0.12969 years 0.0078 0.0062 0.0183 -0.0090 0.9680 0.5717 0.4352 0.125210 years 0.0081 0.0064 0.0190 -0.0094 0.9694 0.5680 0.4424 0.1296

Panel B. at 30th June 2020mean std max min ρ(1) ρ(12) ρ(24) ρ(36)

3 months 0.0001 0.0005 0.0019 -0.0012 0.8652 0.5059 0.2126 0.06346 months 0.0002 0.0011 0.0038 -0.0020 0.9058 0.5484 0.2573 0.09081 year 0.0008 0.0018 0.0064 -0.0025 0.9272 0.6091 0.3514 0.16642 years 0.0026 0.0021 0.0082 -0.0007 0.9454 0.7153 0.5261 0.31963 years 0.0045 0.0020 0.0088 0.0013 0.9578 0.7694 0.6176 0.41544 years 0.0063 0.0019 0.0094 0.0029 0.9609 0.7074 0.5440 0.37745 years 0.0078 0.0019 0.0117 0.0037 0.9559 0.5494 0.3631 0.22686 years 0.0091 0.0021 0.0136 0.0045 0.9502 0.4063 0.2263 0.08677 years 0.0102 0.0024 0.0152 0.0045 0.9476 0.3254 0.1746 0.01028 years 0.0111 0.0027 0.0164 0.0044 0.9478 0.2941 0.1777 -0.01599 years 0.0119 0.0030 0.0175 0.0043 0.9497 0.2930 0.2068 -0.012510 years 0.0125 0.0033 0.0187 0.0040 0.9522 0.3090 0.2457 0.0067

Notes: (1) Sample period 2005:01-2020:06. (2) all calculations performed using system of weights labelled 1 (3) number offactors K = 5, (4) base period for blended approach 2005:01-2014:12.

4.4 Term premia regressions to test the Expectations Hypothesis

Having estimated term premia we now turn to revisiting the expectations hypothesis tests again.However, this time we do not focus on PEH (pure expectations hypothesis) but the lighter version- EH (expectations hypothesis) which claims that the long term rates are determined by the shortrate interest rate path expectations augmented by some premium (being fixed or variable). We didestimated term premia structure for a wide range of models: with different number of factors, NSSweighting systems used and method of forming the time series (full sample and blended from Section4.3.). The idea here is to try to re-establish the fame of expectations hypothesis by proving thatmarket participants behave rationally and take more into consideration than just pure mathematics(as one may say, arithmetic) of spot and forward rates, namely, tenor-specific term premia. In thetesting part we will use monthly data frequency as it was the case in Chapter 3.

We propose and run a regression which derives from Thornton conventional one presented in


Chapter 1. We believe this is the best way to reflect the definition of EH, unlike Fama&Blissregressions of type 1 and 2 which are not directly translatable and extendible to include term premia.We build on Thornton’s idea of rolling realised returns as regressand and correct the regressor (yieldslope) by term premia structure we estimated via ACM model in the following way:

1

k

k−1∑i=0

y(h)t+i×h − y

(h)t = α + β

(y(n)t − y

(h)t − tp

(n)t + tp

(h)t

)+ ϵ

(n)t+h (83)

where tp(x)t is a term premium estimated at time t and prevailing for period x. Recall that

1k

∑k−1i=0 y

(h)t+i×h is an average realised h-month rate in n -months period when we roll the short

term investment k times.

The natural constraint of h < n, we have explained already in Chapter 1, holds here as well.For the sake of clarity and continued comparativeness with the previous results we utilize the samesets of tenors (bond maturities) and investment horizons. The results of all the four regressions arereported in extensive Table 41. As in the original regression PEH testing, here the null hypothesis(EH holds) is equivalent to the claim that H0 : β = 1. The p-values for different choices of n, k andk as a consequence is reported in a form of a heatmap as Figure 39.

Figure 39: Expectations hypothesis testing adjusted by term premia regression of Thorn-ton (2006) for Poland, five factor model

6 12 24 36 48 60 72


1

3

6

12

24

36

h -

horizon in m

onth

s

sys. label: 1.

0

0

-1

-1

-1

-1

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-1

-1

-1

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-1

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-1

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1

Notes: (1) 0 indicates that there is no sufficient evidence to reject null hypothesis of EH, (2) 1 - sufficient evidence (with at0.05 significance) to reject EH, (3) -1 - not applicable (i.e. a horizon is longer or equal bond’s maturity) (4) NSS fit using

weight system labelled 1. (5) Term premia are estimated at for the full sample here

There are striking differences of these results with the ones obtained for pure expectations hy-pothesis in Chapter 2. Namely:

1. large swathe of the domain (h, n) is now labelled no sufficient evidence to reject H0, which isa major change in comparison to PEH testing where the null hypothesis was rejected for vastmajority of longer bonds (starting from 48-months maturity)

2. the two cases in which we theoretically have evidence to reject the null are for the modelswhich did not pass the F-test, hence we do not consider them as exemptions. Having saidthat, for all plausible models with short term rolling bonds’ maturities of 1 to 12 months andlonger term bonds from 6 to 72 months we infer that expectations hypothesis holds


3. R2 values for majority of models that are plausible is much higher in EH testing than in PEH,which means that inclusion of term premia structure helps in explanation of the investmentreturn form rolling short term bonds in longer horizons by slope of the yield curve for longerterm bonds. This phenomenon is illustrated by Figure 40

4. β coefficients for EH regression are much closer to 1 than they are for PEH, which is obviouslyreflected in the p-value for H0 : β = 1.

Figure 88 in Appendix reports robustness checks of the above findings for changes in numberof factors. Generally the results hold for number of factors up to and including five, adding onlyslightly to the domain of null hypothesis’ rejection for higher number of factors the combinationsof (h, n) for n = 72. This means that if we allow for over fitted versions of ACM model we couldfind stronger evidence for rejection of expectations hypothesis for long term bonds, or in anotherwords - term premia structure would not be enough to explain the difference between the rates ofcompounded rolled short investments in bonds and the rate of return of a long term bond. But sincein the previous chapters we inferred that five factors ACM model is fitted enough to the data on onehand and useful in medium term forecasting on the other, we claim here that the optimal numberof factors that also guarantee that EH holds, is five.

Figure 40: R2 values for PEH and EH testing for various longer maturity n and short bond h

1: 6

1:1

2

1:2

4

1:3

6

1:4

8

1:6

0

1:7

2

3: 6

3:1

2

3:2

4

3:3

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3:4

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6:1

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6:2

4

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6:6

0

6:7

2

12:2

4

12:3

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12:4

8

12:6

0

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2

24:4

8

24:7

2

36:7

2

0

0.1

0.2

0.3

0.4

0.5

0.6

EH

PEH

Notes: (1) x-axis labels are formatted: h:n, where h is a short bond maturity (the bond which is rolled k times until theinvestment is of the same length as the longer bond) and n is a longer bond maturity.

The obvious critique of such a method of testing expectations hypothesis would be that the ACMterm premia in the regressions are estimated on the basis of all the history information and hencethey contaminate the expectations’ formation along the history of other variables. If we were to usedifferent time series for each observation of a monthly yield curve we would not be able to check theEH at least via regression developed from Thornton’s formulation. This is why we have proposedand documented the idea of building incrementally a time series of term premia which we calledblended of very simple but desirable property. The term premia adjusted in blended algorithm arenot subject to every day revision but do not contain any information about future (with respect toa certain observation) developments of the interest rate structure. Our robustness checks involvedalso checking if our EH results hold even if we change term premia to the blended time series. AsTable 42 reveals - they do hold except for a small miss for 12-month horizon and 72-month bond.

Bearing all of the above in mind, we claim finally here that expectations hypothesis saying thatthe long term rates are determined by the short rate interest rate path expectations augmentedby some premium holds for the Polish government yield curve in vast majority of plausible and


feasible (due to data series’ length constraints) choices of long and short term investments, andthat existence of term premia structure explains the differences between compounded rates of re-turns from shorter investments and corresponding maturity longer term zero-coupon bond’s yield.Therefore, we have not found here sufficient evidence to reject the Hypothesis 3 of this dissertation.

4.5 Summary

In this chapter we have reported the results of various model set-ups in term premia estimation forPolish government yield curve. The parameters time series for these curves were calculated pre-viously with originally designed optimal system of weights based on liquidity measures for variousbonds and blending information from NBP bills in the ultra-short end.

First three principal components in PCA decomposition, as it is common for most of the marketsglobally, are driving more that 99% of explained variance in any of the four possible pairs betweentwo dimensions: daily, monthly frequency of data and denser, sparser calibration tenors. However,in sample errors analysis, suggested to consider more factors in the benchmark model. With five andmore factors we obtained yield pricing errors within one basis point on average in each consideredtenor. Therefore, in the subsequent part of this thesis we consider only minimum five factor set-ups.Further verification whether models with high number of factors are too tight will be provided in thenext chapter where we conduct extensive out-of-sample research testing forecasting power of varietyof yield curve models.

Term premia estimations and market prices of risk factors were found. Contrary to Adrian etal. (2013) findings for US data, in our modelling environment for Poland the factor responsible forlevel risk was not priced on a statistically significant level in any of the considered model’s variates.All other factors were priced and some of them included one more components in pricing then theself source. For example, the slope risk has two significant components: 2nd and 3rd factors. Ourterm premia are countercyclical and positively correlated with market uncertainty measures, whichare desired characteristics of such results.

Estimated term premia structure are, on average, upward sloping, with premia significantly dif-ferent from zero for at least medium tenor bonds (3-4-year), which is in line with findings of theprevious chapter on the domain in which pure expectations hypothesis operates in Poland. Levelof risk premia in these estimations is very sensitive to many modelling choices, such as beginningand end dates of the sample, short interest rate’s type used in calculation of excess returns, and thenumber of factors used in PCA. We have proposed a solution to a part of these problems in a formof a simple recipe to produce new time series of term premia using a base period data and thenadding every day only a newly estimated premia for that particular date.

We have verified here the key Hypothesis 3 on holding of expectations hypothesis in Poland,which serves as an integrating backbone of the whole thesis. Using previously meticulously preparedyield curves and the extracted term premia times series, we have managed to show that expectationshypothesis holds for almost all feasible choices of long and short term investments, and that existenceof term premia structure explains the differences between compounded rates of returns from shorterinvestments and corresponding maturity longer term zero-coupon bond’s yield.


Fig

ure

41:Extended

Thorn

ton’s

regression

:R

olli

ng

real

ised

retu

rns

onte

rmsl

ope

and

term

pre

mia

regr

essi

ons

hn

αSE(α

)p-val

βSE(β

)p-val

R2

R2adj.

obs

RMSE

F-stat

Fp-val

ρ(6)

ρ(12)

ρ(24)

ρ(36)

R1

1.0000

6.0000

-0.0008

0.0004

0.0000

1.0598

0.2259

0.6045

0.3720

0.3680

179.0000

0.0022

104.8400

0.0000

0.4600

-0.0100

-0.3300

-0.1700

R2

1.0000

12.0000

-0.0016

0.0008

0.0000

1.1695

0.2103

0.7899

0.4430

0.4400

173.0000

0.0038

135.9400

0.0000

0.5800

0.0200

-0.4200

-0.2000

R3

1.0000

24.0000

-0.0029

0.0013

0.0000

1.2153

0.1448

0.9315

0.5250

0.5220

161.0000

0.0053

176.0200

0.0000

0.6600

0.1800

-0.4000

-0.1600

R4

1.0000

36.0000

-0.0040

0.0017

0.0000

1.0920

0.1434

0.7394

0.5270

0.5230

149.0000

0.0058

163.4800

0.0000

0.7100

0.3200

-0.2000

-0.1100

R5

1.0000

48.0000

-0.0053

0.0020

0.0000

0.9193

0.1758

0.3231

0.4790

0.4750

137.0000

0.0060

123.9600

0.0000

0.7700

0.4600

0.0200

-0.0900

R6

1.0000

60.0000

-0.0072

0.0023

0.0000

0.8088

0.1589

0.1144

0.4230

0.4180

125.0000

0.0065

90.1100

0.0000

0.7900

0.5000

0.1200

-0.0700

R7

1.0000

72.0000

-0.0089

0.0029

0.0000

0.7581

0.1642

0.0704

0.3720

0.3660

113.0000

0.0073

65.7700

0.0000

0.8400

0.5800

0.1200

-0.0100

R8

3.0000

6.0000

-0.0005

0.0003

0.0000

1.0881

0.2589

0.6331

0.3100

0.3060

179.0000

0.0014

79.6600

0.0000

0.4100

0.0000

-0.2700

-0.1500

R9

3.0000

12.0000

-0.0013

0.0007

0.0000

1.2077

0.2327

0.8140

0.3930

0.3900

173.0000

0.0033

110.8800

0.0000

0.5400

0.0000

-0.3800

-0.1800

R10

3.0000

24.0000

-0.0026

0.0013

0.0000

1.2515

0.1610

0.9409

0.4680

0.4640

161.0000

0.0051

139.6300

0.0000

0.6400

0.1700

-0.3600

-0.1300

R11

3.0000

36.0000

-0.0037

0.0017

0.0000

1.1134

0.1648

0.7543

0.4550

0.4520

149.0000

0.0060

122.8900

0.0000

0.7100

0.3400

-0.1700

-0.0800

R12

3.0000

48.0000

-0.0050

0.0021

0.0000

0.9094

0.2065

0.3304

0.3870

0.3820

137.0000

0.0063

85.1800

0.0000

0.7700

0.4600

0.0400

-0.0800

R13

3.0000

60.0000

-0.0069

0.0024

0.0000

0.7771

0.1873

0.1170

0.3250

0.3190

125.0000

0.0068

59.1500

0.0000

0.8000

0.5000

0.1200

-0.0600

R14

3.0000

72.0000

-0.0088

0.0030

0.0000

0.7216

0.1926

0.0741

0.2810

0.2740

113.0000

0.0076

43.3000

0.0000

0.8400

0.5700

0.1100

-0.0200

R15

6.0000

12.0000

-0.0009

0.0005

0.0000

1.2488

0.2753

0.8169

0.3100

0.3060

173.0000

0.0024

76.7500

0.0000

0.3900

-0.0400

-0.2700

-0.1500

R16

6.0000

24.0000

-0.0022

0.0012

0.0000

1.2834

0.1919

0.9302

0.3810

0.3770

161.0000

0.0048

97.6900

0.0000

0.6100

0.1300

-0.3000

-0.0800

R17

6.0000

36.0000

-0.0033

0.0017

0.0000

1.1300

0.1998

0.7425

0.3590

0.3550

149.0000

0.0060

82.3400

0.0000

0.7000

0.3300

-0.1400

-0.0500

R18

6.0000

48.0000

-0.0046

0.0022

0.0000

0.8864

0.2542

0.3274

0.2760

0.2700

137.0000

0.0066

51.3700

0.0000

0.7700

0.4600

0.0500

-0.0600

R19

6.0000

60.0000

-0.0066

0.0025

0.0000

0.7192

0.2319

0.1130

0.2100

0.2030

125.0000

0.0072

32.6400

0.0000

0.7900

0.4900

0.1100

-0.0500

R20

6.0000

72.0000

-0.0086

0.0031

0.0000

0.6545

0.2376

0.0730

0.1750

0.1680

113.0000

0.0080

23.5600

0.0000

0.8300

0.5500

0.1000

-0.0200

R21

12.0000

24.0000

-0.0014

0.0009

0.0000

1.2351

0.2742

0.8044

0.2090

0.2040

161.0000

0.0039

42.0900

0.0000

0.5200

-0.0200

-0.2100

0.0200

R22

12.0000

36.0000

-0.0025

0.0015

0.0000

1.1358

0.2728

0.6907

0.2230

0.2180

149.0000

0.0056

42.1900

0.0000

0.6900

0.2900

-0.1500

0.0200

R23

12.0000

48.0000

-0.0038

0.0021

0.0000

0.8343

0.3509

0.3184

0.1440

0.1380

137.0000

0.0067

22.7600

0.0000

0.7700

0.4500

0.0400

-0.0500

R24

12.0000

60.0000

-0.0060

0.0025

0.0000

0.5738

0.3246

0.0945

0.0790

0.0720

125.0000

0.0074

10.6100

0.0015

0.7900

0.4700

0.0900

-0.0500

R25

12.0000

72.0000

-0.0081

0.0031

0.0000

0.4863

0.3318

0.0608

0.0580

0.0500

113.0000

0.0082

6.8900

0.0099

0.8200

0.5200

0.0800

-0.0300

R26

24.0000

48.0000

-0.0023

0.0016

0.0000

0.9025

0.5029

0.4232

0.0710

0.0640

137.0000

0.0055

10.2500

0.0017

0.7700

0.4200

-0.0400

0.0700

R27

24.0000

72.0000

-0.0069

0.0027

0.0000

0.1164

0.4719

0.0306

0.0020

-0.0070

113.0000

0.0074

0.1700

0.6770

0.8300

0.5300

0.0300

0.0300

R28

36.0000

72.0000

-0.0055

0.0020

0.0000

-0.6533

0.7287

0.0116

0.0260

0.0170

113.0000

0.0061

2.9900

0.0863

0.8000

0.5200

0.1400

-0.0800

Reg

ression:

1 k

∑ k−1

i=0y(h

)t+

i×h−

y(h

)t

=α+

β( y

(n)

t−

y(h

)t

−tp

(n)

t+

tp(h

)t

) +ϵ(

n)

t+h

Nullhypoth

esis

(EH

holds):H

0:β=

1.

Standard

errors

are


asticityandauroco

rrelationco

rrectionala

New

ley-W

est

Bold

estimatesofβ

havelower

p-valueth

an0.05(ourassumed

significa

nce

level)andca


esis

redp-values

forF-testindicate

thatth

ereis

no

suffi

cientevid

enceto

reject

nullhypoth

esis

thatth

emodel

withnoindep

enden

tvariablesfits

thedata

aswellasth

emodel

tested

Term

premia

usedare

estimatedforfivefactormodel

withth

ehighestranked

system

ofweights

atth

een

doffullsample’s

scope(June2020)


Fig

ure

42:Extended

Thorn

ton’s

regression

:R

olli

ng

real

ised

retu

rns

onte

rmsl

ope

and

term

pre

mia

regr

essi

ons,

blen

ded

term

pre

mia

tim

ese

ries h

nα

SE(α

)p-val

βSE(β

)p-val

R2

R2adj.

obs

RMSE

F-stat

Fp-val

ρ(6)

ρ(12)

ρ(24)

ρ(36)

R1

1.0000

6.0000

-0.0009

0.0004

0.0000

1.1273

0.2948

0.6670

0.3720

0.3680

179.0000

0.0022

104.7300

0.0000

0.4200

-0.0400

-0.3400

-0.1800

R2

1.0000

12.0000

-0.0018

0.0008

0.0000

1.2336

0.2296

0.8454

0.4290

0.4260

173.0000

0.0039

128.4900

0.0000

0.5500

0.0100

-0.4000

-0.2200

R3

1.0000

24.0000

-0.0037

0.0013

0.0000

1.2757

0.1353

0.9792

0.5350

0.5320

161.0000

0.0052

182.8500

0.0000

0.6500

0.1900

-0.4000

-0.1800

R4

1.0000

36.0000

-0.0053

0.0018

0.0000

1.1339

0.1829

0.7679

0.5480

0.5450

149.0000

0.0057

178.1000

0.0000

0.7200

0.3800

-0.1200

-0.0500

R5

1.0000

48.0000

-0.0067

0.0022

0.0000

0.9523

0.2209

0.4145

0.4980

0.4950

137.0000

0.0059

134.1300

0.0000

0.8000

0.5600

0.1800

0.0100

R6

1.0000

60.0000

-0.0084

0.0025

0.0000

0.8333

0.1903

0.1906

0.4230

0.4190

125.0000

0.0065

90.2700

0.0000

0.8500

0.6300

0.2800

0.0400

R7

1.0000

72.0000

-0.0098

0.0030

0.0000

0.7744

0.1688

0.0907

0.3610

0.3550

113.0000

0.0073

62.6600

0.0000

0.8700

0.6500

0.2400

0.0200

R8

3.0000

6.0000

-0.0005

0.0003

0.0000

1.1782

0.2934

0.7282

0.2970

0.2930

179.0000

0.0015

74.7900

0.0000

0.3900

-0.0200

-0.2500

-0.1800

R9

3.0000

12.0000

-0.0015

0.0007

0.0000

1.2389

0.2289

0.8517

0.3630

0.3590

173.0000

0.0034

97.2700

0.0000

0.5200

-0.0100

-0.3500

-0.2100

R10

3.0000

24.0000

-0.0034

0.0013

0.0000

1.2693

0.1620

0.9518

0.4630

0.4600

161.0000

0.0051

137.0300

0.0000

0.6500

0.1900

-0.3400

-0.1400

R11

3.0000

36.0000

-0.0050

0.0018

0.0000

1.1101

0.2099

0.7002

0.4630

0.4590

149.0000

0.0059

126.8000

0.0000

0.7400

0.4000

-0.0900

-0.0300

R12

3.0000

48.0000

-0.0064

0.0023

0.0000

0.9057

0.2529

0.3546

0.3980

0.3940

137.0000

0.0063

89.3000

0.0000

0.8100

0.5600

0.1600

0.0100

R13

3.0000

60.0000

-0.0081

0.0026

0.0000

0.7723

0.2171

0.1471

0.3250

0.3190

125.0000

0.0068

59.1100

0.0000

0.8500

0.6100

0.2500

0.0300

R14

3.0000

72.0000

-0.0097

0.0031

0.0000

0.7174

0.1949

0.0736

0.2730

0.2660

113.0000

0.0077

41.6100

0.0000

0.8700

0.6300

0.2100

0.0100

R15

6.0000

12.0000

-0.0010

0.0005

0.0000

1.2248

0.2723

0.7955

0.2780

0.2740

173.0000

0.0025

65.8400

0.0000

0.3900

-0.0600

-0.2500

-0.1800

R16

6.0000

24.0000

-0.0029

0.0012

0.0000

1.2448

0.2030

0.8860

0.3730

0.3690

161.0000

0.0048

94.6300

0.0000

0.6200

0.1700

-0.2700

-0.0800

R17

6.0000

36.0000

-0.0045

0.0019

0.0000

1.0696

0.2435

0.6125

0.3640

0.3600

149.0000

0.0060

84.0700

0.0000

0.7400

0.4100

-0.0700

0.0100

R18

6.0000

48.0000

-0.0059

0.0024

0.0000

0.8375

0.2900

0.2876

0.2870

0.2820

137.0000

0.0066

54.4400

0.0000

0.8100

0.5400

0.1500

0.0100

R19

6.0000

60.0000

-0.0077

0.0027

0.0000

0.6862

0.2509

0.1056

0.2190

0.2120

125.0000

0.0071

34.4000

0.0000

0.8400

0.5800

0.2100

0.0200

R20

6.0000

72.0000

-0.0094

0.0032

0.0000

0.6352

0.2336

0.0592

0.1790

0.1710

113.0000

0.0080

24.1800

0.0000

0.8600

0.6100

0.1800

0.0000

R21

12.0000

24.0000

-0.0019

0.0009

0.0000

1.1638

0.2876

0.7156

0.2220

0.2170

161.0000

0.0039

45.3100

0.0000

0.5500

0.0300

-0.1600

0.0200

R22

12.0000

36.0000

-0.0036

0.0017

0.0000

1.0107

0.3029

0.5140

0.2390

0.2340

149.0000

0.0056

46.1300

0.0000

0.7400

0.3800

-0.0700

0.0800

R23

12.0000

48.0000

-0.0049

0.0023

0.0000

0.7394

0.3368

0.2195

0.1660

0.1590

137.0000

0.0066

26.8000

0.0000

0.8100

0.5200

0.1300

0.0100

R24

12.0000

60.0000

-0.0068

0.0026

0.0000

0.5444

0.2911

0.0588

0.1040

0.0960

125.0000

0.0073

14.2200

0.0003

0.8300

0.5500

0.1700

0.0100

R25

12.0000

72.0000

-0.0087

0.0031

0.0000

0.4885

0.2952

0.0416

0.0770

0.0690

113.0000

0.0082

9.2400

0.0030

0.8400

0.5700

0.1600

0.0000

R26

24.0000

48.0000

-0.0030

0.0018

0.0000

0.6709

0.3378

0.1649

0.0830

0.0760

137.0000

0.0054

12.2100

0.0006

0.8000

0.4700

0.0100

0.1000

R27

24.0000

72.0000

-0.0070

0.0024

0.0000

0.2259

0.2971

0.0046

0.0110

0.0020

113.0000

0.0074

1.1900

0.2770

0.8400

0.5700

0.1000

0.0400

R28

36.0000

72.0000

-0.0049

0.0019

0.0000

-0.2768

0.3070

0.0000

0.0110

0.0020

113.0000

0.0061

1.2100

0.2735

0.8000

0.5400

0.1700

-0.0800

Reg

ression:

1 k

∑ k−1

i=0y(h

)t+

i×h−

y(h

)t

=α+

β( y

(n)

t−

y(h

)t

−tp

(n)

t+

tp(h

)t

) +ϵ(

n)

t+h

Nullhypoth

esis

(EH

holds):H

0:β=

1.

Standard

errors

are


asticityandauroco

rrelationco

rrectionala

New

ley-W

est

Bold

estimatesofβ

havelower

p-valueth

an0.05(ourassumed

significa

nce

level)andca


esis

redp-values

forF-testindicate

thatth

ereis

no

suffi

cientevid

enceto

reject

nullhypoth

esis

thatth

emodel

withnoindep

enden

tvariablesfits

thedata

aswellasth

emodel

tested

Term

premia

usedare

estimatedforfivefactormodel

withth

ehighestranked

system

ofweights

usingblen

ded

algorith

mdescribed

inChapter4


5 Chapter 5. Forecasting power of the Polish government yield curve

In the previous chapter we estimated a family of ACM models for Polish yield curve and extractedpure risk-free interest rate structure and term premia (term premia structure being a simple differ-ence between risky yield curves CACM

yc,tmaxand the risk-free ones CACM

rf,tmaxfor each date t0 ≤ t ≤ tmax).

It is worth recalling that we are talking about risk-free and risky layers of yield curve despite that weanalyse government bonds only, which in certain jurisdiction may be as such referred to as risk-free.The risk-free-ness in ACM model means that the rates do not include any premium or discount dueto risk in variation of certain PCA factors, and by no means it is a judgement on the probability ofdefault of the bonds’ issuer.

The chapter is organised in five parts. First, explains our approach towards ex-post models’accuracy evaluation and has introductory character. Second and third part is devoted to extensiveout-of-sample evaluation of different ACM model variants (especially with different number of fac-tors and two initial period of estimation lengths; 5 and 10-years) against naive benchmarks of (1)implied term structure of interest rates and (2) constant interest rates from a forecasting origin.Our main enquiry would focus on the best (judging by the average ex-post errors) variants of statespace generation models and the potential explanations of such their performance. We are verycircumspect in this ex-post model accuracy evaluation and split it onto two subsections: (1) on levelof interest rates forecasts, which are naturally prone to a number of biases due to the ambiguityof degree of integration of the first factor when modelling rates, (2) on slope of interest rates fore-cast, which are usually stationary processes and potentially the prognosis produced ex-post wouldbe much more accurate. The fourth part is dedicated to short term interest rate forecast qualityevaluation, whereas the fifth part provide verification of the final Hypothesis 4: Unlike in thecase of liquid markets, professional forecasters expectations do not help to increase theinformational content of the yield curves in the case of less liquid markets. To do so wewill conduct here an in-depth study of short interest rate paths time series forecasted by the profes-sionals. As we indicated in the Introduction, we have a string suspicion that, professional forecasterssurveys report effectively concurrent implied forward rates rather than the actual expected futureshort interest.

5.1 Forecasting strategy

We build our strategy along two partitions, a distinction between level and slope forecasts as wellas a variety of types of interest rates to be forecasted. Ideally, we would like ACM-family models tocorrectly predict the whole yield curves in the future, hence our primary focus would be on the levelforecasts. However, we acknowledge the findings of the previous chapter, one in particular, that thelevel factor is not priced by ACM model for Poland, and therefore we will additionally investigatethe forecasting power of the Polish government yield curve in the domain of slope forecasts. Thesecond source of the degrees of freedom in our analysis would be type of the variable forecasted.Naturally, we would like to know the forecast characteristics for all the significant for the marketparticipants and economists bond yields in a form of a vector of yields, but on the top of these wewould like to know if ACM model gives some insights into the future paths of risk-free short interestrate, solely. The latter is of a special interest to monetary and fiscal policy makers. Knowing thestructure of ACM model it is obvious to notice that the stripped risk-free yield curve is a forecastper se, and it is straight forward to extract the implied short term interest rate path from the shapeof the risk-free ACM yield curve and no additional assumptions on the dynamics of state space areneeded.

Yet another important dimension to consider when designing forecasting framework is the trade-off in a trio of the maximum tenor on the yield curve we may estimate, the maximum horizon ofprognosis and the minimum statistically reasonable, common out-of-sample size. We will check two


set-ups: for 10-year and for 5-year maximum term premia, because we would like to check if thebest performing model’s parametrisations are aligned in these two different sampling regimes and toexplore impact of different trends of interest rates in both approaches on our results. As one mayrecall from Figure 1 in Chapter 2, in the period of 2005:01-2010:01 Polish interest rates went up anddown without an obvious trend, billowed by the outburst of the Financial Crisis of 2008, whereasin the next 5-year period 2010:01-2015:01 we have experienced a well-established downward trendprimarily driven by falling inflation or even deflation. We expect these two different staring pointsof out-of-sample periods have an impact on the performance of analysed models here.

Having both remarks in mind we propose the following main areas of analysis of the forecastingpower of the Polish government bond yield curve:

1. level forecasts of risky rates with maximum 10-year tenors (CACMyc,10y )

2. level forecasts of risky rates with maximum 5-year tenors (CACMyc,5y )

3. slope forecasts of risky rates with maximum 10-year tenors (CACMyc,10y − CACM

rf,1m)

4. slope forecasts of risky rates with maximum 5-year tenors (CACMyc,10y − CACM

rf,1m)

5. level forecasts of risk-free short term interest rates (CACMrf,1m)

Note that there in no sense of considering slope in the case of short term rates (as it would be simplyzero)70.

5.2 Ex-post level forecasts

The maximum time span of the dataset we generated for Poland is 186 months (01:2005-06:2020)while the minimum number of months for the ACM model with up to 10-year premia is 120 andfor 5-year - 60, hence we have 66 or 126 months out-of-sample range. Due to the constraints for10-year tenors we will use expanding in-sample window (as opposed to: rolling estimation scheme)in forecasting in both regimes. We would move the forecast origin and keep the beginning of estima-tion period the same, equal to the beginning of the full sample, January 2005. The main reason forthat choice is that we will behave in the same manner once the optimal model has been chosen andwe would like to put it in production and we do not want any discrepancies between two the twomaximum tenor regimes. Recursive estimation in the case of ACM yield curve modelling makes lesssense because we are forecasting the whole yield curves (sets of interest rates for different tenors)for various forecasting horizons anyway.

Since we have 66 months in out-of-sample period in 10-year variant and 126 in 5-year case wehave to choose forecasting horizons carefully, or not too greedy to be precise. It would not be a goodidea to choose, for example, horizons up to and including 60-months in 10-year maximum set-up,because we would have only 6 forecasts paths to compare them with realisations. Therefore, wehave decided to use forecasting monthly horizons vector of h10y = {3, 6, 12, 18, 24} in 10-year andh5y = {3, 6, 12, 18, 24, 36, 60} when 5-year maximum tenor of term premium would be considered.

In what follows, we consider both set-ups: 10-year and 5-year maximum tenors carefully in-dicating when any difference should be explained. We conduct a forecasting horse-race of modelsspanning from ACM environment against some simple benchmark models of yield curves:

70Also notice that CACMrf,1m is a scalar representing a short term 1-month interest rate, not a vector of interest

rates as in the case of a full yield curve i.e : CACMyc,5y


1. level persistence (or so called random walk, when density forecasts are made) naive bench-mark (henceforth: Bper) - which simply assume that the yield curve stays the same in thewhole forecasting period as it is just prior to forecasting origin’s date

2. implied level forwards - which is extensively used in pricing and valuations of certainassets and originates from the assumption of no arbitrage on the financial markets (withoutrisk premia) (henceforth: Bimp)

In ACM model we are reconstructing the fitted yields (both: risky and risk-free) from estimatedmatrices A in B and the matrix of market prices of risk factors (Λ) in the affine representation ofinterest rates. Recall that to have it done successfully we need also the short interest rate path(1-month rate in our specification). In forecasting then we would need some assumptions on thebehaviour of the state space of factors (XN

t : Tfo < t ≤ Tfo+h, where Tfo means a forecasts originand Tfo+h indicates the longest forecasting horizon’s date) and the path of short interest rates tt forthe same period. Layered structure of ACM model gives a modeller several possible ways to produceout-of-sample and then ex-ante forecasts. The decision space we propose here is generated by thefollowing choices:

1. on rt short-term interest rate path for Tfo < t ≤ Tfo+h:

(a) choose a risk-free interest rates path that is implied in curves CACMrf,Tfo

by calculating a

series of 1-month forward rates prevailing at Tfo71 (referred to as rf:imp)

(b) assume that the risk-free short interest rates stay the same as at the forecast origin Tfo

(referred to as rf:per)

2. on factors Xt for Tfo < t ≤ Tfo+h:

(a) each of the factor orthogonally follow AR(p) process and we generate forecasts usingthese processes’ estimated in-sample parameters. Lags p are chosen to be p = {1, 3, 6}(referred to as X:AR(p))

(b) factors follow VAR(p) - since our factors in-sample are standardised we also use vectorautoregressive models with µ = 0 model and we generate forecasts using this processestimated in-sample parameters. Lags p are chosen to be p = {1, 3, 6} (referred to asX:VAR(p))

(c) factors follow BVAR(p) - to limit the number of models tested we concentrate on acommon set-up of Bayesian vector autoregressive models, namely, the one with normalconjugate priors (cf. R. B. Litterman (1986)) with fixed innovations covariance matrixand we generate forecasts using this model of factors 72. Lags p are chosen to be p ={1, 3, 6} (referred to as X:BVAR(p))

We would like to underline that the split forecasting approach proposed above (pure short term risk-free interest rate structure forecast and separate forecast of term premia dynamics) is not commonin the literature (to say the least) but we believe it is worth exploiting here in the horse race ofmodels. In order to do it properly we have to investigate a bit more into the dynamics of estimatedterm premia structure to find candidates for models governing solely the term premia.

The above presented strategy with regard to level forecasts results in 18 models per each num-ber of factors tested (we will conduct tests for K = {3, 4, 5, 6, 7}) as combinations of short-terminterest rate choice and factors model (i.e MK=5

Tfo,h(rf:imp, X:VAR(2)) is a five factor forecasting

model estimated in-sample up to Tfo and using implied short risk-free interest rates as of Tfo andVAR(2) model to generate state space of factors for number of horizons from the vector h of months).Each model will be used to produce forecasts for yields of tenors x10y = {3m, 1y, 3y, 5y, 10y} and

71these are, in fact, ACM model’s implicit expectations of the short interest rates (one-month)72Matlab’s function normalbvarm


x5y = {3m, 1y, 3y, 5y} (as seen in Diebold & Li (2006)) for horizons from the vectors h10y and h5y

respectively. The RMSFE (Root Mean Square Forecast Error) and MAFE (Mean Absolute ForecastErrors) out-of-sample fit measures will be calculated in the following manner (for each tenor x andeach horizon h)

RMSFExh =

√√√√√√ 1

N∗

TN∗fo∑

t=T 1fo

(yxt+h − yx,ft,h )2 (84)

MAFExh =

1

N∗

TN∗fo∑

t=T 1fo

|yxt+h − yx,ft,h | (85)

where T 1fo and TN∗

fo are the first and the last forecast origin in out-of-sample range respectively, yxt+h

is a zero coupon rate for tenor x observed at t + h date and yx,ft,h is the forecasted zero coupon ratefor the same tenor and prevailing at the same time t + h.

The choice of these measures is primarily driven by the wish for diversity: we want one atleast one measure that shows potential bias of our forecasts and the other which augments extremeabsolute errors. The measures are arithmetically averaged for all forecasting origins and reportedfor selected horizons h accompanied by the total averages (by horizons). Therefore, each model inparticular will be cast N∗ = 42 and N∗ = 68 times73 and compared ex-post with observed yieldcurves in the full time span of the dataset we use. Since the strategy presented requires more than100 models to be run we present in the main text only the results for five factor model and a selectedtenors of {10y} from the 10-year window (cf. Table 24) and simulations for selected other factorsand tenors are included in the technical appendix (cf. Tables 36 to 50 for 10-year window and Tables51 to 62 for the 5-year window).

Table 24: Forecast errors for 10-year tenor in different models with K = 5, 10-year window,level forecasts

RMSFE MAFEModel variety 3m 6m 12m 18m 24m 3m 6m 12m 18m 24m

Bper 32.40 47.84 46.59 64.19 95.17 25.75 32.69 36.86 53.55 76.74Bimp 38.34 43.84 50.78 78.06 117.18 27.66 33.57 43.55 60.95 93.31

MK=5Tfo,h

(rf:imp, X:AR(1)) 31.02 44.90 46.08 63.22 95.58 24.17 31.12 36.92 54.59 78.91

MK=5Tfo,h

(rf:imp, X:AR(3)) 29.51 42.95 46.19 64.36 94.45 23.09 29.31 34.51 57.03 82.64

MK=5Tfo,h

(rf:imp, X:AR(6)) 30.11 44.93 45.20 61.13 92.45 23.35 30.70 34.04 53.25 79.52

MK=5Tfo,h

(rf:imp, X:VAR(1)) 30.63 42.84 49.29 67.55 102.75 24.11 32.54 43.15 53.11 84.68

MK=5Tfo,h

(rf:imp, X:VAR(3)) 29.51 42.18 51.88 61.93 86.59 24.20 30.60 41.62 56.16 75.75

MK=5Tfo,h

(rf:imp, X:VAR(6)) 34.57 54.27 64.93 69.05 94.61 28.07 37.15 53.05 60.81 84.12

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 43.42 44.36 41.47 59.62 156.52 33.86 37.43 35.84 51.98 117.62

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 43.76 43.16 43.76 75.25 211.35 33.30 35.01 37.04 59.57 167.28

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 44.02 59.46 69.97 161.02 924.38 34.03 41.60 51.59 115.97 672.46

MK=5Tfo,h

(rf:per, X:AR(1)) 30.72 43.35 44.54 62.73 93.06 22.29 30.61 35.05 54.67 78.56

MK=5Tfo,h

(rf:per, X:AR(3)) 29.34 41.90 46.19 65.67 93.21 21.36 29.15 34.25 57.73 83.00

MK=5Tfo,h

(rf:per, X:AR(6)) 29.95 43.58 44.52 61.65 90.27 21.59 30.6 32.91 53.73 78.89

MK=5Tfo,h

(rf:per, X:VAR(1)) 29.92 40.10 46.07 65.21 99.86 22.89 29.54 40.70 51.12 81.19

MK=5Tfo,h

(rf:per, X:VAR(3)) 29.75 40.86 51.01 62.09 84.94 24.61 29.99 39.62 55.65 74.40

MK=5Tfo,h

(rf:per, X:VAR(6)) 34.46 52.67 63.13 68.36 92.64 27.75 36.30 50.76 59.51 81.93

MK=5Tfo,h

(rf:per, X:BVAR(1)) 41.61 41.69 38.75 61.62 162.47 30.91 34.22 33.34 53.92 124.63

MK=5Tfo,h

(rf:per, X:BVAR(3)) 43.00 42.20 42.94 77.08 220.11 30.97 33.30 35.44 59.70 175.76

MK=5Tfo,h

(rf:per, X:BVAR(6)) 44.32 58.59 68.90 163.06 962.10 33.53 40.81 51.95 119.45 703.28

Notes: (1) various horizons are reflected in different columns h = {3, 6, 12, 18, 24}. (2) all errors in bps

73we have 66 out-of-sample months and maximum horizon of 24 months in the 10-year set-up and 128out-of-sample months and maximum horizon of 60 months in the 5-year set-up


Table 43 offers an overview of the best performing models in the space of various factors, tenorsand horizons in the 10-year window set-up, to which we may add the following commentary:

1. for horizons h ≥ 18m both measures RMSFE and MAFE indicate the same best model typein most of the cases,

2. the higher the number of factors K the more frequent is the case that Bper is the best model(especially for h ≥ 18m ), meaning that there is no forecasting power of the yield curve modelsused in that window, as we did not beat even the naive benchmark which assumes no changein interest rates until infinity,

3. the longer the tenor the more chance that the best model would be some version of vectorautoregressive model (normal of Bayesian) with 1 or 3 lags, especially for longer horizons,

4. models with lags of 6 only appeared for 10-year tenors with h = {12m, 18m} and only in theset-ups of multiple autoregressions on factors (AR(6)),

5. the vast majority of the best performing models are casted with assumption of short interestrates rf staying the same as at forecasting origin, hence implied rates paths from the yieldcurve shape did not help in this period of time,

6. Bayesian vector autoregressive models are only appearing in this ranking with lag equal 1, ifany.

Tables 45 and 46 report information of the same structure and qualitative content as the onediscussed above but the models are evaluated with 5-year window. The maximum horizon as well astenor are therefore 5-years as explained before. The results are alike the ones obtained for 10-yearwindow with subtle dissimilarities:

1. the horizon for which Bper is the best model for tenors of at least 3 years is shifted to the righth ≥ 36m from 18-months (in the case of 10-year window),

2. only one model and only for three factor set-up was based on the implied path of risk-freerates, all the others are casted with assumption of short interest rates rf staying the same asat forecasting origin (this is more stark than in the case of 10-year window),

3. the share of VAR and BVAR models in the best model’s table increases with the lengtheningof the forecasting horizon up to and including five factor set-ups,

4. very frequently best models in their categories are with 3 lags (not 1, as in 10-year window).

The corresponding mean error values (RMSFE and MAFE) in these models with 10-year and 5-yearwindows are presented in Tables 44 and 46 respectively. Acknowledging the fact that the tableshave slightly different domains in the space of forecasting horizons and tenors of bonds, even if wecompare the errors in pairs, the results obtained for 5-year windows are considerably worse thanthe ones for longer, 10-year window, roughly by 50− 100%. For example, a 5-year tenor, 12-monthshorizon MAFE in five factor models is approx. 64 basis points for 10-year window and approx. 102basis points calculated for 5-year window. Such a discrepancy is mainly due to different stages ofinterest rates cycles and trend momentum at the end of both initial windows of 10- and 5-years.The latter happens to be the end of relatively high interest rates and just a day before a massiveslide in the interest rates by almost 3 pct points in 3 years on average.

In figures 47, 48, 79 we visually inspect selected data already reported in detail in the tables withMAFE and RMSFE. We propose to look at each model’s category: AR, VAR, BVAR separately andthree factors K = {3, 5, 7} in each one of them against the two naive benchmarks Bper and Bimp forthree selected bond’s tenors xsel = {3m, 3y, 10y}. Clearly,


Fig

ure

43:

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ors,

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and

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Fig

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44:

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AR(1)p

AR(6)p

AR(3)p

AR(3)p

VAR(3)p

VAR(3)p

Bper

45y

AR(1)p

AR(6)p

AR(3)p

BVAR(6)p

BVAR(6)p

Bper

AR(3)p

AR(1)p

AR(6)p

AR(3)p

VAR(3)p

BVAR(6)p

Bper

Bper

53m

Bim

pBim

pVAR(1)p

VAR(1)p

AR(3)p

VAR(3)p

VAR(3)p

Bim

pBim

pVAR(1)p

AR(1)p

AR(3)p

VAR(3)p

VAR(3)p

51y

Bim

pAR(6)p

AR(1)p

AR(3)p

AR(3)p

VAR(3)p

VAR(3)p

Bim

pAR(6)p

AR(1)p

AR(3)p

AR(3)p

VAR(3)p

VAR(3)p

53y

AR(1)p

AR(6)p

AR(3)p

AR(3)p

BVAR(3)p

Bper

Bper

AR(1)p

AR(6)p

AR(3)p

AR(3)p

BVAR(3)p

VAR(3)p

Bper

55y

AR(1)p

AR(6)p

AR(3)p

BVAR(3)p

BVAR(3)p

Bper

AR(3)p

AR(1)p

AR(6)p

AR(3)p

AR(3)p

BVAR(3)p

Bper

Bper

63m

Bim

pBim

pVAR(1)p

VAR(1)p

AR(3)p

AR(3)p

VAR(3)p

Bim

pBim

pVAR(1)p

AR(1)p

AR(3)p

AR(3)p

VAR(3)p

61y

Bim

pAR(6)p

AR(1)p

AR(3)p

AR(3)p

AR(3)p

AR(3)p

Bim

pAR(6)p

AR(1)p

AR(3)p

AR(3)p

AR(3)p

AR(3)p

63y

AR(1)p

AR(6)p

AR(3)p

AR(3)p

Bper

Bper

Bper

AR(1)p

AR(6)p

AR(3)p

AR(3)p

Bper

Bper

Bper

65y

AR(1)p

AR(6)p

AR(3)p

AR(3)p

Bper

Bper

AR(3)p

AR(1)p

AR(6)p

AR(3)p

AR(3)p

BVAR(3)p

Bper

Bper

73m

Bim

pBim

pVAR(1)p

VAR(1)p

AR(3)p

AR(3)p

AR(3)p

Bim

pBim

pVAR(1)p

AR(1)p

AR(3)p

AR(3)p

AR(3)p

71y

Bim

pAR(6)p

AR(1)p

AR(3)p

AR(3)p

AR(3)p

AR(3)p

Bim

pAR(6)p

AR(1)p

AR(3)p

AR(3)p

AR(3)p

AR(3)p

73y

AR(1)p

AR(6)p

AR(3)p

AR(3)p

Bper

Bper

Bper

AR(1)p

AR(6)p

AR(3)p

AR(3)p

Bper

Bper

Bper

75y

AR(1)p

AR(6)p

AR(3)p

AR(3)p

BVAR(6)p

Bper

AR(3)p

AR(1)p

AR(6)p

AR(3)p

AR(3)p

BVAR(6)p

Bper

Bper


Fig

ure

46:

Bes

tp

erfo

rmin

gm

od

els’errors

by

nu

mb

erof

fact

ors,

ten

ors

and

hor

izon

s,5-

year

win

dow

,le

vel

fore

cast

s

RMSFE

MAFE

Kt

3m

6m

12m

18m

24m

36m

60m

3m

6m

12m

18m

24m

36m

60m

33m

19.78

34.14

74.45

104.05

131.53

172.31

219.32

15.09

27.08

59.46

90.57

112.47

146.56

200.00

31y

27.35

44.03

79.49

107.38

136.37

175.26

235.32

21.50

34.40

59.02

88.31

112.57

158.81

215.59

33y

38.36

56.74

94.02

119.82

140.38

187.00

237.46

30.58

45.15

70.92

98.56

122.16

160.64

216.66

35y

39.70

60.91

98.12

114.21

117.76

182.05

226.64

31.36

49.95

73.75

98.81

100.31

152.45

202.09

43m

19.78

34.14

74.45

102.20

129.79

170.28

212.73

15.09

27.08

59.45

88.85

110.58

142.7

190.88

41y

27.35

43.69

78.61

106.82

136.13

182.52

233.30

21.50

34.08

58.08

87.40

112.82

150.92

211.98

43y

38.05

55.29

91.81

117.44

147.58

187.00

237.46

30.41

44.75

69.98

96.61

124.81

170.14

216.66

45y

39.80

61.57

98.28

115.93

124.87

182.05

226.07

32.02

50.50

73.73

97.33

109.32

164.59

202.09

53m

19.78

34.14

74.28

102.37

129.95

170.92

213.80

15.09

27.08

58.70

88.93

110.67

143.32

191.43

51y

27.35

43.94

78.79

107.12

136.44

184.20

234.92

21.50

34.18

58.20

87.53

113.03

152.89

212.08

53y

38.21

55.59

92.04

117.75

140.40

187.00

237.46

30.53

45.00

70.03

96.73

120.70

169.80

216.66

55y

39.78

61.51

98.15

114.21

122.24

182.05

226.21

31.94

50.39

73.45

98.72

102.46

164.59

202.09

63m

19.78

34.14

74.31

102.82

129.99

171.46

217.77

15.09

27.08

58.56

88.93

110.68

145.67

196.32

61y

27.35

43.95

78.82

107.14

136.45

184.86

236.89

21.50

34.17

58.24

87.55

113.04

160.12

217.22

63y

38.16

55.64

92.12

117.83

148.54

187.00

237.46

30.47

45.12

70.03

96.82

131.33

170.99

216.66

65y

39.73

61.60

98.19

119.88

146.63

182.05

226.25

31.83

50.61

73.47

98.67

122.46

164.59

202.09

73m

19.78

34.14

73.24

102.48

129.98

171.44

219.29

15.09

27.08

58.14

88.92

110.67

145.66

200.00

71y

27.35

43.96

78.81

107.14

136.45

184.85

236.88

21.50

34.19

58.23

87.55

113.04

160.12

217.21

73y

38.15

55.68

92.13

117.83

148.54

187.00

237.46

30.48

45.17

70.02

96.82

131.33

170.99

216.66

75y

39.69

61.65

98.18

119.87

138.66

182.05

226.23

31.82

50.66

73.48

98.67

114.89

164.59

202.09


1. the differences in MAFE between models with various K are minuscule (within 2 basis points)for AR category and substantial for VAR and BVAR group, although - as suggested by theliterature - we observe higher errors for versions with higher number of factors K,

2. the shorter the horizon the more chance that a considered model beats both benchmarks,

3. in 3-month tenor most set-ups of AR models with fixed short term rate (rf:per) are better thenbenchmark in all considered horizons but the difference gradually diminishes along the horizonaxis, suggesting that for longer horizons than 2-years no model is better than benchmark,

4. in 3-year tenor the advantage of AR models is still pronounced, but the maximum horizon inwhich they beat benchmarks is approximately 15-months,

5. in 10-year tenor this feature of AR models is less and less visible, which smaller differences tobenchmarks and maximum forecasting horizons of 12 months,

6. for VAR group of models we obtain results satisfactory and systematically better than naivebenchmarks for the version with fixed short term rate (rf:per) and maximum number of lagsup to 3, nevertheless the gaps to benchmarks are less spectacular and consistent than in thecase of AR based models,

7. the most popular no-arbitrage benchmark Bimp used in valuation of many instruments by thefinancial markets’ participants is almost always the worst performer if compared with AR andVAR models with any number of factors K and for any horizon h.

In summary of this section on level forecasts, and taking these findings into account we posit herethat the safest forecasting strategy (when considering 10-year window) would be to use multiple ARprocesses with lags up to 3 as our state space generators and to stick to the fixed short interest ratefor the whole forecasting horizons span. In our ex-post evaluation in this window we have obtainedgood results up to 1 year horizon for all tenor under investigation (up to 10-year). The magnitudeof MAFE for these models (five factors for the sake of clarity, but errors for three or seven factorsdiffer only by 1-2 basis points) in such defined horizons is also reasonable: approx. 14 bps for short3-month tenor, approx. 32-33 bps for 3-years, and approx. 33-34 bps for 10-years.


Figure 47: Mean absolute forecasting errors of autoregressive models in 10-year window set-up, level forecasts

0 5 10 15 20 25

horizon in months

0

50

100

AR

(1)im

p

3m tenor

0 5 10 15 20 25

horizon in months

0

50

100

3y tenor

0 5 10 15 20 25

horizon in months

20

40

60

80

10010y tenor

0 5 10 15 20 25

horizon in months

0

50

100

AR

(3)im

p

0 5 10 15 20 25

horizon in months

0

50

100

0 5 10 15 20 25

horizon in months

20

40

60

80

100

0 5 10 15 20 25

horizon in months

0

50

100

AR

(6)im

p

0 5 10 15 20 25

horizon in months

0

50

100

0 5 10 15 20 25

horizon in months

20

40

60

80

100

0 5 10 15 20 25

horizon in months

0

50

100

AR

(1)p

er

3m tenor

0 5 10 15 20 25

horizon in months

0

50

100

3y tenor

0 5 10 15 20 25

horizon in months

20

40

60

80

10010y tenor

0 5 10 15 20 25

horizon in months

0

50

100

AR

(3)p

er

0 5 10 15 20 25

horizon in months

0

50

100

0 5 10 15 20 25

horizon in months

20

40

60

80

100

0 5 10 15 20 25

horizon in months

0

50

100

AR

(6)p

er

0 5 10 15 20 25

horizon in months

0

50

100

0 5 10 15 20 25

horizon in months

20

40

60

80

100

Notes: (1) all errors are reported in basis points. (2) first 3 rows of tiles consist of the models where risk free short interestrates are implied by the curve (whole path), next 3 rows of tiles - where risk free short interest rate is the one prevailing at themoment of making the forecast. (3) red lines are naive benchmarks: the thicker one is Bper and the thinner one is Bimp. (4)

there are 3 models plotted in each tile for K = {3, 5, 7}, the thicker the line the higher number of factors.


Figure 48: Mean absolute forecasting errors of vector autoregressive models in 10-year windowset-up, level forecasts

0 5 10 15 20 25

horizon in months

0

50

100

VA

R(1

)im

p

3m tenor

0 5 10 15 20 25

horizon in months

0

50

100

3y tenor

0 5 10 15 20 25

horizon in months

20

40

60

80

10010y tenor

0 5 10 15 20 25

horizon in months

0

50

100

VA

R(3

)im

p

0 5 10 15 20 25

horizon in months

0

50

100

0 5 10 15 20 25

horizon in months

20

40

60

80

100

0 5 10 15 20 25

horizon in months

0

50

100

VA

R(6

)im

p

0 5 10 15 20 25

horizon in months

0

50

100

0 5 10 15 20 25

horizon in months

20

40

60

80

100

0 5 10 15 20 25

horizon in months

0

50

100

VA

R(1

)pe

r

3m tenor

0 5 10 15 20 25

horizon in months

0

50

100

3y tenor

0 5 10 15 20 25

horizon in months

20

40

60

80

10010y tenor

0 5 10 15 20 25

horizon in months

0

50

100

VA

R(3

)pe

r

0 5 10 15 20 25

horizon in months

0

50

100

0 5 10 15 20 25

horizon in months

20

40

60

80

100

0 5 10 15 20 25

horizon in months

0

50

100

VA

R(6

)pe

r

0 5 10 15 20 25

horizon in months

0

50

100

0 5 10 15 20 25

horizon in months

20

40

60

80

100




Figure 49: Mean absolute forecasting errors of vector autoregressive models in 5-year windowset-up, level forecasts

0 20 40 60

horizon in months

0

100

200

300

400V

AR

(1)im

p3m tenor

0 20 40 60

horizon in months

0

100

200

3003y tenor

0 20 40 60

horizon in months

0

100

200

3005y tenor

0 20 40 60

horizon in months

0

100

200

300

400

VA

R(3

)im

p

3m tenor

0 20 40 60

horizon in months

0

100

200

3003y tenor

0 20 40 60

horizon in months

0

100

200

3005y tenor

0 20 40 60

horizon in months

0

100

200

300

400

VA

R(6

)im

p

3m tenor

0 20 40 60

horizon in months

0

100

200

3003y tenor

0 20 40 60

horizon in months

0

100

200

3005y tenor

0 20 40 60

horizon in months

0

100

200

300

400

VA

R(1

)per

3m tenor

0 20 40 60

horizon in months

0

100

200

3003y tenor

0 20 40 60

horizon in months

0

100

200

3005y tenor

0 20 40 60

horizon in months

0

100

200

300

400

VA

R(3

)per

3m tenor

0 20 40 60

horizon in months

0

100

200

3003y tenor

0 20 40 60

horizon in months

0

100

200

3005y tenor

0 20 40 60

horizon in months

0

100

200

300

400

VA

R(6

)per

3m tenor

0 20 40 60

horizon in months

0

100

200

3003y tenor

0 20 40 60

horizon in months

0

100

200

3005y tenor




5.3 Ex-post slope forecasts

This subsection provides the results of evaluation of ex-post forecasts of the yield curve slope (againstshort term 1-month interest rate) and extends the tests started in the previous part. We exploithere exactly the same testing environment, including:

1. two window frames: 10-year and 5-year as common introductory in-sample period and ex-panding in-sample window as a method to re-estimate the models in each of the monthlyforecasting origins in the out-of-sample range,

2. usage of two naive benchmarks, this time called: slope persistence (Bslope:per) and impliedslope forwards (Bslope:imp), which are exact analogues of the level naive benchmarks. Asfar as the latter is concerned one remark is necessary. The implied slope is defined as adifference between a forward rate of a given tenor at a given point in the future and a 1-month forward rate also implied by the curve for the same given date in the future (not thecontemporaneous short term interest rate at a given the forecasting origin),

3. usage of use two forecasting monthly horizons vectors of: h10y = {3, 6, 12, 18, 24} in 10-yearand h5y = {3, 6, 12, 18, 24, 36, 60} when 5-year maximum tenor of term premium would beconsidered,

4. exactly the same list of models (18) used for generation of the future state space,

5. the same measures of error to be analysed: RMSFE and MAFE and their standard deviations.

An example of ex-post errors in slope forecasts for five factor models, for 10-year tenor bond ispresented in Table 25 against slope benchmarks mentioned above.

Table 25: Forecast errors for 10-year tenor in different models with K = 5, 10-year window,slope forecasts


Bslope:per 34.88 51.42 49.02 66.03 89.02 24.99 33.69 34.77 53.53 71.54Bslope:imp 41.49 46.1 45.69 67.72 71.96 31.08 37.06 36.8 52.46 58.04

MK=5Tfo,h

(rf:imp, X:AR(1)) 36.01 53.36 51.73 66.73 81.76 25.42 35.34 39.80 56.64 69.14

MK=5Tfo,h

(rf:imp, X:AR(3)) 36.31 53.18 53.25 68.31 81.00 27.28 37.67 44.14 59.75 70.50

MK=5Tfo,h

(rf:imp, X:AR(6)) 38.41 57.96 52.44 66.94 78.31 29.29 41.63 43.46 59.31 69.00

MK=5Tfo,h

(rf:imp, X:VAR(1)) 32.39 42.31 37.67 59.23 78.79 24.65 31.41 32.17 41.02 59.70

MK=5Tfo,h

(rf:imp, X:VAR(3)) 35.71 49.60 48.80 60.84 72.24 28.2 33.6 38.22 49.93 58.24

MK=5Tfo,h

(rf:imp, X:VAR(6)) 42.60 65.66 62.73 62.9 66.61 34.46 45.93 49.83 56.55 53.25

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 43.65 49.81 39.58 65.27 112.69 33.08 33.74 29.3 48.81 92.42

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 52.15 62.88 58.23 70.78 164.98 44.53 49.10 47.29 54.56 140.48

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 90.38 97.57 109.73 163.71 837.70 77.41 85.81 93.01 120.34 574.58

MK=5Tfo,h

(rf:per, X:AR(1)) 36.53 53.62 50.64 65.24 82.26 26.13 35.59 37.10 54.87 68.68

MK=5Tfo,h

(rf:per, X:AR(3)) 36.71 52.67 50.94 65.81 80.43 28.24 36.86 39.78 56.76 69.24

MK=5Tfo,h

(rf:per, X:AR(6)) 39.28 58.12 50.30 64.55 77.69 30.50 40.93 39.67 56.69 67.86

MK=5Tfo,h

(rf:per, X:VAR(1)) 35.47 47.38 44.74 63.82 85.15 28.16 37.18 38.65 45.71 67.88

MK=5Tfo,h

(rf:per, X:VAR(3)) 36.16 51.77 50.85 61.33 75.02 29.73 36.57 42.6 50.33 59.29

MK=5Tfo,h

(rf:per, X:VAR(6)) 43.61 67.91 65.55 63.83 69.92 35.46 48.99 52.68 55.5 56.41

MK=5Tfo,h

(rf:per, X:BVAR(1)) 40.96 49.96 42.81 70.58 124.16 29.99 34.81 34.79 53.68 105.45

MK=5Tfo,h

(rf:per, X:BVAR(3)) 47.36 59.75 55.36 74.39 191.88 39.79 46.12 42.46 57.74 162.43

MK=5Tfo,h

(rf:per, X:BVAR(6)) 89.10 96.67 109.83 165.40 900.41 75.65 83.97 90.95 123.23 619.85


When compared with the level forecasts’ errors in Table 24 we could notice that:


1. significant differences in errors (between level and slope forecasts) start in relatively longertime horizons: 18 and 24-month, which may be explained by the fact that for h ≤ 12 shortterm interest rates were persistent and did not manage to diverge enough from the originalreading at forecasting horizon.

2. judging by MAFE and contrary to the level forecasts, in naive slope benchmarks Bslope:imp

was better than Bslope:per for the longest horizon of 24-months and the difference was 14bps in favour of the first one. For all other horizons, the results were in line with levelbenchmarks, meaning that the implied slope based benchmark was worse than the constantslope benchmark.

3. all the best models’ variants were vector autoregressive (VAR) with one exception for Bayesianvector autoregressive for 12-month horizon, whereas in the level forecasts there were mostlyautoregressive (AR) classes

4. all the best resulst for slope forecasts come from the models with assumption of implied inthe curve risk free rate path, as oppose to level forecasts were constant short risk free interestrates was found in all the best performing models’ versions.

There are substantial differences in all the tenors and horizons if we juxtapose best models’names in Tables 43 and 50 in one pair and best errors in Tables 44 and 51 in a second. In the10-year window set-up, we clearly see that

1. situations in which there are the same models ranked as first in both measures: RMSFE andMAFE are less common in slope than in level forecasts

2. there is a very well pronounced domination of VAR and BVAR types in slope forecasts bestmodels table

3. simple autoregressive set-ups almost never win (except for 2 out of 75 cases)

4. the vast majority of the best performing models are casted with assumption of short interestrates rf are implied from the yield curve shape, in opposition to level forecasts

5. the longer the horizon the higher lags appear in a model’s specification, i.e. up to h ≤ 18 isnot common the see other lags than 1, whereas for h = 24 lags of 1.

The mean error values in out-of-sample forecasting exercises are reported in Table 51. When com-pared with level forecasts ex-post performance in Table 44 we see generally lower errors for slopeset-ups with significantly lower errors readings for 24-month horizon and for 3-month tenor bonds.The differences for the longest horizon are in the range of 9-30 bps for 10-year tenor in favour ofslope forecasting.

Tables 52 and 53 report information of the same structure and qualitative content as the onediscussed above but the models are evaluated with 5-year window. The maximum horizon as wellas tenor are therefore 5-years as explained before. The results are substantially different from theones obtained for 10-year window in that:

1. Bimp is very common to be the best model for horizons of 24-months or more and for tenors of5-years in both measures: RMSFE and MAFE, whereas for 10-year window it only happenedonce (10-year tenor and 24-month horizon)

2. majority of the best performing models are was based on the assumption of short interest ratesrf staying the same as at forecasting origin (this is more stark than in the case of 10-yearwindow),

3. the share of VAR and BVAR models in the best model’s table sudelly drops to zero for horizonsgreater or equal to 36-months and tenors of 5-years.


Fig

ure

50:

Bes

tp

erfo

rmin

gm

od

els

by

nu

mb

erof

fact

ors,

ten

ors

and

hor

izon

s,10

-yea

rw

ind

ow,slope

fore

cast

s

RMSFE

MAFE

Kt

3m

6m

12m

18m

24m

3m

6m

12m

18m

24m

33m

VAR(1)p

BVAR(3)p

Bper

BVAR(1)i

VAR(3)i

VAR(1)p

BVAR(3)p

AR(6)i

BVAR(1)i

VAR(3)i

31y

BVAR(1)i

BVAR(3)i

Bper

Bper

VAR(3)i

BVAR(1)i

BVAR(3)i

Bper

Bper

VAR(3)i

33y

BVAR(1)i

BVAR(1)i

Bper

Bper

VAR(3)i

BVAR(1)i

Bper

Bper

BVAR(1)i

VAR(3)i

35y

VAR(1)i

BVAR(1)i

BVAR(1)i

BVAR(1)i

VAR(3)i

VAR(1)i

BVAR(1)i

Bper

VAR(1)i

VAR(3)i

310y

VAR(1)i

VAR(1)i

VAR(1)i

VAR(3)p

VAR(3)i

Bper

BVAR(1)i

BVAR(1)i

VAR(1)i

VAR(1)i

43m

BVAR(3)p

BVAR(3)p

Bper

VAR(3)i

VAR(3)p

BVAR(3)p

AR(6)i

Bper

VAR(3)i

VAR(3)i

41y

BVAR(3)p

Bper

Bper

Bper

VAR(3)i

Bper

Bper

Bper

Bper

VAR(6)i

43y

BVAR(1)i

BVAR(1)i

BVAR(1)i

Bper

VAR(6)i

BVAR(1)i

Bper

AR(6)p

BVAR(1)i

VAR(6)i

45y

VAR(1)i

VAR(1)i

BVAR(1)i

VAR(1)i

VAR(6)p

VAR(1)i

BVAR(1)i

BVAR(1)i

VAR(1)i

VAR(6)p

410y

VAR(1)i

VAR(1)i

VAR(1)i

VAR(1)i

VAR(6)p

VAR(1)i

VAR(3)i

VAR(1)i

VAR(1)i

Bim

p

53m

BVAR(1)i

BVAR(1)i

Bper

Bper

VAR(3)i

BVAR(1)i

AR(3)p

Bper

BVAR(1)i

VAR(3)i

51y

BVAR(1)i

Bper

Bper

Bper

VAR(3)i

Bper

Bper

Bper

Bper

Bper

53y

BVAR(1)i

BVAR(1)i

BVAR(1)i

Bper

VAR(3)i

BVAR(1)i

BVAR(1)i

AR(6)p

BVAR(1)i

VAR(3)i

55y

VAR(1)i

BVAR(1)i

BVAR(1)i

VAR(1)i

VAR(6)i

VAR(1)i

BVAR(1)i

BVAR(1)i

VAR(1)i

VAR(1)i

510y

VAR(1)i

VAR(1)i

VAR(1)i

VAR(1)i

VAR(6)i

VAR(1)i

VAR(1)i

BVAR(1)i

VAR(1)i

VAR(6)i

63m

BVAR(1)p

BVAR(1)i

Bper

Bper

Bper

BVAR(1)p

Bper

Bper

BVAR(1)i

Bper

61y

BVAR(1)i

BVAR(1)i

Bper

Bper

Bper

BVAR(1)i

Bper

Bper

Bper

Bper

63y

VAR(1)i

BVAR(1)i

BVAR(1)i

BVAR(1)i

VAR(3)i

Bper

BVAR(1)i

BVAR(1)i

BVAR(1)i

VAR(3)i

65y

VAR(1)i

VAR(1)i

BVAR(1)i

VAR(1)i

VAR(6)i

Bper

BVAR(1)i

BVAR(1)i

VAR(1)i

VAR(6)i

610y

Bper

VAR(1)i

VAR(1)i

VAR(6)p

VAR(6)i

Bper

VAR(1)i

VAR(1)i

VAR(1)i

VAR(6)i

73m

BVAR(1)p

BVAR(1)i

Bper

Bper

Bper

BVAR(1)p

BVAR(1)i

Bper

Bper

Bper

71y

BVAR(1)i

BVAR(1)i

Bper

Bper

Bper

Bper

Bper

Bper

Bper

Bper

73y

BVAR(1)p

BVAR(1)i

BVAR(1)i

BVAR(1)i

VAR(3)i

Bper

BVAR(1)i

BVAR(1)i

BVAR(1)i

Bper

75y

VAR(1)i

VAR(1)i

BVAR(1)i

VAR(1)i

VAR(6)i

Bper

BVAR(1)i

BVAR(1)i

VAR(1)i

VAR(6)i

710y

VAR(1)i

VAR(1)i

VAR(1)i

VAR(6)p

VAR(6)i

Bper

VAR(1)i

VAR(1)i

VAR(1)i

VAR(6)i


Fig

ure

51:

Bes

tp

erfo

rmin

gm

od

els’errors

by

nu

mb

erof

fact

ors,

ten

ors

and

hor

izon

s,10

-yea

rw

ind

ow,slope

fore

cast

s

RMSFE

MAFE

Kt

3m

6m

12m

18m

24m

3m

6m

12m

18m

24m

33m

2.45

3.86

4.73

4.14

2.62

1.89

3.09

3.62

3.59

2.06

31y

10.87

16.34

18.97

17.59

12.07

8.43

13.00

15.69

15.05

9.47

33y

20.33

30.90

35.72

35.24

29.00

14.92

23.93

28.30

30.93

24.77

35y

26.56

36.94

37.59

45.24

42.69

20.74

27.21

32.71

36.68

37.71

310y

31.89

42.73

38.66

57.08

64.97

24.99

31.03

31.81

40.90

54.78

43m

3.02

4.49

4.73

4.53

3.03

2.40

3.61

3.88

3.82

2.56

41y

12.42

17.60

18.97

17.59

13.49

9.10

13.47

15.69

15.05

10.90

43y

20.12

30.87

34.48

35.24

30.26

14.68

23.93

27.68

30.40

23.04

45y

26.62

38.44

36.14

44.45

49.76

20.29

26.73

31.04

34.64

42.78

410y

32.54

43.31

37.01

57.47

67.25

24.52

31.67

31.76

40.29

58.04

53m

3.19

4.54

4.73

4.83

3.63

2.65

3.60

3.88

3.96

2.95

51y

11.99

17.60

18.97

17.59

14.15

9.10

13.47

15.69

15.05

11.88

53y

19.92

29.48

33.09

35.24

37.72

14.60

23.46

28.09

30.5

31.96

55y

26.42

37.53

34.02

46.67

55.56

20.48

25.15

29.39

36.55

46.90

510y

32.39

42.31

37.67

59.23

66.61

24.65

31.41

29.30

41.02

53.25

63m

3.04

4.43

4.73

4.83

4.61

2.44

3.65

3.88

3.96

3.43

61y

11.35

17.42

18.97

17.59

18.06

9.08

13.47

15.69

15.05

11.88

63y

21.11

28.64

31.16

33.76

42.65

15.55

21.62

27.72

29.07

35.52

65y

27.79

36.7

34.28

43.59

46.79

21.57

27.78

27.97

34.97

39.73

610y

34.88

45.05

35.63

54.36

57.37

24.99

31.09

28.59

40.93

47.07

73m

3.04

4.37

4.73

4.83

4.61

2.43

3.55

3.88

4.08

3.43

71y

11.52

16.99

18.97

17.59

18.06

9.10

13.47

15.69

15.05

11.88

73y

21.03

28.15

30.82

35.08

44.41

15.55

21.11

27.84

30.55

37.26

75y

28.02

36.15

33.61

43.79

54.70

21.57

26.90

27.71

35.27

46.61

710y

34.02

41.36

33.87

51.15

58.12

24.99

30.25

27.72

40.89

47.49


For the corresponding pairs of tenors and horizons in two Tables 51 and 53 we observe:

1. slightly higher errors in 5-year window than in 10-year window, but the structure is the sameand the magnitude of errors is similar in both set-ups,

2. the errors seem not to be monotone in horizon (for 5-year window), they increase with horizonlength up to 12-24 months range and then they start to decease, reaching more or less the samelevels for 60-month horizons as for 3-month. This phenomenon is due to cyclical characteristicsand tendency of the slope of yield curve. It was not clearly visible in 10-year window exercisesbecause there we had only 24 months of horizon domain,

3. the difference between errors for 3-year and 5-year tenors for the longest forecasting horizonis in very small (1-2 bps).

The ex-post performance testing provided in this subsection for slope forecasts suggest that againwe should rather stick to 10-year window and shorter horizons (up to 24-months), because 5-yearwindow and longer horizons are subject to greater errors and it is harder to point one type of modelthat is systematically better than the others regardless of tenors and horizon choices. Obviously,slope forecasts are more accurate than the level forecasts. Should we wish to stick to 24-monthmaximum horizon for slope prognosis we suggest using VAR with 3 or 6-monthly lags.


Fig

ure

52:

Bes

tp

erfo

rmin

gm

od

els

by

nu

mb

erof

fact

ors,

ten

ors

and

hor

izon

s,5-

year

win

dow

,slope

fore

cast

s

RMSFE

MAFE

Kt

3m

6m

12m

18m

24m

36m

60m

3m

6m

12m

18m

24m

36m

60m

33m

VAR(3)p

VAR(3)p

BVAR(1)p

BVAR(6)p

BVAR(6)p

AR(1)p

Bim

pVAR(3)p

VAR(3)p

BVAR(1)p

BVAR(6)p

BVAR(6)p

AR(1)p

Bim

p

31y

VAR(3)p

VAR(3)p

BVAR(6)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pVAR(3)p

VAR(3)p

BVAR(1)p

BVAR(6)p

BVAR(6)p

Bim

pBim

p

33y

VAR(3)p

Bim

pBVAR(6)p

BVAR(1)p

Bim

pBim

pBim

pVAR(3)p

Bim

pBVAR(6)p

BVAR(1)p

Bim

pBim

pAR(1)p

35y

AR(6)i

AR(6)p

Bim

pBVAR(1)p

Bim

pBim

pAR(1)p

AR(1)p

AR(6)i

Bim

pBim

pBim

pAR(1)p

AR(1)p

43m

VAR(3)p

Bim

pBVAR(6)p

BVAR(6)p

Bim

pBim

pBim

pVAR(3)p

VAR(3)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

p

41y

VAR(3)p

Bim

pBVAR(6)p

BVAR(6)p

Bim

pBim

pBim

pVAR(3)p

VAR(3)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

p

43y

VAR(3)p

Bim

pBVAR(6)p

BVAR(6)p

Bim

pBim

pBim

pVAR(3)p

VAR(3)p

BVAR(6)p

BVAR(1)p

Bim

pBim

pAR(1)p

45y

VAR(3)p

AR(1)p

Bim

pBVAR(1)p

Bim

pBim

pAR(1)p

VAR(3)p

VAR(3)p

Bim

pBim

pBim

pAR(1)p

AR(1)p

53m

VAR(1)p

BVAR(1)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

pVAR(3)p

BVAR(1)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

p

51y

AR(1)p

Bim

pBVAR(6)p

BVAR(6)p

Bim

pBim

pBim

pAR(1)p

BVAR(1)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

p

53y

AR(1)p

Bim

pBVAR(1)p

BVAR(6)p

Bim

pBim

pBim

pAR(1)p

VAR(3)p

BVAR(1)p

BVAR(6)p

Bim

pBim

pAR(1)p

55y

AR(1)p

VAR(3)p

BVAR(1)p

Bim

pBim

pBim

pAR(1)p

AR(1)p

AR(1)i

Bim

pBim

pBim

pAR(1)p

AR(1)p

63m

VAR(1)p

BVAR(1)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

pVAR(3)p

BVAR(1)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

p

61y

AR(1)p

VAR(3)p

BVAR(6)p

Bim

pBim

pBim

pBim

pAR(1)p

VAR(3)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

p

63y

AR(1)p

VAR(3)p

BVAR(6)p

Bim

pBim

pBim

pBim

pAR(1)p

VAR(3)p

BVAR(6)p

Bim

pBim

pBim

pAR(1)p

65y

AR(1)p

VAR(3)p

Bim

pBim

pBim

pBim

pAR(1)p

AR(1)p

AR(1)i

Bim

pBim

pBim

pAR(1)p

AR(1)p

73m

VAR(1)p

Bim

pBVAR(6)p

Bim

pBim

pBim

pBim

pAR(1)p

Bim

pBVAR(6)p

BVAR(6)p

Bim

pBim

pBim

p

71y

AR(1)p

VAR(3)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

pAR(1)p

Bim

pBVAR(6)p

BVAR(6)p

Bim

pBim

pBim

p

73y

AR(1)p

VAR(3)p

BVAR(6)p

BVAR(6)p

Bim

pBim

pBim

pAR(1)p

VAR(3)p

BVAR(1)p

BVAR(6)p

Bim

pBim

pAR(1)p

75y

AR(1)p

AR(1)p

BVAR(1)p

Bim

pBim

pBim

pAR(1)p

AR(1)p

AR(1)i

BVAR(1)p

Bim

pBim

pAR(1)p

AR(1)p


Fig

ure

53:

Bes

tp

erfo

rmin

gm

od

els’errors

by

nu

mb

erof

fact

ors,

ten

ors

and

hor

izon

s,5-

year

win

dow

,slope

fore

cast

s

RMSFE

MAFE

Kt

3m

6m

12m

18m

24m

36m

60m

3m

6m

12m

18m

24m

36m

60m

33m

4.84

7.25

8.22

7.77

8.61

7.59

5.80

3.88

5.44

6.31

5.81

6.35

5.44

4.99

31y

19.72

31.77

34.53

30.50

35.88

30.45

14.73

15.65

24.54

27.19

22.83

26.98

22.76

12.10

33y

36.84

59.03

65.58

56.43

60.31

44.15

41.61

29.20

49.95

52.14

43.37

47.45

33.57

34.71

35y

39.78

66.49

78.99

65.29

62.82

65.56

43.89

31.15

53.58

64.37

53.84

50.85

50.17

35.46

43m

6.09

8.62

9.23

8.72

10.10

8.53

5.80

4.43

6.47

6.85

6.39

7.46

6.59

4.99

41y

21.86

34.13

36.74

33.84

38.98

30.45

14.73

16.71

25.73

28.13

25.44

29.04

22.76

12.10

43y

34.91

59.03

64.43

56.47

60.31

44.15

41.61

27.33

46.11

51.87

43.31

47.45

33.57

33.16

45y

39.38

67.44

78.99

68.21

62.82

65.56

44.54

30.94

54.15

64.37

53.84

50.85

51.26

35.47

53m

5.92

8.42

9.45

9.08

10.10

8.53

5.80

4.69

6.06

7.01

6.30

7.46

6.59

4.99

51y

23.00

34.13

36.39

33.84

38.98

30.45

14.73

17.87

25.78

28.17

23.49

29.04

22.76

12.10

53y

36.44

59.03

64.34

59.54

60.31

44.15

41.61

28.70

47.11

50.96

45.22

47.45

33.57

33.57

55y

40.29

66.76

77.42

68.85

62.82

65.56

44.70

31.49

54.09

64.37

53.84

50.85

51.39

35.60

63m

5.89

8.46

9.05

9.27

10.10

8.53

5.80

4.67

6.22

6.75

7.01

7.46

6.59

4.99

61y

23.00

33.79

35.2

40.44

38.98

30.45

14.73

17.87

26.24

26.62

29.51

29.04

22.76

12.10

63y

36.51

58.42

65.47

65.70

60.31

44.15

41.61

28.73

47.29

51.78

51.97

47.45

33.57

33.59

65y

40.31

67.25

78.99

68.85

62.82

65.56

44.61

31.46

54.29

64.37

53.84

50.85

51.34

35.54

73m

5.93

8.62

9.73

10.22

10.10

8.53

5.80

4.73

6.57

7.23

7.28

7.46

6.59

4.99

71y

22.99

34.12

36.62

35.29

38.98

30.45

14.73

17.86

27.20

27.52

25.90

29.04

22.76

12.10

73y

36.51

58.82

64.44

56.13

60.31

44.15

41.61

28.72

48.47

51.08

45.68

47.45

33.57

33.58

75y

40.31

67.63

77.73

68.85

62.82

65.56

44.60

31.45

54.90

62.60

53.84

50.85

51.34

35.54


5.4 Ex-post short term interest rates forecasts

As signalled at the beginning of this Chapter, with all the preparatory work done, we are now readyto investigate relations between short interest rate forecasts indirectly assumed by risky yield curvewith those estimated via ACM decomposition risk-free where we set all market prices of risk equalto zero. We are interested in answering here to two questions: (1) which curve gave best resultsex-post in Poland and weather the rank is dependent on forecasting horizon or number of factorsin the original ACM model, and (2) do professional forecasters’ data series on official short interestrate consist of some orthogonal information that may be used in forecasting interest rates, which isnot spanned by the yield curve as such. This subsection is split into two parts. First, we will checkex-post performance of implicit 1-month rate forecasts that are hidden in the two: risky and riskfree yield curves. We will use the longer data set-up of 5-year initial window and already introducederror measures: RMSFE and MAFE. Second, we extract and adjust in accordance with our researchgoals, the explicit forecasts of short interest rate (official reference rate of Polish Central Bank - thesame that we have used as an ultra short tenor price of the NSS parsimonious yield curve) frompublicly available data. Such quarterly data will be then compared with what we have alreadyknown by estimating NSS and ACM curves in pursue to answer the second research question of thissubsection and, in consequence, verify Hypothesis 4.

Visual inspection of the forecasted short interest rates paths implied the above mentioned riskyand risk free for selected dates together with naive benchmark of constant interest rates and ex-postrealisation paths on the top of these is provided at Figure 54.

Figure 54: Short risk-free interest rates forecasts for selected dates

0 5 10 15 20 25

horizon in months

0.035

0.04

0.045

0.05

sh

ort

te

rm in

tere

st

rate

30-Apr-2010

realisation

implied by YC

implied by ACM(rf)

no change line

0 5 10 15 20 25

horizon in months

0.034

0.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052

0.054

sh

ort

te

rm in

tere

st

rate

30-Nov-2010

0 5 10 15 20 25

horizon in months

0.015

0.02

0.025

0.03

0.035

0.04

sh

ort

te

rm in

tere

st

rate

31-May-2013

0 5 10 15 20 25

horizon in months

0.01

0.015

0.02

0.025

0.03

0.035

0.04

sh

ort

te

rm in

tere

st

rate

28-Feb-2014

0 5 10 15 20 25

horizon in months

0.014

0.015

0.016

0.017

0.018

0.019

0.02

0.021

sh

ort

te

rm in

tere

st

rate

31-Dec-2014

0 5 10 15 20 25

horizon in months

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

sh

ort

te

rm in

tere

st

rate

31-May-2016

There is no obvious pattern speaking out of these images, except for that in the periods oftight monetary policy the risky yield curve implications tend to overshoot the path of short interestrates (cf. 30-Apr-2010 or 30-Nov-2010). But all other situations depicted in the Figure 54 are not at


all easily interpretable. Hence we turn to classic analysis of ex-post errors of forecasts, which follows.

Table 26: Forecast errors of short (1-month) interest rates in different models, 5-year window

RMSFEModel variety 3m 6m 12m 18m 24m 36m 60m

Per 13.8 39.6 81.8 109.7 131.1 163.3 203.4Imp, K = 3 16.9 27.4 61.1 106.7 159.0 254.7 329.2RF, K = 3 21.1 40.6 81.3 120.6 156.7 215.2 273.9Imp, K = 4 16.6 28.3 65.1 106.8 154.4 246.8 351.4RF, K = 4 20.8 40.2 80.0 118.6 154.5 213.2 272.8Imp, K = 5 16.4 28.8 65.8 107.0 152.9 246.5 354.2RF, K = 5 19.2 37.5 78.3 118.3 154.9 214.1 273.0Imp, K = 6 16.5 28.9 65.8 107.1 153.1 246.4 354.8RF, K = 6 20.1 38.3 79.1 119.4 156.4 215.3 273.9Imp, K = 7 16.5 28.9 65.8 107.0 153.1 246.4 355.0RF, K = 7 21.3 39.4 79.8 120.9 159.1 218.5 276.1

MAFEPer 6.9 25.6 60.7 93.0 114.5 135.4 179.9

Imp, K = 3 12.9 21.6 47.9 88.4 141.1 237.8 311.1RF, K = 3 16.0 33.0 66.0 99.2 133.2 200.7 269.6Imp, K = 4 12.8 21.5 52.8 89.3 136.9 229.2 334.0RF, K = 4 15.4 31.3 64.5 97.8 131.3 198.1 268.1Imp, K = 5 12.6 22.2 53.1 88.8 135.5 230.0 334.0RF, K = 5 14.1 28.7 62.5 97.5 132.1 198.6 268.7Imp, K = 6 12.7 22.2 53.1 88.9 135.6 229.9 334.0RF, K = 6 15.0 29.7 63.4 99.2 134.2 201.0 270.6Imp, K = 7 12.7 22.2 53.0 88.9 135.6 229.9 334.2RF, K = 7 16.0 31.9 65.4 101.0 137.1 205.6 273.5

Notes: (1) various horizons are reflected in different columns h = {3, 6, 12, 18, 24, 36, 60}. (2) Per means a benchmark ofconstant short interest rates, Imp stands for the forecasts implied by the risky yield curve and RF - by the ACM derived risk

free curve (3) all errors in bps

In the Tables 26 and 27 we present the results of the first enquiry, which are pretty self-explanatory. We test ACM models with five different numbers of factors (K = {3, 4, 5, 6, 7}) forhorizons of h = {3, 6, 12, 18, 24, 36, 60} months against naive benchmark of no change in the interestrates at all (called in tables: Per). We found that:

1. no model beats its benchmark equivalent for horizons of 24 months and more, and in thistime-frame the risk free yield curve implied (RF) produces much better results than the riskycurve implied (YC).

2. in shorter horizons, up to 18-months the implied short interest rates in the yield curve beateven the naive benchmark, especially in 6 and 12-months horizons, but the performance ofRF implied forecasts may be interpreted as a slight miss only, i.e. in h = 12 we have MAFEof 93, 89 and 98 bps in four factor model for the forecasts from naive benchmark, implied YCand implied RF short interest rates respectively.

3. increasing number of factors not always, and if so, only mildly (by additional 1-3 bps) aug-mented the errors, with exceptions for MAFEs of horizons: 24 and 36-months.

4. standard deviations of RMSFE and MAFE of RF time-series are substantially lower thenbenchmark for horizons of at least 3-years


Table 27: Standard deviations of forecast errors of short (1-month) interest rates in differentmodels, 5-year window

STD RMSFEModel variety 3m 6m 12m 18m 24m 36m 60m

Per 21.8 51.9 102.6 116.5 123.9 174.3 186.6Imp, K = 3 20.0 33.6 71.2 114.1 144.6 197.6 253.4RF, K = 3 26.7 51.2 89.6 120.1 145.2 167.6 144.7Imp, K = 4 18.3 35.4 73.1 113.5 140.8 195.3 260.8RF, K = 4 25.9 52.8 90.0 118.1 142.9 167.8 147.3Imp, K = 5 18.7 36.0 74.1 114.4 139.6 193.4 269.0RF, K = 5 24.3 49.2 88.6 118.2 143.7 169.1 145.5Imp, K = 6 18.5 36.0 74.2 114.6 139.8 193.1 271.2RF, K = 6 24.9 49.7 89.1 118.6 144.1 168.0 140.2Imp, K = 7 18.5 36.0 74.2 114.5 139.8 193.2 271.4RF, K = 7 26.2 48.9 88.0 118.9 145.7 166.7 135.0

STD MAFEPer 12.0 30.4 55.3 58.6 64.5 92 95.6

Imp, K = 3 10.9 17.0 38.3 60.3 73.9 92.0 108.3RF, K = 3 13.9 23.9 47.8 69.1 83.0 78.2 48.3Imp, K = 4 10.6 18.4 38.5 59.1 71.8 92.4 110.0RF, K = 4 14.1 25.4 47.6 67.6 82 79.4 50.6Imp, K = 5 10.6 18.6 39.2 60.1 71.4 89.4 118.7RF, K = 5 13.1 24.2 47.5 67.5 81.6 80.6 48.9Imp, K = 6 10.6 18.6 39.2 60.1 71.6 89.2 120.6RF, K = 6 13.5 24.3 47.6 67.1 80.9 77.8 43.1Imp, K = 7 10.6 18.6 39.2 60.0 71.6 89.3 120.7RF, K = 7 14.1 23.2 46 67.0 81.3 74.4 38.3

Notes: (1) various horizons are reflected in different columns h = {3, 6, 12, 18, 24, 36, 60}. (2) Per means a benchmark ofconstant short interest rates, Imp stands for the forecasts implied by the risky yield curve and RF - by the ACM derived risk

free curve (3) all errors in bps

5.5 Professional forecasters short interest rates time series

As far as the second enquiry is concerned we have to start with explanation on how the datacollected in National Bank of Poland is adjusted to our purpose. The main, and so far the onlylonger time series of professional forecasters survey gathered regularly using consistent methods isthe NBP Survey of Professional Forecasters74, which aims at collecting macroeconomic forecasts ofCPI inflation and GDP growth for different time horizons. Experts who participate in the surveyare asked to consider various scenarios of economic developments and to provide - on the basis ofthe conducted analysis - the range of possible values and a central point forecast for those variables.The survey is conducted at the end of a certain quarter from 3Q2011 untill now75. In the additionalquestions section we find, among others the forecasts of the NBP reference rate which is of a specialinterest to us in this analysis. It seems that the fact that the questionnaire was designed to collectinformation on macroeconomic variables that are quarterly changes of some indices is not very handyin straightforward extraction of the forecasted NBP interest rate path. For example, in the researchcarried out at the end of 3Q2011 the pool of forecasters was asked to provide prognosis of NBP rate:

1. average in a quarter for: 3Q2012 and 3Q2013

2. average in a year for: 2011, 2012, 2013

3. average during the period of the nearest 5 years: 2011-2015

We are interested in the longest possible path of NBP interest rates that is implied in these results.Using the central values in all the answers provided by the forecasters we follow the rules listedbelow:

1. we disregard the quarterly data, because they do not form a continuous set and are sometimeshard to reconcile with the medium and long term forecasts provided

74available at: https://amakro.nbp.pl/amakro-forecaster/pages/about.nbp75at the moment of writing this Chapter last available data were coming from 2Q2020 questionnaire


2. we adjust the data for what is already known at the moment of formulating a prognosis. Forexample, following the example of the questionnaire at 3Q2011 we had already known thepath of the short term interest rates for 75% of the year 2011 and hence we assume thata particular forecaster must had been taking this history into his/hers consideration whenproviding an average annual rate for 2011 or the average for 2011-2015. But we are interestedin new information hidden within these data, not the history, and based on that reasonableassumption we have cleared the raw time series using the financial calculus presented inChapter 1.

3. we assume that the long term quarterly forwards are constant to produce the 4-year and 5-yeartime series.

Table 28: Polish professional forecasters implied averaged annually short term interest rates

Raw averages Full year adjusted averagesDate 1Y* 2Y* 3Y* 4Y* 5Y* 1Y 2Y 3Y 4Y 5Y

30-Sep-2011 4.68 4.26 4.11 4.19 4.19 4.37 4.15 4.17 4.19 4.1931-Dec-2011 4.26 4.28 4.16 4.07 4.07 4.26 4.28 4.16 4.07 4.0731-Mar-2012 4.46 4.18 4.23 4.26 4.26 4.39 4.19 4.24 4.26 4.2630-Jun-2012 4.86 4.47 4.28 4.33 4.33 4.67 4.37 4.30 4.33 4.3330-Sep-2012 4.46 4.04 4.09 4.46 4.46 4.14 4.08 4.37 4.46 4.4631-Dec-2012 4.53 3.65 3.73 3.94 3.94 4.53 3.65 3.73 3.94 3.9431-Mar-2013 2.83 3.37 3.84 4.24 4.24 2.96 3.49 3.94 4.24 4.2430-Jun-2013 1.62 2.64 3.35 4.23 4.23 2.13 2.99 3.79 4.23 4.2330-Sep-2013 2.27 2.66 3.45 3.85 3.85 2.56 3.25 3.75 3.85 3.8531-Dec-2013 2.90 2.59 3.39 3.86 3.86 2.90 2.59 3.39 3.86 3.8631-Mar-2014 2.50 3.11 3.63 3.61 3.61 2.65 3.24 3.63 3.61 3.6130-Jun-2014 2.43 2.64 3.2 3.28 3.28 2.53 2.92 3.24 3.28 3.2830-Sep-2014 2.11 1.97 2.51 3.09 3.09 2.00 2.37 2.95 3.09 3.0931-Dec-2014 2.37 1.92 2.31 2.95 2.95 2.37 1.92 2.31 2.95 2.9531-Mar-2015 1.29 1.75 2.47 2.89 2.89 1.41 1.93 2.58 2.89 2.8930-Jun-2015 1.23 1.70 2.35 2.91 2.91 1.47 2.03 2.63 2.91 2.9130-Sep-2015 1.39 1.62 2.17 2.70 2.70 1.57 2.04 2.57 2.70 2.7031-Dec-2015 1.56 1.37 1.64 2.59 2.59 1.56 1.37 1.64 2.59 2.5931-Mar-2016 1.43 1.60 2.07 2.46 2.46 1.48 1.72 2.17 2.46 2.4630-Jun-2016 1.49 1.61 1.96 2.28 2.28 1.55 1.78 2.12 2.28 2.2830-Sep-2016 1.52 1.58 1.94 2.28 2.28 1.57 1.85 2.20 2.28 2.2831-Dec-2016 1.50 1.57 1.95 2.60 2.60 1.50 1.57 1.95 2.60 2.6031-Mar-2017 1.53 1.86 2.22 2.47 2.47 1.62 1.95 2.29 2.47 2.4730-Jun-2017 1.51 1.66 2.11 2.32 2.32 1.59 1.89 2.22 2.32 2.3230-Sep-2017 1.51 1.59 2.09 2.30 2.30 1.57 1.96 2.25 2.30 2.3031-Dec-2017 1.50 1.60 2.04 2.60 2.60 1.50 1.60 2.04 2.60 2.6031-Mar-2018 1.50 1.62 1.92 2.18 2.18 1.53 1.69 1.99 2.18 2.1830-Jun-2018 1.50 1.58 1.90 2.17 2.17 1.54 1.74 2.04 2.17 2.1730-Sep-2018 1.50 1.56 1.94 2.16 2.16 1.54 1.84 2.10 2.16 2.1631-Dec-2018 1.50 1.53 1.83 2.23 2.23 1.50 1.53 1.83 2.23 2.2331-Mar-2019 1.51 1.59 1.74 2.00 2.00 1.53 1.63 1.81 2.00 2.0030-Jun-2019 1.50 1.53 1.59 1.80 1.08 1.51 1.56 1.70 1.80 1.8030-Sep-2019 1.50 1.52 1.52 1.77 1.77 1.51 1.52 1.70 1.77 1.7731-Dec-2019 1.50 1.52 1.58 1.85 1.85 1.50 1.52 1.58 1.85 1.8531-Mar-2020 0.78 0.91 1.13 1.46 1.46 0.82 0.97 1.22 1.46 1.4630-Jun-2020 -0.66 0.19 0.43 1.56 1.56 -0.24 0.31 1.00 1.56 1.56

Notes: (1) own calculations based on the median of answers to the NBP professional forecasters survey - on the official interestrate in Poland at: https://amakro.nbp.pl/amakro-forecaster/pages/datasets.nbp

As a result we extracted the implied (usually) annual averages of forecasted NBP reference rateswith one special feature that we have to adjust for as well. The construction of the questionnaireand the method we used to extract the implied path at the end means that the first period it isreferring to has variable length: for the data from 1Q - 0.75-year, 2Q - 0.5-year, 3Q - 0.25-year andfor 4Q - full 1-year. The following annual rates are also in fact shifted, i.e. the ones produced in 3Qwould concern the periods of 1.25-2.25, 2.25-3.25, 3.25-4.25 years from now. For comparison reasons(with ACM model and implied paths from the yield curve) we can easily adjust these forecasts insuch a way that we get true yearly averages for the years starting at the end of the quarter a certainquestionnaire refers to. The results of such a procedure applied on the whole available data set are


reported in the Table 28.

In order to compare these forecasts with our ACM models implied path for risk free interest rateand the risky 1-month rate implied by the yield curve we have to bear in mind that the adjusted fullyear averages (in columns) we calculated are in fact: 1-year spot rate followed by four consecutive 1-year forward rates. To verify the Hypothesis 4 of this dissertation, which claims that unlike in thecase of liquid markets, professional forecasters expectations do not help to increase the informationalcontent of the yield curves in less liquid market we will conduct two studies:

1. an evaluation of how the structure of professional forecasters’ median path (henceforth: PF)of short term interest rates compares to (a) the one implied by the yield curve (henceforth:YC) at the forecasting origin and (b) the one implied by ACM model for risk-free rates(henceforth: RF) also at the same time. This inquiry will be conducted with the maximumpossible sample set because we are only limited by the starting and ending quarter of theprofessional forecasters’ time series, namely: 2011:Q3 to 2020:Q2

2. a study of forecasting errors of the mentioned three time series: professional forecasters path,the risky yield curve’s implied path and the ACM implied risk free rates path. Here we wouldhave to adjust the end date in out-of-sample period because we have to leave at least theperiod which is equivalent to the maximum forecasting horizon, arbitrarily chosen here to be36 months (12 quarters). The end of 5-year window of ACM model (January 2010) falls beforethe starting date of the professional forecasters time series (3rd quarter 2011), hence thereis no other adjustment necessary. Our out-of-sample consists of 24 observations: 2011:Q3 to2017:Q2.

Table 29: Time series comparison statistics

times series 1Yspot 2Yfwd 3Yfwd 4YfwdMean

yield curve implied (YC) 0.0215 0.0240 0.0286 0.0330prof. forecasters (PF) 0.0211 0.0219 0.0252 0.0289

ACM RF implied (RF) 0.0224 0.0250 0.0282 0.0307Spread (in bps) YC-PF 4.4 20.6 33.9 41.1Spread (in bps) YC-RF -8.4 -10.4 4.1 23.2Spread (in bps) PF-RF -12.8 -31.0 -29.8 -17.9

Standard deviationyield curve implied (YC) 0.0113 0.0107 0.0105 0.0107

prof. forecasters (PF) 0.0125 0.0106 0.0100 0.0091ACM RF implied (RF) 0.0112 0.0109 0.0108 0.0101Spread (in bps) YC-PF 37.0 28.5 43.0 64.6Spread (in bps) YC-RF 12.7 27.9 42.8 55.2Spread (in bps) PF-RF 35.8 26.8 25.5 31.7

Notes: (1) own calculations based on the median of answers to the NBP professional forecasters survey - on the official interestrate in Poland at: https://amakro.nbp.pl/amakro-forecaster/pages/datasets.nbp . (2) RF and YC series are estimated using five

factor ACM model and NSS curves with system of weights labelled 1

The descriptive statistics of the three different time series from the first enquiry are reportedin Table 29. All time series have averages that are increasing with the distance of a certain rate’sreference period beginning from the forecasting origin. For example, professional forecasters meanreading for 1-year average NBP reference rate starting at the moment of making this forecast was2.15%, in one year horizon 2.40%, in two year horizon 2.86% and in three year horizon 3.30%. Suchdefined monotonicity prevails for YC and RF time series. The distances between the time seriesreveal that the professional forecasters behave as if there is some time-related premia in the yieldcurve and these spreads resemble term premia but shifted by some 15 bps. To test if the PF timeseries gives any additional information over the load already implied by yield curve (directly in YCor indirectly in RF series) we have carried over several co-integration tests, which are reported inTable 30. PF forecasts up to two years are co-integrated with YC information and even more so


with RF series (up to 3 years). These findings are robust for changing the number of factors inACM model for K = {3, 4, 5, 6} and for K = 7 we even observe a co-integration of all rates (up andincluding 4 years) from PF with the risk free rates from ACM decomposition. It may be interpretedthat professional forecasters are not unfolding any new information other than already spanned byyield curve. Given the fact that the forecasts are only formulated once in a quarter and we have dailyobservations and estimation results of NSS fitting and ACM term premia calculation, PF time serieswith regard to short interest rate are very impractical. Hence, we fail to reject the Hypothesis 4here.

Table 30: Co-integration tests of three groups of time series: professional forecasters (PF),yield curve implied (YC) and ACM model implied risk free (RF)

test result p-valuex x(PF ) ∼ Y C x(PF ) ∼ RF x(RF ) ∼ Y C x(PF ) ∼ Y C x(PF ) ∼ RF x(RF ) ∼ Y C

1Yspot 1 1 0 0.0442 0.0048 0.08102Yfwd 1 1 0 0.0046 0.0019 0.77903Yfwd 0 1 0 0.8733 0.0282 0.91614Yfwd 0 0 0 0.8669 0.0671 0.9354

Notes: (1) Engle-Granger tests for unit root in residuals from a regression x ∼ Y , the null hypothesis H0 is that there in noco-integration. (2) 1 - indicates rejection of H0 in favour of the alternative of co-integration, 0 - a failure to reject H0 (3) RF

and YC series are estimated using five factor ACM model and NSS curves with system of weights labelled 1.

The second study of professional forecasters’ performance evaluates the ex-post errors as such,not the innovative character of the times series. Due to time series constraints again we can onlyevaluate the professional forecasters performance once in a quarter starting from 2011:Q3 and endingat 2017:Q2 if we would like to examine 36-month horizon errors.

Figure 55: Various short risk-free interest rates forecasts and its ex-post realisation path

2012 2013 2014 2015 2016 2017

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.051Yspot

PF

YC

RF

realisation

2012 2013 2014 2015 2016 2017

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.052Yfwd

2012 2013 2014 2015 2016 2017

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.0553Yfwd

Notes: (1) RF and YC series are estimated using five factor ACM model and NSS curves with system of weights labelled 1. (2)quarterly observations from 2011:Q3 to 2017:Q2 (3) 1YSpot indicates an average short term interest rate in the first year

starting at forecasting origin, 2Yfwd represent the average of short interest rates in the second year etc.

As Figure 55 shows our out-of-sample period is, in essence, a monetary policy loosening partof a cycle, which on the top of a small sample size (23 forecasting origins) had an impact on theperformance measures’ magnitude reported in Table 31. During the reported period on average PFforecast ex-post deviated from the observed yearly averages of short interest rates by 37 bps for thefirst year, 77 for the second and 137 for the third year from the forecasting origins. The impliedYC paths were better only for 1-year averages prognosis whereas RF implied paths were very closeto their PF counterpart for that horizon. Longer term forecasts casted both by YC and RF where


worse than PF. MAFE results are robust for changes in number of factors.

Table 31: Mean absolute forecast errors: professional forecasters (PF), yield curve implied(YC) and ACM model implied risk free (RF)

1YSpot 2Yfwd 2YfwdPF 36.69 76.56 136.65YC 27.61 95.39 161.78RF 37.58 116.57 177.33

Notes: (1) RF and YC series are estimated using five factor ACM model and NSS curves with system of weights labelled 1.

In summary of the two studies of the professional forecasters’ short term interest rate time series(PF) available for Poland carried over in this section we claim that on average the central path ofthese forecasts mimic the risky and risk free yield curve implied rates (YC and RF respectively).As we have shown the time-series are co-integrated in such a way that PF is easily representedby a combination of different time horizon averages from YC and RF series. Nevertheless, PFproven to be the best forecasts ex-post in our second study, but due to short and less frequentsample, predominantly falling into a period of a downward trend in interest rates, these results areto be taken with caution, and are subject to revision once we add observations to the history andout-of-sample length increases.

5.6 Summary

As we have shown in this chapter, the best variants of ACM models when forecasting levels ofinterest rates of zero-coupon bonds up to 10-year mark for up to 24-months horizons are the onesinvolving multiple AR processes with lags up to 3 as our state space generators, with the assump-tion of constant short interest rate from the moment of forecasting origin. Ex-post out-of-sampleevaluation carried over with 10-year of estimation period (window) revealed reasonable magnitudeerrors up to 12-months horizons of approx. 14 bps for short 3-month bonds and approx. 32-34 forlonger tenors. In the case of slope forecasts, the errors are - obviously - much lower especially forlonger term horizons, but the recommended best ex-post performing state space generating modelare vector autoregressive with 3-6 months lags. Additionally, we have shown that models with 10-year estimation period produce more systematically lower errors than 5-year window.

This part of dissertation was also the place to finally show that the professional forecasters’ shortterm interest rate time series are co-integrated with different time horizon averages implied from arisky yield curve and the risk-free curve estimated by ACM, that is to verify Hypothesis 4, claimingthat that the professional forecasters time series on short term rates do not carry any additionalinformation that it already reflected in the yield curve shape.

The results presented in this Chapter are subject to some challenges that we already signalledin the Introduction, namely, a relatively small sample and an extraordinary mode of monetary andfiscal policies during the dominant period of time in our sample, which impacted the real economiesand financial markets globally. As we have shown both term premia (Chapter 4) and the inferenceon forecasting (Chapter 5) are sensitive to changes in the sample starting and ending dates, which- of course - as a problem can only be soften, not - solved, by the usage of the blended method ofcreating term premia series and by the assurance that the period of 2005:01-2020:06 is the periodof the best quality, consistent daily data available for Poland (we could not have done better, wereckon). The integrated method of estimations of the yield curve and term premia presented hereis therefore open for further testing and evaluation in the future, but it seems it is an immanentproperty of any research and finding. Broader discussion on that matter follows in Conclusions ofthis thesis.


Summary and and conclusions

Recall that the main goal of this thesis was to understand what expectations of the future interestrates evolution are confined in the government yield curves of less liquid markets as Polish one andto explore methods to improve their forecasting power. During this enquiry into the implied expec-tations about the future we verified, as planned, the following four hypotheses.

The first hypothesis, that there exists a class of weighting schemes which improves fitrelative to conventionally used methods in the Polish government yield curve estima-tion we have verified positively. After we carefully analysed, collected and integrated data on thePolish government bonds for the period which, we posit, guarantees high quality of the database cre-ated, we have conduced an extensive inquiry into the morphology and dynamics of yield curves andtheir components - particular bonds or their types. We have explained more than ten stylised factson Polish government bonds market and used them in filtering rules and weighting system design.Key features under investigation were: a notion of liquidity during bond’s lifespan, heterogeneity ofliquidity in different yield curve segments, bid-ask spreads behaviour as well as price distortion inthe very short end of the curve due to switch auctions. During the detailed checks we have foundthat the maximum timespan all segment-wise average yield time series are trend stationary whencorrected for long term variance, which potentially could have strong implications later in the thesis,when testing various expectations hypotheses. We propose to avoid the problem of choosing xIBORtype rate on the short end of the curve (which is not comparable in creditworthiness to governmentbond’s risk) and a method to blend it with the whole government bond yield curve, by means ofusing the National Bank of Poland’s bill rate, which importance in the default free instrumentsmarket in PLN cannot be overlooked. Having tested 28 different weight systems and ranked themin the space of goodness-of-fit and smoothness we confirmed that there exists a class of weights thatsystematically gives better results than the classic approach of all equal weights. The highest rankedsystems have at least the same weight for the short end of the curve as a sum for all other tenorsof bonds, eligible-for-switch bonds were excluded from the estimation and weights were based on atleast outstanding amounts. As postulated in the Introduction we have proven that inferring fromthe liquidity in particular maturities raises the information content of yield curve estimation.

In the course of this research we decomposed the previously estimated yield curves into therisk-free layer and the term premia structure using the method of three stepped linear regressionsproposed by Adrian et al. (2013). In the Principal Component Analysis we have found out that, asit is common for most of the markets globally, three first components are driving more that 99%of explained variance in any of the four possible pairs between two dimensions: daily, monthly fre-quency of data and denser, sparser calibration tenors. However, in sample errors analysis, suggestedto consider more factors in the benchmark model. With five and more factors we obtained yieldpricing errors within one basis point on average in each considered tenor. Contrary to the authors ofthis method Adrian et al. (2013) findings for US data, in estimations for Poland the factor respon-sible for level risk was not priced on a statistically significant level in any of the considered model’svariates. All other factors were priced and some of them included one more component in pricingthen the self source (i.e.: the slope risk has two significant components: 2nd and 3rd factors). Suchestimated term premia for Poland proved to be countercyclical and positively correlated with marketuncertainty measures. The term premia are, on average, upward sloping, and significantly differentfrom zero for medium and longer term bonds (starting from 3-year tenor). The level of risk premiais sensitive to choices of the beginning and end dates of the sample. To mend this problem, wepropose a solution in a form of a simple recipe to produce new time series of term premia usinga base period data and then adding every day only a newly estimated premia for that particulardate (for the purpose of this thesis we referred to it as : blended algorithm). The advantage of suchan approach is the following. When estimating term premia for a certain date within a full samplerange we only use the information up to and including that date and we do not, in a sense, sneakpeek into the future. The final result is a time series which would not change should we add more


data in the future, but still it will be up-to-date and include the recent developments.

The second hypothesis we have verified posits that Pure Expectations Hypothesis (PEH)does not hold universally in Poland. The evidence presented in Chapter 3 suggests that indeedPEH does not hold universally for Polish government bonds yield curve and contrary to the researchconducted on the US markets where PEH is almost always rejected, we have found that for Polandthere is, although limited, domain (in a space spanned by different investment horizons and selectedbond tenors) where PEH cannot be ruled out. The scope where pure expectations hypothesis holdsin Poland is bounded by the investment horizon of approximately 12 months on one hand and bymaturity of the bond of circa 36 months, on the other. At this stage we were not sure what causesthe rejection of PEH for all other combinations of horizon length and maturity: existence of somekind of risk premia or unexpected excess yield (we have only a mixture of these two contained inβ coefficient estimators). This led us to using the previously estimated term premia structure intesting the third hypothesis whether Expectations hypothesis (EH) holds. Yet again, basedon the previously meticulously curated yield curves and the extracted term premia times series, weshowed that EH holds for almost all choices of long and short term investments, and that existenceof term premia structure explains the differences between compounded rates of returns from shorterinvestments and corresponding maturity longer term zero-coupon bond’s yield.

During this study we found that the professional forecasters’ short term interest rate time seriesin Poland are co-integrated with different time horizon averages implied from a risky yield curveand the risk-free curve estimated using Adrian et al. (2013) term premia extraction recipe, there-fore verifying the fourth hypothesis, that unlike in the case of liquid markets, professionalforecasters expectations do not help to increase the informational content of the yieldcurves in the case of Poland.

Our detailed tests revealed that when forecasting levels of interest rates of zero-coupon bondsup to 10-year mark for up to 24-months horizons, the best variants of ACM models are the onesinvolving multiple auroregressive (AR) processes with 1-3 months lags as our state space generators,accompanied by the assumption of constant short interest rates from the moment of forecastingorigin. The ex-post, out-of-sample evaluation carried over with 10-year estimation period (window)revealed reasonable (acceptable) magnitude errors up to 12-months horizons. As far as the slopeforecasts are concerned we showed that, the errors are - obviously - much lower especially for longerterm horizons, but the recommended best ex-post performing state space generating model are vec-tor autoregressive (VAR) with 3-6 months lags, with additional assumption of short interest ratesbeing implied from the yield curve shape (not constant as in the level forecasts). Additionally, wehave shown that models with 10-year estimation period produce more systematically lower errorsthan 5-year window. Short term interest rate forecasting power of the estimated two-layeredyield curves models’ studies were also conducted. No model version has beaten its naive benchmarkfor horizons longer than 18 months, and in this time-frame the risk free yield curve implied producedmuch better results than the risky curve. However, in shorter horizons, up to 18-months, the impliedshort interest rates in the yield curve beats the naive benchmark (especially in 6 and 12-months).

At least two caveats to these results should be invoked here. First, the majority of our sample’sspan covers the period of the Financial Crisis, its aftermath and the ripple effects including non-standard or even - extraordinary - monetary and fiscal policy responses, the pandemic of COVID-19and the ongoing international tectonic shifts in the public debt and fiscal deficit financing. Thisis the time of major quantitative easing programmes especially in the US, the Euro Zone, the UKand all other major economies which one may claim artificially influence the slopes of the curves inmajor currencies and eventually distort the picture of implied structure of short interest rates’ paths.Yet another phenomenon that resurfaced from the history of financial markets during the aftermathof the Financial Crisis and now during the Pandemic is the possibility of negative nominal rates,both the market rates and the rates set in the framework of monetary policy. Even the proximity


of zero rates let alone the intuitively assumed limit on absolute values of negative rates, which maybe called in general zero lower bound problem (ZLB) are making the monetary policy cumbersomeand call for new, out-of-the-box thinking and courage in implementation of the newly designed in-struments. Therefore our sample may seem to be very special, judging by the historical standards ofcomprehensive and stable monetary and fiscal toolboxes and hence the results may not be robust fortime passage during back-to-normal shift, but on the other hand, we cannot rule out the possibilitythat this new normal set-up will stay for much longer, given the state of the global public financesduring and after the Pandemic of COVID-19. If this is the case, our everyday appended databaseof the government bond yield curves and estimated term premia would become more valuable witheven more capacity to stand the test of time.

The second caveat builds on the same narrative but from a different angle. As Kolasa &Weso lowski (2020) showed recently, the unconventional responses of monetary and fiscal policiesof the major economies strongly undermine international price competitiveness of smaller economies(which is vaguely the same division as between developed and less liquid markets, respectively) andin consequence exert great impact on their local government yield curves and term premia struc-ture. In this light, further studies should be conducted on how the different curves and impliedterm premia interact with each other and what forecasting frameworks are better suited in suchglobal, interconnected financial markets. We may hypothesise, that there exist an additional termpremia that spans from the perceived risk of the major currencies’ monetary and fiscal policy possi-ble shocks, hence the implied forward rates should be adjusted not only by the local component ofterm premia but also by the global analogue as well.

Acknowledgements

The author would like to express his gratitude to the supervisor of this doctoral thesis Prof. MarcinKolasa for his open-mindedness since our first meeting in winter of 2017, the highest degree ofprofessionalism in all aspects of academic and research activities and his patience for me as hisstudent. I thank also prof. Joanna Tyrowicz for her belief in me as a scientist-to-be and her greathelp in receiving a National Science Centre PRELUDIUM grant to work further, deeper and broader(geographically) on the topics covered in this thesis. Last but not least to mention is the importanceof author’s contact in April 2020 and exchange e-mails full of technical details and thoughts withone of the authors of the method heavily employed in this research, namely Richard K. Crump fromFederal Reserve Bank of New York.


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APPENDIX. AUXILIARY FIGURES.

DETAILED TABLES


Figure 56: Selected characteristics of Polish fixed coupon government bonds traded onBondSpot in 2005:01-2020:06 - short term bonds < 1.5 Y

2006 2008 2010 2012 2014 2016 2018 20200

5

10

15

outs

tandin

g

am

ount in

PLN

104

= 70943

2006 2008 2010 2012 2014 2016 2018 20200

2

4

6

nr

of bonds = 3.9

2006 2008 2010 2012 2014 2016 2018 20200

500

1000

1500

Vol in

mln

PLN

= 78

2006 2008 2010 2012 2014 2016 2018 20200.5

1

1.5

yr

frac

= 0.93

2006 2008 2010 2012 2014 2016 2018 20200

50

100

150

200

bid

ask s

pre

ad

= 20.83

2006 2008 2010 2012 2014 2016 2018 20200

0.5

1

ZT

D

= 0.61

[ 0, 1.5 ]


Figure 57: Selected characteristics of Polish fixed coupon government bonds traded onBondSpot in 2005:01-2020:06 - bonds with time to maturity between 1.5 and 3.5

2006 2008 2010 2012 2014 2016 2018 2020

0.5

1

1.5

2

outs

tandin

g

am

ount in

PLN

105

= 103012

2006 2008 2010 2012 2014 2016 2018 20203

4

5

6

7

nr

of bonds

= 4.9

2006 2008 2010 2012 2014 2016 2018 20200

500

1000

1500

2000

Vol in

mln

PLN

= 141

2006 2008 2010 2012 2014 2016 2018 2020

2

2.2

2.4

2.6

yr

frac = 2.35

2006 2008 2010 2012 2014 2016 2018 20200

50

100

150

200

bid

ask s

pre

ad

= 10.37

2006 2008 2010 2012 2014 2016 2018 20200

0.5

1

ZT

D

= 0.61

[ 1.5, 3.5 ]


Figure 58: Selected characteristics of Polish fixed coupon government bonds traded onBondSpot in 2005:01-2020:06 - bonds with time to maturity between 3.5 and 6.0

2006 2008 2010 2012 2014 2016 2018 2020

5

10

15

outs

tandin

g

am

ount in

PLN

104

= 82391

2006 2008 2010 2012 2014 2016 2018 20202

3

4

5

6

nr

of bonds

= 3.7

2006 2008 2010 2012 2014 2016 2018 20200

500

1000

1500

2000

Vol in

mln

PLN

= 239

2006 2008 2010 2012 2014 2016 2018 20204

4.5

5

5.5

yr

frac

= 4.53

2006 2008 2010 2012 2014 2016 2018 20200

50

100

150

200

bid

ask s

pre

ad

= 6.93

2006 2008 2010 2012 2014 2016 2018 20200

0.5

1

ZT

D

= 0.61

[ 3.5, 6 ]


Figure 59: Selected characteristics of Polish fixed coupon government bonds traded onBondSpot in 2005:01-2020:06- bonds with time to maturity between 6.0 and 12.0

2006 2008 2010 2012 2014 2016 2018 20200

5

10

15

outs

tandin

g

am

ount in

PLN

104

= 84243

2006 2008 2010 2012 2014 2016 2018 20202

3

4

5

6

nr

of bonds

= 3.7

2006 2008 2010 2012 2014 2016 2018 20200

500

1000

1500

2000

Vol in

mln

PLN

= 227

2006 2008 2010 2012 2014 2016 2018 2020

8

8.5

9

9.5

10

yr

frac

= 8.63

2006 2008 2010 2012 2014 2016 2018 20200

50

100

150

bid

ask s

pre

ad

= 6.18

2006 2008 2010 2012 2014 2016 2018 20200

0.5

1

ZT

D

= 0.61

[ 6, 12 ]


Figure 60: Selected characteristics of Polish fixed coupon government bonds traded onBondSpot in 2005:01-2020:06 - ultra long bonds > 12 Y

2006 2008 2010 2012 2014 2016 2018 20200

1

2

3

outs

tandin

g

am

ount in

PLN

104

= 9819

2006 2008 2010 2012 2014 2016 2018 20200

0.5

1

1.5

2

nr

of bonds

= 0.9

2006 2008 2010 2012 2014 2016 2018 20200

200

400

600

Vol in

mln

PLN

= 14

2006 2008 2010 2012 2014 201612

14

16

18

20

yr

frac

= 15.49

2006 2008 2010 2012 2014 20160

50

100

150

200

bid

ask s

pre

ad

= 13.17

2006 2008 2010 2012 2014 20160

0.5

1

ZT

D

= 0.61

[ 12, 30 ]


Figure 61: Averaged turnover share of fixed coupon government bonds of 2, 5, 10Y typesduring their lifespan

0 20 40 60 80 100 120 140


0

0.05

0.1

0.15

0.2

0.25

share

in m

ark

et tu

rnover

10Y bonds

Max = 0.145

Month

: 16

= 0.053

0 10 20 30 40 50 60 70


0

0.1

0.2

0.3

0.4

share

in m

ark

et tu

rnover

5Y bonds

Max = 0.19

Month

: 8

= 0.067

0 5 10 15 20 25 30 35


0

0.05

0.1

0.15

0.2

0.25

share

in m

ark

et tu

rnover

2Y bonds

Max = 0.073

Month

: 3

= 0.039

Notes: (1) thinner red lines indicate one standard deviation up and down from the mean on a certain month, (2) (2) Onlybonds with full life history contained in the period 2005.Q1: 2020.Q3 were taken into consideration.


Figure 62: Selected averaged liquidity measures of 2Y fixed coupon government bonds

0 5 10 15 20 250

2

4

Mo

nth

ly a

ve

rag

e o

f

da

ily n

r o

f d

ea

ls

= 1.42

0 5 10 15 20 2510

20

30

Bid

-ask s

pre

ad

in b

ps

= 14.88 bps

0 5 10 15 20 25

20

40

60

Mo

nth

ly a

ve

rag

e o

f

da

ily v

olu

me

on

Bo

nd

Sp

ot

in m

ln P

LN

= 30.38 mln PLN

0 5 10 15 20 25

0.6

0.7

0.8

Mo

nth

ly p

ct

of

ze

ro t

rad

ing

da

ys

= 0.7

0 5 10 15 20 25


0.5

1

1.5

Ou

tsta

nd

ing

am

ou

nt

104

= 14175.3

2 YR Bonds

Notes: (1) Only bonds with full life history contained in the period 2005.Q1: 2020.Q3 were taken into consideration.



0 10 20 30 40 50 600

5

10

Mo

nth

ly a

ve

rag

e o

f

da

ily n

r o

f d

ea

ls

= 2.7

0 10 20 30 40 50 60

10

15

20

Bid

-ask s

pre

ad

in b

ps

= 9.8 bps

0 10 20 30 40 50 600

50

100

150

Mo

nth

ly a

ve

rag

e o

f

da

ily v

olu

me

on

Bo

nd

Sp

ot

in m

ln P

LN

= 49.61 mln PLN

0 10 20 30 40 50 60

0.2

0.4

0.6

0.8

Mo

nth

ly p

ct

of

ze

ro t

rad

ing

da

ys

= 0.6

0 10 20 30 40 50 60


0

1

2

3

Ou

tsta

nd

ing

am

ou

nt

104

= 23584.6

5 YR Bonds




0 20 40 60 80 100 1200

2

4

Mo

nth

ly a

ve

rag

e o

f

da

ily n

r o

f d

ea

ls

= 2.24

0 20 40 60 80 100 120

10

20

30

Bid

-ask s

pre

ad

in b

ps

= 8.57 bps

0 20 40 60 80 100 1200

50

100

Mo

nth

ly a

ve

rag

e o

f

da

ily v

olu

me

on

Bo

nd

Sp

ot

in m

ln P

LN

= 42.29 mln PLN

0 20 40 60 80 100 1200.2

0.4

0.6

0.8

Mo

nth

ly p

ct

of

ze

ro t

rad

ing

da

ys

= 0.5

0 20 40 60 80 100 120


1

2

3

Ou

tsta

nd

ing

am

ou

nt

104

= 22553.9

10 YR Bonds



Figure 65: Selected liquidity measures of Polish fixed coupon government bonds traded onBondSpot in 2005:01-2020:06 - by segments (in years)

2006 2008 2010 2012 2014 2016 2018 2020

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Am

ihud

0-1.5

1.5-3.5

3.5-6

6-12

12+

2006 2008 2010 2012 2014 2016 2018 2020

-2

-1.5

-1

-0.5

0

Roll

10-3

2006 2008 2010 2012 2014 2016 2018 2020

-5

0

5

1010

-7

Notes: Amihud’s illiquidity measure (yield change as a proxy of return, volume taken from BS, when ZTD: volume of 0.1 wasimputed to avoid division by zero) , Roll’s effective spread measure, and γ a measure proposed by Bao et al. (2011)


Table 32: Descriptive statistics of switch spread of Polish fixed coupon government bonds in2005:01-2020:06

Segment mean std

[0.2, 0.3] 22.35 17.15(0.3, 0.4] 23.95 20.37(0.4, 0.5] 21.49 19.03(0.5, 0.6] 10.06 20.50(0.6, 0.7] 10.35 18.83(0.7, 0.8] 6.41 16.75(0.8, 0.9] 4.74 14.72(0.9, 1.0] 4.15 12.73

Notes: (1) Switch spread is a difference between linearly interpolated rate between NBP rate and the average ytm in thesegment [1.0, 1.5] and the ytm of a particular shorter than 1.25 years bonds

Figure 66: Timeseries of the best and worst performer in the ranking

2006 2008 2010 2012 2014 2016 2018 20200

0.5

1

1.5

2

2.5

3

3.5

MA

E in b

ps

best set in ranking

worst set in ranking

Notes: (1) the highest and the lowest ranked weighting system in Table ?? were (2) data timespan: 2005:01-2020:06


Input: ∀t prices pi,t, weights Wi,t, rates rnbpt and a list starting values ΘOutput: time series of optimal Θ∗ for each tfor t ∈ DS do

retrieve from database for date t: pi and chosen set of weights Wi for every i-bondand W nbp, rnbp

for s ∈ Θ dowhile tolerance conditions not met do

retrieve cash-flow schedule ⊞i for each bond icalculate Pi(Θk) for each bond using ⊞i and current iteration spot ratesy(x; Θk) , R(Θk)calculate objective function value in k-th iteration

endreturn Θ∗

s

endchoose Θ∗

s with the lowest objective function’s value (O)store Θ∗ ≡ mins O(Θ∗

s) for the date tend

Algorithm 2: Calculating time series of the optimal Θ∗ (slower version)

Figure 67: NSS parameters in estimated Polish zero coupon yield curve

2006 2008 2010 2012 2014 2016 2018 2020

0

0.05

0.1

0.15

2006 2008 2010 2012 2014 2016 2018 2020

-0.15

-0.1

-0.05

0

0.05

2006 2008 2010 2012 2014 2016 2018 2020

-0.1

0

0.1

0.2

2006 2008 2010 2012 2014 2016 2018 2020

-1

-0.5

0

0.5

2006 2008 2010 2012 2014 2016 2018 2020

0

5

10

2006 2008 2010 2012 2014 2016 2018 2020

0

100

200

300

Notes: the highest ranked weight system labelled 1 was used


Figure 68: Robustness checks on null hypothesis decisions by system of weights - regressiontype 1 Fama & Bliss

24 30 36 42 48 54 60 72


6

12

24

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n in

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nth

s

sys. label: 1.

0.0026

0.1362

0.0004

0.1103

0.0479

0.0002

0

0.0692

0.0165

0.0001

0

0.0341

0.0008

0.0001

0

0

0.3878

0.6234

0.3745

0.5333

0.8086

0.3494

0.3993

0.5435

0.3732

0.3428

0.3244

0.3361

0.2175

0.2769 0.2315 0.2086

24 30 36 42 48 54 60 72


6

12

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n in

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sys. label: 2.

0.0049

0.1397

0.0005

0.1161

0.0535

0.0002

0

0.0724

0.0189

0.0001

0

0.0359

0.0008

0.0001

0

0

0.5083

0.7062

0.4517

0.5909

0.7183

0.4014

0.4431

0.5087

0.413

0.3783

0.3146

0.364

0.233

0.2901 0.2354 0.1978

24 30 36 42 48 54 60 72


6

12

24

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ho

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n in

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s

sys. label: 3.

0.0027 0.0003

0.1242

0.0702

0.0001

0

0.08

0.0253

0

0

0.0405

0.0011

0

0

0

0.3941

0.6277

0.3773

0.5399

0.823

0.3515

0.41

0.5835

0.3779

0.358

0.3814

0.3439

0.234

0.182

0.2879 0.2449 0.2253

24 30 36 42 48 54 60 72


6

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ho

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nth

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sys. label: 4.

0.0043 0.0004

0.1322

0.0798

0.0001

0

0.0854

0.029

0

0

0.0434

0.0011

0

0

0

0.4969

0.6734

0.4311

0.5573

0.7383

0.3809

0.4224

0.5661

0.4008

0.3765

0.3905

0.3597

0.2463

0.1958

0.2948 0.2451 0.2096

24 30 36 42 48 54 60 72


6

12

24

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nth

s

sys. label: 5.

0.0023

0.1347

0.0003

0.1094

0.0476

0.0002

0

0.0688

0.0165

0.0001

0

0.034

0.0008

0.0001

0

0

0.3778

0.6128

0.3686

0.5299

0.8068

0.346

0.398

0.5392

0.3712

0.3429

0.321

0.336

0.2167

0.2783 0.2335 0.2118

24 30 36 42 48 54 60 72


6

12

24

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ho

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nth

s

sys. label: 6.

0.0047

0.1272

0.0004

0.117

0.0487

0.0001

0

0.0735

0.0175

0

0

0.0367

0.0008

0

0

0

0.5089

0.7001

0.4529

0.5822

0.7067

0.4025

0.435

0.4889

0.415

0.3732

0.2946

0.3657

0.2323

0.291 0.2358 0.196

24 30 36 42 48 54 60 72


6

12

24

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ho

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nth

s

sys. label: 7.

0.0031 0.0005

0.1251

0.0691

0.0002

0

0.0803

0.0246

0.0001

0

0.0404

0.001

0.0001

0

0

0.4014

0.6313

0.3849

0.5465

0.8239

0.3575

0.4159

0.5841

0.3824

0.3627

0.3822

0.3477

0.2363

0.1813

0.2907 0.2467 0.2261

24 30 36 42 48 54 60 72


6

12

24

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ho

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nth

s

sys. label: 8.

0.0045 0.0004

0.1331

0.0838

0.0001

0

0.0856

0.0304

0

0

0.0432

0.0011

0

0

0

0.5038

0.6763

0.4392

0.5623

0.746

0.3889

0.4279

0.5772

0.4059

0.3794

0.4028

0.3643

0.2483

0.205

0.2986 0.2479 0.2113

24 30 36 42 48 54 60 72


6

12

24

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ho

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nth

s

sys. label: 9.

0.002

0.09

0.0002

0.0893

0.0298

0.0001

0

0.0576

0.0104

0

0

0.0292

0.0006

0.0001

0

0

0.345

0.5189

0.3377

0.4415

0.7636

0.3222

0.3295

0.4745

0.3456

0.2792

0.2457

0.3131

0.1764

0.2599 0.2201 0.2052

24 30 36 42 48 54 60 72


6

12

24

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ho

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nth

s

sys. label: 10.

0.0046

0.1045

0.0003

0.1085

0.0391

0.0001

0

0.0687

0.014

0

0

0.0333

0.0008

0.0001

0

0

0.4455

0.6387

0.4043

0.5343

0.6787

0.3672

0.4053

0.4555

0.3734

0.3332

0.2526

0.3338

0.2107

0.2712 0.2262 0.2057

24 30 36 42 48 54 60 72


6

12

24

36

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60h -

ho

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nth

s

sys. label: 11.

0.0023

0.1439

0.0002

0.1182

0.0525

0.0001

0

0.0773

0.0185

0

0

0.039

0.0008

0

0

0

0.341

0.5341

0.3394

0.4686

0.7886

0.3313

0.367

0.5418

0.3631

0.3251

0.3279

0.3375

0.2189

0.2874 0.2476 0.2318

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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nth

s

sys. label: 12.

0.0039

0.1591

0.0003

0.1317

0.0642

0.0001

0

0.0848

0.0238

0

0

0.0414

0.0011

0

0

0

0.4601

0.6417

0.4201

0.5442

0.7068

0.386

0.4254

0.5188

0.3965

0.3636

0.3301

0.3618

0.2435

0.3013 0.2543 0.2294

24 30 36 42 48 54 60 72


6

12

24

36

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60h -

ho

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n in

mo

nth

s

sys. label: 13.

0.0021

0.0948

0.0002

0.0897

0.0312

0.0001

0

0.0577

0.0107

0.0001

0

0.029

0.0006

0.0001

0

0

0.3477

0.5269

0.3387

0.4479

0.7742

0.3229

0.3341

0.4891

0.346

0.2821

0.2561

0.3145

0.1775

0.2622 0.2231 0.2089

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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nth

s

sys. label: 14.

0.0045

0.108

0.0004

0.1086

0.0403

0.0002

0

0.0686

0.0145

0.0001

0

0.0334

0.0008

0.0001

0

0

0.4485

0.6378

0.4112

0.5328

0.6959

0.3777

0.4046

0.4673

0.3889

0.3399

0.2616

0.3496

0.2142

0.2852 0.2372 0.2117

24 30 36 42 48 54 60 72


6

12

24

36

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60h -

ho

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n in

mo

nth

s

sys. label: 15.

0.0024

0.1482

0.0003

0.1175

0.0551

0.0001

0

0.0775

0.0196

0

0

0.0393

0.0008

0.0001

0

0

0.3375

0.532

0.3363

0.4652

0.7905

0.328

0.362

0.5404

0.3595

0.3214

0.3305

0.3339

0.2163

0.2835 0.2434 0.2264

24 30 36 42 48 54 60 72


6

12

24

36

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60h -

ho

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n in

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nth

s

sys. label: 16.

0.0033 0.0002

0.1341

0.0656

0.0001

0

0.0867

0.0241

0

0

0.0425

0.001

0

0

0

0.4395

0.6304

0.403

0.5358

0.712

0.3734

0.421

0.5253

0.3885

0.3646

0.3389

0.3557

0.2451

0.1638

0.2972 0.2506 0.2255

24 30 36 42 48 54 60 72


6

12

24

36

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60h -

ho

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mo

nth

s

sys. label: 17.

0.0025

0.1279

0.0004

0.1018

0.0451

0.0002

0

0.0646

0.0159

0.0001

0

0.0326

0.0008

0.0001

0

0

0.3733

0.5886

0.3627

0.5014

0.7948

0.3408

0.3736

0.5255

0.3638

0.3185

0.3073

0.3277

0.1997

0.2704 0.2261 0.2053

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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nth

s

sys. label: 18.

0.0051

0.1469

0.0005

0.1156

0.0571

0.0002

0

0.0728

0.0205

0.0001

0

0.0365

0.0009

0

0

0

0.5059

0.6919

0.4492

0.5774

0.7193

0.3979

0.4341

0.5131

0.4047

0.367

0.3208

0.3557

0.2277

0.2839 0.2307 0.196

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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nth

s

sys. label: 19.

0.0025 0.0003

0.1249

0.0693

0.0001

0

0.0807

0.0251

0

0

0.0405

0.0011

0

0

0

0.3736

0.5951

0.3665

0.5152

0.8101

0.3487

0.3956

0.5721

0.3785

0.3513

0.3742

0.3467

0.233

0.1789

0.2903 0.2457 0.2232

24 30 36 42 48 54 60 72


6

12

24

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ho

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nth

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sys. label: 20.

0.0049 0.0004

0.1347

0.0748

0.0001

0

0.0863

0.0273

0

0

0.0429

0.0011

0

0

0

0.5078

0.676

0.4498

0.5623

0.7277

0.4021

0.43

0.5468

0.4192

0.3818

0.372

0.3759

0.2502

0.1849

0.3075 0.2539 0.2151

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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nth

s

sys. label: 21.

0.0026

0.1363

0.0004

0.1039

0.0492

0.0002

0

0.066

0.0175

0.0001

0

0.0333

0.0009

0.0001

0

0

0.3784

0.59

0.3684

0.5078

0.7997

0.3459

0.3814

0.5359

0.3662

0.3226

0.3173

0.3311

0.2033

0.2746 0.2308 0.2118

24 30 36 42 48 54 60 72


6

12

24

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ho

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nth

s

sys. label: 22.

0.0046

0.1351

0.0005

0.1159

0.0523

0.0002

0

0.0727

0.0191

0.0001

0

0.0365

0.0009

0.0001

0

0

0.4997

0.6921

0.4455

0.5799

0.7221

0.3984

0.4392

0.5043

0.4052

0.3703

0.3049

0.358

0.2293

0.2871 0.2338 0.1973

24 30 36 42 48 54 60 72


6

12

24

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ho

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nth

s

sys. label: 23.

0.0027 0.0004

0.1293

0.0733

0.0001

0

0.0829

0.0267

0

0

0.0411

0.0011

0.0001

0

0

0.3844

0.6099

0.3788

0.5342

0.8183

0.3601

0.415

0.5845

0.3866

0.3647

0.3846

0.354

0.2417

0.1871

0.2963 0.2507 0.2276

24 30 36 42 48 54 60 72


6

12

24

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ho

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nth

s

sys. label: 24.

0.005 0.0005

0.1381

0.0797

0.0002

0

0.0882

0.0288

0

0

0.0438

0.0011

0

0

0

0.5005

0.6793

0.4438

0.5712

0.7443

0.3993

0.4431

0.5709

0.4158

0.3917

0.392

0.3735

0.2566

0.1972

0.3062 0.2537 0.2183

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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nth

s

sys. label: 25.

0.0027

0.1396

0.0004

0.1225

0.0499

0.0002

0

0.076

0.0174

0.0001

0

0.0368

0.0008

0.0001

0

0

0.4293

0.6995

0.401

0.6041

0.8175

0.3619

0.4523

0.547

0.3821

0.3827

0.328

0.3448

0.2412

0.287 0.2418 0.2224

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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nth

s

sys. label: 26.

0.0042

0.1226

0.0004

0.122

0.0435

0.0002

0

0.0781

0.0149

0.0001

0

0.0399

0.0006

0.0001

0

0

0.5003

0.7359

0.4335

0.6047

0.7307

0.3817

0.4425

0.5126

0.4

0.3754

0.3014

0.3591

0.2354

0.2917 0.2413 0.2037

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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nth

s

sys. label: 27.

0.0029 0.0004

0.1316

0.0768

0.0001

0

0.0845

0.0273

0

0

0.0427

0.0009

0

0

0

0.4068

0.6614

0.3767

0.5763

0.8212

0.342

0.4385

0.5842

0.3686

0.3817

0.394

0.3399

0.248

0.1953

0.2896 0.2497 0.2369

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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n in

mo

nth

s

sys. label: 28.

0.0041 0.0004

0.134

0.0748

0.0001

0

0.087

0.026

0

0

0.0448

0.0007

0

0

0

0.4678

0.6955

0.3986

0.5747

0.7603

0.3515

0.4321

0.5906

0.375

0.3796

0.4013

0.3427

0.249

0.1943

0.2851 0.2409 0.2099

Notes: (1) p-values are reported in the boxes (2) grey areas indicate non-feasible choices of pairs (h,n) (3) system labels are asin the ranking used in Chapter 2.


Figure 69: Robustness checks on null hypothesis decisions by system of weights - regressiontype 2 Fama & Bliss

24 30 36 42 48 54 60 72


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s

sys. label: 1.

0.0024

0.1312

0.0004

0.1009

0.0467

0.0002

0

0.0622

0.0162

0.0001

0

0.0292

0.0007

0.0001

0

0

0.3742

0.6096

0.3544

0.5158

0.7976

0.3242

0.3812

0.5273

0.344

0.3264

0.3129

0.3022

0.2043

0.2409 0.196 0.1697

24 30 36 42 48 54 60 72


6

12

24

36

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60h -

ho

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nth

s

sys. label: 2.

0.0048

0.1349

0.0005

0.1065

0.0522

0.0002

0

0.0653

0.0186

0.0001

0

0.1597

0.0311

0.0008

0.0001

0

0

0.491

0.6919

0.4278

0.5717

0.7046

0.373

0.423

0.4928

0.3809

0.36

0.3038

0.3273

0.219

0.252 0.1986

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

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n in

mo

nth

s

sys. label: 3.

0.0025 0.0003

0.1145

0.069

0.0001

0

0.0728

0.0251

0

0

0.0353

0.001

0

0

0

0.3802

0.6142

0.3572

0.5228

0.8127

0.3264

0.3923

0.5681

0.3484

0.3419

0.3699

0.3096

0.2209

0.1766

0.2511 0.2081 0.1841

24 30 36 42 48 54 60 72


6

12

24

36

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ho

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nth

s

sys. label: 4.

0.0042 0.0004

0.1223

0.0784

0.0001

0

0.0779

0.0288

0

0

0.0382

0.001

0

0

0

0.4792

0.6586

0.4071

0.5382

0.7254

0.3526

0.4033

0.5509

0.3684

0.3593

0.3792

0.3223

0.2327

0.1902

0.2558 0.2068 0.1695

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

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n in

mo

nth

s

sys. label: 5.

0.0021

0.1298

0.0003

0.0999

0.0464

0.0002

0

0.0617

0.0162

0.0001

0

0.0291

0.0008

0.0001

0

0

0.3646

0.5988

0.3489

0.5122

0.7957

0.3213

0.3797

0.5229

0.3424

0.3262

0.3096

0.3024

0.2032

0.2426 0.1981 0.1728

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 6.

0.0046

0.1225

0.0004

0.1076

0.0474

0.0001

0

0.0665

0.0171

0

0

0.1586

0.0319

0.0008

0.0001

0

0

0.4915

0.6856

0.4287

0.5628

0.6926

0.374

0.415

0.4726

0.3829

0.3553

0.2838

0.3291

0.2185

0.2533 0.1993

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 7.

0.0029 0.0005

0.1155

0.0677

0.0002

0

0.073

0.0243

0.0001

0

0.0353

0.001

0.0001

0

0

0.3877

0.6179

0.365

0.5295

0.8135

0.3326

0.3982

0.5685

0.3533

0.3466

0.3706

0.3135

0.2232

0.1757

0.2539 0.2099 0.1848

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 8.

0.0044 0.0004

0.1231

0.0823

0.0001

0

0.0781

0.0302

0

0

0.0379

0.0011

0

0

0

0.486

0.6616

0.415

0.5433

0.7334

0.3604

0.4088

0.5622

0.3735

0.3621

0.3914

0.3269

0.2346

0.1993

0.2595 0.2095 0.1712

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 9.

0.0019

0.0852

0.0002

0.0808

0.0286

0.0001

0

0.0512

0.0101

0.0001

0

0.0247

0.0005

0.0001

0

0

0.3329

0.5044

0.3195

0.424

0.7505

0.2991

0.3123

0.4563

0.3179

0.2637

0.2336

0.2807

0.1641

0.2253 0.1854 0.1662

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 10.

0.0045

0.0998

0.0003

0.0995

0.0378

0.0001

0

0.062

0.0137

0

0

0.0286

0.0007

0.0001

0

0

0.4293

0.624

0.3822

0.5158

0.6641

0.3408

0.3866

0.439

0.3437

0.3166

0.2416

0.3

0.1979

0.236 0.1914 0.1674

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 11.

0.0021

0.1384

0.0002

0.1091

0.0511

0.0001

0

0.0703

0.0182

0

0

0.0339

0.0008

0.0001

0

0

0.3289

0.5203

0.3213

0.452

0.777

0.3079

0.3506

0.5252

0.335

0.3102

0.3157

0.3042

0.2067

0.2512 0.211 0.1899

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 12.

0.0038

0.1538

0.0003

0.1224

0.0628

0.0001

0

0.0776

0.0236

0

0

0.0363

0.001

0

0

0

0.444

0.6275

0.398

0.5264

0.6935

0.3594

0.4076

0.5034

0.3661

0.3477

0.319

0.3267

0.2308

0.2639 0.217 0.1881

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 13.

0.0019

0.09

0.0002

0.0811

0.03

0.0001

0

0.0513

0.0104

0.0001

0

0.0245

0.0005

0.0001

0

0

0.3356

0.5124

0.3205

0.4305

0.7614

0.2999

0.3168

0.4709

0.3187

0.2665

0.2438

0.2822

0.1652

0.2275 0.1883 0.1695

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 14.

0.0043

0.1033

0.0004

0.0995

0.039

0.0002

0

0.0618

0.0142

0.0001

0

0.0287

0.0007

0.0001

0

0

0.4324

0.6229

0.389

0.514

0.6814

0.3512

0.3857

0.4504

0.3588

0.3229

0.2503

0.3149

0.201

0.2485 0.2007 0.1719

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 15.

0.0022

0.1428

0.0003

0.1083

0.0538

0.0001

0

0.0704

0.0193

0

0

0.0343

0.0008

0.0001

0

0

0.3254

0.518

0.3182

0.4484

0.7788

0.3045

0.3453

0.5239

0.3312

0.3062

0.3184

0.3003

0.2038

0.2471 0.2068 0.1848

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 16.

0.0031

0.1584

0.0002

0.1246

0.0642

0.0001

0

0.0794

0.0238

0

0

0.0373

0.001

0

0

0

0.4236

0.616

0.3811

0.5179

0.6985

0.347

0.4032

0.5097

0.3582

0.3487

0.3275

0.3205

0.2323

0.2597 0.2131 0.1839

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 17.

0.0023

0.1229

0.0004

0.0927

0.0438

0.0002

0

0.0578

0.0156

0.0001

0

0.0279

0.0008

0.0001

0

0

0.3603

0.5746

0.3432

0.484

0.7831

0.3162

0.3558

0.5088

0.3351

0.3024

0.2957

0.2942

0.1869

0.2348 0.1911 0.1668

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 18.

0.005

0.1421

0.0005

0.1061

0.0557

0.0002

0

0.0657

0.0203

0.0001

0

0.1583

0.0317

0.0009

0.0001

0

0

0.4888

0.6777

0.4255

0.5586

0.7058

0.3698

0.4146

0.4975

0.373

0.3495

0.3101

0.3194

0.2141

0.2462 0.1941

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 19.

0.0023 0.0003

0.1154

0.068

0.0001

0

0.0734

0.0249

0

0

0.0354

0.0011

0.0001

0

0

0.3608

0.5816

0.3472

0.4984

0.7993

0.3241

0.3784

0.5565

0.3496

0.3356

0.3626

0.3127

0.2201

0.1734

0.2536 0.2091 0.1826

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 20.

0.0048 0.0004

0.125

0.0733

0.0001

0

0.079

0.027

0

0

0.0378

0.0011

0

0

0

0.4907

0.6619

0.4263

0.544

0.7146

0.3739

0.4114

0.5315

0.3872

0.3651

0.3607

0.3387

0.2369

0.1794

0.2682 0.2154 0.1748

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 21.

0.0024

0.1313

0.0004

0.0947

0.0479

0.0003

0

0.0591

0.0172

0.0001

0

0.0285

0.0008

0.0001

0

0

0.3655

0.5761

0.3491

0.4905

0.7884

0.3215

0.3637

0.5195

0.3377

0.3065

0.3059

0.2977

0.1904

0.2388 0.1953 0.1724

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 22.

0.0045

0.1303

0.0005

0.1063

0.0509

0.0002

0

0.0656

0.0188

0.0001

0

0.1594

0.0316

0.0009

0.0001

0

0

0.4828

0.6779

0.4222

0.5611

0.7084

0.3707

0.4198

0.4882

0.374

0.3528

0.2941

0.3221

0.2156

0.2495 0.1972

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 23.

0.0025 0.0004

0.1197

0.072

0.0001

0

0.0756

0.0265

0.0001

0

0.0359

0.0011

0.0001

0

0

0.3713

0.5966

0.3591

0.5176

0.808

0.3352

0.3978

0.5691

0.3574

0.3489

0.373

0.3195

0.2287

0.1816

0.2591 0.2136 0.1862

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 24.

0.0049 0.0005

0.1282

0.0782

0.0002

0

0.0806

0.0286

0.0001

0

0.0385

0.0011

0.0001

0

0

0.4836

0.6652

0.4206

0.553

0.7317

0.3714

0.4245

0.5559

0.3839

0.3749

0.3807

0.3363

0.2431

0.1915

0.2667 0.2149 0.1773

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 25.

0.0025

0.1348

0.0005

0.1127

0.0487

0.0002

0

0.0688

0.0172

0.0001

0

0.0319

0.0007

0.0001

0

0

0.4147

0.6868

0.3804

0.587

0.8072

0.3371

0.4338

0.5316

0.3532

0.3658

0.3171

0.3111

0.2276

0.2508 0.2057 0.1821

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 26.

0.0042

0.1177

0.0004

0.112

0.0422

0.0002

0

0.0706

0.0146

0.0001

0

0.0347

0.0005

0.0001

0

0

0.4816

0.7221

0.4084

0.5854

0.7175

0.3524

0.4223

0.4964

0.3672

0.3571

0.29

0.3217

0.2211

0.2528 0.2033 0.1649

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 27.

0.0027 0.0004

0.1217

0.0754

0.0001

0

0.0771

0.0271

0

0

0.0375

0.0009

0

0

0

0.3925

0.6482

0.3567

0.5591

0.8111

0.3175

0.4202

0.5695

0.3399

0.3649

0.3828

0.3058

0.2344

0.1897

0.2524 0.212 0.1937

24 30 36 42 48 54 60 72


6

12

24

36

48

60h -

ho

rizo

n in

mo

nth

s

sys. label: 28.

0.004 0.0004

0.1237

0.0732

0.0001

0

0.0791

0.0257

0

0

0.0393

0.0007

0

0

0

0.4488

0.6808

0.3736

0.5554

0.7481

0.3226

0.4124

0.5753

0.3423

0.3617

0.3892

0.305

0.2347

0.1882

0.2458 0.2022 0.1695



Figure 70: Regression coefficients of one year excess returns on forward rates - Cochrane &Piazzesi

1 2 3 4 5-50

0

50

reg

ressio

n c

oeffic

ients

1 2 3 4 5

-50

0

50

reg

ressio

n c

oeffic

ients

1 2 3 4 5

-50

0

50

reg

ressio

n c

oeffic

ients

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oeffic

ients

1 2 3 4 5-40

-20

0

20

40

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-200

-100

0

100

regre

ssio

n c

oeffic

ien

ts

1 2 3 4 5

-100

-50

0

50

100

regre

ssio

n c

oeffic

ien

ts

1 2 3 4 5-100

-50

0

50

regre

ssio

n c

oeffic

ien

ts

1 2 3 4 5-100

-50

0

50

regre

ssio

n c

oeffic

ien

ts

1 2 3 4 5-200

-100

0

100

200

reg

ressio

n c

oe

ffic

ients

1 2 3 4 5-200

-100

0

100

reg

ressio

n c

oe

ffic

ients

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oe

ffic

ients

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oe

ffic

ients

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5

-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5

-100

-50

0

50

100

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-100

-50

0

50

reg

ressio

n c

oe

ffic

ien

ts

1 2 3 4 5-50

0

50

regre

ssio

n c

oeff

icie

nts

1 2 3 4 5

-100

-50

0

50

100

regre

ssio

n c

oeff

icie

nts

1 2 3 4 5-50

0

50

regre

ssio

n c

oeff

icie

nts

1 2 3 4 5-100

-50

0

50

regre

ssio

n c

oeff

icie

nts



Figure 71: Robustness checks on null hypothesis decisions by system of weights in rollingrealised returns on term premia regressions - Thornton - conventional

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 1.

0.1017 0.1571 0.0479

0.0552

0.0713

0.1353

0.0001

0

0.0002

0

0

0

0

0.0001

0.247

0.2918

0.4783

0.613

0.2389

0.4345

0.6072

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 2.

0.074 0.0582

0.0716

0.0799

0.1391

0.0002

0.0002

0.0005

0

0

0

0

0.0001

0.2021

0.4227

0.4271

0.6544

0.6953

0.3313

0.5178

0.6331

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 3.

0.0915 0.1531 0.0522

0.0636

0.089

0.0001

0.0001

0.0004

0

0

0

0

0.0001

0.252

0.2953

0.4883

0.6176

0.2458

0.4396

0.6158

0.1809

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 4.

0.0784 0.0594

0.0746

0.0946

0.0002

0.0003

0.0008

0

0

0

0

0.0001

0.22

0.4513

0.4221

0.6365

0.6622

0.3201

0.4977

0.6232

0.1948

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 5.

0.1045 0.1526 0.0487

0.0572

0.0662

0.1338

0.0001

0

0.0001

0

0

0

0

0.0001

0.2365

0.281

0.457

0.6024

0.2291

0.4192

0.5992

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 6.

0.0764 0.0564

0.0678

0.0753

0.1267

0.0001

0.0001

0.0005

0

0

0

0

0.0001

0.2014

0.4196

0.427

0.6523

0.6891

0.3261

0.5082

0.6202

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 7.

0.0902 0.1526 0.0519

0.0635

0.0887

0.0001

0.0001

0.0004

0

0

0

0

0.0001

0.2544

0.2937

0.487

0.6213

0.245

0.4392

0.619

0.18

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 8.

0.0825 0.0592

0.0744

0.0993

0.0002

0.0003

0.0008

0

0

0

0

0.0001

0.2225

0.4548

0.4267

0.6396

0.6651

0.3245

0.5045

0.6323

0.2039

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 9.

0.113 0.1577 0.0346

0.0322

0.029

0.0888

0

0

0

0

0

0

0

0.0001

0.2295

0.2497

0.3809

0.5078

0.1754

0.3259

0.5139

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 10.

0.0671 0.04

0.0404

0.0472

0.1036

0

0.0001

0.0002

0

0

0

0

0.0001

0.1768

0.3536

0.3523

0.5623

0.6273

0.2582

0.4274

0.5641

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 11.

0.1034 0.1471 0.0397

0.0405

0.0519

0.1427

0.0001

0

0.0001

0

0

0

0

0.0001

0.2199

0.2431

0.3836

0.5236

0.1836

0.3456

0.5508

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 12.

0.0641 0.0442

0.0501

0.0686

0.1581

0.0001

0.0001

0.0005

0

0

0

0

0.0001

0.1796

0.3728

0.3512

0.5635

0.6306

0.2615

0.4329

0.5827

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 13.

0.1148 0.1601 0.0364

0.037

0.0324

0.0937

0

0

0

0

0

0

0

0.0001

0.2346

0.2525

0.3882

0.5159

0.1786

0.3342

0.5247

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 14.

0.0702 0.0422

0.0415

0.0492

0.1071

0

0.0001

0.0002

0

0

0

0

0.0001

0.1764

0.3489

0.3509

0.5599

0.6262

0.2625

0.4373

0.5777

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 15.

0.1023 0.1439 0.0411

0.0438

0.052

0.147

0.0001

0

0.0001

0

0

0

0

0.0001

0.2148

0.2404

0.3809

0.5214

0.182

0.3434

0.5469

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 16.

0.0664 0.0421

0.0469

0.0656

0.0001

0.0001

0.0005

0

0

0

0

0.0001

0.1725

0.3494

0.3333

0.544

0.6192

0.248

0.4191

0.578

0.1627

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 17.

0.1062 0.1565 0.0437

0.0495

0.0572

0.1269

0.0001

0

0.0001

0

0

0

0

0.0001

0.2382

0.2732

0.439

0.5778

0.2091

0.3898

0.5748

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 18.

0.0702 0.0523

0.066

0.0788

0.1463

0.0001

0.0002

0.0005

0

0

0

0

0.0001

0.1985

0.4142

0.4093

0.6355

0.681

0.3098

0.4965

0.6206

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 19.

0.0971 0.1495 0.0476

0.0554

0.076

0.0001

0

0.0003

0

0

0

0

0.0001

0.233

0.2682

0.4402

0.5848

0.2143

0.3967

0.5892

0.1777

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 20.

0.0708 0.0578

0.0725

0.0907

0.0002

0.0003

0.0008

0

0

0

0

0.0001

0.2101

0.4367

0.4137

0.6311

0.6652

0.3132

0.4891

0.6156

0.184

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 21.

0.1048 0.1543 0.0444

0.0517

0.0597

0.1353

0.0001

0

0.0001

0

0

0

0

0.0001

0.2375

0.2701

0.4356

0.5794

0.2098

0.391

0.5779

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 22.

0.0727 0.0573

0.0709

0.0782

0.1343

0.0001

0.0001

0.0005

0

0

0

0

0.0001

0.1982

0.4095

0.4132

0.6377

0.6811

0.3244

0.5117

0.6293

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 23.

0.0919 0.1465 0.0482

0.055

0.0804

0.0001

0.0001

0.0003

0

0

0

0

0.0001

0.233

0.2701

0.4493

0.5998

0.2204

0.4069

0.6041

0.186

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 24.

0.0702 0.0575

0.0728

0.0968

0.0002

0.0003

0.0009

0

0

0

0

0.0001

0.2089

0.4333

0.4123

0.6306

0.6683

0.3186

0.5014

0.6338

0.1961

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 25.

0.1061 0.0568

0.068

0.0859

0.1388

0.0001

0.0001

0.0003

0

0

0

0

0.0001

0.1819

0.2995

0.387

0.5956

0.6902

0.3262

0.5247

0.6604

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 26.

0.1372 0.0666

0.0709

0.0726

0.1217

0.0002

0.0002

0.0003

0

0

0

0

0.0001

0.2893

0.485

0.5613

0.7239

0.7245

0.4229

0.5861

0.6679

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 27.

0.0958 0.0557

0.0702

0.0915

0.0002

0.0002

0.0005

0

0

0

0

0

0.1753

0.2957

0.3513

0.5492

0.6518

0.2901

0.4763

0.6333

0.1941

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 28.

0.1235 0.0609

0.0731

0.0916

0.0002

0.0003

0.0006

0

0

0

0

0.0001

0.2844

0.4828

0.5101

0.6738

0.6835

0.3743

0.5379

0.658

0.1929



Figure 72: Robustness checks on null hypothesis decisions by system of weights in rollingrealised returns on term premia regressions - Thornton - contrarian

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 1.

0.0018 0.0035 0.0041

0.7141 0.9999

0.8378

1

1

0.9803

1

1

0.9986

0.8424

1

1

0.9996

0.6944

1

1

0.9938

0.4169

1

1

0.9879

0.3186

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 2.

0.0019 0.0029 0.0035

0.7104 1

0.9325

1

1

0.9846

1

1

0.9989

0.8154

1

1

0.9997

0.6883

1

1

0.9945

0.431

1

1

0.9883

0.3399

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 3.

0.0016 0.0023 0.0031

0.7101 0.9999

0.842

1

1

0.9769

1

1

0.9984

0.8594

1

1

0.9996

0.7396

1

1

0.993

0.4671

1

1

0.986

0.3603

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 4.

0.0015 0.002 0.0029

0.7315 1

0.9394

1

1

0.9802

1

1

0.9984

0.8443

1

1

0.9996

0.7511

1

1

0.9937

0.4897

1

1

0.9873

0.3875

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 5.

0.0015 0.0033 0.004

0.7216 0.9999

0.8249

1

1

0.9793

1

1

0.9987

0.8395

1

1

0.9996

0.6905

1

1

0.994

0.415

1

1

0.9878

0.3178

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 6.

0.0015 0.0022 0.0031

0.7107 1

0.932

1

1

0.9839

1

1

0.9987

0.8048

1

1

0.9997

0.673

1

1

0.9937

0.4184

1

1

0.9873

0.3331

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 7.

0.0023 0.0032 0.0037

0.7076 0.9999

0.8418

1

1

0.9774

1

1

0.9985

0.8609

1

1

0.9996

0.7436

1

1

0.993

0.4695

1

1

0.986

0.3614

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 8.

0.0016 0.002 0.003

0.7433 1

0.9414

1

1

0.9802

1

1

0.9985

0.8498

1

1

0.9996

0.7611

1

1

0.9939

0.4991

1

1

0.9876

0.3958

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 9.

0.0008 0.0027 0.0041

0.7264 0.9999

0.8208

1

1

0.9741

1

1

0.9982

0.8134

1

1

0.9994

0.6414

1

1

0.9921

0.3635

1

1

0.9865

0.2769

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 10.

0.0012 0.0022 0.0034

0.6927 1

0.9112

1

1

0.9776

1

1

0.9982

0.78

1

1

0.9995

0.6406

1

1

0.993

0.3908

1

1

0.9889

0.305

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 11.

0.001 0.0021 0.0033

0.7179 0.9999

0.808

1

1

0.9709

1

1

0.9981

0.8401

1

1

0.9993

0.7065

1

1

0.9913

0.4267

1

1

0.9851

0.3227

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 12.

0.0011 0.0016 0.0028

0.6883 1

0.9117

1

1

0.9753

1

1

0.998

0.8139

1

1

0.9994

0.7089

1

1

0.993

0.4535

1

1

0.9884

0.3555

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 13.

0.001 0.0034 0.0047

0.7274 0.9999

0.8225

1

1

0.9753

1

1

0.9984

0.8244

1

1

0.9995

0.655

1

1

0.9929

0.374

1

1

0.9878

0.2831

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 14.

0.0016 0.0033 0.0042

0.6913 1

0.9086

1

1

0.9793

1

1

0.9985

0.7928

1

1

0.9996

0.646

1

1

0.9942

0.3883

1

1

0.99

0.3015

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 15.

0.0014 0.0026 0.0037

0.7144 0.9999

0.8049

1

1

0.9704

1

1

0.9982

0.8398

1

1

0.9994

0.7094

1

1

0.9914

0.4349

1

1

0.9851

0.3315

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 16.

0.0009 0.0016 0.0029

0.679 1

0.9028

1

1

0.975

1

1

0.9981

0.819

1

1

0.9994

0.7168

1

1

0.9934

0.4596

1

1

0.989

0.3592

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 17.

0.0016 0.0038 0.0045

0.7265 0.9999

0.8277

1

1

0.9773

1

1

0.9984

0.8353

1

1

0.9995

0.6825

1

1

0.9933

0.4098

1

1

0.9865

0.3171

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 18.

0.0019 0.0028 0.0034

0.7003 1

0.9283

1

1

0.9821

1

1

0.9987

0.8162

1

1

0.9997

0.6944

1

1

0.9947

0.4423

1

1

0.989

0.3526

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 19.

0.0015 0.0027 0.0036

0.7181 0.9999

0.8213

1

1

0.9744

1

1

0.9982

0.8537

1

1

0.9995

0.7375

1

1

0.9924

0.4681

1

1

0.9854

0.3632

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 20.

0.0017 0.0022 0.0031

0.7112 1

0.9324

1

1

0.9787

1

1

0.9981

0.8328

1

1

0.9995

0.7392

1

1

0.9933

0.4829

1

1

0.9868

0.3833

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 21.

0.0019 0.0043 0.0047

0.7266 0.9999

0.8264

1

1

0.9758

1

1

0.9983

0.8381

1

1

0.9996

0.6895

1

1

0.9936

0.4206

1

1

0.9875

0.3276

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 22.

0.0017 0.003 0.0037

0.7099 1

0.9316

1

1

0.9827

1

1

0.9988

0.8132

1

1

0.9997

0.6746

1

1

0.9948

0.4226

1

1

0.989

0.3385

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 23.

0.0018 0.0029 0.0037

0.7108 0.9999

0.8225

1

1

0.9751

1

1

0.9982

0.8586

1

1

0.9995

0.7431

1

1

0.9925

0.4753

1

1

0.9855

0.3692

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 24.

0.0021 0.0029 0.0035

0.7086 1

0.9316

1

1

0.9795

1

1

0.9984

0.8454

1

1

0.9996

0.7497

1

1

0.9942

0.4901

1

1

0.9882

0.3874

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 25.

0.0017 0.0031 0.0038

0.7487 1

0.8787

1

1

0.9848

1

1

0.9988

0.8388

1

1

0.9996

0.6889

1

1

0.9939

0.4172

1

1

0.9881

0.3237

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 26.

0.0012 0.0028 0.0036

0.8324 1

0.946

1

1

0.9887

1

1

0.9991

0.8186

1

1

0.9997

0.676

1

1

0.9943

0.409

1

1

0.9886

0.3267

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 27.

0.0016 0.0015 0.0024

0.7544 1

0.8733

1

1

0.977

1

1

0.9987

0.8561

1

1

0.9997

0.7534

1

1

0.994

0.4887

1

1

0.9873

0.3803

6 12 24 36 48 60 72


3

6

12

24

36h -

ho

rizo

n in

mo

nth

s

sys. label: 28.

0.0014 0.0021 0.0029

0.8329 1

0.94

1

1

0.9839

1

1

0.9989

0.8612

1

1

0.9997

0.7688

1

1

0.9951

0.4972

1

1

0.9893

0.3978



Table 33: Market prices of risk for N=3, 4, 5, 6 and 7 factor ACM model of Polish termstructure

λ0 λ1,1 λ1,2 λ1,3 WΛ Wλ1

PC1 -0.015 -0.006 0.005 -0.006 2.697 0.742t-stat1 -1.398 -0.535 0.413 -0.533 0.610 0.863PC2 0.007 -0.024 -0.050 0.060 9.258 9.180t-stat2 0.279 -0.888 -1.876 2.207 0.055 0.027PC3 0.092 -0.004 -0.014 -0.109 11.231 6.587t-stat3 2.155 -0.084 -0.320 -2.545 0.024 0.086

λ0 λ1,1 λ1,2 λ1,3 λ1,4 WΛ Wλ1

PC1 -0.016 -0.006 0.005 -0.006 0.003 2.876 0.876t-stat1 -1.414 -0.578 0.427 -0.518 0.302 0.719 0.928PC2 0.005 -0.031 -0.048 0.062 0.037 12.408 12.366t-stat2 0.205 -1.182 -1.831 2.369 1.417 0.030 0.015PC3 0.082 -0.038 -0.001 -0.097 0.090 14.885 10.993t-stat3 1.973 -0.912 -0.027 -2.337 2.168 0.011 0.027PC4 0.033 -0.009 0.056 -0.002 -0.230 23.258 22.824t-stat4 0.659 -0.179 1.133 -0.041 -4.638 0.000 0.000

λ0 λ1,1 λ1,2 λ1,3 λ1,4 λ1,5 WΛ Wλ1

PC1 -0.016 -0.006 0.005 -0.006 0.004 0.012 4.022 1.979t-stat1 -1.429 -0.566 0.418 -0.506 0.313 1.063 0.674 0.852PC2 0.003 -0.031 -0.049 0.064 0.038 -0.003 13.163 13.150t-stat2 0.113 -1.178 -1.898 2.445 1.474 -0.106 0.041 0.022PC3 0.068 -0.035 -0.010 -0.087 0.097 0.050 15.496 12.659t-stat3 1.684 -0.863 -0.245 -2.148 2.395 1.227 0.017 0.027PC4 0.003 -0.003 0.037 0.019 -0.216 0.040 20.792 20.788t-stat4 0.065 -0.066 0.762 0.395 -4.402 0.819 0.002 0.001PC5 0.015 0.088 -0.016 0.095 -0.086 -0.409 49.019 48.960t-stat5 0.244 1.408 -0.254 1.527 -1.374 -6.534 0.000 0.000

λ0 λ1,1 λ1,2 λ1,3 λ1,4 λ1,5 λ1,6 WΛ Wλ1

PC1 -0.016 -0.006 0.005 -0.006 0.004 0.012 -0.001 4.053 2.004t-stat1 -1.431 -0.570 0.416 -0.504 0.315 1.070 -0.091 0.774 0.919PC2 0.003 -0.032 -0.049 0.064 0.038 -0.002 0.033 14.999 14.988t-stat2 0.104 -1.230 -1.905 2.457 1.488 -0.086 1.262 0.036 0.020PC3 0.067 -0.041 -0.010 -0.085 0.100 0.053 0.068 19.243 16.472t-stat3 1.665 -1.016 -0.256 -2.116 2.506 1.330 1.687 0.007 0.011PC4 -0.001 -0.018 0.036 0.026 -0.205 0.048 0.140 28.842 28.841t-stat4 -0.012 -0.363 0.763 0.535 -4.289 1.006 2.904 0.000 0.000PC5 0.001 0.042 -0.022 0.125 -0.041 -0.374 -0.026 41.616 41.616t-stat5 0.011 0.665 -0.350 2.022 -0.658 -6.030 -0.410 0.000 0.000PC6 0.034 -0.001 0.024 0.068 0.122 -0.025 -0.786 91.980 91.797t-stat6 0.427 -0.011 0.309 0.855 1.527 -0.306 -9.410 0.000 0.000

λ0 λ1,1 λ1,2 λ1,3 λ1,4 λ1,5 λ1,6 λ1,7 WΛ Wλ1

PC1 -0.016 -0.006 0.004 -0.005 0.004 0.012 -0.001 0.017 6.564 4.484t-stat1 -1.442 -0.540 0.407 -0.485 0.329 1.094 -0.109 1.573 0.584 0.723PC2 0.002 -0.032 -0.050 0.064 0.039 -0.002 0.033 0.019 15.689 15.680t-stat2 0.094 -1.221 -1.917 2.472 1.497 -0.077 1.271 0.737 0.047 0.028PC3 0.066 -0.041 -0.011 -0.084 0.101 0.054 0.069 0.026 19.825 17.110t-stat3 1.647 -1.018 -0.275 -2.106 2.518 1.344 1.730 0.649 0.011 0.017PC4 -0.002 -0.018 0.035 0.027 -0.205 0.049 0.144 0.046 30.584 30.582t-stat4 -0.047 -0.372 0.727 0.562 -4.307 1.029 3.014 0.952 0.000 0.000PC5 -0.006 0.036 -0.029 0.128 -0.042 -0.373 -0.003 -0.002 44.069 44.060t-stat5 -0.095 0.591 -0.492 2.117 -0.698 -6.205 -0.045 -0.035 0.000 0.000PC6 0.001 -0.030 -0.021 0.081 0.114 -0.020 -0.658 0.033 86.231 86.231t-stat6 0.007 -0.413 -0.293 1.127 1.585 -0.274 -9.051 0.449 0.000 0.000PC7 0.020 -0.011 -0.051 0.012 -0.064 -0.027 0.127 -0.744 97.870 97.799t-stat7 0.266 -0.149 -0.676 0.158 -0.850 -0.358 1.663 -9.679 0.000 0.000

Note: this table reports λ coefficient estimates paired with their t-statistic value based on standard error computed as in ACM.Last two columns provide Wald statistics to test if the whole row of Λk is statistically different from 0. Certain factor is notpriced if Wald test shows we cannot reject this null hypothesis. Bold text is used to annote significant coefficients (at 5%).

PC1, PC2,... are factor names from PCA


Table 34: Mean and standard deviation of term premia datasets for different systems ofweights for the Polish yield curve for K = 3

Panel A. Mean term premia by tenors (in basis points)label n=1 n=3 n=6 n=12 n=24 n=36 n=48 n=60 n=84 n=1201 0.0 -0.4 0.1 4.3 20.6 41.2 61.6 79.6 106.1 125.72 0.0 0.5 2.2 8.1 25.9 46.1 65.0 81.2 105.0 126.13 0.0 0.2 1.3 6.0 21.7 41.1 60.6 78.3 105.6 126.74 0.0 1.4 4.0 11.1 28.9 47.7 65.1 80.3 104.0 126.95 0.0 -0.2 0.4 4.5 20.3 40.4 60.7 78.8 106.1 125.96 0.0 0.9 3.0 9.6 27.9 47.6 65.8 81.3 104.9 127.27 0.0 -0.2 0.4 4.3 19.3 39.0 59.2 77.8 106.0 125.88 0.0 1.2 3.7 10.6 28.5 47.7 65.5 80.9 104.5 126.89 0.0 -0.5 -0.0 4.1 20.1 40.2 60.3 78.2 105.3 125.010 0.0 0.4 1.9 7.4 25.0 45.5 65.1 82.0 106.8 126.611 0.0 -0.2 0.3 4.0 18.5 37.7 57.7 76.2 105.1 125.812 0.0 0.7 2.5 8.2 25.1 44.7 63.9 81.0 106.8 126.913 0.0 -0.5 -0.3 3.5 19.1 39.3 59.8 78.3 106.2 126.114 0.0 0.1 1.4 7.0 25.2 46.2 65.6 82.0 105.7 125.615 0.0 -0.2 0.3 4.0 18.6 37.7 57.5 76.0 104.9 125.916 0.0 1.1 3.3 9.8 27.2 46.5 64.9 81.0 105.3 126.517 0.0 -0.6 -0.3 3.6 19.5 40.0 60.4 78.5 105.5 125.218 0.0 0.6 2.2 7.7 24.8 45.0 64.7 81.8 106.5 125.819 0.0 -0.1 0.5 4.3 18.8 38.2 58.6 77.4 106.2 125.520 0.0 1.0 3.1 9.3 26.4 45.9 64.6 81.1 106.0 126.721 0.0 -0.5 -0.2 3.9 20.1 40.5 60.8 78.8 105.9 126.322 0.0 0.6 2.3 8.1 25.5 45.7 65.0 81.6 106.0 126.323 0.0 0.5 2.0 6.9 22.3 40.9 59.7 77.0 104.2 125.924 0.0 1.4 4.0 10.6 27.4 46.1 64.3 80.7 106.1 127.525 0.0 1.0 3.1 9.4 26.5 45.2 62.6 77.8 101.3 124.426 0.0 1.5 4.5 12.2 31.1 50.8 68.7 83.9 107.2 130.127 0.0 0.5 2.1 7.8 25.0 44.5 63.2 79.5 104.1 124.828 0.0 1.8 5.1 13.0 31.8 51.1 68.8 84.1 107.6 130.0



Table 35: Mean and standard deviation of term premia datasets for different systems ofweights for the Polish yield curve for K = 7

Panel A. Mean term premia by tenors (in basis points)label n=1 n=3 n=6 n=12 n=24 n=36 n=48 n=60 n=84 n=1201 0.0 0.6 2.4 8.5 26.1 45.3 63.3 79.0 103.4 126.62 0.0 -1.8 -1.0 6.5 26.9 47.3 65.6 81.0 104.6 128.03 0.0 0.7 2.5 8.6 26.1 45.3 63.2 78.9 103.0 125.64 0.0 -1.9 -1.0 6.5 27.0 47.4 65.6 81.0 104.5 127.65 0.0 0.4 2.2 8.4 26.2 45.5 63.4 79.1 103.4 126.76 0.0 -2.0 -1.1 6.8 27.4 47.7 65.7 81.1 104.6 128.37 0.0 0.6 2.4 8.5 26.0 45.2 63.3 78.9 103.0 125.28 0.0 -1.7 -0.9 6.8 27.6 47.9 66.1 81.4 104.9 127.99 0.0 0.8 2.7 9.0 26.8 46.0 63.9 79.5 103.5 126.210 0.0 -1.4 -0.3 7.2 27.4 47.7 65.8 81.3 104.9 128.111 0.0 0.8 2.8 9.0 26.6 45.9 63.8 79.4 103.4 125.612 0.0 -1.3 -0.1 7.2 27.1 47.4 65.7 81.2 104.9 127.313 0.0 0.7 2.7 9.1 26.8 46.1 64.0 79.5 103.6 126.514 0.0 -1.2 0.1 7.5 27.4 47.6 65.8 81.4 105.2 128.115 0.0 0.5 2.5 9.0 26.8 46.1 64.1 79.7 103.7 125.716 0.0 -1.1 0.1 7.4 27.2 47.4 65.6 81.2 104.9 127.617 0.0 0.7 2.6 8.7 26.0 45.1 63.1 78.8 103.1 126.018 0.0 -1.6 -0.6 6.6 26.5 46.7 65.0 80.6 104.4 127.719 0.0 0.6 2.4 8.4 25.8 45.1 63.2 78.9 102.9 125.220 0.0 -1.5 -0.6 6.8 27.0 47.3 65.5 80.9 104.6 127.721 0.0 0.6 2.4 8.6 26.1 45.3 63.2 79.0 103.4 126.622 0.0 -1.6 -0.8 6.6 26.4 46.6 65.1 80.7 104.5 128.423 0.0 0.7 2.5 8.6 25.9 45.1 63.1 78.8 103.0 126.124 0.0 -1.9 -1.1 6.5 27.0 47.4 65.7 81.1 104.6 128.025 0.0 0.2 1.7 7.8 25.7 45.3 63.5 79.2 103.5 126.426 0.0 -2.0 -0.4 8.8 30.6 50.9 68.7 83.8 107.3 130.527 0.0 0.2 1.8 8.2 26.1 45.5 63.7 79.4 103.4 125.628 0.0 -2.1 -0.5 9.0 30.7 51.0 68.9 84.1 107.6 130.2



Figure 73: Decomposition of expected 1, 2 and 3-year excess returns in ACM model forPoland

2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

-4

-3

-2

-1

0

1

2

310

-3

1

2

3

4

5

6

7

2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

-6

-4

-2

0

2

4

610

-3

1

2

3

4

5

6

7

2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

-8

-6

-4

-2

0

2

4

610

-3

1

2

3

4

5

6

7



Figure 74: Decomposition of expected 5 and 8-year excess returns in ACM model for Poland

2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

-12

-10

-8

-6

-4

-2

0

2

4

6

810

-3

1

2

3

4

5

6

7

2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1

2

3

4

5

6

7




2006 2008 2010 2012 2014 2016 2018 2020

-4

-2

0

2

4

6

8

1010

-3

As of 30th Jun 2020

Blended series

As of 2nd Jan 2015



2006 2008 2010 2012 2014 2016 2018 2020

-4

-2

0

2

4

6

8

10

1210

-3

As of 30th Jun 2020

Blended series

As of 2nd Jan 2015




2006 2008 2010 2012 2014 2016 2018 2020

-6

-4

-2

0

2

4

6

8

10

1210

-3

As of 30th Jun 2020

Blended series

As of 2nd Jan 2015



2006 2008 2010 2012 2014 2016 2018 2020

-0.01

-0.005

0

0.005

0.01

0.015

0.02

As of 30th Jun 2020

Blended series

As of 2nd Jan 2015



Table 36: Level forecast errors for 3-month tenor in different models with K = 3, 10-yearwindow


Bper 20.74 22.83 24.83 23.99 29.52 17.46 19.3 21.05 20.05 23.39Bimp 10.35 14.77 28.56 51.55 91.99 8.25 12.54 22.65 43.16 82.1

MK=3Tfo,h

(rf:imp, X:AR(1)) 18.72 25.13 32.39 36.23 43.76 15.69 21.74 27.42 30.61 37.59

MK=3Tfo,h

(rf:imp, X:AR(3)) 20.57 25.52 31.61 33.37 38.21 16.92 21.87 25.76 26.65 29.86

MK=3Tfo,h

(rf:imp, X:AR(6)) 21.81 27.07 36.13 42.32 48.87 16.7 22.51 29.23 34.79 40.18

MK=3Tfo,h

(rf:imp, X:VAR(1)) 19.44 21.09 28.96 36.36 48.54 15.94 17.67 23.13 29.79 41.37

MK=3Tfo,h

(rf:imp, X:VAR(3)) 19.33 22.42 31.39 36.26 42.44 16.97 18.83 25.08 29.84 35.83

MK=3Tfo,h

(rf:imp, X:VAR(6)) 13.1 21.75 44.59 64.66 84.17 10.75 18.16 37.7 53.59 65.96

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 60.74 57.27 38.47 33.14 130.93 54.89 51.77 31.9 27.83 123.66

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 69.66 71.38 57.47 45.34 250.58 63.73 65.33 49.72 30.69 205.71

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 80.51 83.56 86.76 70.38 255.08 57.15 63.47 68.94 50.6 199.13

MK=3Tfo,h

(rf:per, X:AR(1)) 13.76 14.92 19.51 22.58 31.05 8.49 10.46 15.54 18.06 24.82

MK=3Tfo,h

(rf:per, X:AR(3)) 16.8 15.94 19.27 20.86 27.22 10.68 11.28 14.25 15.29 19.36

MK=3Tfo,h

(rf:per, X:AR(6)) 20.26 19.85 23.89 29.51 36.79 13.3 14.55 18.14 22.58 28.12

MK=3Tfo,h

(rf:per, X:VAR(1)) 15.33 13.53 18.09 23.1 35.08 8.95 9.03 13.86 17.89 28.56

MK=3Tfo,h

(rf:per, X:VAR(3)) 15.04 12.63 20.47 25.81 32.77 10.11 9.21 16.13 22.27 27.17

MK=3Tfo,h

(rf:per, X:VAR(6)) 15.21 13.97 32.59 54.16 75.81 11.24 11.3 26.72 43.63 56.36

MK=3Tfo,h

(rf:per, X:BVAR(1)) 48.89 44.19 26.15 47.5 155.91 38.84 35.64 17.62 43.6 151.38

MK=3Tfo,h

(rf:per, X:BVAR(3)) 56.59 57.54 42.88 54.91 274.23 49.05 50.93 35.61 45.42 235.11

MK=3Tfo,h

(rf:per, X:BVAR(6)) 73.48 75.35 77.5 63.94 270.9 47.62 52.68 57.66 46.57 210.63


Table 37: Level forecast errors for 1-year tenor in different models with K = 3, 10-yearwindow


Bper 21.85 27.92 33.7 30.85 34.54 18.52 23.2 28.04 24.38 27.4Bimp 14.69 25.51 49.75 76.17 118.21 12.36 20.7 39.96 66.76 106.82

MK=3Tfo,h

(rf:imp, X:AR(1)) 16.64 24.58 34.78 37.29 48.97 14.27 21.5 28.57 29.86 39.44

MK=3Tfo,h

(rf:imp, X:AR(3)) 18.16 24.54 34.11 36.04 45.89 14.91 21.2 28.35 29.63 36.83

MK=3Tfo,h

(rf:imp, X:AR(6)) 18.73 25.24 38.12 42.03 52.24 14.81 21.59 31.02 33.72 42.08

MK=3Tfo,h

(rf:imp, X:VAR(1)) 22.47 29.27 43.85 49.47 63.17 19.17 25.08 36.78 41.04 55.63

MK=3Tfo,h

(rf:imp, X:VAR(3)) 20.75 28.04 43.31 44.57 46.45 17.81 24.16 37.96 37.86 39.66

MK=3Tfo,h

(rf:imp, X:VAR(6)) 14.16 25.82 53.69 70.73 86.72 11.68 22.15 45.33 56.12 67.65

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 63.81 63.06 50.72 33.99 130.72 58.41 57.34 42.36 28.43 118.65

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 72.21 75.55 67.58 48.11 249.8 65.65 67.14 56.55 36.63 194.79

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 83.22 88.87 93.49 71.75 294.16 60.17 66.62 72.97 54.61 221.8

MK=3Tfo,h

(rf:per, X:AR(1)) 11.71 14.89 24.35 28.04 40.4 9.01 13.1 20.56 23.06 30.62

MK=3Tfo,h

(rf:per, X:AR(3)) 14.4 15.89 25.6 30.59 41.46 10.59 14.19 21.42 26.22 33.8

MK=3Tfo,h

(rf:per, X:AR(6)) 16.43 17.02 28.21 33.37 44.33 11.74 15.29 23.23 27.74 35

MK=3Tfo,h

(rf:per, X:VAR(1)) 15.1 17.54 31.47 36.37 49.98 9.69 15.26 27.31 29.58 42.69

MK=3Tfo,h

(rf:per, X:VAR(3)) 14.21 17.91 34.72 36.97 37.71 9.89 15.74 30.26 31.25 31.68

MK=3Tfo,h

(rf:per, X:VAR(6)) 12.79 17.21 44.64 62.92 79.49 9.76 14.75 37.96 49.68 59.8

MK=3Tfo,h

(rf:per, X:BVAR(1)) 51.41 48.84 36.44 43.29 153.92 42.59 41.57 29.15 37.49 145.54

MK=3Tfo,h

(rf:per, X:BVAR(3)) 58.86 61.36 53.2 52.56 268.92 51.08 52.92 43.73 40.87 222.13

MK=3Tfo,h

(rf:per, X:BVAR(6)) 76.27 80.99 84.98 65.61 302.88 50.98 56.74 63.92 47.47 231.22





Bper 24.66 37.62 44.05 41.61 56.64 20.89 31.52 37.78 36.16 45.62Bimp 25.49 40.69 65.57 91.73 135.34 21.31 36.31 55.16 74.84 117.52

MK=3Tfo,h

(rf:imp, X:AR(1)) 20.08 34.34 42.47 47.33 70.16 16.53 28.43 37.56 42.78 56.02

MK=3Tfo,h

(rf:imp, X:AR(3)) 19.74 32.64 42.58 49.24 70.5 15.64 26.98 36.41 45.94 59.85

MK=3Tfo,h

(rf:imp, X:AR(6)) 20.03 33.64 44.27 48.24 69.79 15.7 27.87 38.63 44.46 57.92

MK=3Tfo,h

(rf:imp, X:VAR(1)) 26.3 39.31 57.61 62.54 83.25 21.95 32.54 49.36 53.58 74.09

MK=3Tfo,h

(rf:imp, X:VAR(3)) 24.4 37.77 57.47 53.33 53.11 19.85 32.27 49.1 45.02 46.46

MK=3Tfo,h

(rf:imp, X:VAR(6)) 22.74 40.68 66.14 73.84 81.33 19.71 34.13 54.36 57.25 66.41

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 59.63 62.09 58.4 39 130.28 52.45 54.75 49.88 31.67 107.76

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 63.52 68.25 69.64 50.65 238.09 54.98 58.56 57.5 40.69 169.19

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 69.34 79.91 86.57 86.84 411.41 53 61.26 70.7 64.77 286.1

MK=3Tfo,h

(rf:per, X:AR(1)) 18.53 30.81 37.95 44.92 66.65 12.62 23.05 32.04 41.59 53.57

MK=3Tfo,h

(rf:per, X:AR(3)) 18.43 29.8 40.26 50.44 70.46 12.79 22.14 31.85 45.68 60.38

MK=3Tfo,h

(rf:per, X:AR(6)) 18.75 30.02 40.03 46.6 67.15 13.5 22.56 33.28 42.21 55.99

MK=3Tfo,h

(rf:per, X:VAR(1)) 19.7 30.06 47.68 52.69 73.4 14.98 24.91 41.64 44.37 64.23

MK=3Tfo,h

(rf:per, X:VAR(3)) 19.42 31.7 52.57 49.55 46.77 15.07 25.66 44 41.02 40.43

MK=3Tfo,h

(rf:per, X:VAR(6)) 21.11 36.58 61.44 70.13 76.05 15.94 29.14 49.76 52.93 60.05

MK=3Tfo,h

(rf:per, X:BVAR(1)) 49.51 50.4 46.98 42.47 147.13 40.97 44.21 40.44 33.47 128.42

MK=3Tfo,h

(rf:per, X:BVAR(3)) 53 57.1 59.46 51.74 249.76 44.8 48.93 49.36 41.64 190.98

MK=3Tfo,h

(rf:per, X:BVAR(6)) 63.97 74.27 80.96 87.5 421.58 45.04 54.5 64.82 64.7 295.86




Bper 28.67 43.96 46.92 53.25 79.02 23.42 33.25 41.17 49.29 66.28Bimp 32.6 43.72 60.2 86.55 130.5 24.67 37.88 51.92 65.59 107.51

MK=3Tfo,h

(rf:imp, X:AR(1)) 24.14 39.2 43.74 52.49 81.4 18.92 28.87 37.66 48.26 66.43

MK=3Tfo,h

(rf:imp, X:AR(3)) 23.17 37.33 44.35 55.07 81.93 17.97 27.21 35.65 51.18 71.47

MK=3Tfo,h

(rf:imp, X:AR(6)) 23.65 39 44.83 52.15 79.87 17.94 29.01 37.59 48.09 68

MK=3Tfo,h

(rf:imp, X:VAR(1)) 27.42 40.59 55.46 64 91.41 21.45 33.38 47.9 53.58 79.57

MK=3Tfo,h

(rf:imp, X:VAR(3)) 26.37 40.39 56.4 53.39 59.42 20.15 32.72 45.59 46.75 53.08

MK=3Tfo,h

(rf:imp, X:VAR(6)) 27.84 46.79 63.87 67.86 75.05 22.89 36.55 50.25 53.4 65.04

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 51.83 53.89 52.62 43.1 129.45 44.04 47.18 45.03 36.95 98.42

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 51.71 55.76 59.9 49.6 220.89 43.93 46.98 48.91 41.38 149.73

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 52.74 66.06 72.13 97.93 466.19 41.89 51.27 59.26 72.03 304.86

MK=3Tfo,h

(rf:per, X:AR(1)) 23.88 37.73 41.63 52.09 79.22 17.45 26.34 33.58 48.01 65.89

MK=3Tfo,h

(rf:per, X:AR(3)) 22.99 36.49 44.19 57.56 82.28 16.65 25.06 32.31 51.68 72.34

MK=3Tfo,h

(rf:per, X:AR(6)) 23.34 37.48 43.19 52.85 78.46 16.96 26.3 33.05 47.53 67.74

MK=3Tfo,h

(rf:per, X:VAR(1)) 23.48 34.63 48.31 57.45 84.88 17.38 27.37 41.86 47.84 72.85

MK=3Tfo,h

(rf:per, X:VAR(3)) 23.55 36.9 53.38 51.43 55.04 17.97 28.36 42.6 44.77 48.55

MK=3Tfo,h

(rf:per, X:VAR(6)) 27.61 44.96 61.13 66.19 71.1 20.63 33.61 46.32 51.39 60.73

MK=3Tfo,h

(rf:per, X:BVAR(1)) 44.71 45.73 44.74 45.84 141.49 35.54 39.82 38.42 37.24 113.72

MK=3Tfo,h

(rf:per, X:BVAR(3)) 44.93 48.62 53.76 51.47 228.98 37.01 41.08 44.26 41.18 165.65

MK=3Tfo,h

(rf:per, X:BVAR(6)) 49.44 63.03 69.34 102.89 479.5 37.57 48.48 55.66 74.23 317.27





Bper 32.4 47.84 46.59 64.19 95.17 25.75 32.69 36.86 53.55 76.74Bimp 38.34 43.84 50.78 78.06 117.18 27.66 33.57 43.55 60.95 93.31

MK=3Tfo,h

(rf:imp, X:AR(1)) 30.04 42.39 43.27 60.57 93.43 23.15 29.5 34.22 52.34 79.03

MK=3Tfo,h

(rf:imp, X:AR(3)) 29.03 41.16 44.25 62.62 93.02 22.41 27.99 32.1 55.1 82.8

MK=3Tfo,h

(rf:imp, X:AR(6)) 29.76 43.49 44.1 60.09 91.35 22.53 30.07 33.37 52.1 80.11

MK=3Tfo,h

(rf:imp, X:VAR(1)) 30.86 41.21 48.59 67.19 101.65 23.79 31.83 42.09 53.34 83

MK=3Tfo,h

(rf:imp, X:VAR(3)) 30.76 42.73 50.27 57.2 75.34 23.98 31.61 40.08 50.13 65.41

MK=3Tfo,h

(rf:imp, X:VAR(6)) 34.38 51.84 57.32 63.89 79.46 27.48 37.51 43.47 52.45 70.17

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 45.26 44.62 43.73 52.96 126.29 34.54 36.91 36.21 45.14 100.11

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 42.22 44.07 48.38 55.19 189.94 33.03 35.75 38.48 47.06 130.92

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 43.97 58.06 64.45 115.01 500.05 34.03 44.92 48.61 84.42 310

MK=3Tfo,h

(rf:per, X:AR(1)) 30.55 42.44 42.98 61.34 92.27 22.09 29.53 32.66 53.49 79.61

MK=3Tfo,h

(rf:per, X:AR(3)) 29.32 41.62 45.56 65.38 93.25 21.64 28.99 32.83 57.49 84.07

MK=3Tfo,h

(rf:per, X:AR(6)) 29.97 43.54 44.62 62.04 90.49 22.11 30.66 32.71 54.27 80.32

MK=3Tfo,h

(rf:per, X:VAR(1)) 30.01 39.2 45.03 64.62 98.97 22.69 29.33 39.09 50.62 79.75

MK=3Tfo,h

(rf:per, X:VAR(3)) 30.11 41.74 48.9 56.74 73.15 23.9 30.39 37.94 50.05 63.27

MK=3Tfo,h

(rf:per, X:VAR(6)) 36.11 52.51 56.81 64.35 77.29 26.42 37.2 41.3 51.97 67.64

MK=3Tfo,h

(rf:per, X:BVAR(1)) 41.84 41.12 40.15 55.37 133.74 30.21 33.57 32.92 47.16 108.13

MK=3Tfo,h

(rf:per, X:BVAR(3)) 40.26 42.36 46.75 57.38 194.98 31.08 33.46 36.86 48.17 140.74

MK=3Tfo,h

(rf:per, X:BVAR(6)) 44.59 59.17 65.7 123.02 516.38 34.21 45.25 48.7 91.1 324.75





Bper 20.74 22.83 24.83 23.99 29.52 17.46 19.3 21.05 20.05 23.39Bimp 10.35 14.77 28.56 51.55 91.99 8.25 12.54 22.65 43.16 82.1

MK=5Tfo,h

(rf:imp, X:AR(1)) 13.83 20.17 28.22 32.35 41.47 11.48 17.08 23.22 26.53 34.2

MK=5Tfo,h

(rf:imp, X:AR(3)) 15.75 21.01 27.96 30.56 37.51 12.88 17.43 21.87 23.62 28.85

MK=5Tfo,h

(rf:imp, X:AR(6)) 16.87 22.35 32.7 39.41 47.19 12.58 18.08 26.01 31.26 37.86

MK=5Tfo,h

(rf:imp, X:VAR(1)) 16.18 16.1 24.74 31.51 41.6 13.24 13.16 19.55 24.77 33.18

MK=5Tfo,h

(rf:imp, X:VAR(3)) 20.88 18.63 25.12 31.9 37.05 17.66 15.75 19.59 25.77 30.37

MK=5Tfo,h

(rf:imp, X:VAR(6)) 19.08 20.15 39.28 55.04 67.75 15.65 16.67 30.06 45.81 58.27

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 67.84 62.21 38.97 41.58 166.02 60.32 54.75 30.91 35.86 158.24

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 76.88 76.67 64.68 59.96 222.22 66.98 66.62 53.89 42.5 195.99

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 101.75 101.81 103.66 96.49 324.97 87.93 86.55 83.66 81.15 265.32

MK=5Tfo,h

(rf:per, X:AR(1)) 12.18 13.25 18.1 21.03 31.01 6.02 8.47 14.2 16.24 23.19

MK=5Tfo,h

(rf:per, X:AR(3)) 15.54 14.72 18.38 20.87 29.25 9.16 10.21 14.4 15.93 21.9

MK=5Tfo,h

(rf:per, X:AR(6)) 19.05 18.47 22.68 28.52 37.06 11.69 12.81 17.38 21.26 27.9

MK=5Tfo,h

(rf:per, X:VAR(1)) 16.38 13.57 17.93 22.01 30.92 8.19 8.58 14.45 17.24 23.61

MK=5Tfo,h

(rf:per, X:VAR(3)) 19.27 18.49 22.17 29.02 34.21 11.22 12.82 19.26 25.05 27.31

MK=5Tfo,h

(rf:per, X:VAR(6)) 19.5 19.38 33.4 49.31 62.86 15.65 14.53 25.01 41.55 54.14

MK=5Tfo,h

(rf:per, X:BVAR(1)) 59.12 52.62 29.88 55.14 190.15 47.62 41.8 20.16 51.11 183.47

MK=5Tfo,h

(rf:per, X:BVAR(3)) 68.26 67.62 55.39 65.9 252.66 55.5 55.44 44.16 50.54 218.64

MK=5Tfo,h

(rf:per, X:BVAR(6)) 96.21 96.04 98.05 94.67 328.71 80.61 78.37 77.15 78.13 275.62




Bper 21.85 27.92 33.7 30.85 34.54 18.52 23.2 28.04 24.38 27.4Bimp 14.69 25.51 49.75 76.17 118.21 12.36 20.7 39.96 66.76 106.82

MK=5Tfo,h

(rf:imp, X:AR(1)) 13.71 23.81 34.79 39.27 50.68 11.85 20.72 28.72 31.47 39.33

MK=5Tfo,h

(rf:imp, X:AR(3)) 13.92 22.58 33.7 38.42 48.31 12.05 19.3 28.15 31.27 37.03

MK=5Tfo,h

(rf:imp, X:AR(6)) 13.97 22.8 37.31 43.26 53.39 11.99 19.39 30.41 34.65 41.37

MK=5Tfo,h

(rf:imp, X:VAR(1)) 20.2 28.28 43.53 48.18 59.46 17.73 23.47 35.59 38.61 50.69

MK=5Tfo,h

(rf:imp, X:VAR(3)) 21.25 24.33 41.37 44.08 44.48 17.92 20.21 35.41 36.92 37.09

MK=5Tfo,h

(rf:imp, X:VAR(6)) 22.39 31.46 62.85 81.53 101.12 18.15 26.22 53.28 68.85 81.51

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 69.71 68.59 53.27 39.15 162.03 63.07 61.45 44.38 33.76 149.27

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 82.33 83.61 78.13 70.65 224.59 71.33 70.84 64.7 50.92 195.15

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 106.07 109.69 114.68 117.25 657.3 92.71 94.92 95.11 96.34 468.72

MK=5Tfo,h

(rf:per, X:AR(1)) 11.41 16.32 26.59 31.21 42.34 9.4 13.75 22.14 25.6 31.45

MK=5Tfo,h

(rf:per, X:AR(3)) 13.08 16.01 27.25 33.62 43.52 10.27 13.69 22.94 28.25 34.39

MK=5Tfo,h

(rf:per, X:AR(6)) 14.88 16.45 29.27 35.42 45.44 11.51 14.22 24.22 28.93 34.48

MK=5Tfo,h

(rf:per, X:VAR(1)) 15.82 19.26 34.34 37.88 48.44 10.95 16.75 29.64 29.7 40.16

MK=5Tfo,h

(rf:per, X:VAR(3)) 17.49 18.96 37.3 41.38 41.08 11.1 15.89 31.16 35.15 33.17

MK=5Tfo,h

(rf:per, X:VAR(6)) 19.15 23.88 57.46 77.69 98.11 13.87 19.98 47.89 65.14 78.02

MK=5Tfo,h

(rf:per, X:BVAR(1)) 60.53 57.84 42.14 47.38 183.58 50.8 49.16 33.7 41.69 173.82

MK=5Tfo,h

(rf:per, X:BVAR(3)) 73.66 74.24 68.79 71.43 248.51 60.37 60.02 55.97 50.65 216.65

MK=5Tfo,h

(rf:per, X:BVAR(6)) 100.63 103.75 108.99 114.36 668.2 85.13 88.24 90 92.81 486.52





Bper 24.66 37.62 44.05 41.61 56.64 20.89 31.52 37.78 36.16 45.62Bimp 25.49 40.69 65.57 91.73 135.34 21.31 36.31 55.16 74.84 117.52

MK=5Tfo,h

(rf:imp, X:AR(1)) 20.49 35.83 43.35 49.48 72.67 16.57 28.94 37.96 44.02 57.05

MK=5Tfo,h

(rf:imp, X:AR(3)) 19.19 33.74 43.27 51.44 73.24 15.27 27.25 36.43 47.31 60.83

MK=5Tfo,h

(rf:imp, X:AR(6)) 19.04 34.32 44.39 49.88 72.07 14.98 27.81 38.56 45.3 58.53

MK=5Tfo,h

(rf:imp, X:VAR(1)) 24.92 38.51 55.31 58.97 78.47 20.57 32.14 47.39 48.97 68.7

MK=5Tfo,h

(rf:imp, X:VAR(3)) 23.69 35.25 57.51 57.23 61.77 18.2 29.52 49.04 49.76 51.49

MK=5Tfo,h

(rf:imp, X:VAR(6)) 27.54 45.31 74.82 87.63 105.59 20.46 39.36 64.97 74.04 90.19

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 63.61 65.03 57.74 43.27 160.21 55.77 56.27 48.78 34.2 134.13

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 72.88 75.07 76.51 81.18 226.37 62.56 64.25 66.66 58.86 188.15

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 85.67 92.64 100.69 131.17 919.65 72.22 82.3 86.36 103.38 667.2

MK=5Tfo,h

(rf:per, X:AR(1)) 19.04 31.82 38.97 46.45 67.92 13.44 23.75 33.11 42.91 54.34

MK=5Tfo,h

(rf:per, X:AR(3)) 18.47 30.49 40.92 51.56 71.43 13.12 22.71 32.82 46.69 60.8

MK=5Tfo,h

(rf:per, X:AR(6)) 18.59 30.48 40.48 47.6 67.98 13.36 22.87 33.79 43.13 56.36

MK=5Tfo,h

(rf:per, X:VAR(1)) 20.97 31.39 48.37 51.91 70.38 15.62 25.93 42.46 43.77 60.33

MK=5Tfo,h

(rf:per, X:VAR(3)) 21.38 31.14 55.19 56.79 59.33 15.86 24.76 46.18 48.83 47

MK=5Tfo,h

(rf:per, X:VAR(6)) 24.7 40.27 71.8 86.51 104.07 18.8 33.84 62.49 71.67 87.06

MK=5Tfo,h

(rf:per, X:BVAR(1)) 56.68 56.57 49.52 48.18 175.6 47.32 48.74 42.18 38.32 153.84

MK=5Tfo,h

(rf:per, X:BVAR(3)) 66.69 68.16 70.34 81.09 242.06 54.09 57.21 60.5 58.05 204.26

MK=5Tfo,h

(rf:per, X:BVAR(6)) 81.39 87.68 96.31 131.28 945.75 66.17 77.48 81.39 103.71 690.19




Bper 28.67 43.96 46.92 53.25 79.02 23.42 33.25 41.17 49.29 66.28Bimp 32.6 43.72 60.2 86.55 130.5 24.67 37.88 51.92 65.59 107.51

MK=5Tfo,h

(rf:imp, X:AR(1)) 25.62 41.3 45.11 55.16 85.59 19.47 29.94 37.72 50.82 70.83

MK=5Tfo,h

(rf:imp, X:AR(3)) 24.32 39.72 46.43 58.92 87.4 18.73 28.67 36.04 54.06 76.45

MK=5Tfo,h

(rf:imp, X:AR(6)) 24.58 40.97 46.4 55.56 84.83 18.89 29.84 37.8 50.79 72.8

MK=5Tfo,h

(rf:imp, X:VAR(1)) 26.91 39.94 52.21 59.39 86.99 20.52 31.75 45.44 50.54 73.38

MK=5Tfo,h

(rf:imp, X:VAR(3)) 26.32 39.1 58.05 62.33 76.36 20.65 30.19 47.3 55.95 66.03

MK=5Tfo,h

(rf:imp, X:VAR(6)) 30.57 48.92 70.58 79.02 97.08 23.65 38.19 59.78 66.93 87.21

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 54.14 54.67 50.21 51.16 163.85 45.41 46.61 42.71 42.64 127.11

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 59.21 59.84 63.02 81.7 220.56 48.8 51.83 55.67 63.04 183.42

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 63.43 72.33 82 141.31 952.39 51.49 63.24 68.13 107.21 701.12

MK=5Tfo,h

(rf:per, X:AR(1)) 25.54 39.22 42.94 54.5 82.46 18.65 27.34 33.83 50.21 69.94

MK=5Tfo,h

(rf:per, X:AR(3)) 24.63 38.36 46.15 60.8 86.51 17.87 26.32 33.01 54.36 76.69

MK=5Tfo,h

(rf:per, X:AR(6)) 24.95 39.13 44.96 55.93 82.39 18.36 27.42 33.66 50.27 71.93

MK=5Tfo,h

(rf:per, X:VAR(1)) 25.38 35.47 47.42 55.08 81.49 19.17 27.33 41.4 47.78 67.36

MK=5Tfo,h

(rf:per, X:VAR(3)) 26.29 37.17 57.14 63.19 74.85 21.23 27.49 44.21 55.67 64.23

MK=5Tfo,h

(rf:per, X:VAR(6)) 29.87 46.34 68.88 79.23 95.99 24.03 34.64 57.01 65.33 85.41

MK=5Tfo,h

(rf:per, X:BVAR(1)) 49.79 48.97 45.11 55.97 175.11 39.36 41.69 39.24 44.88 141.83

MK=5Tfo,h

(rf:per, X:BVAR(3)) 55.63 55.7 59.77 83.26 233.13 43.8 47.24 51.72 62.72 195.06

MK=5Tfo,h

(rf:per, X:BVAR(6)) 60.82 68.84 79.11 143 985.6 47.48 60.17 65.85 110.74 725.89





Bper 32.4 47.84 46.59 64.19 95.17 25.75 32.69 36.86 53.55 76.74Bimp 38.34 43.84 50.78 78.06 117.18 27.66 33.57 43.55 60.95 93.31

MK=5Tfo,h

(rf:imp, X:AR(1)) 31.02 44.9 46.08 63.22 95.58 24.17 31.12 36.92 54.59 78.91

MK=5Tfo,h

(rf:imp, X:AR(3)) 29.51 42.95 46.19 64.36 94.45 23.09 29.31 34.51 57.03 82.64

MK=5Tfo,h

(rf:imp, X:AR(6)) 30.11 44.93 45.2 61.13 92.45 23.35 30.7 34.04 53.25 79.52

MK=5Tfo,h

(rf:imp, X:VAR(1)) 30.63 42.84 49.29 67.55 102.75 24.11 32.54 43.15 53.11 84.68

MK=5Tfo,h

(rf:imp, X:VAR(3)) 29.51 42.18 51.88 61.93 86.59 24.2 30.6 41.62 56.16 75.75

MK=5Tfo,h

(rf:imp, X:VAR(6)) 34.57 54.27 64.93 69.05 94.61 28.07 37.15 53.05 60.81 84.12

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 43.42 44.36 41.47 59.62 156.52 33.86 37.43 35.84 51.98 117.62

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 43.76 43.16 43.76 75.25 211.35 33.3 35.01 37.04 59.57 167.28

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 44.02 59.46 69.97 161.02 924.38 34.03 41.6 51.59 115.97 672.46

MK=5Tfo,h

(rf:per, X:AR(1)) 30.72 43.35 44.54 62.73 93.06 22.29 30.61 35.05 54.67 78.56

MK=5Tfo,h

(rf:per, X:AR(3)) 29.34 41.9 46.19 65.67 93.21 21.36 29.15 34.25 57.73 83

MK=5Tfo,h

(rf:per, X:AR(6)) 29.95 43.58 44.52 61.65 90.27 21.59 30.6 32.91 53.73 78.89

MK=5Tfo,h

(rf:per, X:VAR(1)) 29.92 40.1 46.07 65.21 99.86 22.89 29.54 40.7 51.12 81.19

MK=5Tfo,h

(rf:per, X:VAR(3)) 29.75 40.86 51.01 62.09 84.94 24.61 29.99 39.62 55.65 74.4

MK=5Tfo,h

(rf:per, X:VAR(6)) 34.46 52.67 63.13 68.36 92.64 27.75 36.3 50.76 59.51 81.93

MK=5Tfo,h

(rf:per, X:BVAR(1)) 41.61 41.69 38.75 61.62 162.47 30.91 34.22 33.34 53.92 124.63

MK=5Tfo,h

(rf:per, X:BVAR(3)) 43 42.2 42.94 77.08 220.11 30.97 33.3 35.44 59.7 175.76

MK=5Tfo,h

(rf:per, X:BVAR(6)) 44.32 58.59 68.9 163.06 962.1 33.53 40.81 51.95 119.45 703.28





Bper 20.74 22.83 24.83 23.99 29.52 17.46 19.3 21.05 20.05 23.39Bimp 10.35 14.77 28.56 51.55 91.99 8.25 12.54 22.65 43.16 82.1

MK=7Tfo,h

(rf:imp, X:AR(1)) 13.32 19.75 28.14 32.57 41.93 11.14 16.49 22.83 26.81 34.47

MK=7Tfo,h

(rf:imp, X:AR(3)) 15.12 20.51 27.88 30.82 38.08 12.55 17.1 21.78 23.94 29.37

MK=7Tfo,h

(rf:imp, X:AR(6)) 16.04 21.81 32.8 39.76 47.73 12.24 17.64 25.93 31.73 38.48

MK=7Tfo,h

(rf:imp, X:VAR(1)) 17.77 17.21 25.01 31.82 41.7 14.99 14.08 19.78 25.37 33.62

MK=7Tfo,h

(rf:imp, X:VAR(3)) 21.36 19.73 26.99 34.66 43.66 18.22 16.91 21.52 28.8 37.57

MK=7Tfo,h

(rf:imp, X:VAR(6)) 22.81 25.55 55.61 64.14 72.28 17.37 21.1 48.55 55.36 62.25

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 70.78 63.84 37.75 45.59 173.24 64.26 57.51 30.13 40.2 166.52

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 75.53 75.73 67.95 73.18 260.98 66.25 65.2 55.77 47.39 209.83

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 109.71 114.04 121.76 150.17 346.8 94.94 100.97 104.64 113.11 278.61

MK=7Tfo,h

(rf:per, X:AR(1)) 11.89 12.85 18.02 21.05 31.15 6.07 8.42 14.09 16.26 23.16

MK=7Tfo,h

(rf:per, X:AR(3)) 15.23 14.32 18.31 20.97 29.54 9.05 10.1 14.37 16.05 22.26

MK=7Tfo,h

(rf:per, X:AR(6)) 18.71 18.08 22.65 28.59 37.26 11.59 12.69 17.29 21.28 28.05

MK=7Tfo,h

(rf:per, X:VAR(1)) 17.15 13.84 18.01 22.25 30.74 8.25 8.59 14.48 17.18 23.23

MK=7Tfo,h

(rf:per, X:VAR(3)) 20.02 16.34 21.24 29.28 38.55 11.46 9.75 17.47 24.08 31.63

MK=7Tfo,h

(rf:per, X:VAR(6)) 24.55 22.39 47.94 58 68.04 19.95 16.25 42.78 50.52 58.3

MK=7Tfo,h

(rf:per, X:BVAR(1)) 62.38 54.86 30.65 60.11 195.21 51.36 44.48 19.45 56.22 191.21

MK=7Tfo,h

(rf:per, X:BVAR(3)) 67.88 68 60.44 75.62 279.25 55.36 54.78 45.93 56.94 237.62

MK=7Tfo,h

(rf:per, X:BVAR(6)) 107.37 111.07 117.97 139.7 345.35 91.96 97 100.05 109.96 286.28




Bper 21.85 27.92 33.7 30.85 34.54 18.52 23.2 28.04 24.38 27.4Bimp 14.69 25.51 49.75 76.17 118.21 12.36 20.7 39.96 66.76 106.82

MK=7Tfo,h

(rf:imp, X:AR(1)) 14.12 24.32 35.32 40.22 51.87 11.97 21.04 29.18 32.04 40.1

MK=7Tfo,h

(rf:imp, X:AR(3)) 14.04 22.96 34.15 39.34 49.46 11.96 19.55 28.59 31.79 37.58

MK=7Tfo,h

(rf:imp, X:AR(6)) 13.87 23.12 37.83 44.19 54.53 11.69 19.67 30.89 35.17 42.12

MK=7Tfo,h

(rf:imp, X:VAR(1)) 20.85 28.14 42.62 47.78 59.17 17.99 22.85 34.51 37.25 51.2

MK=7Tfo,h

(rf:imp, X:VAR(3)) 22.29 28.73 47.03 49.98 56.72 18.71 23.85 41.35 42.69 50.64

MK=7Tfo,h

(rf:imp, X:VAR(6)) 27.01 41.46 81.44 91.96 109.18 20.43 33.06 72.06 78.54 88.71

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 71.89 68.36 50.29 44.06 172.62 66.12 62.57 41.46 37.96 160.13

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 81.07 82.9 82.33 85.67 314.5 70.11 71.28 68.56 52.65 238.62

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 116.68 121.83 133.85 185.95 816.2 102.69 109.92 115.02 137.79 649.97

MK=7Tfo,h

(rf:per, X:AR(1)) 11.4 16.65 27 31.72 42.88 9.46 14.04 22.53 26.01 31.82

MK=7Tfo,h

(rf:per, X:AR(3)) 12.97 16.25 27.61 34.06 43.97 10.29 13.91 23.31 28.63 34.7

MK=7Tfo,h

(rf:per, X:AR(6)) 14.68 16.64 29.66 35.89 45.91 11.48 14.32 24.6 29.29 34.77

MK=7Tfo,h

(rf:per, X:VAR(1)) 16.48 19.13 33.53 37.4 47.77 11.42 16.57 28.52 28.45 39.86

MK=7Tfo,h

(rf:per, X:VAR(3)) 17.71 21.24 41.42 44.92 51.05 10.26 18.52 36.25 38.23 43.8

MK=7Tfo,h

(rf:per, X:VAR(6)) 24.94 34.42 76.12 88.24 106.79 17.28 28.01 67.79 74.6 85.45

MK=7Tfo,h

(rf:per, X:BVAR(1)) 63.12 58.16 40.47 53.58 191.4 53.65 50.18 31.44 47.86 184.12

MK=7Tfo,h

(rf:per, X:BVAR(3)) 73.25 74.62 74.74 84.7 338.68 59.61 61.15 61.32 52.67 267.39

MK=7Tfo,h

(rf:per, X:BVAR(6)) 114.18 118.72 130.39 178.93 841.45 99.24 105.84 110.8 132.6 675.45





Bper 24.66 37.62 44.05 41.61 56.64 20.89 31.52 37.78 36.16 45.62Bimp 25.49 40.69 65.57 91.73 135.34 21.31 36.31 55.16 74.84 117.52

MK=7Tfo,h

(rf:imp, X:AR(1)) 20.94 36.26 43.43 49.92 73.42 16.82 29.05 38.08 44.23 57.44

MK=7Tfo,h

(rf:imp, X:AR(3)) 19.55 34.1 43.25 51.82 73.99 15.51 27.24 36.54 47.5 61.15

MK=7Tfo,h

(rf:imp, X:AR(6)) 19.38 34.7 44.4 50.29 72.86 15.27 27.84 38.69 45.53 58.86

MK=7Tfo,h

(rf:imp, X:VAR(1)) 24.8 36.65 51.9 56.69 77.11 19.64 30.25 43.05 46.73 68.52

MK=7Tfo,h

(rf:imp, X:VAR(3)) 26.44 40.42 64.95 64.52 76.55 20.25 35.5 56.17 55.39 68.19

MK=7Tfo,h

(rf:imp, X:VAR(6)) 30.13 53.27 86.59 91.04 113.52 22.59 42.48 75.51 78.92 95.96

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 64.63 62.5 52.63 47.83 172.37 57.01 54.63 43.51 38.45 147.49

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 72.7 75.68 81.91 101.86 363.48 61 63.34 72.16 67.95 248.65

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 100.6 101.81 114.78 206.3 1146.01 85.19 85.33 91.86 154.21 905.59

MK=7Tfo,h

(rf:per, X:AR(1)) 19.07 31.97 39.05 46.51 68.11 13.4 23.91 33.2 42.99 54.57

MK=7Tfo,h

(rf:per, X:AR(3)) 18.48 30.63 41 51.62 71.63 13.06 22.79 32.86 46.77 61.05

MK=7Tfo,h

(rf:per, X:AR(6)) 18.62 30.65 40.57 47.65 68.18 13.35 23.01 33.88 43.19 56.58

MK=7Tfo,h

(rf:per, X:VAR(1)) 21.24 29.4 44.99 49.51 68.55 15.61 23.44 38.62 40.9 59.87

MK=7Tfo,h

(rf:per, X:VAR(3)) 22.84 35.48 62.1 62.49 72.55 17.04 29.36 52.52 52.03 62.84

MK=7Tfo,h

(rf:per, X:VAR(6)) 28.01 48.87 84.2 90.01 112.32 22 37.47 73.26 77.01 93.03

MK=7Tfo,h

(rf:per, X:BVAR(1)) 58.24 54.43 45.54 54.13 184.9 48.1 47.05 37.48 44.93 167.07

MK=7Tfo,h

(rf:per, X:BVAR(3)) 66.69 69.39 76.87 101.63 395.13 53.31 55.99 67.13 66.56 272.1

MK=7Tfo,h

(rf:per, X:BVAR(6)) 98.85 99.29 112.21 204.93 1192.84 82.99 81.81 89.42 152.87 943.67




Bper 28.67 43.96 46.92 53.25 79.02 23.42 33.25 41.17 49.29 66.28Bimp 32.6 43.72 60.2 86.55 130.5 24.67 37.88 51.92 65.59 107.51

MK=7Tfo,h

(rf:imp, X:AR(1)) 26.15 41.89 45.4 55.74 86.37 19.81 30.17 38.02 51.21 71.04

MK=7Tfo,h

(rf:imp, X:AR(3)) 24.77 40.25 46.59 59.38 88.1 19.09 28.83 36.32 54.42 76.67

MK=7Tfo,h

(rf:imp, X:AR(6)) 24.98 41.51 46.56 56.04 85.57 19.26 30.09 38.11 51.16 73

MK=7Tfo,h

(rf:imp, X:VAR(1)) 27.27 38.58 48.46 56.66 85.65 21.14 30.51 40.88 48.27 73.48

MK=7Tfo,h

(rf:imp, X:VAR(3)) 28.55 42.03 64.96 69.44 89.9 22.18 33.16 53.6 59.05 79.93

MK=7Tfo,h

(rf:imp, X:VAR(6)) 31.68 55.23 77.72 76.7 101.01 24.68 41.87 65.21 65.92 88.98

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 55.27 52.05 45.57 54.36 173.49 45.93 44.82 36.7 41.54 138.6

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 58.82 60.46 68.33 107.54 343.36 47.2 48.94 59.12 74.81 244.14

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 79.84 77.34 89.64 204 1229.85 65.35 62.48 70.62 155.73 953

MK=7Tfo,h

(rf:per, X:AR(1)) 25.62 39.44 43.19 54.77 82.78 18.68 27.57 34.13 50.5 70.1

MK=7Tfo,h

(rf:per, X:AR(3)) 24.71 38.57 46.36 61.03 86.78 17.92 26.54 33.35 54.65 76.89

MK=7Tfo,h

(rf:per, X:AR(6)) 25.01 39.36 45.18 56.18 82.69 18.41 27.63 34.01 50.56 72.13

MK=7Tfo,h

(rf:per, X:VAR(1)) 26 33.93 43.56 52.2 79.64 20.5 26.53 36.6 44.89 67.18

MK=7Tfo,h

(rf:per, X:VAR(3)) 27.5 39.57 63.82 69.21 87.28 22 29.87 50.15 57.78 76

MK=7Tfo,h

(rf:per, X:VAR(6)) 31.22 52.86 76.7 77.09 100.14 25.43 39.83 63.14 63.91 87.62

MK=7Tfo,h

(rf:per, X:BVAR(1)) 51.47 46.7 41.56 59.95 181.93 39.94 39.66 33.77 46.57 153.57

MK=7Tfo,h

(rf:per, X:BVAR(3)) 55.08 56.68 65.9 108.57 367.27 42.16 45.29 56.6 75.59 261.7

MK=7Tfo,h

(rf:per, X:BVAR(6)) 79.09 75.69 87.83 204.97 1277.67 64.4 60.84 69.32 156.34 992.67





Bper 32.4 47.84 46.59 64.19 95.17 25.75 32.69 36.86 53.55 76.74Bimp 38.34 43.84 50.78 78.06 117.18 27.66 33.57 43.55 60.95 93.31

MK=7Tfo,h

(rf:imp, X:AR(1)) 31.58 45.69 46.79 63.97 96.41 24.5 31.73 37.45 55.21 79.18

MK=7Tfo,h

(rf:imp, X:AR(3)) 30 43.67 46.75 64.98 95.16 23.44 29.87 35.01 57.49 82.89

MK=7Tfo,h

(rf:imp, X:AR(6)) 30.54 45.62 45.68 61.71 93.15 23.63 31.25 34.45 53.7 79.71

MK=7Tfo,h

(rf:imp, X:VAR(1)) 30.83 42.02 46.39 65 101.13 23.88 31.96 39.81 51.78 85.05

MK=7Tfo,h

(rf:imp, X:VAR(3)) 30.83 42.43 57.2 67.9 99.02 24.71 31.69 45.12 60.1 85.92

MK=7Tfo,h

(rf:imp, X:VAR(6)) 35.47 58.8 66.65 61.73 94.68 28.27 42.26 53.88 53.96 84.52

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 44.22 41.8 38.33 59.88 161.89 33.63 34.58 31.35 48.81 119.12

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 43.46 42.7 50.38 112.89 328.42 33.03 32.75 41.24 80.86 242.28

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 57.71 55.41 65.92 199.05 1277.72 47.65 46.82 54.91 156.02 1010.85

MK=7Tfo,h

(rf:per, X:AR(1)) 30.89 43.7 45.02 63.21 93.51 22.45 30.98 35.56 55.14 78.74

MK=7Tfo,h

(rf:per, X:AR(3)) 29.5 42.24 46.62 66.12 93.65 21.49 29.52 34.66 58.24 83.31

MK=7Tfo,h

(rf:per, X:AR(6)) 30.08 43.92 44.89 62.1 90.74 21.76 30.85 33.18 54.24 79.19

MK=7Tfo,h

(rf:per, X:VAR(1)) 30.17 38.87 42.69 62.32 97.68 23.6 28.94 36.83 49.48 81.22

MK=7Tfo,h

(rf:per, X:VAR(3)) 30.87 41.25 56.43 67.51 96.87 25.11 31.36 42.95 59.24 83.55

MK=7Tfo,h

(rf:per, X:VAR(6)) 35.46 57.03 65.26 61.17 92.86 28.55 41.21 52.26 51.87 82.22

MK=7Tfo,h

(rf:per, X:BVAR(1)) 42.82 39.19 35.96 61.74 164.73 30.65 31.54 29.04 51.99 126.43

MK=7Tfo,h

(rf:per, X:BVAR(3)) 42.48 41.76 50.36 113.12 333.26 31.07 31.24 39.82 81.7 249.74

MK=7Tfo,h

(rf:per, X:BVAR(6)) 58.63 55.23 64.91 200.84 1315.55 48 46.06 53.93 157.87 1033.19



RMSFE MAFEModel variety 3m 6m 12m 18m 24m 36m 60m 3m 6m 12m 18m 24m 36m 60m

Bper 62 74 102 128 150 189 235 50 61 84 108 131 173 223Bimp 20 34 74 117 164 256 357 15 27 59 100 147 241 337

MK=3Tfo,h

(rf:imp, X:AR(1)) 41 63 97 124 150 195 245 33 53 82 104 129 179 232

MK=3Tfo,h

(rf:imp, X:AR(3)) 52 71 103 127 150 192 240 41 60 87 109 130 176 225

MK=3Tfo,h

(rf:imp, X:AR(6)) 37 56 99 130 156 200 248 30 48 82 108 133 183 235

MK=3Tfo,h

(rf:imp, X:VAR(1)) 38 50 84 117 148 199 252 31 44 70 99 127 183 244

MK=3Tfo,h

(rf:imp, X:VAR(3)) 44 61 103 132 156 197 244 34 54 88 114 135 180 232

MK=3Tfo,h

(rf:imp, X:VAR(6)) 48 59 107 152 182 210 255 29 47 88 127 154 195 244

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 126 131 145 162 179 209 263 103 105 119 136 157 199 242

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 105 115 139 160 179 208 368 88 95 115 135 157 198 303

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 107 118 143 165 183 208 984 89 98 118 139 162 198 468

MK=3Tfo,h

(rf:per, X:AR(1)) 36 49 81 107 133 176 225 31 37 62 91 114 152 207

MK=3Tfo,h

(rf:per, X:AR(3)) 38 55 86 109 132 172 219 31 42 67 94 112 147 200

MK=3Tfo,h

(rf:per, X:AR(6)) 38 48 86 116 142 183 228 31 39 65 98 121 158 211

MK=3Tfo,h

(rf:per, X:VAR(1)) 38 49 76 104 133 181 232 31 40 60 91 115 159 221

MK=3Tfo,h

(rf:per, X:VAR(3)) 40 52 91 119 141 179 224 32 40 73 104 123 155 206

MK=3Tfo,h

(rf:per, X:VAR(6)) 51 57 97 138 166 192 235 39 44 74 113 137 170 220

MK=3Tfo,h

(rf:per, X:BVAR(1)) 112 116 131 148 164 185 278 89 92 109 130 149 169 222

MK=3Tfo,h

(rf:per, X:BVAR(3)) 86 98 126 149 166 186 357 69 79 106 131 150 171 277

MK=3Tfo,h

(rf:per, X:BVAR(6)) 89 103 130 153 170 185 936 69 82 109 135 155 169 443

Notes: (1) various horizons are reflected in different columns h = {3, 6, 12, 18, 24, 36, 60}. (2) all errors in bps


Table 52: Level forecast errors for 1-year tenor in different models with K = 3, 5-year window


Bper 59 70 99 124 148 190 241 47 58 79 103 129 178 227Bimp 27 46 92 136 184 269 359 22 36 77 121 167 255 340

MK=3Tfo,h

(rf:imp, X:AR(1)) 37 56 95 126 156 205 259 30 45 77 105 134 189 243

MK=3Tfo,h

(rf:imp, X:AR(3)) 46 62 97 125 153 201 254 37 51 79 105 132 184 237

MK=3Tfo,h

(rf:imp, X:AR(6)) 35 52 98 133 164 214 266 29 43 80 112 142 198 251

MK=3Tfo,h

(rf:imp, X:VAR(1)) 41 59 103 141 175 226 279 33 48 83 117 150 213 271

MK=3Tfo,h

(rf:imp, X:VAR(3)) 42 61 105 137 164 209 261 32 50 85 116 142 193 247

MK=3Tfo,h

(rf:imp, X:VAR(6)) 49 64 115 162 195 231 276 32 50 94 140 171 217 265

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 120 128 149 167 183 212 715 93 102 123 144 163 200 461

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 100 113 141 161 175 193 1396 78 89 114 137 156 181 846

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 106 119 147 167 182 215 5329 83 95 119 145 164 200 1678

MK=3Tfo,h

(rf:per, X:AR(1)) 37 47 79 110 140 189 240 31 35 59 90 117 165 222

MK=3Tfo,h

(rf:per, X:AR(3)) 37 49 80 107 136 184 235 30 38 61 88 113 159 216

MK=3Tfo,h

(rf:per, X:AR(6)) 38 44 83 117 150 198 248 32 34 61 94 123 174 230

MK=3Tfo,h

(rf:per, X:VAR(1)) 40 52 88 124 158 208 260 32 40 69 104 133 190 251

MK=3Tfo,h

(rf:per, X:VAR(3)) 40 51 91 122 149 193 243 33 40 72 103 125 169 225

MK=3Tfo,h

(rf:per, X:VAR(6)) 52 59 101 146 179 213 256 42 48 79 120 147 190 241

MK=3Tfo,h

(rf:per, X:BVAR(1)) 108 116 136 154 169 192 970 81 91 115 135 149 176 539

MK=3Tfo,h

(rf:per, X:BVAR(3)) 86 100 130 150 162 175 1456 63 77 108 131 145 159 920

MK=3Tfo,h

(rf:per, X:BVAR(6)) 92 107 136 156 169 200 5706 68 83 113 138 154 181 1902




Bper 52 71 104 124 149 187 237 41 56 81 104 131 171 217Bimp 40 65 112 151 193 258 323 32 52 90 132 176 238 302

MK=3Tfo,h

(rf:imp, X:AR(1)) 41 67 109 138 170 215 261 32 52 87 117 150 195 245

MK=3Tfo,h

(rf:imp, X:AR(3)) 46 68 107 133 164 210 256 37 54 86 114 146 190 238

MK=3Tfo,h

(rf:imp, X:AR(6)) 41 65 111 145 179 224 271 32 52 91 126 160 205 254

MK=3Tfo,h

(rf:imp, X:VAR(1)) 48 79 134 172 205 246 291 39 62 110 148 181 231 285

MK=3Tfo,h

(rf:imp, X:VAR(3)) 45 72 120 148 176 219 266 35 55 96 125 156 200 251

MK=3Tfo,h

(rf:imp, X:VAR(6)) 52 76 126 166 199 239 280 37 60 104 144 177 222 270

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 105 117 140 156 171 209 1336 82 98 121 135 148 193 747

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 87 102 128 142 150 197 2587 70 83 109 122 131 175 1580

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 97 110 131 144 155 288 15443 78 93 111 125 136 198 4166

MK=3Tfo,h

(rf:per, X:AR(1)) 38 60 97 125 158 203 247 31 48 73 102 137 180 228

MK=3Tfo,h

(rf:per, X:AR(3)) 39 59 94 120 153 200 243 31 47 71 99 134 177 223

MK=3Tfo,h

(rf:per, X:AR(6)) 39 57 98 132 167 214 258 31 45 75 111 147 191 239

MK=3Tfo,h

(rf:per, X:VAR(1)) 43 71 122 159 193 234 279 34 54 98 133 167 217 273

MK=3Tfo,h

(rf:per, X:VAR(3)) 41 63 108 135 164 209 253 33 47 85 111 142 186 236

MK=3Tfo,h

(rf:per, X:VAR(6)) 53 68 111 149 182 223 260 43 54 90 126 156 199 248

MK=3Tfo,h

(rf:per, X:BVAR(1)) 102 113 134 149 163 204 1785 83 96 119 132 140 186 920

MK=3Tfo,h

(rf:per, X:BVAR(3)) 85 98 121 134 140 191 2751 68 82 106 118 122 161 1744

MK=3Tfo,h

(rf:per, X:BVAR(6)) 94 106 125 136 145 316 19527 79 92 109 123 127 200 5069





Bper 49 71 106 123 147 182 228 39 56 82 99 130 165 202Bimp 45 72 115 144 179 229 283 38 59 90 122 164 205 258

MK=3Tfo,h

(rf:imp, X:AR(1)) 40 69 111 136 165 205 243 32 54 86 113 146 185 228

MK=3Tfo,h

(rf:imp, X:AR(3)) 44 70 108 131 160 201 238 35 54 84 110 143 181 219

MK=3Tfo,h

(rf:imp, X:AR(6)) 40 67 111 141 173 213 252 31 54 88 120 154 193 236

MK=3Tfo,h

(rf:imp, X:VAR(1)) 46 81 136 171 201 233 273 36 63 110 145 177 216 267

MK=3Tfo,h

(rf:imp, X:VAR(3)) 44 73 120 144 169 208 247 33 55 94 120 151 189 234

MK=3Tfo,h

(rf:imp, X:VAR(6)) 52 77 121 153 182 222 257 38 60 98 129 162 202 248

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 101 110 128 138 153 196 1489 83 96 113 121 130 177 822

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 86 96 114 121 126 193 2925 72 82 99 104 107 162 1872

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 97 106 115 120 130 364 22712 83 92 99 103 107 192 5778

MK=3Tfo,h

(rf:per, X:AR(1)) 40 65 102 125 156 196 230 33 53 78 103 137 173 213

MK=3Tfo,h

(rf:per, X:AR(3)) 40 64 98 120 152 194 227 33 52 74 99 134 173 206

MK=3Tfo,h

(rf:per, X:AR(6)) 40 61 100 130 164 205 241 33 50 77 110 146 183 224

MK=3Tfo,h

(rf:per, X:VAR(1)) 44 76 128 162 193 226 265 35 58 102 135 168 207 259

MK=3Tfo,h

(rf:per, X:VAR(3)) 42 67 112 135 161 202 238 34 51 86 110 142 179 222

MK=3Tfo,h

(rf:per, X:VAR(6)) 55 71 107 137 167 207 238 43 57 87 116 145 182 227

MK=3Tfo,h

(rf:per, X:BVAR(1)) 104 112 126 135 150 200 2021 88 99 113 120 125 176 1015

MK=3Tfo,h

(rf:per, X:BVAR(3)) 91 99 111 114 118 191 3110 75 83 98 99 100 152 2049

MK=3Tfo,h

(rf:per, X:BVAR(6)) 100 107 112 114 123 431 30953 88 96 99 101 100 207 7251





Bper 62 74 102 128 150 189 235 50 61 84 108 131 173 223Bimp 20 34 74 117 164 256 357 15 27 59 100 147 241 337

MK=5Tfo,h

(rf:imp, X:AR(1)) 37 59 95 122 148 194 244 30 50 79 102 127 178 230

MK=5Tfo,h

(rf:imp, X:AR(3)) 48 67 100 126 149 191 240 37 56 84 107 128 175 225

MK=5Tfo,h

(rf:imp, X:AR(6)) 34 53 96 129 155 199 248 27 44 79 107 132 182 234

MK=5Tfo,h

(rf:imp, X:VAR(1)) 37 47 81 115 147 199 250 31 41 68 97 126 182 242

MK=5Tfo,h

(rf:imp, X:VAR(3)) 60 70 110 137 157 193 237 44 57 93 119 137 176 221

MK=5Tfo,h

(rf:imp, X:VAR(6)) 49 62 106 144 180 214 252 33 50 87 118 152 200 243

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 122 126 143 162 179 206 647 100 102 118 135 156 194 368

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 109 114 135 155 174 205 1140 91 94 112 131 153 190 619

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 106 115 138 162 182 215 6262 86 94 114 136 159 202 2209

MK=5Tfo,h

(rf:per, X:AR(1)) 35 47 79 105 131 175 225 30 36 60 89 112 152 206

MK=5Tfo,h

(rf:per, X:AR(3)) 37 53 84 108 130 171 219 30 41 65 92 111 146 200

MK=5Tfo,h

(rf:per, X:AR(6)) 37 46 84 115 141 182 229 31 38 63 96 119 158 211

MK=5Tfo,h

(rf:per, X:VAR(1)) 38 48 74 102 131 179 229 30 39 59 89 114 157 218

MK=5Tfo,h

(rf:per, X:VAR(3)) 45 57 94 119 137 171 214 32 45 76 103 118 143 191

MK=5Tfo,h

(rf:per, X:VAR(6)) 51 59 95 127 161 189 228 38 49 75 102 132 168 216

MK=5Tfo,h

(rf:per, X:BVAR(1)) 108 113 130 148 163 182 791 87 89 107 129 147 167 350

MK=5Tfo,h

(rf:per, X:BVAR(3)) 92 99 122 141 159 181 1242 73 79 102 124 145 160 594

MK=5Tfo,h

(rf:per, X:BVAR(6)) 90 101 127 152 170 194 6023 71 82 106 132 152 176 2245




Bper 59 70 99 124 148 190 241 47 58 79 103 129 178 227Bimp 27 46 92 136 184 269 359 22 36 77 121 167 255 340

MK=5Tfo,h

(rf:imp, X:AR(1)) 36 55 95 126 157 206 260 29 44 75 105 134 189 244

MK=5Tfo,h

(rf:imp, X:AR(3)) 46 61 97 125 154 202 255 36 50 78 105 132 184 238

MK=5Tfo,h

(rf:imp, X:AR(6)) 34 52 99 134 166 215 268 28 42 79 112 143 198 252

MK=5Tfo,h

(rf:imp, X:VAR(1)) 42 60 104 142 177 227 279 35 49 83 119 152 214 271

MK=5Tfo,h

(rf:imp, X:VAR(3)) 58 70 108 134 158 201 252 43 56 88 114 137 183 234

MK=5Tfo,h

(rf:imp, X:VAR(6)) 51 70 116 162 204 247 281 36 56 97 142 181 233 272

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 118 126 150 169 185 221 2228 96 103 125 147 167 212 966

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 106 113 138 155 171 213 4461 83 90 112 133 153 188 2019

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 104 116 143 166 182 392 26100 81 92 117 143 162 264 8204

MK=5Tfo,h

(rf:per, X:AR(1)) 36 46 79 109 140 189 241 30 34 58 90 117 166 223

MK=5Tfo,h

(rf:per, X:AR(3)) 36 48 80 107 136 185 237 29 37 60 88 113 160 217

MK=5Tfo,h

(rf:per, X:AR(6)) 37 44 84 118 150 199 250 31 34 61 94 124 176 232

MK=5Tfo,h

(rf:per, X:VAR(1)) 40 53 89 124 158 209 260 32 39 69 104 134 192 251

MK=5Tfo,h

(rf:per, X:VAR(3)) 44 57 92 117 140 184 235 33 45 72 97 115 153 212

MK=5Tfo,h

(rf:per, X:VAR(6)) 49 60 101 146 186 225 259 37 47 80 124 159 205 247

MK=5Tfo,h

(rf:per, X:BVAR(1)) 107 115 138 156 170 206 2639 84 92 115 136 151 192 1060

MK=5Tfo,h

(rf:per, X:BVAR(3)) 93 101 126 143 157 193 4223 70 80 105 126 142 166 1900

MK=5Tfo,h

(rf:per, X:BVAR(6)) 91 105 133 155 170 355 23221 70 84 113 137 151 243 7744





Bper 52 71 104 124 149 187 237 41 56 81 104 131 171 217Bimp 40 65 112 151 193 258 323 32 52 90 132 176 238 302

MK=5Tfo,h

(rf:imp, X:AR(1)) 40 66 108 137 169 213 260 31 52 86 115 149 194 244

MK=5Tfo,h

(rf:imp, X:AR(3)) 46 68 107 133 164 209 255 36 54 86 114 146 190 237

MK=5Tfo,h

(rf:imp, X:AR(6)) 39 64 110 144 178 223 270 31 51 91 125 159 204 254

MK=5Tfo,h

(rf:imp, X:VAR(1)) 50 80 134 173 206 245 291 41 65 111 150 182 231 284

MK=5Tfo,h

(rf:imp, X:VAR(3)) 55 75 114 135 161 207 254 43 58 93 115 139 185 237

MK=5Tfo,h

(rf:imp, X:VAR(6)) 55 78 126 168 208 256 289 42 60 103 149 185 237 277

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 102 115 143 160 175 236 3182 82 95 121 140 157 215 1509

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 91 103 128 139 150 244 9463 70 84 107 119 131 201 3873

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 93 106 132 148 169 702 50552 78 92 113 129 147 315 14015

MK=5Tfo,h

(rf:per, X:AR(1)) 38 59 95 123 156 201 244 31 48 72 101 135 178 226

MK=5Tfo,h

(rf:per, X:AR(3)) 38 58 92 118 151 198 241 31 47 70 97 132 175 220

MK=5Tfo,h

(rf:per, X:AR(6)) 39 56 96 129 165 211 255 32 45 74 109 145 188 237

MK=5Tfo,h

(rf:per, X:VAR(1)) 46 71 118 157 190 232 277 37 53 97 132 165 215 271

MK=5Tfo,h

(rf:per, X:VAR(3)) 45 64 102 121 149 197 243 36 51 81 100 126 170 221

MK=5Tfo,h

(rf:per, X:VAR(6)) 50 68 113 156 195 239 270 38 52 92 137 169 216 257

MK=5Tfo,h

(rf:per, X:BVAR(1)) 97 109 135 151 165 233 3604 78 91 117 132 146 203 1662

MK=5Tfo,h

(rf:per, X:BVAR(3)) 86 98 121 131 140 215 8159 67 82 104 115 121 179 3525

MK=5Tfo,h

(rf:per, X:BVAR(6)) 88 102 125 140 157 622 44597 75 89 112 125 136 295 13725




Bper 49 71 106 123 147 182 228 39 56 82 99 130 165 202Bimp 45 72 115 144 179 229 283 38 59 90 122 164 205 258

MK=5Tfo,h

(rf:imp, X:AR(1)) 40 69 112 136 166 205 243 32 54 86 112 146 185 228

MK=5Tfo,h

(rf:imp, X:AR(3)) 45 70 109 132 161 202 238 36 55 84 110 143 181 219

MK=5Tfo,h

(rf:imp, X:AR(6)) 41 68 113 143 174 214 254 33 54 89 121 156 195 238

MK=5Tfo,h

(rf:imp, X:VAR(1)) 50 84 139 175 206 237 277 40 67 112 150 182 221 271

MK=5Tfo,h

(rf:imp, X:VAR(3)) 53 76 115 131 155 199 239 41 58 90 109 135 177 221

MK=5Tfo,h

(rf:imp, X:VAR(6)) 56 81 125 158 192 238 274 44 61 98 137 168 218 260

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 87 101 127 141 156 236 3318 71 85 109 121 134 206 1687

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 80 92 113 119 129 269 12487 65 78 95 100 109 202 4837

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 87 98 116 127 152 809 58884 76 87 101 111 124 319 15628

MK=5Tfo,h

(rf:per, X:AR(1)) 40 65 102 125 155 196 230 32 53 77 103 136 173 213

MK=5Tfo,h

(rf:per, X:AR(3)) 40 64 98 120 151 193 226 33 52 73 99 133 172 205

MK=5Tfo,h

(rf:per, X:AR(6)) 41 62 101 130 163 205 241 33 50 76 109 145 183 224

MK=5Tfo,h

(rf:per, X:VAR(1)) 46 76 126 162 194 228 267 39 58 101 137 170 209 261

MK=5Tfo,h

(rf:per, X:VAR(3)) 45 68 107 122 147 193 231 37 53 82 100 127 170 208

MK=5Tfo,h

(rf:per, X:VAR(6)) 52 73 115 150 182 224 257 40 56 90 129 157 201 244

MK=5Tfo,h

(rf:per, X:BVAR(1)) 86 100 123 135 149 236 3644 71 84 108 118 128 199 1836

MK=5Tfo,h

(rf:per, X:BVAR(3)) 80 92 110 114 122 237 10705 64 78 95 99 102 184 4365

MK=5Tfo,h

(rf:per, X:BVAR(6)) 88 98 113 121 142 720 52410 74 86 101 109 116 297 15294





Bper 62 74 102 128 150 189 235 50 61 84 108 131 173 223Bimp 20 34 74 117 164 256 357 15 27 59 100 147 241 337

MK=7Tfo,h

(rf:imp, X:AR(1)) 43 64 98 125 150 195 246 34 54 83 106 130 182 233

MK=7Tfo,h

(rf:imp, X:AR(3)) 53 72 104 128 151 193 241 41 60 88 110 132 178 226

MK=7Tfo,h

(rf:imp, X:AR(6)) 40 58 99 131 157 201 249 31 49 83 110 135 186 237

MK=7Tfo,h

(rf:imp, X:VAR(1)) 37 49 83 118 151 202 253 30 43 71 100 131 188 247

MK=7Tfo,h

(rf:imp, X:VAR(3)) 57 70 107 134 156 201 246 40 55 93 117 135 188 236

MK=7Tfo,h

(rf:imp, X:VAR(6)) 48 57 99 146 183 236 273 35 46 86 125 157 223 269

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 125 127 140 157 175 273 9457 105 102 116 132 154 215 1494

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 102 108 131 153 172 256 12553 89 89 108 130 151 214 2995

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 110 120 143 165 185 239 13035 89 97 118 139 163 210 3857

MK=7Tfo,h

(rf:per, X:AR(1)) 35 47 79 105 131 175 225 30 35 60 89 112 152 206

MK=7Tfo,h

(rf:per, X:AR(3)) 37 53 84 108 130 171 219 30 40 65 92 111 146 200

MK=7Tfo,h

(rf:per, X:AR(6)) 37 46 84 115 141 182 229 31 38 63 96 119 158 211

MK=7Tfo,h

(rf:per, X:VAR(1)) 36 47 73 102 133 181 230 30 39 58 89 116 160 221

MK=7Tfo,h

(rf:per, X:VAR(3)) 44 59 94 117 137 178 223 33 47 75 101 115 154 206

MK=7Tfo,h

(rf:per, X:VAR(6)) 55 63 92 133 164 204 245 43 51 76 110 133 186 239

MK=7Tfo,h

(rf:per, X:BVAR(1)) 109 110 123 140 156 252 9289 90 87 102 122 141 181 1488

MK=7Tfo,h

(rf:per, X:BVAR(3)) 80 89 116 139 156 249 17122 67 71 96 123 139 183 3821

MK=7Tfo,h

(rf:per, X:BVAR(6)) 96 106 131 154 172 258 25053 74 84 110 134 154 192 5567




Bper 59 70 99 124 148 190 241 47 58 79 103 129 178 227Bimp 27 46 92 136 184 269 359 22 36 77 121 167 255 340

MK=7Tfo,h

(rf:imp, X:AR(1)) 41 59 97 128 158 207 260 33 47 79 108 136 192 245

MK=7Tfo,h

(rf:imp, X:AR(3)) 50 65 99 127 155 203 255 39 53 82 108 134 187 238

MK=7Tfo,h

(rf:imp, X:AR(6)) 39 56 101 135 167 216 269 32 46 83 116 145 201 253

MK=7Tfo,h

(rf:imp, X:VAR(1)) 43 61 106 146 181 231 282 36 50 87 123 158 221 277

MK=7Tfo,h

(rf:imp, X:VAR(3)) 58 73 111 140 169 221 268 43 58 97 121 149 208 257

MK=7Tfo,h

(rf:imp, X:VAR(6)) 48 63 119 171 222 288 307 38 53 106 148 198 273 293

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 120 127 147 168 205 783 36868 97 103 123 148 178 311 5428

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 100 109 136 157 185 680 40868 81 86 111 137 167 292 10020

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 109 121 148 170 189 388 40522 85 96 122 145 167 274 11991

MK=7Tfo,h

(rf:per, X:AR(1)) 36 46 79 109 140 189 241 30 34 58 90 117 166 223

MK=7Tfo,h

(rf:per, X:AR(3)) 36 48 80 107 136 185 237 29 37 60 88 113 160 217

MK=7Tfo,h

(rf:per, X:AR(6)) 37 44 84 118 150 199 250 31 34 61 94 124 176 232

MK=7Tfo,h

(rf:per, X:VAR(1)) 38 51 88 126 161 211 262 31 37 69 106 137 196 254

MK=7Tfo,h

(rf:per, X:VAR(3)) 45 59 94 121 149 201 248 34 48 79 102 124 178 231

MK=7Tfo,h

(rf:per, X:VAR(6)) 53 62 108 157 202 257 280 42 52 97 133 176 240 266

MK=7Tfo,h

(rf:per, X:BVAR(1)) 105 112 132 152 188 770 36347 84 90 110 134 159 288 5554

MK=7Tfo,h

(rf:per, X:BVAR(3)) 83 93 122 143 169 700 51976 64 72 101 125 151 288 12369

MK=7Tfo,h

(rf:per, X:BVAR(6)) 98 111 139 160 178 538 70848 75 87 117 140 156 291 17085





Bper 52 71 104 124 149 187 237 41 56 81 104 131 171 217Bimp 40 65 112 151 193 258 323 32 52 90 132 176 238 302

MK=7Tfo,h

(rf:imp, X:AR(1)) 43 67 110 138 169 214 260 34 53 88 118 149 194 245

MK=7Tfo,h

(rf:imp, X:AR(3)) 49 70 108 134 164 209 255 39 56 88 116 146 189 237

MK=7Tfo,h

(rf:imp, X:AR(6)) 42 66 112 145 178 224 271 34 53 93 127 159 205 255

MK=7Tfo,h

(rf:imp, X:VAR(1)) 51 81 137 178 210 248 294 42 66 115 156 187 237 290

MK=7Tfo,h

(rf:imp, X:VAR(3)) 57 77 118 146 177 227 271 44 61 99 127 155 209 259

MK=7Tfo,h

(rf:imp, X:VAR(6)) 48 73 130 179 232 300 322 38 61 111 155 204 280 296

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 102 116 142 169 243 1245 58819 80 94 120 150 188 392 8844

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 89 101 130 150 199 1043 54346 69 79 109 132 160 354 13928

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 99 111 135 151 165 453 45118 83 96 117 129 144 291 14260

MK=7Tfo,h

(rf:per, X:AR(1)) 38 59 95 123 156 201 244 30 48 72 101 135 178 226

MK=7Tfo,h

(rf:per, X:AR(3)) 38 58 92 118 151 198 241 31 47 70 97 132 175 220

MK=7Tfo,h

(rf:per, X:AR(6)) 39 56 96 129 165 211 255 32 45 74 109 145 188 237

MK=7Tfo,h

(rf:per, X:VAR(1)) 44 70 119 159 192 233 278 35 53 98 135 168 218 273

MK=7Tfo,h

(rf:per, X:VAR(3)) 46 64 103 129 163 214 256 37 51 84 111 136 189 241

MK=7Tfo,h

(rf:per, X:VAR(6)) 51 69 119 166 216 276 300 40 58 103 144 188 253 274

MK=7Tfo,h

(rf:per, X:BVAR(1)) 92 108 132 158 235 1244 58725 74 90 115 139 173 382 9077

MK=7Tfo,h

(rf:per, X:BVAR(3)) 78 91 119 139 187 1017 60810 60 73 102 122 148 375 16430

MK=7Tfo,h

(rf:per, X:BVAR(6)) 95 107 129 144 161 666 86193 81 94 116 127 137 328 21616




Bper 49 71 106 123 147 182 228 39 56 82 99 130 165 202Bimp 45 72 115 144 179 229 283 38 59 90 122 164 205 258

MK=7Tfo,h

(rf:imp, X:AR(1)) 43 70 112 136 165 205 244 34 55 87 114 146 184 229

MK=7Tfo,h

(rf:imp, X:AR(3)) 47 72 110 132 161 201 239 37 56 86 111 143 181 220

MK=7Tfo,h

(rf:imp, X:AR(6)) 43 69 113 143 174 214 254 34 55 91 123 156 194 239

MK=7Tfo,h

(rf:imp, X:VAR(1)) 51 85 142 180 209 240 281 42 69 116 155 186 225 276

MK=7Tfo,h

(rf:imp, X:VAR(3)) 54 77 118 141 170 215 253 42 60 94 120 147 193 242

MK=7Tfo,h

(rf:imp, X:VAR(6)) 48 76 128 167 214 275 312 39 61 102 142 186 253 282

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 86 102 127 153 241 1297 60799 67 84 109 132 172 394 9358

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 76 89 115 136 196 1157 56023 59 70 96 114 140 372 14412

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 93 102 119 127 139 431 34795 82 92 104 108 119 258 11984

MK=7Tfo,h

(rf:per, X:AR(1)) 40 65 102 124 155 196 230 32 53 77 103 136 173 213

MK=7Tfo,h

(rf:per, X:AR(3)) 40 64 98 120 151 193 226 33 52 73 99 133 172 206

MK=7Tfo,h

(rf:per, X:AR(6)) 41 62 101 130 163 205 241 33 51 76 109 145 182 224

MK=7Tfo,h

(rf:per, X:VAR(1)) 45 76 127 165 195 228 268 38 57 102 139 171 210 263

MK=7Tfo,h

(rf:per, X:VAR(3)) 46 68 107 130 160 206 242 39 53 84 110 137 180 229

MK=7Tfo,h

(rf:per, X:VAR(6)) 51 73 118 155 200 253 292 41 60 97 135 173 230 262

MK=7Tfo,h

(rf:per, X:BVAR(1)) 81 99 121 147 238 1308 61292 64 82 106 126 162 390 9643

MK=7Tfo,h

(rf:per, X:BVAR(3)) 70 83 108 127 183 1089 57381 54 66 91 108 132 397 16437

MK=7Tfo,h

(rf:per, X:BVAR(6)) 93 101 116 123 139 624 75036 80 90 104 107 115 300 19261



Figure 79: Mean absolute level forecasting errors of Bayesian vector autoregressive modelsin 10-year window set-up

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Figure 80: Mean absolute level forecasting errors of autoregressive models in 5-year windowset-up

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Figure 81: Mean absolute level forecasting errors of Bayesian vector autoregressive modelsin 5-year window set-up

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Table 63: Slope forecast errors for 3-month tenor in different models with K = 3, 10-yearwindow


Bper 3.65 4.66 4.73 4.83 4.61 2.82 3.65 3.88 4.08 3.43Bimp 3.76 4.85 7.3 8.26 8.81 2.97 3.96 6.27 7.48 8.06

MK=3Tfo,h

(rf:imp, X:AR(1)) 2.66 4.48 4.8 5.4 6.33 2.04 3.38 3.81 4.52 4.54

MK=3Tfo,h

(rf:imp, X:AR(3)) 2.56 4.29 4.94 5.76 6.73 1.96 3.25 3.69 4.64 4.84

MK=3Tfo,h

(rf:imp, X:AR(6)) 2.69 4.53 4.85 5.25 6.08 1.97 3.42 3.62 4.23 4.39

MK=3Tfo,h

(rf:imp, X:VAR(1)) 2.49 4.11 5.53 4.82 4.7 1.92 3.47 4.89 4.07 4.23

MK=3Tfo,h

(rf:imp, X:VAR(3)) 2.53 4.03 5.9 4.73 2.62 2.06 3.28 4.75 3.89 2.06

MK=3Tfo,h

(rf:imp, X:VAR(6)) 2.88 4.96 7.18 6.4 3.99 2.47 3.8 5.52 4.87 2.93

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 2.63 4.09 5.22 4.14 4.67 2.05 3.44 4.65 3.59 3.83

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 2.85 3.88 5.69 5.3 9.99 2.2 3.11 4.73 4.51 7.28

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 3.27 4.61 6.56 9.26 34.45 2.7 3.62 5.16 6.8 23.1

MK=3Tfo,h

(rf:per, X:AR(1)) 2.67 4.49 4.8 5.38 6.3 2.04 3.4 3.83 4.52 4.54

MK=3Tfo,h

(rf:per, X:AR(3)) 2.56 4.29 4.93 5.74 6.71 1.96 3.26 3.69 4.63 4.83

MK=3Tfo,h

(rf:per, X:AR(6)) 2.71 4.55 4.85 5.24 6.08 2 3.43 3.63 4.23 4.4

MK=3Tfo,h

(rf:per, X:VAR(1)) 2.45 4.1 5.54 4.85 4.73 1.89 3.48 4.91 4.1 4.24

MK=3Tfo,h

(rf:per, X:VAR(3)) 2.59 4.09 5.93 4.77 2.65 2.12 3.32 4.78 3.93 2.1

MK=3Tfo,h

(rf:per, X:VAR(6)) 2.94 4.99 7.17 6.4 4 2.53 3.85 5.5 4.85 2.94

MK=3Tfo,h

(rf:per, X:BVAR(1)) 2.58 4.1 5.25 4.18 4.74 2.01 3.44 4.7 3.62 3.9

MK=3Tfo,h

(rf:per, X:BVAR(3)) 2.79 3.86 5.7 5.37 10.62 2.15 3.09 4.72 4.58 7.65

MK=3Tfo,h

(rf:per, X:BVAR(6)) 3.35 4.67 6.59 9.52 36.53 2.79 3.7 5.22 6.88 24.35


Table 64: Slope forecast errors for 1-year tenor in different models with K = 3, 10-yearwindow


Bper 12.43 17.6 18.97 17.59 18.06 9.1 13.47 15.69 15.05 11.88Bimp 14.15 21.41 32.18 35.64 37.27 11.57 18.37 27.33 31.47 33.43

MK=3Tfo,h

(rf:imp, X:AR(1)) 12.19 20.55 21.61 24.49 29.37 9.03 14.96 16.7 20.31 21.64

MK=3Tfo,h

(rf:imp, X:AR(3)) 11.72 19.73 22.33 26.16 31.08 8.69 14.58 16.3 20.33 22.6

MK=3Tfo,h

(rf:imp, X:AR(6)) 12.39 20.96 21.93 23.94 28.29 8.84 15.5 16.15 18.46 20.66

MK=3Tfo,h

(rf:imp, X:VAR(1)) 10.9 18.02 23.77 20.96 21.56 8.44 15.22 20.88 17.87 19.34

MK=3Tfo,h

(rf:imp, X:VAR(3)) 11.25 17.99 25.84 20.69 12.07 9.12 14.53 20.62 16.68 9.47

MK=3Tfo,h

(rf:imp, X:VAR(6)) 13.03 22.47 31.56 27.97 17.25 11.01 17.1 24.01 21.42 13.11

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 10.87 17.3 21.99 18.1 22.75 8.43 14.67 19.56 15.85 18.69

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 11.77 16.34 24.05 22.71 45.82 8.95 13 19.82 19.4 35.25

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 13.23 19.28 27.83 42.7 161.05 11 14.97 21.41 30.29 108.56

MK=3Tfo,h

(rf:per, X:AR(1)) 12.18 20.56 21.6 24.33 29.19 8.99 15.11 16.88 20.39 21.67

MK=3Tfo,h

(rf:per, X:AR(3)) 11.72 19.69 22.23 25.93 30.84 8.65 14.67 16.26 20.37 22.57

MK=3Tfo,h

(rf:per, X:AR(6)) 12.5 21.03 21.86 23.81 28.18 9.01 15.6 16.17 18.45 20.69

MK=3Tfo,h

(rf:per, X:VAR(1)) 11 18.32 24.23 21.51 22.13 8.54 15.48 21.36 18.35 19.86

MK=3Tfo,h

(rf:per, X:VAR(3)) 11.66 18.4 26.11 20.89 12.31 9.6 14.91 20.96 17.01 9.82

MK=3Tfo,h

(rf:per, X:VAR(6)) 13.4 22.68 31.5 27.94 17.26 11.4 17.43 23.96 21.22 13.14

MK=3Tfo,h

(rf:per, X:BVAR(1)) 10.99 17.74 22.62 18.81 23.39 8.49 15.04 20.2 16.4 19.3

MK=3Tfo,h

(rf:per, X:BVAR(3)) 11.87 16.66 24.47 23.46 49.39 8.97 13.27 20.21 20.09 37.77

MK=3Tfo,h

(rf:per, X:BVAR(6)) 13.67 19.65 28.03 44.09 170.93 11.24 15.57 21.73 30.91 114.7





Bper 21.96 34.06 35.72 35.24 46.48 15.55 23.93 28.3 31.71 37.26Bimp 27.82 40.14 51.64 58.12 60 22.46 36.43 42.12 45.15 49.4

MK=3Tfo,h

(rf:imp, X:AR(1)) 26 42.63 42.93 49.82 62.13 18.07 28.98 31.12 41.64 48.5

MK=3Tfo,h

(rf:imp, X:AR(3)) 25.27 41.52 44.82 53.22 64.75 17.87 28.86 32.08 42.81 50.67

MK=3Tfo,h

(rf:imp, X:AR(6)) 26.86 44.67 44.16 49.73 60.29 19.18 31.23 31.82 39.84 47.5

MK=3Tfo,h

(rf:imp, X:VAR(1)) 21.55 33.92 40.58 38.9 46.01 16.67 28.18 35 33.18 39.44

MK=3Tfo,h

(rf:imp, X:VAR(3)) 22.79 35.44 46.74 38.76 29 17.83 28.33 36.22 32.14 24.77

MK=3Tfo,h

(rf:imp, X:VAR(6)) 26.87 45.59 58.3 52.76 37.11 21.56 34.57 43.21 41.13 28.52

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 20.33 30.9 36.3 35.62 55.35 14.92 25.48 32.16 30.93 44.31

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 24.88 32.5 42.21 41.72 98.23 18.83 25.17 32.92 36.55 82.65

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 30.8 39.61 53.54 92.73 346.95 24.48 31.49 40.09 64.35 233.85

MK=3Tfo,h

(rf:per, X:AR(1)) 25.7 42.42 42.49 48.87 61.49 17.9 29.57 31.45 41.44 48.68

MK=3Tfo,h

(rf:per, X:AR(3)) 24.98 41.08 43.93 51.89 63.68 17.78 28.83 30.79 42.12 50.27

MK=3Tfo,h

(rf:per, X:AR(6)) 26.98 44.67 43.5 48.73 59.55 19.74 31.84 31.24 38.99 47.31

MK=3Tfo,h

(rf:per, X:VAR(1)) 22.79 36.08 43.51 41.88 49.27 17.83 30.12 37.62 35.86 42.78

MK=3Tfo,h

(rf:per, X:VAR(3)) 24.04 36.8 47.63 39.13 29.93 19.32 29.51 37.3 33.12 25.21

MK=3Tfo,h

(rf:per, X:VAR(6)) 27.96 46.25 57.93 52.27 37.01 22.62 35.36 42.24 40.45 29.01

MK=3Tfo,h

(rf:per, X:BVAR(1)) 20.95 33 39.51 39.19 59.11 15.74 27.42 35.19 33.59 47.66

MK=3Tfo,h

(rf:per, X:BVAR(3)) 24.98 33.51 43.84 44.93 109.09 18.56 25.85 34.67 39.26 92.18

MK=3Tfo,h

(rf:per, X:BVAR(6)) 30.7 39.91 53.6 96.07 368.5 23.82 31.55 39.76 66.58 248.49




Bper 28.94 44.29 43.84 51.93 71.54 21.57 28.61 32.71 46.35 61.33Bimp 35.41 44.75 49.94 62.44 66.43 27.37 38.74 40.79 46.1 53.36

MK=3Tfo,h

(rf:imp, X:AR(1)) 32.34 50.98 50.12 60.02 75.26 23.19 33.17 37.27 51.38 61.86

MK=3Tfo,h

(rf:imp, X:AR(3)) 31.84 50.45 52.67 63.7 77.27 23.67 34.31 40.75 54.71 64.26

MK=3Tfo,h

(rf:imp, X:AR(6)) 33.74 54.6 52.3 61.14 73.46 25.38 37.42 40.21 52.58 61.58

MK=3Tfo,h

(rf:imp, X:VAR(1)) 26.56 38.89 40.91 45.61 58.51 20.74 31.56 35.2 36.68 46.79

MK=3Tfo,h

(rf:imp, X:VAR(3)) 28.09 41.79 49.52 45.59 42.69 22.57 31.66 37.15 38.95 37.71

MK=3Tfo,h

(rf:imp, X:VAR(6)) 32.78 54.08 64.03 62.36 54.28 25.55 40.48 50.14 51.38 47.89

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 27.59 36.94 37.59 45.24 75.46 21.37 27.21 32.88 37.55 58.91

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 37.7 45.22 49.32 50.76 124.43 28.96 33.97 37.9 42.68 108.05

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 51.37 58.25 71.26 118.1 422.61 39.96 45.68 55.36 84.13 280.08

MK=3Tfo,h

(rf:per, X:AR(1)) 31.78 50.41 48.84 58.28 74.49 23.17 33.47 35.74 50.55 61.53

MK=3Tfo,h

(rf:per, X:AR(3)) 31.29 49.5 50.72 61.35 75.74 23.77 33.64 37.47 52.49 63.38

MK=3Tfo,h

(rf:per, X:AR(6)) 33.77 54.34 50.73 59.26 72.29 25.96 37.8 38.4 50.82 61.01

MK=3Tfo,h

(rf:per, X:VAR(1)) 28.49 42.28 45.75 50.08 63.75 22.29 35.21 39.48 40.61 52.67

MK=3Tfo,h

(rf:per, X:VAR(3)) 29.67 43.46 50.38 45.64 44.19 23.93 33.99 37.91 39.03 37.81

MK=3Tfo,h

(rf:per, X:VAR(6)) 34.28 54.91 63.11 61.26 54.14 27.03 40.97 48.28 50.27 47.79

MK=3Tfo,h

(rf:per, X:BVAR(1)) 26.84 38.78 41.79 50.48 82.67 20.81 30.26 37.36 41.84 66.36

MK=3Tfo,h

(rf:per, X:BVAR(3)) 35.75 44.67 49.91 54.75 140.36 26.33 32.26 38.63 45.59 123.36

MK=3Tfo,h

(rf:per, X:BVAR(6)) 49.95 57.47 70.23 121.79 448.68 37.64 43.57 53.6 86.84 299.06





Bper 34.88 51.42 49.02 66.03 89.02 24.99 33.69 34.77 53.53 71.54Bimp 41.49 46.1 45.69 67.72 71.96 31.08 37.06 36.8 52.46 58.04

MK=3Tfo,h

(rf:imp, X:AR(1)) 37.05 55.15 54.09 70.34 86.6 27.84 38.64 43.54 62.05 75.76

MK=3Tfo,h

(rf:imp, X:AR(3)) 36.92 55.78 57.02 73.31 86.91 28.75 40.48 47.9 65.2 77.16

MK=3Tfo,h

(rf:imp, X:AR(6)) 38.9 60.8 57.67 73.3 85.07 30.59 44.57 47.37 64.69 76.39

MK=3Tfo,h

(rf:imp, X:VAR(1)) 31.89 42.73 38.66 57.44 75.21 25.89 32.19 32.38 40.9 54.78

MK=3Tfo,h

(rf:imp, X:VAR(3)) 33.3 46.47 48.02 57.35 64.97 27.77 33.12 38.77 49.65 56.12

MK=3Tfo,h

(rf:imp, X:VAR(6)) 38.18 59.75 66.96 77.18 83.64 30.9 42.81 53.68 67.48 76.65

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 37.65 45.57 40.55 61.17 101.35 28.56 31.03 31.81 45.7 77.85

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 55.04 63.07 61.44 67.32 156.62 43.66 50.78 47.21 54.56 132.23

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 81.47 86.79 100.15 145.45 474.06 59.51 67.29 81.25 112.97 304.05

MK=3Tfo,h

(rf:per, X:AR(1)) 37.05 54.5 51.67 68.23 86.53 28.34 38.06 39.81 59.07 75.4

MK=3Tfo,h

(rf:per, X:AR(3)) 36.87 54.61 53.85 70.5 85.78 29.61 39.38 43.1 62.3 76.05

MK=3Tfo,h

(rf:per, X:AR(6)) 39.33 60.42 55.02 71.05 84.12 31.54 44.09 44.05 62.64 75.3

MK=3Tfo,h

(rf:per, X:VAR(1)) 34.78 46.89 44.23 62 81.89 27.69 37.65 37.2 45.38 61.31

MK=3Tfo,h

(rf:per, X:VAR(3)) 35.66 48.37 48.08 57.08 67.49 29.74 36.25 37.69 47.89 56.47

MK=3Tfo,h

(rf:per, X:VAR(6)) 40.6 61.03 65.49 76.03 84.17 32.61 44.1 51.77 66.05 76.61

MK=3Tfo,h

(rf:per, X:BVAR(1)) 34.77 45.37 43.13 67.52 113.82 27.6 31.28 35.88 51.91 91.66

MK=3Tfo,h

(rf:per, X:BVAR(3)) 49.93 59.34 58.23 70.76 178 37.33 44.22 43.43 56 155.11

MK=3Tfo,h

(rf:per, X:BVAR(6)) 79.01 84.92 97.8 148.91 503.1 55.91 63.24 77.58 113.68 323.83





Bper 3.65 4.66 4.73 4.83 4.61 2.82 3.65 3.88 4.08 3.43Bimp 3.76 4.85 7.3 8.26 8.81 2.97 3.96 6.27 7.48 8.06

MK=5Tfo,h

(rf:imp, X:AR(1)) 4 5.75 5.91 6.57 6.89 2.94 3.98 4.88 5 4.64

MK=5Tfo,h

(rf:imp, X:AR(3)) 3.72 5.19 5.58 6.75 7.19 2.82 3.63 4.54 5.25 5.24

MK=5Tfo,h

(rf:imp, X:AR(6)) 3.77 5.16 5.06 6.01 6.45 2.89 3.67 4.04 4.79 4.69

MK=5Tfo,h

(rf:imp, X:VAR(1)) 3.32 4.95 6.57 6.23 6.57 2.84 4.21 5.51 5.42 6.02

MK=5Tfo,h

(rf:imp, X:VAR(3)) 3.46 5.13 6.74 5.3 3.63 2.96 4.47 5.91 4.38 2.95

MK=5Tfo,h

(rf:imp, X:VAR(6)) 4.63 7.82 11.1 13.3 16.76 3.78 6.63 9.17 9.24 10.01

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 3.19 4.54 5.74 4.9 5.31 2.65 3.85 4.72 3.96 3.88

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 5.12 5.81 6.7 8.89 25.72 3.7 4.48 5.83 6.48 15.44

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 5.33 6.91 8.7 15.54 149.36 4.2 5.72 7.31 11.82 81.13

MK=5Tfo,h

(rf:per, X:AR(1)) 3.97 5.72 5.89 6.54 6.85 2.92 3.98 4.88 4.98 4.62

MK=5Tfo,h

(rf:per, X:AR(3)) 3.68 5.14 5.54 6.7 7.13 2.81 3.6 4.51 5.22 5.22

MK=5Tfo,h

(rf:per, X:AR(6)) 3.74 5.11 5.02 5.97 6.4 2.89 3.65 4.01 4.77 4.67

MK=5Tfo,h

(rf:per, X:VAR(1)) 3.32 5.03 6.72 6.38 6.73 2.86 4.28 5.63 5.55 6.16

MK=5Tfo,h

(rf:per, X:VAR(3)) 3.51 5.2 6.8 5.33 3.65 3.04 4.56 5.98 4.41 2.96

MK=5Tfo,h

(rf:per, X:VAR(6)) 4.75 8.07 11.48 13.61 17.02 3.85 6.82 9.48 9.58 10.37

MK=5Tfo,h

(rf:per, X:BVAR(1)) 3.2 4.64 5.91 5.02 5.31 2.69 3.94 4.86 4.05 3.88

MK=5Tfo,h

(rf:per, X:BVAR(3)) 5.09 5.84 6.79 9.13 25.62 3.69 4.54 5.89 6.66 15.75

MK=5Tfo,h

(rf:per, X:BVAR(6)) 5.47 7.09 9.03 16.65 154.77 4.27 5.9 7.49 12.52 85.39




Bper 12.43 17.6 18.97 17.59 18.06 9.1 13.47 15.69 15.05 11.88Bimp 14.15 21.41 32.18 35.64 37.27 11.57 18.37 27.33 31.47 33.43

MK=5Tfo,h

(rf:imp, X:AR(1)) 15.29 23.58 24.39 26.87 29.45 10.56 16.03 20.28 20.21 18.89

MK=5Tfo,h

(rf:imp, X:AR(3)) 14.26 21.23 23.22 27.51 30.49 10.1 13.94 18.36 21.25 20.66

MK=5Tfo,h

(rf:imp, X:AR(6)) 14.72 21.74 21.36 24.27 27.25 10.4 14.71 16.53 19.24 18.49

MK=5Tfo,h

(rf:imp, X:VAR(1)) 12.86 20.72 27.52 25.91 27.93 11 17.84 23.32 23.3 26.66

MK=5Tfo,h

(rf:imp, X:VAR(3)) 13.22 21.83 29.16 21.97 14.15 11.3 19.13 25.47 18.43 11.95

MK=5Tfo,h

(rf:imp, X:VAR(6)) 18.38 32.57 42.52 47.97 58.75 15.81 28.1 35.75 33.57 35.87

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 11.99 18.47 23.51 20.54 24.77 9.84 15.98 19.53 16.98 19.82

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 18.22 22.64 27.07 35.92 95.79 13.43 17.87 23.5 26.56 60.05

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 19.68 25.99 33.4 59.17 549.53 15.16 20.89 27.97 46.48 311.05

MK=5Tfo,h

(rf:per, X:AR(1)) 15.17 23.51 24.35 26.72 29.28 10.53 16.23 20.3 20.18 18.78

MK=5Tfo,h

(rf:per, X:AR(3)) 14.1 21.03 23.04 27.23 30.19 10.03 13.82 18.28 21.09 20.43

MK=5Tfo,h

(rf:per, X:AR(6)) 14.64 21.62 21.2 24.04 27 10.47 14.75 16.54 19.09 18.31

MK=5Tfo,h

(rf:per, X:VAR(1)) 13.05 21.38 28.56 26.91 28.96 11.31 18.42 24.13 24.2 27.62

MK=5Tfo,h

(rf:per, X:VAR(3)) 13.43 22.26 29.58 22.14 14.3 11.57 19.66 25.94 18.59 12.11

MK=5Tfo,h

(rf:per, X:VAR(6)) 18.73 33.65 44.16 49.24 59.88 16.18 28.95 37.05 34.98 37.35

MK=5Tfo,h

(rf:per, X:BVAR(1)) 12.24 19.26 24.69 21.52 25.22 10.28 16.69 20.43 17.77 20.32

MK=5Tfo,h

(rf:per, X:BVAR(3)) 18.27 22.98 27.8 37.44 98.64 13.67 18.41 24.1 28.01 64.79

MK=5Tfo,h

(rf:per, X:BVAR(6)) 20.14 26.71 34.93 63.84 568.56 15.5 21.59 29.02 50.19 324.56





Bper 21.96 34.06 35.72 35.24 46.48 15.55 23.93 28.3 31.71 37.26Bimp 27.82 40.14 51.64 58.12 60 22.46 36.43 42.12 45.15 49.4

MK=5Tfo,h

(rf:imp, X:AR(1)) 25.62 41.67 41.63 47.55 58.24 16.84 26.34 30.28 38.94 44.34

MK=5Tfo,h

(rf:imp, X:AR(3)) 24.8 39.67 42.11 49.77 59.94 17.07 25.83 30.32 39.46 45.7

MK=5Tfo,h

(rf:imp, X:AR(6)) 26.43 42.56 40.56 45.65 55.37 18.11 28.21 29.04 36.39 42.48

MK=5Tfo,h

(rf:imp, X:VAR(1)) 21.45 34.11 41.28 40.87 48.22 16.96 28.83 36.14 34.71 43.45

MK=5Tfo,h

(rf:imp, X:VAR(3)) 22.88 39.31 50.9 44 37.72 18.62 32.56 42.18 39.89 31.96

MK=5Tfo,h

(rf:imp, X:VAR(6)) 30.34 53.87 62.17 60.58 61.95 24.65 44.91 50.35 43.85 46.27

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 19.92 29.48 33.09 36.15 57.62 14.6 23.46 30.33 30.5 47.83

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 24.59 35.06 41.78 59.26 136.41 19.94 25.9 33.56 45.12 107.05

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 36.25 42.29 56.89 100.46 812.68 28.93 34.57 45.79 79.72 526.28

MK=5Tfo,h

(rf:per, X:AR(1)) 25.47 41.66 41.42 46.92 57.99 16.82 26.87 30.82 39.08 44.44

MK=5Tfo,h

(rf:per, X:AR(3)) 24.62 39.34 41.38 48.7 59.22 17.03 25.46 28.9 38.89 45.18

MK=5Tfo,h

(rf:per, X:AR(6)) 26.54 42.55 39.96 44.73 54.82 18.59 28.53 28.09 35.58 42.07

MK=5Tfo,h

(rf:per, X:VAR(1)) 22.88 36.71 44.69 43.88 51.6 18.68 31.2 38.85 37.73 46.97

MK=5Tfo,h

(rf:per, X:VAR(3)) 23.42 40.7 52.15 44.32 38.47 19.73 34.39 43.77 40.51 32.34

MK=5Tfo,h

(rf:per, X:VAR(6)) 31.12 56.01 65.13 62.46 64.15 25.53 46.97 52.57 45.81 48.93

MK=5Tfo,h

(rf:per, X:BVAR(1)) 20.35 31.74 36.76 39.5 61.23 15.36 26.18 33.43 33.02 51.8

MK=5Tfo,h

(rf:per, X:BVAR(3)) 24.35 35.76 43.46 63.3 151.01 19.83 27.14 35.85 49.26 119.64

MK=5Tfo,h

(rf:per, X:BVAR(6)) 36.56 43.06 59.28 106.33 857.42 28.77 34.67 48.15 83.93 560.41




Bper 28.94 44.29 43.84 51.93 71.54 21.57 28.61 32.71 46.35 61.33Bimp 35.41 44.75 49.94 62.44 66.43 27.37 38.74 40.79 46.1 53.36

MK=5Tfo,h

(rf:imp, X:AR(1)) 31.81 49.98 49.36 59.21 73.76 22.66 31.97 36.93 50.75 61.09

MK=5Tfo,h

(rf:imp, X:AR(3)) 31.4 49.29 51.48 62.5 75.49 23.28 33.52 40.95 54.34 63.5

MK=5Tfo,h

(rf:imp, X:AR(6)) 33.32 53.19 50.73 59.94 71.84 25.06 36.59 39.7 52.47 60.84

MK=5Tfo,h

(rf:imp, X:VAR(1)) 26.42 38.06 40.25 46.67 59.44 20.48 29.89 35.05 36.55 46.9

MK=5Tfo,h

(rf:imp, X:VAR(3)) 29.25 45.38 54.66 56.05 57.07 24.16 33.13 44.17 49.61 49.53

MK=5Tfo,h

(rf:imp, X:VAR(6)) 35.71 59.85 64.27 60.97 55.56 29.78 45.84 51.75 52.24 47.84

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 29.81 37.53 34.02 47.01 80.36 22.86 25.15 29.39 38.61 64.51

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 34.82 45.73 48.48 66.87 143.98 29.69 33.91 36.66 52.01 121.54

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 58.94 64.81 79.63 127.19 851.18 49.22 54.2 65.98 98.86 576.71

MK=5Tfo,h

(rf:per, X:AR(1)) 31.42 49.7 48.42 57.85 73.5 22.54 31.87 35.07 49.85 60.94

MK=5Tfo,h

(rf:per, X:AR(3)) 31.03 48.59 49.86 60.56 74.54 23.31 32.73 37.86 52.34 62.67

MK=5Tfo,h

(rf:per, X:AR(6)) 33.41 53 49.32 58.21 71.05 25.35 36.28 37.89 50.72 60.23

MK=5Tfo,h

(rf:per, X:VAR(1)) 28.2 41.44 44.83 50.13 63.99 21.84 33.25 38.97 39.59 52.34

MK=5Tfo,h

(rf:per, X:VAR(3)) 29.43 46.88 55.95 56 58.2 24.64 36.02 46.69 49.37 49.51

MK=5Tfo,h

(rf:per, X:VAR(6)) 36.51 62.03 67.22 62.22 58.07 30.27 48.36 54.15 52.49 50.65

MK=5Tfo,h

(rf:per, X:BVAR(1)) 28.52 38.76 37.54 50.63 86.68 22.01 27.57 33.88 41.64 71.5

MK=5Tfo,h

(rf:per, X:BVAR(3)) 32.37 44.75 48.49 70.79 164.62 27.36 32.56 37.9 56.75 138.4

MK=5Tfo,h

(rf:per, X:BVAR(6)) 58.52 64.73 81.05 131.16 906.64 48.34 53.12 67.32 102.86 616.67





Bper 34.88 51.42 49.02 66.03 89.02 24.99 33.69 34.77 53.53 71.54Bimp 41.49 46.1 45.69 67.72 71.96 31.08 37.06 36.8 52.46 58.04

MK=5Tfo,h

(rf:imp, X:AR(1)) 36.01 53.36 51.73 66.73 81.76 25.42 35.34 39.8 56.64 69.14

MK=5Tfo,h

(rf:imp, X:AR(3)) 36.31 53.18 53.25 68.31 81 27.28 37.67 44.14 59.75 70.5

MK=5Tfo,h

(rf:imp, X:AR(6)) 38.41 57.96 52.44 66.94 78.31 29.29 41.63 43.46 59.31 69

MK=5Tfo,h

(rf:imp, X:VAR(1)) 32.39 42.31 37.67 59.23 78.79 24.65 31.41 32.17 41.02 59.7

MK=5Tfo,h

(rf:imp, X:VAR(3)) 35.71 49.6 48.8 60.84 72.24 28.2 33.6 38.22 49.93 58.24

MK=5Tfo,h

(rf:imp, X:VAR(6)) 42.6 65.66 62.73 62.9 66.61 34.46 45.93 49.83 56.55 53.25

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 43.65 49.81 39.58 65.27 112.69 33.08 33.74 29.3 48.81 92.42

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 52.15 62.88 58.23 70.78 164.98 44.53 49.1 47.29 54.56 140.48

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 90.38 97.57 109.73 163.71 837.7 77.41 85.81 93.01 120.34 574.58

MK=5Tfo,h

(rf:per, X:AR(1)) 36.53 53.62 50.64 65.24 82.26 26.13 35.59 37.1 54.87 68.68

MK=5Tfo,h

(rf:per, X:AR(3)) 36.71 52.67 50.94 65.81 80.43 28.24 36.86 39.78 56.76 69.24

MK=5Tfo,h

(rf:per, X:AR(6)) 39.28 58.12 50.3 64.55 77.69 30.5 40.93 39.67 56.69 67.86

MK=5Tfo,h

(rf:per, X:VAR(1)) 35.47 47.38 44.74 63.82 85.15 28.16 37.18 38.65 45.71 67.88

MK=5Tfo,h

(rf:per, X:VAR(3)) 36.16 51.77 50.85 61.33 75.02 29.73 36.57 42.6 50.33 59.29

MK=5Tfo,h

(rf:per, X:VAR(6)) 43.61 67.91 65.55 63.83 69.92 35.46 48.99 52.68 55.5 56.41

MK=5Tfo,h

(rf:per, X:BVAR(1)) 40.96 49.96 42.81 70.58 124.16 29.99 34.81 34.79 53.68 105.45

MK=5Tfo,h

(rf:per, X:BVAR(3)) 47.36 59.75 55.36 74.39 191.88 39.79 46.12 42.46 57.74 162.43

MK=5Tfo,h

(rf:per, X:BVAR(6)) 89.1 96.67 109.83 165.4 900.41 75.65 83.97 90.95 123.23 619.85





Bper 3.65 4.66 4.73 4.83 4.61 2.82 3.65 3.88 4.08 3.43Bimp 3.76 4.85 7.3 8.26 8.81 2.97 3.96 6.27 7.48 8.06

MK=7Tfo,h

(rf:imp, X:AR(1)) 4.52 6.33 6.69 7.4 7.56 3.21 4.33 5.62 5.62 4.86

MK=7Tfo,h

(rf:imp, X:AR(3)) 4.14 5.69 6.33 7.59 7.91 2.97 3.86 5.11 5.85 5.52

MK=7Tfo,h

(rf:imp, X:AR(6)) 4.18 5.58 5.71 6.78 7.12 2.99 3.66 4.46 5.36 4.95

MK=7Tfo,h

(rf:imp, X:VAR(1)) 3.2 4.62 6.4 6.27 6.78 2.56 3.78 5.24 5.24 6.14

MK=7Tfo,h

(rf:imp, X:VAR(3)) 3.64 5.59 7.51 6.13 5.25 2.91 4.9 6.59 5.17 4.38

MK=7Tfo,h

(rf:imp, X:VAR(6)) 5.84 9.71 13.24 16.06 19.96 4.55 7.46 10.42 10.75 12.04

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 3.07 4.37 5.72 5.31 6.99 2.44 3.55 4.64 4.08 4.39

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 4.72 5.27 7.53 11.8 45.36 3.75 4.62 6.6 8.73 27.32

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 6.72 6.63 9.06 51.65 240.48 4.89 5.19 7.3 34.89 182.72

MK=7Tfo,h

(rf:per, X:AR(1)) 4.48 6.3 6.67 7.37 7.52 3.18 4.33 5.61 5.61 4.84

MK=7Tfo,h

(rf:per, X:AR(3)) 4.09 5.63 6.28 7.54 7.86 2.93 3.8 5.08 5.83 5.49

MK=7Tfo,h

(rf:per, X:AR(6)) 4.13 5.52 5.66 6.73 7.06 2.98 3.63 4.43 5.33 4.92

MK=7Tfo,h

(rf:per, X:VAR(1)) 3.15 4.62 6.51 6.39 6.9 2.54 3.79 5.33 5.34 6.25

MK=7Tfo,h

(rf:per, X:VAR(3)) 3.6 5.58 7.57 6.18 5.3 2.86 4.91 6.63 5.21 4.39

MK=7Tfo,h

(rf:per, X:VAR(6)) 6.01 10.07 13.66 16.25 20.05 4.61 7.69 10.78 11.04 12.36

MK=7Tfo,h

(rf:per, X:BVAR(1)) 3.04 4.39 5.83 5.32 6.8 2.43 3.58 4.72 4.09 4.27

MK=7Tfo,h

(rf:per, X:BVAR(3)) 4.66 5.24 7.59 12.23 47.95 3.69 4.61 6.61 9.1 28.56

MK=7Tfo,h

(rf:per, X:BVAR(6)) 6.83 6.71 9.11 52.82 249.23 4.9 5.31 7.38 35.26 189.74




Bper 12.43 17.6 18.97 17.59 18.06 9.1 13.47 15.69 15.05 11.88Bimp 14.15 21.41 32.18 35.64 37.27 11.57 18.37 27.33 31.47 33.43

MK=7Tfo,h

(rf:imp, X:AR(1)) 15.63 23.99 25.09 27.52 29.82 10.72 16.61 21.01 20.7 18.76

MK=7Tfo,h

(rf:imp, X:AR(3)) 14.56 21.54 23.85 28.09 30.83 10.2 14 18.98 21.62 20.31

MK=7Tfo,h

(rf:imp, X:AR(6)) 15.03 22.06 21.93 24.79 27.58 10.36 14.63 17.05 19.6 18.17

MK=7Tfo,h

(rf:imp, X:VAR(1)) 12.38 18.75 25.73 25.11 27.74 9.88 15.53 21.25 21.81 26.32

MK=7Tfo,h

(rf:imp, X:VAR(3)) 14.98 23.89 31.83 24.8 20.93 12.59 21.28 27.65 20.92 18.43

MK=7Tfo,h

(rf:imp, X:VAR(6)) 20.98 36.28 44.68 51.52 64.5 17.4 29.56 36.2 35.29 39.44

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 11.52 16.99 22.23 20.92 28.57 9.18 14.18 18.4 16.67 20.16

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 16.8 20.15 29.7 45.08 172.12 13.42 17.85 25.99 34.61 107.7

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 21.2 22.71 28.53 146.2 745.9 16.06 17.78 22.09 107.32 577.12

MK=7Tfo,h

(rf:per, X:AR(1)) 15.51 23.94 25.08 27.41 29.71 10.65 16.71 21.05 20.7 18.7

MK=7Tfo,h

(rf:per, X:AR(3)) 14.4 21.35 23.68 27.83 30.57 10.18 13.9 18.98 21.48 20.11

MK=7Tfo,h

(rf:per, X:AR(6)) 14.92 21.92 21.77 24.57 27.35 10.42 14.68 17.06 19.45 18.01

MK=7Tfo,h

(rf:per, X:VAR(1)) 12.38 19.22 26.68 26.06 28.75 9.99 15.94 21.97 22.68 27.24

MK=7Tfo,h

(rf:per, X:VAR(3)) 14.88 24.13 32.32 25.17 21.32 12.67 21.52 28.08 21.32 18.71

MK=7Tfo,h

(rf:per, X:VAR(6)) 21.69 37.99 46.38 52.25 64.88 17.68 30.83 37.53 36.45 40.72

MK=7Tfo,h

(rf:per, X:BVAR(1)) 11.65 17.58 23.28 21.69 28.45 9.39 14.75 19.15 17.25 20.41

MK=7Tfo,h

(rf:per, X:BVAR(3)) 16.71 20.36 30.27 46.92 186.3 13.5 18.13 26.42 36.28 113.12

MK=7Tfo,h

(rf:per, X:BVAR(6)) 21.62 23.12 28.86 150.19 777.03 16.29 18.23 22.33 109.24 597.97





Bper 21.96 34.06 35.72 35.24 46.48 15.55 23.93 28.3 31.71 37.26Bimp 27.82 40.14 51.64 58.12 60 22.46 36.43 42.12 45.15 49.4

MK=7Tfo,h

(rf:imp, X:AR(1)) 25.77 41.87 42.05 47.82 58.28 16.9 26.76 30.69 39.17 44.2

MK=7Tfo,h

(rf:imp, X:AR(3)) 24.94 39.86 42.54 50.04 60.01 17.1 26.11 30.54 39.57 45.6

MK=7Tfo,h

(rf:imp, X:AR(6)) 26.58 42.76 40.98 45.91 55.45 18.04 28.31 29.12 36.38 42.39

MK=7Tfo,h

(rf:imp, X:VAR(1)) 21.75 30.92 36.92 38.25 46.95 16.52 25.16 31.41 33.07 42.53

MK=7Tfo,h

(rf:imp, X:VAR(3)) 26.57 42.03 54.57 46.74 44.41 21.78 35.74 44.88 40.38 37.8

MK=7Tfo,h

(rf:imp, X:VAR(6)) 31.57 54.78 57.63 57.39 67.42 25.51 44.14 46.32 38.75 47.65

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 21.12 28.15 30.82 35.08 58.79 15.88 21.11 27.84 30.55 45.04

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 23.72 31.75 46.13 73.58 278.78 18.37 24.87 36.63 58.69 176.46

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 31.95 36.49 39.41 193.6 1102.2 26.01 29.36 31.54 149.19 848.72

MK=7Tfo,h

(rf:per, X:AR(1)) 25.64 41.9 41.84 47.29 58.17 16.88 27.3 31.25 39.38 44.34

MK=7Tfo,h

(rf:per, X:AR(3)) 24.77 39.55 41.79 49.03 59.4 17 25.69 29.16 39.08 45.12

MK=7Tfo,h

(rf:per, X:AR(6)) 26.7 42.75 40.32 45.02 54.97 18.56 28.68 28.24 35.58 41.99

MK=7Tfo,h

(rf:per, X:VAR(1)) 22.66 33.07 40.09 41.2 50.28 17.34 27.14 33.81 35.9 46.02

MK=7Tfo,h

(rf:per, X:VAR(3)) 26.87 43.05 55.75 47.36 45.56 22.46 37.24 46.4 40.91 39.22

MK=7Tfo,h

(rf:per, X:VAR(6)) 33.17 57.74 60.3 58.12 68.41 26.52 46.4 48.36 40.6 49.91

MK=7Tfo,h

(rf:per, X:BVAR(1)) 21.03 29.74 33.98 38.13 61.4 16.06 22.88 30.6 32.66 48.64

MK=7Tfo,h

(rf:per, X:BVAR(3)) 23.53 32.49 47.54 76.71 299.31 18.34 26.44 38.39 61.61 186.74

MK=7Tfo,h

(rf:per, X:BVAR(6)) 32.28 36.71 39.27 198.53 1150.54 26.18 29.2 31.05 152.8 883.89




Bper 28.94 44.29 43.84 51.93 71.54 21.57 28.61 32.71 46.35 61.33Bimp 35.41 44.75 49.94 62.44 66.43 27.37 38.74 40.79 46.1 53.36

MK=7Tfo,h

(rf:imp, X:AR(1)) 32.02 50.16 49.79 59.46 73.76 22.64 32.28 37.12 50.78 60.94

MK=7Tfo,h

(rf:imp, X:AR(3)) 31.59 49.45 51.88 62.69 75.48 23.25 33.68 41 54.26 63.26

MK=7Tfo,h

(rf:imp, X:AR(6)) 33.47 53.32 51.06 60.07 71.82 24.93 36.59 39.57 52.33 60.61

MK=7Tfo,h

(rf:imp, X:VAR(1)) 28.02 36.15 35.62 43.79 58.27 22.82 28.28 30.18 35.27 46.83

MK=7Tfo,h

(rf:imp, X:VAR(3)) 32.08 47.13 57.89 58.01 62.09 26.84 35.47 46.19 48.74 51.72

MK=7Tfo,h

(rf:imp, X:VAR(6)) 35.52 60.03 57.57 52.2 54.7 28.55 46.13 45.78 40.69 46.61

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 32.4 38.37 33.61 45.37 79.78 25.45 26.9 27.71 38.33 61.39

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 35.25 43.26 52.9 83.6 277.9 28.52 30.38 40.47 67.25 191.28

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 54.03 62.65 63.49 204.61 1198.65 45.49 53.26 55.3 162.99 908.18

MK=7Tfo,h

(rf:per, X:AR(1)) 31.66 49.93 48.83 58.2 73.64 22.56 32.18 35.23 50.11 60.71

MK=7Tfo,h

(rf:per, X:AR(3)) 31.25 48.77 50.19 60.82 74.65 23.29 32.86 37.88 52.22 62.48

MK=7Tfo,h

(rf:per, X:AR(6)) 33.58 53.15 49.54 58.36 71.12 25.28 36.42 37.64 50.58 60.05

MK=7Tfo,h

(rf:per, X:VAR(1)) 29.05 38.79 39.75 47.1 62.72 23.48 30.74 33.81 38.03 52.17

MK=7Tfo,h

(rf:per, X:VAR(3)) 32.27 48.06 58.86 58.14 63.56 27.27 37.26 48.38 48.87 53.45

MK=7Tfo,h

(rf:per, X:VAR(6)) 37.15 62.72 59.67 51.89 56.33 29.58 48.05 46.62 39.4 48.58

MK=7Tfo,h

(rf:per, X:BVAR(1)) 30.55 38.63 36.03 48.66 85.09 24.68 27.88 30.81 41.19 68.06

MK=7Tfo,h

(rf:per, X:BVAR(3)) 33.51 42.58 53.24 85.8 289.18 25.95 29.37 41.65 69.93 201.49

MK=7Tfo,h

(rf:per, X:BVAR(6)) 53.82 62.28 62.68 207.6 1246.66 45.13 52.82 54.61 166.11 942.32





Bper 34.88 51.42 49.02 66.03 89.02 24.99 33.69 34.77 53.53 71.54Bimp 41.49 46.1 45.69 67.72 71.96 31.08 37.06 36.8 52.46 58.04

MK=7Tfo,h

(rf:imp, X:AR(1)) 36.35 53.72 52.46 67.13 81.87 25.52 35.46 40.1 56.74 69.07

MK=7Tfo,h

(rf:imp, X:AR(3)) 36.58 53.46 53.89 68.61 81.05 27.21 37.86 44.42 59.78 70.31

MK=7Tfo,h

(rf:imp, X:AR(6)) 38.57 58.16 52.95 67.14 78.32 29.15 41.6 43.66 59.27 68.74

MK=7Tfo,h

(rf:imp, X:VAR(1)) 34.02 41.36 33.87 56.57 77.49 26.97 30.25 27.72 40.89 59.39

MK=7Tfo,h

(rf:imp, X:VAR(3)) 37.51 50.43 51.93 61.74 75.54 30.59 36.71 37.8 51.75 60.8

MK=7Tfo,h

(rf:imp, X:VAR(6)) 41.18 65.94 57.09 51.83 58.12 32.06 47.54 44.28 42.96 47.49

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 46.25 51.54 40.65 62.71 110.46 36.68 37.15 28.47 47.12 89.32

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 55.48 63.58 64.9 93.18 296.02 46.51 52.81 50.15 69.93 223.76

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 88.79 100.68 102.3 220.06 1260.31 78.32 89.39 92.87 173.74 983.9

MK=7Tfo,h

(rf:per, X:AR(1)) 36.8 54.03 51.33 65.78 82.52 26.17 36.19 37.31 55.18 68.48

MK=7Tfo,h

(rf:per, X:AR(3)) 36.93 52.98 51.48 66.22 80.67 28.25 37.05 39.96 56.77 69.14

MK=7Tfo,h

(rf:per, X:AR(6)) 39.43 58.33 50.63 64.8 77.9 30.31 41.12 39.76 56.54 67.77

MK=7Tfo,h

(rf:per, X:VAR(1)) 35.88 45.35 40.23 60.94 83.72 28.18 34.64 33.66 45.47 67.52

MK=7Tfo,h

(rf:per, X:VAR(3)) 37.64 51.23 52.71 61.93 78.41 31.03 37.32 40.96 52.3 62.4

MK=7Tfo,h

(rf:per, X:VAR(6)) 42.83 68.37 58.34 51.15 61.53 32.94 49.11 44.27 41.05 51.17

MK=7Tfo,h

(rf:per, X:BVAR(1)) 42.75 50.63 42.6 67.93 121.19 32.87 35.85 32.97 52.51 102.45

MK=7Tfo,h

(rf:per, X:BVAR(3)) 51.75 60.85 63.01 93.85 297.35 41.44 48.55 47.24 72.41 234.37

MK=7Tfo,h

(rf:per, X:BVAR(6)) 88.29 99.99 101.31 219.77 1296.61 77.88 88.65 91.31 174.61 995.04





Bper 6 9 12 12 13 11 9 5 7 10 9 9 8 7Bimp 7 9 10 10 10 9 6 5 7 8 8 7 7 5

MK=3Tfo,h

(rf:imp, X:AR(1)) 41 63 97 124 150 195 245 33 53 82 104 129 179 232

MK=3Tfo,h

(rf:imp, X:AR(3)) 52 71 103 127 150 192 240 41 60 87 109 130 176 225

MK=3Tfo,h

(rf:imp, X:AR(6)) 37 56 99 130 156 200 248 30 48 82 108 133 183 235

MK=3Tfo,h

(rf:imp, X:VAR(1)) 38 50 84 117 148 199 252 31 44 70 99 127 183 244

MK=3Tfo,h

(rf:imp, X:VAR(3)) 44 61 103 132 156 197 244 34 54 88 114 135 180 232

MK=3Tfo,h

(rf:imp, X:VAR(6)) 48 59 107 152 182 210 255 29 47 88 127 154 195 244

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 126 131 145 162 179 209 263 103 105 119 136 157 199 242

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 105 115 139 160 179 208 368 88 95 115 135 157 198 303

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 107 118 143 165 183 208 984 89 98 118 139 162 198 468

MK=3Tfo,h

(rf:per, X:AR(1)) 5 8 10 9 10 8 7 4 7 8 7 7 5 6

MK=3Tfo,h

(rf:per, X:AR(3)) 5 8 10 10 10 9 8 4 7 9 8 8 6 6

MK=3Tfo,h

(rf:per, X:AR(6)) 5 8 10 11 11 10 8 4 6 8 8 8 7 7

MK=3Tfo,h

(rf:per, X:VAR(1)) 5 8 11 12 12 11 9 4 6 9 10 10 10 9

MK=3Tfo,h

(rf:per, X:VAR(3)) 5 7 10 10 10 8 7 4 5 8 7 7 6 6

MK=3Tfo,h

(rf:per, X:VAR(6)) 6 8 10 10 10 10 8 4 6 8 8 8 8 7

MK=3Tfo,h

(rf:per, X:BVAR(1)) 7 8 8 8 9 15 218 6 6 6 6 7 10 105

MK=3Tfo,h

(rf:per, X:BVAR(3)) 8 8 9 9 9 22 339 7 7 7 7 7 15 202

MK=3Tfo,h

(rf:per, X:BVAR(6)) 8 9 8 8 9 22 939 7 7 7 6 6 13 361




Bper 24 36 48 49 51 45 39 19 29 41 36 37 33 31Bimp 25 34 41 40 39 30 15 20 27 32 31 29 23 12

MK=3Tfo,h

(rf:imp, X:AR(1)) 37 56 95 126 156 205 259 30 45 77 105 134 189 243

MK=3Tfo,h

(rf:imp, X:AR(3)) 46 62 97 125 153 201 254 37 51 79 105 132 184 237

MK=3Tfo,h

(rf:imp, X:AR(6)) 35 52 98 133 164 214 266 29 43 80 112 142 198 251

MK=3Tfo,h

(rf:imp, X:VAR(1)) 41 59 103 141 175 226 279 33 48 83 117 150 213 271

MK=3Tfo,h

(rf:imp, X:VAR(3)) 42 61 105 137 164 209 261 32 50 85 116 142 193 247

MK=3Tfo,h

(rf:imp, X:VAR(6)) 49 64 115 162 195 231 276 32 50 94 140 171 217 265

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 120 128 149 167 183 212 715 93 102 123 144 163 200 461

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 100 113 141 161 175 193 1396 78 89 114 137 156 181 846

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 106 119 147 167 182 215 5329 83 95 119 145 164 200 1678

MK=3Tfo,h

(rf:per, X:AR(1)) 21 35 43 40 42 34 27 16 28 37 30 33 24 23

MK=3Tfo,h

(rf:per, X:AR(3)) 21 35 44 44 46 39 32 17 28 38 34 35 29 26

MK=3Tfo,h

(rf:per, X:AR(6)) 22 35 44 47 49 42 35 17 28 38 35 36 30 29

MK=3Tfo,h

(rf:per, X:VAR(1)) 20 35 52 56 56 48 41 17 26 42 48 48 43 38

MK=3Tfo,h

(rf:per, X:VAR(3)) 20 32 45 44 45 38 33 16 25 39 32 34 30 28

MK=3Tfo,h

(rf:per, X:VAR(6)) 22 34 43 43 44 41 33 18 27 38 34 31 33 30

MK=3Tfo,h

(rf:per, X:BVAR(1)) 32 35 35 33 38 66 956 26 27 27 25 30 44 463

MK=3Tfo,h

(rf:per, X:BVAR(3)) 35 36 36 37 40 100 1493 29 30 29 27 30 67 904

MK=3Tfo,h

(rf:per, X:BVAR(6)) 36 37 35 31 36 112 5728 29 29 28 23 27 61 1906





Bper 42 63 84 87 90 80 75 34 52 72 69 70 63 59Bimp 42 59 71 66 60 44 42 34 50 57 52 47 34 35

MK=3Tfo,h

(rf:imp, X:AR(1)) 41 67 109 138 170 215 261 32 52 87 117 150 195 245

MK=3Tfo,h

(rf:imp, X:AR(3)) 46 68 107 133 164 210 256 37 54 86 114 146 190 238

MK=3Tfo,h

(rf:imp, X:AR(6)) 41 65 111 145 179 224 271 32 52 91 126 160 205 254

MK=3Tfo,h

(rf:imp, X:VAR(1)) 48 79 134 172 205 246 291 39 62 110 148 181 231 285

MK=3Tfo,h

(rf:imp, X:VAR(3)) 45 72 120 148 176 219 266 35 55 96 125 156 200 251

MK=3Tfo,h

(rf:imp, X:VAR(6)) 52 76 126 166 199 239 280 37 60 104 144 177 222 270

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 105 117 140 156 171 209 1336 82 98 121 135 148 193 747

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 87 102 128 142 150 197 2587 70 83 109 122 131 175 1580

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 97 110 131 144 155 288 15443 78 93 111 125 136 198 4166

MK=3Tfo,h

(rf:per, X:AR(1)) 37 63 82 74 77 64 43 29 52 72 59 59 46 35

MK=3Tfo,h

(rf:per, X:AR(3)) 38 63 84 82 85 72 50 30 51 74 66 64 53 40

MK=3Tfo,h

(rf:per, X:AR(6)) 39 63 82 87 90 76 57 31 52 72 69 66 55 43

MK=3Tfo,h

(rf:per, X:VAR(1)) 38 70 103 110 107 84 71 31 55 83 93 91 69 62

MK=3Tfo,h

(rf:per, X:VAR(3)) 37 63 90 86 86 74 58 29 50 79 69 66 56 48

MK=3Tfo,h

(rf:per, X:VAR(6)) 38 63 83 76 74 67 50 30 53 73 60 51 51 41

MK=3Tfo,h

(rf:per, X:BVAR(1)) 76 75 67 56 69 129 1773 63 60 54 43 53 87 865

MK=3Tfo,h

(rf:per, X:BVAR(3)) 76 76 70 68 78 195 2792 63 61 56 53 62 132 1764

MK=3Tfo,h

(rf:per, X:BVAR(6)) 76 76 66 57 75 303 19543 63 63 52 48 62 141 5106




Bper 51 73 98 101 104 98 93 40 61 83 84 83 81 73Bimp 48 68 79 69 63 66 86 39 58 64 54 51 58 76

MK=3Tfo,h

(rf:imp, X:AR(1)) 40 69 111 136 165 205 243 32 54 86 113 146 185 228

MK=3Tfo,h

(rf:imp, X:AR(3)) 44 70 108 131 160 201 238 35 54 84 110 143 181 219

MK=3Tfo,h

(rf:imp, X:AR(6)) 40 67 111 141 173 213 252 31 54 88 120 154 193 236

MK=3Tfo,h

(rf:imp, X:VAR(1)) 46 81 136 171 201 233 273 36 63 110 145 177 216 267

MK=3Tfo,h

(rf:imp, X:VAR(3)) 44 73 120 144 169 208 247 33 55 94 120 151 189 234

MK=3Tfo,h

(rf:imp, X:VAR(6)) 52 77 121 153 182 222 257 38 60 98 129 162 202 248

MK=3Tfo,h

(rf:imp, X:BVAR(1)) 101 110 128 138 153 196 1489 83 96 113 121 130 177 822

MK=3Tfo,h

(rf:imp, X:BVAR(3)) 86 96 114 121 126 193 2925 72 82 99 104 107 162 1872

MK=3Tfo,h

(rf:imp, X:BVAR(6)) 97 106 115 120 130 364 22712 83 92 99 103 107 192 5778

MK=3Tfo,h

(rf:per, X:AR(1)) 40 68 89 78 79 66 44 31 56 79 64 62 50 35

MK=3Tfo,h

(rf:per, X:AR(3)) 41 68 91 87 88 75 50 32 56 81 73 68 57 39

MK=3Tfo,h

(rf:per, X:AR(6)) 41 66 88 92 93 77 57 33 55 78 75 70 57 43

MK=3Tfo,h

(rf:per, X:VAR(1)) 41 76 113 117 111 81 70 33 61 91 99 95 61 60

MK=3Tfo,h

(rf:per, X:VAR(3)) 40 70 101 94 92 79 61 32 57 88 79 71 60 52

MK=3Tfo,h

(rf:per, X:VAR(6)) 43 70 93 83 78 68 49 33 59 81 68 57 53 40

MK=3Tfo,h

(rf:per, X:BVAR(1)) 104 97 81 65 83 150 2006 86 80 66 55 63 102 965

MK=3Tfo,h

(rf:per, X:BVAR(3)) 104 101 91 86 99 222 3149 84 82 77 72 82 160 2084

MK=3Tfo,h

(rf:per, X:BVAR(6)) 104 102 88 83 106 440 30965 87 84 71 71 87 196 7296





Bper 6 9 12 12 13 11 9 5 7 10 9 9 8 7Bimp 7 9 10 10 10 9 6 5 7 8 8 7 7 5

MK=5Tfo,h

(rf:imp, X:AR(1)) 37 59 95 122 148 194 244 30 50 79 102 127 178 230

MK=5Tfo,h

(rf:imp, X:AR(3)) 48 67 100 126 149 191 240 37 56 84 107 128 175 225

MK=5Tfo,h

(rf:imp, X:AR(6)) 34 53 96 129 155 199 248 27 44 79 107 132 182 234

MK=5Tfo,h

(rf:imp, X:VAR(1)) 37 47 81 115 147 199 250 31 41 68 97 126 182 242

MK=5Tfo,h

(rf:imp, X:VAR(3)) 60 70 110 137 157 193 237 44 57 93 119 137 176 221

MK=5Tfo,h

(rf:imp, X:VAR(6)) 49 62 106 144 180 214 252 33 50 87 118 152 200 243

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 122 126 143 162 179 206 647 100 102 118 135 156 194 368

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 109 114 135 155 174 205 1140 91 94 112 131 153 190 619

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 106 115 138 162 182 215 6262 86 94 114 136 159 202 2209

MK=5Tfo,h

(rf:per, X:AR(1)) 6 10 12 11 11 10 8 5 7 10 8 9 7 7

MK=5Tfo,h

(rf:per, X:AR(3)) 6 10 12 12 13 11 10 5 8 10 10 10 8 8

MK=5Tfo,h

(rf:per, X:AR(6)) 7 10 12 13 13 12 10 5 8 10 9 10 8 9

MK=5Tfo,h

(rf:per, X:VAR(1)) 6 9 13 15 15 14 11 5 6 11 12 13 12 11

MK=5Tfo,h

(rf:per, X:VAR(3)) 6 9 13 12 13 12 13 5 7 10 10 10 10 10

MK=5Tfo,h

(rf:per, X:VAR(6)) 8 10 15 16 18 20 14 6 8 13 13 14 15 12

MK=5Tfo,h

(rf:per, X:BVAR(1)) 7 8 10 10 11 30 819 6 6 7 7 8 16 270

MK=5Tfo,h

(rf:per, X:BVAR(3)) 8 9 10 10 12 38 1251 6 7 8 7 8 23 546

MK=5Tfo,h

(rf:per, X:BVAR(6)) 9 9 9 9 13 87 6049 7 7 7 6 9 39 2224




Bper 24 36 48 49 51 45 39 19 29 41 36 37 33 31Bimp 25 34 41 40 39 30 15 20 27 32 31 29 23 12

MK=5Tfo,h

(rf:imp, X:AR(1)) 36 55 95 126 157 206 260 29 44 75 105 134 189 244

MK=5Tfo,h

(rf:imp, X:AR(3)) 46 61 97 125 154 202 255 36 50 78 105 132 184 238

MK=5Tfo,h

(rf:imp, X:AR(6)) 34 52 99 134 166 215 268 28 42 79 112 143 198 252

MK=5Tfo,h

(rf:imp, X:VAR(1)) 42 60 104 142 177 227 279 35 49 83 119 152 214 271

MK=5Tfo,h

(rf:imp, X:VAR(3)) 58 70 108 134 158 201 252 43 56 88 114 137 183 234

MK=5Tfo,h

(rf:imp, X:VAR(6)) 51 70 116 162 204 247 281 36 56 97 142 181 233 272

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 118 126 150 169 185 221 2228 96 103 125 147 167 212 966

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 106 113 138 155 171 213 4461 83 90 112 133 153 188 2019

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 104 116 143 166 182 392 26100 81 92 117 143 162 264 8204

MK=5Tfo,h

(rf:per, X:AR(1)) 23 37 47 44 46 38 30 18 29 40 33 35 27 25

MK=5Tfo,h

(rf:per, X:AR(3)) 24 38 49 48 50 43 35 19 29 42 37 38 32 29

MK=5Tfo,h

(rf:per, X:AR(6)) 25 39 48 51 53 47 39 19 30 41 38 39 33 31

MK=5Tfo,h

(rf:per, X:VAR(1)) 23 38 57 64 63 56 47 19 29 47 55 54 48 44

MK=5Tfo,h

(rf:per, X:VAR(3)) 23 35 51 50 53 50 52 18 27 42 39 40 39 37

MK=5Tfo,h

(rf:per, X:VAR(6)) 28 39 59 61 64 70 53 21 31 52 49 51 54 43

MK=5Tfo,h

(rf:per, X:BVAR(1)) 29 34 38 38 43 107 2677 24 26 29 28 33 63 1009

MK=5Tfo,h

(rf:per, X:BVAR(3)) 33 36 40 39 42 121 4259 26 28 33 28 32 83 1906

MK=5Tfo,h

(rf:per, X:BVAR(6)) 36 38 36 34 52 337 23245 28 30 28 23 35 140 7760





Bper 42 63 84 87 90 80 75 34 52 72 69 70 63 59Bimp 42 59 71 66 60 44 42 34 50 57 52 47 34 35

MK=5Tfo,h

(rf:imp, X:AR(1)) 40 66 108 137 169 213 260 31 52 86 115 149 194 244

MK=5Tfo,h

(rf:imp, X:AR(3)) 46 68 107 133 164 209 255 36 54 86 114 146 190 237

MK=5Tfo,h

(rf:imp, X:AR(6)) 39 64 110 144 178 223 270 31 51 91 125 159 204 254

MK=5Tfo,h

(rf:imp, X:VAR(1)) 50 80 134 173 206 245 291 41 65 111 150 182 231 284

MK=5Tfo,h

(rf:imp, X:VAR(3)) 55 75 114 135 161 207 254 43 58 93 115 139 185 237

MK=5Tfo,h

(rf:imp, X:VAR(6)) 55 78 126 168 208 256 289 42 60 103 149 185 237 277

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 102 115 143 160 175 236 3182 82 95 121 140 157 215 1509

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 91 103 128 139 150 244 9463 70 84 107 119 131 201 3873

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 93 106 132 148 169 702 50552 78 92 113 129 147 315 14015

MK=5Tfo,h

(rf:per, X:AR(1)) 36 62 81 73 76 63 42 29 51 71 58 58 46 34

MK=5Tfo,h

(rf:per, X:AR(3)) 38 62 83 81 83 71 49 30 50 73 65 63 53 38

MK=5Tfo,h

(rf:per, X:AR(6)) 39 62 81 86 88 75 55 31 51 71 68 65 54 41

MK=5Tfo,h

(rf:per, X:VAR(1)) 39 69 103 113 110 89 75 32 55 84 97 94 72 64

MK=5Tfo,h

(rf:per, X:VAR(3)) 37 59 89 86 89 85 81 30 47 76 69 67 65 55

MK=5Tfo,h

(rf:per, X:VAR(6)) 45 65 100 95 88 91 77 34 53 89 74 66 72 61

MK=5Tfo,h

(rf:per, X:BVAR(1)) 61 65 64 60 71 170 3637 50 53 51 46 56 112 1628

MK=5Tfo,h

(rf:per, X:BVAR(3)) 63 66 69 64 73 207 8201 51 53 57 50 62 144 3540

MK=5Tfo,h

(rf:per, X:BVAR(6)) 72 72 65 60 98 632 44618 58 59 52 45 66 252 13760




Bper 51 73 98 101 104 98 93 40 61 83 84 83 81 73Bimp 48 68 79 69 63 66 86 39 58 64 54 51 58 76

MK=5Tfo,h

(rf:imp, X:AR(1)) 40 69 112 136 166 205 243 32 54 86 112 146 185 228

MK=5Tfo,h

(rf:imp, X:AR(3)) 45 70 109 132 161 202 238 36 55 84 110 143 181 219

MK=5Tfo,h

(rf:imp, X:AR(6)) 41 68 113 143 174 214 254 33 54 89 121 156 195 238

MK=5Tfo,h

(rf:imp, X:VAR(1)) 50 84 139 175 206 237 277 40 67 112 150 182 221 271

MK=5Tfo,h

(rf:imp, X:VAR(3)) 53 76 115 131 155 199 239 41 58 90 109 135 177 221

MK=5Tfo,h

(rf:imp, X:VAR(6)) 56 81 125 158 192 238 274 44 61 98 137 168 218 260

MK=5Tfo,h

(rf:imp, X:BVAR(1)) 87 101 127 141 156 236 3318 71 85 109 121 134 206 1687

MK=5Tfo,h

(rf:imp, X:BVAR(3)) 80 92 113 119 129 269 12487 65 78 95 100 109 202 4837

MK=5Tfo,h

(rf:imp, X:BVAR(6)) 87 98 116 127 152 809 58884 76 87 101 111 124 319 15628

MK=5Tfo,h

(rf:per, X:AR(1)) 40 68 91 80 80 68 45 31 56 80 65 62 51 36

MK=5Tfo,h

(rf:per, X:AR(3)) 42 68 93 88 89 76 50 33 56 82 74 68 58 39

MK=5Tfo,h

(rf:per, X:AR(6)) 43 68 90 93 93 79 57 34 56 79 76 70 58 43

MK=5Tfo,h

(rf:per, X:VAR(1)) 44 78 115 124 119 90 79 35 64 93 106 101 67 65

MK=5Tfo,h

(rf:per, X:VAR(3)) 42 67 101 97 96 94 84 33 54 87 81 73 72 60

MK=5Tfo,h

(rf:per, X:VAR(6)) 53 74 113 103 85 82 78 39 62 98 80 65 63 60

MK=5Tfo,h

(rf:per, X:BVAR(1)) 82 83 77 69 84 196 3672 68 69 65 57 68 138 1804

MK=5Tfo,h

(rf:per, X:BVAR(3)) 83 83 83 79 92 252 10740 67 68 70 65 77 173 4387

MK=5Tfo,h

(rf:per, X:BVAR(6)) 96 94 85 81 124 740 52429 78 78 70 64 87 296 15329





Bper 6 9 12 12 13 11 9 5 7 10 9 9 8 7Bimp 7 9 10 10 10 9 6 5 7 8 8 7 7 5

MK=7Tfo,h

(rf:imp, X:AR(1)) 43 64 98 125 150 195 246 34 54 83 106 130 182 233

MK=7Tfo,h

(rf:imp, X:AR(3)) 53 72 104 128 151 193 241 41 60 88 110 132 178 226

MK=7Tfo,h

(rf:imp, X:AR(6)) 40 58 99 131 157 201 249 31 49 83 110 135 186 237

MK=7Tfo,h

(rf:imp, X:VAR(1)) 37 49 83 118 151 202 253 30 43 71 100 131 188 247

MK=7Tfo,h

(rf:imp, X:VAR(3)) 57 70 107 134 156 201 246 40 55 93 117 135 188 236

MK=7Tfo,h

(rf:imp, X:VAR(6)) 48 57 99 146 183 236 273 35 46 86 125 157 223 269

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 125 127 140 157 175 273 9457 105 102 116 132 154 215 1494

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 102 108 131 153 172 256 12553 89 89 108 130 151 214 2995

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 110 120 143 165 185 239 13035 89 97 118 139 163 210 3857

MK=7Tfo,h

(rf:per, X:AR(1)) 6 10 12 11 11 10 8 5 8 10 8 9 7 7

MK=7Tfo,h

(rf:per, X:AR(3)) 7 10 12 12 13 11 10 5 8 10 10 10 8 8

MK=7Tfo,h

(rf:per, X:AR(6)) 7 10 12 13 13 12 10 5 8 10 10 10 8 9

MK=7Tfo,h

(rf:per, X:VAR(1)) 6 9 13 15 15 14 11 5 7 11 13 13 12 11

MK=7Tfo,h

(rf:per, X:VAR(3)) 7 9 12 11 12 12 10 5 7 10 8 8 9 9

MK=7Tfo,h

(rf:per, X:VAR(6)) 10 11 17 18 24 32 24 7 10 14 15 20 23 16

MK=7Tfo,h

(rf:per, X:BVAR(1)) 7 9 10 14 29 194 9290 6 7 8 8 12 40 1398

MK=7Tfo,h

(rf:per, X:BVAR(3)) 9 9 11 14 28 204 17156 7 7 8 8 12 60 3821

MK=7Tfo,h

(rf:per, X:BVAR(6)) 9 9 10 10 17 168 25083 7 7 7 7 10 57 5533




Bper 24 36 48 49 51 45 39 19 29 41 36 37 33 31Bimp 25 34 41 40 39 30 15 20 27 32 31 29 23 12

MK=7Tfo,h

(rf:imp, X:AR(1)) 41 59 97 128 158 207 260 33 47 79 108 136 192 245

MK=7Tfo,h

(rf:imp, X:AR(3)) 50 65 99 127 155 203 255 39 53 82 108 134 187 238

MK=7Tfo,h

(rf:imp, X:AR(6)) 39 56 101 135 167 216 269 32 46 83 116 145 201 253

MK=7Tfo,h

(rf:imp, X:VAR(1)) 43 61 106 146 181 231 282 36 50 87 123 158 221 277

MK=7Tfo,h

(rf:imp, X:VAR(3)) 58 73 111 140 169 221 268 43 58 97 121 149 208 257

MK=7Tfo,h

(rf:imp, X:VAR(6)) 48 63 119 171 222 288 307 38 53 106 148 198 273 293

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 120 127 147 168 205 783 36868 97 103 123 148 178 311 5428

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 100 109 136 157 185 680 40868 81 86 111 137 167 292 10020

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 109 121 148 170 189 388 40522 85 96 122 145 167 274 11991

MK=7Tfo,h

(rf:per, X:AR(1)) 23 37 47 44 46 38 30 18 29 40 33 35 27 25

MK=7Tfo,h

(rf:per, X:AR(3)) 24 38 49 48 50 43 35 19 29 42 37 38 32 29

MK=7Tfo,h

(rf:per, X:AR(6)) 25 39 48 51 53 47 38 19 30 41 38 39 33 31

MK=7Tfo,h

(rf:per, X:VAR(1)) 24 39 58 65 63 56 47 20 29 49 56 55 48 44

MK=7Tfo,h

(rf:per, X:VAR(3)) 24 34 48 46 48 45 40 19 28 41 35 33 35 35

MK=7Tfo,h

(rf:per, X:VAR(6)) 30 43 61 68 87 112 87 22 37 53 54 72 80 58

MK=7Tfo,h

(rf:per, X:BVAR(1)) 29 36 40 57 115 761 36349 24 28 31 34 48 162 5495

MK=7Tfo,h

(rf:per, X:BVAR(3)) 33 35 42 54 105 711 52014 25 28 33 35 47 213 12418

MK=7Tfo,h

(rf:per, X:BVAR(6)) 37 37 37 35 55 499 70878 29 28 28 26 35 188 17081





Bper 42 63 84 87 90 80 75 34 52 72 69 70 63 59Bimp 42 59 71 66 60 44 42 34 50 57 52 47 34 35

MK=7Tfo,h

(rf:imp, X:AR(1)) 43 67 110 138 169 214 260 34 53 88 118 149 194 245

MK=7Tfo,h

(rf:imp, X:AR(3)) 49 70 108 134 164 209 255 39 56 88 116 146 189 237

MK=7Tfo,h

(rf:imp, X:AR(6)) 42 66 112 145 178 224 271 34 53 93 127 159 205 255

MK=7Tfo,h

(rf:imp, X:VAR(1)) 51 81 137 178 210 248 294 42 66 115 156 187 237 290

MK=7Tfo,h

(rf:imp, X:VAR(3)) 57 77 118 146 177 227 271 44 61 99 127 155 209 259

MK=7Tfo,h

(rf:imp, X:VAR(6)) 48 73 130 179 232 300 322 38 61 111 155 204 280 296

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 102 116 142 169 243 1245 58819 80 94 120 150 188 392 8844

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 89 101 130 150 199 1043 54346 69 79 109 132 160 354 13928

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 99 111 135 151 165 453 45118 83 96 117 129 144 291 14260

MK=7Tfo,h

(rf:per, X:AR(1)) 37 62 81 73 76 63 42 29 51 71 58 58 46 34

MK=7Tfo,h

(rf:per, X:AR(3)) 38 63 83 81 83 71 49 30 50 73 65 63 53 39

MK=7Tfo,h

(rf:per, X:AR(6)) 39 63 81 85 88 75 55 31 51 71 68 65 54 41

MK=7Tfo,h

(rf:per, X:VAR(1)) 39 71 105 115 109 87 74 33 57 87 99 94 71 63

MK=7Tfo,h

(rf:per, X:VAR(3)) 38 59 88 86 81 69 60 31 48 77 70 58 52 49

MK=7Tfo,h

(rf:per, X:VAR(6)) 46 71 96 102 117 140 133 36 60 84 84 100 97 88

MK=7Tfo,h

(rf:per, X:BVAR(1)) 59 66 67 95 196 1243 58726 49 52 51 59 85 275 9041

MK=7Tfo,h

(rf:per, X:BVAR(3)) 55 60 70 92 168 1042 60848 44 49 56 64 88 336 16484

MK=7Tfo,h

(rf:per, X:BVAR(6)) 73 71 64 56 85 645 86222 60 59 51 46 62 264 21615




Bper 51 73 98 101 104 98 93 40 61 83 84 83 81 73Bimp 48 68 79 69 63 66 86 39 58 64 54 51 58 76

MK=7Tfo,h

(rf:imp, X:AR(1)) 43 70 112 136 165 205 244 34 55 87 114 146 184 229

MK=7Tfo,h

(rf:imp, X:AR(3)) 47 72 110 132 161 201 239 37 56 86 111 143 181 220

MK=7Tfo,h

(rf:imp, X:AR(6)) 43 69 113 143 174 214 254 34 55 91 123 156 194 239

MK=7Tfo,h

(rf:imp, X:VAR(1)) 51 85 142 180 209 240 281 42 69 116 155 186 225 276

MK=7Tfo,h

(rf:imp, X:VAR(3)) 54 77 118 141 170 215 253 42 60 94 120 147 193 242

MK=7Tfo,h

(rf:imp, X:VAR(6)) 48 76 128 167 214 275 312 39 61 102 142 186 253 282

MK=7Tfo,h

(rf:imp, X:BVAR(1)) 86 102 127 153 241 1297 60799 67 84 109 132 172 394 9358

MK=7Tfo,h

(rf:imp, X:BVAR(3)) 76 89 115 136 196 1157 56023 59 70 96 114 140 372 14412

MK=7Tfo,h

(rf:imp, X:BVAR(6)) 93 102 119 127 139 431 34795 82 92 104 108 119 258 11984

MK=7Tfo,h

(rf:per, X:AR(1)) 40 68 91 80 80 68 45 31 56 80 65 62 51 36

MK=7Tfo,h

(rf:per, X:AR(3)) 42 68 93 88 89 76 50 33 56 82 74 68 58 39

MK=7Tfo,h

(rf:per, X:AR(6)) 43 68 90 92 93 79 57 34 56 79 76 70 57 42

MK=7Tfo,h

(rf:per, X:VAR(1)) 44 80 117 125 117 88 77 36 66 95 107 100 65 64

MK=7Tfo,h

(rf:per, X:VAR(3)) 43 68 102 98 88 71 61 35 56 90 82 66 53 51

MK=7Tfo,h

(rf:per, X:VAR(6)) 55 83 107 106 113 120 139 43 69 92 90 96 85 91

MK=7Tfo,h

(rf:per, X:BVAR(1)) 79 82 78 104 214 1309 61292 67 67 63 69 100 306 9618

MK=7Tfo,h

(rf:per, X:BVAR(3)) 67 72 83 108 189 1119 57418 55 60 67 80 111 383 16498

MK=7Tfo,h

(rf:per, X:BVAR(6)) 97 94 86 77 107 616 75063 81 79 72 65 84 276 19273



Figure 82: Mean absolute slope forecasting errors of autoregressive models in 10-year windowset-up

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Notes: (1) all errors are reported in basis points. (2) first 3 rows of tiles consist of the models where risk free short interestrates are implied by the curve (whole path), next 3 rows of tiles - where risk free short interest rate is the one prevailing at the

moment of making the forecast. (3) red lines are naive benchmarks: the thicker one is Bslope:per and the thinner one isBslope:imp. (4) there are 3 models plotted in each tile for K = {3, 5, 7}, the thicker the line the higher number of factors.


Figure 83: Mean absolute slope forecasting errors of vector autoregressive models in 10-yearwindow set-up

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Figure 84: Mean absolute slope forecasting errors of Baysian vector autoregressive modelsin 10-year window set-up

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Figure 85: Mean absolute slope forecasting errors of autoregressive models in 5-year windowset-up

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Figure 86: Mean absolute slope forecasting errors of vector autoregressive models in 5-yearwindow set-up

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Figure 87: Mean absolute slope forecasting errors of Baysian vector autoregressive modelsin 5-year window set-up

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Figure 88: Expectations hypothesis testing adjusted by term premia regression of Thorn-ton (2006) for Poland - robustness checks - various numbers of factors

K=36 12 24 36 48 60 72


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Notes: (1) 0 indicates that there is no sufficient evidence to reject null hypothesis of EH, (2) 1 - sufficient evidence (with at0.05 significance) to reject EH, (3) -1 - not applicable (i.e. a horizon is longer or equal bond’s maturity) (4) NSS fit using

weight system labelled 1. (5) Term premia are estimated at for the full sample here


List of Figures

1 Average yields in segments daily in the period of 2005:01-2020:06 . . . . . . . . . . . 343 Proposed segmentation of bonds on BondSpot in the period 2005:01-2020:06 . . . . . 364 Turnover of fixed coupon bonds versus NBP bills monthly sales . . . . . . . . . . . . 362 Selected characteristics of Polish fixed coupon government bonds traded on BondSpot

in 2005:01-2020:06 by segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Selected averaged liquidity measures of 2Y, 5Y and 10Y fixed coupon government

bonds traded on BondSpot platform . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Averaged turnover of fixed coupon 2, 5, 10Y government bonds during their lifespan

in months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Maximum time to maturity of bonds eligible to switch . . . . . . . . . . . . . . . . . 408 Switch spread of Polish fixed coupon government bonds in 2005:01-2020:06 . . . . . . 419 Share of BondSpot in total market turnover 2005:01-2020:06 . . . . . . . . . . . . . . 4210 Selected liquidity measures of Polish fixed coupon government bonds traded on BondSpot

in 2005:01-2020:06 by segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311 Selected characteristics of Polish fixed coupon government bonds traded on BondSpot

in 2005:01-2020:06 - all segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412 Goodness-of-fit and smoothness of all 28 tested weight systems . . . . . . . . . . . . 5113 Full history of estimated zero coupon yield curves in Poland . . . . . . . . . . . . . . 5414 An example of estimated NSS family of zero coupon forward curves . . . . . . . . . . 5415 An example of estimated par curve with rich/cheap analysis . . . . . . . . . . . . . . 5516 Level, slope and curvature of NSS estimated Polish zero coupon yield curve . . . . . 5617 Distributions of one month excess returns for Polish bonds . . . . . . . . . . . . . . . 5718 Pure expectations hypothesis: regression of type 1 and 2 (Fama & Bliss) for Poland 6019 Type 1 regression: a modo Fama & Bliss (1987) -Term premium regressions, Poland

2005:01-2020:06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6220 Type 2 regression: a modo Fama & Bliss (1987) - Forecasts of the change in the spot

rate, Poland 2005:01-2020:06) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421 Pure expectations hypothesis testing: conventional regression a modo Thornton (2006)

for Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6722 Smile and smirk in coefficients of conventional regressions a modo Thornton (2006)

for Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823 Rolling realised returns on term premia regressions, Poland 2005:01-2020:06 . . . . . 6924 Smirks in coefficients of contrarian regressions a modo Thornton (2006) for Poland . 7025 Pure expectations hypothesis testing: contrarian regression a modo Thornton (2006)

for Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7026 Realised spread on pro rata temporis current spread regressions, Poland 2005:01-2020:06 7127 Correlation structure of the Polish zero coupon rates . . . . . . . . . . . . . . . . . . 7428 Loadings in 7 factor PCA of the Polish yield curve . . . . . . . . . . . . . . . . . . . 7529 Time series of factors from PCA of the Polish yield curve . . . . . . . . . . . . . . . 7630 Decomposition of expected 10y excess returns in ACM model for Poland . . . . . . 8031 Model implied loadings in ACM five factor model for Poland in five factor model . . 8132 Regression coefficients compared with recursive pricing parameters in ACM seven

factor model for Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8233 The average term premia with one standard deviation bands by tenors for the Polish

yield curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8434 Evolution of term premia for selected tenors of Polish yield curve . . . . . . . . . . . 8535 Term premia (10Y) for different short interest rates in ACM model for Poland . . . 8836 Term premia (10Y) for different end dates of a sample in ACM model for Poland . 8937 Term premia (10Y) for different start dates of a sample in ACM model for Poland . 8938 Term premia in 10Y tenor - blended time series for Poland . . . . . . . . . . . . . . 90


39 Expectations hypothesis testing adjusted by term premia regression of Thornton(2006) for Poland, five factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

40 R2 values for PEH and EH testing for various longer maturity n and short bond h . 9341 Extended Thornton’s regression: Rolling realised returns on term slope and term

premia regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9542 Extended Thornton’s regression: Rolling realised returns on term slope and term

premia regressions, blended term premia time series . . . . . . . . . . . . . . . . . . . 9643 Best performing models by number of factors, tenors and horizons, 10-year window,

level forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10244 Best performing models’ errors by number of factors, tenors and horizons, 10-year

window, level forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10345 Best performing models by number of factors, tenors and horizons, 5-year window,

level forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10446 Best performing models’ errors by number of factors, tenors and horizons, 5-year

window, level forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10547 Mean absolute forecasting errors of autoregressive models in 10-year window set-up,

level forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10748 Mean absolute forecasting errors of vector autoregressive models in 10-year window

set-up, level forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10849 Mean absolute forecasting errors of vector autoregressive models in 5-year window

set-up, level forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10950 Best performing models by number of factors, tenors and horizons, 10-year window,

slope forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11251 Best performing models’ errors by number of factors, tenors and horizons, 10-year

window, slope forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11352 Best performing models by number of factors, tenors and horizons, 5-year window,

slope forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11553 Best performing models’ errors by number of factors, tenors and horizons, 5-year

window, slope forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11654 Short risk-free interest rates forecasts for selected dates . . . . . . . . . . . . . . . . 11755 Various short risk-free interest rates forecasts and its ex-post realisation path . . . . 12256 Selected characteristics of Polish fixed coupon government bonds traded on BondSpot

in 2005:01-2020:06 - short term bonds < 1.5 Y . . . . . . . . . . . . . . . . . . . . . 14257 Selected characteristics of Polish fixed coupon government bonds traded on BondSpot

in 2005:01-2020:06 - bonds with time to maturity between 1.5 and 3.5 . . . . . . . . 14358 Selected characteristics of Polish fixed coupon government bonds traded on BondSpot

in 2005:01-2020:06 - bonds with time to maturity between 3.5 and 6.0 . . . . . . . . 14459 Selected characteristics of Polish fixed coupon government bonds traded on BondSpot

in 2005:01-2020:06- bonds with time to maturity between 6.0 and 12.0 . . . . . . . . 14560 Selected characteristics of Polish fixed coupon government bonds traded on BondSpot

in 2005:01-2020:06 - ultra long bonds > 12 Y . . . . . . . . . . . . . . . . . . . . . . 14661 Averaged turnover share of fixed coupon government bonds of 2, 5, 10Y types during

their lifespan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14762 Selected averaged liquidity measures of 2Y fixed coupon government bonds . . . . . 14863 Selected averaged liquidity measures of 5Y fixed coupon government bonds . . . . . 14964 Selected averaged liquidity measures of 10Y fixed coupon government bonds . . . . . 15065 Selected liquidity measures of Polish fixed coupon government bonds traded on BondSpot

in 2005:01-2020:06 - by segments (in years) . . . . . . . . . . . . . . . . . . . . . . . 15166 Timeseries of the best and worst performer in the ranking . . . . . . . . . . . . . . . 15267 NSS parameters in estimated Polish zero coupon yield curve . . . . . . . . . . . . . . 15368 Robustness checks on null hypothesis decisions by system of weights - regression type

1 Fama & Bliss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154


69 Robustness checks on null hypothesis decisions by system of weights - regression type2 Fama & Bliss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

70 Regression coefficients of one year excess returns on forward rates - Cochrane & Piazzesi 15671 Robustness checks on null hypothesis decisions by system of weights in rolling realised

returns on term premia regressions - Thornton - conventional . . . . . . . . . . . . . 15772 Robustness checks on null hypothesis decisions by system of weights in rolling realised

returns on term premia regressions - Thornton - contrarian . . . . . . . . . . . . . . 15873 Decomposition of expected 1, 2 and 3-year excess returns in ACM model for Poland 16274 Decomposition of expected 5 and 8-year excess returns in ACM model for Poland . 16375 Term premia in 1Y tenor - blended time series for Poland . . . . . . . . . . . . . . . 16476 Term premia in 2Y tenor - blended time series for Poland . . . . . . . . . . . . . . . 16477 Term premia in 3Y tenor - blended time series for Poland . . . . . . . . . . . . . . . 16578 Term premia in 5Y tenor - blended time series for Poland . . . . . . . . . . . . . . . 16579 Mean absolute level forecasting errors of Bayesian vector autoregressive models in

10-year window set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18180 Mean absolute level forecasting errors of autoregressive models in 5-year window set-up18281 Mean absolute level forecasting errors of Bayesian vector autoregressive models in

5-year window set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18382 Mean absolute slope forecasting errors of autoregressive models in 10-year window

set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19983 Mean absolute slope forecasting errors of vector autoregressive models in 10-year

window set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20084 Mean absolute slope forecasting errors of Baysian vector autoregressive models in

10-year window set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20185 Mean absolute slope forecasting errors of autoregressive models in 5-year window

set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20286 Mean absolute slope forecasting errors of vector autoregressive models in 5-year win-

dow set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20387 Mean absolute slope forecasting errors of Baysian vector autoregressive models in

5-year window set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20488 Expectations hypothesis testing adjusted by term premia regression of Thornton

(2006) for Poland - robustness checks - various numbers of factors . . . . . . . . . . 205

List of Tables

1 Dates with less then 10 eligible bonds on BondSpot fixing . . . . . . . . . . . . . . . 352 Statistics of liquidity measures of Polish fixed coupon government bonds traded on

BondSpot in 2005:01-2020:06 - by segments . . . . . . . . . . . . . . . . . . . . . . . 403 KPSS tests with different lags in 2005:01-2020:06 - by segments . . . . . . . . . . . . 424 List of starting vectors of parameters Θ . . . . . . . . . . . . . . . . . . . . . . . . . 505 Volatility, error and smoothness statistics for all 28 tested weight systems . . . . . . 526 Descriptive statistics of zero coupon rates for different tenors and systems . . . . . . 537 NSS parameters yearly averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 Descriptive statistics of NSS fitted yields in Poland, 2005:01-2020:06 . . . . . . . . . 599 One year excess return on average one year forward rates regressions, a modo Cochrane

(2005) , Poland 2005:01-2020:06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6510 Multiregressions of excess one year return on return-forecasting error and Fama &

Bliss term premium, Poland 2005:01-2020:06 . . . . . . . . . . . . . . . . . . . . . . . 6511 Regression coefficients of one-year excess returns on forward rates, Poland 2005:01-

2020:06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612 PCA explained variance (in pct points) by factors for the Polish yield curve . . . . . 7413 Correlation of factors from PCA with selected economic indicators . . . . . . . . . . 77


14 Market prices of risk for seven factor ACM model of the Polish interest rates . . . . 7915 Response of Polish government yield curve to shocks to factors . . . . . . . . . . . . 7916 Response of one-month excess returns on Polish government bonds to shocks to factors 8017 Fit diagnostics for seven factor ACM model of the Polish yield curve . . . . . . . . . 8318 Yield pricing errors for different ACM model specifications . . . . . . . . . . . . . . . 8319 Identification tests for various numbers of factors K in ACM model for Poland . . . 8420 Correlation of term premia with selected economic indicators in Poland . . . . . . . 8621 Mean and standard deviation of term premia datasets for different systems of

weights for the Polish yield curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8722 Mean and standard deviation of term premia in models with different number of

factors for the Polish yield curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8823 Descriptive statistics of term premia by tenors - comparison of blended and normal

approaches for Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9124 Forecast errors for 10-year tenor in different models with K = 5, 10-year window,

level forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025 Forecast errors for 10-year tenor in different models with K = 5, 10-year window,

slope forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11026 Forecast errors of short (1-month) interest rates in different models, 5-year window . 11827 Standard deviations of forecast errors of short (1-month) interest rates in different

models, 5-year window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11928 Polish professional forecasters implied averaged annually short term interest rates . . 12029 Time series comparison statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12130 Co-integration tests of three groups of time series: professional forecasters (PF), yield

curve implied (YC) and ACM model implied risk free (RF) . . . . . . . . . . . . . . 12231 Mean absolute forecast errors: professional forecasters (PF), yield curve implied (YC)

and ACM model implied risk free (RF) . . . . . . . . . . . . . . . . . . . . . . . . . . 12332 Descriptive statistics of switch spread of Polish fixed coupon government bonds in

2005:01-2020:06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15233 Market prices of risk for N=3, 4, 5, 6 and 7 factor ACM model of Polish term structure15934 Mean and standard deviation of term premia datasets for different systems of weights

for the Polish yield curve for K = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16035 Mean and standard deviation of term premia datasets for different systems of weights

for the Polish yield curve for K = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16136 Level forecast errors for 3-month tenor in different models with K = 3, 10-year window16637 Level forecast errors for 1-year tenor in different models with K = 3, 10-year window 16638 Level forecast errors for 3-year tenor in different models with K = 3, 10-year window 16739 Level forecast errors for 5-year tenor in different models with K = 3, 10-year window 16740 Level forecast errors for 10-year tenor in different models with K = 3, 10-year window16841 Level forecast errors for 3-month tenor in different models with K = 5, 10-year window16942 Level forecast errors for 1-year tenor in different models with K = 5, 10-year window 16943 Level forecast errors for 3-year tenor in different models with K = 5, 10-year window 17044 Level forecast errors for 5-year tenor in different models with K = 5, 10-year window 17045 Level forecast errors for 10-year tenor in different models with K = 5, 10-year window17146 Level forecast errors for 3-month tenor in different models with K = 7, 10-year window17247 Level forecast errors for 1-year tenor in different models with K = 7, 10-year window 17248 Level forecast errors for 3-year tenor in different models with K = 7, 10-year window 17349 Level forecast errors for 5-year tenor in different models with K = 7, 10-year window 17350 Level forecast errors for 10-year tenor in different models with K = 7, 10-year window17451 Level forecast errors for 3-month tenor in different models with K = 3, 5-year window17452 Level forecast errors for 1-year tenor in different models with K = 3, 5-year window 17553 Level forecast errors for 3-year tenor in different models with K = 3, 5-year window 17554 Level forecast errors for 5-year tenor in different models with K = 3, 5-year window 17655 Level forecast errors for 3-month tenor in different models with K = 5, 5-year window177


56 Level forecast errors for 1-year tenor in different models with K = 5, 5-year window 17757 Level forecast errors for 3-year tenor in different models with K = 5, 5-year window 17858 Level forecast errors for 5-year tenor in different models with K = 5, 5-year window 17859 Level forecast errors for 3-month tenor in different models with K = 7, 5-year window17960 Level forecast errors for 1-year tenor in different models with K = 7, 5-year window 17961 Level forecast errors for 3-year tenor in different models with K = 7, 5-year window 18062 Level forecast errors for 5-year tenor in different models with K = 7, 5-year window 18063 Slope forecast errors for 3-month tenor in different models with K = 3, 10-year window18464 Slope forecast errors for 1-year tenor in different models with K = 3, 10-year window 18465 Slope forecast errors for 3-year tenor in different models with K = 3, 10-year window 18566 Slope forecast errors for 5-year tenor in different models with K = 3, 10-year window 18567 Slope forecast errors for 10-year tenor in different models with K = 3, 10-year window18668 Slope forecast errors for 3-month tenor in different models with K = 5, 10-year window18769 Slope forecast errors for 1-year tenor in different models with K = 5, 10-year window 18770 Slope forecast errors for 3-year tenor in different models with K = 5, 10-year window 18871 Slope forecast errors for 5-year tenor in different models with K = 5, 10-year window 18872 Slope forecast errors for 10-year tenor in different models with K = 5, 10-year window18973 Slope forecast errors for 3-month tenor in different models with K = 7, 10-year window19074 Slope forecast errors for 1-year tenor in different models with K = 7, 10-year window 19075 Slope forecast errors for 3-year tenor in different models with K = 7, 10-year window 19176 Slope forecast errors for 5-year tenor in different models with K = 7, 10-year window 19177 Slope forecast errors for 10-year tenor in different models with K = 7, 10-year window19278 Slope forecast errors for 3-month tenor in different models with K = 3, 5-year window19379 Slope forecast errors for 1-year tenor in different models with K = 3, 5-year window 19380 Slope forecast errors for 3-year tenor in different models with K = 3, 5-year window 19481 Slope forecast errors for 5-year tenor in different models with K = 3, 5-year window 19482 Slope forecast errors for 3-month tenor in different models with K = 5, 5-year window19583 Slope forecast errors for 1-year tenor in different models with K = 5, 5-year window 19584 Slope forecast errors for 3-year tenor in different models with K = 5, 5-year window 19685 Slope forecast errors for 5-year tenor in different models with K = 5, 5-year window 19686 Slope forecast errors for 3-month tenor in different models with K = 7, 5-year window19787 Slope forecast errors for 1-year tenor in different models with K = 7, 5-year window 19788 Slope forecast errors for 3-year tenor in different models with K = 7, 5-year window 19889 Slope forecast errors for 5-year tenor in different models with K = 7, 5-year window 198


morphology, dynamics and forecasting power of …

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