Testing the monetary model of exchange rate determination: new evidence from a century of data

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1 Journal of Inernaional Economics 58 () locae/ econbase Tesing he moneary model of exchange rae deerminaion: new evidence from a cenury of daa David E. Rapach *, Mark E. Wohar a, b a Albers School of Business and Economics, Seale Universiy, 9 Broadway, Seale, WA , USA b Deparmen of Economics, Universiy of Nebraska a Omaha, CBA 51K, Omaha, NE , USA Received 4 July 1; received in revised form 1 Ocober 1; acceped 1 November 1 Absrac We es he long-run moneary model of exchange rae deerminaion for a collecion of 14 indusrialized counries using daa spanning he lae nineeenh or early wenieh cenury o he lae wenieh cenury. Ineresingly, we find suppor for a simple form of he long-run moneary model in over half of he counries we consider. For hese counries, we esimae vecor error-correcion models o invesigae he adjusmen process o he long-run moneary equilibrium. In he spiri of Meese and Rogoff [Journal of Inernaional Economics 14 (1983) 3 4], we also compare nominal exchange rae forecass based on moneary fundamenals o hose based on a naıve random walk model. Elsevier Science B.V. All righs reserved. Keywords: Nominal exchange rae; Moneary model; Coinegraion; Forecasing JEL classificaion: C; C3; C53; F31; F47 1. Inroducion The moneary model of exchange rae deerminaion posis a srong link beween he nominal exchange rae and a simple se of moneary fundamenals. *Corresponding auhor. Tel.: ; fax: addresses: rapachd@sealeu.edu (D.E. Rapach), wohar@unomaha.edu (M.E. Wohar) //$ see fron maer Elsevier Science B.V. All righs reserved. PII: S-1996(1)17-

2 36 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () The moneary model s clear-cu inuiion ha a counry s price level is deermined by is supply and demand for money and ha he price level in differen counries should be he same when expressed in he same currency makes i an aracive heoreical ool for undersanding flucuaions in exchange raes over ime. I also provides a long-run benchmark for he nominal exchange beween wo currencies and hus a clear crierion for deermining wheher a currency is significanly overvalued or undervalued. Despie is heoreical appeal, he moneary model did no escape he Meese and Rogoff (1983) rap ha seemingly ensnared all models of exchange rae de- erminaion. In heir seminal paper, Meese and Rogoff (1983) find ha a naıve random walk model ouperforms an array of srucural models, including hose based on moneary fundamenals, in predicing U.S. dollar exchange raes a horizons of up o 1onhs during he lae 197s and early 198s. Mark (1995) rekindled hope for he moneary model by showing ha deviaions from a simple se of moneary fundamenals relaive money supplies and relaive real oupu levels are useful in predicing U.S. dollar exchange raes a longer horizons over 1 he period. However, Berben and van Dijk (1998) and Berkowiz and Giorgianni (1) show ha Mark s (1995) findings hinge criically on he assumpion of a sable coinegraing relaionship among nominal exchange raes, relaive money supplies, and relaive oupu levels. When his assumpion is relaxed, he evidence for exchange-rae predicabiliy in Mark (1995) is grealy diminished, and, in fac, Mark (1995) fails o find evidence of coinegraion among nominal exchange raes and moneary fundamenals in preliminary esing. A number of oher sudies also find lile evidence of coinegraion among nominal exchange raes and moneary fundamenals during he pos-breon Woods floa; see, for example, Meese (1986), Baillie and Selover (1987), McNown and Wallace 3 (1989), Baillie and Pecchenino (1991), and Saranis (1994). The lack of empirical evidence for a sable long-run relaionship among nominal exchange raes and moneary fundamenals renders he moneary model a seemingly plausible heoreical model wih lile pracical relevance. A ready explanaion for he failure o find coinegraion beween nominal exchange raes and moneary fundamenals in much of he exan lieraure is he 1 Chinn and Meese (1995) also find ha moneary fundamenals are helpful in predicing U.S. dollar exchange raes over he period. Chinn and Meese (1995) also fail o find srong evidence of coinegraion among nominal exchange raes and moneary fundamenals. 3 MacDonald and Taylor (1994) find evidence of coinegraion beween he U.S. dollar U.K. pound exchange rae and a se of moneary fundamenals from 1976 o 199, bu heir coinegraing vecor is difficul o inerpre heoreically. They only claim ha i does no, in fac, do grea violence o he moneary model. Cushman () finds evidence of coinegraion beween he U.S. dollar Canadian dollar exchange rae and a se of moneary fundamenals during he modern floa, bu he esimaed coinegraing coefficiens differ widely from hose prediced by he moneary model. Cushman () hus concludes ha he U.S. Canadian daa do no suppor he moneary model.

3 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () relaively shor spans of daa ypically employed, which cover only he pos- Breon Woods floa. Sandard ess ake no coinegraion as he null hypohesis, and he power o rejec his null is exremely low using daa from he pos-breon Woods period alone, which spans 5 years or less. I does no help ha he daa are ofen sampled a monhly or quarerly frequencies, as he power of uni roo and coinegraion ess depends on he daa s span, raher han is frequency (Shiller and Perron, 1985; Hakkio and Rush, 1991). A similar siuaion exiss in he empirical purchasing power pariy (PPP) lieraure. Long-run PPP posis a sable long-run relaionship beween nominal exchange raes and relaive price levels, bu 4 empirical suppor for such a relaionship is scan using daa from he modern floa. Again, his can be aribued o he low power of sandard ess for samples as shor as he modern floa. Given ha PPP is a building block of he moneary model, i is no surprising ha i is difficul o find evidence of coinegraion beween nominal exchange raes and moneary fundamenals during he modern floa. Two responses o he problem of low power appear in he PPP lieraure. Firs, a number of sudies employ panels from he pos-breon Woods floa. As iniially shown by Levin and Lin (199), panel echniques can grealy improve he power of uni roo and coinegraion ess. Indeed, many panel sudies find suppor for long-run PPP for he pos-breon Woods era, including Frankel and Rose (1996), Oh (1996), Wu (1996), Papell (1997), and Taylor and Sarno (1998). The second response o low power in he PPP lieraure is he use of long spans of daa, ofen covering more han a cenury. For example, Abuaf and Jorion (199), Glen (199), Lohian and Taylor (1996, ), and Taylor (1a) all use long spans of daa o es long-run PPP. These sudies also find considerable suppor for long-run PPP. Boh he panel and long spanning daa sudies show ha deviaions from long-run PPP are quie persisen and display near-uni-roo behavior, precisely he ype of saionary behavior ha will be difficul for sandard single-counry ess o deec for samples as shor as he modern floa. In regard o he moneary model, wo recen sudies by Groen () and Mark and Sul (1) follow he firs response in he PPP lieraure and es for a sable long-run relaionship beween nominal exchange raes and moneary fundamenals using panel coinegraion ess for he pos-breon Woods floa. Ineresingly, hese sudies boh find srong evidence of coinegraion among nominal exchange raes, relaive money supplies, and relaive real oupu levels using panel coinegraion ess. Mark and Sul (1) acually find suppor for a very simple long-run moneary model ha imposes basic homogeneiy resricions. They also find ha nominal exchange rae forecass based on he moneary model are generally superior o forecass of a naıve random walk model. Given ha he main criicisms of Mark (1995) are based on he lack of coinegraion among nominal 4 See Rogoff (1996) and Sarno and Taylor (1) for recen surveys of he PPP lieraure.

4 36 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () exchange raes and moneary fundamenals, he recen findings of Groen () and Mark and Sul (1) again rekindle hope in he abiliy of moneary fundamenals o rack nominal exchange raes. While Groen () and Mark and Sul (1) follow he firs response in he PPP lieraure and use panel daa from he modern floa, no sudy pursues he second response in he PPP lieraure and ess he moneary model using long spans of daa. In his paper, we pursue his second response. Jus as Groen () and Mark and Sul (1) es he moneary model in a panel framework, moivaed by he findings of PPP in panel sudies, we es he moneary model using long spans of daa, moivaed by he findings of PPP in sudies uilizing long spans of daa. In paricular, we apply a baery of uni roo and coinegraion ess o annual daa daing back o he lae nineeenh or early wenieh cenury for 14 indusrialized counries in order o es he long-run moneary model of exchange rae deerminaion. By using long spans of daa, we are able o side-sep some of he problems ha poenially plague panel-esing procedures. Of paricular concern is he possibiliy of concluding ha all counries in a panel saisfy he long-run moneary model when, in fac, some individual counries are no well 5 characerized by he moneary model. Our esimaion resuls exhibi considerable suppor for a simple long-run moneary model of U.S. dollar exchange rae deerminaion for France, Ialy, he Neherlands, and Spain; moderae suppor for Belgium, Finland, and Porugal; and weaker suppor for Swizerland. For hese eigh counries, we hus find a leas some evidence of a heoreically consisen long-run link beween nominal exchange raes and a simple se of moneary fundamenals. Along wih Groen () and Mark and Sul (1), our findings are noeworhy given he lack of empirical suppor in much of he exan lieraure for he long-run relaionship among exchange raes and moneary fundamenals implied by he moneary model. In conras, we fail o find suppor for he long-run moneary model for Ausralia, Canada, Denmark, Norway, Sweden, and he Unied Kingdom using long spans of daa. For he counries for which we find suppor for he simple long-run moneary model, we consider wo addiional opics. Firs, we esimae vecor-error correcion models for nominal exchange raes and moneary fundamenals in order o es for weak exogeneiy. This gives us insigh ino he adjusmen process hrough which he long-run equilibrium relaionship beween exchange raes and moneary fundamenals is mainained. Second, in he spiri of Meese and Rogoff (1983) and 5 Of course, here is he poenial problem of srucural insabiliy when using long spans of daa. We follow he PPP lieraure ha uilizes long spans of daa and assume ha he dynamics are relaively sable over he sample period. In order o employ more powerful ess of PPP or he long-run moneary model, we have o assume eiher a subsanial degree of homogeneiy across counries (in order o employ panel ess) or relaively sable dynamic processes over long periods (in order o employ long spans of daa).

5 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () Mark (1995), we compare ou-of-sample exchange rae forecass from a naıve random walk model wih forecass based on moneary fundamenals. In line wih he recen work of Berben and van Dijk (1998) and Berkowiz and Giorgianni (1), we find ha here is a close connecion beween he ou-of-sample forecas performance of he moneary model and he weak exogeneiy es resuls. The res of he paper is organized as follows. Secion presens a simple heoreical moneary model and oulines our esing sraegy. Secion 3 repors our es resuls for he long-run moneary model, including uni roo and coinegraion ess. Secion 4 analyzes error-correcion models suggesed by our coinegraion es resuls. Secion 5 compares ou-of-sample forecass of nominal exchange raes based on moneary fundamenals wih hose of a naıve random walk model. Secion 6 summarizes our main findings.. Theoreical framework and esing sraegy A number of relaionships underlie he basic varian of he moneary model. We emphasize ha we have in mind a long-run equilibrium relaionship. Firs, sable 6 money demand funcions are assumed for he domesic and foreign counries: m p5 a1i 1 a, (1) m* p* 5 a1i* 1 a *, () where m is he money supply, p is he price level, i is he nominal ineres rae, and y is real oupu (all a ime ). Wih he excepion of he nominal ineres rae, lower-case leers denoe log-levels. Aserisks denoe a foreign variable. Noe ha he money demand parameers, a and a (a, and a. ), are assumed o be 1 1 idenical in he domesic and foreign counries. In our empirical work below, he U.S. serves as he domesic counry. Second, purchasing power pariy is assumed: e5 p* p, (3) where e is he nominal exchange rae measured in he number of unis of foreign currency per uni of domesic currency. Solving (1) and () for p and p* and subsiuing he resuling expressions ino (3) yields e5 (m * ) a 1(i * i ) a ( y* ). Finally, he moneary model ypically assumes uncovered ineres pariy: i* i5 E(De11 u I ), where E(? u I ) is he expecaions operaor condiional on informaion available a 6 Consan erms are suppressed for exposiional convenience.

6 364 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () ime. Ife is I() or I(1), hen De11 will be equal o zero in he seady sae (absracing away from any deerminisic rend growh in e ), so ha i* 5 i. This leaves e5 (m * ) a ( y* ). (4) Eq. (4) is a basic form of he moneary model ha esablishes a long-run relaionship beween he nominal exchange rae and a simple se of moneary fundamenals. Mark and Sul (1, p. 3) emphasize ha (4) can be viewed as a generic represenaion of he long-run equilibrium exchange rae implied by modern heories of exchange rae deerminaion, as a relaionship like (4) can be also derived from he Lucas (198) and Obsfeld and Rogoff (1995) equilibrium models. Mark (1995) and Mark and Sul (1) impose he addiional resricion ha a 5 1 in (4), yielding he simple form of he moneary model: e5 (m * ) ( y* ). Tesing he long-run moneary model enails esing for he exisence of a sable long-run relaionship among e, m*, and y*, or, equivalenly, esing wheher deviaions of e from a linear combinaion of m* and y* are saionary. Our firs sep in esing he basic long-run moneary model is hus o examine he inegraion properies of e, m*, and y* using he uni roo ess from Ng and Perron (), which have goods size and power properies. If e I(), hen m* and y* mus also boh be I() in order for he nominal 8 exchange rae deviaions o be I(). In fac, if e, m*, y* I(), his is sufficien o esablish he saionariy of nominal exchange rae deviaions from any linear combinaion of he relaive money supply and relaive oupu level. If e I(1), a necessary condiion for he long-run moneary model is ha one of, or boh of, m* and y* also be I(1) (and neiher can be inegraed of an order greaer han one). When e, m*, y* I(1), he long-run moneary model requires hese hree variables o be coinegraed, and so we esimae he following coinegraing relaionship: e 5 b1 b 1(m * ) 1 b ( y* ). (5) We esimae (5) using OLS, fully modified OLS (Phillips and Hansen, 199; FM-OLS), dynamic OLS (Saikkonen, 1991; Sock and Wason, 1993; DOLS), and he mulivariae maximum likelihood procedure of Johansen (1988, 1991; JOH- ML). As is now well known, OLS esimaes of b1 and b in (5) are superconsisen. However, hey are no asympoically efficien, and he OLS covariance marix for he esimaed coefficiens is inappropriae for inference, as i is 7 In our uni roo es resuls repored below, we find ha e is eiher I() or I(1) for all of he counries we consider. 8 An excepion is if e I() bu m*, y* CI(1,1).

7 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () asympoically biased. In conras, he FM-OLS, DOLS, and JOH-ML esimaes are asympoically efficien and yield covariance marices appropriae for inference. We es for coinegraion among e, m*, and y* using he Phillips and Ouliaris (199), Hansen (199), and Shin (1994) single-equaion procedures, as well as he Johansen (1988, 1991) sysem-based procedure, which are based on he OLS, FM-OLS, DOLS, and JOH-ML esimaes, respecively. In addiion, we es he simple form of he moneary model ha implies b15 1 and b51by esing he saionariy of e [(m * ) ( y* )] using he same uni roo ess ha we use for he individual series, as well as he Horvah and Wason (1995) es for coinegraion wih a pre-specified coinegraing vecor. Noe ha, for a few counries, our uni roo es resuls indicae ha e, m* I(1), while y* I(). For hese counries, we proceed wih he coinegraion analysis as described above bu wih b Moneary model es resuls 3.1. Daa The daa used in his sudy consis of annual observaions for he nominal exchange rae (foreign currency per U.S. dollar), he money supply relaive o he U.S., and real GDP relaive o he U.S. for 14 counries: Ausralia, Belgium, Canada, Denmark, Finland, France, Ialy, he Neherlands, Norway, Porugal, Spain, Sweden, Swizerland, and he Unied Kingdom. The nominal exchange rae series are from Taylor (1a), and he money supply and real GDP series are from Bordo and Jonung (1998) and Bordo e al. (1998). The counries considered are deermined by daa availabiliy. The daa run from he lae nineeenh or early wenieh cenury o he lae wenieh cenury and hus cover a variey of inernaional moneary arrangemens, including he classical gold sandard, he Breon Woods era, and he modern floa. The exac sample period for each 9 counry is repored in he ables below. All variables are measured in log-levels. 3.. Uni roo es resuls For he 14 counries considered, we firs invesigae he inegraion properies of e, m*, and y* using he Ng and Perron () DF-GLS and MZa uni roo ess, which are varians of he well-known Dickey and Fuller (1979) and Phillips and Perron (1988) ess, respecively. Boh of hese ess use GLSderending (as in Ellio e al., 1996) in order o maximize power and a modified 9 For some counries, daa are missing for cerain series for a few warime years. We follow Taylor (1a) and fill in he missing daa using linear inerpolaion. We do no include Germany and Japan, as here are a large number of missing observaions for hese counries.

8 366 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () informaion crierion o selec he lag runcaion parameer in an effor o minimize size disorions. Ng and Perron () find ha he DF-GLS and MZa saisics have good size and power properies in exensive Mone Carlo simulaions. 1 Table 1 repors he resuls for he DF-GLS and MZa ess for our daa. Columns (1) and (5) of Table 1 show he counry, ime period, variable esed, and wheher a linear rend was included in he uni roo ess. The inclusion of a linear rend is indicaed 11 by visual inspecion of he series, as well as formal saisical ess. Because differen ess yield conradicory resuls on a few occasions, we designae he variables in Table 1 as I(1), I(), or I(1) or I() according o he following simple decision rule. We designae a variable as I() if boh of he ess rejec he null hypohesis of nonsaionariy a convenional significance levels or if a leas one es rejecs a he 5 percen significance level. If neiher es rejecs he null hypohesis of nonsaionariy a convenional significance levels, we designae he variable as I(1). Finally, if only one es rejecs a he 1 percen level, we designae he series as I(1) or I(). Based on he uni roo es resuls in Table 1, we conclude ha all hree of he variables, e, m*, and y*, are I(1) for Ausralia, Belgium, France, Ialy, Spain, and he Unied Kingdom. All hree variables are found o be I() for he Neherlands. For Finland and Porugal, we find ha e and m* are each I(1), while y* I(). For Denmark and Norway, we find ha e I(), while m* I(1) (y* is inconclusive for Denmark and y* I() for Norway). For Sweden, our es resuls indicae ha e I() and m*, y* y I(1). Finally, for Canada and Swizerland, our uni roo es resuls are 1 inconclusive for e, while hey indicae ha m*, y* I(1). Nex, we discuss he implicaions of our uni roo es resuls for esing he long-run moneary model using coinegraion procedures. For he Neherlands, all hree variables are saionary, so we conclude on he basis of he uni roo es resuls alone ha deviaions of e from any linear combinaion of m* and y* are saionary for he Neherlands. For Ausralia, Belgium, France, Ialy, Spain, and he Unied Kingdom, e, m*, and y* are all I(1). In he nex subsecion, we hus proceed o es for a coinegraing relaionship among hese hree variables, as required by he long-run moneary model. Finland and Porugal appear o be an inermediae case, wih he nominal exchange rae and he relaive money supply being I(1), bu wih he relaive oupu level being saionary. For hese counries, he long-run moneary model requires a coinegraing relaionship 1 In erms of relaive performance, he DF-GLS saisic appears more powerful, while he MZa es has beer size properies. We use he GAUSS program available from Serena Ng s home page (hp:// ngse/ research.hml) o generae he DF-GLS and MZa saisics. 11 All of our specificaions include a consan erm. 1 Addiional uni roo es resuls (no repored o conserve space) overwhelmingly indicae ha he firs differences for every variable and every counry are saionary, so ha no variable for any counry appears I().

9 Table 1 Uni roo es resuls (1) () (3) (4) (5) (6) (7) (8) Variable a DF-GLS b MZa Decision Variable DF-GLS MZa Decision Ausralia ( ) Neherlands (19 199) e (rend) I(1) e * I() m* I(1) m*.81** 13.66* I() y* I(1) y* y.1* 1.17* I() Belgium ( ) Norway ( ) e (rend) I(1) e (rend).97* 18.83* I() m* I(1) m* (rend) I(1) y* (rend) I(1) y* y.1* 8.73* I() Canada ( ) Porugal ( ) e (rend) I(1) or I() e (rend) I(1) m* (rend) I(1) m* (rend) I(1) y* (rend) I(1) y* y.7* 3.3 I() Denmark ( ) Spain ( ) e (rend) 3.14* 1.93* I() e (rend) I(1) m* I(1) m* (rend) I(1) y* (rend) I(1) or I() y* y (rend) I(1) Finland ( ) Sweden ( ) e (rend) I(1) e (rend) * I() m* I(1) m* I(1) y* (rend) 3.77**.73* I() y* y (rend) I(1) France ( ) Swizerland ( ) e (rend) I(1) e (rend) I(1) or I() m* (rend) I(1) m* (rend) I(1) y* (rend) I(1) y* y (rend) I(1) Ialy ( ) Unied Kingdom ( ) e (rend) I(1) e (rend) I(1) m* (rend) I(1) m* (rend).4.45 I(1) y* I(1) y* y (rend) I(1),*,** indicae significance a he 1, 5, and 1 percen levels, respecively; (rend) indicaes ha he es allows for saionariy around a linear rend. a Ng and Perron () one-sided (lower-ail) es of H : Nonsaionariy; 1, 5, and 1 percen criical values equal 1.6, 1.98, and.58, respecively; when a linear rend is included, 1, 5, and 1 percen criical values equal.6,.91, and 3.4, respecively. b Ng and Perron () one-sided (lower-ail) es of H : Nonsaionariy; 1, 5, and 1 percen criical values equal 5.7, 8.1, and 13.8, respecively; when a linear rend is included, 1, 5, and 1 percen criical values equal 14., 17.3, and 3.8, respecively. D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 ()

10 368 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () beween only e and m*, as here are no long-run changes in y*. In he nex subsecion, we hus es for coinegraion beween he nominal exchange rae and he relaive money supply for Finland and Porugal. For Denmark, Norway, and Sweden, e is saionary, bu m* I(1). We can hus conclude on he basis of he uni roo es resuls alone ha he long-run moneary model does no 13 hold in hese counries over our sample. For Canada and Swizerland, i is difficul o ell wheher e is I(1) or I(). If e I(), we have direc evidence agains he long-run moneary model, as i requires e I(1) if m*, y* I(1). For hese wo counries, we give he moneary model a chance and es for coinegraion among e, m*, and y* under he assumpion ha e I(1) Coinegraion es resuls We repor coinegraing coefficien esimaes for 1 counries in Table (excluding he Neherlands, Denmark, Norway, and Sweden on he basis of he uni roo es resuls in Table 1). Column (1) of Table gives he counry, sample 14 period, and wheher a rend is included in he coinegraing vecor. Based on he uni roo es resuls in Table 1, we esimae he coinegraing relaionship e 5 b1 b 1(m * ) 1 b ( y* ) for Ausralia, Belgium, Canada, France, Ialy, Spain, Swizerland, and he Unied Kingdom, while we esimae he coinegraing relaionship e 5 b1 b 1(m * ) for Finland and Porugal. Table includes OLS, FM-OLS, DOLS, and JOH-ML esimaes. Following he applicaions in Hansen (199), we use he quadraic kernel and he Andrews (1991) auomaic bandwidh selecor wih Andrews and 15 Monohan (199) prewhiening when compuing he FM-OLS esimaes. Following Sock and Wason (1993), we se he number of leads and lags in he DOLS esimaor equal o wo, and we use an auoregressive procedure o compue robus sandard errors. We also repor a Sock and Wason (1993) Wald es (SW-Wald) of he join hypohesis ha b15 1 and b51, as implied by he simple moneary model ha ses he common income elasiciy of money demand o uniy (see 13 As noed above, an excepion is when e I() bu m*, y* CI(1,1). We do no find any evidence of his. 14 We include a rend in he coinegraing vecor if he deviaions of he exchange rae from he moneary fundamenals exhibi a srong rend ha is confirmed by formal saisical ess of he significance of he linear rend in he coinegraing vecor. Noe ha for he JOH-ML esimaes in Table and he race es in Table 3 for France and Swizerland, we resric he linear rend o appear only in he coinegraing vecor (case * in Oserwald-Lenum, 199). A linear rend in he coinegraing vecor allows for a deerminisic Balassa Samuelson effec in real exchange raes. However, he inclusion of a linear rend could be viewed as a weaker form of he long-run moneary model. 15 We use he GAUSS program available from Bruce Hansen s home page (hp:// bhansen/ ) o generae he FM-OLS esimaes.

11 Table Coinegraing coefficien esimaes, e 5 b 1 b (m* ) 1 b ( y* ) 1 (1) () (3) (4) (5) (6) (7) (8) (9) (1) (11) OLS esimaes FM-OLS esimaes DOLS esimaes JOH-ML esimaes Counry a b1 b b1 b b1 b SW-Wald b1 b JOH-x Ausralia ( ) (.9) (.1) (.9) (.63) (.35) (.73) (.3) (.63) Belgium ( ) (.9) (.8) (.38) (.34) (.5) (.3) (.5) (.3) Canada ** ** ( ) (.3) (.4) (.13) (.15) (.8) (.8) (.8) (.8) Finland ( ) (.3) (.13) (.7) (.11) France (rend) ( ) (.4) (.1) (.11) (.5) (.1) (.31) (.14) (.54) Ialy ** ( ) (.1) (.9) (.3) (.) (.3) (.39) (.) (.5) Porugal ( ) (.4) (.9) (.6) (.9) Spain ** ** ( ) (.3) (.9) (.6) (.) (.5) (.16) (.4) (.14) Swizerland (rend) ** ( ) (.15) (.15) (.55) (.53) (.46) (.43) (.5) (.61) Unied Kingdom ** ** ( ) (.4) (.4) (.14) (.16) (.19) (.4) (.13) (.16),*,** indicae significance a he 1, 5, and 1 percen levels, respecively; for Finland and Porugal, b is consrained o be equal o zero; sandard errors for he coefficien esimaes are given in parenheses; (rend) indicaes ha a linear rend is included in he coinegraing relaionship. a Sock and Wason (1993) one-sided (upper-ail) es of H : b15 1, b51; 1, 5, and 1 percen criical values for a x () equal 4.61, 5.99, and 9.1, respecively; for Finland and Porugal, he es is for H : b15 1; for Finland and Porugal, he 1, 5, and 1 percen criical values for a x (1) equal.71, 3.84, and 6.64, respecively. b Johansen (1991) one-sided (upper-ail) es of H : b15 1, b51; 1, 5, and 1 percen criical values for a x () equal 4.61, 5.99, and 9.1, respecively; for Finland and Porugal, he es is for H : b 5 1; for Finland and Porugal, he 1, 5, and 1 percen criical values for a x (1) equal.71, 3.84, and 6.64, respecively. 1 b D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 ()

12 37 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () column (8) of Table ). We selec he lag order for he JOH-ML esimaor by sequenially esing a VAR in levels using he Sims (198) modified likelihood- 16 raio saisic, a maximum lag order of five, and he 1 percen significance level. We also presen a x es due o Johansen (1991), labeled JOH-x, ha we use o es he join null hypohesis ha b15 1 and b51 in he coinegraing vecor (see column (11) of Table ). For Belgium, Ialy, and Spain, all four esimaion procedures generally yield parameer esimaes close o he heoreical values implied by he simple moneary 17 model ( b1 5 1 and b 5 1). The coinegraing coefficien esimaes for Belgium are very close o hose implied by he simple moneary model, and he SW-Wald and JOH-x ess canno rejec he join resricion ha b1 5 1 and b 51 for Belgium. For Ialy, he SW-Wald es also canno rejec he join resricion ha b15 1 and b51. For Spain, he join resricion is rejeced by boh he SW-Wald and JOH-x ess, apparenly due o b1 esimaes ha are significanly less han one. While less han one in saisical erms, he b1 esimaes are sill relaively close o heir heoreical value of uniy in magniude. On he whole, he esimaed coinegraing relaionships for Belgium, Ialy, and Spain are consisen wih he simple long-run moneary model. The OLS, FM-OLS, and DOLS coefficien esimaes for France are also very close o hose implied by he simple moneary model, and he SW-Wald es canno rejec he null hypohesis ha b 5 1 and b 51. The JOH-ML esimaor for 1 France yields a b esimae very close o uniy, and while he esimae for b has 1 18 he correc sign, i is quie small in magniude and saisically insignifican. Phillips (1994) provides a possible explanaion for he discrepancy beween he JOH-ML b esimae and he FM-OLS and DOLS b esimaes. Phillips (1994) shows ha coinegraion coefficien esimaes based on reduced rank regressions (such as JOH-ML) can have Cauchy-like ails and no finie ineger momens in finie samples, so ha ouliers can be expeced o occur more frequenly han for oher asympoically efficien esimaors such as he FM-OLS and DOLS 19 esimaors. The JOH-ML b esimae appears o be an oulier for France. Turning o he resuls for Swizerland in Table, we see some suppor for he moneary model. The FM-OLS esimaes are reasonably close, and he DOLS esimaes are very close, o he values implied by he simple moneary model. Using he SW-Wald es, we canno rejec he null hypohesis ha b1 5 1 and 16 We obain he following lag orders for he VAR in levels: Ausralia, 5; Belgium, 4; Canada, ; Denmark, 5; Finland, 4; France, ; Ialy, 3; Norway, 3; Porugal, 4; Spain, 4; Sweden, ; Swizerland, ; Unied Kingdom,. Box Ljung Q-saisics give no indicaion of serial correlaion in any of he VAR equaions for hese lag orders. 17 Recall ha he OLS sandard errors canno be used for valid inference. 18 Neverheless, he JOH-x es canno rejec he join resricion ha b1 5 1 and b 51 for France. 19 In Mone Carlo experimens, Sock and Wason (1993) also find ha ouliers can be expeced o occur more ofen for he JOH-ML esimaor han he FM-OLS and DOLS esimaors.

13 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () b 51. However, he same null is rejeced using he JOH-x es, apparenly due o he JOH-ML esimae of b. As wih France, he resuls in Phillips (1994) sugges ha he DOLS esimae of b and he SW-Wald es are more reliable han he JOH-ML esimae and he JOH-x es. There is lile suppor in he coinegraing coefficien esimaes for he moneary model for Ausralia, Canada, and he Unied Kingdom. For Canada, all four esimaion procedures yield esimaed b coefficiens ha are of he wrong sign and saisically insignifican. While he coefficien esimaes for Ausralia have he correc sign, hey are ypically insignifican, and while he coefficien esimaes are he correc sign for he Unied Kingdom, he b1 esimaes are all more han wo sandard errors below heir prediced heoreical value of uniy. Recall ha we consider he coinegraing relaionship e 5 b1 b 1(m * ) for Finland and Porugal, as i appears ha y* I() in hese counries. For Finland, hree of he four b1 esimaes are close o heir prediced value of uniy, 1 while all four b esimaes are close o heir prediced value of uniy for Porugal. 1 Overall, we are able o obain coinegraing coefficiens esimaes consisen wih he long-run moneary model in Table for Belgium, Finland, France, Ialy, Porugal, Spain, and Swizerland. The nex sep is o deermine if a coinegraing relaionship exiss beween he nominal exchange rae and he moneary fun- damenals in hese seven counries. Table 3 repors he resuls from four differen coinegraion ess. The firs is he well-known Phillips and Ouliaris (199) es (PO-Z ) ha ess he saionariy of a he OLS residuals using a Phillips and Perron (1988)-ype procedure. We use he quadraic specral kernel and he Andrews (1991) auomaic bandwidh selecor wih prewhiening when compuing he semi-parameric adjusmen for he PO-Z a saisic. We also repor resuls for he popular Johansen (1988, 1991) race es. 3 The PO-Za and race ess boh ake no coinegraion as he null hypohesis and coinegraion as he alernaive hypohesis. We consider wo addiional ess, due o Hansen (199) and Shin (1994), ha es he null hypohesis of coinegraion agains he alernaive of no coinegraion. If we ake he moneary model as our Using daa from he modern floa, Cushman () also finds ha esimaed coinegraing coefficiens for Canada do no accord wih he moneary model. 1 Noe ha we use a longer sample for Porugal in Table han in Table 1. Real oupu daa is only available beginning in 199 for Porugal, so we use a sample beginning in 199 for Porugal in Table 1. We use a sample beginning in 189 for Porugal in Table, as nominal exchange rae and money supply daa are available beginning in 189 for Porugal. The uni roo es resuls repored in Table 1 for he nominal exchange rae and relaive money supply for Porugal are qualiaively unchanged if we use a sample beginning in 189. In he working paper aneceden o he presen paper, we es he sabiliy of he coinegraing vecors and find lile evidence of srucural change for Belgium, Finland, Ialy, Porugal, Spain, and Swizerland. There is more evidence for France. 3 Given he well-documened poenial for size disorions when using asympoic criical values for he race es (see, for example, Cheung and Lai, 1993), we also calculaed boosrapped p-values. The unrepored resuls indicae ha inferences for he race es are largely unchanged.

14 Table 3 Coinegraion es resuls, e 5 b 1 b (m* ) 1 b ( y* ) 1 (1) () (3) (4) (5) (6) (7) (8) (9) (1) a b c d Counry PO-Za Trace Lc Cm Counry PO-Za Trace Lc Cm Belgium Porugal ** ( ) ( ) Finland 3.89* 16.6*.1.4 Spain **.7.8 ( ) ( ) France (rend) Swizerland (rend) * ( ) ( ) Ialy 33.36* 53.6**.17.9 ( ),*,** indicae significance a he 1, 5, and 1 percen levels, respecively; for Finland and Porugal, b is consrained o be equal o zero; (rend) indicaes ha a linear rend is included in he coinegraing relaionship. a Phillips and Ouliaris (199) one-sided (lower-ail) es of H : No coinegraion; 1, 5, and 1 percen criical values equal.7, 6.7, and 35., respecively; for Finland and Porugal, he 1, 5, and 1 percen criical values equal 17.1,.6, and 8.3, respecively; for France, Sweden, and Swizerland, he 1, 5, and 1 percen criical values equal 8.5, 3.8, and 4., respecively. b Johansen (1991) one-sided (upper-ail) es of H : No coinegraion; Oserwald-Lenum (199) 1, 5, and 1 percen criical values equal 6.79, 9.68, and 35.65, respecively; for Finland and Porugal, he 1, 5, and 1 percen criical values equal 13.33, 15.41, and.4, respecively; for France and Swizerland, he 1, 5, and 1 percen criical values equal 39.6, 4.44, and 48.45, respecively. c Hansen (199) one-sided (upper-ail) es of H : Coinegraion; significance is based on p-values repored in Hansen (199). d Shin (1994) one-sided (upper-ail) es of H : Coinegraion; 1, 5, and 1 percen criical values equal.163,.1, and.38, respecively; for Finland and Porugal, he 1, 5, and 1 percen criical values equal.31,.314, and.533, respecively; for France, Sweden, and Swizerland, he 1, 5, and 1 percen criical values equal.81,.11, and.15, respecively. 37 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 ()

15 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () mainained hypohesis, he null of coinegraion may be more appropriae han he null of no coinegraion. The Hansen (199) Lc saisic is based on he FM-OLS residuals, while he Shin (1994) Cm saisic is consruced from he DOLS 4 residuals. For France, Ialy, and Spain, all four ess indicae he exisence of a coinegraing relaionship a convenional significance levels, and coinegraion is indicaed by hree of he four ess for Finland and Porugal. The Lc saisic poins o coinegraion for Belgium and Swizerland. As he esimaed coinegraing coefficiens are close o b15 1 and b51in Table for Belgium, Finland, France, Ialy, Porugal, Spain, and Swizerland, a complemenary es of he simple long-run moneary model for hese counries is o es wheher e [(m* ) ( y* )], he deviaion of he exchange rae from he level prediced by he simple moneary model, is saionary. This is anamoun o esing for coinegraion beween he exchange rae and he moneary fundamenals wih pre-specified coinegraing coefficiens of b 5 1 and 1 b 51. We es he saionariy of he deviaions using he wo uni roo ess used for he individual series in Table 1, and he resuls are repored in Table 4. Table 4 also repors resuls for he mulivariae Horvah and Wason (1995) es of he null hypohesis of no coinegraion agains he alernaive hypohesis of coinegraion wih pre-specified coinegraing coefficiens b1 5 1 and b 51. As discussed in Horvah and Wason (1995), his es is poenially more powerful han univariae uni roo ess when esing for coinegraion wih a known coinegraing vecor. Noe ha we consider he deviaion e (m * ) for Finland and Porugal, in line wih he previous resuls. Also noe ha we include he deviaion e [(m * ) ( y* )] for he Neherlands in Table 4. The uni roo ess in Table 1 clearly indicae ha each componen of he deviaion is saionary for he Neherlands, and so we expec he deviaion o be saionary. We include he Neherlands as a robusness check of he Table 1 resuls. For Belgium, Ialy, he Neherlands, and Spain, he DF-GLS and MZa ess boh indicae ha he deviaion is saionary a convenional significance levels. The Horvah and Wason (1995) es suppors he simple long-run moneary model for six of he seven counries for which i is relevan. (I is no relevan for he Neherlands.) Fig. 1 presens graphs of he nominal exchange rae deviaions from he level prediced by he simple moneary model for he eigh counries examined in Table 4. Verical lines are drawn for he years 1913, 1946, and 197 o roughly depic differen inernaional exchange rae regimes: classical gold sandard, inerwar period, Breon Woods era, and he modern floa. 6 5 A endency for mean-reversion 4 We use a Barle kernel (as in Shin, 1994) and se he lag runcaion o four in calculaing C. m 5 We implemen he Horvah and Wason (1995) es using a GAUSS program available from Mark Wason s home page (hp:// mwason/ ). Criical values for he case where a linear rend is included in he coinegraing vecor are no available in Horvah and Wason (1995), so we base inferences on boosrapped criical values. 6 We are following he divisions used in Taylor (1a).

16 374 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () Table 4 Uni roo es resuls, e [(m* ) ( y* )] (1) () (3) (4) (5) (6) (7) (8) Counry a DF-GLS b MZ c HW Counry DF-GLS MZ HW a Belgium Neherlands.17* 8.6* ( ) (19 199) Finland Porugal ** ( ) ( ) France (rend) * Spain.16* 8.61* 14.66* ( ) ( ) Ialy 3.17**.77**.37** Swizerland (rend) ( ) ( ),*,** indicae significance a he 1, 5, and 1 percen levels, respecively; for Finland and Porugal, he uni roo ess are for e (m* ); (rend) indicaes ha he es allows for saionariy around a linear rend. a Ng and Perron () one-sided (lower-ail) es of H : Nonsaionariy; 1, 5, and 1 percen criical values equal 1.6, 1.98, and.58, respecively; when a linear rend is included, 1, 5, and 1 percen criical values equal.6,.91, and 3.4, respecively. b Ng and Perron () one-sided (lower-ail) es of H : Nonsaionariy; 1, 5, and 1 percen criical values equal 5.7, 8.1, and 13.8, respecively; when a linear rend is included, 1, 5, and 1 percen criical values equal 14., 17.3, and 3.8, respecively. c Horvah and Wason (1995) one-sided (upper-ail) es of H : No coinegraion among e, m*, y* vs. H : Coinegraion wih prespecified coinegraing relaionship e 5 b 1 (m* ) ( y* 1 y ); 1, 5, and 1 percen criical values equal 9.7, 11.6, and 15.41, respecively; 1, 5, and 1 percen boosrapped criical values for France (Swizerland) equal 1.89 (1.7), (14.6), and.16 (17.4), respecively; for Finland and Porugal, one-sided (upper-ail) es of H : No coinegraion among e, m* vs. H : Coinegraion wih prespecified coinegraing relaionship e 5 b 1 (m* 1 m ); for Finland and Porugal, 1, 5, and 1 percen criical values equal 8.3, 1.18, and 13.73, respecively. a in each deviaion is eviden in Fig. 1. (For France and Swizerland, he deviaions appear saionary around a rend.) I is also eviden from Fig. 1 ha deviaions from he moneary fundamenals can be quie subsanial and persisen. Because of his, i will be difficul o deec he long-run relaionship beween he nominal exchange rae and moneary fundamenals using daa from he modern floa 7 alone. Also noe ha he early 198s sand ou in Fig. 1 as a period where U.S. dollar exchange raes diverge considerably from he underlying moneary fundamenals. The U.S. dollar appears subsanially overvalued during his period, as is widely believed. To summarize he resuls of his secion, we find considerable suppor for a very simple form of he long-run moneary model of U.S. dollar exchange rae deerminaion for France, Ialy, he Neherlands, and Spain using long daa spans. 7 Taylor (1b) argues ha i may also be difficul o deec equilibrium relaionships if he adjusmen o he equilibrium occurs more quickly han he frequency of he available daa.

17 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () Fig. 1. Deviaions from he simple moneary model. Noe: verical lines appear a 1913, 1946, and 197.

18 376 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () We find moderae suppor for Belgium, Finland, and Porugal and weaker suppor for Swizerland. From Fig. 1, we see ha long spans of daa will generally be required o deec he long-run equilibrium relaionship implied by he moneary model. 4. Error-correcion models In order o gain insigh ino how he long-run equilibrium is resored beween nominal exchange raes and moneary fundamenals, we esimae he following bivariae vecor error-correcion model (VECM) in e and f, where f5 (m * m ) ( y* ): p p De 5 g 1O g De 1O g Df 1 l z 1, (6) 1i i i i De,z 1 1 i51 i51 p p Df 5 d 1O d De 1O d Df 1 l z 1, (7) 1i i i i Df,z 1 i51 i51 where z5 e f. We esimae he VECM for all of he counries in Table 4, wih he excepion of he Neherlands, due o he saionariy of he nominal exchange rae and moneary fundamenals in he Neherlands. Following he lead of Tables 4, we use f5 m* for Finland and Porugal. Table 5 repors OLS esimaes of he error-correcion coefficiens, lde,z and l Df,z, ha govern he adjusmen o he long-run equilibrium. For Belgium, Finland, and Ialy, he error-correcion coefficien in he exchange rae equaion ( l De,z) is significan, while he error-correcion coefficien in he fundamenals Table 5 Error-correcion coefficien esimaes (1) () (3) (4) (5) (6) Counry l l Counry l l De,z Df,z De,z Df,z Belgium.9*.3 Porugal.6.6** ( ) (.4) (.3) ( ) (.4) (.) Finland.17**.1 Spain.8.1* ( ) (.5) (.3) ( ) (.6) (.4) France.1.9* Swizerland.6*.5* ( ) (.6) (.3) ( ) (.3) (.) Ialy.**.3 ( ) (.6) (.6),*,** indicae significance a he 1, 5, and 1 percen levels, respecively; Whie (198) heeroskedasiciy-consisen sandard errors for he coefficien esimaes are given in parenheses.

19 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () equaion ( l Df,z) is insignifican. This implies ha he moneary fundamenals are weakly exogenous for hese counries (see Engle e al., 1983). In oher words, when deviaions from he long-run equilibrium occur in Belgium, Finland, and Ialy, i is primarily he exchange rae ha adjuss o resore long-run equilibrium over our sample, raher han he moneary fundamenals. For Porugal and Spain, he resuls are reversed: lde,z is insignifican, while ldf,z is significan, so ha he exchange rae is weakly exogenous for hese counries over our sample. When deviaions from he long-run equilibrium occur in Porugal and Spain, i is he 8 moneary fundamenals ha bear he brun of adjusmen over our sample. For France and Swizerland, an inermediae resul obains, as boh errorcorrecion coefficiens, lde,z and l Df,z, are significan (and have he correc sign), so ha neiher he exchange rae nor he moneary fundamenals are weakly exogenous. Boh he nominal exchange rae and he moneary fundamenals adjus o resore long-run equilibrium for hese wo counries over our sample. The differen adjusmen mechanisms a work in he differen counries over he las cenury likely reflec varying degrees of commimen o nominal exchange rae 9 sabiliy. 5. Nominal exchange rae forecasing An imporan srand of he exan lieraure invesigaes he forecasing performance of he moneary model of exchange rae deerminaion. In heir seminal paper, Meese and Rogoff (1983) repor ha ou-of-sample forecass of moneary models canno ouperform a naıve random walk model for U.S. dollar exchange raes for Germany, Japan, and he Unied Kingdom during he period. However, in a well-known paper, Mark (1995) shows ha pas nominal exchange rae deviaions from he level prediced by he simple moneary model, z5 e [(m * ) ( y* )], are useful in predicing U.S. dollar exchange raes a longer horizons for he period The Mark (1995) finding is noeworhy, given he pessimism generaed by Meese and Rogoff 8 We combine he moneary fundamenals in (6) and (7) o faciliae he inerpreaion of he forecasing resuls repored in Secion 5. In he working paper aneceden o he presen paper, we also consider a rivariae VECM ha separaes ou he moneary fundamenals ino he relaive money supply and relaive income level. We find no counry for which he error-correcion coefficien on he relaive income level is significan, so i is he relaive money supply ha adjuss in order o bring he moneary fundamenals in line wih he exchange rae. 9 In he working paper aneceden o he presen paper, we es he sabiliy of he individual equaions of he VECM. The nominal exchange rae equaion for Porugal and he fundamenals equaion for Finland and France exhibi he greaes evidence of srucural change. 3 Surveys by Frankel and Rose (1995) and Taylor (1995) conclude ha srucural exchange rae models in general have no done well a explaining or predicing exchange rae movemens during he modern floaing exchange rae period.

20 378 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () (1983). In he spiri of Meese and Rogoff (1983) and Mark (1995), we examine he ou-of-sample forecasing performance of he simple moneary model using our long spans of daa for he counries in Table 5. Mark (1995) compues recursive ou-of-sample forecass a he k-horizon based on moneary fundamenals. He esimaes he following equaion hrough period, T, where T is he size of he available sample, in order o generae he firs k-horizon forecas for he moneary model: eˆ e 5 a(k; ˆ ) 1 b(k; ˆ )z. (8) 1k Eq. (8) is hen re-esimaed using daa hrough period 1 1 in order o generae a second k-horizon forecas for he moneary model, and his process is coninued hrough period T k. These k-horizon forecass are hen compared o he k- horizon forecass from a naıve random walk model. The forecass are compared using Theil s U, he raio of he roo mean squared predicion error (RMSE) for he moneary model o he RMSE for he random walk model, and he Diebold and Mariano (1995) es for equal predicive abiliy based on he MSE crierion. Mark (1995) finds ha forecass from he moneary model are ofen superior o hose of he naıve random walk model, especially a longer horizons. Berkowiz and Giorgianni (1) challenge he robusness of Mark s (1995) findings by showing ha hey hinge criically on he assumpion ha z is saionary (ha is, ha nominal exchange raes and moneary fundamenals are coinegraed). This is problemaic for Mark (1995), as he fails o find evidence of coinegraion beween nominal exchange raes and moneary fundamenals for his pos-breon Woods daa. However, we do find evidence of coinegraion for he counries in Table 4, 31 so he saionariy of z is much less of an issue for our daa. We follow Mark (1995) and compare forecass from he moneary model, (8), 3 wih hose obained from a simple random walk wih drif model. Recen heoreical work by McCracken (1999) and Clark and McCracken (1) is relevan for our forecasing exercise. McCracken (1999) shows ha while he popular Diebold and Mariano (1995) saisic has a sandard asympoic disribuion when i is used o compare one-sep-ahead forecass beween nonnesed models, i has a nonsandard disribuion when used o compare forecass beween wo nesed models. When comparing (8) agains he random walk wih drif model, we are, of course, comparing nesed models. Clark and McCracken (1) show ha similar resuls hold for he Ericsson (199) and Harvey e al. (1998) forecas encompassing ess. While here are no heoreical resuls for ess beyond 31 Berben and van Dijk (1998) and Kilian (1999) also quesion he robusness of Mark s (1995) resuls. 3 Acually, Mark (1995) compares forecass from (8) wih hose of a random walk wihou drif. Kilian (1999) argues ha forecass from (8) should be compared o hose of a random walk wih drif, as we do.

21 D.E. Rapach, M.E. Wohar / Journal of Inernaional Economics 58 () he one-sep-ahead horizon for nesed models (a he ime of he wriing of his paper), Berben and van Dijk (1998) and Berkowiz and Giorgianni (1) show ha he one-sep-ahead horizon is he mos imporan horizon for he predicive regression (8). We use five ess from Clark and McCracken (1) o compare recursive ou-of-sample one-sep-ahead forecass from (8) o hose of a random walk wih drif for he counries considered in Table 5. The firs wo ess, MSE-F and MSE-T, are versions of he popular Diebold and Mariano (1995) and Wes (1996) ess. They are used o es he null hypohesis ha he MSE of he moneary model (MSE MF) is equal o he MSE of he random walk wih drif model (MSE RW) agains he alernaive hypohesis ha MSE MF,MSE RW. The oher hree ess are he ENC-T es of Harvey e al. (1998), he ENC-REG es of Ericsson (199), and he ENC-NEW es developed by Clark and McCracken (1). The null hypohesis for each of hese ess is ha forecass from he random walk wih drif model encompass he forecass from (8). Forecas encompassing is based on opimally consruced composie forecass. If he forecass from he random walk wih drif model encompass forecass based on (8), his essenially means ha forecass from (8) provide no addiional informaion ha is valuable in forecasing exchange raes apar from he informaion already conained in he random walk wih drif model. If we can rejec he null of forecas encompassing, hen forecass from (8) provide informaion above and beyond he informaion already in forecass from he random walk wih drif model. For all five ess, he firs recursive forecas is generaed using he firs half of he available sample. Inferences are based on he asympoic criical values in McCracken (1999) and Clark and McCracken (1). Clark and McCracken (1) find ha hese asympoic criical values work well in finie samples in exensive Mone Carlo simulaions. They also esablish he following ranking of he power of he various ess based on exensive Mone Carlo simulaions: ENC-NEW. MSE-F, ENC-T, ENC-REG. MSE-T. The forecasing resuls are repored in Table 6. Column () gives he forecas period for each counry, and column (3) repors Theil s U (RMSE MF / RMSE RW). There is considerable evidence of exchange rae predicabiliy based on moneary fundamenals for Belgium, Ialy, and Swizerland. These resuls are consisen wih hose in Table 5, where he error-correcion coefficien in he exchange rae equaion is significan for Belgium, Ialy, and Swizerland. For hese hree counries, we expec he exchange rae o adjus o resore he long-run moneary equilibrium, and hus moneary fundamenals should be helpful in predicing fuure exchange raes. In conras, here is no evidence ha moneary fundamenals improve exchange rae forecass for France, Porugal, and Spain. Again, his is consisen wih he resuls in Table 5, where he error-correcion coefficien in he exchange rae equaion is insignifican for Porugal and Spain and only significan