Forecasting exchange rates using panel model and model averaging
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Economic Modelling 37 (2014) 3240
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j ourna l homepage: www.e lset al. (2014) in their analysis of forecasting ination and measuringoutput gap uncertainty. For each model specication, we use the h-step ahead forecast variance to construct the forecast density. We then Financial support from the ESRC (grant No. RES-062-23-1753) is gratefullydensities provides information that are more helpful for makingeconomic decisions (see for example Granger and Pesaran, 2000b;
each component model's predictive power by usPool (LOP), following (among others) Jore et athe true model. The second feature is the relative scarcity of studiesexamining predictive densities as opposed to point forecasts, at lower(quarterly) data frequencies. This is surprising given the widespreadrecognition that evaluation of point forecasts is only relevant for highlyrestricted loss functions, and that examining the features of predictive
considered. The resulting ensemble predictive approximates the manyunknown relationships between the exchange rates and a set offundamentals using time-varying weights across component densities.
In order to constructweights attached to componentmodels, each ofwhich is likely to be mis-specied at some point in time, we consideracknowledged. Corresponding author.
E-mail address: email@example.com (E. Mise).1 In addition papers such asMark (1995), Abhyankar et
(2012) nd evidence, of varying degrees of robustnessvariables at long horizons. Carriero et al. (2009) nd weat both long and short horizons using a panel based large
0264-9993/$ see front matter 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.econmod.2013.10.017ncies, across time and inor may not perform welled as approximations to
recursion in our application) to produce a forecast for the aggregate.Each component forecast is produced from a xed effect linear panelmodel, with differing fundamentals, for the set of exchange ratesrelation to structural change. Each model mayin any particular period and as such can be view1. Introduction
Forecasting exchange rates has a loby poor performance when comparedmodels. The lack of success is not howMark and Sul (2001), Engle et al. (Stavrakova (2008) [RS] report favofrequency) forecast horizons, using a ramodels.1 However a number of featurout. The rst is the uncertainty associaacross types of fundamental, for differeory largely characterisedenchmark random walkniversal, studies such as[EMW], and Rogoff andresults for long (low
panel based fundamentalis limited success standh the model specication
Timmermann, 2006 and for the exchange rate Abhyankar et al., 2005;Garratt and Lee, 2010; Della Corte et al. (2012)).
In this paper we seek to address these issues by extending the use ofpanel-based fundamentalmodels to examine interval forecasts aswell aspoint forecasts and at the same time address the issues of model andbreak date uncertainty. To accommodate a range ofmodel specications,we adopt an ensemble or combination approach to construct forecasts.The focus is on combined models, as opposed to individual models,wherewe formulate the forecasting problemas one inwhich a forecasterselects a linear combination of component forecast densities (at eachPanel modelsForecasting exchange rates using panel mo
Anthony Garratt a, Emi Mise b,a University of Warwick, United Kingdomb Department of Economics, University of Leicester, University Road, Leicester LE1 7RH, United
a b s t r a c ta r t i c l e i n f o
Article history:Accepted 8 October 2013Available online xxxx
Keywords:Exchange rate forecastingPoint and interval forecastsModel averaging
We propose to produce accuknown fundamental based pa linear mixture of experts'spredictive performance. As wthe models into consideratiothe period 1990q12008q4,forecasts that outperform eqcombining forecasts is particso for interval forecasts. Calcor ensemblemodel showa gouncertainty of whether the eal. (2005) and Della Corte et al., of predictability using macroak point forecast predictabilityBayesian VAR.
ghts reserved.el and model averaging
point and interval forecasts of exchange rates by combining a number of welll models. Combination of each model utilizes a set of weights computed usingework, where weights are determined by log scores assigned to each model'sas model uncertainty, we take potential structural break in the parameters ofn our application, to quarterly data for ten currencies (including the Euro) forshow that the forecasts from ensemble models produce mean and intervalweight, and to a lesser extent random walk benchmark models. The gain fromrly pronounced for longer-horizon forecasts for central forecasts, but much lessions of the probability of the exchange rate rising or falling using the combinedcorrespondencewith known events and potentially provide a usefulmeasure forange rate is likely to rise or fall.
2013 Elsevier B.V. All rights reserved.
ev ie r .com/ locate /ecmoduse the KullbarkLeiblerdistance between each model's density andthe true but unknown density to construct weights, and as such theweights reect each model's forecasting ability. The aim of thistype of combination (irrespective of the types of weights used) isto approximate the unknown process with a large number of likelymis-specied forecasting models. Hence the modelling strategy is
based on a Bayesian perspective, and although here we estimate andcombine the models based on a more frequentist approach, could beimplemented as such.
In addition, this paper explicitly models the Euro-dollar exchangerate, as one of a system of exchange rates considered. Unlike most ofthe literature, we model the Euro bloc both before and after unicationand are therefore able tomake a number of interesting probability eventforecast statements regarding its movements. Many studies either
2.2. Fundamental models of exchange rate disequilibrium
The fundamentals which motivate the set of models we combineare:
Monetary Fundamentals [MF] Purchasing Power Parity [PPP] Efcient Market Hypothesis [EMH]
33A. Garratt, E. Mise / Economic Modelling 37 (2014) 3240estimate up to 1999 or treat the Euro countries separately then combinethem (using some trade based weights), despite their being noexchange rate against which to evaluate their models.
To anticipate our main results, we nd that by accounting for modeland break date uncertainty recursive log score weighted (RLSW)models generally outperform random walk benchmark. The evidencefavours RLSW over equally weighted models (EQW) for both pointand interval forecasts. While the evidence does not always supportthe use of RLSW over the RW benchmark for interval forecasts,particularly for long horizons, our results for point forecasts revealthat, the longer the horizons, the greater the gain from combiningforecasts using log scores. This is demonstrated in the signicantreduction in the root mean square error of forecasts.
The remainder of this paper is organised as follows. In Section 2 wedescribe our basic model structure and the forms of the fundamentalmodels of exchange rate disequilibrium considered. In Section 3 weoutline our methodology for combining different model specications.Section 4 describes our quarterly data set and provides additionaldetails of the application. Section 5 reports our evaluation of the pointand interval forecasts as compared with the benchmark random walkmodel. Section 6 considers the models' ability to forecast probabilisticevents. Section 7 concludes.
2. The model space
In this section we rst describe the xed effects panel modelstructure adopted, used by (among others) Mark and Sul (2001),EMW and RS, which seeks to explain movements in the exchangerate via an error-correcting framework based on deviations fromfundamentals. We then outline the set of models we use to deneexchange rate fundamentals.
2.1. Basic model structure
We consider the linear xed effects panel model:
si;th jih jth jhz jit jith; 1
where si,t + h= si,t + h sit; with sit dened as the natural log of the(nominal) exchange rate measured in terms of foreign currency perunit of the base currency (the US dollar) for country i, i=1, 2,, N, attime t, t = 1, 2,.., T where h is the forecast horizon. The term zitj
represents, for country i, a deviation from an equilibrium, determinedaccording to a range of j exchange rate fundamentals based models,j=1, 2, , J, described in the next sub-section.2 The terms ih
j and thj
are country specic and time effect dummies respectively for the jthmodel and h forecast horizon.
2 Following the literaturewe adopt axed as opposed to a randomeffectsmodel, wherewe note that this formulation, the two-way error component xed effect model (seeBaltagi, 2008) is equivalent to using cross-section averages in the panel regression asadvocated by Pesaran (2006). Note also Baillie and Baltagi (1999), document the goodperformance of a one-way xed effects predictors relative to an ordinary optimal
predictor. Taylor Rule(s) [TR]
Based on these four types of fundamentals, we dene alternativespecications of the error-correction term:
z jit fjitsit ; 2
in Eq. (1), representing the deviation of the exchange rate from thefundamentals fitj for country i. These specications, combined with theneed to accommodate structural change, give rise to our total of Mmodels. This form for the error-correction term assumes stationarityand a cointegrating vector with coefcients (1, 1), but where wealso consider specications which relax this assumption.3
In the pure MF model, j= 1, for zit1 we dene fit1 = (m0t mit)(y0t yit) and set =1, where m0t and y0t denote the log-levels ofUS (the base country) money supply and output and mit and yit arethe corresponding foreign money supply and output respectively. Thisspecication has a long tradition in the analysis of exchange ratedetermination, and has been the subject of much debate (as in Mark(1995), Mark and Sul(2001), Berkowitz and Giorgianni (2001), forexample).
We then consider two adjustments to theMFmodel following DellaCorte et al. (2009) [DST]. In the rst, we allow for something other thana one to one relationship between the fundamentals and the exchangerate (but we are still assuming they cointegrate). Hence when j= 2,for zit2 we dene f
2it ^ i0 ^ i1fun1it where the coefcients are from the
regression sit = i0 + i1fit1 + uit,2. For the second, j= 3, we allow forthe possibility that cointegration occurs only after a deterministic timetrend (t) adjustment. Hence for zit3, we dene f
3it ^ i2 ^ i3 f 1it ^ i4t
where the coefcients are from the regression sit = i2 + i3 fit1 +i4t+uit,3.
In the pure PPPmodel, j=4, for zit4 we dene fit4=p0tpitwhere p0tand pit denote the logarithm of the US and foreign price levelrespectively and where the real exchange rate is assumed to bestationary. This theory is often viewed as an arbitrage condition ininternational goods and is considered to be an integral to many openeconomy views of the world. The literature considering the empiricalvalidity of PPP is well developed and the conclusions are mixed, butthere is some recent evidence that it may hold in the long-run (seeGarratt et al. (2006), for example).
We then consider the same two adjustments made to the MFfundamentals as applied to the PPP fundamentals. Hence when j=5,for zit5 we dene f
5it ^ i5 ^ i6 f 4it where the coefcients are from the
regression sit=i5+i6 fit4+uit,5. For the second, j=6, for zit6 we denef 6it ^ i7 ^ i8 f 4it ^ i9t where the coefcients are from the regressionsit=i7+i8 fit4+i9t+uit,6.
In the pure EMHmodel j=7, for zit7 we dene fit7= fsitwhere fsit is thelogarithm of the forward (end-of-period) nominal bilateral exchangerate. This model relates to the literature on foreign exchange marketefciency which tests whether the forward rate is an optimal predictor
3 Anticipating the empirical section and bearing in mind the small sample, panel unitroot tests (in amajority of cases) failed to reject the null of a unit root in levels but rejectedthe null in rst differences. We therefore proceed in the analysis assuming the variablesare I(1). Analysis of the long run relationships that exist among the variables providedgood evidence to support pairwise cointegration of the exchange rate with our candidatefundamental models. There was however weak evidence to support the one-to-onerelationships suggested by the theories. However the precise forms these take in themodel average and across different recursions make the issue of stationarity complex in
of the future spot exchange rate. Although the empirical evidence ismixed regarding the optimality of the forward rate as a predictor of
since, in effect, each candidate break date denes a new componentpanel model. If we have J panel model specications dened aroundthe deviation from equilibrium terms, zitj , and for any given zitj , wehave K variants dened over different values of the break date location,then in total we have M = J K models, and therefore M associatedforecasts of each country's nominal exchange rate.
34 A. Garratt, E. Mise / Economic Modelling 37 (2014) 3240the spot rate, there is evidence that some information is contained inthe term structure of the forward rate; see Clarida and Taylor (1997),for example.Moreover the EMH specicationwe adopt does not requireefcient markets to hold at all points in time. We also consider anadjusted efcient market model, j=8, where for zit8 we dene the f
^ i10 ^ i11 f 7it where the coefcients are from the regression sit =i10+i11 fsit7+uit,8.
Finally, we also consider fundamentals based on the Taylor Rule(TR). This involves developing an error-correction formulation for theTR through replacing the interest differential term in the uncoveredinterest parity condition (UIP) with the components of a (relative) TR.Here we consider three specications for zitj which result from differentforms of the TR. UIP is dened as Etsit+1=r0trit+sit, wherewe seek toreplace r0t rit, with the components of the relative TR:
r0trit 1:5 0tit 0:1 y0tgapyitgap 0:1 sit pitp0t ;
as used in EMW, where r0t and rit are US and foreign short term interestrates, 0t and it US and foreign ination rates, y0tgap and yitgap are US andforeign output gaps, computed using the HP lter. The coefcients arenot estimated but are xed at EMW's specied values. Substitutingthis term into the UIP condition and equating the fundamental termwith the expected future exchange rate we derive the fundamentalterm:
f 9it 1:5 0tit 0:1 y0tgapyitgap 0:1 sit pitp0t sit;
and therefore, using Eq. (2):
z9it 1:5 0tit 0:1 y0tgapyitgap 0:1 sit pitp0t :
Adopting the same argument, but with different TR specications(dening fit10 and fit11) which introduce lagged interest rate differentialsor smoothing terms, following Molodtsova et al. (2008) [MNP] andMolodtsova and Papell (2009) [MP], two alternative specications forthe error correction term are:
z10it 1:5 0tit 0:1 y0tgapyitgap 0:1 sit pitp0t
z11it 1:5 0tit 0:1 y0tgapyitgap
0:1 sit pitp0t 0:55 i0t1iit1 ;
where the zit11 error correction term, with the exception of thecoefcient on the real exchange rate, reverses the signs of zit10. For anexplanation of this difference and for the estimates of US-German TR,which suggest the value of 0.55 for the interest rate differentialcoefcient see MNP and MP.
In total we have eleven models, J=11, but in addition we estimate,as benchmark models, random walks with and without drift terms.4
A number of papers, for example Engle et al. (2007) and Rogoff andStavrakova (2008), have argued that the relationships betweenexchange rates and fundamentals are subject to structural change.Hence, it would be of interest for our model space to accommodatestructural breaks of unknown timing in a computationally convenientmanner. We take a pragmatic response and consider every feasiblesingle break date, assuming a single coincident break in all theestimated parameters of the model i.e. the conditional mean andvariance. This raises the potential number of models dramatically
4 In the empirical section, recursive and equal all fourmodels are assessed relative to the
standard random walk model without drift.3. Methodology
Here we describe our methodology for measuring model un-certainty for exchange rates. We then describe how we broaden thespace of these models to allow scope for a single structural break ofunknown timing and how to construct the ensemble or combinedpredictive densities from the component densities.
3.1. Combination construction
We construct the exchange rate predictives based on the ability ofeach component model's predictive density to produce the observedexchange rate movement. We combine the component forecastdensities using the linear mixture of experts, also known as the linearopinion pool; see Timmermann (2006). At every recursion, theestimated component panel model of each exchange rate is scored forthe KullbackLeibler distance between the h-step ahead density andthe true but unknown density. Density combination, via the linearopinion pool, then constructs the ensemble forecast densities for theexchange rate using KullbackLeibler distance weights, based on thelogarithmic scores of the forecast densities. The resulting exchangerate densities reect the model uncertainty in the many componentspecications.
More formally, we consider a situation where it is not clear whichmodel of the exchange rate a forecaster might wish to adopt and thuswe seek to aggregate forecasts supplied by experts, each of whichuses a different panel specication to produce forecast densities forthe range of exchange rates. Given m= 1, , M panel specications(where M= J K), the combined densities are dened by the convexcombination5:
m1wim;;hg si;;h Im;
; ;; ;
where g(si,,h|Im,) is the h-step ahead forecast density of model m, ofthe h-step change in the ith exchange rate si,,h, conditional oninformation available at time . The non-negative weights, wim,,h, inthis nite mixture sum to unity.6 Furthermore, the weights changewith each recursion in the evaluation period ;; .
Each component panel model considered can be estimated bymaximum likelihood for the Gaussian linear model to provide eachcomponent forecast density g(). However, the ensemble densitydened by Eq. (3) will be a mixture of the component densities.
While some in the literature favours the measure of the t of eachmodel to the data such as Bayesian Information Criterion (BIC) toconstruct the weightswim,,h, we propose the weights to be determinedby the cumulative probability of observing si,,h from the predictivedensity of the m-th model. Following Amisano and Giacomini (2007),Hall and Mitchell (2007) and Jore et al. (2010), the logarithmic scoremeasures density t for each component model through the evaluationperiod. The logarithmic scoring rule is intuitively appealing as it gives a
5 Morris (1974, 1977), Winkler (1981), Lindley (1983) and Genest and McConway(1990) discuss linear opinion pools and expert combination. Wallis (2005) proposes thelinear opinion pool as a tool to aggregate forecast densities from survey participants.Mitchell and Hall (2005) combine two ination density forecasts but do not considerensemble macroeconometric systems.
6 The restriction that eachweight is positive could be relaxed; for discussion see Genest
and Zidek (1986).
high score to a model that predicts a high probability to the realizedvalue. The logarithmic score for the m-th density forecast, lng(tasi,,h|Im,), is the logarithm of the probability density function g(.|Im,),evaluated at si,,h. Specically, the recursive weights for the h-stepahead densities take the form:
X1h lng si;;h
Im; h i
X1h lng si;;h
Im; h i ; ;; 4
35A. Garratt, E. Mise / Economic Modelling 37 (2014) 3240where to 1 comprises the training period used to initialize theweights, for a given choice of .
As indicated by (4), the weighting scheme we have chosen has theadvantage that it is a function of the t of eachmodel's entire predictivedensity to the true density of si,,h, and not limited to the model'sability to match its central forecast to the outturn. Also, at eachrecursion, new set of weights are constructed for every exchange rate,reecting the predictive performance of each model estimated forevery break dates, with respect to each exchange rate, although thecumulative nature of the weights means that it also takes intoconsideration the m-th model's predictive performance for the last h time periods. This contrasts with alternative weighting schemessuch as the multivariate log score-based approach discussed inAdolfson et al. (2005).7 Weights chosen according to the forecastperformance of the density combination, based on recursive logarithmicscore weights, have many similarities with an approximate predictivelikelihood approach.8
We conclude this section by remarking on a number of interestingfeatures of our ensemble modelling strategy for measuring theuncertainty in forecasting the exchange rate. First, our methodologyinvolves combining forecasting densities from a potentially largenumber of locally linear Gaussian components. In this regard, we aremotivated by our desire to account for uncertainty over the correctfundamental model. Selection of any single empirical specica-tion inevitably gives rise to the uncertain instabilities problemsdocumented by, for example Engle et al. (2007) and Rogoff andStavrakova (2008). That is, our empirical methodology utilizes amodel space which could be described as incomplete; see Geweke(2009). Given that we attach a negligible probability to our modelspace containing the true empirical specication, we approximatethe unknown model using our entire ensemble system.9
Second,we combine the predictives based on out-of-sample forecastdensity performance for each of the N exchange rates through theevaluation period, even though the panel components are esti-mated individually by conventional (in-sample) maximum likelihoodmethods. This feature limits the extent of over-tting, and permitscombinations of panel models using different sample lengths forparameter estimation.
Third, recursive updating of the KullbackLeibler distance basedweights, wim,,h, occurs through the evaluation period. That is, theensemble density has time varying weights, and can approximate(highly) non-linear processes, even though the component modelsthemselves are (locally) linear.
7 Multivariate log score weights are computed using a log likelihood constructed fromthe forecast errors of all the variables in the system, the more conventional approach.Where the evaluation period is small this has the advantage of increasing the sample sizeonwhichweights are computed but the disadvantage of not focusing on the performanceof the single variables, which may be of interest. In our case we seek to forecast all theexchange rates in the system, but choose to allow different weights for each, so as to tailorthe weights in each case.
8 In applications with h N 1, the product of the h-step ahead forecast densities does notcorrespond to the marginal likelihood.
9 Morley and Piger (2012) use Bayesian model averaging to construct point forecasts
but not forecast densities for the US business cycle.Finally, we note that ensemble forecasting methods, such as thoseused in this paper, have been found to be effective in producing well-calibrated forecast densities outside the economics literature. Forexample, meteorologists commonly construct ensemble densities todeal with uncertainty in initial conditions (auxiliary assumptions).The Ensemble Prediction System developed by the European Centrefor Medium-Range Weather Forecasts follows the same generalensemble principles to forecast weather densities effectively. For anearly description of weather ensemble forecasting see Molteni et al.(1996).
4. Application: a recursive forecasting exercise
In our application, we construct point and interval forecasts for a setof bilateral nominal exchange rates using, recursively, the methodsdescribed in the previous section. Here we provide some details of ourdata set, sample and application.
In our empirical work, we use a panel of quarterly observations forthe period 1990q12008q4 (T = 76 observations) for ten (N = 10)country or bloc pairings. The United States is the base country, wherethe other ten countries or blocs are: the Euro area, United Kingdom,Japan, Canada, Denmark, Norway, Sweden, Switzerland, Australia andKorea. We choose to model the Euro area as a single bloc, in contrastto much of the literature (examples of which include EMW), whichmodels the Euro countries separately and therefore have a panel overlarger number of countries. This has the advantage of being able toforecast a currency of major interest, the Euro, post 1998q4 using anobserved variable for evaluation. The disadvantage is that it limits thenumber of observations which we can use in light of the lack ofavailability of Euro data before 1990.
For each country or bloc we dene ve series: s the natural log of the(nominal, end period) exchange rate measured in terms of foreigncurrency per one unit of the base currency (the US dollar) for country,i the nominal representative short term interest rate, m the moneysupply (mostly narrow money), p consumer prices and y real GDP.Note we construct the forward exchange rate used in the EMH modelassuming covered interest rate parity and therefore use the exchangerate and interest rates. All the data used in the analysis are in naturallogarithms, the main source of data is the International MonetaryFund where the precise denitions, sources and transformations aredescribed in the Data appendix.
4.2. Weights, forecasting and components
We consider two sets of weights. First, we use the RLSW schemediscussed in Section 3.1, applied recursively, following Jore et al.(2010). Second, we use equal weights (EQW), dened as 1/M.
The exact computational methods for the forecasts are provided inAppendix A. Briey, each individual model is used to produce h-stepahead forecasts via the direct approach; see the discussion byMarcellino et al. (2003). Given si,t + h is our variable of interest, weforecast using information available at time t, where analyticalexpressions provide us with a predictive density for each modelg(si,,h|Im,). These are then combined using the aforementioned setsof weights. All of the features of interest in the tables below arefunctions of these predictive densities (i.e. point forecasts are themeans of these densities, etc.). Note in Eq. (1) the time effect dummy,htj , will enter the predictive regression contemporaneously. Hence inorder to construct the h-period ahead forecast of the exchange ratechange we forecast this term, following EMW, by using the recursivemean 14
We also allow for a single structural break of unknown timing in
each panel component. In order to reduce the computational burden,
the break date is restricted to occur before the start of the evaluationperiod, , with at least 30% of the sample used for post-break in-sample estimation of each component.10 The break occurs in theconditional mean and the variance for both equations. For forecast
36 A. Garratt, E. Mise / Economic Modelling 37 (2014) 3240horizon h=1we consider K=33 component models for eachmeasureof the J=11 fundamental models considered. Hence the predictives forthe exchange rate ensembles combine M = 363 componentspecications for each observation in the evaluation period. For forecasthorizons h=4, 8M equals 330 and 276 respectively.
For h=1, 4 and 8 the rst recursions begins in 1990q2, 1991q1 and1992q1 respectively and all end in 1999q4. The last recursions begin inthe same periods and end for h= 1, 4 and 8 in 2008q3, 2007q4 and2006q4 respectively. Given the relatively small numbers of evaluationperiods this allow (36, 33 and 29 respectively) we set =4, allowinga training period of four quarters. Note, we also have computed resultsusing a rolling window of 40 and the results are qualitatively verysimilar. Therefore for h = 1, our out-of-sample density forecastevaluation period is: ;; where 2001q1 and 2008q4(32 observations). For h = 4, 2001q4 and 2008q4 (29observations) and for h = 8, 2002q4 and 2008q4 (25observations).
5. Empirical results: point and interval forecasts
We begin our evaluation by examining some root mean squarederror (RMSE) statistics for our central forecasts. We shall then discussthe combined models' ability to predict an interval. We will also see ifthe variance of si,tau,h is forecast accurately by examining the 70%coverage rate of our ensemble models, relative to that of the RWbenchmark. Finally we report hit rates with respect to the probabilityevent of the change in the exchange rate being greater than zero i.e. isthe exchange rate going to rise or fall over the specied forecast horizon.
5.1. Point forecasts
Table 1 presents RMSE and their ratios for the quarterly forecasthorizons 1, 4 and 8. The left block of the tables presents RMSE for arandom walk with no drift (RW) in the rst column, followed bycolumns reporting the RLSW and EQW models RMSEs relative to therandom walk. Entries with values less than one mean a forecast ismore accurate than the random walk benchmark. The right handblock of the tables uses the random walk with drift (RWD) as thebenchmark. To provide a guide as to the statistical signicance, wealso report the p-values for the null hypothesis that the RMSE of therandom walk models are equal to those of the average model, againstthe one-sided alternative that the RMSE of the average model is lower.The p-values are obtained by comparing the Diebold and Mariano(1996) test against the standard normal critical values11
In Table 1 we rst note that the RMSE for the random walk modelsare lower than the RMSE for the random walk with drift models, forall currencies and forecast horizons and as such represents a moredemanding benchmark. Therefore we opt to focus on the randomwalk results in left hand block, where we observe a number of features.First, the RLSW model's RMSEs are generally lower than those derivedusing the EQW and the random walk models. At the h = 1 forecasthorizon, for the RLSW model, seven of the ten currency pairs haveRMSE ratios less than one, as compared to four out of ten for EQWmodels. For the RLSW model, the differences of the ratio from onetypically are not large (0.933 is the lowest), but where the DM statisticp-values (reported in the parentheses) suggest that (at the 90%
10 In larger samples this is often set at 15%, but in light of our small sample sizewe chooseto set it at 30%.11 Monte Carlo results in Clark and McCracken (2009) indicate that the use of a normaldistribution for testing equal accuracy in a nite sample can be viewed as a conservative
guide to models that are nested, as they (partly at least) are here.condence level or higher) three of the seven are signicantly differentfrom the random walk. For most part, the RMSEs from the twoensemble models are not signicantly different from those generatedby the random walk. Nonetheless, the potential for the ensemblemodels to do better than the random walk, on this somewhat limitedmetric, is apparent. As the horizon increases, from h= 1 to h= 4, 8,we observe larger numbers of RMSEs less than one, whilst in starkcontrast all ten RMSE ratios for the EQW average model exceed one.Moreover, the DM test statistics suggest that, at forecast horizons ofh= 4 and h= 8, these RMSE differences are often highly signicant.For example, for horizon h=8, seven out of ten RLSW model's RMSEsare signicantly below the random walk benchmark at the 95%condence level (with the exception of one which is signicant at 90%condence level).
Individual currency pair results worth noting are the Euro and Japan,where the RLSWmodel does well at all forecast horizons relative to therandom walk and Australia, which has a ratio of 0.580 for forecasthorizon h=8. For UK and Korea, the RLSW model is outperformed bythe random walk at horizons h = 1 and h = 4 but not at the longerh = 8 horizon. Overall, we observe RMSE results consistent with theliterature, suggesting predictive power at longer horizons. Clear supportfor the RLSW models is shown, where the EQW models, the type ofmodel that often performs well in the model averaging literature, areclearly outperformed here. But we note, following Gneiting (2011),the limitations of RMSE as a forecast evaluation metric.
5.2. Interval forecasts
We shall evaluate the ensemble models' power to predict theprobability of future exchange rate movements by looking at theircoverage rates, relative to that of random walk (see recent studiessuch as Giordani and Vallani (2009), Clark (2011) describing intervalforecasts). Table 2 reports the frequency with which actual outcomesfor our exchange rates fall inside 70% posterior density intervalsgenerated byRLSW, EQW, and the benchmark RWwithout driftmodels.Accurate intervals should result in frequencies of about 70%. Afrequency of more (less) than 70% means that, on average over agiven sample, the posterior density is too wide (narrow). Reported inthe parentheses are the p-values for the null of correct coverage, i.e.that the empirical density has 70% coverage, based on t-statistics,which provide a guide as to the statistical signicance of the deviationof the empirical coverage rate from that under the null.
For the forecast horizon h= 1, we observe (with few exceptions,most notably the Korean exchange rate) coverage rateswhich liewithin5% of 70%. The average coverage rates, for all currencies, are 72.4%,72.0% and 72.6% for the RLSW, EQW and random walk modelsrespectively (and are almost exactly 70% when excluding Korea fromthe average) with no systematic tendency to be either too narrow orwide. A large majority of these deviations from the 70% coverage rateare not statistically signicant, but the RW model has a too narrowdensity for Canada and too wide for Switzerland. Korea, for all threemodels considered, has exchange rate densities which are signicantlywider than the 70% coverage rate (in excess of 90%).
As the forecast horizon increases to h=4 and h=8,we see a gradualdecline in coverage rates, suggesting that the predictive densities are alltoo narrow. For h=4, the average coverage rates are 65.6%, 62% and64.1% for the RLSW, EQW and RW models respectively. Here we notethat the standard deviation across coverage rates has increased from0.0925, 0.1303 and 0.1127 for the RLSW, EQW and the RW modelsrespectively, for h=1, to 0.1605, 0.1958 and 0.1428 for h=4. Koreacontinues to have, for all three models, distributions that aresignicantly above the null of 70% coverage rates i.e. are too wide. Thisis also true for the Japanese h=4 exchange rate density.
However, it is still the case that a majority of the deviations for thecoverage rate from 70% are not statistically signicant (at the 5% level
and bearing in mind the small sample). For h= 8, we note a further
3 (0.8 (0.1 (0.8 (0.5 (0.7 (0.8 (0.0 (0.3 (0.9 (0.
2 (0.5 (0.1 (0.9 (0.2 (0.7 (0.7 (0.5 (0.1 (0.8 (0.
0 (0.9 (0.7 (0.1 (0.
37A. Garratt, E. Mise / Economic Modelling 37 (2014) 3240Table 1Root mean square error (RMSE) ratios.
RW RLSW EQW
h=1, 2001q12008q4Euro 0.0493 0.933 (0.09) 0.97UK 0.0512 1.023 (0.69) 1.03Japan 0.0535 0.991 (0.40) 1.01Canada 0.0450 0.951 (0.00) 0.96Denmark 0.0515 0.969 (0.24) 1.00Norway 0.0625 1.001 (0.52) 1.03Sweden 0.0603 0.969 (0.14) 0.99Switzerland 0.0548 0.993 (0.36) 1.02Australia 0.0658 0.933 (0.01) 0.96Korea 0.0488 1.077 (0.78) 1.07
h=4, 2001q42008q4Euro 0.1053 0.943 (0.22) 1.12UK 0.0969 1.103 (0.81) 1.18Japan 0.0982 0.836 (0.21) 0.96Canada 0.0896 0.826 (0.02) 0.97Denmark 0.1084 0.909 (0.19) 1.10Norway 0.1169 1.014 (0.57) 1.15Sweden 0.1300 0.936 (0.08) 1.06Switzerland 0.1023 0.948 (0.41) 1.11Australia 0.1335 0.891 (0.15) 1.07Korea 0.1100 1.028 (0.59) 1.06
h=8, 2002q42008q4Euro 0.1732 0.816 (0.07) 1.20UK 0.1305 0.930 (0.03) 1.15Japan 0.1321 0.865 (0.30) 1.15Canada 0.1361 0.904 (0.19) 1.13largemovement down in coverage rates. The average coverage rates arenow too narrow at 52%, 52.9% and 54.2% for the RLSW, EQW and RWmodels respectively (with across currency coverage standard deviationsof 0.1282, 0.1867 and 0.1878 respectively). Hence longer forecastdensities seem to narrow considerably, althoughwe note that as before,a majority of these deviations from the coverage rate of 70% are notstatistically signicant (only one for the RLSW model, one for theEQWmodel and three for the RWmodel).
Although the results in Table 2 indicate that the predictive densitiesare too narrow for all threemodels for both h=4 and 8, in contrast withthe RMSE results of Table 1, there is no indication that RLSW modeloutperforms the RW benchmark. For the majority of countries forh=8, there was strong evidence against the null of equal predictabilitywhen we considered point forecasts alone. It is interesting that, forNorway for example, the null was rejected at 1% signicance levelhowever, this was achieved from a predictive distribution whosecorrect 70% correct coverage is emphatically rejected. It thereforeseems that the ensemble models correctly predict the centre of thedensities of si,,h, but not its variance and/or shape.
Overall, we observe a reasonable set of coverage rates for our threemodels, but where no one model type appears to be the dominant orbest tting model. The EQW average model do less well at horizonsh= 4 and h= 8, but the choice between RLSW and the RW modelswould have to be on the basis of differences that are not statisticallysignicant. In terms of the worsening performance of coverage ratesas the forecast horizon increases, one possible explanation is samplingerror. Even if we were to believe that a model is correctly specied,
Denmark 0.1767 0.773 (0.01) 1.122 (0.Norway 0.1625 0.733 (0.01) 1.121 (0.Sweden 0.1886 0.731 (0.03) 1.041 (0.Switzerland 0.1576 0.772 (0.05) 1.192 (0.Australia 0.1965 0.580 (0.03) 1.103 (0.Korea 0.1400 0.951 (0.38) 1.077 (0.
Therst three columns are: (i) RWrootmean square error (RMSE) for a randomwalkwithoutdrift and (iii) EQWequal weight models RMSE ratios, relative to the RWwithout drift. Columfollowed by (ii) RLSWand (iii) EQW,which are RMSEs dened relative to the randomwalkwithof equal RMSE with the respective benchmark RWmodel (without and with drift).RWD RLSW EQW
29) 0.0503 0.913 (0.08) 0.952 (0.21)81) 0.0520 1.006 (0.57) 1.021 (0.70)62) 0.0565 0.938 (0.21) 0.957 (0.29)03) 0.0458 0.936 (0.05) 0.953 (0.10)55) 0.0524 0.951 (0.17) 0.986 (0.38)95) 0.0639 0.979 (0.13) 1.014 (0.90)47) 0.0618 0.945 (0.12) 0.973 (0.29)81) 0.0556 0.979 (0.04) 1.006 (0.70)02) 0.0674 0.911 (0.01) 0.941 (0.04)80) 0.0516 1.019 (0.58) 1.020 (0.59)
89) 0.1204 0.826 (0.10) 0.982 (0.40)95) 0.1058 1.011 (0.54) 1.086 (0.84)43) 0.1154 0.711 (0.11) 0.818 (0.21)41) 0.0925 0.800 (0.10) 0.948 (0.33)85) 0.1218 0.808 (0.06) 0.981 (0.37)90) 0.1307 0.907 (0.12) 1.035 (0.96)91) 0.1483 0.969 (0.11) 0.998 (0.28)76) 0.1103 0.878 (0.04) 1.034 (0.73)91) 0.1526 0.780 (0.11) 0.937 (0.29)69) 0.1187 0.953 (0.33) 0.990 (0.46)
90) 0.2371 0.596 (0.07) 0.876 (0.08)83) 0.1505 0.806 (0.02) 1.005 (0.54)73) 0.1391 0.822 (0.23) 1.100 (0.66)77) 0.1538 0.800 (0.15) 1.001 (0.50)random variation in a given data sample could cause the empiricalcoverage to differ from the nominal values.
6. Probability event forecast analysis
We shall now consider our ensemble densities' ability to forecastan event. Specically, we are interested in our ensemble models'ability to predict a rise or fall in si,,h correctly. Such informationmay well be regarded as being useful for a policy maker or for avariety of investment decisions as it helps convey uncertaintysurrounding a forecast.
We compute three statistics to this end. First is the most intuitivelyobvious hit rate (HR). If the central forecast of si,,h N 0 at time t, andif the exchange rate does indeed rise at time t+ h from its level at t,this is a case of a correct prediction, denoted UU, indicating that theprediction and realisation are both in the upward direction. Similarly,if the prediction and realisation are both in the downwards direction,this makes another correct prediction, and denoted DD. UD and DUdenote incorrect directional prediction: the former indicates upwardprediction and downward realisation, and the latter downwardprediction and upward realisation. High values for UU and DD indicatean ability of the model to forecast upward and downward movementscorrectly, while high values of UD and DU suggest poor forecastingability. HR is calculated as HR=(DD+UU)/(UD+DD+DU+UU).
The second is the Kuipers Score (KS) statistic, dened byHF, whereH= UU/(UU UD) is the proportion of correct upward forecasts andF=DU/(DU+DD) is the proportion of incorrect downward forecasts.
80) 0.2374 0.576 (0.02) 0.835 (0.04)76) 0.2265 0.526 (0.02) 0.805 (0.06)69) 0.2594 0.531 (0.06) 0.757 (0.05)86) 0.2037 0.598 (0.06) 0.922 (0.20)86) 0.2765 0.412 (0.05) 0.784 (0.05)63) 0.1910 0.697 (0.00) 0.789 (0.01)
drift (ii) RLSWrecursive log scoreweightmodels RMSE ratios, relative to the RWwithoutns four to six are: (i) RWDroot mean square error (RMSE) for a random walk with drift,drift. The numbers in parentheses are the p-values for theDieboldMariano test of the null
Australia 55.2 0.16 0.92 37.9 0.30 1.90Korea 51.7 0.02 0.12 41.4 0.22 1.35All 60.7 0.17 0.31 50.3 0.11 2.10
Table 2Ensemble forecast density coverage rates.
% of outcomes within 70% interval
RLSW EQW RW
h=1, 2001q12008q4Euro 0.62 (0.12) 0.62 (0.17) 0.69 (0.80)
38 A. Garratt, E. Mise / Economic Modelling 37 (2014) 3240UK 0.75 (0.45) 0.75 (0.45) 0.75 (0.45)Japan 0.74 (0.05) 0.84 (0.05) 0.75 (0.35)Canada 0.75 (0.45) 0.72 (0.76) 0.50 (0.01)Denmark 0.65 (0.25) 0.52 (0.25) 0.75 (0.25)Norway 0.69 (0.84) 0.66 (0.58) 0.78 (0.24)Sweden 0.66 (0.39) 0.59 (0.19) 0.69 (0.84)Switzerland 0.66 (0.58) 0.75 (0.25) 0.81 (0.03)Australia 0.78 (0.26) 0.78 (0.26) 0.62 (0.32)Korea 0.94 (0.00) 0.97 (0.00) 0.92 (0.00)
h=4, 2001q42008q4Euro 0.52 (0.13) 0.38 (0.02) 0.62 (0.52)UK 0.66 (0.69) 0.59 (0.38) 0.55 (0.18)Japan 0.93 (0.00) 0.93 (0.00) 0.86 (0.04)Canada 0.66 (0.71) 0.76 (0.60) 0.45 (0.04)Denmark 0.52 (0.11) 0.45 (0.04) 0.59 (0.27)Norway 0.62 (0.49) 0.55 (0.26) 0.69 (0.93)Sweden 0.55 (0.14) 0.48 (0.06) 0.69 (0.89)Switzerland 0.69 (0.92) 0.66 (0.67) 0.72 (0.80)Australia 0.48 (0.08) 0.48 (0.13) 0.60 (0.61)Korea 0.93 (0.00) 0.93 (0.00) 0.93 (0.00)
h=8, 2002q42008q4Euro 0.40 (0.06) 0.44 (0.11) 0.52 (0.24)This statistic provides a measure of the directional forecasts of themodel, with a high positive number indicating high predictive accuracy.The third statistic is a formal test provided by Pesaran and Timmermann(1992), which as shown by Granger and Pesaran (2000a), is equivalentto a test based on the Kuipers Score.12
Table 3 reports the HR, KS and PT statistics for the central forecastsfrom our ensemble models. We report the RSLW and EQW models forboth the individual countries and for the countries aggregated together,so as to increase the number of observations.
For h=1, we observe that RLSW and EQW models have aggregatehit rates of 58.4% and 56.3% respectively. By comparison, the RWwithout drift achieves (not shown) aggregate hit rate of 45.9%. Positiveaggregate KS statistic suggests a decent forecast performance as whenhalf the predicted directions are incorrect, the KS score is 0. The PTstatistic is normally distributed under the null hypothesis that forecastsand realisations are independently distributed. Hence PT statistics of2.47 and 1.6 for RLSW and EQW respectively, suggest a signicantdifference from independence for the former, but not for the latter, at5% signicance level.
12 The PT statistic proposed in Pesaran and Timmermann (1992) which is denedby PTn = (Pn Pn)/V(Pn) V(Pn)(1/2) where n is the number of events considered,Pn is the proportions of correctly predicted events, Pn is the estimate of thisproportion under the null hypothesis that forecasts and realizations areindependently distributed, and V(Pn) and V(Pn) are the consistent estimates of thevariances of Pn and Pn , respectively. Under the null hypothesis, the PT statistic has astandard normal distribution.
UK 0.56 (0.30) 0.64 (0.69) 0.52 (0.23)Japan 0.76 (0.60) 0.84 (0.09) 0.88 (0.04)Canada 0.44 (0.05) 0.56 (0.36) 0.28 (0.00)Denmark 0.48 (0.14) 0.44 (0.11) 0.52 (0.24)Norway 0.48 (0.01) 0.44 (0.10) 0.56 (0.34)Sweden 0.52 (0.23) 0.48 (0.16) 0.48 (0.16)Switzerland 0.44 (0.12) 0.44 (0.12) 0.56 (0.37)Australia 0.60 (0.40) 0.48 (0.16) 0.56 (0.36)Korea 0.76 (0.44) 0.96 (0.00) 0.92 (0.01)
The columns report the frequencies with which the actual outcomes fall within the 70%intervals, computed from the forecast densities, where RLSW and EQW are respectivelythe recursive log score and equal weight models and RW is the random walk withoutdrift. In parentheses are the p-values for the null of correct coverage (empirical:70%),based on t-statistics using standard errors computed using a NeweyWest estimator.Table 3Diagnostic statistics for central forecasts of Pr(sit+ h b 0).
HR KS PT HR KS PT
h=1, 2001q12008q4Euro 68.8 0.33 1.95 56.3 0.03 0.17UK 56.3 0.17 0.83 56.3 0.17 0.83Japan 46.9 0.13 0.62 46.9 0.13 0.23Canada 68.8 0.40 2.23 68.8 0.40 2.23Denmark 59.4 0.21 1.10 59.4 0.21 1.04Norway 62.5 0.17 0.86 59.4 0.10 0.52Sweden 65.6 0.28 1.53 59.4 0.13 0.63Switzerland 43.8 0.12 0.63 43.8 0.12 0.63Australia 65.6 0.25 1.34 62.5 0.20 1.15Korea 46.9 0.08 0.40 50.0 0.00 0.00All 58.4 0.15 2.47 56.3 0.10 1.60
h=4, 2001q42008q4Euro 62.1 0.04 0.29 51.7 0.03 0.24UK 51.7 0.13 0.70 55.2 0.10 0.54Japan 58.6 0.23 1.03 55.2 0.12 0.55Canada 69.0 0.30 1.89 72.4 0.34 2.13Denmark 72.4 0.19 0.93 51.7 0.10 0.59Norway 75.9 0.09 0.45 51.7 0.13 0.92Sweden 48.3 0.22 1.31 48.3 0.24 1.51Switzerland 62.1 0.07 0.43 51.7 0.13 0.92As forecast horizon gets longer, the statistics reported here indicate astrong forecasting performance of RLSW model, but a markeddeterioration in the predictive ability of EQW. For h=4, the aggregatehit rate for the former is 60.7% compared to 50.3% for the latter. This issurpassed by the benchmark RW model, as it achieves a hit rate of54.8%. As for the PT statistics, the null of independence is not rejectedfor any of the countries at 5% level. However, for EQW, the statistic isnegative and signicant, meaning that the prediction and realisationsare strongly negatively correlated, indicating a poor forecastingperformance.
For h = 8, we observe both a further improvement in predictiveperformance of RLSW and a further deterioration in that of EQW. Theaggregate hit rate for RLSW is 63.2%, and the PT statistic is positive butno longer signicant. By contrast for EQW, the null hypothesis ofindependence between prediction and realisation is strongly rejectedin favour of a negative association, at 0.1% signicance level.
Overall, we observe probability event forecasts which perform witha high degree of accuracy in terms of hit rates, KS and PT statistics for theRLSWmodels, both in absolute terms and relative to the EQWmodel. Atshorter horizons the gain is also observed relative to the RWbenchmarkbut this is not so pronounced at longer horizonsthis is consistent withour ndings in Table 2, but contrasts to the results of point forecasts inTable 1.
h=8, 2002q42008q4Euro 40.0 0.17 1.18 28.0 0.19 2.31UK 60.0 0.07 0.46 48.0 0.25 1.67Japan 56.0 0.08 0.26 32.0 0.45 2.26Canada 68.0 0.07 0.58 44.0 0.02 0.18Denmark 68.0 0.16 1.20 44.0 0.21 1.67Norway 76.0 0.14 0.70 44.0 0.21 1.67Sweden 68.0 0.01 0.05 24.0 0.43 2.54Switzerland 68.0 0.13 0.86 40.0 0.29 1.97Australia 84.0 0.24 1.24 44.0 0.03 0.26Korea 44.0 0.22 1.39 56.0 0.27 1.82All 63.2 0.05 1.04 40.4 0.19 4.07
HR is the hit rate, dened as the proportion of ups and downs correctly forecast to occur.KS is the Kuipers-Score statistic, dened in the text and the PT statistic is the Pesaran andTimmermann (1992) test, which is normally distributed under the null that the forecastsof the probabilities and the realizations of the proportions of correctly predictedmovements are independently distributed i.e. the null is that there is no predictivecontent.
39A. Garratt, E. Mise / Economic Modelling 37 (2014) 32407. Conclusion
In this paper we have adopted an ensemble or model combinationapproach, applied to a range of exchange rate fundamental models, inorder to generate point and interval exchange rate forecasts.We assumethat each of the exchange ratemodels based on fundamentals is likely tobe mis-specied at some point in time. We have incorporated thismodel uncertainty by taking weighted average of predictive densities,where the weights are derived from the predictive power of eachmodel. We have also taken into consideration the possibility ofstructural break by allowing for structural breaks within each recursivesub-samples.
For point forecasts, we nd that our RLSW models perform well,when compared to the EQW and RW (without drift) models. Inparticular, at the longer horizons h=4 and h=8, we nd statisticallysignicant evidence that for the RLSW models, RMSE are lower thanthose of the EQW and RW models. This nding is consistent withprevious literature. A notable feature of the point forecasts, which isalso true for density forecasts, is the poor performance of the EQWmodels. This result contrasts with much of the model forecastcombination literature, which often nds equal weight models to dowell in comparison with more sophisticated weighting methods. Thisprovides some support in favour of the LOP weights used in this paper.
The results for interval forecasts and probability event forecastssuggest that the coverage is reasonably good for RLSW model.However, the results for the various horizons and model types aremore mixed, in the sense that whilst there is good evidence thatthe RLSW model is the preferred model, the distinction betweenmodels is less clear. For the h=1 forecast horizon, we have strongevidence that the forecast densities have good coverage and providereasonable hit rates with signicant effects for the probability eventconsidered. As the forecast horizon increases, coverage and themodels' ability to predict a rise/fall in exchange rate deterioraterapidly, particularly for EQW.
In terms of the differences between types of model averaging, weobserved a clear distinction between RLSW and EQW models for bothinterval forecasts and the evaluation of the probability events. Coupledwith the point forecast evaluation, this lends strong support to theLOP which underlies the RLSW. Regarding the differences we observebetween the RLSW and RWmodels, the results aremixed. The coveragerates, pits tests for h=1 and strong hit rates for the RW (subject to thequalication that RWwith drift does badly) suggest little gain from theRLSW averaging approach. However, the small sample points towardsthe log score statistics having the most power in this context and theysuggest strong and in some cases (especially at h=8) signicant supportin favour of the RLSW over the RW.
The sources and transformations for the data are as follows:
 sit: the natural log of the nominal, end period, exchange ratemeasured in terms of foreign currency per one unit of the basecurrency (the US dollar). Except the Euro, Source: InternationalFinancial Statistics (IFS), code AE.ZF market rate, end of period.For the Euro exchange rate, 1990q11998q4, we use the dollar-ECU end period rate, Source: IFS, code: EA.ZF. For the period1999q12008q4, we use data downloaded from the FREDdatabase, US-Euro Foreign Exchange Rate, Averages of dailygures, U.S. Dollars to One Euro, code: EXUSEU. These seriesweremonthly series, wherewe took the lastmonth in the quarter.
 r0t: US threemonth treasury bill rates expressed as a quarterly rate,r0t = 0.25 ln[1 + (R0t/100)] where R0t is the annualised rate.Source: IFS, code 60C..ZF.
 rit: foreign short term interest rates, expressed as a quarterly rate,
rit = 0.25 ln[1 + (Rit/100)] where Rit is the annualised threerepresentative rates. For Canada, Switzerland and the UK, weused the Treasury bill rate, Source: IFS, Code: 60C..ZF. ForNorway, Japan, Denmark, Sweden and Korea, we use the discountrate, Source: IFS, Code: 60..ZF. For Australia we used theaverage rate on money market funds, Source: IFS, Code: 60B..ZF.For the Euro short term interest rate, 1990q11998q4, weused the German Treasury Bill Rate, Source: IFS, Code: 60C.ZF.From 1999q1 we use the Main Renancing Operations ECBInterest Rate, Source: http://www.ecb.int/sats/monetary/rates/html/index.en.html, ECBWebsite. This is a document stating levelsand times of rate changes.
 fit: the natural logarithm of the end period forward discount rate,which we construct assuming covered interest parity i.e. fit =r0t rit+ sit.
 y0t: the natural logarithm of US real GDP, Source: IFS, Code:99BVRZF Vol. (2000 = 100), seasonally adjusted (Units: IndexNumbers).
 yit: the natural logarithm of foreign real GDP, Source: IFS, Code:99BVRZF seasonally adjusted, Vol. (2000 = 100) (Units: IndexNumber. Where not seasonally adjusted we used the seasonaladjusted procedure X12 in Eviews. For Euro area we use theseasonally adjusted total economy GDP series in 2000 prices,Source: Eurostat, Euro area 12. For the period 1990q11990q4,we use percentage change in the IFS quarterly industrialproduction series. For 1991q11994q4, we use the last vintagefrom the EACBN real time data set (yer.xls le).
 p0t: the natural logarithmofUS consumer prices, Source: IFS, Code:64ZF CPI All Items City Average, seasonally adjusted, 2000 =100.
 pit: the natural logarithm of foreign consumer prices, Source: IFS,Code: 64ZF CPI: All, seasonally adjusted, 2000 = 100. Wherenot seasonally adjusted we used the seasonal adjusted procedureX12 in Eviews.
 m0t: the natural logarithm of USmoney, Source: IFS, Code: 34..BZFseasonally adjusted (Units: US Dollars, Billions of dollars).
 mit: the natural logarithm of foreign narrow money, seasonallyadjusted. For Japan, Korea, Denmark and Australia, Source: IFS,Code: 34..BZFmoney seasonally adjustedUnits: National Currency.For Norway we have used Broad Money M2, Source: IFS, Code:59MB.ZF unadjusted (we adjusted it using X12). For Switzerland,we used the M1 money seasonally adjusted denition, averageon 3months, Source: Swiss National Bank Website. For Sweden,we use M0 money supply, notes and coins held by Swedish non-bank public, month ending stock, SEK millions, then average ofthe three months in the quarter and seasonally adjusted usingX12. Source: Riksbank Website. For the UK, we used notes andcoins, seasonally adjusted, Source: downloaded from NationalStatistics Website, Code: AVAB. For Canada, we use the M1Money Supply, average of monthly data, Source: OECD MainEconomic Indicators. For the Euro area, 1990q11997q2, we useaverage of the three months in each quarter, of M1 Euro area,Outstanding amounts at the end of the period (stocks), seasonallyadjusted, Millions of Euros. Source: EACBN real time data set(M1.xls, last vintage). For 1997q32009q2, we have updated thisseries from ECB website (Euro-Money-M1-update.xls), where wetake the average of the three months in each quarter.
Appendix A. Predictive densities for xed effect panels
Let the panel be written as:
Y XB U; 5
where Y is a TN 1 matrix of observations on the N variables in thepanel. X is an appropriately dened matrix of fundamental dis-
equilibrium terms, deterministic terms, etc. B are the estimated
coefcients and U is the error matrix, characterised by error covariancematrix .
Based on these T observations, Zellner (1971, pages 233236)derives the predictive distribution (using a common non-informativeprior) for out-of-sample observations. Crucially, X is assumed to beknown. In this setup, the predictive distribution is multivariateStudent's-t (see page 235 of Zellner, 1971). Analytical results forpredictive means, variances and probabilities such as Pr(si,,h b a|I)can be directly obtained using the properties of the multivariateStudent's-t distribution. For other predictive features of interest,predictive simulation, involving simulating from this multivariateStudent's-t can be done in a straightforward manner. The previousmaterial assumed X is known. In the case of one period aheadprediction, h=1, then X is known. That is, in (4), if Y is the T+1 changeavailable at time t, then X will contain information dated t or earlier.Where h N 1, following common practice, we can simply estimate adifferent panel model for each h, dening the dependant variable as
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Forecasting exchange rates using panel model and model averaging1. Introduction2. The model space2.1. Basic model structure2.2. Fundamental models of exchange rate disequilibrium
3. Methodology3.1. Combination construction
4. Application: a recursive forecasting exercise4.1. Data4.2. Weights, forecasting and components
5. Empirical results: point and interval forecasts5.1. Point forecasts5.2. Interval forecasts
6. Probability event forecast analysis7. ConclusionData appendixAppendix A. Predictive densities for fixed effect panelsReferences