EUROPEAN CENTRAL BANK WORKING PAPER SERIES WORKING PAPER NO 248 EXCHANGE RATES AND FUNDAMENTALS BY CHARLES ENGEL AND KENNETH D WEST Augus 2003
EUROPEAN CENTRAL BANK WORKING PAPER SERIES WORKING PAPER NO 248 EXCHANGE RATES AND FUNDAMENTALS 1 BY CHARLES ENGEL 2 AND KENNETH D WEST 3 Augus 2003 1 We hank Shiu-Sheng Chen, Akio Masumoo and Yu Yuan for research assisance, he Naional Science Foundaion for financial suppor, and many seminar audiences for helpful commens Porions of his paper were compleed while Wes was in he Research Visior Program a he Direcorae General Research for he European Cenral Bank, a Houblon-Norman Fellow a he Bank of England and he Professional Fellow in Moneary Economics a Vicoria Universiy and he Reserve Bank of New Zealand The opinions expressed herein are hose of he auhor(s) and do no necessarily represen hose of he European Cenral Bank This paper can be downloaded wihou charge from hp://wwwecb in or from he Social Science Research Nework elecronic library a: hp://ssrncom/ absrac_id=457320. 2 Universiy of Wisconsin and NBER, e-mail: cengel@sscwiscedu 3 Universiy of Wisconsin and NBER, e-mail: kdwes@wiscedu
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Conens Absrac 4 Non-echnical summary 5 1. Inroducion 6 2. Models 10 3. Empirical findings 16 4. Random walk in s as b --> 1 25 5. Conclusions 31 References 34 Tables 36 Appendix 43 European Cenral Bank working paper series 47 ECB Working Paper No 248 Augus 2003 3
Absrac Sandard economic models hold ha exchange raes are influenced by fundamenal variables such as relaive money supplies, oupus, inflaion raes and ineres raes. Noneheless, i has been well documened ha such variables lile help predic changes in floaing exchange raes ha is, exchange raes follow a random walk. We show ha he daa do exhibi a relaed link suggesed by sandard models ha he exchange rae helps predic fundamenals. We also show analyically ha in a raional expecaions presen value model, an asse price manifess near random walk behavior if fundamenals are I(1) and he facor for discouning fuure fundamenals is near one. We sugges ha his may apply o exchange raes. Keywords: exchange raes, random walk, presen value, moneary model, asse price JEL classificaions: F310, F370, G150, G120 4 ECB Working Paper No 248 Augus 2003
NON-TECHNICAL SUMMARY A longsanding puzzle in inernaional economics is he difficuly of ying floaing exchange raes o macroeconomic fundamenals such as money supplies, prices, oupus, and ineres raes. Economic heories sae ha he exchange rae is deermined by such fundamenal variables, bu in pracice fundamenal variables have no proved helpful in predicing fuure changes in exchange raes. In his paper we ake a new line of aack on he quesion of he link beween exchange raes and fundamenals. We do no aemp o develop or esimae a paricular exchange rae model. Insead, we work wih an exising and convenional class of economic models, in which he exchange rae is deermined by curren and expeced fuure values of observable fundamenals and unobservable shocks. We firs pu aside he quesion of why fundamenals have no been very helpful in predicing changes in exchange raes. We ask insead if he convenional models have implicaions for wheher he exchange rae helps predic fundamenals. I is plausible o look in his direcion. Surely much of he shor-erm flucuaions in exchange raes is driven by changes in expecaions abou he fuure. If he models are good approximaions, and expecaions reflec informaion abou fuure fundamenals, exchange rae changes will likely be useful in forecasing hese fundamenals. Using quarerly bilaeral dollar exchange raes, 1974-2001, for he dollar versus he six oher G7 counries, we find some evidence of such predicabiliy, especially for nominal variables. For example, depreciaions of he dollar relaive o he Deuschmark ended o forecas a rise in he U.S. price level relaive o he German price level. We hen ask how one can reconcile our new finding on he abiliy of exchange raes o predic fundamenals wih he well-esablished failure of fundamenals o predic exchange rae changes. We show ha under some empirically plausible condiions, sample sizes are oo small o allow reliable use of fundamenals o predic changes in exchange raes, even when hese fundamenals deermine he exchange rae. We explain ha our resul is no an applicaion of he simple efficien markes model ha argues ha changes in asse prices are inherenly unpredicable. Indeed, when ha model is applied o exchange raes, i implies ha cross-counry ineres rae differenials will predic exchange rae changes and hus ha exchange rae changes will be predicable by ineres rae differenials as well as variables ha help predic ineres rae differenials. To preven confusion, we noe ha he wo findings summarized in he wo preceding paragraphs are disinc. While we hope ha he reader will find boh persuasive, i is possible o agree wih one finding bu no he oher. We conclude ha he evidence is consisen wih he view ha exchange raes are deermined by curren and expeced fuure values of observable economic fundamenals and unobservable shocks. ECB Working Paper No 248 Augus 2003 5
A longsanding puzzle in inernaional economics is he difficuly of ying floaing exchange raes o macroeconomic fundamenals such as money supplies, oupus, and ineres raes. Our heories sae ha he exchange rae is deermined by such fundamenal variables, bu floaing exchange raes beween counries wih roughly similar inflaion raes are in fac well-approximaed as random walks. Fundamenal variables do no help predic fuure changes in exchange raes. Meese and Rogoff (1983a, 1983b) firs esablished his resul. They evaluaed he ou-of-sample fi of several models of exchange raes, using daa from he 1970s. They found ha by sandard measures of forecas accuracy, such as he mean-squared deviaion beween prediced and acual exchange rae, accuracy generally increased when one simply forecas he exchange rae o remain unchanged compared o when one used he predicions from he exchange rae models. While a large number of sudies have subsequenly claimed o find success for various versions of fundamenals-based models, someimes a longer horizons, and over differen ime periods, he success of hese models has no proven o be robus. A recen comprehensive sudy by Cheung, Chinn, and Pascual (2002) concludes, he resuls do no poin o any given model/specificaion combinaion as being very successful. On he oher hand, i may be ha one model will do well for one exchange rae, bu no for anoher. In his paper, we ake a new line of aack on he quesion of he link beween exchange raes and fundamenals. We work wih a convenional class of exchange models, in which he exchange rae is he expeced presened discouned value of a linear combinaion of observable fundamenals and unobservable shocks. Linear driving processes are posied for fundamenals and shocks. We firs pu aside he quesion of why fundamenals seem no o help predic changes in exchange raes. We ask insead if hese convenional models have implicaions for wheher he exchange rae helps predic fundamenals. I is plausible o look in his direcion. Surely much of he shor-erm flucuaions in exchange raes is driven by changes in expecaions abou he fuure. If he models are good approximaions, and expecaions reflec informaion abou fuure fundamenals, he exchange rae 6 ECB Working Paper No 248 Augus 2003
changes will likely be useful in forecasing hese fundamenals. So hese models sugges ha exchange raes Granger-cause he fundamenals. Using quarerly bilaeral dollar exchange raes, 1974-2001, for he dollar versus he six oher G7 counries, we find some evidence of such causaliy, especially for nominal variables. The saisical significance of he predicabiliy is no uniform, and suggess a link beween exchange raes and fundamenals ha perhaps is modes in comparison wih he links beween oher ses of economic variables. Bu in our view, he saisical predicabiliy is noable in ligh of he far weaker causaliy from fundamenals o exchange raes. For counries and daa series for which here is saisically significan evidence of Granger causaliy, we nex gauge wheher he Granger causaliy resuls are consisen wih our models. We compare he correlaion of exchange rae changes wih wo esimaes of he change in he presen discouned value of fundamenals. One esimae uses only he lagged value of fundamenals. The oher uses boh he exchange rae and own lags. We find ha he correlaion is subsanially higher when he exchange rae is used in esimaing he presen discouned value. We hen ask how one can reconcile he abiliy of exchange raes o predic fundamenals wih he failure of fundamenals o predic exchange rae changes. We show analyically ha in he class of presen value models ha we consider, exchange raes will follow a process arbirarily close o a random walk if (1) a leas one forcing variable (observable fundamenal or unobservable shock) has a uni auoregressive roo, and (2) he discoun facor is near uniy. So, in he limi, as he discoun facor approaches uniy, he change in he ime exchange rae will be uncorrelaed wih informaion known a ime -1. We explain below ha our resul is no an applicaion of he simple efficien markes model of Samuelson (1965) and ohers. When ha model is applied o exchange raes, i implies ha cross-counry ineres rae differenials will predic exchange rae changes and hus ha exchange raes will no follow a random walk. ECB Working Paper No 248 Augus 2003 7
Inuiively, as he discoun facor approaches uniy, he model pus relaively more weigh on fundamenals far ino he fuure in explaining he exchange rae. Transiory movemens in he fundamenals become relaively less imporan compared o he permanen componens. Imagine performing a Beveridge-Nelson decomposiion on he linear combinaion of fundamenals ha drive he exchange rae, expressing i as he sum of a random walk componen and a ransiory componen. The class of heoreical models we are considering hen express he exchange rae as he discouned sum of he curren and expeced fuure fundamenals. As he discoun facor approaches one, he variance of he change of discouned sum of he random walk componen approaches infiniy, while he variance of he change of he saionary componen approaches a consan. So he variance of he change of he exchange rae is dominaed by he change of he random walk componen as he discoun facor approaches one. We view as unexcepionable he assumpion ha a forcing variable has a uni roo, a leas as a working hypohesis for our sudy. The assumpion abou he discoun facor is, however, open o debae. We noe ha in reasonable calibraions of some exchange rae models, his discoun facor in fac is quie near uniy. Of course our analyical resul is a limiing one. Wheher a discoun facor of.9 or.99 or.999 is required o deliver a process saisically indisinguishable from a random walk depends on he sample size used o es for random walk behavior, and he enire se of parameers of he model. Hence we presen some correlaions calculaed analyically in a simple sylized model. We assume a simple univariae process for fundamenals, wih parameers chosen o reflec quarerly daa from he recen floaing period. We find ha discoun facors above 0.9 suffice o yield near zero correlaions beween he period exchange rae and period -1 informaion. We do no aemp o verify our heoreical conclusion ha large discoun facors accoun for random walk behavior in exchange raes using any paricular fundamenals model from he lieraure. Tha is, we do no pick specific models ha we claim saisfy he condiions of our heorem, and hen esimae hem and verify ha hey produce random walks. 8 ECB Working Paper No 248 Augus 2003
To preven confusion, we noe ha our finding ha exchange raes predic fundamenals is disinc from our finding ha large discoun facors raionalize a random walk in exchange raes. I may be reasonable o link he wo findings. When expecaions of fuure fundamenals are very imporan in deermining he exchange rae, i seems naural o pursue he quesion of wheher exchange raes can forecas hose fundamenals. Bu one can be persuaded ha exchange raes Granger cause fundamenals, and sill argue ha he approximae random walk in exchange raes is no subsanially aribuable o a large discoun facor. In he class of models we consider, all our empirical resuls are consisen wih a leas one oher explanaion, namely, ha exchange rae movemens are dominaed by unobserved shocks ha follow a random walk. The plausibiliy of his explanaion is underscored by he fac ha we generally fail o find coinegraion beween he exchange rae and observable fundamenals, a failure ha is raionalized in our class of models by he presence of an I(1) (hough no necessarily random walk) shock. As well, he random walk also can arise in models ha fall ouside he class we consider. I does so in models ha combine nonlineariies/hreshold effecs wih small sample biases (see Taylor, Peel, and Sarno (2002), and Kilian and Taylor (2001).) Exchange raes will sill predic fundamenals in such models, hough a nonlinear forecasing process may be required. Our suggesion ha he exchange rae will nearly follow a random walk when he discoun facor is close o uniy means ha forecasing changes in exchange rae is difficul, bu perhaps sill possible. Some recen sudies have found success a forecasing changes in exchange raes a longer horizons, or using nonlinear mehods, and furher research along hese lines may prove fruiful. Mark (1995), Chinn and Meese (1995), and MacDonald and Taylor (1994) have all found some success in forecasing exchange raes a longer horizons imposing long-run resricions from moneary models. Groen (2000) and Mark and Sul (2001) find greaer success using panel mehods. Kilian and Taylor (2001) sugges ha models ha incorporae nonlinear mean-reversion can improve he forecasing accuracy of fundamenals models, hough i will be difficul o deec he improvemen in ou-of-sample forecasing exercises. ECB Working Paper No 248 Augus 2003 9
The paper is organized as follow. Secion 2 describes he class of linear presen value models ha we use o organize our houghs. Secion 3 presens evidence ha changes in exchange raes help predic fundamenals. Secion 4 discusses he possibiliy ha he random walk in exchange raes resuls from a discoun facor near one. Secion 5 concludes. An Appendix has some algebraic deails. An addiional appendix conaining empirical resuls omied from he paper o save space is available on reques. 2. MODELS Exchange rae models since he 1970s have emphasized ha nominal exchange raes are asse prices, and are influenced by expecaions abou he fuure. The asse-marke approach o exchange raes refers o models in which he exchange rae is driven by a presen discouned sum of expeced fuure fundamenals. Obsfeld and Rogoff (1996, p. 529) say, One very imporan and quie robus insigh is ha he nominal exchange rae mus be viewed as an asse price. Like oher asses, he exchange rae depends on expecaions of fuure variables. [Ialics in he original.] Frenkel and Mussa s (1985) survey explains he asse-marke approach (p. 726): These facs sugges ha exchange raes should be viewed as prices of durable asses deermined in organized markes (like sock and commodiy exchanges) in which curren prices reflec he marke s expecaions concerning presen and fuure economic condiions relevan for deermining he appropriae values of hese durable asses, and in which price changes are largely unpredicable and reflec primarily new informaion ha alers expecaions concerning hese presen and fuure economic condiions. A variey of models relae he exchange rae o economic fundamenals and o he expeced fuure exchange rae. We wrie his relaionship as: (2.1) s f z be s1. Here, we define he exchange rae s as he home currency price of foreign currency (dollars per uni of foreign currency, if he U.S. is he home counry.) f and z are economic fundamenals ha ulimaely 10 ECB Working Paper No 248 Augus 2003
drive he exchange rae, such as money supplies, money demand shocks, produciviy shocks, ec. We differeniae beween fundamenals observable o he economerician, f, and hose ha are no observable, z. One possibiliy is ha he rue fundamenal is measured wih error, so ha f is he measured fundamenal and he z include he measuremen error; anoher is z is unobserved shocks. In equaion (2.1), 0 b 1. The value of he currency is lower ( s is higher) when he currency is expeced o depreciae ( E s s 0.) 1 Upon imposing he no bubbles condiion ha presen value relaionship j b E s j goes o zero as j, we have he j (2.2) s b E ( f j z ) j0 j We now ouline some models ha fi ino he framework of equaions (2.1) and (2.2). We will no aemp o esimae direcly he models ha we are abou o ouline. Raher, we use hese o moivae alernaive measures of observable fundamenals, f. A. Money-Income Model Consider firs he familiar moneary models of Frenkel (1976), Mussa (1976), and Bilson (1978); and heir close cousins, he sicky-price moneary models of Dornbusch (1976) and Frankel (1979). Assume in he home counry here is a money marke relaionship given by: (2.3) m p y i vm. Here, m is he log of he home money supply, p is he log of he home price level, i is he level of he home ineres rae, y is he log of oupu, and v m is a shock o money demand. Here and hroughou we use he erm shock in a somewha unusual sense. Our shocks poenially include consan and rend ECB Working Paper No 248 Augus 2003 11
erms, may be serially correlaed, and may include omied variables ha in principle could be measured. Assume a similar equaion holds in he foreign counry. The analogous foreign variables are m, p, i, counry s parameers. y, and v m, and he parameers of he foreign money demand are idenical o he home The nominal exchange rae equals is purchasing power pariy value plus he real exchange rae: (2.4) s p p q. In financial markes, he ineres pariy relaionship is (2.5) E s s i i 1 Here is he deviaion from raional expecaions uncovered ineres pariy. I can be inerpreed as a risk premium or an expecaional error. Puing hese equaions ogeher and rearranging, s 1 m 1 m ( y y (2.6) m m 1 ) q ( v v ) E s 1. This equaion akes he form of equaion (2.1) when he discoun facor is given by b, he 1 observable fundamenals are given by f m m ( y y ), and he unobservables are: z 1 q ( vm vm ). 1 Equaion (2.6) is implied by boh he flexible-price and sicky-price versions of he moneary model. In he flexible-price monearis models of Frenkel (1976), Mussa (1976), and Bilson (1978), oupu, y, and he real exchange rae, q, are exogenous. In he sicky-price models of Dornbusch (1976) and Frankel (1979), hese wo variables are endogenous. Because nominal prices adjus slowly, he real exchange rae is influenced by changes in he nominal exchange rae. Oupu is demand deermined, and may respond o changes in he real exchange rae, income and real ineres raes. 12 ECB Working Paper No 248 Augus 2003
Noneheless, since equaion (2.3) (and is foreign counerpar), (2.4), and (2.5) hold in he Dornbusch- Frankel model, one can derive relaionship (2.6) in hose models. Dornbusch and Frankel each consider special cases for he exogenous moneary processes (in Dornbusch, all shocks o he money supply are permanen; Frankel considers permanen shocks o he level and o he growh rae of money.) As a resul of heir assumpion ha all shocks are permanen, hey each can express he exchange rae purely in erms of curren fundamenals, which may obscure he general implicaion ha exchange raes depend on expeced fuure fundamenals. Following Mark (1995), our empirical work ses 1. Under some condiions, he model implies ha he exchange rae should Granger cause m m ( y y ) in a bivariae Granger causaliy es namely, if he opimal forecas of m m ( y y ) does no depend only on own lags. Failure o find such a relaionship is no, however, inconsisen wih equaion (2.6), because he presence of he shocks q and breaks wha would oherwise be a singular relaionship. (I may help readers familiar wih Campbell and Shiller s (1987) work on equiy and bond markes o sress ha he presence of he unobservable shocks relaxes many resricions of a presen value model, including he one jus noed relaing o Granger causaliy.) In addiion o considering he bivariae relaionship beween s and m m ( y y ), we will also invesigae he relaionship beween s and m m. Tha is, we also use (2.6) o moivae seing f m m, and moving y y o z. We do so largely because we wish o conduc a relaively unsrucured invesigaion ino he link beween exchange raes and various measures of fundamenals. Bu we could argue ha we focus on m m because financial innovaion has made sandard income measures poor proxies for he level of ransacions. Similarly, we invesigae he relaionship beween s and y y. ECB Working Paper No 248 Augus 2003 13
We noe here ha some recen exchange-rae models developed from he new open economy macroeconomics yield very similar relaionships o he ones we describe in his secion. For example, in Obsfeld and Rogoff (1998), he exchange rae is given by (heir equaion (30): (2.7) j s b E (1 )( ) j 0 b m j m j b j, where we have ranslaed heir noaion o be consisen wih ours. Equaion (2.7) is in fac he forward soluion o a special case of equaion (2.6) above. The discoun facor, b, in Obsfeld and Rogoff (1998) is relaed o he semi-elasiciy of money demand exacly as in equaion (2.6). However, heir money demand funcion is derived from a uiliy-maximizing framework in which real balances appear in he uiliy funcion, and heir risk premium is derived endogenously from firs principles. B. Taylor-Rule Model Here we draw on he burgeoning lieraure on Taylor rules. Le p p1 denoe he inflaion rae, and g y be he oupu gap. We assume ha he home counry (he U.S. in our empirical work) follows a Taylor rule of he form: g 1. (2.8) i y 2 v In (2.8), 1 0, 1 2, and he shock v conains omied erms. The foreign counry follows a Taylor rule ha explicily includes exchange raes: (2.9) i g 0 ( s s ) 1y 2 v. In (2.9), 0 0 1, and arge he PPP level of he exchange rae: s is a arge for he exchange rae. We will assume ha moneary auhoriies (2.10) s p p. 14 ECB Working Paper No 248 Augus 2003
Since s is measured in dollars per uni of foreign currency, he rule indicaes ha ceeris paribus he foreign counry raises ineres raes when is currency depreciaes relaive o he arge. Clarida, Gali and Gerler (1998) esimae moneary policy reacion funcions for Germany and Japan (using daa from 1979-1994) of a form similar o equaion (2.9). They find ha a one percen real depreciaion of he mark relaive o he dollar led he Bundesbank o increase ineres raes (expressed in annualized erms) by five basis poins, while he Bank of Japan increased raes by 9 basis poins in response o a real yen depreciaion relaive o he dollar. As he nex equaion makes clear, our argumen sill follows if he U.S. were also o arge exchange raes. We omi he exchange rae arge in (2.8) on he inerpreaion ha U.S. moneary policy has virually ignored exchange raes excep, perhaps, as an indicaor. Subracing he foreign from he home money rule, we obain (2.11) i i g g 0 ( s s ) 1( y y ) 2 ( ) v v rae arge: Use ineres pariy (2.5) o subsiue ou for i i, and (2.10) o subsiue ou for he exchange (2.12) 1 1 s p p y y v v E s 0 g g ( ) [ 1( ) 2( ) ] 1 10 10 10. This equaion is of he general form (2.1) of he expeced discouned presen value models. The model provides a moivaion for why he exchange rae migh Granger cause p p (reaing g g 1 y y ) 2 ( ) ( v v as unobserved forcing variables.) Equaion (2.11) can be expressed anoher way, again using ineres pariy (2.5), and he equaion for he arge exchange rae, (2.10): (2.13) s g g 0 ( i i ) 0 ( p p ) 1( y y ) 2 ( ) v v (1 0 ) (1 0 ) E s1 ECB Working Paper No 248 Augus 2003 15
This equaion is very much like (2.12), excep ha i shows ha he exchange rae may be useful in forecasing fuure i i. The inuiion is ha when he exchange rae is above is arge, for example, he gap beween he exchange rae and arge will be eliminaed only gradually. As long as he gap persiss, ceeris paribus i i will be above average. So, high s may predic high fuure values of i i. As wih he money-income model, we will no esimae explicily he Taylor-rule model. We do no ake a sand on he paricular form of he Taylor rule. We use equaions (2.12) and (2.13) merely o moivae our unsrucured empirical work in he nex secion. A. Daa and Basic Saisics 3. EMPIRICAL FINDINGS We use quarerly daa, usually 1974:1-2001:3 (wih excepions noed below). Wih one observaion los o differencing, he sample size is T 110. We sudy bilaeral US exchange raes versus he oher six members of he G7: Canada, France, Germany, Ialy, Japan and he Unied Kingdom. The Inernaional Financial Saisics (IFS) CD-ROM is he source for he end of quarer exchange rae s and consumer prices p. The OECD s Main Economic Indicaors CD-ROM is he source for our daa on he seasonally adjused money supply, m (M4 in he U.K., M1 in all oher counries; 1978:1-1998:4 for France, 1974:1-1998:4 for Germany, 1975:1-1998:4 for Ialy). The OECD is also he source for real, seasonally adjused GDP, y, for all counries bu Germany, which we obain by combining IFS (1974:1-2001:1) and OECD (2001:2-2001:3) daa, and Japan, which combines daa from he OECD (1974:1-2002) wih 2002:3 daa from he web sie of he Japanese Governmen s Economic and Social Research Insiue. Daasream is he source for he ineres raes, i, which are 3 monh Euro raes (1975:1-2001:3 for Canada, 1978:3-2001:3 for Ialy and Japan). 16 ECB Working Paper No 248 Augus 2003
We conver all daa bu ineres raes by aking logs and muliplying by 100. Through he res of he paper, he symbols defined in his paragraph ( s, m, y, p ) refer o he ransformed daa. Le f denoe a measure of fundamenals in he U.S. relaive o abroad (for example, f m m.) Using Dickey-Fuller ess wih a ime rend included, we were generally unable o rejec he null of a uni roo in f wih he following measures of f : analysis presens saisics on m, p, i, y, m y. Hence our f for all measures of fundamenals. Even hough we fail o rejec uni roos for ineres differenials, we are uneasy using ineres differenials only in differenced form. So we presen saisics for boh levels and differences of ineres raes. Some basic saisics are presened in Table 3.1. Row 1 is consisen wih much evidence ha changes in exchange raes are serially uncorrelaed, and quie volaile. The sandard deviaion is 5 o 10 imes he size of he mean. Firs order auocorrelaions are small, under 0.15 in absolue value. Under he null of no serial correlaion, he sandard error on he esimaor of he auocorrelaion is approximaely 1/ T 0.1, so none of he esimaes are significan a even he 10 percen level. Rows 2 hrough 7 presen saisics on our measures of fundamenals. A posiive value for he mean indicaes ha he variable has been growing faser in he U.S. han abroad. For example, he figure of -0.92 for he mean value of he U.S.- Ialy inflaion differenial means ha quarerly inflaion was, on average, 0.92 percenage poins lower in he U.S. han in Ialy during he 1974-2001 period. Of paricular noe is ha he vas majoriy of esimaes of firs order auocorrelaion coefficiens sugges a rejecion of he null of no serial correlaion a he 10% level, and mos do a he 5% level as well (again using an approximae sandard error of 0.1). An excepion o his paern is in oupu differenials in row (7). None of he auocorrelaions are significan a he 5% level, and only one (France, for which he esimae is 0.19) a he 10% level. ECB Working Paper No 248 Augus 2003 17
For each counry we conduced four coinegraion ess, beween s and each of our measures of fundamenals, m m, p i p, i, y y and m y ( m y ). We used Johansen s (1991) race and maximum eigenvalue saisics, wih criical values from Oserwald-Lenum (1992). Each bivariae VAR conained four lags. Of he 30 ess (6 counries, 5 fundamenals), we rejeced he null of no coinegraion a he 5 percen level in 5 insances using he race saisic. These were for m m, p p, and i i for Ialy, and, p p, and i i for he U.K. Of he 30 ess using he maximum eigenvalue saisic, he null was rejeced only once, for he U.K. for p p. We conclude ha i will probably no do grea violence o assume lack of coinegraion, recognizing ha a complemenary analysis using coinegraion would be useful. We ake he lack of coinegraion o be evidence ha unobserved variables such as real demand shocks, real money demand shocks, or possibly even ineres pariy deviaions have a permanen componen, or a leas are very persisen. Alernaively, i may be ha he daa we use o measure he economic fundamenals of our model have some errors wih permanen or very persisen componens. For example, i may be ha he appropriae measure of he money supply has permanenly changed because of numerous financial innovaions over our sample, so ha he M1 money supply series vary from he rue money supply by some I(1) errors. B. Granger-Causaliy Tess Campbell and Shiller (1987) observe ha when a variable s is he presen value of a variable x, hen eiher (1) s Granger causes x relaive o he bivariae informaion se consising of lags of s and x, or (2), s is an exac disribued lag of curren and pas values of x. Tha is, as long as s embodies 18 ECB Working Paper No 248 Augus 2003
some informaion in addiion o ha included in pas values of x, s Granger causes x. 1 As was emphasized in he previous secion, however, exchange rae models mus allow for unobservable fundamenals he possibiliy ha x is a linear combinaion of unobservable as well as observable variables, and hus x iself is unobservable. Failure o find Granger causaliy from s o observable variables no longer implies an obviously unenable resricion ha he exchange rae is an exac disribued lag of observables. I is clear, hough, ha a finding of Granger causaliy is supporive of a view ha exchange raes are deermined as a presen value ha depends in par on observable fundamenals. Table 3.2 summarizes he resuls of our Granger causaliy ess on he full sample. We see in panel A ha a he five percen level of significance, he null ha ha s fails o Granger cause m p i i y ( m ), ( p ), i, ( i ), ( y ), and [ m y ( m y )], can be rejeced in 9 cases a he 5 percen level, and 3 more cases a he 10 percen level. There are no rejecions for Canada and he U.K., bu rejecions in 12 of he 24 ess for he oher four counries. The sronges rejecions are for prices, where he null is rejeced in hree cases a he one percen level. In a sense, his is no paricularly srong evidence ha exchange raes predic fundamenals. Afer all, even if here were zero predicabiliy, one would expec a handful of significan saisics jus by chance. We accordingly are cauions in assering ha he posied link is well esablished. Bu one saisical (as opposed o economic) indicaion ha he resuls are noeworhy comes from conrasing hese resuls wih ones for Granger causaliy ess running in he opposie direcion. We see in panel B of Table 3.2 ha he null ha he fundamenals fail o Granger cause s can be rejeced a he 5 percen level in only one es, and a he 10 percen level in only one more es. So, however modes is he 1 In he appendix, his addiional informaion is formalized as addiional random variables ha are used by privae agens in forecasing fuure fundamenals. ECB Working Paper No 248 Augus 2003 19
evidence ha exchange raes help o predic fundamenals, he evidence is disincly sronger han ha on he abiliy of fundamenals o predic exchange raes. There were some major economic and non-economic developmens during our sample ha warran invesigaion of sub-samples. Several of he European counries exchange raes and moneary policies became more ighly linked in he 1990s because of he evoluion of he European Moneary Union. Germany s economy was ransformed dramaically in 1990 because of reunificaion. We herefore look a causaliy resuls for wo subsamples. Table 3.3 presens resuls for 1974:1-1990:2, and Table 3.4 for he remaining par of he sample (1990:3-2001:2). The resuls generally go he same direcion as for he whole sample. In Table 3.3A, we see ha for he firs par of he sample, we rejec he null of no Granger causaliy from exchange raes o fundamenals a he one or five percen level in 10 cases, and a he en percen level in 2 more cases. Table 3.3 B indicaes ha here are no cases in which we can rejec he null of no Granger causaliy from fundamenals o exchange raes a he five percen level, and only 2 cases a he en percen level. Table 3.4 repors resuls for he second par of he sample. Panel A shows we rejec he null of no Granger causaliy from exchange raes o fundamenals in 9 cases a he one or five percen level, and five more cases a he 10 percen level. Bu for he es of no causaliy from fundamenals o exchange raes, Panel B shows we rejec nine imes a he one or five percen level, once a he 10 percen level. In he 1990s, hen, here appears o be more evidence of exchange-rae predicabiliy. This perhaps is no enirely surprising given he effor by he European counries o sabilize exchange raes. We noe, however, ha several of he rejecions of he null are for he yen/dollar rae. In addiion o he causaliy ess we repor from bivariae VARs, we also performed causaliy ess based on some mulivariae VARs. We chose several differen combinaions of variables o include in hese VARs, based on he models oulined in Secion 2. There are five groupings: ( s, ( y y ), ( p p ), i i ), ( s, ( m m ), ( y y )), ( s, ( p p ), ( y y )), 20 ECB Working Paper No 248 Augus 2003
( s, ( m m ), ( y y ), ( p p )), and ( s, ( y y ), ( p p ), ( i i )). We performed causaliy ess for he null ha null ha each of he fundamenals x does no cause s does no cause x for each of he fundamenals x, and he es wheher all of he fundamenals (in each grouping) joinly Granger cause s, again for each of he fundamenals. We also s. The resuls are very much like he resuls from he bivariae VARs. There is almos no evidence of causaliy from he fundamenals o he exchange rae. Of all of he ess we performed, here are no cases (ou of 108 ess performed) in which we could rejec a he 5 percen level he hypohesis of no causaliy from fundamenals o exchange raes, and only four cases where ha hypohesis is rejeced a he 10 percen level. We presen deails for he Granger causaliy ess on he fundamenals as a group in Table 3.5, relegaing o he addiional appendix deails on he oher ess. As Table 3.5 demonsraes, here were no cases in which we rejeced he join null of no causaliy from he group of fundamenals o he exchange rae. In conras, in 35 ess (ou of 108 performed) we rejeced he null of no causaliy from exchange raes o fundamenals a he 10 percen level, and hese were significan a he 5 percen level in 16 cases. Noable are he ess for wheher he exchange rae does no Granger cause any of he economic fundamenals. Table 3.5 repors ha we rejec he null of no causaion in 16 of he 30 ess performed a he 10 percen level, and 12 of hose were significan rejecions a he 5 percen level. Noneheless, here were many more cases in which he exchange rae could no help predic fundamenals. The exchange rae was found o be useful in forecasing real oupu in only wo cases. To summarize, while he evidence is far from overwhelming, here does appear o be a link from exchange raes o fundamenals, going in he direcion ha exchange raes help forecas fundamenals. C. Correlaion beween s and he Presen Value of Fundamenals Here we propose a saisic similar o one developed in Campbell and Shiller (1987). The modificaion of he Campbell-Shiller saisic is necessary for wo reasons. Firs is ha, unlike Campbell ECB Working Paper No 248 Augus 2003 21
and Shiller, our variables are no well approximaed as coinegraed. Second is ha we allow for unobservable forcing variables, again in conras o Campbell and Shiller. Wrie he presen value relaionship (2.4) as j j (3.1) s b E f j b E z j F U. j 0 j0 j 1 j Now b E ( 1 ) 0 j f j f b E 1 0 j f j. Thus b 1 1 j (3.2) s f 1 b E j f j U b b 0. 1 1 Our uni roo ess indicae ha f, and hence b E f j are I(0), and ha s and f are no j 0 j coinegraed. For (3.2) o be consisen wih lack of coinegraion beween s and f, we mus have U ~ I(1). A saionary version of (3.1) is hen (3.3) s F U. Le F i be he presen value of fuure he i subscrip. The wo informaion ses we use are univariae and bivariae: j 1 0 1 ) j j (3.4) F E( b f f, f,, f s compued relaive o an informaion se indexed by j j 0 1 (3.5) F 2 E( b f j s, f, s1, f, ). We hope o ge a feel for wheher eiher of hese informaion ses yield economically meaningful presen values by esimaing corr( F i, s ), he correlaion beween Fi and s. We esimae corr( F i, s ) using esimaes of Fi consruced from univariae auoregressions ( F 1 ) or bivariae vecor auoregressions ( F 2 ). If he esimaed correlaion is subsanially sronger using he bivariae esimae, we ake ha as evidence ha he coefficiens of s in he VAR equaion for f are economically reasonable and imporan. We limi our analysis o he variables in which here is a 22 ECB Working Paper No 248 Augus 2003
saisically significan relaionship beween f and s, as indicaed by he Granger causaliy ess in Table 3.2. Noe ha a low value of he correlaion is no necessarily an indicaion ha s is lile affeced by he presen value of f. A low correlaion will resul from a small covariance beween Fi and s Bu since cov( F, s ) cov( F, F ) cov( F, U ), his covariance migh be small because a i i i. sharply negaive covariance beween Fi and U offses a posiive covariance beween Fi and F. Conversely, of course, a high correlaion migh reflec a igh relaionship beween Fi and U wih lile connecion beween Fi and F. 2 We do, however, ake as reasonable he noion ha if he correlaion is higher for he bivariae han for he univariae informaion se, he coefficiens on lags of s in he f equaion are economically meaningful. We consruc Fˆ 1 from esimaes of univariae auoregressions, and Fˆ 2 from bivariae VARs, imposing a value of he discoun facor b. The lag lengh is four in boh he univariae and bivariae esimaes. We hen esimae he correlaions corr( F i, s ) using hese esimaed Fˆ i. We repor resuls only for daa ha show Granger causaliy from s o f a he 10 percen level or higher in he whole sample (Table 3.2, panel A). When f is measured by he ineres rae differenial, we consruc F 2 wih a VAR in he level bu no difference of i i i and ( i ). F 1 and i i and hus we do no repor separae resuls for 2 Since s is an elemen of he bivariae informaion se, projecion of boh sides of (3.1) ono his informaion se yields s F2 E( U s, f, s1, f1, ). I may help readers familiar wih Campbell and Shiller (1987) o noe ha because our models include unobserved forcing variables (i.e., because U is presen), we may no have s F 2 F. These equaliies hold only if E U s, f, s, f, ) 0. ( 1 1 ECB Working Paper No 248 Augus 2003 23
We ried hree values of he discoun facor, b 0. 5, b 0. 9, and b 0. 98, and repor resuls for he firs wo of hese values of he discoun facor in Panels A, and B, respecively, of Table 3.6. For he univariae informaion se ( F 1 ), he hree discoun facors give very similar resuls. Of he 10 esimaed correlaions, only wo are posiive for each value of b. (All of he relaions should be posiive for he four m p i variables repored in Table 3.6 -- ( m ), ( p ), ( i ), and [ m y ( m y )] -- according o he models of secion 2, if he conribuion of U is sufficienly small.) So if one relies on univariae esimaes of he presen value, one would find lile suppor for he noion ha changes in exchange raes reflec changes in he presen value of fundamenals. The bivariae esimaes lend raher more suppor for his noion, especially for b 0. 9. The esimaed correlaion beween F 2 and s is posiive in 6 of he 10 cases for b 0. 5 ; 7 of he 10 cases for b 0. 9. The median correlaions can be summarized as: Informaion se b 0.5 b 0.9 b 0.98 (3.6) F 1-0.04-0.05-0.05 F 2 0.10 0.24 0.30 I is clear ha using lags of s o esimae he presen value of fundamenals resuls in an esimae ha is more closely ied o s iself han when he presen value of fundamenals is based on univariae esimaes. Bu even limiing ourselves o daa in which here is Granger causaliy from f p s o, he larges single correlaion in he full sample is 0.59 (Germany, for ( p ), when b 0. 98.) A correlaion less han one may be due o omied forcing variables, U. In addiion, we base our presen values on he expeced presen discouned value of fundamenal variables one a a ime, insead of rying o find he appropriae linear combinaion (excep when we use no be surprised ha he correlaions are sill subsanially below one. m y as a fundamenal.) So we should 24 ECB Working Paper No 248 Augus 2003
The long lieraure on random walks in exchange raes causes us o inerpre he correlaions in Table 3.6 as new evidence ha exchange raes are ied o fundamenals. We recognize, however, ha hese esimaes leave a vas par of he movemens in exchange raes no ied o fundamenals. The resuls may sugges a direcion for fuure research ino he link beween exchange raes and fundamenals looking for improvemens in he definiion of fundamenals used o consruc F 2. Bu why is i so difficul o find a link going he oher direcion using he fundamenals o forecas exchange raes? We urn o ha quesion in he nex secion. 4. RANDOM WALK IN s AS b 1 In he class of models ha we consider, one simple and direc explanaion for s following a random walk is ha he observable fundamenals variables, f, and he unobservable forcing variables, z, each follow random walks. We saw in Table 3.1 ha his is no an appealing argumen for our candidaes for f, since Table 3.1's esimaes indicae ha mos of our measures of f have significan auocorrelaion. Noneheless, i is possible he exchange rae is dominaed by unobservable shocks ha are well-approximaed by random walks ha is, ha z is well-approximaed by a random walk, and he variance of s is dominaed by he changes in z raher han by changes in f. In such a case i may be difficul o rejec he null of a random walk in small samples. We pu his possibiliy aside o consider a more appealing (o us) explanaion an explanaion ha is less relian on assumpions abou unobservable shocks. A. Theoreical Saemen We begin by spelling ou he sense in which he exchange rae should be expeced o follow a random walk for a discoun facor b ha is near 1. We assume ha f and z are forecas using curren ECB Working Paper No 248 Augus 2003 25
and lagged values of an ( n 1) I(1) vecor money-income in secion 2.A, for example, x would include any oher variables used by privae agens o forecas f and z. x whose Wold innovaion is he ( n 1 m, m, y, y, v m, ) vecor. In he v m, q,, and Our proof disinguishes for echnical reasons beween wo ypes of fundamenals, depending on he specificaion of equaion (2.1). We recas (2.1) as: (4.1) s 1 b)( f1 z1 ) b( f 2 z2 ) be s 1 ( Our resul requires ha eiher (1) f z1 1 ~ I(1), f 2 z2 0, or (2) f2 z2 ~ I(1), wih he order of inegraion of f z1 1 essenially unresriced (I(0), I(1) or idenically 0). In eiher case, for b near 1, s will be well approximaed by a linear combinaion of he elemens of he unpredicable innovaion. In a sense made precise in he Appendix, his approximaion is arbirarily good for b arbirarily near 1. This means, for example, ha any and all auocorrelaions of s will be very near zero for b very near 1. Of course, here is coninuiy in he auocorrelaions in he following sense: for b near 1, he auocorrelaions of s will be near zero if he previous paragraph s condiion ha cerain variables are I(1) is replaced wih he condiion ha hose variables are I(0) bu wih an auoregressive roo very near one. For a given auoregressive roo less han one, he auocorrelaions will no converge o zero as b approaches 1. Bu hey will be very small for b very near 1. Table 4.1 gives an indicaion of jus how small small is. The able gives correlaions of wih ime -1 informaion when x follows a scalar univariae AR(2). (One can hink of x f 2 z2 x f1 z1. One can hink of hese wo possibiliies inerchangeably since for given b 1, he auocorrelaions of s are no affeced by wheher or no a facor of 1-b muliplies he presen value of fundamenals.) Lines (1)-(9) assume ha x ~ I(1) specifically, s, or x ~ AR(1) wih parameer. We 26 ECB Working Paper No 248 Augus 2003
see ha for b 0. 5 he auocorrelaions in columns (4)-(6) and he cross-correlaions in columns (7)-(9) are appreciable. Specifically, suppose ha one uses he convenional sandard error of 1 / T. Then when 0. 5, a sample size larger han 55 will likely suffice o rejec he null ha he firs auocorrelaion of s is zero (since row (2), column (5) gives corr ( s, s 1) 0. 269, and 0.269 /[1/ 55] 2.0 ). (In his argumen, we absrac from sampling error in esimaion of he auocorrelaion.) Bu for b 0. 9, he auocorrelaions are dramaically smaller. For b 0. 9, 0. 5, a sample size larger han 1600 will be required, since 0.051/[1/ 1600] 2. 0. We see in lines (10)-(13) in he able ha if he uni roo in x is replaced by an auoregressive roo of 0.9 or higher, he auo- and cross-correlaions of s are no much changed. To develop inuiion on his hypohesis, consider he following example. Suppose he exchange rae is deermined by a simple equaion such as ha of he moneary model (wih suiable redefiniions): s (1 bm ) b be( s ). 1 Assume he firs-differences of he fundamenals follow firs order auoregressions: m. m 1 m ; 1 Then he no-bubble soluion o his model is given by: (1 b ) 1 b s m b 1b 1b 1 b (1 b)(1 b) 1 m 1 1 Consider firs he special case of 0. Then as b 1, s m 1.. In his case, he variance of he change in he exchange rae is finie as b 1. If 0, hen as b 1, s consan. In his case, as b increases, he variance of he change in he exchange rae ges large, bu he variance is dominaed by he i.i.d. erm. ECB Working Paper No 248 Augus 2003 27
B. Discussion We begin by noing ha he classic efficien markes model of Samuelson (1965) and ohers does no predic a random walk in exchange raes. The essence of his model is ha here are no predicable profi opporuniies for a risk-neural invesor o exploi. If he U.S. ineres rae i is higher han foreign ineres rae i by x%, hen he U.S. dollar mus be expeced o fall by x% over he period of he invesmen if here is o be no such opporuniies. In erms of equaion (2.5), hen, he classic efficien markes model says ha he risk premium is zero, and ha a populaion regression of s 1 on i i will yield a coefficien of 1. (For equiies, he parallel predicion is ha he day a sock goes ex-dividend is price should fall by he amoun of he dividend (e.g., Elon and Gruber (1970).) Our explanaion yields a random walk approximaion even when, as in he previous paragraph, uncovered ineres pariy holds. The reader may wonder how he daa can simulaneously saisfy: (1) a regression of 1 on s i i yields a nonzero coefficien, and (2) s is arbirarily well approximaed as a random walk (i.e., s 1 is arbirarily well approximaed as whie noise). The answer is ha when b is arbirarily close o 1, he R 2 of he regression of 1 on s i i will be arbirarily close o zero, and he correlaion of s 1 wih i i will be arbirarily small. I is in hose senses ha he random walk approximaion will be arbirarily good. The key quesion is no he logic of our resul bu he empirical validiy of he assumpions needed for i. We do no require uncovered ineres pariy, which was mainained in he previous wo paragraphs merely o clarify he relaion of our resul o he sandard efficien markes resul. Insead, wo condiions are required. The firs is ha fundamenals variables be very persisen I(1) or nearly so. This is arguably he case wih our daa. We saw in secion 3 ha we canno rejec he null of a uni roo in any of our daa. Furher, here is evidence in oher research ha he unobservable variable z is very persisen. For he money-income model (equaion (2.6)), his is suggesed for v m, q, and 28 ECB Working Paper No 248 Augus 2003
by he lieraure on money demand, e.g., Sriram (2000); purchasing power pariy, e.g., Rogoff (1996); and, ineres pariy, e.g., Engel, (1996). (We recognize ha heory suggess ha a risk premium like is I(0); our inerpreaion is ha if is I(0), i has a very large auoregressive roo.) A second condiion for s o follow an approximae random walk is ha b is sufficienly close o 1. We ake Table 3.1's esimaes of firs order auocorrelaions as suggesing ha he lines in Table 4.1 mos relevan o our daa are hose wih 0. 3 or 0. 5. If so, Table 4.1 suggess ha he second condiion holds if b is around 0.9 or above. This condiion seems plausible in he models skeched in secion 2. In he money-income models presened in secion 2, b is relaed o he ineres semi-elasiciy of money demand: b. Bilson (1978) esimaes 60 in he moneary model, while Frankel 1 (1979) finds 29. The esimaes from Sock and Wason (1993, Table 2, panel I, page 802) give us 40. 3 They imply a range for b of 0.97 o 0.98 for quarerly daa. To ge a sense of he plausibiliy of his discoun facor, compare i o he discoun facor implied in a heoreical model in which opimal real balance holdings are derived from a money-in-he-uiliyfuncion framework. Obsfeld and Rogoff (1998) derive a money demand funcion ha is very similar o equaion (2.3), when uiliy is separable over consumpion and real balances, and money eners he uiliy funcion as a power funcion: 1 M 1 P 1. They show ha 1/ i, where i is he seady-sae nominal ineres rae in heir model. They sae (p. 27), Assuming ime is measured in years, hen a value beween 0.04 and 0.08 seems reasonable for i. I is usually hough ha is higher han one, 3 Bilson uses quarerly ineres raes ha are annualized and muliplied by 100 in his empirical sudy. So his acual esimae of 0. 15 should be muliplied by 400 o consruc a quarerly discoun rae. MacDonald and Taylor (1993) esimae a discouned sum of fundamenals and es for equaliy wih he acual exchange rae following he mehods of Campbell and Shiller (1987) for equiy prices. MacDonald and Taylor rely on he esimaes of Bilson o calibrae heir discoun facor, bu misakenly use 0.15 insead of 60 as he esimae of. Sock and Wason s daa esimaes also use annualized ineres raes muliplied by 100, so we have muliplied heir esimae by 400. ECB Working Paper No 248 Augus 2003 29