Forecasting exchange rates using panel model and model averaging

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<ul><li><p>dKing</p><p>rateaneframelln. Iweualulaulatodxch</p><p>ng histwith bever u2007)urablenge ofes of thted witnt curre</p><p>ing the Linear Opinionl. (2010), and Garratt</p><p>Economic Modelling 37 (2014) 3240</p><p>Contents lists available at ScienceDirect</p><p>Economic M</p><p>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;</p><p>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</p><p>considered. The resulting ensemble predictive approximates the manyunknown relationships between the exchange rates and a set offundamentals using time-varying weights across component densities.</p><p>In order to constructweights attached to componentmodels, each ofwhich is likely to be mis-specied at some point in time, we consideracknowledged. Corresponding author.</p><p>E-mail address: em92@leicester.ac.uk (E. Mise).1 In addition papers such asMark (1995), Abhyankar et</p><p>(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</p><p>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</p><p>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</p><p>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</p><p>panel based fundamentalis limited success standh the model specication</p><p>Timmermann, 2006 and for the exchange rate Abhyankar et al., 2005;Garratt and Lee, 2010; Della Corte et al. (2012)).</p><p>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</p><p>Anthony Garratt a, Emi Mise b,a University of Warwick, United Kingdomb Department of Economics, University of Leicester, University Road, Leicester LE1 7RH, United</p><p>a b s t r a c ta r t i c l e i n f o</p><p>Article history:Accepted 8 October 2013Available online xxxx</p><p>JEL classication:C32C53E37</p><p>Keywords:Exchange rate forecastingPoint and interval forecastsModel averaging</p><p>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.</p><p>ghts reserved.el and model averaging</p><p>dom</p><p>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.</p><p> 2013 Elsevier B.V. All rights reserved.</p><p>odelling</p><p>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</p></li><li><p>based on a Bayesian perspective, and although here we estimate andcombine the models based on a more frequentist approach, could beimplemented as such.</p><p>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</p><p>2.2. Fundamental models of exchange rate disequilibrium</p><p>The fundamentals which motivate the set of models we combineare:</p><p> Monetary Fundamentals [MF] Purchasing Power Parity [PPP] Efcient Market Hypothesis [EMH]</p><p>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.</p><p>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.</p><p>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.</p><p>2. The model space</p><p>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.</p><p>2.1. Basic model structure</p><p>We consider the linear xed effects panel model:</p><p>si;th jih jth jhz jit jith; 1</p><p>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</p><p>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</p><p>j and thj</p><p>are country specic and time effect dummies respectively for the jthmodel and h forecast horizon.</p><p>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</p><p>predictor. Taylor Rule(s) [TR]</p><p>Based on these four types of fundamentals, we dene alternativespecications of the error-correction term:</p><p>z jit fjitsit ; 2</p><p>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</p><p>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).</p><p>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</p><p>2it ^ i0 ^ i1fun1it where the coefcients are from the</p><p>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</p><p>3it ^ i2 ^ i3 f 1it ^ i4t</p><p>where the coefcients are from the regression sit = i2 + i3 fit1 +i4t+uit,3.</p><p>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).</p><p>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</p><p>5it ^ i5 ^ i6 f 4it where the coefcients are from the</p><p>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.</p><p>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</p><p>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</p><p>this context.</p></li><li><p>of the future spot exchange rate. Although the empirical evidence ismixed regarding the optimality of the forward rate as a predictor of</p><p>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.</p><p>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</p><p>8it </p><p>^ i10 ^ i11 f 7it where the coefcients are from the regression sit =i10+i11 fsit7+uit,8.</p><p>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:</p><p>r0trit 1:5 0tit 0:1 y0tgapyitgap 0:1 sit pitp0t ;</p><p>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:</p><p>f 9it 1:5 0tit 0:1 y0tgapyitgap 0:1 sit pitp0t sit;</p><p>and therefore, using Eq. (2):</p><p>z9it 1:5 0tit 0:1 y0tgapyitgap 0:1 sit pitp0t :</p><p>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. (...</p></li></ul>

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