Exchange Rate Models Are Not as Bad as You Think

Transcription

1 Exchange Rae Models Are No as Bad as You Think Charles Engel Universiy of Wisconsin and NBER Nelson C. Mark Universiy of Nore Dame and NBER Kenneh D. Wes Universiy of Wisconsin and NBER April 10, 2007 Absrac Sandard models of exchange raes, based on macroeconomic variables such as prices, ineres raes, oupu, ec., are hough by many researchers o have failed empirically. We presen evidence o he conrary. Firs, we emphasize he poin ha beaing a random walk in forecasing is oo srong a crierion for acceping an exchange rae model. Typically models should have low forecasing power of his ype. We hen propose a number of alernaive ways o evaluae models. We examine in-sample fi, bu emphasize he imporance of he moneary policy rule, and is effecs on expecaions, in deermining exchange raes. Nex we presen evidence ha exchange raes incorporae news abou fuure macroeconomic fundamenals, as he models imply. We demonsrae ha he models migh well be able o accoun for observed exchange-rae volailiy. We discuss sudies ha examine he response of exchange raes o announcemens of economic daa. Then we presen esimaes of exchange-rae models in which expeced presen values of fundamenals are calculaed from survey forecass. Finally, we show ha ouof-sample forecasing power of models can be increased by focusing on panel esimaion and long-horizon forecass. Prepared for he NBER Macroeconomics Annual, Conference in Cambridge, Massachuses, March 30-31, We hank Joong Shik Kang for very able research assisance. We hank Paolo Peseni and paricipans in he Second Annual Workshop on Global Inerdependence, a Triniy College, Dublin for very helpful commens. We also hank Barbara Rossi and Ken Rogoff for heir helpful commens a he Macro Annual conference. This version does no ye reflec a full revision in ligh of heir commens, hough i does correc some misakes ha hey noed.

2 There appears o be a consensus among researchers in exchange-rae economics ha he sandard models ha relae exchange raes o moneary variables, prices, ineres raes, ec., are off he mark. For example, Sarno and Taylor (2002, pp ) sae, Overall, he conclusion emerges ha, alhough he heory of exchange rae deerminaion has produced a number of plausible models, empirical work on exchange raes sill has no produced models ha are sufficienly saisically saisfacory o be considered reliable and robus In paricular, alhough empirical exchange rae models occasionally generae apparenly saisfacory explanaory power in-sample, hey generally fail badly in ou-of-sample forecasing ess in he sense ha hey fail o ouperform a random walk. Bacchea and van Wincoop (2006, p. 552) observe, The poor explanaory power of exising heories of he nominal exchange rae is mos likely he major weakness of inernaional macroeconomics. Richard A. Meese and Kenneh Rogoff (1983a) and he subsequen lieraure have found ha a random walk predics exchange raes beer han macroeconomic models in he shor run. Evans and Lyons (2002, pp ) asser, Macroeconomic models of exchange raes perform poorly a frequencies higher han one year. Indeed, he explanaory power of hese models is essenially zero (Meese and Rogoff 1983a, Meese 1990). In he words of Frankel and Rose (1995, p. 1704), his negaive resul has had a pessimisic effec on he field of empirical exchange rae modeling in paricular and inernaional finance in general. The pessimisic effec has been wih us 20 years. We presen evidence ha exchange rae models are no so bad afer all. We approach he problem from several angles, bu all of he approaches are linked by he observaion ha shor-run movemens in exchange raes are primarily deermined by changes in expecaions exacly as he sandard models say. We begin in secion 1 by demonsraing ha sandard models imply near random walk behavior in exchange raes, so ha heir power o bea he random walk in ou-of-sample forecass is low. We hen offer various alernaive means for evaluaing exchange-rae models: (1) using in-sample fi of he models, bu highlighing he role of endogeneiy of moneary policy in explaining nominal and real exchange rae behavior (secion 2); (2) examining wheher exchange raes incorporae news ha helps o predic he fuure macroeconomic fundamenals, as implied by he models (secion 3); (3) re-examining he quesion of wheher he models can accoun for he volailiy of exchange raes (also secion 3); (4) reviewing he recen lieraure ha has examined he response of exchange raes o announcemens of macroeconomic news (secion 4); (5) presening esimaes of he model in which expecaions of fundamenals are drawn from survey daa (also secion 4); and, (6) demonsraing ha he predicive power 1

3 of he models can be grealy increased by using panel echniques and forecasing exchange raes a longer horizons (secion 5). Conclusions are in secion 6. We begin by examining he heorem in Engel and Wes (2005) ha demonsraes ha under plausible assumpions, he models acually imply ha he exchange rae should nearly follow a random walk. Therefore, i should no be surprising ha he exchange rae models canno provide beer forecass han he random walk model. As we will elucidae, he key insigh behind he heorem is ha curren economic fundamenals have relaively lile weigh in deermining he exchange rae in sandard models. Much greaer weigh is pu on expecaions of fuure fundamenals, even fundamenals several years ino he fuure. We elaborae on he implicaions of he heorem. We show ha in a sandard parameerizaion of he famous Dornbusch overshooing model, he exchange rae nearly follows a random walk. We make his characerizaion in spie of he fac ha he bes-known feaure of his model is ha exchange-rae changes should be predicable ha he exchange rae overshoos is long-run value in response o moneary shocks. We also argue ha Meese and Rogoff s (1983a) exercise, in which he model forecass use acual ex-pos (raher han forecased) values of he fundamenals, is poenially flawed, because he ou-of-sample fi of he models can be made arbirarily worse or beer by algebraic ransformaions of he model. Indeed, he ou-of-sample fi of he sandard models can be made much beer (under he Meese- Rogoff mehodology) if he models are wrien in a way ha emphasizes he imporance of expecaions in deermining exchange raes. If we do no use he crierion of ouperforming he random walk model in ou-of-sample forecasing power, how hen should we evaluae exchange rae models? We offer a number of alernaives. Firs, we can look a he in-sample fi of he models. Mos empirical moneary models of he 1970s and 1980s paid lile aenion o he endogeneiy of moneary policy. Bu if exchange raes are primarily driven by expecaions, hen correcly modeling moneary policy is criical. Changes in curren economic fundamenals, for example, may have a greaer impac on exchange raes indirecly hrough he induced changes in expecaions of moneary policy han hrough any direc channel. Engel and Wes (2006), Mark (2007), Clarida and Waldman (2006) and Molodsova and Papell (2007) have explored he empirical performance of models based on Taylor rules for moneary policy. In a radiional flexible price model, an increase in curren inflaion depreciaes he currency. Bu i is imporan o undersand he policy reacion o higher inflaion. In he new Taylor-rule models, higher inflaion leads o an appreciaion in inflaionargeing counries, because higher inflaion induces expecaions of igher fuure moneary policy. 2

4 We review hese models, and also make noe of he imporan resul of Benigno (2004): he Taylorrule models offer a poenial soluion o he purchasing power pariy puzzle. In paricular, he models offer he possibiliy ha persisen real exchange raes do no require unrealisic assumpions abou he sickiness of nominal price seing, and poenially de-link he wo alogeher. Engel and Wes (2004, 2005) propose esing wo implicaions of he presen value models. They emphasize ha because we acknowledge ha here are unobserved fundamenals (e.g., money demand shocks, risk premiums), he exchange rae may no be exacly he expeced presen value of observed fundamenals. Bu if exchange raes reac o news abou fuure economic fundamenals, hen perhaps exchange raes can help forecas he (observed) fundamenals. If he observed fundamenals are he primary drivers of exchange raes, hen he exchange raes should incorporae some useful informaion abou fuure fundamenals. We verify his proposiion using Granger causaliy ess. Engel and Wes (2004) also develop a echnique for measuring he conribuion of he presen discouned sum of curren and expeced fuure observed fundamenals o he variance of changes in he exchange rae, which is valid even when he economerician does no have he full informaion se ha agens use in making forecass. Here we find ha he observed fundamenals can accoun for a relaively large fracion of acual exchangerae volailiy, a leas under some specificaions of he models. Sandard ess of forward looking models under raional expecaions make he assumpion ha he sample disribuion of ex pos realizaions of economic variables provides a good approximaion of he disribuion used by agens in making forecass. Bu (as Rossi (2005) has recenly emphasized) when agens are rying o forecas levels of variables ha are driven by persisen or permanen shocks, he economerician migh ge a very poor measure of he agens probabiliy disribuion by using realized ex pos values. The problem is enhanced when he daa generaing process is subjec o long-lasing regime shifs (caused, for example, by changes in he moneary policy regime.) For an economic variable such as he exchange rae which is primarily driven by expecaions i migh be useful o find alernaive ways of measuring he effec of expecaion changes. Several recen papers (Andersen, Bollerslev, Diebold, and Vega (2002), Faus, Rogers, Wang, and Wrigh (2007), Clarida and Waldman (2006)) have looked a he effecs of news announcemens on exchange raes, using high frequency daa. We review hese sudies, and argue ha he response of exchange raes o news is precisely in line wih he predicions of he Taylor rule models. We also provide new evidence of anoher sor. We direcly measure expecaions of inflaion and oupu using surveys of professional forecasers. The paricular survey we employ asks forecasers wice a 3

5 year o provide predicions for inflaion and oupu growh in a dozen advanced counries for he curren year, each of he nex five years, and an average for years From hese surveys, we are able o consruc a presen value of curren and expeced fuure fundamenals implied by he Taylor rule model. We find srong confirmaion of he model higher oupu growh and higher inflaion in he US relaive o he oher counries leads o an appreciaion of he dollar relaive o he oher currencies. While we argue ha heoreically he models may have low power o produce forecass of changes in he exchange rae ha have a lower mean-squared-error han he random walk model, we also explore ways of increasing he forecasing power. Mark and Sul (2001) and Groen (2005) have used panel errorcorrecion models o forecas exchange raes a long horizons (16 quarers, for example.) We find ha wih he increased efficiency from panel esimaion, and wih he focus on longer horizons, he macroeconomic models consisenly provide forecass of exchange raes ha are superior o he no change forecas from he random walk model. We find ourselves in he uncomforable posiion of boh poining ou ha common formulaions of moneary models imply ha he models should have lile power o produce beer forecass han a random walk, while a he same ime finding more forecasing power for he models han many previous sudies (hough in line wih he findings of he sudies ha employ panel echniques, cied above.) There are wo possible resoluions o his conflic. Firs, in secion 5, we demonsrae in an example ha if he foreign exchange risk premium whose behavior is no well undersood, and which is no observable o he economerician is saionary wih an innovaion variance ha is relaively small (compared o variances in innovaions of sandard observed fundamenals), hen he models migh have predicive power relaive o he random walk a long horizons in he error-correcion framework. So i may be ha indeed, using panel echniques, we have confirmed he usefulness of he models in forecasing a horizons of 16 quarers. The oher resoluion o he conflic is ha our predicion resuls migh prove fleeing: Hisory has shown ha models ha seem o fi well over some ime periods end up no holding up as he sample exends. Tha migh hold rue for forecasing power as well. While i is encouraging ha our forecasing resuls confirm he findings of Mark and Sul (2001) on an exended sample, hese resuls may ulimaely prove no o be robus. Bu he Engel and Wes (2005) heorem ells us ha ou-of-sample predicion power relaive o a random walk is no a reliable gauge o judge exchange rae models. 4

6 1. Presen-Value Models and Random Walks Le s denoe he log of he exchange rae, measured as he log of he domesic currency price of foreign currency. Thus, depreciaion of he currency implies an increase in s. Consider models of he exchange rae ha relae he value of he currency o economic fundamenals, and o he expeced fuure exchange rae: (1) s = (1 b) a1' x + ba2' x + bes + 1, 0< b < 1. x is a vecor of economic fundamenals ha ulimaely drive exchange-rae behavior. Many familiar exchange-rae models based on macroeconomic fundamenals ake his general form, as subsequen examples will demonsrae. Equaion (1) is an expecaional difference equaion wih a no-bubbles forward soluion given by: (2) (1 ) j j s = b E b a1' x+ j + be b a2' x+ j. j= 0 j= 0 The log of he exchange rae is deermined as he expeced presen discouned value of curren and fuure fundamenals. As in many models of oher asse prices, if he discoun facor is large (close o one), expeced fuure fundamenals maer a lo more han he curren value of he fundamenal. For example, if he fundamenals were expeced o change beween period and +1 o new permanen values, exchange rae would be a weighed average of he curren and he fuure fundamenal bu wih much more weigh placed on he fuure fundamenal: b b s = (1 )[ + ] + [ + b a 'x a 'x b a 'x a 'x ]. 1 1 b b 2 +1 Tha is, in a model such as his, when fundamenals are very persisen we can say ha he exchange rae is primarily deermined by he expeced fuure pah of he fundamenals, wih lile weigh given o he curren fundamenal. The moneary exchange-rae models of he 1970s and 1980s ake he form given by equaion (1). They are based on a Cagan-syle money demand model. For he home counry we can wrie: m p = + y i + v, (3) α γ λ where m is he log of he money supply, p is he log of he home consumer price level, x +1, he y is he log of oupu, i is he home ineres rae (in levels), and v is a sochasic shif erm. Defining he real exchange 5

7 rae as q = + s p p, and assuming he foreign money demand equaion has he same parameers as he home, we can wrie: (4) m m s + q = γ( y y ) λ( i i ) + v v. Foreign variables are denoed wih a. (5) Now we will inroduce he relaionship: i i Es+ 1 s = + ρ. This relaionship defines ρ, he deviaion from uncovered ineres pariy. As is well known, a vas empirical lieraure has rejeced he hypohesis ha ρ = 0. Bu so far here is no consensus on a model for ρ. Perhaps i is a risk premium, a shor-run deviaion from raional expecaions, or some oher marke imperfecion. I is possible ha movemens in ρ are imporan in explaining exchange rae movemens (as Obsfeld and Rogoff (2003) have suggesed), bu we do no explore ha avenue in his paper. We rea ρ as an unobserved fundamenal an economic variable ha migh drive he exchange rae, bu a variable for which we do no have direc observaions. The money deman d shifs, reaed as unobserved fundamenals. v and v are also When we combine equaions (4) and (5), we ge an equaion ha akes he form of equaion (1): 1 λ λ = + y y v v + ρ + Es 1+ λ 1+ λ 1+ λ (6) s ( m m q γ( ) ( )) In his case, he discoun facor, b, from equaion (1) corresponds o 1 λ + λ in equaion (6). The linear combinaion of fundamenals o (7) a'x 1 is given by a'x 2. The no-bubbles forward soluion o equaion (6), hen, is: + 1 m + γ m q ( y y ) ( v v ), while ρ corresponds. s = j j 1 λ λ E ( m m q γ( y y ) ( v v + )) + E ρ λ. + j + j + j + j + j + j + j + j 1+ λ j = 0 1+ λ 1+ λ j = 0 1+ λ Equaion (7) is represenaive of he ype of model ha we conend is beer han you hink. Tha is, i is a raional-expecaions model based on macroeconomic fundamenals. I is he radiional models ha were explored in deph in he 1970s and 1980s. Moreover, a model such as his is derived in a 6

8 sraighforward way by log-linearizing equaions from modern opimizing macroeconomic models. In fac, he money demand equaion (3) can be obained direcly from a dynamic model in which agens maximize uiliy of consumpion and real balances. Obsfeld and Rogoff (2003) derive such an equaion when consumpion and real balances ener separably ino uiliy as power funcions: 1 V M ϖ 1 σ C +, σ ϖ P where V is a random shif facor in preferences for real balances. The only difference beween he money demand equaion derived from he firs-order condiion in he money-in-he-uiliy funcion model and he ad hoc money demand equaion is ha he log of consumpion, demand, raher han he log of income, c, appears as he aciviy variable in money y, as in equaion (4). Using equaion (5), Obsfeld and Rogoff (2003) derive an expression for he exchange rae analogous o equaion (7). rae, q Equaion (7) is no enirely saisfacory as an exchange rae equaion because he real exchange, appears as an explanaory variable on he righ-hand-side of he equaion. Equilibrium models wih flexible goods prices would elaborae on his equaion by relaing he equilibrium real exchange rae o underlying economic variables such as produciviy and curren accoun balances. Or, he simples such model, which assumes purchasing power pariy, reas oupu differenials o oher economic driving variables. q as a consan. These models migh also relae The sicky-nominal-price models of Dornbusch (1976) and subsequen auhors rea he real exchange rae and perhaps he oupu differenial as endogenous variables whose dynamics are in par deermined by he sochasic process for money supplies. We noe also ha moneary policy migh be endogenous, so ha he relaive money supplies are se in response o he realizaions of macroeconomic variables. We reurn o discussion of his in much greaer deail laer. One migh argue ha equaion (6) holds by definiion. Tha is, equaion (4) defines he money demand errors, v v, and equaion (5) defines he deviaion from ineres pariy, ρ. The exchange-rae equaion (6) mus hold if we allow a role for hese suiably defined unobserved fundamenals. Bu while we need o acknowledge a role for unobserved fundamenals, his class of models is only ineresing if he observed fundamenals do a good job of explaining exchange raes. As he quoes ha we begin his paper wih sugges, a sandard way of evaluaing exchange-rae models is o compare heir ou-of-sample forecasing power o ha of he random walk model. There are 7

9 many varians of he sandard model, which migh depend on he way he fundamenals are measured, or he se of fundamenals ha are included in order o accoun for he behavior of q, y y, v, and/or ρ. Because of he possibiliy of overfiing or daa mining (by a single researcher, or by exchange-rae researchers collecively), in-sample fi is considered an unreliable benchmark. Good ou-of-sample forecasing power is a higher hurdle, and has become he sandard by which exchange-rae models are judged. Engel and Wes (2005) (hereinafer referred o as EW05), however, demonsrae ha under some plausible condiions, hese models acually have he implicaion ha he exchange rae is nearly a random walk. In ypical samples, he models acually imply ha he change in he exchange rae is no predicable. The heorem saes ha as he discoun facor, b, goes o one, he change in he log of he exchange rae beween ime -1 and becomes uncorrelaed wih informaion in he ime 1 informaion se. The condiions under which ha holds are eiher (i) a'x 1 is inegraed of order 1, and a'x 2 is zero, or (ii) a'x 2 is I(1). Noe firs ha his heorem does no require a'x 1 or a'x 2 o be pure random walks. If i did, he heorem would be rivial, since he weighed sum of random walk processes ha would appear in he presen-value formulas are also random walks. Second, his heorem does no say ha for discoun facors less han one, he log of he exchange rae is exacly a random walk. I says, in essence, ha for large values of he discoun facor, he log of he exchange rae is approximaely a random walk. To illusrae he heorem, suppose a'x 1 is a scalar x, and ha a'x 2 shown below, his is a special case of he moneary model. Suppose furher ha no a random walk. Assume x x = φ( x x ) + ε, ε i.i.d Then he soluion for he change in he exchange rae is given by: φ(1 b) 1 s s 1 = ( x 1 x 2) + ε. 1 bφ 1 bφ v is idenically zero. As x has a uni roo, bu is I is clear from his example ha he change in he exchange rae is predicable from he lagged change in he fundamenal x. Bu as exchange rae approaches a random walk. b 1, he coefficien on he lagged money supply goes o zero, and he 8

10 The heorem is proved in EW05. Inuiively, suppose firs ha a'x 2 is idenically zero. Consider 1 +j j= 0 j he discouned sum, b a ' x. Using a Beveridge-Nelson decomposiion, we can wrie a ' x as he sum of wo componens, a pure random walk permanen componen, and a ransiory componen, τ. We have ha var ( τ + approaches a consan as j ges large, bu he condiional variance of he random walk j) componen grows in proporion o j. So from he perspecive of ime, he permanen componen becomes more and more imporan in accouning for he ex ane variaion in a x as j ges large. When he ' 1 +j 1 discoun facor is close o one, he discouned sum pus a lo of weigh on values of ' 1 +j a x in he fuure. As b 1, he discouned sum begins o look more and more like a sum of pure random walk variables. Now allow non-zero values for boh a'x 1 and a'x 2. Because he presen value of a'x 1 is muliplied by (1-b), movemens in s are dominaed by he presen value of a'x 2 as b 1. For reasons skeched in he previous paragraph s will behave like a random walk for b near 1 if a, 2'x has a uni roo. If his heorem is applicable o exchange rae models, i suggess ha we should no evaluae he models by he crierion of beaing a random walk in ou-of-sample forecasing power. How close he exchange rae is o a random walk, if i is generaed by a presen-value model as in equaion (2), depends in pracice on how close he discoun facor is o one, and how persisen is he ransiory componen of he economic fundamenals. EW05 calibrae hese for some sandard exchange-rae models, and show ha apparenly he models imply near-random-walk exchange-rae behavior. For example, consider a simple moneary model in which uncover ed ineres pariy holds ( ρ = 0 ), purchasing power pariy holds ( q = 0 ), here are no money demand errors ( v = v = 0 ) and in which he income elasiciy of money demand is uniy ( γ = 1). In his case, he exchange rae model simplifies o j 1 λ 1 λ. + j = 0 1+ λ (8) s = E ( m+ j m+ j ( y+ j y+ j) ) A quie conservaively low esimae of λ for quarerly daa, from sudies of money demand and exchange raes, is λ = 10, which implies b In he daa for he U.S. relaive o each of he oher G7 1 I is imporan o recognize ha he ineres semi-elasiciy of money demand depends on he unis in which ineres raes are expressed. For example, a value of λ of esimaed wih quarerly daa when ineres raes are annualized and expressed in percen erms mus be muliplied by 400 o ge he relevan esimae when ineres raes are in he same unis as he change in he log of he exchange rae. 9

11 counries, he highes serial correlaion for ( m m ( y y )) is Bu he compuaions of EW05 show ha for b = 0.90 and a serial correlaion of ( m m ( y y )) equal o 0.50, ha he correlaion of and is only 0.05, and he correlaion of s s 1 s wih ( m 1 m 1 ( y 1 y 1 ) ) is Tha is, if he exchange rae were generaed from equaion (8), i would exhibi near-random-walk behavior. The exchange rae is predicable, bu (as EW05 discuss) we would no be likely o rejec he random walk in sample sizes ha are ypically available o open-economy researchers. We noe ha he echnical condiions for he random walk rule ou a saionary process for a'x. In he moneary model, his means ha he risk premium, ρ, if i is presen, mus no be saionary. (See equaion (7).) In pracice, however, if his erm has nearly a uni roo, he random walk will nearly follow, as illusraed in he compuaions in EW05. Neverheless, in pracice, i is a criical quesion as o wheher arguably saionary erms such as risk premia can be exploied o make predicions ha bea he random walk. We discuss his furher in secion 5 below, when we presen resuls from panel predicion exercises. We observe ha a discoun facor close o one is helpful o reconcile he observaion ha he variance in innovaions in exchange raes is large relaive o he variance of innovaions in ineres differenials. Using equaion (5), and if we associae a'x 2 wih he risk premium ( rewrie equaion (1) as: (9) (1 b)( s a1' x) = bρ + b( i i ) Typically he volailiy of innovaions in he fundamenals raes. Bu innovaions in differenial, s a 1 ' x. a'x 2 2 = ρ ), we can are small compared o ha of exchange have a much higher variance han innovaions in he ineres. In a model such as his, reconciliaion of hese facs could be accomplished eiher by having he unobserved risk premium being highly volaile, or by having he discoun facor b close o one. I is of course much more saisfying no o have o rely on he volailiy of an unobserved variable o accoun for exchange rae volailiy, so models wih he discoun facor near one are appealing on his score. Example 1 i i per Japanese yen ( ) and HK$ per exchange raes are coinegraed (wih coinegraing vecor (1,-1).) a'x 1 We urn now o wo examples o help elucidae he EW05 heorem. The Hong Kong dollar (HK$) per U.S. dollar (US$) nominal exchange rae is apparenly a saionary random variable. I flucuaes beween 7.75 and 7.85 HK$ per US$. Tha means ha he US$ 10

12 The Engle-Granger represenaion heorem ells us ha if he US$/ and HK$/ exchange raes are coinegraed, hen a leas one of hem is predicable. UJ s We use his example o illusrae he EW05 heorem. Suppose, hen ha he US$/ exchange rae, is generaed by a presen value model, UJ j UJ (1 ) j = 0 s = b b x, where HJ x is an I(1) fundamenal. The HK$/ exchange rae,, is deermined by an analogous model, UJ s wih x as he I(1) fundamenal. If s UJ and s HJ are coinegraed, hen x UJ and x HJ mus also be HJ coinegraed, say x UJ = x HJ + z, for some saionary z. For simpliciy, assume z has a mean of zero Then (10) where HJ j HJ j UJ j UJ j= 0 j= 0 j= 0 z, s = (1 b) b x = (1 b) b x + (1 b) b z = s + (1 b) j z b z is saionary and has finie variance even in he limi as b 1. Thus, we can wrie for j = 0 he HK$/US$ exchange rae, s. HU (11) s = s s = (1 b) z, HU HJ UJ HU and a s b 1, s 0 (i.e., a consan, equal o zero here since z has a zero mean.) In oher words, he EW05 heorem implies ha while s UJ and s HJ each approach random walks as b 1, approaches a consan. HU To ge a beer sense of he behavior of hese exchange raes for simple example. Suppose x = e, e i.i.d. (ha is, UJ HJ UJ also be i.i.d. Then z = z, and s = s + (1 b) z, implying b < 1, consider he following UJ x is a random walk), and le z defined above (12) s = s + b) z = + (1 )( z ) = e + (1 ) ( ). HJ UJ HJ UJ (1 e b z 1 b z s 1 s 1 So, u pon defining he i.i.d. variable v = e + (1 b) z, we can wrie he VECM for UJ HJ s a nd s as: s (13a) UJ s = e HJ HJ UJ (13b) s = v ( s 1 s 1). 11

13 According o equaion (13b), HJ s is predicable using he lagged HK$/US$ exchange rae, HU HJ UJ HJ UJ HJ s 1 = s 1 s 1. Bu as b 1, s 1 s 1 0, and we ge he EW05 resul ha s follows a random walk. Example 2 Probably he bes cied exchange rae model ever is Dornbusch s (1976) overshooing model. A firs glance, i seems as hough EW05 s heorem could no apply o Dornbusch s model. The mos celebraed aspec of he model he fac ha in response o a permanen money supply shock, he exchange rae overshoos, responding more in he shor run han in he long run implies ha exchange rae changes are predicable. When he currency depreciaes in response o a domesic moneary expansion, we can predic ha i will appreciae oward is long-run equilibrium value. The EW05 heorem acually is no designed o answer he quesion of wheher he exchange rae in he Dornbusch model heoreically is nearly a random walk. Since equaions (3), (4) and (5) hold in he Dornbusch model, hen he presen value relaionship (6) also holds. The EW05 heorem akes he daa generaing processes for he fundamenals as given, and asks wha he implied exchange rae behavior is for large values of he discoun facor. Tha is subly differen han asking wha happens in he model o he behavior of he exchange rae as he discoun facor goes o one, because a change in he discoun facor may change he implied daa-generaing process for he fundamenals. In oher words, he EW05 heorem suggess ha if he exchange rae is deermined by he model (6), wih he observed DGPs for he fundamenals, and he discoun facor is close o one, hen he exchange rae will be nearly a random walk. Here we briefly examine he heoreical behavior of he exchange rae in he Dornbusch model when he discoun facor is nearly uniy, allowing for he fac ha he daa generaing process for he fundamenals paricularly he real exchange rae is affeced by he discoun facor. We look a a version of he model very close in spiri o Dornbusch s original model. We use equaions (3), (4), and (5), and, as in Dornbusch, assume ha uncovered ineres pariy holds exacly, ρ = 0. As in he overshooing analysis of Dornbusch, we will ake oupu as exogenous. Dornbusch examined he impac of a permanen change in he money supply in a non-sochasic model. In he sochasic seing, his is equivalen o looking a a random walk process for he money supply. Since oupu shocks and money demand shocks have idenical effecs on he exchange rae as money supply shocks (up o he sign of he effec), we will simply assume ha he fundamenals follow a random walk: ( ) (14) m m γ ( y y ) ( v v ) = u, u i.i.d. 12

14 We need o supplemen he model wih a price adjusmen equaion. The open-economy macro lieraure of he 1970s and 1980s assumed a backward looking elemen o price seing. The log of he domesic price level for ime, p, is prese in ime 1, and adjused o eliminae par of he deviaion of 1 p 1 from is long-run equilibrium level. As in Dornbusch, we will assume purchasing power pariy holds in he long run, so p eliminaes par of he ime 1 PPP deviaion, p 1 s 1+ p 1 p 1. In addiion, here is a forward looking rend erm o price adjusmen. Obsfeld and Rogoff (1984) emphasize ha priceadjusmen equaions ha do no include he forward looking elemen lead o couner-inuiive dynamics when considering expeced fuure changes in policy, or non-saionary dynamics in he fundamenals. Here, we implemen a version of wha hey call a Mussa rule he rend erm is he expeced change in he marke-clearing exchange rae. We have: (15) p p = θ ( s + p p ) + E s + p ( s + p ) This pricing rule is symmeric, in ha he analogous pricing rule for he foreign counry yields equaion (15) as well. Equaion (15) implies ha he real exchange rae follows a firs-order auoregressive process: (16) E 1q = (1 θ ) q 1. Noe ha he persisence of he real exchange rae is enirely deermined by he speed of adjusmen of nominal prices. We reurn o his poin below when we discuss he PPP puzzle. However, he real exchange rae does depend on moneary shocks and on he discoun facor, which work hrough heir effec on innovaions in he real exchange rae. If we subsiue equaions (14) and (16) ino he presen value formula (7), we derive: θ 1 + λθ (1 b) θ 1 b+ bθ (17) s s 1 = q 1+ u = q 1+ u, 1+ λθ λθ 1 b+ bθ bθ where, recall, he discoun facor is resul. In response o a shock, λ b =. Equaion (17) demonsraes he famous overshooing 1 + λ, he exchange rae jumps more han one-for-one, u 1+ λθ. The volailiy λθ is greaer he sickier are prices (he smaller is θ.) Bu he change in he exchange rae is predicable. When s is above is PPP value in period 1 (so q 1 is posiive), hen E 1( s s 1) < 0. We can see, however, ha he EW05 resul holds in his model. As b 1, s s 1 u. As noed above, he real exchange rae behavior does depend on he value of b: 13

15 1 λθ 1 b bθ (18) q (1 θ) q + 1 u (1 θ) q + = + = 1+ λθ bθ Assume ha λ = 10, a conservaively low value if calibraed o quarerly daa. Also, assume θ = 0.25, which implies a half life of price adjusmen of 2.4 quarers. This speed of price adjusmen is in line wih he ypical calibraion of modern sicky-price macroeconomic models. However, i would imply from equaion (16) real exchange rae convergence ha is much faser han wha is ypically observed among advanced counries. Wih hese parameers, how well could we predic nominal exchange rae changes using he lagged real exchange rae? The implied R-squared from he regression in equaion (17) is 0.012, or slighly greaer han 1 per cen. While he exchange rae change is predicable in heory, in pracice i would no be predicable in any reasonably sized sample. One migh expec ha if price adjusmen were slower (ha is, θ lower), ha perhaps exchange rae changes would be more predicable. Since he overshooing is greaer, he exchange rae has furher o adjus o reach is long-run value, and perhaps we can predic ha change. In fac, lower values of θ reduce he predicabiliy of exchange rae changes. Afer a shock o he fundamenals, i is rue ha he gap beween he exchange rae and is long-run value is wider he smaller is θ. The overshooing for a given moneary shock is proporional o 1/θ. Bu, he predicable percenage change in he exchange rae oward is long-run value is smaller in proporion o θ. These wo effecs precisely offse each oher. Bu he oher effec of small values of θ is o make he variance of innovaions o he exchange rae larger he overshooing is larger. So he variance of he unpredicable componen of changes in he exchange rae grows relaive o he variance of he predicable componen as θ ges smaller. To furher develop inuiion, we can ask abou longer-run changes in he exchange rae. We find: u. (19) k k j 1 (1 (1 θ) ) (1 θ) + λθ s+ k s= q+ u+ k 1 j 1+ λθ j = 1 λθ. + There are wo effecs as he horizon for forecass ges longer. Firs, for higher k, he variance of he predicable par of he exchange rae change increases, since k (1 (1 θ )) 1+ λθ is increasing in k. Bu, since he exchange rae has a uni roo, he variance of he unpredicable par grows wihou bound as k grows. Tha can be seen from he second erm on he righ-hand-side of equaion (19), which has a variance ha is greaer han k imes he innovaion variance. In pracice, for he parameers considered above ( λ = 10 and θ = 0.25 ), he maximum R-squared in his regression comes a 6 quarers, and has a value of A 16 quarers, he implied R-squared is

16 Using fuure values o forecas : Meese-Rogoff revisied While recen lieraure has aemped o validae exchange rae models by comparing heir ou-ofsample forecasing power o ha of he random walk, Meese and Rogoff (1983a) did somehing differen. They apparenly gave he exchange rae models an advanage relaive o he random walk model. Forecasing from he exchange rae model requires forecasing he values of he fundamenals. Bu Meese and Rogoff evaluaed he exchange rae models using he acual realized values of he fundamenals, raher han forecasing hem. The more recen lieraure has no used he Meese-Rogoff echnique, and insead compared moneary exchange rae models o he random walk model using rue ou-of-sample forecasing power. By his sandard, moneary models have no fared well in general. See, for example, Cheung, Chinn and Garcia-Pascual (2005), who find ha he models generally do no have significanly beer forecasing power han he random walk model. Below, we noe ha moneary model forecass based on panel esimaion echniques a long horizons (such as in Mark and Sul (2001)) do seem o have greaer forecasing power han he random walk. We also examine he forecasing power of models in which moneary policy is endogenized (as in Molodsova and Papell (2007).) Bu here we wan o reconsider he Meese-Rogoff echnique. As an example (his is one model acually considered by Meese and Rogoff), use equaion (4) o solve for he log of he exchange rae, under he assumpion of PPP ( q = 0 ): (20) s = m m γ( y y ) + λ( i i ) + u, where here he error erm is associaed wih he error erms in money demand, v v. correlaion in u Rossi (2005) emphasizes ha Meese and Rogoff may no have fully accouned for he serial u. She noes ha if he exchange rae is coinegraed wih he economic fundamenals included in equaion (20), hen u is saionary, bu i migh be serially correlaed. Suppose u = ρu 1 + ε, ε i.i.d. Then he model forecas for ime + 1 should be m m γ ( y y ) + λ( i i ) + ρu Rossi argues ha even wih his addendum, he models sill migh be a a disadvanage relaive o he random walk. This is because ρ is plausibly near one, in which case esimaes of ρ end o be biased downwards. Hence imposing a uni roo (imposing ρ = 1) migh resul in long-horizon forecass superior o ones ha rely on an esimae of ρ ha is far below is rue value. Moreover, when ρ is esimaed, 15

17 disribuions of es saisics end o be nonsandard, a resul capured by Rossi by modeling ρ as local o uniy. Suppose we impose ρ = 1, hen our forecas of s + 1 is given by: m m γ( y y ) + λ( i i ) + u m+ 1 m+ 1 γ( y+ 1 y+ 1) λ( i+ 1 i+ 1) s. = + + In oher words, aking ino accoun he serial correlaion of he residual, we should be using he change in he fundamenals o forecas he change in he exchange rae. Noe however ha Meese and Rogoff (1983b) allow for a grid of possible values for ρ, including ρ = 1, and sill find ha he models do no improve he ou-of-sample fi compared o he random walk. We make a differen observaion abou he Meese and Rogoff sandard for evaluaing he exchange rae models. There is a sense in which he ou-of-sample fi of he model is arbirary he model can be rewrien o make he fi arbirarily good or bad. Tha is, suppose ha x is he model for some 2 variable y, and we have: y = x + ω. The variance of ω, σ ω, serves as a measure of he goodness of fi of he model under he Meese-Rogoff approach. Now, consider rewriing he model as: y = ( x (1 a) y)/ a+ ω / a = z + ϖ, where a is an arbirary consan, z ( x (1 a) y)/ a, and ϖ = ω / a. The model is no changed he second equaion is simply an algebraic manipulaion of he 2 firs. Bu he variance of he error in he rewrien model he variance of ϖ -- is (1/ a ) imes he variance in he firs model. For example, using he log approximaion o he pure arbirage covered ineres pariy condiion, we have f s = i i, where f is he log of he one-period ahead forward exchange rae. Subsiue his expression ino equaion (20), and rearrange erms o ge: (21) 1 λ 1 s = ( m m γ ( y y )) + f + 1+ λ 1+ λ 1+ λ This represenaion of he model is very similar in spiri o equaions (1) or (6): he exchange rae is a weighed average of he (observed) economic fundamenals, and he expeced fuure exchange rae, here f measured as. This way of wriing he exchange rae equaion emphasizes he weigh of expecaions of he fuure relaive o curren fundamenals. Bu i also gives us an error erm wih a much lower variance. u. The variance here is λ 2 imes he variance of he error erm in equaion (20). We have been using a 16

18 value of λ = 10 for quarerly daa in our examples so far, which would imply ha he variance of he error erm is lower by a facor of Which is he correc way o wrie he model? Since hey are algebraically equivalen, i is hard o argue for one in favor of he oher, and indeed ha is exacly he problem wih he Meese and Rogoff mehodology. Moreover, boh ways of wriing he model have naural economic inerpreaions. If he model is rue wih no error, hen he righ-hand-sides of equaions (20) and (21) are equal: (22) 1 λ. 1+ λ 1+ λ m m γ( y y ) + λ( i i ) = ( m m γ( y y )) + f When here is an error erm, using he formulaion in (20) magnifies he error because i includes an explanaory variable. λs as To be clear, his criique does no apply o genuine ou-of-sample forecas comparisons. A some level, i is obvious ha we canno simply rewrie he model and produce ou-of-sample forecass ha have arbirarily lower variance. If we could, we would no be wriing or reading his paper, and insead would be ou using his echnique o ge very rich. While we can rewrie he righ-hand-side of he model o arbirarily change he in-sample fi, i follows from equaion (22) ha our forecas of he righ-hand-side is he same no maer which way i is wrien. Finally, we noe ha i is ofen assered ha he forecas using he acual realized values of he explanaory variables mus produce beer forecass of he exchange rae han when he righ-hand-side variables mus be forecas. However, his is no in general rue if he explanaory fundamenal variables are correlaed wih he unobserved variables. Unless we ake his correlaion ino accoun, he fi could poenially be worse using he ex pos fundamenals. Our general poin, hen, is ha he Meese-Rogoff procedure of using realized values of he explanaory variables is no invarian o he way he model is wrien. Plausible ways of rewriing he model can give much lower mean-squared-errors for he model. Of course, we canno conclude ha if he model is no useful in forecasing exchange rae changes, we have suppor for he model. Any model can fail o forecas exchange raes. We urn now o alernaive means of assessing exchange-rae models. 17

19 2. Taylor-rule models Overview Meese and Rogoff originally suggesed he ou-of-sample fi crierion as a check on empirical sudies ha found good in-sample fi. Here, we reurn o examinaion of he in-sample fi, bu wih more aenion paid o he marke s expecaions of fuure values of he macroeconomic fundamenals. The moneary models ha we have explored so far have been formulaed in such a way ha he endogeneiy of moneary policy has been essenially compleely ignored. We have used money supply o capure he moneary fundamenal, and have focused on formulaions in which nominal ineres raes move o equilibrae money supply and money demand. We have no ried o relae movemens of he money supply o he macroeconomic variables ha policymakers migh arge. Bu modern moneary macroeconomic models formulae he deerminaion of ineres raes and moneary equilibrium quie differenly. Firs, hey emphasize he endogeneiy of moneary policy. Advanced counries have managed o sabilize inflaion and apparenly esablish moneary policy credibiliy over he pas weny or weny-five years. If our models of exchange raes are o capure expeced fuure fundamenals, we need o recognize ha marke forecass of he fuure incorporae heir assumpions abou moneary policy reacions o changes in he macro environmen. Second, since he mid-1980s, cenral banks have used shor-erm ineres raes as heir policy insrumen, raher han he money supply. Engel and Wes (2006) and Mark (2007) (hereinafer, EW06 and M07) specify he moneary policy rules for he home and foreign counry as ineres-rae reacion funcions for he cenral bank. Specifically, hey assume he home counry (in heir empirical sudies, he home counry is Germany, prior o he adopion of he euro) ses he nominal ineres rae o arge he deviaion of expeced inflaion from he cenral bank s arge, Eπ + 1 ; he oupu gap, y ; and, possibly, he deviaion of he nominal exchange rae from is purchasing power pariy value ha is, he real exchange rae, q. The laer em is included o capure he noion ha he moneary auhoriies in some counries end o raise ineres raes when heir currency depreciaes. For example, Clarida, Gali and Gerler (1998) find empirical suppor for his noion in Japan and some oher counries. We summarize he moneary policy rule in equaion (23): (23) i = γ qq + γ Eπ+ 1+ γ yy + δi 1+ um. π We assume γ q > 0, γ π > 1, γ y > 0, and 0 δ < 1. Here, represens an error or shif in he moneary policy rule. The foreign counry (aken o be he U.S. in EW06 and M07) follows a similar policy rule: u m 18

20 (24) i = γ Eπ + γ y + δi + u m. π + 1 y 1 Here, we assume ha he parameers on he inflaion deviaion and he oupu gap are he same in he home and foreign counry, bu assume ha he foreign cenral bank is passive wih respec o exchange rae flucuaions. Using he uncovered ineres pariy relaionship (including he u.i.p. deviaion, i i Es+ 1 s = + ρ, we can manipulae hese equaions o ge a forward-looking expression for he real exchange rae: j (25) q = b Ez+ j, j = 0 ρ ), where 1 b, + γ 1 q 1 z = ( γ π 1)( Eπ + 1 Eπ + 1) + γ y ( y y ) + δ( i 1 i 1) + ( um um ρ ) 1 γ. + q M07 s formulaion acually ses γ q = 0, because his esimaes of he Bundesbank s moneary policy reacion funcion yielded insignifican esimaes for his parameer. Wih ha change, we can sill represen he real exchange rae wih a presen value expression, bu he discoun facor, b, is equal o one. We can do ha if z has a zero (uncondiional) mean implying q has a zero mean, or long-run purchasing power pariy holds and saisfies a mild summabiliy condiion. Any ARMA process saisfies he condiion, hough we noe ha fracionally inegraed processes do no. In his bu no all conexs, here will be lile difference in empirical resuls ha impose γ = 0 and empirical resuls ha, consisen wih he esimaes in Clarida e al. (1998), impose small bu posiive values such as γ q = Through mos of he empirical work below, we assume ha in quarerly daa, γ q is a small posiive number, less han 0.10 bu sricly greaer han zero. The equaion for he real exchange rae given by equaion (25) does no solve a full general equilibrium model in erms of exogenous variables. The fundamenals inflaion and he oupu gap are deermined by underlying driving variables, such as produciviy disurbances and cos-push shocks. We noe, noneheless, ha equaion (25) has an ineresing implicaion ceeris paribus, an increase in home relaive o foreign inflaion leads o a home real appreciaion. This predicion sands in conras o he usual q 19

21 inerpreaion of he effec of an increase in expeced inflaion in he moneary models we invesigae above. In hose, an increase in expeced inflaion a home lowers home money demand, leading o a home depreciaion. Here, when γ > 1, and increase in expeced inflaion leads he home cenral bank o raise he home real ineres rae, leading o an appreciaion. π EW06 and M07 esimae he model as summarized by equaion (25). The presen-value model requires a measure of expeced inflaion, oupu gap, and ineres raes for all periods in he fuure. I also requires an esimae of fuure expeced values of u u ρ. The empirical researcher faces a severe m m problem in esimaing such a model, because he marke forms expecaions based on many sources of informaion ha are no measurable by he economerician. hey ignore EW06 and M07 handle esimaion of he expeced presen value sum, (25), in similar ways. Firs, u u ρ, reaing hese as unobservable deerminans of he real exchange rae. We m m now describe EW06 s mehodology, hen noe he differences beween EW06 and M07. Firs, EW06 do no include he lagged ineres rae in he moneary policy rule, so hey have se δ =0. EW06 hen mus measure E ( γπ 1)( π+ j+ 1 π+ j+ 1) + γ y( y+ j y+ j) for all j. EW06 do no esimae he parameers γ π and γ y -- insead hey base hem on esimaes of he Taylor rule on pos-1979 daa in Clarida, Gali, and Gerler (1998). EW06 also use Clarida e al. s esimae of γ q, from which he discoun facor, b, is calculaed. Then, EW06 esimae a VAR in π π, y y, and i i. From he VAR, expeced values of π π, and y y for all periods can be consruced, and hen he presen value can be calculaed. EW06 hen compare he model real exchange rae he value of he righ-hand-side of equaion (yy3), where expecaions are calculaed as jus described o he behavior of he acual deuschemark/dollar real exchange rae. EW06 esimae he model on pos-1979 daa, using monhly daa, 1979: :12. The correlaion of he model real exchange rae and acual real exchange rae is Tha is no exremely high, bu i is no oo bad, and represens a promising sar in his lieraure. We noe, however, ha he model real exchange rae esimaed by EW06 has a sandard deviaion abou one-fifh of ha of he acual real exchange rae. The approach of M07 differs from EW06 in a number of ways. Firs, M07 esimaes he parameers of he Taylor rule, for wo periods 1960:II-1979:II, and 1979:III-2003:IV, using quarerly daa. Second, as noed above, Mark does no include he real exchange rae in he ineres-rae rule for Germany. Third, 20

22 M07 does include he lagged ineres rae in he policy rule. Fourh, M07 compares he behavior of he model real exchange rae o he acual real exchange rae over a longer period, 1976:II-2003:IV. The correlaion of he model and acual real exchange rae in M07 is quie similar o ha in EW06, equal o (when he oupu gap is measured using deviaions he HP-filer). However, he model volailiy of he real exchange rae is much larger in M07, and more nearly maches ha of he real exchange rae. The variance of 1-quarer changes in he real exchange rae from he model is percen from he model, compared o percen in he daa. Purchasing power pariy puzzle In advanced economies, real and nominal exchange rae changes are highly correlaed. A plausible model of real exchange rae behavior mus accoun for his correlaion. For an inernaional macroeconomis, a model of nominal exchange raes ha canno be reconciled wih real exchange rae behavior is no appealing, and vice-versa. Some exising lieraure (as exemplified by Rogoff (1996)) argues ha boh sicky and flexible price models fail o replicae some imporan exchange rae characerisics. As explained below, such auhors argue ha flexible price models have a hard ime explaining volailiy of real exchange raes, while sicky price models have difficuly explaining persisence of real exchange raes. Bu recen work by Benigno (2004) and ohers shows ha wih suiable modeling of price sickiness and moneary policy, real exchange rae persisence can be plausibly explained as coming from persisence in ineres raes. I is possible o undersand he high correlaion of real and nominal exchange raes in an environmen in which nominal goods prices adjus quickly (no price sickiness), if we assume ha moneary auhoriies sabilize nominal prices. In ha case, we should hink of nominal exchange raes as being driven by underlying real shocks ha drive he real exchange rae. Tha is, since s = q + p p, if nominal prices are flexible bu and p are sabilized by moneary policy, hen movemens in s will be p highly correlaed wih hose in q. However, as Rogoff (1996) emphasized, while flexible-price models can accoun for he exreme persisence of real exchange raes ha we see among advanced counries, hey are unable o explain he high volailiy. Alernaively, we migh consider models in which and have low volailiy (ha is, low innovaion variance) a leas in par because of nominal price sickiness. In models wih price sickiness, Rogoff noes, we have beer explanaions of real exchange rae volailiy. Specifically, we can appeal o he Dornbusch overshooing model (ha we have described in Secion 1). Moneary shocks cause volaile p p 21