Exchange Rate Equations Based on Interest Rate Rules: In-Sample and Out-of-Sample Performance

Transcription

1 Exchange Rae Equaions Based on Ineres Rae Rules: In-Sample and Ou-of-Sample Performance Mahir Binici and Yin-Wong Cheung * Cenral Bank of Turkey and Universiy of California, Sana Cruz Absrac Using exchange rae daa on five currencies vis-à-vis he US dollar, his paper examines he insample and ou-of-sample performance of exchange rae equaions derived from alernaive empirical and opimal ineres rae rules. These rules could have eiher homogeneous or heerogeneous response coefficiens. Our exercise shows ha hese exchange rae equaions do no offer good in-sample explanaory power consisenly across currencies and over ime. The relaive forecasing performance of hese exchange rae equaions end o vary across currencies and over ime and bears limied relaionship wih he relaive in-sample performance. When he forecas performance is compared wih a random walk model, hese exchange rae equaions offer no beer performance under he usual MSFE crierion bu are beer when he abiliy of predicing he direcion of change is considered. Key Words: Taylor Rule, Exchange Rae Deerminaion, Forecas Comparison, Mean Squared Forecas Error, Direcion of Change JEL Classificaion: F31, E52, C52 * We hank an anonymous referee and he paricipans of ICEF-2011, TOBB-ETU Economics Deparmen Seminar and he CESifo Area Conference on Macro, Money & Inernaional Finance for heir commens and suggesions. Cheung acknowledges he financial suppor of faculy research funds of he Universiy of California, Sana Cruz. The views expressed here are hose of he auhor(s) and do no reflec hose of he Cenral Bank of Turkey. Conac Informaion: Mahir Binici (corresponding auhor), Cenral Bank of Turkey, Isiklal Cad. No:10. Ulus- Ankara, Turkey. mahir.binici@cmb.gov.r. Yin-Wong Cheung, Economics Deparmen, Universiy of California, Sana Cruz, CA95064, USA. cheung@ucsc.edu.

2 1. Inroducion The challenge of modeling exchange raes is well aesed by he difficuly of overurning he well-known Meese and Rogoff (1983a) resul. These auhors showed ha he srucural exchange rae models do no ouperform a naïve random walk exchange rae specificaion. Specifically, Meese and Rogoff find ha mean squared forecas errors generaed from srucural exchange rae models are no consisenly and significanly smaller han hose from a random walk model. I is asonishing, and a imes, frusraing ha he Meese and Rogoff resul has survived largely inac he numerous aemps using differen specificaions and esimaion echniques o bea a random walk. In general, i is hard o find a model ha could ouperform a random walk exchange rae specificaion on a consisen basis. 1 Recenly, Engel and Wes (2005, 2006) explore he implicaions of moneary policy endogeneiy for exchange rae deerminaion. By endogenizing moneary policy and explicily inroducing he ineres rae rule, hese auhors advanced a new and promising approach o model exchange rae behavior. In he conex of open economy DSGE model, Benigno (2004), Groen and Masumoo (2004), Gali (2008) illusrae he effec of moneary policy shocks on exchange rae dynamics. Indeed he repored performance of exchange rae equaions based on ineres rae rules is quie posiive. For insance, Chinn (2008), Mark (2009), Clarida and Waldman (2008), Molodsova and Papell (2009), Molodsova e al. (2008), and Wang and Wu (2008) consider various ineres rae rule based exchange rae equaions, and presen favorable findings on heir empirical performance ha include ou-forecasing a random walk. 2 Mos of hese sudies are based on an empirical or operaional ineres rae rule from exan empirical moneary policy evaluaion sudies. A imes, he ineres rae rules are allowed o have differen response coefficiens or include a real exchange rae erm. A quick review shows ha empirical ineres rae rules come in differen forms and varians for differen counries a differen hisorical ime periods. When he resuls based on a paricular rule are encouraging, hey do no necessarily shed ligh on he general relevance of he ineres rae rule approach. 1 See, for example, Cheung e al. (2005a, b). 2 Rogoff and Savrakeva (2008) express some reservaions on he repored superior forecasing resuls. 2

3 A relaed issue is ha he chosen empirical rule may be relevan for empirical policy analysis for a seleced daa sample, bu ha rule is no necessarily he opimal one in heory. Under a heoreical consruc, an opimal ineres rule is ypically expeced o mee he equilibrium deerminacy and sabiliy condiions. 3 While policymakers operae in he real world ha is far more complicaed han a ypical heoreical model, an opimal ineres rae rule is always a useful benchmark for discussing policy rules. In addiion o inellecual curiosiy, sudies based on opimal ineres rae policy rules should complemen hose based on empirical and operaional rules. In he curren exercise, we assess he performance of exchange rae equaions derived from boh empirical and heoreically opimal ineres rae policy rules. As observed by he pioneering work of Meese and Rogoff (1983a) and echoed by Cheung e al. (2005b), here is lile correspondence beween an exchange rae model s in-sample performance and is forecasing abiliy. This leads o he quesion, should he assessmen be based on in-sample or ou-of-sample properies? One moivaion for using ou-of-sample forecas performance is o minimize he effec of daa mining. I is conceived ha researchers end o repor a model ha offers he bes explanaory power. Thus, he repored explanaory power may be spurious and reflec only researchers conscienious or sub-conscienious search over, say, alernaive specificaions and sample periods. On he oher hand, he ou-of-sample forecas exercise subjecs a model o he es of real daa, and hus, alleviaes he daa mining effec. However, daa mining is no limied o in-sample analyses. Usually, only models performing well in ou-of-sample analyses are published. Daa mining comes in because, a leas in principle, researchers are free o explore differen ways o generae forecass in various sample periods and for differen currencies. Indeed, i is no easy o replicae good forecasing performance repored in he lieraure using differen forecasing horizons, forecasing periods, and currencies. Insead of choosing one of he wo, he curren sudy repors boh in-sample and ou-ofsample resuls. The in-sample measures include informaion crieria and R-squared saisics (Inoue and Kilian, 2004, 2006; Clemens and Hendry, 2001, 2005). For ou-of-sample performance, we follow he Meese and Rogoff radiion and use mean squared forecas errors (MSFEs). In addiion, we also consider he direcion of change (DOC) saisics. 3 See, for example, Evans and Honkapohja (2003) on learnabiliy. 3

4 In he nex secion, we briefly discuss he empirical and heoreical ineres rae rules used o derive exchange rae equaions in our exercise. Specifically, we consider hree ypes of empirical ineres rae rules, namely (a) he conemporaneous rule, (b) he backward-looking rule, and (c) he forward-looking rule. The opimal ineres rules are derived from a canonical new Keynesian framework, which is he workhorse of moneary policy analysis. The specificaion of an opimal rule depends on he model srucure, he policy objecive, and he assumed adjusmen mechanism. In his exercise, we consider opimal ineres rae rules consruced under: (a) learning, (b) ineres rae ineria, (c) inflaion ineria, (d) inflaion argeing, and (e) he consan money growh. We examine he in-sample and ou-of-sample performance of exchange rae equaions derived from alernaive empirical and opimal ineres rae rules using he US dollar exchange rae of he Briish pound, Canadian dollar, Japanese yen, German mark, and Swiss franc. The sample period is 1974: :12, wih he excepion of he German mark of 1974: :12. Our esimaion resuls indicae ha hese exchange rae equaions do no offer subsanial explanaory power. The ou-of-sample resuls are a mix. These exchange rae equaions do no ou-forecas a random walk specificaion based on he MSFE comparison bu hey have he abiliy o correcly predic he direcion of exchange rae movemen. In addiion, he rolling regression resuls are used o assess he relaive in-sample and ou-of-sample performance. I is found ha he relaive performance of hese exchange rae equaions varies over ime. Furher, when an exchange rae equaion displays a good in-sample performance, i does no necessarily offer a good forecas. The remainder of he paper is srucured as follows. In he nex secion we discuss he empirical and opimal ineres rae rules and he relaed exchange rae equaions. Secion 3 describes he daa, esimaion mehods, and he forecas experience. The in-sample and ou-ofsample resuls are presened in Secion 4. Secion 5 offers some concluding remarks. 2. Ineres Rae Rules and Exchange Rae Dynamics In his 1993 seminal work, John Taylor showed ha he US moneary policy could be characerized by a deerminisic rule i = r + p + b ( p - p ) + b y, (1) p y 4

5 where i is he arge shor erm ineres rae, r is he equilibrium real ineres rae, p and p are acual and arge inflaion rae, respecively, y is he oupu gap. In he 1993 specificaion, p and r are assumed o be 2 percen and he response coefficiens b y and b p are 0.5. See Taylor (1993) for a deailed discussion of he rule and he relaed issues. Since hen, he Taylor ineres rae rule has been modified in various empirical and heoreical sudies. 2.1 Empirical Ineres Rae Rules and Exchange Raes Empirical ineres rae rules come in differen forms and wih differen explanaory variables. In he subsequen analysis, we consider hree represenaive empirical rules. Readers who are familiar wih empirical ineres rae rules and are only ineresed in exchange rae equaions implied by hese rules could refer o Table 1 and skip his subsecion Conemporaneous Rule One common feaure of differen varians of he ineres rae rule is ineres rae smoohing. Using he US daa Sack (1998) repors significan evidence on ineres rae smoohing, which is esimaed by he coefficien on he lagged ineres rae. Sack and Wieland (2000) review empirical sudies and offer various facors ha accoun for ineres rae smoohing. Levin e al. (1999), Roemberg and Woodford (1999), and Giannoni and Woodford (2002), on he oher hand, discuss he ineres-rae smoohing from he heoreical perspecive. 4 To incorporae ineres rae smoohing, we follow Clarida e al. (1998) and assume he acual ineres rae adjuss o he arge rae according o i = (1 - a) i + a( L) i + e, (2) -1 n-1 where a( L) = a + a L +... a L, a º a(1), and e is an exogenous random shock o he 1 2 n ineres rae, which is i.i.d. In mos policy rule esimaions, as discussed below in deail, we conclude ha he lag lengh o be n=2. Thus, wih ineres rae smoohing, he ineres rae rule akes he form i = (1 - a)( b + b p + b y ) + ai + a i + e, (3) Rudebusch (2002, 2006) argues ha he significan lagged ineres rae is induced by shock persisence raher han moneary policy ineria. Consolo and Favero (2009) also find ha he esimaed degree of ineres rae smoohing is significanly lower han he common value in he empirical lieraure once conrolled for he weak insrumens problem in policy rule esimaions. 5

6 where a = a 1 + a 2, b 0 = r - bp p, and b1 = 1 + bp, and b = b 2 y. Equaion (3) gives a conemporaneous rule in which he moneary policy insrumen, i.e. shor-erm ineres rae, reacs o curren values of inflaion and oupu gap in addiion o he ineres rae smoohing erms. Cenral banks have access o inflaion and oupu daa before he public hus, hey could adjus ineres raes according o he conemporaneous ineres rae rule (3). Benhabib e al. (2003), for insance, argue ha a policy ineres rae rule ha responds o pas ineres raes and curren inflaion could ensure global sabiliy provided ha he ineres rae smoohing erms have coefficiens ha are greaer han uniy in oal. Assuming ha boh he home and foreign counries adop he same ineres rae rule wih he same coefficiens, he ineres rae differenial is * * * i - i = (1 - a) éb0 + b1( p - p ) + b2( y -y ) ù êë úû * * 1( i -1 i -1) 2( i -2 i -2) + a - + a - + e, (4) where he foreign variables are indicaed by *. The exchange rae equaion under uncovered ineres pariy i - i = ED s 1 is * + E * * D s + 1 = (1 - a) éb0 + b1( p - p ) + b2( y - y ) ù êë úû * * 1( i -1 i -1) 2( i -2 i -2) + a - + a - + e, (5) where s is he log nominal exchange rae expressed as unis of domesic currency per one uni of foreign currency. Equaion (5) gives he exchange rae equaion under he homogenous coefficien assumpion. We use he noaion E1 o denoe he empirical specificaion based on he conemporaneous rule (3). If we relax he assumpion and allow he domesic and foreign auhoriies o adjus he ineres rae differenly in response o inflaion and oupu, hen he exchange rae will respond differenly o domesic and foreign shocks. Under he heerogeneous coefficien assumpion, he response coefficiens of he ineres rae rules could be differen and he resuling exchange rae equaion is E D s = w + (1 - a)( b p + b y ) + ai + a i f f * f * f * f * (1 - a )( b p + b y ) - a i - a i + e, (6) 6

7 where w is he composie inercep erm and he coefficiens of he foreign counry ineres rae rule are indicaed by he superscrip f. By incorporaing ineres rae seing behavior, equaions (5) and (6) highligh he role of cenral bank preferences in deermining exchange raes. One sriking difference beween exchange rae equaions derived from he ineres rule approach and he sandard moneary framework is he absence of money variables in (5) and (6). By endogenizing moneary policy, he ineres rule approach does no direcly link exchange rae dynamics o money. Under he homogenous coefficien assumpion, boh domesic and foreign moneary auhoriies se heir ineres raes he same way. The resuling exchange rae responds symmerically o variaions in domesic and foreign deerminans. On he oher hand, he srong assumpion ha he domesic and foreign moneary auhoriies have idenical reacions o inflaion, oupu gap, and lagged ineres raes is relaxed under he heerogeneous coefficien seing. In his case, exchange raes could respond differenly o domesic and foreign variables Backward-Looking Rule The ime srucure buil ino he ineres rae rule is subjec of exensive invesigaion. One concern is relaed o inflaion indeerminacy and sunspo muliple equilibria. To ensure deerminacy and enhance sabiliy, Carlsrom and Fuers (2000), for example, advocae he adopion of a backward-looking ineres rae rule which responds aggressively o pas inflaion raes. The inuiion is ha linking ineres raes o fuure inflaion forecass could lead o aggregae flucuaions riggered by self-fulfilling expecaions. Benhabib e al. (2003) and Eusepi (2005, 2007) presen he sabilizing propery of a backward-looking ineres rae rule, which could be specified as i = (1 - a)( b + b p + b y ) + ai + a i + e. (7) The exchange rae equaions under he homogenous and heerogeneous coefficien assumpions, respecively, are given by * * ED s+ 1 = (1 - a) éb0 + b1( p-1 - p-1) + b2( y-1 -y-1) ù ë û and * * 1( i-1 i-1) 2( i-2 i-2) + a - + a - + e, (8) E D s = w + (1 - a)( b p + b y ) + ai + a i

8 f f * f * f * f * (1 - a )( b p + b y ) -a i - a i + e. (9) The empirical specificaion based on he backward-looking rule will be labeled E Forward-Looking Rule Insead of a backward-looking rule, Bernanke and Woodford (1997), Clarida e al. (1998), Baini and Haldane (1999) argue ha cenral banks should respond aggressively o expeced inflaion o avoid real indeerminacy. Counries adoping he inflaion argeing policy, including Canada, New Zealand, and he UK, are perceived o conemplae heir ineres rae policies based on inflaion forecass. The empirical relevancy of forward-looking rules is presened in Clarida e al. (1998). In he subsequen analysis, we consider he following empirical forward-looking ineres rae rule i = (1 - a)( b + b Ep + b y ) + ai + a i +, (10) and label he relaed exchange rae specificaions by E3. The exchange rae equaions under he homogenous and heerogeneous coefficien assumpions, respecively, are given by a é * * E D + = - b + b p + - p + + b - ù s 1 (1 ) êë 0 1( E 12 E 12) 2( y y ) úû and * * 1( i-1 i-1) 2( i-2 i-2) + a - + a - + e, (11) E D s = w + (1 - a)( b Ep + b y ) + a i + a i f f * f * f * f * (1 - a )( b Ep + b y ) -a i - a i + e. (12) For easy reference, we collec hese exchange rae equaions in Table 1. The hree ypes of ineres rae rules give hree differen exchange rae formulaions. These exchange rae equaions embody he complicaions and heoreical argumens underlying he differen ineres rae rules. The relevan quesion is wheher he cenral bank should reac o lag, curren, or fuure economic condiions. These are imporan quesions for boh academic and pracical moneary policy reasons. Apparenly, he verdic is sill ou. In his sudy, however, our focus is no on he relevance of alernaive ineres rae rules bu on comparing he performance of he implied exchange rae equaions. 8

9 The hree ypes of exchange rae dynamics may reflec marke s percepions abou he cenral bank behaviors. For insance, he exchange rae migh respond srongly o any surprises o inflaion or oupu gap if marke players perceive ha cenral banks adjus ineres raes conemporaneously. The backward looking rule migh be relevan for policy evaluaion and yield exchange rae dynamics ha is in accordance wih he adapive expecaions approach, hough is relevance is under debae. The exchange rae equaion under forward looking rule, on he oher hand, is consisen wih he forward looking asse model approach o exchange rae deerminaion. Under he sandard forward looking rule, fuure inflaion expecaions affec he conemporaneous ineres rae, and hence he exchange rae. 2.2 Opimal Rules and Exchange Rae Dynamics The lieraure on opimal rule design is voluminous. The formulaion of an opimal ineres rule depends on, among oher hings, he srucure of he economy under consideraion, he cenral bank s objecive funcion, and he assumpion abou he adjusmen mechanism. In he following, we consider a few opimal ineres rae rules derived from a canonical Keynesian macroeconomic framework, which is commonly used in moneary policy lieraure. The rules have he desirable properies of equilibrium deerminacy, sabiliy and learnabiliy (Evans and Honkapohja, 2003). Specifically, hese opimal ineres rae rules are derived under (a) learnabiliy condiion, (b) ineres rae ineria assumpion, (c) inflaion ineria assumpion, (d) sric inflaion argeing, and (e) consan money growh condiion. Furher, we consider opimal rules derived under commimen ha is, he rules ha incorporae rade-offs across all possible fuure scenarios. 5 To conserve space, he derivaions of hese opimal ineres rae rules are presened in he Appendix A1. Again, he exchange rae equaion follows from he ineres rae rule and he uncovered ineres pariy. The implied exchange rae equaions wih he homogenous and heerogeneous ineres rae rule assumpions are presened in Table 2. The response coefficiens of he opimal ineres rae rules and, hence, he coefficiens of he implied exchange rae equaions are funcions of he srucural parameers of he underlying objecive funcion and macroeconomic model. 5 Theoreically, he opimal rule under commimen should yield an equilibrium ha is superior in erms of welfare. Also, wih commimen policy, he cenral bank could enhance moneary policy credibiliy and affec he privae secor s inflaion expecaions (Clarida e al., 1999; Walsh, 2003). 9

10 To faciliae discussion, we label T1 he exchange rae equaions based on he opimal rule incorporaing learning, T2 hose based on he opimal rule incorporaing ineres rae ineria, T3 hose based on he opimal rule incorporaing inflaion ineria, T4 hose based on he opimal rule under sric inflaion argeing, and T5 hose based on he opimal rule under consan money growh. 2.3 Discussion In he previous subsecions, we inroduced exchange rae equaions implied by empirical and heoreically opimal ineres rae rules. In he moneary policy lieraure, differen sudies consider differen empirical and heoreically opimal ineres rae rules. There is no consensus on which is he mos appropriae empirical or heoreical formulaion. The empirical rule could change across counries or hisorical ime periods. The heoreically opimal rule, on he oher hand, varies wih changes in he model srucure and policy preferences. Furher, i is known ha he opimal ineres rae rule could be differen from he rules esimaed from empirical daa, or used in empirical sudies. We noe ha some sudies inroduced an exchange rae variable o he ineres rae equaion o capure he possible policy response o exchange rae variabiliy. The exchange rae variable could ake differen forms he change in he nominal, real, effecive, or real effecive exchange rae, or he deviaion from he nominal, real, effecive, or real effecive equilibrium exchange rae. Taylor (2001), for example, expresses some skepicism on such a modificaion because he original rule already allows for exchange rae reacion via responding o boh inflaion and oupu variaions. 6 Recenly, Engel (2009) shows ha, even for a policy arge ha includes currency misalignmens, he ineres rae insrumen ha responds o he CPI inflaion rae could suppor he policy. In our pilo sudy, we found ha he real exchange rae erm was usually insignifican. Thus, we did no include an exchange rae in he ineres rae rule in he previous subsecions. In sum, an ineres rule based exchange rae equaion depends on how he ineres rae rule is defined. The performance of he implied exchange rae equaion and he relevancy of he 6 Baini e al. (2003) and Leiemo and Södersröm (2005), for insance, show ha including he exchange rae in policy rule may no improve welfare. The evidence on he empirical relevance of he exchange rae variable is mixed and could be counry- and ime-period- specific. See, for example, references cied in Taylor (2001), Clarida e al. (1998) and Mark (2009). 10

11 policy rule based approach could depend on which empirical or heoreically opimal ineres rae rule is under consideraion. Wihou knowing which rule is he appropriae one, a sudy based on some represenaive ineres rae rules is deemed uninformaive. While he empirical and opimal rules share some similariies, hey have differen bases. In our comparison, we rea wo groups of implied exchange rae equaions; hose in Table 1 and hose in Table 2, separaely. 3. Daa and Mehodology In his sudy, we consider he US dollar exchange raes of he Briish pound, Canadian dollar, Japanese yen, German mark, and Swiss franc. The sample period is 1974: :12, wih he excepion of German s sample period of 1974: :12. The daa on exchange raes, money marke raes, consumer price indexes, indusrial producion indexes, and money supplies are drawn from IMF and OECD daabases. Inflaion is given by he year-over-year change in log consumer price indexes. Oupu gap is he difference beween he indusrial producion index (in logs) and he index s rend, which is obained using he Hodrick-Presco filer wih lambda equals o Appendix A2 gives addiional informaion on he daa descripion and sources. The in-sample performance is firs assessed using he 1974: :12 sample period. The sample 1984: :12 is reserved for he subsequen in-sample and ou-of-sample performance analysis. Depending on heir funcional forms, he empirical- and opimal-rule-based exchange rae equaions are esimaed by ordinary leas squares (OLS), non-linear leas squares (NLLS), and general mehods of momens (GMM) procedures. For he GMM esimaion, we follow Clarida e al. (1998) and use he lags of he ineres raes, inflaion, oupu gap, and money supply growh as insrumens. In he esimaion and forecasing exercises, he acual realizaions of regressors are used in place of he expecaions variables, he conemporaneous and fuure variables. The approach will alleviae he uncerainy abou he values of hese regressors on model performance (Meese and Rogoff, 1983a, 1983b). We consider differen merics for in-sample and ou-of-sample comparisons. For insample comparison, we use he Akaike Informaion Crierion (AIC) and he Schwarz Bayesian Crierion (SBC) in addiion o he usual adjused R-squared measure (Inoue and Kilian, 2004). To evaluae he ou-of-sample performance, we adop he rolling regression approach o 11

12 generae he MSFE and DOC saisics, and use he random walk benchmark (Meese and Rogoff, 1983a, 1983b). Specifically, for each exchange rae equaion, he firs se of parameer esimaes are obained from he iniial 1974: :12 sample period; ha is, from he firs 120 daa poins. The esimaed equaion is used o generae he one-period-ahead forecas. Then he sample is rolled forward by dropping he firs daa poin and adding an addiional one a he end of he iniial sample period. The esimaion and forecasing procedures are repeaed. The rolling procedure is repeaed unil all he ou-of-sample observaions from 1984: :12 are exhaused. In he case of German mark, he forecas sample ends a 1998:12. For each sep of he rolling regression procedure, we record boh in-sample and ou-of-sample performance measures. 4. In-Sample and Ou-of-Sample Performance Analyses This secion presens he esimaion resuls for assessing he in-sample and ou-of-sample performance. Firs we focus on resuls peraining o he exchange rae equaions based on empirical ineres rae rules. Then we discuss he resuls from he exchange rae equaions derived from heoreically opimal ineres rae rules. In each case, we compare he esimaion resuls from he iniial sample. Then we summarize he in-sample and ou-of-sample performance measures from rolling regression. The forecas performance agains he random walk benchmark is evaluaed using he MSFE and he direcion of change saisics (Diebold and Mariano, 1995; Rogoff and Savrakeva, 2008). 4.1 Exchange Rae Equaions Based on Empirical Ineres Rae Rules Homogenous Specificaion In Table 3, we repor he resuls of esimaing he exchange rae equaions based on empirical ineres rae rules ha have he same reacion coefficiens for he domesic and foreign counries. The esimaion period is 1974: :12; roughly he firs decade of he free-floaing exchange rae regime. This period parially overlaps wih he sample period in Meese and Rogoff (1983a). In general, he adjused R-squared esimaes are small and, hus, are indicaive of he limied explanaory power of hese exchange rae equaions. The resul is in accordance wih he 12

13 difficuly of esimaing exchange raes using fundamenals repored in lieraure. The inflaion and oupu gap differenial effecs revealed by he b 1 and b 2 esimaes display differen signs across currencies and exchange rae equaions. I is noed ha hese esimaes have a posiive sign in heory. Thus, he empirical esimaes do no lend srong suppor on he effecs of inflaion and oupu gap differenials on exchange rae changes. While i is encouraging o observe ha he saisically significan b 1 and b 2 esimaes are all posiive, here are only wo significan b 1 esimaes and wo significan b 2 esimaes. Ineresingly, hree of he four significan esimaes are generaed under E3, he exchange rae equaion based on he forward looking ineres rae rule given by equaion (11). One of he characerisics of he empirical ineres rae rule is ineres rae smoohing, which is capured by a 1 and a 2, he coefficiens of lagged ineres raes. These ineres rae smoohing parameers are incorporaed in he exchange rae equaions hrough an uncovered ineres pariy. The a 1 and a 2 esimaes in Table 3, however, are quie small in magniude and do no indicae subsanial ineres rae persisence. In fac, hese esimaes do no have he same sign across currencies and are saisically significan for only a handful of cases. For Swizerland, he a 1 esimae is significanly negaive under specificaions E1 and E2 he exchange rae equaions are based on he conemporaneous and backward looking ineres rae rules given by equaions (5) and (8). The oher case of a significan ineres rae smoohing is observed for Canada - a 2 esimae is significanly posiive under all hree selecion exchange rae equaions. Apparenly, he resuls do no suppor a srong and consisen ineres rae smoohing effec in hese exchange rae specificaions. In passing, we noe ha when we esimaed he empirical ineres rae rules (3), (7), and (10), he a 1 and a 2 esimaes are large and significanly posiive. The resuls are in accordance wih he usual ineres rae smoohing behavior repored in sudies esimaing policy rules. The uncovered ineres pariy ha links ineres rae rules and exchange rae equaions seems o be he culpri of differences in he a 1 and a 2 esimaes from he empirical ineres rae rules and heir implied exchange rae equaions. The use of uncovered ineres pariy is no unconroversial. The empirical relevancy of he pariy condiion is frequenly challenged by he so-called forward premium puzzle, which suggess ha he exchange rae changes are negaively, insead of 13

14 posiively, relaed o ineres rae differenials. 7 The resuls in Table 3, noneheless, offer no consisen evidence for eiher a significan posiive or negaive ineres rae effec. As noed earlier, he in-sample performance as gauged by he adjused R-squared esimaes is quie weak. I is nooriously difficul o explain exchange raes using fundamenals; especially a high frequency like monhly daa. The low adjused R-squared esimaes aes o he apparen disconnec beween exchange raes and heir fundamenals. In mos cases, parameer esimaes are saisically insignifican and could have signs differen from heir heoreical values. The rankings based on AIC are lised near he boom of Table 3. The rankings based on SBC and adjused R-squared esimaes are similar o hose repored. According o boh informaion crieria, he preferred model is eiher he one based on he backward looking rule (E2) or he forward looking (E3) rule. Table 4 presens he in-sample and ou-of-sample performance measures obained from he rolling regression scheme described in Secion 3. The in-sample AIC, SBC and R-squared esimaes and he ou-of-sample one-sep ahead squared forecas errors are summarized. Specifically, for each currency and each exchange rae equaion, he means of individual measures are given under he column labeled M, he column R gives a model s rankings based on he average values of he measures lised in he firs column, and he column F gives he frequencies a which a model is seleced as he bes one among he hree exchange rae specificaions during he rolling regression exercise. For each currency under consideraion, he average in-sample performance measures; ha is, he AIC, SBC and R-squared esimaes, end o selec he same model specificaion. In he case of he Briish pound, for example, he hree measures favor he E1 specificaion; ha is he exchange rae equaion based on he conemporaneous ineres rae rule. The seleced model specificaion, noneheless, varies across currencies. These seleced models are quie differen from hose seleced from he iniial sample of he rolling regression in Table 3. Indeed, only he Canadian dollar has he same specificaion he model based on he backward looking rule ranked number one in boh ables. The numbers repored under he column F show ha he ranking of hese models could 7 Molodsova and Papell (2009) and Chinn (2008), for example, imposed a negaive ineres rae (and, hence, a negaive inflaion) effec on heir policy rule based exchange rae equaions. 14

15 swich during he rolling regression exercise. Tha is, he relaive explanaory power of hese models is no consan during he forecas period of 1984:01 o 2008:12. In he case of Briish pound, he average AIC selecs model E1, which is ranked he bes 60% of he ime. However, in he case of Swiss franc, he average AIC selecs model E1, which is ranked he bes only 25% of he ime. In checking hrough he individual AIC esimaes, i is found ha when he model E1 is ranked number 1, i is much beer han he alernaive; when i is no seleced, i is only slighly worse. Thus, he average AIC is beer han ohers. Therefore, he relaive in-sample performance of hese models could vary quie subsanially across rolling samples and currencies. The rankings based on he one-sep ahead mean squared forecas errors (MSFEs) are no all he same as hose based on he hree in-sample performance measures. In hree cases under consideraion, he model ranked he bes by MSFEs is differen from he one ranked he firs by, say, he average AIC. Tha is he model ha gives he smalles MSFE may no have he bes average AIC value. There is a discrepancy beween he one-sep ahead forecas performance and he in-sample performance. I is of ineres o noe ha he MSFE crierion selecs model E2, which is based on he backward looking rule, for hree of he five currencies. In sum, he resuls in Tables 3 and 4 show ha none of he hree exchange rae equaions is consisenly beer han oher wo hroughou he rolling sample periods across all he seleced performance measures Heerogeneous Specificaion In Table 5, we repor esimaion resuls from heerogeneous models. Arguably, differen cenral banks could assign differen response coefficiens for heir ineres rae policies. To allow for differen reacion funcions, we esimae he exchange rae equaions based on empirical ineres rae rules wih heerogeneous coefficiens. The se of parameers ( b 1, b 2, a 1, a 2 ) is for he home counry, while he second se of parameer ( 1 f b, 2 f b, 1 f a, 2 f a ) is for he foreign counry, in his case he US. The inflaion effec is mainly found in he UK daa. The b 1 and b 1 f esimaes under he BP/$ heading are significan and wih he expeced signs. The b 2 and b 2 f esimaes ha capure he oupu effec are only significan in wo BP/$ cases. The wo significan cases, however, 15

16 display negaive oupu effec ha is opposie o he prediced posiive sign. Similar o he homogenous coefficien resuls in Table 3, he a 1, a 2, a 1 f, and a 2 f do no offer serious evidence on ineres rae smoohing behavior even hough he smoohing behavior is quie commonly found in esimaing individual ineres rae rules. Somehow he smoohing behavior peraining o he ineres rae rule dissipaes in he exchange rae equaions. According o he adjused R-squared esimaes, he BP/$ and DM/$ models wih ineres rae rules ha allow for heerogeneous coefficiens have an explanaory power beer han he corresponding models wih homogenous coefficiens. The CAN$/$ model, however, gives he opposie resuls. For he remaining wo cases, he relaive explanaory power depends on he underlying ineres rae rule. Thus, allowing he home and foreign counries o have differen response coefficiens does no always improve he in-sample performance. Among E1, E2 and, E3, he informaion crieria end o favor he model derived from he backward-looking rule E2, which is ranked firs in hree ou of five cases. Table 6 summaries he in-sample and ou-of-sample performance saisics compiled from he rolling regression exercise. Similar o he cases wih homogenous coefficiens, for each currency, he in-sample AIC, SBC and R-squared measures give same rankings for alernaive specificaions. Compared wih Table 4, he MSFE rankings in Table 6 are slighly more in line wih he in-sample measure rankings. However, here are sill wo of he five cases in which he specificaion ha gives he smalles MSFE is no he one ha yields he bes average in-sample performance. Again, hese in-sample and ou-of-sample measures do no offer srong evidence ha any one of he hree specificaions consisenly ouperform ohers. The resuls on he differen in-sample and ou-of-sample performance rankings and he spread of firs ranked specificaion are similar o hose observed in Table Forecas Evaluaion In he previous subsecions, i is observed ha he forecas abiliy of hese models could differ from heir in-sample performance and be worse han he simple random walk hypohesis. Here, we provide addiional saisical evidence on hese wo observaions. Table 7 assesses he associaion beween in-sample and ou-of sample performance. For each currency, we ranked individual exchange rae equaions based on he MSFE and AIC crieria a each round of he rolling regression exercise. Then, we calculaed he correlaion 16

17 coefficien of he wo series of MSFE and AIC ranks. The use of rank correlaions alleviaes he implicaion of he disribuions of hese performance measures on he inference. The rank correlaion esimaes in he able are small (in absolue value) and mosly saisically insignifican. The occurrence of negaive rank correlaions slighly ounumbers he posiive rank correlaions. The posiive and negaive rank correlaions are quie evenly disribued across he homogenous and heerogeneous specificaions. Among he hree cases ha he rank correlaion esimaes are significan a he 10% level, wo have a negaive sign. The evidence, hus, is weakly suggesive ha he ou-of-sample performance is disconneced wih he in-sample performance. Indeed, a specificaion ha has a good in-sample performance according o AIC ends o give a no-so-good forecasing resul. Table 8 evaluaes he ou-of-sample forecas performance using he Diebold and Mariano (1995) es saisic. The focus is on he MSFE and he abiliy o predic he direcion of change. Individual exchange rae equaions are compared again a random walk model, which is common benchmark model used in he exchange rae forecasing exercises. For MSFE, we compare he equaion s one-sep ahead forecas errors o he random walk specificaion. The null hypohesis of no difference in forecasing accuracy is rejeced in favor of he exchange rae equaion (random walk specificaion) if he Diebold and Mariano es saisic is negaive (posiive) and saisically significan. These saisics and heir saisical significance based on boosrapping disribuions are repored under he heading MSFE. The MSFE differenials do no offer any evidence ha hese exchange rae equaions yield beer one-sep ahead forecass han he random walk model. Mos of he differenials and hence he saisics are posiive; indicaing ha he forecas errors are usually more volaile han he observed changes in exchange raes. Indeed, some of hese posiive Diebold and Mariano es saisics are saisically significan based on boosrapped criical values. For some limied cases, we obained negaive bu insignifican Diebold and Mariano es saisics. Thus, while here are some cases in which he random walk ou-forecass an exchange rae equaion, here is no saisically significan evidence ha any of hese exchange rae equaions ou-forecas he random walk specificaion. In addiion o he meric based on mean squared forecas errors, we assess he forecasing abiliy using he DOC crierion. In essence, he DOC focuses on he abiliy o forecas he exchange rae s direcion of change. The DOC saisic is based on he normalized raio of he 17

18 number of forecass ha correcly predic he direcion of exchange rae movemen o he oal number of forecass. If he raio is saisically larger han ½, hen he corresponding exchange rae equaion conains useful informaion abou exchange rae movemens. We implicily assume ha a random walk specificaion assign a probabiliy of ½ for eiher an upward or a downward movemen. The DOC saisics calculaed according o Diebold and Mariano (1996) and heir significance are presened under he columns labeled DOC. While hese exchange rae equaions yield forecass ha are more volaile han he random walk specificaion, hey end o do well in predicing he direcion of movemen. The DOCs are all posiive and mos of hem are saisically significan. Tha is, more han one half of hese forecass correcly predic he exchange rae movemen. The direcion forecas abiliy is significan in he saisical sense and is no likely o be spurious. The exchange rae equaions based on heerogeneous ineres rae rules garner more significan DOC saisics han hose based on homogeneous ineres rae rules and, hus, display a sronger direcion forecas abiliy. 4.2 Opimal Rule Resuls In his subsecion, we examine he in-sample and ou-of sample performance of exchange rae equaions based on policy ineres rae rules ha are derived from a canonical macroeconomic model. As in he previous subsecion, we presen resuls from specificaions ha allow for homogeneous and heerogeneous ineres rae response coefficiens. Table 9 and Table 10 repor he in-sample esimaion resuls for exchange rae equaions derived from homogeneous and heerogeneous ineres rae rules. These resuls are quie compared o hose based on empirical ineres rae rules (Tables 3 and 5) in he sense ha he insample fi is weak as indicaed by he adjused-square esimaes. The esimaed inflaion effec, oupu effec, and smoohing behavior vary across currencies and are quie differen from heir prediced values. There is some evidence ha ha he equaion fis slighly beer when he home and foreign coefficiens were no consrained o be he same however, he improvemen is no uniform across currencies and specificaions. Furher, according o adjused-square esimaes, here specificaions are no necessarily beer han hose based on empirical ineres rae rules. In Table 9, he AIC selecs T4 he exchange rae specificaion under inflaion argeing, as he bes model for wo currencies, he Canadian dollar and Japanese yen. For hose wih heerogeneous response coefficiens in Table 10, he AIC selecs he specificaion T1 ha allows 18

19 for learning as he bes model for hree currencies; he Briish pound, Swiss franc and German mark. In oal, model T1 is chosen as he bes model in 4 ou of 10 cases in Tables 9 and 10. The in-sample and ou-of-sample performance measures for models wih homogeneous and heerogeneous response coefficiens obained from he rolling regression exercise are presened in Tables 11 and 12. Compared wih resuls in Tables 4 and 6, he average in-sample performance measures in Tables 11 and 12 do no necessarily selec he same equaion specificaion. There are only wo cases each in hese wo Tables ha he hree in-sample measures AIC, SBC and R-squared esimae seleced he same specificaion. On he rankings based on he in-sample and ou-sample performance measures, here are several cases in which he average MSFE and a leas one of he average in-sample measures selec he same specificaion. I is hard o asser hese resuls represen a srong link beween in-sample and ouof-sample performance since hese in-sample measures could selec differen specificaions. In Table 13, he rank correlaion esimaes show ha, compared wih he resuls in Table 7, he AIC and MSFE performance measures display a sronger degree of associaion. There are 10 significan rank correlaion esimaes. However, only hree of hem are posiive and he oher seven are negaive. Thus, he significance does no mean ha a model s in-sample performance is necessarily aligned wih is ou-of-sample performance. The forecasing performance of hese exchange rae equaions relaive o he random walk specificaion presened in Table 14 is qualiaively similar o he resuls in Table 8 for hose derived from empirical ineres rae rules. Specifically, he MSFE crierion suggess ha hese exchange rae equaions do no ou-forecas a random walk model. There are cases where a random walk model is significanly beer han he exchange rae equaion, bu here is no a case where an exchange rae equaion is beer. The DOC saisic, on he oher hand, suggess hese exchange rae equaions predic he exchange rae movemen quie well. All hese saisics indicae he exchange rae equaion forecass correcly predic he direcion more han half of he cases. Indeed, 11 cases for models wih homogeneous coefficiens and 17 cases for models wih heerogeneous coefficiens have saisically significan DOC saisics indicaing significan direcion predicion power. I is ineresing o noe ha none of he DOC saisics are significan for he Japanese yen. 19

20 5. Conclusions In his paper, we examine he performance of exchange rae equaions ha are derived from empirical and heoreically opimal ineres rae rules. The domesic and foreign ineres rae rules may have he same or differen response coefficiens. The US dollar exchange raes of he Briish pound, Canadian dollar, Japanese yen, German mark, and Swiss franc are considered. The sample period is from 1974:01 o 2008:12; wih he excepion of he German sample period from 1974:01 o 1998:12. Resuls from rolling regressions are used o assess in-sample and ouof-sample performance. The esimaion resuls sugges ha hese exchange rae equaions have limied in-sample explanaory power. There is no srong evidence ha a specific exchange rae equaion based on eiher heoreically opimal or empirical ineres rae rules offers a consisenly beer explanaory power han oher specificaions across all he five currencies under consideraion. Furher, he relaive explanaory power of hese models varies over ime. The relaive rankings of hese models forecas performance as measured by heir MSFEs change over ime. There is no one model ha consisenly ouperforms ohers over ime and across currencies. Furher, he correlaion beween a model s ranks based on MSFE and AIC is quie weak. There is lile associaion beween an exchange rae equaion s explanaory power and is forecas abiliy an equaion ha offers beer in-sample performance does no necessarily have good forecas abiliy. The comparison of he forecas performance beween hese exchange rae equaions and a random walk offers some mixed evidence. The MSFEs of hese exchange rae equaions are usually no smaller han he random walk specificaion. The Diebold-Mariano es affirms ha he random walk model is no ou-performed in erm of MSFEs. Insead, he es resuls poin o he possibiliy ha hese exchange rae equaions offer forecass han are worse han he random walk model. The DOC saisic, on he oher hand, suggess ha hese exchange rae equaions have abiliy o predic he direcion of exchange rae variaions. Mos exchange rae equaions correcly predic he exchange rae s direcion of change in a saisically significan manner. To shed some ligh on he change in forecas abiliy of alernaive exchange rae equaions, we repor he forecas error for empirical and opimal rule based exchange rae equaion in Graph 1 and 2, respecively. The forecas errors obained from exchange rae equaions based on homogenous coefficiens for boh empirical and opimal rules do no display 20

21 any consisen paern across currencies and daa sample. For insance, forecas errors from empirical rule based equaions increase for Canada afer early 2000s while here is a decline for he UK and Swizerland, if he recen observaions are excluded. Thus, we are no able o make any assessmen on wheher he forecasing power of hese equaions has improved over ime. Three counries in our sample; namely Canada, he U.K. and Swizerland, adoped he inflaion argeing policy a differen poins of ime hroughou he 1990s. The swich o inflaion argeing may affec he implemenaion of ineres rae rules, he corresponding implied exchange rae equaions, and hence he exchange rae behavior. Therefore, for hese counries, we repeaed he exercise and assessed he in-sample and ou-of-sample performance using daa from he respecive pre- and pos-inflaion argeing periods. The in-sample performance, ou-ofsample performance, and heir degree of associaion are qualiaively he same as hose repored in Secion 4. These resuls are available upon reques. In sum, our exercise offers very limied evidence on he abiliy of ineres rae rule based exchange rae equaions o explain exchange rae behavior or o forecas exchange raes according o he commonly used MSFE crierion. While hese resuls do no preclude a specific ineres rae rule based exchange rae equaion o offer good explanaory and forecasing resuls in a specific seing, hey allude o he generaliy of using ineres rae rule based exchange rae equaions o model exchange rae behavior. 21

22 Appendix A1: Opimal Ineres Rae Rules The opimal rules in his sudy are based on he canonical New Keynesian model ha includes an expecaions-augmened Phillips curve and a forward-looking IS curve. The model could be wrien as (Clarida e al., 1999): Phillips curve: p = be p + ky + e, + 1 IS curve: y = Ey+ 1 -jéi Ep ù ë - + 1û + u, where p is inflaion rae, y is he oupu gap given by he deviaion of he acual oupu from is naural level in logs, i is he nominal ineres rae, and E is he expecaions operaor. The parameers k and j are assumed o be posiive. b is he discoun facor beween zero and one. The cos push shock is given by e and u is a preference or demand shock. 8 The policy objecive is o minimize he expeced discouned sum of he losses given by j E é o j= 0b L ù ês ë úû, where he loss funcion relevan o he cenral bank s decision is given by L 2 2 = p + ly, in which, l is he relaive weigh placed on sabilizing he oupu gap versus inflaion. The lossfuncion defined here is also known as he period loss funcion. Given he policy objecive for he cenral bank, in general for he moneary policy implemenaions, he shor erm ineres raes are he policy insrumens. Therefore, he opimal ineres rae rule can be obained by minimizing he objecive funcion subjec o he srucure of economy represened by he IS and Phillips curves. Anoher relevan quesion in moneary policy analysis is wheher he opimal rule should be derived under discreion or commimen? Under discreion, he cenral bank s job is o minimize he loss in he nex period, and is no bounded by happenings in any subsequen periods. Under commimen, he cenral bank has o choose a policy rule (and, hence he implied inflaion and oupu gap pahs) which minimizes he ineremporal loss-funcion. In addiion o he heoreical aribue ha he opimal rule under commimen should yield an equilibrium ha is superior in erms of welfare, i also argued ha he cenral bank could enhance moneary policy credibiliy and affec he privae secor s inflaion expecaions if i follows commimen rule (Clarida e al., 1999; Walsh, 2003). In curren exercise, we only consider he opimal rule under commimen policy, and hence he exchange rae equaions based on hese rules. T1- Learning By subsiuing he Phillips curve and he IS curve ino he ineremporal loss funcion, he objecive funcion of he opimizing exercise can be wrien as j j= 0b p 2 + j l + j = E S {( + y ) + q+ j[ y+ j - y+ j+ 1 + j( i+ j -p+ j+ 1) - u+ j] 8 If he aim is o find he soluions of p and y in erm of shocks, hen i is cusomary o assume he shocks follow an auoregressive process like e = r e e -1 + e and u e, = ruu- 1 + eu, ; where 0 r, r e u < 1, and e e, and e u, are whie noises. 22

23 + y+ i[ p+ j -bp+ j+ 1 -ky+ j - e+ j] }. Under commimen, he firs order condiions are j ( q + ) = 0 if j ³ 0, p E j + y = 0 if j = 0, j j-1 E + j + j + j j -1 b ( p + y -y )- jb q = 0 if j ³ 1, ly + q - ky = 0 if j = 0, j j-1 y+ j + j + j + j-1 b E ( l + q - ky ) + b q = 0 if j ³ 1. From he firs order condiion, he IS curve does no impose any consrain since q = 0. In his case, we have a compac form given by p + y = 0 = 0, j E ( p + y - y ) = 0 j ³ 1, + j + j + j-1 E( ly+ j - ky+ j) = 0 j ³ 0. Suppose he expecaions/decision peraining o he las equaion was formed in disan pas, which Woodford (1999) calls he imeless-perspecive, we have y l = y k. Therefore, l p =- ( y - y 1) k - gives he firs order opimal condiion. Using he inflaion equaion, he soluion for he oupu gap (no imposing raional expecaions) is kb y E l =- + y - k e. p l+ k + 1 l+ k - 1 l+ k Thus, he rule under commimen can be derived by subsiuing his equaion ino he IS curve and be wrien as k l ( ) j i 1 1 = 1 + be p ( ) y ( ) E y u k e jl k + - jl k j jl ( + k) As shown in Table 2, we rewrie he rule under learning in esimaing equaion as in he following form; i = b + b E p + b E y + b y + e, where he response coefficiens are given by b = + kb j l + k /[ ( )], b2 = 1/ j, b =- l j l + k 2 3 /[ ( )], and e, is given by e = / j + k / j( l + k 2 u e ). 9 The exchange rae equaions based on he opimal rule under learning wih homogenous and heerogeneous coefficiens can be wrien as * b b p p b * * ED s+ 1 = 0 + 1E( ) + 2E( y+ 1 - y+ 1) + b3( y- 1 - y -1) + e and 9 Noe ha we add inercep erm b o he esimaing equaion corresponding o saisfy sandard assumpions. 0 23