Out-of-Sample Exchange Rate Predictability with Taylor Rule Fundamentals

Transcription

1 Ou-of-Sample Exchange Rae Predicabiliy wih Taylor Rule Fundamenals Tanya Molodsova * Emory Universiy David H. Papell ** Universiy of Houson November 25, 2008 Absrac An exensive lieraure ha sudied he performance of empirical exchange rae models following Meese and Rogoff s (1983a) seminal paper has no convincingly found evidence of ou-of-sample exchange rae predicabiliy. This paper exends he convenional se of models of exchange rae deerminaion by invesigaing predicabiliy of models ha incorporae Taylor rule fundamenals. We find evidence of shorerm predicabiliy for 11 ou of 12 currencies vis-à-vis he U.S. dollar over he pos-breon Woods floa, wih he sronges evidence coming from specificaions ha incorporae heerogeneous coefficiens and ineres rae smoohing. The evidence of predicabiliy is much sronger wih Taylor rule models han wih convenional ineres rae, purchasing power pariy, or moneary models. We are graeful o Menzie Chinn, Todd Clark, Charles Engel, Marin Evans, Luz Kilian, Nelson Mark, Mike McCracken, John Rogers, Moo Shinani, Vadym Volosovych, Jian Wang, Ken Wes, wo anonymous referees, and paricipans a he 2006 Economeric Sociey Summer Meeings, Inernaional Economics and Finance Sociey sessions a he 2006 SEA and 2007 ASSA meeings, 2007 NBER Summer Insiue, 2007 Bank of Canada-ECB Workshop on Exchange Rae Modeling, and Texas Camp Economerics XI for helpful commens and discussions. * Deparmen of Economics, Emory Universiy, Alana, GA Tel: +1 (404) molod@emory.edu ** Deparmen of Economics, Universiy of Houson, Houson, TX Tel: +1 (713) dpapell@uh.edu

2 1. Inroducion The failure of open-economy macro heory o explain exchange rae behavior using economic fundamenals has prevailed in he inernaional economics lieraure since he seminal papers by Meese and Rogoff (1983a, 1983b), who examine he ou-of-sample performance of hree empirical exchange rae models during he pos-breon Woods period and conclude ha economic models of exchange rae deerminaion of he 1970 s vinage do no perform beer han a random walk model. While, saring wih Mark (1995), a number of sudies have found evidence of greaer predicabiliy of economic exchange rae models a longer horizons, hese findings have been quesioned by Kilian (1999). The recen comprehensive sudy by Cheung, Chinn and Pascual (2005) examines he ou-of-sample performance of he ineres rae pariy, moneary, produciviy-based and behavioral exchange rae models and concludes ha none of he models consisenly ouperforms he random walk a any horizon. There is a disconnec beween mos research on ou-of-sample exchange rae predicabiliy, which is based on empirical exchange rae models of he 1970s, and he lieraure on moneary policy evaluaion, which is based on some varian of he Taylor (1993) rule. A recen lieraure uses Taylor rules o model exchange rae deerminaion. The Taylor rule specifies ha he cenral bank adjuss he shor-run nominal ineres rae in response o changes in inflaion and he oupu gap. By specifying Taylor rules for wo counries and subracing one from he oher, an equaion is derived wih he ineres rae differenial on he lef-hand-side and he inflaion and oupu gap differenials on he righ-hand-side. If one or boh cenral banks also arge he purchasing power pariy (PPP) level of he exchange rae, he real exchange rae will also appear on he righ-hand-side. Posiing ha he ineres rae differenial equals he expeced rae of depreciaion by uncovered ineres rae pariy (UIRP) and solving expecaions forward, an exchange rae equaion is derived. Engel and Wes (2005) use he Taylor rule model as an example of presen value models where asse prices (including exchange raes) will approach a random walk as he discoun facor approaches one. Engel and Wes (2006) consruc a model-based real exchange rae as he presen value of he difference beween home and foreign oupu gaps and inflaion raes, and find a posiive correlaion beween he model-based rae and he acual dollar-mark real exchange rae. Mark (2007) considers Taylor rule ineres rae reacion funcions for Germany and he U.S. and esimaes he real dollar-mark exchange rae pah assuming ha he exchange rae is priced by uncovered ineres rae pariy. He provides evidence ha he ineres rae differenial can be modeled as a Taylor rule differenial and he real dollar-mark exchange rae is linked o he Taylor rule fundamenals, which may provide a resoluion for he exchange rae disconnec puzzle. Groen and Masumoo (2004) and Gali (2008) embed Taylor rules in open economy dynamic sochasic general equilibrium models and race ou he effecs of moneary policy shocks on real and nominal exchange raes, respecively. 1

3 In his paper, we examine ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals. The saring poin for our analysis is he same as for he Taylor rule model of exchange rae deerminaion, he Taylor rule for he foreign counry is subraced from he Taylor rule for he Unied Saes (he domesic counry). There are a number of differen specificaions ha we consider. While each specificaion has he ineres rae differenial on he lef-hand-side, here are a number of possibiliies for he righ-hand-side variables. 1. In Taylor s (1993) original formulaion, he rule posis ha he Fed ses he nominal ineres rae based on he curren inflaion rae, he inflaion gap - he difference beween inflaion and he arge inflaion rae, he oupu gap - he difference beween GDP and poenial GDP, and he equilibrium real ineres rae. Assuming ha he foreign cenral bank follows a similar rule, we consruc a symmeric model wih inflaion and he oupu gap on he righ-hand-side. Following Clarida, Gali, and Gerler (1998), (hereafer CGG), we can also posi ha he foreign cenral bank includes he difference beween he exchange rae and he arge exchange rae, defined by PPP, in is Taylor rule and consruc an asymmeric model where he real exchange rae is also included. 2. I has become common pracice, following CGG, o posi ha he ineres rae only parially adjuss o is arge wihin he period. In his case, we consruc a model wih smoohing so ha he lagged ineres rae differenial appears on he righ-hand-side. Alernaively, we can derive a model wih no smoohing ha does no include he lagged ineres rae differenial. Models wih and wihou smoohing can be symmeric or asymmeric If he wo cenral banks respond idenically o changes in inflaion and he oupu gap and heir ineres rae smoohing coefficiens are equal, so ha he coefficiens in heir Taylor rules are equal, we derive a homogeneous model where relaive (domesic minus foreign) inflaion, he relaive oupu gap, and he lagged ineres rae differenial are on he righ-hand-side. If he response coefficiens are no equal, a heerogeneous model is consruced where he variables appear separaely. The homogeneous and heerogeneous models can be eiher symmeric or asymmeric, wih or wihou smoohing. 4. If, in addiion o having he same inflaion response and ineres rae smoohing coefficiens, he wo cenral banks have idenical arge inflaion raes and equilibrium real ineres raes, here is no consan on he righ-hand-side. Oherwise, here is a consan. The models wih and wihou a consan can be eiher symmeric or asymmeric, wih or wihou smoohing. The models we have specified all have he ineres rae differenial on he lef-hand-side. If UIRP held wih raional expecaions, an increase in he ineres rae would cause an immediae appreciaion of he exchange rae followed by forecased (and acual) depreciaion in accord wih Dornbusch s (1976) overshooing model. Empirical research on he forward premium and delayed overshooing puzzles, 1 Benigno (2004) shows ha, in he conex of a model incorporaing a Taylor rule, real exchange rae persisence requires ineres rae smoohing. 2

4 however, is no supporive of UIRP in he shor run. Gourinchas and Tornell (2004) propose an explanaion for boh puzzles based on a disorion in beliefs abou fuure ineres raes, and use survey daa o documen he exen of he disorion. We assume ha invesors use his heoreical and economeric evidence for forecasing, so ha an increase in inflaion and/or he oupu gap will increase he counry s ineres rae, cause immediae exchange rae appreciaion, and produce a forecas of furher exchange rae appreciaion. The relevan lieraure on exchange rae predicabiliy compares ou-of-sample predicabiliy of wo models (linear fundamenal-based model and a random walk) on he basis of differen measures. The mos commonly used measure of predicive abiliy is mean squared predicion error (MSPE). In order o evaluae ou-of-sample performance of he models based on he MSPE comparison, ess for equal predicabiliy of wo non-nesed models, inroduced by Diebold and Mariano (1995) and Wes (1996), are ofen used (henceforh, DMW ess). While he DMW ess are appropriae for non-nesed models, i is by now well-known ha, when comparing MSPE s of wo nesed models, mechanical applicaion of he DMW procedures leads o nonnormal es saisics and he use of sandard normal criical values usually resuls in very poorly sized ess, wih far oo few rejecions of he null. 2 This is a problem for ou-of-sample exchange rae predicabiliy because, since he null is a random walk, all ess wih fundamenal-based models are nesed and he ypical resul is ha he random walk null canno be rejeced in favor of he model-based alernaive. In addiion o being severely undersized, he sandard DMW procedure demonsraes very low power, which makes his saisic ill-suied for deecing deparures from he null. Rossi (2005) documens he exisence of size disorions of he DMW ess by revisiing he Meese and Rogoff puzzle. While her paper suggess a possible way o solve his problem by adjusing criical value of he ess, he resuling saisic has low power. We apply a recenly developed inference procedure for esing he null of equal predicive abiliy of a linear economeric model and a maringale difference model proposed by Clark and Wes (2006, 2007), which we call he CW procedure. This mehodology is preferable o he sandard DMW procedure when he wo models are nesed. The es saisic akes ino accoun ha under he null he sample MSPE of he alernaive model is expeced o be greaer han ha of he random walk model and adjuss for he upward shif in he sample MSPE of he aleraive model. The simulaions in Clark and Wes (2006) sugges ha he inference made using asympoically normal criical values resuls in properly-sized ess for rolling regressions. 3 There is an imporan disincion, emphasized by Inoue and Kilian (2004) and Rogoff and Savrakeva (2008), beween forecasing and predicabiliy. If we were evaluaing forecass from wo non-nesed models, 2 McCracken (2007) shows ha using sandard normal criical values for he DMW saisic resuls in severely undersized ess, wih ess of nominal 0.10 size generally having acual size less han An alernaive sraegy, used by Mark (1995) and Kilian (1999), is o calculae boosrapped criical values for he DMW es o consruc an accuraely sized es. While his solves he mos egregious problems wih he applicaion of he DMW es o nesed models, he advanage of he CW es is ha i has somewha greaer power. Wes (2006) provides a summary of recen lieraure on asympoic inference abou forecasing abiliy. 3

5 we could compare he MSPE s from he wo models by he DMW saisic and deermine wheher one model forecass beer han he oher. In our case, however, he null hypohesis is a random walk, all alernaive models are nesed, and we use he CW adjusmen of he DMW saisic o achieve correc size. Predicabiliy, wheher he vecor of coefficiens on he Taylor rule fundamenals is joinly significanly differen from zero in a regression wih he change in he exchange rae on he lef-hand-side, is herefore no equivalen o forecasing conen, wheher he MSPE from he alernaive model is significanly smaller han he MSPE from he null model. Pu differenly, we are using ou-of-sample mehods o evaluae he Taylor rule exchange rae model, no invesigaing wheher he model would poenially be useful o currency raders. We evaluae he ou-of-sample exchange rae predicabiliy of models wih Taylor rule fundamenals using he CW saisic for 12 OECD counries vis-à-vis he Unied Saes over he pos-breon Woods period saring in March 1973 and ending in December 1998 for he European Moneary Union counries and June 2006 for he ohers. In order o consruc Taylor rule fundamenals, we need o define he oupu gap, and we use deviaions from a linear rend, deviaions from a quadraic rend, and he Hodrick-Presco filer. Recen work on esimaing Taylor rules for he Unied Saes, noably Orphanides (2001), has emphasized he imporance of using real-ime daa, he daa available o cenral banks a he poin ha policy decisions are made. Since real-ime daa is no available for mos of hese counries over his period, we define poenial oupu using quasi-real ime rends which, alhough using revised daa, are updaed each period so ha ex-pos daa is no used o consruc he rends. 4 Orphanides and van Norden (2002), using a variey of derending echniques, show ha mos of he difference beween fully revised and real-ime daa comes from using ex pos daa o consruc poenial oupu and no from he daa revisions hemselves. The resuls provide srong evidence of shor-run exchange rae predicabiliy using Taylor rule fundamenals. A he one-monh horizon, we find saisically significan evidence of exchange rae predicabiliy a he 5 percen level for 11 of he 12 currencies. The models wih heerogeneous coefficiens, smoohing, and/or a consan provide subsanially more evidence of predicabiliy han he models wih homogeneous coefficiens, no smoohing, and/or no consan. The symmeric models (no exchange rae argeing) provide more evidence of predicabiliy han he asymmeric models for he specificaions ha include a consan, bu less evidence for he specificaions ha do no include a consan. Overall, he specificaion ha produced he mos evidence of exchange rae predicabiliy was a symmeric model wih heerogeneous coefficiens, smoohing, and a consan. For ha model, he no predicabiliy null was rejeced a he five percen level for 10 of he 12 counries for a leas one of he hree oupu gap measures, and a he 10 percen level for a leas wo of he hree measures. One issue concerning hese resuls is ha, because we are esimaing numerous models, inference based on he p-values of he mos saisically significan models is likely o be oversaed. This is paricularly 4 While real-ime OECD daa is available since 1999, his period is oo shor for comparabiliy wih previous work over he pos-breon Woods period. 4

6 imporan because we use hree oupu gap measures for each specificaion. In order o correc for daa snooping, we implemen Hansen s (2005) es of superior predicive abiliy o compare he MSPE s from he null (random walk) model o he CW-adjused MSPE s of he alernaive (Taylor rule fundamenals) models. While, as expeced, he level of saisical significance falls, here is sill subsanial evidence of exchange rae predicabiliy. In order o compare our resuls wih Taylor rule fundamenals wih oher models, we use he CW saisic o evaluae he ou-of-sample performance of exchange rae models based on ineres rae differenials, purchasing power pariy fundamenals, and hree varians of moneary fundamenals, for he same currencies and ime period. The evidence of predicabiliy is much weaker for hese models han for he models wih Taylor rule fundamenals. A he one-monh horizon, we find saisically significan evidence of exchange rae predicabiliy a he 5 percen level for only 3 of he 12 currencies using a leas one of he models and a he 10 percen level for one addiional currency. For all four currencies, he sronges evidence is provided by he model based on ineres rae differenials ha includes a consan erm. The final quesion ha we invesigae is wheher our evidence of predicabiliy comes from moneary policy characerized by a Taylor rule raher han from an ad hoc forecasing equaion. Using he symmeric model wih heerogeneous coefficiens, smoohing, and a consan, we examine he evoluion of he coefficiens on U.S. and foreign inflaion for he mos significan oupu gap measure. Because he daa sars in March 1973 and we use rolling regressions wih 10 years of daa, he firs forecas is for March The inflaion coefficien for he U.S. follows he same paern for he preponderance of exchange raes. I sars near zero, falls sharply around 1991, and says negaive for he remainder of he sample. Since mos of he empirical evidence is consisen wih he hypohesis ha he Fed adoped some varian of he Taylor rule saring in he mid-1980s, our findings indicae ha an increase in U.S. inflaion caused forecased appreciaion saring a he poin when approximaely one-half of he observaions in he forecasing regression were from periods where U.S. moneary policy is generally characerized by a Taylor rule. While he inflaion coefficiens for he foreign counries, where he evidence ha moneary policy can be characerized by a Taylor rule is weaker, do no follow as sharp a paern, some commonaliies emerge. The coefficiens are generally posiive, consisen wih an increase in inflaion causing forecased appreciaion (depreciaion of he dollar), and hey become more posiive in he laer par of he sample. 2. Exchange Rae Models 2.1 Taylor Rule Fundamenals We examine he linkage beween he exchange raes and a se of fundamenals ha arise when cenral banks se he ineres rae according o he Taylor rule. Following Taylor (1993), he moneary policy rule posulaed o be followed by cenral banks can be specified as * * * = π + φ( π π ) + γy + r (1) i 5

7 where inflaion, and * i is he arge for he shor-erm nominal ineres rae, π is he inflaion rae, * π is he arge level of y is he oupu gap, or percen deviaion of acual real GDP from an esimae of is poenial level, * r is he equilibrium level of he real ineres rae. I is assumed ha he arge for he shor-erm nominal ineres rae is achieved wihin he period so here is no disincion beween he acual and arge nominal ineres rae. According o he Taylor rule, he cenral bank raises he arge for he shor-erm nominal ineres rae if inflaion rises above is desired level and/or oupu is above poenial oupu. The arge level of he oupu deviaion from is naural rae y is 0 because, according o he naural rae hypohesis, oupu canno permanenly exceed poenial oupu. The arge level of inflaion is posiive because i is generally believed ha deflaion is much worse for an economy han low inflaion. Taylor assumed ha he oupu and inflaion gaps ener he cenral bank s reacion funcion wih equal weighs of 0.5 and ha he equilibrium level of he real ineres rae and he inflaion arge were boh equal o 2 percen. The parameers * π and which leads o he following equaion, * r in equaion (1) can be combined ino one consan erm i * = µ + λπ + γy * * = r, µ φπ where λ =1 + φ. Because λ > 1, he real ineres rae is increased when inflaion rises and so he Taylor principle is saisfied. 5 While i seems reasonable o posulae a Taylor rule for he Unied Saes ha includes only inflaion and he oupu gap, i is common pracice, following CGG, o include he real exchange rae in specificaions for oher counries, i * = µ + λπ + γy + δq where q is he real exchange rae. The raionale for including he real exchange rae in he Taylor rule is ha he cenral bank ses he arge level of he exchange rae o make PPP hold and increases (decreases) he nominal ineres rae if he exchange rae depreciaes (appreciaes) from is PPP value. I has also become common pracice o specify a varian of he Taylor rule which allows for he possibiliy ha he ineres rae adjuss gradually o achieve is arge level. Following CGG, we assume ha he acual observable ineres rae i parially adjuss o he arge as follows: * = ( 1 ) i + ρi 1 Subsiuing (3) ino (4) gives he following equaion, i ρ + v (4) (2) (3) 5 While i is quie possible for he arge inflaion rae and/or he equilibrium real ineres rae o vary over ime, his is less of a problem han in esimaion of Taylor rules because he rolling regressions allow for changes in he consan. Cogley, Primiceri, and Sargen (2008) presen evidence of a ime-varying inflaion arge for he U.S. 6

8 where δ = 0 for he Unied Saes. = ( 1 )( µ + λπ + γy + δq ) + ρi 1 i ρ + v (5) To derive he Taylor-rule-based forecasing equaion, we consruc he ineres rae differenial by subracing he ineres rae reacion funcion for he foreign counry from ha for he U.S.: ~ i i = α + α π α ~ π + α y α ~ y α q~ ~ + ρ i ρ i η uπ fπ uy fy q u 1 f 1 + where ~ denoes foreign variables, u and f are coefficiens for he Unied Saes and he foreign counry, α is a consan, = λ( 1 ρ) and αy = γ( 1 ρ) for boh counries, and αq = δ( 1 ρ) for he foreign counry. 6 α π Suppose ha U.S. inflaion rises above arge. If here is no smoohing, all ineres rae adjusmens are immediae. The Fed will raise he ineres rae by λ π, where π is he change in he inflaion rae. If here is smoohing, he adjusmen is gradual. The Fed will raise he ineres rae by (6) ( 1 ρ) λ π in he firs period. In he second period, he ineres rae will be 2 (1 ρ ) λ π above is original level, followed by 3 (1 ρ ) λ π, and so on. The maximum impac on he ineres rae will be approximaely λ π, he same as wih no smoohing. Clarida and Waldman (2008) show ha, under opimal moneary policy where he Taylor principle is saisfied, a surprise increase in U.S. inflaion will appreciae he exchange rae. 7 How will he increase in he ineres rae differenial affec exchange rae forecass? Under UIRP and raional expecaions, he immediae appreciaion of he dollar will be followed by forecased (and acual) depreciaion. In ha case, we could derive an exchange rae forecasing equaion by replacing he ineres rae differenial wih he expeced rae of depreciaion and use he variables from he wo counries Taylor rules o forecas exchange rae changes, so ha an increase in inflaion would produce a forecas of exchange rae depreciaion. There is overwhelming evidence, however, ha UIRP does no hold in he shor run. This is eviden in boh he exensive lieraure on he forward premium puzzle, a recen example being Chinn (2006), and he delayed overshooing lieraure on he response of exchange raes o moneary policy shocks iniiaed by Eichenbaum and Evans (1995). Neiher of hese wo srands of research, however, provides a complee answer o our quesion. 8 Gourinchas and Tornell (2004) show ha an increase in he ineres rae can cause susained exchange rae appreciaion if invesors sysemaically underesimae he persisence of ineres rae shocks. Suppose he federal funds rae increases, hen reurns gradually o is equilibrium value. This would occur wih a Taylor rule if inflaion rises above arge and is gradually brough down. If invesors know he exac 6 As shown by Engel and Wes (2005), his specificaion would sill be applicable if he U.S. had an exchange rae arge in is ineres rae reacion funcion. 7 Engel (2008) argues ha his resul appeared earlier in Engel and Wes (2006). 8 The lieraure on he forward premium puzzle provides evidence of he failure of uncondiional UIRP, he response of he exchange rae o all shocks on average. The lieraure on he response of exchange raes o moneary policy shocks does no capure he sysemaic aspec of policy. 7

9 naure of he ineres rae pah, he exchange rae will immediaely appreciae up o he poin where he ineres rae differenial equals he expeced depreciaion. They call his he forward premium effec. If invesors misperceive ha he increase is ransiory and will rever o is equilibrium value fairly quickly, he dollar will only appreciae moderaely. In he following period, he ineres rae will be higher han invesors originally expeced, leading hem o revise heir beliefs abou he persisence of he ineres rae shock upward and cause furher appreciaion of he dollar. They call his he updaing effec. If he updaing effec dominaes he forward premium effec, he dollar will appreciae unil he rue degree of persisence is revealed, a which poin he dollar will depreciae o is equilibrium value. They suppor heir heory wih survey evidence ha invesors overesimae he relaive imporance of ransiory ineres rae shocks. Wih ineres rae smoohing, higher inflaion no only raises he curren ineres rae, i causes expecaions of furher ineres rae increases in he fuure. Under UIRP and raional expecaions, ineres rae smoohing does no affec he resuls. An increase in he ineres rae, wheher curren or expeced in he fuure, will cause an immediae appreciaion of he dollar, followed by forecased (and acual) depreciaion. In he conex of he Gourinchas and Tornell (2004) model, i seems reasonable o assume ha, since he iniial change in he ineres rae is smaller han he maximal impac, he degree of under-predicion of ineres rae persisence will be more severe when he cenral bank smoohes is response o higher inflaion over ime. Wih a higher degree of under-predicion, he updaing effec will be sronger relaive o he forward premium effec, srenghening he link beween higher inflaion and forecased exchange rae appreciaion. If he foreign counry also follows a Taylor rule, an increase in foreign inflaion above is arge will cause forecased dollar depreciaion. The link beween higher inflaion and forecased exchange rae appreciaion poenially characerizes any counry where he cenral bank uses he ineres rae as he insrumen in an inflaion argeing policy rule. In he conex of he Taylor rule, hree addiional predicions can be made. Firs, if he U.S. oupu gap increases, he Fed will raise ineres raes and cause he dollar o appreciae. If he foreign counry also follows a Taylor rule, an increase in he foreign oupu gap will raise he foreign ineres rae and cause he dollar o depreciae. Second, if he real exchange rae for he foreign counry depreciaes and i is included in is cenral bank s Taylor rule, he foreign cenral bank will raise is ineres rae, causing he foreign currency o appreciae and he dollar o depreciae. Third, if here is ineres rae smoohing, a higher lagged ineres rae will increase curren and expeced fuure ineres raes. Under UIRP and raional expecaions, any even ha causes he Fed o raise he federal funds rae will produce immediae dollar appreciaion and forecased dollar depreciaion. Based on he empirical and heoreical evidence discussed above, however, we believe i is more reasonable o posulae ha hese evens will produce boh immediae and forecased dollar appreciaion. Similarly, any even ha causes he foreign cenral bank o raise is ineres rae will produce immediae and forecased dollar depreciaion. These predicions can be combined wih (6) o produce an exchange rae forecasing equaion: 8

10 s ω ω π ω ~ π ω y ω ~ y ω q~ ω i ~ ω i + 1 = uπ + fπ uy + fy + q ui 1 + fi 1 + η (7) The variable s is he log of he U.S. dollar nominal exchange rae deermined as he domesic price of foreign currency, so ha an increase in s is a depreciaion of he dollar. The reversal of he signs of he coefficiens beween (6) and (7) reflecs he presumpion ha anyhing ha causes he Fed and/or oher cenral banks o raise he U.S. ineres rae relaive o he foreign ineres rae will cause boh immediae and forecased dollar o appreciaion. Since we do no know by how much a change in he ineres rae differenial will cause he exchange rae o adjus, we do no have a link beween he magniudes of he coefficiens in he wo equaions. 9 A number of differen models can be nesed in Equaion (7). If he foreign cenral bank doesn arge he exchange rae δ α = 0 and we call he specificaion symmeric. Oherwise, i is asymmeric. If = q he ineres rae adjuss o is arge level wihin he period ρ ρ = 0 and he model is specified wih no smoohing. Alernaively, here is smoohing. If he coefficiens on inflaion, he oupu gap, and ineres rae smoohing are he same in he U.S. and he foreign counry, so ha α u = α f, α uy = α fy, and ρ u = ρ f, inflaion, oupu gap, and lagged ineres rae differenials are on he righ-hand-side of Equaion (7) and we call he model homogeneous. Oherwise, i is heerogeneous. Finally, if he coefficiens on inflaion, ineres rae smoohing coefficiens, inflaion arges, and equilibrium real ineres raes are he same beween he U.S. u = f and he foreign counry, α = 0. Oherwise, a consan erm is included in Equaion (7). 2.2 Ineres Rae Fundamenals Under UIRP, he expeced change in he log exchange rae is equal o he nominal ineres rae differenial. If we were willing o assume ha UIRP held, we could use i as a forecasing equaion. Since empirical evidence indicaes ha, while exchange rae movemens may be consisen wih UIRP in he longrun, i clearly does no hold in he shor-run, we need a more flexible specificaion. Following Clark and Wes (2006), we use he ineres rae differenial in a forecasing equaion, ~ s = α + ω( i i ) (8) + 1 Since we do no resric ω = 1, or even is sign, (8) can be consisen wih UIRP, where a posiive ineres rae differenial produces forecass of exchange rae depreciaion, and he forward premium puzzle lieraure, where a posiive ineres rae differenial produces forecass of exchange rae appreciaion. 2.3 Moneary Fundamenals Following Mark (1995), mos widely used approach o evaluaing exchange rae models ou of sample is o represen a change in (he logarihm of) he nominal exchange rae as a funcion of is deviaion from is π π 9 Chinn (2008) uses a similar equaion for in-sample esimaion. 9

11 fundamenal value. Thus, he h-period-ahead change in he log exchange rae can be modeled as a funcion of is curren deviaion from is fundamenal value. where z s + h s h h + + h, = f s = α + β z ν, (9) and f is he long-run equilibrium level of he nominal exchange rae deermined by macroeconomic fundamenals. We selec he flexible-price moneary model as represenaive of 1970 s vinage models. The moneary approach deermines he exchange rae as a relaive price of he wo currencies, and models exchange rae behavior in erms of relaive demand for and supply of money in he wo counries. The longrun money marke equilibrium in he domesic and foreign counry is given by: m m = p + ky hi (10) * * * * * * = p + k y h i (11) where m, p, and y are he logs of money supply, price level and income and i is he level of ineres rae in period ; aserisks denoe foreign counry variables. Assuming purchasing power pariy, UIRP, and no raional speculaive bubbles, he fundamenal value of he exchange rae can be derived. f * * = ( m m ) k( y y ) (12) We consruc he moneary fundamenals wih a fixed value of he income elasiciy, k, which can equal o 0, 1, or 3. We subsiue he moneary fundamenals (12) ino (9), and use he resulan equaion for forecasing. 2.4 Purchasing Power Pariy Fundamenals As a basis of comparison, we examine he predicive power of PPP fundamenals. There has been exensive research on PPP in he las decade, and a growing body of lieraure finds ha long-run PPP holds in he pos-1973 period 10. Since he moneary model is build upon PPP bu assumes addiional resricions, comparing he ou-of-sample performance of he wo models is a logical exercise. Mark and Sul (2001) use panel-based forecass and find evidence ha he linkage beween exchange raes and moneary fundamenals is igher han ha beween exchange raes and PPP fundamenals. where Under PPP fundamenals, f * = ( p p ) (13) p is he log of he naional price level. We use he CPI as a measure of naional price levels. We subsiue he PPP fundamenals (13) ino (9), and use he resulan equaion for forecasing. 10 See Papell (2006) for a recen example. 10

12 3. Empirical Resuls The models are esimaed using monhly daa from March 1973 hrough December 1998 for Euro Area counries and June 2006 for he ohers. 11 The currencies we consider are he Japanese yen, Swiss franc, Ausralian dollar, Canadian dollar, Briish pound, Swedish kronor, Danish kroner, Deusche mark, French franc, Ialian lira, Duch guilder, and Poruguese escudo. Our choice of counries reflecs our inenion o examine exchange rae behavior for major indusrialized economies wih flexible exchange raes over he sample. The exchange rae is defined as he US dollar price of a uni of foreign currency, so ha an increase in he exchange rae is a depreciaion of he dollar. 3.1 Daa The primary source of daa used o consruc macroeconomic fundamenals is he IMF's Inernaional Financial Saisics (IFS) daabase. 12 We use M1 o measure he money supply for mos of he counries. We use M0 for he U.K. and M2 for Ialy and Neherlands, because M1 daa is unavailable for hese counries. Using M2 as a measure of he money supply provides similar resuls. We use he seasonally adjused indusrial producion index (IFS line 66) as a proxy for counries naional income since GDP daa are available only a he quarerly frequency. 13 The price level in he economy is measured by consumer price index (IFS line 64). The inflaion rae is he annual inflaion rae, measured as he 12-monh difference of he CPI. 14 We use money marke rae (or call money rae, IFS line 60B) as a measure of he shor-erm ineres rae ha he cenral bank ses every period. The exchange raes are aken from he Federal Reserve Bank of Sain Louis daabase. The oupu gap depends on he measure of poenial oupu. Since here is no presumpion abou which definiion of poenial oupu is used by cenral banks in heir ineres rae reacion funcions, we consider percenage deviaions of acual oupu from a linear ime rend, a quadraic ime rend, and a Hodrick-Presco (1997) (HP) rend as alernaive definiions. 15 In order o mimic as closely as possible he 11 Some of he models are esimaed using shorer spans of daa because of daa unavailabiliy. The foonoes for he ables lis hese excepions. 12 The complee Daa Appendix and daa files are available a he auhor s web-sie: 13 The indusrial producion series for Ausralia and Swizerland, and he CPI series for Ausralia, which were available only quarerly, are ransformed ino monhly periodiciy using he quadraic-mach average opion in Eviews 4.0. This conversion mehod fis a local quadraic polynomial for each observaion of he quarerly frequency by aking ses of hree adjacen poins from he source series. Then, his polynomial is used o fill in all monhly observaions so ha he average of he monhly observaions corresponds o he quarerly daa acually observed. For mos poins, one poin before and one poin afer he period currenly being inerpolaed are used o provide he hree adjacen poins. For end poins, he wo periods are boh aken from he one side where daa is available. 14 An imporan focus of Taylor rule esimaion for he U.S. has been he forward-looking naure of policymaking, eiher by using ex pos realized values of inflaion as in CGG or by using Greenbook forecass as in Orphanides (2001). Since, for he purpose of evaluaing ou-of-sample predicabiliy, i is inappropriae o use ex pos daa and cenral bank forecass are no available for oher counries, we use acual inflaion raes. 15 We use a smoohing parameer equal o o derend he monhly oupu series using he HP filer. While i would be desirable, following Orphanides (2001) for he U.S., o use cenral bank generaed esimaes of he oupu gap, hese are neiher available for our enire sample nor available for oher counries. 11

13 informaion available o he cenral banks a he ime he decisions were made, we use quasi-real ime daa in he oupu gap esimaion. For a given period, we use only he daa poins up o -1 o consruc he rend. Thus, in each period he OLS regression is re-esimaed adding one addiional observaion o he sample Esimaion and Forecasing We consruc one-monh-ahead forecass for he linear regression models wih each of he fundamenals described above. We use daa over he period March February 1982 for esimaion and reserve he remaining daa for ou-of-sample forecasing. To evaluae he ou-of-sample performance of he models, we esimae hem by OLS in rolling regressions and consruc CW saisics. Each model is iniially esimaed using he firs 120 daa poins and he one-period-ahead forecas is generaed. We hen drop he firs daa poin, add an addiional daa poin a he end of he sample, and re-esimae he model. A one monh-ahead forecas is generaed a each sep. The CW saisic is described in he Appendix Taylor Rule Fundamenals Wih a choice beween symmeric and asymmeric, homogeneous and heerogeneous, wih and wihou smoohing, and wih and wihou a consan, we esimae 16 models wih hree measures of he oupu gap, for a oal of 48 models for each counry. 18 Two overall resuls are apparen. Firs, models wih heerogeneous coefficiens provide sronger evidence of exchange rae predicabiliy in all eigh cases. Second, models wih a consan provide sronger evidence of exchange rae predicabiliy in six of he eigh cases. We herefore focus on he models wih heerogeneous coefficiens ha include a consan. Table 1 presens he resuls for 1-monh-ahead forecass of exchange raes using asymmeric Taylor rule fundamenals wih no smoohing, wih linear, quadraic and HP rends o esimae poenial oupu. The model significanly ouperforms he random walk for 4 ou of 12 counries wih a linear rend (Ialy a he 1% significance level, Canada and Sweden a he 5% significance level, and he U.K. a he 10% significance level), for 2 ou of 12 counries wih a quadraic rend (Ialy a he 1% and Canada a he 5% significance level), and for 7 ou of 12 counries wih an HP rend (Canada a he 1%, Ialy, Japan, Sweden, and he U.K. a he 5%, and Neherlands and Swizerland a he 10% significance level). The model significanly ouperforms he random walk in 13 ou of 36 cases and wih a leas one of he oupu gap specificaions for 7 ou of 12 counries. 16 We call his quasi-real ime daa because, while he rend is updaed each period, he daa incorporae revisions ha were no available o he cenral banks a he ime decisions were make. True real ime daa is no available for mos of he counries ha we sudy over he enire floaing rae period. The oupu gap for he firs period is calculaed using oupu series from 1971:1 o 1973:3. 17 We use ou-of-sample raher han in-sample mehods and esimae rolling raher han recursive regressions for comparison wih he exensive lieraure following Meese and Rogoff (1983a), and choose a rolling window of 120 observaions o esimae alernaive forecas models following he empirical exercise in Clark and Wes (2006). Inoue and Kilian (2004) advocae using in-sample raher han ou-of-sample mehods and using recursive mehods for ou-ofsample forecasing. 18 Wih heerogeneous coefficiens, i would require a paricular combinaion of coefficiens, arge inflaion raes, and equilibrium real ineres raes for he erms ha comprise he consan o cancel ou. Neverheless, he consan could be small if he smoohing coefficiens were large, and so we include he heerogeneous model wihou a consan. 12

14 Table 2 depics he resuls for he asymmeric Taylor rule model wih smoohing. The model significanly ouperforms he random walk for 4 ou of 12 counries wih a linear rend (Ialy a he 1% significance level, Canada and Japan a he 5% significance level, and Ausralia a he 10% significance level), for 6 ou of 12 counries wih a quadraic rend (Canada, Ialy, and Japan a he 1%, and Neherlands, Swizerland, and he U.K. a he 10% significance level), and for 8 ou of 12 counries wih an HP rend (Ialy and Japan a he 1%, Canada, Neherlands, and Swizerland a he 5%, and Ausralia, France, and he U.K. a he 10% significance level). The model significanly ouperforms he random walk in 18 ou of 36 cases and wih a leas one of he oupu gap specificaions for 8 ou of 12 counries. Shor-erm predicabiliy increases when we use he Taylor rule where he foreign counry does no arge he exchange rae. Table 3 presens he resuls for he symmeric Taylor rule model wih no smoohing. The model wih Taylor rule fundamenals significanly ouperforms he random walk for 8 ou of 12 counries wih a linear rend (Canada a he 1% significance level, Ausralia, Denmark, France, Ialy, Sweden, and he U.K. a he 5% significance level, and Germany a he 10% significance level), for 6 ou of 12 counries wih a quadraic rend (Canada and Ialy a he 1%, France, Germany, and he U.K. a he 5%, and Swizerland a he 1% significance level), and for 6 ou of 12 counries wih an HP rend (Canada a he 1%, France, Ialy, Sweden, and he U.K. a he 5%, and Swizerland a he 10% significance level). The model significanly ouperforms he random walk in 20 ou of 36 cases and wih a leas one of he oupu gap specificaions for 9 ou of 12 currencies. The sronges resuls are found for he symmeric Taylor rule model wih smoohing. As depiced in Table 4, he model wih Taylor rule fundamenals significanly ouperforms he random walk for 10 ou of 12 counries wih a linear rend (Canada and Ialy a he 1% significance level, Ausralia, France, Japan, Neherlands, and he U.K. a he 5% significance level, and Denmark, Germany, and Swizerland a he 10% significance level), for 9 ou of 12 counries wih a quadraic rend (Canada, Ialy, and Japan a he 1%, Ausralia, France, Germany, Neherlands, and he U.K. a he 5%, and Swizerland a he 1% significance level), and for 9 ou of 12 counries wih an HP rend (France, Ialy, and Neherlands a he 1%, Ausralia, Canada, Denmark, Japan, Swizerland, and he U.K. a he 5% significance level). The model significanly ouperforms he random walk in 28 ou of 36 cases and wih a leas one of he oupu gap specificaions for 10 ou of 12 currencies. 19 Combining he four Taylor rule models, evidence of shor-erm predicabiliy is found for 11 ou of 12 counries, five counries a he 1% level and six addiional counries a he 5% level. No evidence of predicabiliy is found for Porugal. More evidence is found wih symmeric models han wih asymmeric models and wih models wih smoohing han wih models wih no smoohing. The sronger evidence wih 19 We invesigae robusness of he resuls by spliing he sample in half. The symmeric specificaion wih heerogeneous coefficiens and a consan, bu no smoohing, provides he sronges evidence of predicabiliy in boh subsamples. There is evidence of predicabiliy in each subsample, which is relaively sronger in he earlier subsample. 13

15 smoohing is consisen wih he model of Gourinchas and Tornell (2004). Overall, he sronges resuls are found wih he symmeric Taylor rule model wih heerogeneous coefficiens, smoohing, and a consan. For ha model alone, evidence of shor-erm predicabiliy is found for 10 ou of 12 counries, four counries a he 1% level and six addiional counries a he 5% level Ineres Rae, Moneary, and PPP Fundamenals Table 5 conains he resuls for one-monh-ahead forecass of he exchange raes using he ineres rae, moneary, and PPP fundamenals described in Secion 2. We do no find much evidence of exchange rae predicabiliy wih any of he models. The sronges evidence comes from ineres rae fundamenals wih a consan, where he model significanly ouperforms he random walk for 4 ou of 12 counries (Japan a he 1% significance level, Swizerland a he 5% significance level, and Ausralia and Canada a he 10% significance level). Wihou a consan, he model wih ineres rae fundamenals significanly ouperforms he random walk for 2 counries (Ausralia and Canada a he 10% significance level). The evidence is weaker for moneary fundamenals. Wih he coefficien on relaive oupu k equal o 0, he model significanly ouperforms he random walk for 2 ou of 12 counries wih a consan (Canada and Japan a he 5% significance level) and 1 counry wihou a consan (Japan a he 10% significance level). The evidence wih k = 1 and k = 3 is weaker wih a consan and he same wihou a consan. The weakes evidence is found wih PPP fundamenals, where he model significanly ouperforms he random walk for 1 counry (Japan a he 10% significance level) wihou a consan and for no counries wih a consan Tesing for Superior Predicive Abiliy Since we are simulaneously esing muliple hypoheses, inference based on convenional p-values is likely o be conaminaed. This issue arises because we have 58 differen models of 12 bilaeral exchange raes yielding 696 es saisics. As a resul of an exensive specificaion search, i is possible o misake he resuls ha could be generaed by chance for genuine evidence of predicive abiliy. To increase he reliabiliy of our resuls, we perform he es of superior predicive abiliy (SPA) proposed by Hansen (2005). The SPA es is designed o compare he ou-of-sample performance of a benchmark model o ha of a se of alernaives. This approach is a modificaion of he realiy check for daa snooping developed by Whie (2000). The advanages of he SPA es are ha i is more powerful and less sensiive o he inroducion of poor and irrelevan alernaives Engel, Mark, and Wes (2007), using a specificaion of Taylor rule fundamenals from an earlier version of his paper, find lile evidence of predicabiliy. They use an asymmeric model wih no smoohing, a consan, homogeneous coefficiens, and HP filered oupu which, in Table 1, produces only four rejecions a he 5 percen level. In addiion, hey impose φ = γ = 0.5 for boh counries and δ = 0.1 for he foreign counry, which furher resrics he forecass. 21 We also invesigaed longer (3, 6, 12, 24, and 36 monh) horizons. A he hree-monh horizon, we found some evidence of predicabiliy for Canada and Ialy wih Taylor rule fundamenals and Japan wih ineres rae fundamenals. A longer horizons, we found no evidence of exchange rae predicabiliy for eiher he Taylor rule or he oher models. 22 Hansen (2005) provides deails on he consrucion of he es saisic and confirms he advanages of he es by Mone Carlo simulaions. We use he publicly available sofware package MULCOM o consruc he SPA-consisen p- 14

16 We are ineresed in comparing he ou-of-sample performance of linear exchange rae models o a naïve random walk benchmark. The SPA es can be used for comparing he ou-of-sample performance of wo or more models. I ess he composie null hypohesis ha he benchmark model is no inferior o any of he alernaives agains he alernaive ha a leas one of he linear economic models has superior predicive abiliy. In he conex of using he CW saisic o evaluae ou-of-sample predicabiliy, he null hypohesis is ha he random walk has an MSPE which is smaller han or equal o he adjused MSPE s of he linear models, as described by Equaion (A2) in he Appendix. 23 Therefore, rejecing he null indicaes ha a leas one linear model is sricly superior o he random walk. Tables 6-8 repor he SPA p-values ha ake ino accoun he search over models ha preceded he selecion of he model being compared o he benchmark. A low p-value suggess ha he benchmark model is inferior o a leas one of he compeing models. A high p-value indicaes ha he daa analyzed do no provide srong evidence ha he benchmark is ouperformed. The SPA es is designed o guard agains evidence of predicabiliy obained by esimaing a large number of models and focusing on he one wih he mos significan resuls. Wih Taylor rule fundamenals, he mos arbirary choice is he measure of he oupu gap, and we need o evaluae how esimaing models wih linear, quadraic, and HP derending for each specificaion affecs our evidence of predicabiliy. The Taylor rule specificaions hemselves, in conras, are no arbirary. The choice among consan/no consan, homogeneous/heerogeneous, symmeric/asymmeric, and smoohing/no smoohing are guided by economic heory and previous empirical research. Table 6 repors he resuls for he 16 Taylor rule specificaions, where he benchmark model is he random walk and he alernaives are he hree oupu gap measures. The SPA p-values srongly confirm he resuls in Tables 1-4. Combining he 16 models, evidence of shor-erm predicabiliy is again found for 11 ou of 12 counries (Canada a he 1% significance level, Ausralia, France, Ialy, Japan, Neherlands, Sweden, and he U.K. a he 5% significance level, and Denmark, Germany, and Swizerland a he 10% significance level). The models wih heerogeneous coefficiens provide more evidence of exchange rae predicabiliy han he models wih homogeneous coefficiens and he models wih a consan provide more evidence of predicabiliy han he models wihou a consan, wih he mos evidence provided by models wih boh heerogeneous coefficiens and a consan. As above, he sronges resuls are found wih he symmeric Taylor rule model wih heerogeneous coefficiens, smoohing, and a consan. For ha model alone, evidence of shor-erm predicabiliy is again found for 10 ou of 12 counries (Canada a he 1% significance level, Ausralia, France, Ialy, Japan, and Neherlands a he 5% significance level, and Denmark, Germany, Swizerland, he U.K. a he 10% significance level). While, as expeced, he SPA p-values are higher han he mos significan single-oupu-gap p-values, he resuls show ha he evidence of exchange rae predicabiliy values for each counry. The code, deailed documenaion, and examples can be found a hp:// 23 We use he adjused MSPE s from he linear models so ha he ess have correc size. Hubrich and Wes (2007) develop a similar procedure based on he Whie (2000) es. 15

17 repored above is no an arifac of picking he oupu gap specificaion wih he lowes p-value for each model. Table 7 repors SPA p-values wih a larger se of alernaives for he Taylor rule specificaions wih heerogeneous coefficiens and a consan. While hese specificaions are he ones for which he mos evidence of predicabiliy was found, here seems o be no compelling reason o hink ha he Fed and foreign cenral banks followed he same quaniaive ineres rae reacion funcion in response o inflaion and oupu deviaions, much less, in addiion, had he same inflaion arges and equilibrium real ineres raes. The firs four columns es he random walk benchmark agains six alernaives. For example, symmeric would denoe smoohing and no smoohing for he hree oupu gap measures. The SPA p- values again confirm our previous resuls. Combining he 4 models, evidence of shor-erm predicabiliy is found for 10 ou of 12 counries (Canada, France, Ialy, Japan, and Neherlands a he 5% significance level and Ausralia, Denmark, Sweden, Swizerland, and he U.K. a he 10% significance level). The symmeric models provide more evidence of ou-of-sample exchange rae predicabiliy han he asymmeric models (9 versus 3 ou of 12 counries a he 10 percen significance level or higher) and he models wih smoohing provide more evidence of predicabiliy han he models wih no smoohing (9 versus 4 ou of 12 counries a he 10 percen significance level or higher). The fifh column, denoed all, ess he random walk benchmark agains 12 alernaives: symmeric wih smoohing, symmeric wih no smoohing, asymmeric wih smoohing, and asymmeric wih no smoohing for he hree oupu gap measures. While, as expeced, he SPA p-values decline wih he inclusion of he asymmeric and no smoohing specificaions, evidence of shor-erm exchange rae predicabiliy is found for 7 ou of 12 counries (Canada and Japan a he 5 percen significance level and Ausralia, France, Ialy, Neherlands, and Sweden a he 10 percen significance level). Table 8 repors SPA p-values for he hree measures of he oupu gap. The firs hree columns es he random walk agains all 16 possible alernaives in Tables 1 4. Since hese include specificaions wih eiher homogeneous coefficiens and/or no consan, i is no surprising ha much less evidence of predicabiliy is found han in Table 7. The HP filer provides he mos evidence of predicabiliy, wih he no predicabiliy null rejeced a he 5% significance level for Ialy, Japan, and he Unied Kingdom and a he 10% significance level for Canada and Neherlands. The linear rend provides he nex mos evidence, wih wo rejecions a he 5% level and wo more a he 10% level, followed by he quadraic rend wih wo rejecions a he 5% level and one more a he 10% level. The fourh column, denoed all, ess he random walk benchmark agains all 48 possible Taylor rule alernaives. Evidence of predicabiliy is found a he 5% significance level for Canada and Japan and a he 10% significance level for Ialy and he Unied Kingdom. For he purpose of comparison, Table 9 repors SPA p-values for he ineres rae, PPP, and moneary models. There are wo alernaives for he ineres rae and PPP models, wih and wihou a consan, and six alernaives for he moneary models, k=0, k=1, and k=3 wih a consan and no consan. Evidence of shor-run exchange rae predicabiliy is found for 3 ou of 12 counries wih he ineres rae 16