Taylor Rule Deviations and Out-of-Sample Exchange Rate Predictability

Transcription

1 Taylor Rule Deviaions and Ou-of-Sample Exchange Rae Predicabiliy Onur Ince Appalachian Sae Universiy David H. Papell Universiy of Houson Tanya Molodsova Appalachian Sae Universiy April 30, 2015 Absrac The Taylor rule has become he dominan model for academic evaluaion of ou-of-sample exchange rae predicabiliy. Two versions of he Taylor rule model are he Taylor rule fundamenals model, where he variables ha ener he Taylor rule are used o forecas exchange rae changes, and he Taylor rule differenials model, where a Taylor rule wih posulaed coefficiens is used in he forecasing regression. We use daa from 1973 o 2014 o evaluae shor-run ou-of-sample predicabiliy for eigh exchange raes vis-à-vis he U.S. dollar, and find srong evidence in favor of he Taylor rule fundamenals model alernaive agains he random walk null. The evidence of predicabiliy is weaker wih he Taylor rule differenials model, and sill weaker wih he radiional ineres rae differenial, purchasing power pariy, and moneary models. The evidence of predicabiliy for he fundamenals model is no relaed o deviaions from he original Taylor rule for he U.S., bu is relaed o deviaions from a modified Taylor rule for he U.S. wih a higher coefficien on he oupu gap. The evidence of predicabiliy is also unrelaed o deviaions from Taylor rules for he foreign counries and adherence o he Taylor principle for he U.S. Deparmen of Economics, Appalachian Sae Universiy, Boone, NC Tel: +1 (828) inceo@appsae.edu Deparmen of Economics, Appalachian Sae Universiy, Boone, NC Tel: +1 (828) molodsova@appsae.edu Deparmen of Economics, Universiy of Houson, Houson, TX Tel/Fax: +1 (713) / dpapell@uh.edu

2 1. Inroducion The Taylor rule has become he dominan model for academic evaluaion of ou-ofsample exchange rae predicabiliy. Papers by Engel, Mark, and Wes (2008, 2015), Molodsova and Papell (2009, 2013), Molodsova, Nikolsko-Rzhevskyy, and Papell (2008, 2011), and Ince (2014) repor superior ou-of-sample exchange rae predicabiliy wih Taylor rule models han wih he random walk model. Rossi (2013) surveys he lieraure and concludes ha Taylor rule models perform beer han a number of alernaives. Ou-of-sample exchange rae forecasing became a prominen academic opic following Meese and Rogoff (1983), who argued ha empirical exchange rae models which appeared o fi well in-sample did no forecas beer han a random walk ou-of-sample. Their meric was he roo mean squared forecas error (RMSE), where he forecas error is he difference beween he realized and forecased exchange rae for he models and, since a random walk forecas is simply a naïve no change forecas, he realized exchange rae change for he random walk. Because he random walk forecas could be performed by anyone who read a newspaper, his received considerable aenion. The firs modern analysis of ou-of-sample exchange rae forecasing was by Mark (1995), who used error correcion mehods o evaluae Purchasing Power Pariy (PPP), Ineres Rae Pariy (IRP), and moneary models vis-à-vis he random walk model wih DMW saisic developed by Diebold and Mariano (1995) and Wes (1996). Mark found ha, while some evidence of predicabiliy could be found a long horizons of up o four years, no sysemaic evidence of predicabiliy could be found a shor horizons of one quarer. While he longhorizon resuls have been boh criicized and confirmed, he shor-horizon resuls have held up over ime. In a comprehensive paper, Cheung, Chinn, and Pascual (2005) found ha none of he sandard models could sysemaically forecas beer han he random walk a shor ime horizons. Ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals was iniiaed by Molodsova and Papell (2009). The idea is o subrac a Taylor rule for he foreign counry from a Taylor rule for he domesic counry, in his case he Unied Saes. The resulan equaion has he ineres rae differenial on he lef-hand-side and he variables ha comprise he Taylor rule, domesic and foreign inflaion, oupu gaps, and (depending on he specificaion) lagged ineres raes and/or he real exchange rae, on he righ-hand-side. If uncovered ineres rae 1

3 pariy (UIRP) held in he shor run, you would simply replace he ineres rae differenial wih he expeced rae of depreciaion o derive a forecasing equaion. However, here is overwhelming evidence, boh heoreical and empirical, ha UIRP no only does no hold in he shor run, bu ha he shor-run effecs are opposie of he UIRP predicions. The resulan forecasing equaion, herefore, reverses he signs of he coefficiens of he righ-hand-side variables from wha would be prediced by UIRP. Using he CW saisic developed by Clark and Wes (2006), Molodsova and Papell (2009) repor saisically significan evidence of exchange rae predicabiliy a he 5 percen level for 11 of he 12 currencies sudied a he one-monhahead horizon. An alernaive model of ou-of-sample exchange rae predicabiliy wih Taylor rules was developed by Engel, Mark, and Wes (2008). They subrac he Taylor rule for he base counry from he Taylor rule for he foreign counry, bu use posied raher han esimaed coefficiens and include he real exchange rae in he forecasing equaion. We call his he Taylor rule differenials model. They use boh single-equaion and panel mehods a one quarer and 16 quarer-ahead horizons, and repor some evidence of ou-of-sample predicabiliy using he CW saisic. They find sronger evidence a he 16-quarer han a he one-quarer horizon and sronger evidence when he random walk wih drif is used for he null hypohesis insead of he random walk wihou drif. Ince (2014) uses heir mehods wih real-ime daa and repors somewha sronger resuls. The financial crisis, Grea Recession, and slow recovery for he U.S. raise quesions abou wheher Taylor rule exchange rae forecasing is sill relevan in an environmen where he federal funds rae has been a he zero lower bound from he end of 2008 hrough he end of As early as December 2008, Chinn (2008) posed his quesion, concluding ha, wih policy raes near zero for Japan and he U.S. and prediced o be near-zero for he Unied Kingdom and he Euro Area, prospecs for coninued Taylor rule exchange rae forecasing were bleak. A second heme, however, was ha reurning o he moneary model, even in a ime of quaniaive easing, did no seem promising. Molodsova and Papell (2013) used LIBOR-OIS spreads, TED spreads, Bloomberg financial condiions indexes, and OECD financial condiions indexes for he U.S. and he Euro Area o augmen Taylor rule exchange rae forecasing for he dollar/euro exchange rae from 2007:Q1 o 2012:Q1. The Taylor rule fundamenals and 2

4 differenials models wih financial variables provided more evidence of ou-of-sample exchange rae predicabiliy han he models wihou financial variables. This paper has wo objecives. The firs is o updae he analysis in Molodsova and Papell (2009) and see wheher he resuls hold up when he daa is exended hrough We invesigae ou-of-sample exchange rae predicabiliy wih Taylor rule models for U.S. dollar exchange raes for seven non-euro counries ha were considered by Molodsova and Papell (2009), Ausralia, Canada, Denmark, Japan, Sweden, Swizerland, and he Unied Kingdom. Because we are ineresed in he recen period, we do no consider he Euro counries, France, Ialy, Neherlands, and Porugal, bu include he deusche mark/euro exchange rae for Germany. We esimae 20 specificaions of he Taylor rule fundamenals model. Four classes of he model, wih heerogeneous or homogenous coefficiens on inflaion and he oupu gap for he U.S. and he foreign counry and he model wih or wihou he real exchange rae, are esimaed using five specificaions. One of he models does no allow for ineres rae smoohing and, herefore, does no include lagged ineres raes. Among he four specificaions wih smoohing, wo include he lagged ineres rae differenial and wo include individual lagged ineres raes. The second division is ha wo models wih smoohing incorporae he federal funds rae for he U.S. and he oher wo models use a measure of he shadow federal funds rae, he policy rae adjused o incorporae he effecs of quaniaive easing and forward guidance, from 2009 o All of he models include a consan. Ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals does no fall apar afer he financial crisis. Overall, he models wih heerogeneous coefficiens provide subsanially more evidence of predicabiliy han he models wih homogeneous coefficiens. The models ha do no include real exchange rae, which we call symmeric, provide more evidence of predicabiliy han he models ha include exchange rae argeing, which we call asymmeric. As in Molodsova and Papell (2009), he Taylor rule fundamenals models ha produce he sronges evidence of exchange rae predicabiliy are he models wih heerogeneous coefficiens ha include ineres rae smoohing and don include he real exchange rae. For ha model wih he five Taylor rule fundamenals specificaions, he no predicabiliy null of he random walk model wihou drif can be rejeced in favor of he Taylor rule fundamenals model a he 1 percen level for 4 of he 8 counries for all specificaions, a he 5 percen level for 3 3

5 addiional counries for a leas hree specificaions, and a he 10 percen for he remaining counry for hree specificaions ou of five. We find much less evidence of ou-of-sample exchange rae predicabiliy wih he Taylor rule differenials model. We esimae 15 specificaions, he original Taylor (1993) rule, a modified Taylor rule wih a higher oupu gap coefficien for boh he U.S. and he foreign counry, and a hybrid Taylor rule wih a higher oupu gap coefficien only for he U.S., for each of he five models described above. The mos successful resuls are for models wih smoohing where individual lagged ineres raes and he shadow federal funds rae were used. For his model, he no predicabiliy null can be rejeced for 4 of he 8 counries wih he original and modified Taylor rules, and for 5 of he 8 counries wih he hybrid Taylor rule model. The convenional exchange rae models fare even worse. The no predicabiliy null can be rejeced for 3 of he 8 counries wih he ineres rae model ha incorporaes he shadow federal funds rae, 2 ou of 8 counries wih he ineres rae model ha uses he money marke rae, 2 ou of 8 counries wih he moneary model ha assumes he coefficien on relaive oupu equal o 0, 1 ou of 8 counries wih he moneary model ha ses he coefficien on relaive oupu equal o 1, and no counries wih he PPP model. The second objecive of he paper is o invesigae wheher ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals is sronger during he periods in which adherence o he Taylor rule is closer. This quesion arises because, since he Taylor rule fundamenals model includes he variables ha ener in he Taylor rule, bu does no consrain heir coefficiens, i is possible o find evidence of predicabiliy ha is unrelaed o adherence o he Taylor rule. Molodsova and Papell (2009) addressed his by examining he coefficiens on U.S. and foreign inflaion in he forecasing regressions bu, as shown by Nikolsko-Rzhevskyy, Papell, and Prodan (2015), here is no a one-o-one correspondence beween he coefficien on inflaion and adherence o he Taylor rule. Nikolsko-Rzhevskyy, Papell, and Prodan (2014) esimae srucural change models on Taylor rule deviaions, he absolue value of he difference beween he federal funds rae and he rae prescribed by he original Taylor rule, for he U.S. using real-ime daa, and idenify periods of high and low deviaions. Over he span of daa for which we conduc ou-of-sample exchange rae forecass, 1985:M4 2000:M12 is a low deviaions era and 2001:M1 2014:M12 is a high deviaions era. We divide he sample beween high and low deviaions periods, and calculae CW 4

6 saisics for each period. The resuls are no supporive of he hypohesis ha ou-of-sample exchange rae predicabiliy is relaed o adherence o he original Taylor rule, as he evidence agains he no predicabiliy null is sronger for he high deviaions eras for four counries, mixed for hree counries, and sronger for he low deviaions era for one counry. We nex consider modified Taylor rule deviaions, he absolue value of he difference beween he federal funds rae and he rae prescribed by he modified Taylor rule, which are also calculaed by Nikolsko-Rzhevskyy, Papell, and Prodan (2014). Over he span of daa for which we conduc ou-of-sample exchange rae forecass, 1985:M1 1999:M3 and 2006:M :M12 are low deviaions eras and 1999:M4 2006:M9 is a high deviaions era. The congruence beween finding evidence of ou-of-sample predicabiliy and being in a low deviaions era is much greaer for he modified Taylor rule han for he original Taylor rule. The evidence agains he no predicabiliy null hypohesis is sronger for he low deviaions eras for seven counries and mixed for one counry. Among he models, he differenial is larger for he smoohing models and somewha larger for he models wih he shadow federal funds rae. Alhough he Taylor rule fundamenals model incorporaes boh U.S. and foreign variables, he high and low deviaions eras used above are defined solely in erms of U.S. deviaions. Teryoshin (2014) uses he mehods in Nikolsko-Rzhevskyy, Papell, and Prodan (2014) o calculae original and modified Taylor rule deviaions for six counries in our sample: Ausralia, Canada, Japan, Sweden, Swizerland, and he Unied Kingdom. There is no much congruence beween finding evidence of ou-of-sample predicabiliy and being in a low deviaions era for eiher he original or he modified Taylor rule. Moneary policy analysis using Taylor rules is ypically conduced in erms of he Taylor principle ha he nominal ineres rae should increase by more han poin-for-poin when inflaion rises so ha he real ineres rae increases. Nikolsko-Rzhevskyy, Papell, and Prodan (2015) esimae Taylor rules for he U.S. over moneary policy eras defined by several Taylor rule varians. Wihin our sample, he Taylor principle holds from 1983:M3 1999:M3 because he coefficien on inflaion is significanly greaer han one, he evidence is mixed from 1999:M4 2007:M6 because he coefficien on inflaion is greaer han one bu no significan, and he Taylor principle fails o hold from 2007:M7 2014:M12 because he coefficien on inflaion is less han one. Adherence o he Taylor principle, however, is no posiively correlaed wih finding evidence of ou-of-sample predicabiliy, as he evidence of predicabiliy is sronger 5

7 during eras where he Taylor principle does no hold for four counries and sronger during eras where he Taylor principle holds for only one counry. The relaion beween Taylor rule deviaions and ou-of-sample exchange rae predicabiliy closely follow Fed policy as ariculaed by Yellen (2012). The sronges resuls are for deviaions calculaed from a modified Taylor rule wih a specificaion incorporaing ineres rae smoohing ha reflecs quaniaive easing and forward guidance. In conras, srong resuls are no obained from deviaions calculaed from he original Taylor rule and foreign Taylor rules, or from moneary policy eras based on adherence o he Taylor principle insead of he Taylor rule. 2. Exchange Rae Forecasing Models 2.1 Taylor Rule Fundamenals Model 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 where i is he arge for he shor-erm nominal ineres rae, arge level of inflaion, i ( ) y r (1) is he inflaion rae, is he y is he oupu gap, or percen deviaion of acual real GDP from an esimae of is poenial level, and 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 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. erm The parameers r and r in equaion (1) can be combined ino one consan, which leads o he following equaion, i y (2) 6

8 where 1. Because 1, he real ineres rae is increased when inflaion rises and so he Taylor principle is saisfied. Following Clarida, Gali, and Gerler (1998), i has become common pracice o specify varians of he Taylor rule which allow for he possibiliy ha he ineres rae adjuss gradually o achieve is arge level and/or include he real exchange rae in addiion o inflaion and he oupu gap. The raionale for including he real exchange rae 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. We assume ha he acual observable ineres rae i parially adjuss o he arge as follows: Subsiuing (2) ino (3) gives he following equaion, i i ( 1 ) i i v (3) 1 ( 1 )( y q ) i v (4) 1 where q is he real exchange rae. 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 u f uy y fy y q q i u 1 i where denoes foreign variables, u and f are coefficiens for he Unied Saes and he foreign counry. Alhough equaion (5) only includes he real exchange rae in he Taylor rule for he foreign counry, his specificaion would be unchanged if he U.S. also had an exchange rae arge in is ineres rae reacion funcion. 1 Based on empirical research on he forward premium and delayed overshooing puzzles by Eichenbaum and Evans (1995), Faus and Rogers (2003) and Scholl and Uhlig (2008), and he resuls in Gourinchas and Tornell (2004) and Bacchea and van Wincoop (2010), who show ha an increase in he ineres rae can cause susained exchange rae appreciaion if invesors eiher sysemaically underesimae he persisence of ineres rae shocks or make infrequen porfolio decisions, we posulae he following exchange rae forecasing equaion: 2 f 1 (5) 1 This was shown by Engel and Wes (2005). 2 A more exensive discussion can be found in Molodsova and Papell (2009). 7

9 e y y q i i 1 u f uy fy q ui 1 fi 1 (6) The variable e is he log of he U.S. dollar nominal exchange rae deermined as he domesic price of foreign currency, so ha an increase in e is a depreciaion of he dollar. The reversal of he signs of he coefficiens beween (5) and (6) 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 he dollar o appreciae (a decrease in e ). 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 (5) and (6). 2.2 Taylor Rule Differenials Model Engel, Mark, and Wes (2008, 2015) propose an alernaive Taylor rule based model, which we call he Taylor rule differenials model o differeniae i from boh he ineres rae differenials model and he Taylor rule fundamenals model. The difference beween he Taylor rule differenials and fundamenals models is ha he former posis, raher han esimaes, he coefficiens for he Taylor rule. Using Taylor s original coefficiens and subracing he ineres rae reacion funcion for he foreign counry from ha for he U.S., we obain implied ineres rae differenials, where is a consan. 3 i i a 1.5( ) 0.5( y y ) (7) The implied ineres rae differenial can be used o consruc an exchange rae forecasing equaion, e (1.5( ) 0.5( y y )) 1 i (8) where, as in he Taylor rule fundamenals model, he signs of he coefficiens are assumed o swich and we do no have a link beween he magniudes of he coefficiens in (7) and (8). Rudebusch (2010) and Yellen (2012) argue ha he appropriae oupu gap coefficien in he Taylor rule for he U.S. should be double he coefficien in Taylor s original rule. While here has been an acive policy debae on he normaive quesion of wheher prescribed Taylor rule ineres raes should be calculaed using Taylor s original specificaion or wih larger 3 Engel, Mark, and Wes (2008) use single equaion and panel models wih coefficiens of 2.0 on inflaion, 0.5 on he oupu gap, and 0.1 on he real exchange rae. Engel, Mark, and Wes (2015) use panel models ha incorporae exchange rae facors wih Taylor s original coefficiens. 8

10 coefficiens, i is clear ha he laer provide a beer fi for Fed policy in he 2000s. In order o differeniae his rule from he original Taylor rule, we call i he modified Taylor rule and incorporae he higher oupu gap coefficien in he forecasing equaion, e (1.5( ) 1.0( y y )) 1 i (9) Since he same argumen has no ypically been made for he oher counries in our sample, we also esimae a hybrid Taylor rule differenials model wih a coefficien of 1.0 on he U.S. oupu gap and 0.5 on foreign oupu gap, e (1.5( ) 1.0y 0.5y ) 1 i (10) The forecasing equaions for he Taylor rule fundamenals and differenials models include a consan erm. The absence of a consan would require he equilibrium real ineres raes, arge inflaion, and he coefficiens on inflaion o be idenical in he wo counries. Since here is no empirical evidence o suppor his for he counries in our sample, we include a consan in all Taylor rule specificaions Ineres Rae Differenials Model We posulae he following exchange rae forecasing equaion, e 1 i where e is he exchange rae, i is he domesic ineres rae, ( i i ) (11) i is he foreign ineres rae, and an increase in he domesic ineres rae relaive o he foreign ineres rae produces forecased exchange rae appreciaion. This is no consisen wih uncovered ineres rae pariy (UIRP), where i would equal one, bu i is consisen wih he carry rade lieraure and wih he empirical evidence in Chinn (2006), who shows ha, while UIRP may hold in he long-run, i clearly does no hold in periods of less han one year. This is he exchange rae forecasing equaion used by Clark and Wes (2006). While hey did no specify a sign for, heir successful resuls were consisen wih a negaive coefficien. 2.4 Moneary and Purchasing Power Pariy Fundamenals Models 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 fundamenal value. Thus, he one-period-ahead change in he i 4 Taylor rule fundamenals models wihou a consan produced much less evidence of predicabiliy han he models wih a consan in Molodsova and Papell (2009). 9

11 log exchange rae can be modeled as a funcion of is curren deviaion from is fundamenal value. e z, (12) 1 z where z f e and f is he long-run equilibrium level of he nominal exchange rae deermined by macroeconomic fundamenals. The moneary fundamenals model specifies exchange rae behavior in erms of relaive demand for and supply of money in he wo counries. Assuming purchasing power pariy, UIRP, and no raional speculaive bubbles, he fundamenal value of he exchange rae can be derived. where m and f ( m m ) k( y y ) (13) y are he logs of money supply and income in period ; aserisks denoe foreign counry variables. We consruc he moneary fundamenals wih a fixed value of he income elasiciy, k, which can equal o 0 or 1. We subsiue he moneary fundamenals (13) ino (12), and use he resulan equaion for forecasing. The Purchasing Power Pariy (PPP) fundamenals model posulaes ha he exchange rae will adjus over ime o eliminae deviaions from long-run PPP. Under PPP fundamenals, where f ( p p ) (14) p is he log of he naional price level. We subsiue he PPP fundamenals (14) ino (12), and use he resulan equaion for forecasing. 3. Forecasing and Predicabiliy When Meese and Rogoff wroe heir paper, he saisical mehodology for evaluaing wheher a smaller RMSE was significanly differen from a larger RMSE did no exis. Meese and Rogoff (1983) recognized his, repored ha he RMSEs from he models were almos all larger han he RMSEs from he random walk, and saed ha, while hey could conclude ha he forecass from he models were no superior o hose from he random walk, hey could no conclude ha he forecass from he random walk were superior o hose from he models. I was no unil more han a decade laer ha Diebold and Mariano (1995) and Wes (1996) developed 10

12 he mehodology, known joinly as he DMW saisic, o evaluae he significance of he difference beween larger and smaller RMSEs. An imporan issue wih he applicabiliy of DMW ess o ou-of-sample exchange rae forecasing is ha hey are only applicable o non-nesed models where he variables in one model are no a subse of he variables in he oher model. Since he random walk model conains no righ-hand-side variables, i is nesed in all linear models. Wha exacerbaes he problem is ha, if he null hypohesis is correc and he exchange rae is a random walk, esimaes of linear models wih (exraneous) righ-hand-side variables will have higher RMSEs han he random walk model. Since he RMSEs should be equal under he null, his produces undersized ess which will no rejec ofen enough. The magniude of he problem was documened by McCracken (2007), who showed ha using sandard normal criical values for he DMW saisic resuls in ess wih nominal size of 0.10 generally having acual size of less han Clark and Wes (2006) propose an adjusmen o he DMW saisic, called he CW saisic, which adjuss he DMW saisic o achieve correc size wih sandard normal criical values. Wih he DMW es, he null hypohesis is ha he wo models have he same RMSE, while he alernaive hypohesis is ha he RMSE of he linear model is smaller han he RMSE of he random walk model. Wih he CW es, he null hypohesis is ha he regression coefficiens in he linear model equal zero so ha he exchange rae follows a random walk, while he alernaive hypohesis is ha he regression coefficiens are differen from zero so ha he exchange rae can be described by a linear model. I is possible, herefore, o rejec he random walk null in favor of he linear model even if he RMSE is smaller for he random walk han for he linear model. Tha is why hese mehods are ess of predicabiliy, no of forecasing abiliy, as hey are no minimum RMSE ess. 5 The CW saisic has become he sandard mehod o es exchange rae models ou-of-sample, and was used by Engel, Mark, and Wes (2008, 2015) and Molodsova and Papell (2009, 2013). 4. Ou-of-Sample Exchange Rae Predicabiliy The models are esimaed using monhly daa from March 1973 hrough December 2014 for seven non-euro counries ha were considered by Molodsova and Papell (2009); Ausralia, 5 Rogoff and Savrakeva (2008) criicize he use of CW ess because hey are no minimum RMSE ess. 11

13 Canada, Denmark, Japan, Sweden, Swizerland, and he Unied Kingdom, as well as Germany. 6 Our choice of counries is dicaed by our inenion o examine exchange rae behavior for major currencies over he recen period. The exchange rae is defined as he domesic currency (U.S. dollar) price of a uni of foreign currency, so ha an increase in he exchange rae is a depreciaion of he dollar. 4.1 Daa The primary source of daa is he IMF's Inernaional Financial Saisics (IFS) daabase. We updae he daa in Molodsova and Papell (2009) unil December 2014, preserving he same variable definiions. The price level in he counry is measured by consumer price index (IFS line 64). The inflaion rae is he annual inflaion rae, measured as he 12-monh percenage difference of he CPI. We use seasonally adjused indusrial producion index as a measure of a counry s economic aciviy. We use M1 o measure he money supply for all counries, excep he U.K. for which M0 is used because M1 daa is unavailable. The oupu gap is esimaed as a percenage deviaion of acual oupu from a quadraic ime rend. In order o mimic he real-ime forecasing environmen as closely as possible when real-ime daa is unavailable, we use quasi-real-ime oupu gap esimaion, where only he daa poins up o period -1 are used o consruc he rend for a given period. Orphanides and van Norden (2002) find ha he correlaions beween real-ime and revised oupu gap esimaes are low while he correlaions beween real-ime and quasi-real-ime oupu gap esimaes are high for he U.S. Ince and Papell (2013) exend heir findings for he U.S. o 9 addiional OECD counries, 6 of which are included in our sample. These resuls sugges ha mos of he difference beween real-ime and revised oupu gap esimaes comes from using ex-pos daa o esimae he rend, no from he revisions hemselves, and reliable oupu gap esimaes can be consruced wih quasi-real-ime daa when real-ime daa is unavailable. We use he money marke rae (IFS line 60B) as a measure of he shor-erm ineres rae ha he cenral bank ses every period. The money marke rae for he U.S. is he federal funds rae (FFR). Alernaively, we replace he FFR for he U.S. wih he Wu and Xia (2014) shadow FFR afer 2009:M1. The shadow rae is a beer measure of he policy ineres rae when he FFR is consrained by a zero lower bound. The shadow raes are calculaed using a nonlinear erm 6 Some of he models are esimaed using shorer ime spans of daa because of daa unavailabiliy. The foonoes for he ables lis hese excepions. 12

14 srucure model and are consisenly negaive from July 2009 onward. 7 The exchange raes are end-of-monh nominal exchange raes from he Federal Reserve Bank of Sain Louis daabase. The exchange rae for Germany afer 1998 is replaced wih a synheic Deusche mark/dollar rae, which is calculaed, as in Engel, Mark, and Wes (2008) and Ince (2014), using he rae of depreciaion of he dollar/euro rae. 4.2 Forecass We evaluae one-monh-ahead exchange rae forecass wih Taylor rule fundamenals and Taylor rule differenials. For he purpose of comparison, we also evaluae he ou-of-sample performance of he ineres rae differenials, moneary, and PPP models. We use daa over he period March 1973 February 1983 for esimaion and reserve he remaining daa for ou-ofsample forecasing exercise. To evaluae he ou-of-sample performance of he models, we esimae hem by OLS in rolling regressions wih a 120-monh window, consruc 381 forecass, and calculae he CW saisics o ess for equal predicive abiliy beween he drifless random walk and he alernaive linear model. 4.3 Taylor Rule Fundamenals We esimae 20 specificaions of he Taylor rule fundamenals model wih a consan, wih heerogeneous or homogenous coefficiens on inflaion and he oupu gap for he U.S. and he foreign counry, and wih or wihou he real exchange rae. Table 1 repors he resuls for 1- monh-ahead forecass of exchange raes using symmeric Taylor rule fundamenals wih homogenous (Panel A) and heerogeneous coefficiens (Panel B). For each class of models, we esimae five specificaions of he Taylor rule fundamenals model. Column 1 of Table 1 repors he CW saisics for he model wih no smoohing. Columns 2 and 4 include lagged ineres raes differenial in addiion o he U.S. and foreign inflaion and oupu gaps, and Columns 3 and 5 include individual lagged ineres raes. Columns 2 and 3 use money marke raes and Columns 4 and 5 replace he FFR wih he Wu and Xia (2014) shadow FFR for he U.S. Panel A of Table 1 repors he resuls for symmeric Taylor rule fundamenals model wih homogenous coefficiens. The model significanly ouperforms he random walk for 4 ou of 8 counries wih no smoohing (Canada and Germany a he 5 percen significance level and Japan and Swizerland a he 10 percen level), for he same 4 counries when he lagged money marke rae differenial is included (Canada and Japan a he 1 percen, Swizerland a he 5 percen, and 7 Wu and Xia (2014) shadow rae can be accessed a hps:// 13

15 Germany a he 10 percen level), and when he lagged shadow federal funds rae is used in differenial form (Canada a he 1 percen and Germany, Japan and Swizerland a he 5 percen level). The evidence of predicabiliy is sronger wih individual lagged ineres raes. The models wih Taylor rule fundamenals ouperform he random walk for 5 counries wih individual money marke raes (Canada and Japan a he 1 percen level and Ausralia, Germany and Swizerland a he 5 percen level) and for 6 counries wih shadow federal funds rae for he U.S. (Canada, Germany, and Japan a he 1 percen, Ausralia a he 5 percen, and Swizerland and he U.K a he 10 percen level). Overall, he model ouperforms he random walk for 6 ou of 8 counries wih a leas one specificaion. Panel B of Table 1 repors he resuls for symmeric Taylor rule fundamenals model wih heerogeneous coefficiens. The model wih no smoohing significanly ouperforms he random walk for 7 ou of 8 counries (Ausralia, Canada, Sweden, and he U.K. a he 1 percen, Swizerland and Germany a he 5 percen, and Denmark a he 10 percen significance level). When smoohing is inroduced using ineres rae differenial, he model significanly ouperforms he random walk 7 counries wih money marke rae (Ausralia, Canada, Sweden, and he U.K. a he 1 percen, Japan and Swizerland a he 5 percen, and Germany a he 10 percen level) and for all 8 counries wih shadow federal funds rae for he U.S. (Ausralia, Canada, Sweden, and he U.K. a he 1 percen, Germany a he 5 percen, and Denmark, Japan, and Swizerland a he 10 percen level). The model wih individual lagged ineres raes significanly ouperforms he random walk for all 8 counries wih money marke rae (Ausralia, Canada, Japan, Sweden, and he U.K. a he 1 percen, Germany and Swizerland a he 5 percen, and Denmark a he 10 percen level) and for 7 counries wih shadow federal funds rae (Ausralia, Canada, Sweden, Japan, and he U.K. a he 1 percen and Germany and Swizerland a he 5 percen level). Overall, he symmeric Taylor rule fundamenals model wih heerogeneous coefficiens ouperforms he random walk for all 8 counries wih a leas hree specificaions. This model was also found o be he bes-performing model in Molodsova and Papell (2009), where significan evidence of exchange rae predicabiliy was found for 9 ou of 12 counries. Table 2 shows he resuls for asymmeric Taylor rule fundamenals models ha incorporae he real exchange rae. Compared o he resuls in Table 1, he evidence of predicabiliy is weaker for he asymmeric han for he symmeric Taylor rule fundamenals 14

16 models. The asymmeric model wih homogenous coefficiens significanly ouperforms he random walk for 3 counries wih no smoohing (Canada a he 5 percen and Germany and Japan a he 10 percen level) and for 2 counries when eiher he lagged money marke rae or he shadow federal funds rae differenial is included (Canada and Japan a he 1 percen level). The evidence of predicabiliy is slighly sronger wih individual lagged ineres raes. The asymmeric model wih Taylor rule fundamenals ouperforms he random walk wihou drif for 5 counries wih individual lagged money marke raes (Ausralia, Canada and Japan a he 1 percen, Sweden a he 5 percen, and Germany a he 10 percen level) and wih individual lagged shadow federal funds raes (Ausralia and Canada a he 1 percen and Germany, Japan, and Sweden a he 5 percen level). Overall, he model ouperforms he random walk wihou drif for 5 ou of 8 counries wih a leas wo specificaions. The evidence of predicabiliy is again sronger wih heerogeneous coefficiens. Panel B of Table 2 repors he resuls for asymmeric Taylor rule fundamenals model wih heerogeneous coefficiens. The model wih no smoohing significanly ouperforms he random walk for 6 counries (Ausralia, Canada, Sweden, and he U.K. a he 1 percen and Swizerland and Germany a he 10 percen level). When smoohing is inroduced using he lagged ineres rae differenial, he model significanly ouperforms he random walk for 5 counries wih he money marke rae (Ausralia, Canada, and he U.K. a he 1 percen, Japan a he 5 percen, and Swizerland a he 10 percen level) and for 4 ou of 8 counries wih shadow federal funds rae for he U.S. (Ausralia, Canada, and he U.K. a he 1 percen, and Japan a he 10 percen level). The model wih individual lagged ineres raes significanly ouperforms he random walk for he same 6 counries wih money marke rae and wih shadow federal funds rae (Ausralia, Canada, and he U.K. a he 1 percen, Japan and Sweden a he 5 percen, and Swizerland a he 10 percen level). Overall, he model ouperforms he random walk for 7 ou of 8 counries wih a leas one specificaion. 4.4 Taylor Rule Differenials We esimae 15 specificaions for hree Taylor rule differenials models described in Secion 2, he original Taylor (1993) rule, he modified Taylor rule wih a higher oupu gap coefficien, and a hybrid Taylor rule wih a higher oupu gap coefficien for he U.S. bu no for he foreign counry, for each of he five specificaions described above. Table 3 depics he resuls for 1-monh-ahead forecass of exchange raes using symmeric Taylor rule differenials 15

17 models. 8 The bes-performing model is he hybrid Taylor rule model (Panel C), for which he random walk null is rejeced for 5 ou of 8 counries wih a leas one specificaion. For he hybrid Taylor rule differenials model wih no smoohing, he no predicabiliy null can be rejeced for 4 counries (Ausralia, Canada, Germany, and Japan a he 10 percen significance level). When smoohing is inroduced, he evidence of predicabiliy is somewha sronger, especially when he shadow federal funds rae is used. The hybrid Taylor rule differenials model significanly ouperforms he random walk 4 counries wih he lagged money marke rae differenial and wih he shadow federal funds rae differenial (Japan a he 1 percen, Canada a he 5 percen, and Ausralia and Swizerland a he 10 percen level). The model ouperforms he random walk for 4 counries wih individual lagged money marke raes (Japan a he 1 percen, Ausralia and Canada a he 5 percen, and Swizerland a he 10 percen level) and for 5 counries wih individual lagged shadow federal funds raes (Japan a he 1 percen, Ausralia and Canada a he 5 percen, and Germany and Swizerland a he 10 percen level). The oher wo Taylor rule models significanly ouperform he random walk for 4 ou of 8 counries wih a leas 2 specificaions. For he original Taylor rule differenials model wih no smoohing, he no predicabiliy null can be rejeced for 3 counries (Canada a he 5 percen and Japan and Sweden a he 10 percen level). The original Taylor rule differenials model significanly ouperforms he random walk for 2 counries wih a lagged ineres rae differenial (Ausralia and Japan a he 1 percen level). When smoohing is inroduced using individual lagged ineres raes, he evidence of predicabiliy is slighly sronger. The original Taylor rule differenials model ouperforms he random walk for 4 counries wih individual lagged money marke raes (Canada and Japan a he 1 percen, Ausralia a he 5 percen, and Sweden a he 10 percen significance level) and wih individual lagged shadow federal funds raes (Canada and Japan a he 1 percen and Ausralia and Sweden a he 5 percen level). For he modified Taylor rule differenials model wih no smoohing, he no predicabiliy null can be rejeced for 2 of he 8 counries (Japan a he 5 percen and Canada a he 10 percen level). The modified Taylor rule differenials model significanly ouperforms he random walk for 3 ou of 8 counries wih a lagged money marke rae differenial (Canada and Japan a he 1 percen and Swizerland a he 10 percen level) and wih a lagged shadow federal funds rae for 8 The resuls for asymmeric specificaions ha inroduce he real exchange rae wih a coefficien of 0.1, as in Engel, Mark, and Wes (2008), produce similar resuls. 16

18 he U.S. (Japan a he 1 percen, Canada a he 5 percen, and Swizerland a he 10 percen level). When smoohing is inroduced using individual ineres raes, he evidence of predicabiliy is slighly sronger. The original Taylor rule differenials model ouperforms he random walk for 4 counries (Japan a he 1 percen, Ausralia and Canada a he 5 percen, and Swizerland a he 10 percen level). 4.5 Ineres Rae, PPP, and Moneary Fundamenals Table 4 conains he resuls for one-monh-ahead forecass of exchange raes using he ineres rae, PPP, and moneary models described in Secion 2. The evidence of predicabiliy is much weaker wih he convenional models. The sronges evidence of predicabiliy is found wih ineres rae models, where he model ouperforms he random walk for 2 ou of 8 counries when he money marke rae is used (Japan a he 1 percen and Swizerland a he 5 percen level) and for 3 counries when shadow federal funds rae is used (Japan and Swizerland a he 5 percen level and Canada a he 10 percen level). The evidence of predicabiliy is even weaker for he moneary models. Wih he coefficien on relaive oupu k equal o 0, he no predicabiliy null can be rejeced for 2 ou of 8 counries (Swizerland a he 5 percen, and Japan a he 10 percen level). Wih k=1, he evidence of predicabiliy is found only for Swizerland a he 5 percen significance level. No evidence of one-monh-ahead predicabiliy is found wih he PPP model. 5. Taylor Rules and Taylor Rule Predicabiliy We have presened evidence ha he Taylor rule fundamenals model of Molodsova and Papell (2009) coninues o provide evidence of ou-of-sample exchange rae predicabiliy when he daa is exended o include he financial crisis, he Grea Recession, and he zero lower bound on he federal funds rae. The Taylor rule fundamenals model provides more evidence of predicabiliy han he Taylor rule differenials model of Engel and Wes (2008), and much more evidence of predicabiliy han he radiional ineres rae, Purchasing Power Pariy, and moneary models. Since he Taylor rule fundamenals model uses daa on he variables ha ener Taylor rules, inflaion raes, oupu gaps, and (depending on he specificaion) he real exchange rae and/or lagged ineres raes, bu does no use coefficiens from eiher posulaed or esimaed Taylor rules, i leaves open he quesion of wheher finding evidence of ou-of-sample exchange 17

19 rae predicabiliy wih he Taylor rule fundamenals model is relaed o cenral banks following he Taylor rule. Wha does i mean for a cenral bank o follow a Taylor rule? Nikolsko-Rzhevskyy, Papell, and Prodan (2014) esimae Bai and Perron (1998) ess for muliple srucural breaks on Taylor rule deviaions, he absolue value of he difference beween he federal funds rae and he rae prescribed by he original Taylor rule, for he U.S. using real-ime daa from 1965:4 o 2013:4, and idenify periods of high and low deviaions. The oupu gap is he percenage deviaion of GDP/GNP from a quadraic rend, and inflaion is he percenage change in he GDP/GNP deflaor. 9 The Federal funds rae is used for he policy rae unil 2008:Q4 and he shadow Federal funds rae of Wu and Xia (2014) hereafer. The ess idenify significan breaks in 1974:Q3, 1985:Q1, and 2000:Q4, producing low deviaions eras from 1965:Q4 1974:Q3 and 1985:Q2 2000:Q4 and high deviaions eras from 1974:Q4 1985:Q1 and 2001:Q1 2013:Q4. 10 Table 5 repors CW saisics for he Taylor rule fundamenals model when he daa is divided ino periods based on low and high original Taylor rule deviaions for he U.S. We repor saisics for he symmeric model wih heerogeneous coefficiens, which provided he sronges full-sample evidence of predicabiliy. Because our firs forecas is in March 1983 and here is a break in 1985:Q1, we call 1985:M4 2000:M12 a low deviaions period and 2001:M1 2014:M12 a high deviaions period. The resuls do no suppor he hypohesis ha ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals is more successful during periods wih closer adherence o he original Taylor rule. Swizerland is he only counry for which he evidence of exchange rae predicabiliy is sronger during periods of low Taylor rule deviaions. For Ausralia, Denmark, Germany, and Sweden, he evidence of predicabiliy is sronger during he period of high Taylor rule deviaions and, for Canada, Japan, and he Unied Kingdom, he evidence of predicabiliy is abou he same in he high and low deviaions periods. 11 Nikolsko-Rzhevskyy, Papell, and Prodan (2014) also calculae deviaions from a modified Taylor rule wih an oupu gap coefficien of 1.0 insead of 0.5. The ess idenify 9 The daa is from he Real-Time Daa Se for Macroeconomiss, originaed by Croushore and Sark (2001). Nikolsko-Rzhevskyy, Papell, and Prodan (2014) show ha real-ime quadraic derending corresponds much more closely o U.S. recessions and expansions han real-ime linear or Hodrick-Presco derending. 10 The breaks and eras are he same if he daa is exended o 2014:Q4. 11 Since we do no know of a es o formally compare he CW saisic across he same model and differen ime periods, he saemens of sronger and weaker evidence are based on visual examinaion of he resuls. 18

20 significan breaks in 1977:Q4, 1984:Q4, and 1999:Q1, and 2006:Q3, producing low deviaions eras from 1965:Q4 1977:Q4, 1985:Q1 1999:Q1, and 2006:Q4 2013:Q4 and high deviaions eras from 1978:Q1 1984:Q4 and 1999:Q2 2006:Q3. Wihin he period covered by our forecass, his produces low deviaions eras from 1985:M1 1999:M3 and 2006:M :M12 and a high deviaions era from 1999:M4 2006:M9. Table 6 repors CW saisics for he Taylor rule fundamenals model when he daa is divided ino periods based on low and high modified Taylor rule deviaions for he U.S. Table 6A repors he CW saisics for he hree sub-periods and Table 6B combines he wo low deviaions periods ino one era. The resuls srongly suppor he hypohesis ha ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals is more successful during periods wih closer adherence o he modified Taylor rule. The resuls are cleares in Table 6B. For Ausralia, Canada, Germany, Japan, Sweden, Swizerland, and he Unied Kingdom, he evidence of predicabiliy is sronger during he periods of low Taylor rule deviaions han during he period of high Taylor rule deviaions for virually every model. 12 For Denmark, here is weak (10 percen) evidence of predicabiliy during he 2006:M :M12 low deviaions period across all models ha disappears when i is combined wih he 1985:M1 1999:M3 low deviaions period. 13 The resul ha evidence of ou-of-sample exchange rae predicabiliy is sronger during periods of low deviaions han during periods of high deviaions for he modified, bu no he original, Taylor rule is based on deviaions calculaed for he U.S. Teryoshin (2014), using realime daa from Fernandez, Koenig, and Nikolsko-Rzhevskyy (2011), uses Bai and Perron (1998) o idenify periods of high and low deviaions from boh he original and modified Taylor rule for 10 addiional counries, including six counries, Ausralia, Canada, Japan, Sweden, Swizerland, and he Unied Kingdom, sudied in his paper. We use his counry-by-counry resuls o invesigae wheher he resuls obained using U.S. daa exend o oher counries daa. The resuls for he original Taylor rule are repored in Table 7. The daes of he high and low deviaions eras are differen counry-by-counry and, unlike for he U.S., here are also some inermediae eras. The resuls do no suppor he hypohesis ha ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals is more successful during periods wih closer 12 The only excepion among he 35 cases (7 counries imes 5 models) is for he no smoohing model for Japan. 13 The excepion o his saemen is ha, for he specificaion wih smoohing, lagged ineres rae differenials, and he shadow federal funds rae, he rejecion is a he 5 percen level for he 2006:M :M12 period. 19

21 adherence o he original Taylor rule. Japan is he only counry for which he evidence of exchange rae predicabiliy is sronger during periods of low and inermediae Taylor rule deviaions and Ausralia is he only counry for which he evidence of predicabiliy is sronger during periods of high deviaions. The evidence is mixed for Canada, Sweden, and he Unied Kingdom, and here are no significan srucural breaks, and herefore no disinc eras, for Swizerland. The resuls for he modified Taylor rule are repored in Table 8. The resuls do no suppor he hypohesis ha ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals is more successful during periods wih closer adherence o he modified Taylor rule. Swizerland is he only counry for which he evidence of exchange rae predicabiliy is sronger during periods of low and inermediae Taylor rule deviaions and Japan is he only counry for which he evidence of predicabiliy is sronger during periods of high deviaions. The evidence is mixed for Ausralia, Canada, Sweden, and he Unied Kingdom. Moneary policy evaluaion wih Taylor rules is ypically conduced in erms of he Taylor principle ha he nominal ineres rae is raised more han poin-for-poin when inflaion increases. This is boh necessary and sufficien for saionariy of inflaion in a exbook IS curve, Phillips curve, and Taylor rule model and necessary and almos sufficien for deerminacy of inflaion in he forward-looking IS curve, New Keynesian Phillips curve, and Taylor rule model of Woodford (2003). Nikolsko-Rzhevskyy, Papell, and Prodan (2015) idenify low, posiive, and negaive deviaions eras from he original and modified Taylor rule by conducing srucural change ess on he difference beween he acual and prescribed federal funds rae. They use he difference, raher han he absolue value of he difference, beween he raes in order o esimae Taylor rules over he sub-periods defined by he ess. For he period covered by our ou-ofsample exchange rae forecasing, he Taylor principle holds during 1983:M3 1999:M3 because he coefficien on inflaion is significanly greaer han one, he evidence is mixed beween 1999:M4 2007:M6 because, while he coefficien on inflaion is greaer han one, i is no significanly greaer han one, and he Taylor principle does no hold for 2007:M7 2014:M12 because he coefficien on inflaion is less han one. Table 9 repors CW saisics for he Taylor rule fundamenals model when he daa is divided ino periods based on adherence o he Taylor principle for he U.S. Ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals is no more successful during periods 20

22 wih closer adherence o he Taylor principle. Swizerland is he only counry for which he evidence of exchange rae predicabiliy is sronger during periods of sronger adherence o he Taylor principle. For Ausralia, Denmark, Germany, and Sweden, he evidence of predicabiliy is sronger during he periods of less adherence and, for Canada, Japan, and he Unied Kingdom, he evidence of predicabiliy is abou he same in he high and low adherence periods. This paern exacly maches he resuls for he original Taylor rule because he period where he Taylor principle holds closely overlaps he low deviaions era and he periods where he evidence is mixed and where he Taylor principle does no hold closely overlap he high deviaions era. The paern is very differen from he resuls wih he modified Taylor rule because he overlap is much lower. 6. Conclusion The Taylor rule fundamenals model of Molodsova and Papell (2009) was moivaed by he shif in policy evaluaion over he pas weny-five years from money supplies o ineres raes as he insrumen of moneary policy. Using daa from he sar of he pos-breon Woods floaing exchange rae era in 1973 hrough he end of 2014, he model provides evidence of ouof-sample exchange rae predicabiliy for all of he eigh counries in our sample. The Taylor rule fundamenals model provides sronger evidence of predicabiliy han he Taylor rule differenials model of Engel, Mark, and Wes (2008) and much sronger evidence of predicabiliy han he radiional ineres rae, Purchasing Power Pariy, and moneary models. The mos successful specificaions allow for heerogeneous coefficiens on domesic and foreign inflaion and oupu gaps, bu do no include he real exchange rae. These were also he mos successful specificaions in Molodsova and Papell (2009) using daa from 1983 hrough mid-2006, and demonsrae ha ou-of-sample exchange rae predicabiliy wih Taylor rule fundamenals has survived he financial crisis, he Grea Recession, and he zero lower bound on he federal funds rae Because he Taylor rule fundamenals model uses he variables included in he Taylor rule, U.S. and foreign inflaion, oupu gaps, and (depending on he specificaion) lagged ineres raes and/or he real exchange rae, bu does no impose eiher posulaed or esimaed coefficiens on he variables, finding evidence of ou-of-sample exchange rae predicabiliy does no, by iself, provide a link beween he Taylor rule and he findings of predicabiliy. In order o 21