Deusches Insiu für Wirschafsforschung www.diw.de Discussion Papers 944 Joscha Beckmann Ansgar Belke Michael Kühl How Sable Are Moneary Models of he Dollar-Euro Exchange Rae? A Time-varying Coefficien Approach Berlin, November 2009
Opinions expressed in his paper are hose of he auhor and do no necessarily reflec views of he insiue. IMPRESSUM DIW Berlin, 2009 DIW Berlin German Insiue for Economic Research Mohrensr. 58 07 Berlin Tel. +49 (30) 897 89-0 Fax +49 (30) 897 89-200 hp://www.diw.de ISSN prin ediion 433-020 ISSN elecronic ediion 69-4535 Available for free downloading from he DIW Berlin websie. Discussion Papers of DIW Berlin are indexed in RePEc and SSRN. Papers can be downloaded free of charge from he following websies: hp://www.diw.de/english/producs/publicaions/discussion_papers/27539.hml hp://ideas.repec.org/s/diw/diwwpp.hml hp://papers.ssrn.com/sol3/jeljour_resuls.cfm?form_name=journalbrowse&journal_id=07999
How Sable Are Moneary Models of he Dollar-Euro Exchange Rae? A Time-varying Coefficien Approach by Joscha Beckmann*, Ansgar Belke** and Michael Kühl*** * Universiy of Duisburg-Essen, ** Universiy of Duisburg-Essen and IZA Bonn, *** Universiy of Goeingen Absrac This paper examines he significance of differen fundamenal regimes by applying various moneary models of he exchange rae o one of he poliically mos imporan exchange raes, he exchange rae of he US dollar vis-à-vis he euro (he DM). We use monhly daa from 975:0 o 2007:2. Applying a novel ime-varying coefficien esimaion approach, we come up wih ineresing properies of our empirical models. Firs, here is no sable long-run equilibrium relaionship among fundamenals and exchange raes since he breakdown of Breon Woods. Second, here are no recurring regimes, i.e. across differen regimes eiher he coefficien values for he same fundamenals differ or he significance differs. Third, here is no regime in which no fundamenals ener. Fourh, he deviaions resuling from he sepwise coinegraing relaionship ac as a significan error-correcion mechanism. In oher words, we are able o show ha fundamenals play an imporan role in deermining he exchange rae alhough heir impac differs significanly across differen sub-periods. JEL codes: E44, F3, G2 Keywords: Srucural exchange rae models, coinegraion, srucural breaks, swiching regression, ime-varying coefficien approach Corresponding auhor: Professor Dr. Ansgar Belke, Universiy of Duisburg-Essen, Deparmen of Economics, Chair for Macroeconomics, D-457 Essen, and IZA Bonn, e-mail: ansgar.belke@uni-due.de, phone: + 49 2083 2277, fax: + 49 20 83 48. Acknowledgmens: We are graeful for valuable commens from he paricipans of he XII. Applied Economics Meeing Conference, held from June 4-6, 2009, Madrid, and he paricipans of he 2 h Goeinger Workshop for Inernaional Economic Relaions, March 2.-4, 2009, Goeingen.
. Inroducion Disenangling he main drivers of exchange raes is sill one of he mos conroversial research areas in economics. Afer he firs generaion models of exchange rae deerminaion which see he exchange rae as he relaive price of domesic and foreign monies (Dornbusch, 976; Frenkel, 976; Kouri, 976; Mussa, 976) was brough o he daa, i became clear ha exchange rae models can only parly be used o explain pas exchange raes wih he help of fundamenals and perform poorly in forecasing, in paricular (Meese and Rogoff, 983 and 988). The resuls of he seminal sudy by Meese and Rogoff (983) sill represen he benchmark: exchange rae forecass by srucural models can hardly ouperform naïve random walk forecass (Rogoff, 2009). Since hen many conribuions have ried o refue heir resuls. Sicking o he implici assumpion ha exchange raes and fundamenals are coinegraed and implemening exogenous parameer resricions, a couple of auhors find predicabiliy in he long run for a similar period as in Meese and Rogoff (Mark, 995; Chinn and Meese, 995). However, exending he esimaion period yields mosly conrary findings (Kilian, 999; Abhyankar, Sarno and Valene, 2005). A criical poin is he implici assumpion of coinegraion which leads o biased conclusions if a sable long run relaion does no exis (Berkowiz and Giorgianni, 200). While he empirical models of he lae 980s mosly neglec he poenial exisence of a long-run relaionship beween he fundamenals and he exchange rae, srucural models which es explicily for a long-run relaionship among exchange raes and fundamenals were applied a he beginning of he 990s. These kinds of empirical models which are based upon coinegraion relaionships can indeed improve he evidence in favour of predicabiliy in he long run when periods up o he end of he 990s are covered (MacDonald and Taylor, 993, 994). 2 However, any exension of he sample period ypically yields a breakdown in coinegraion relaionships (Groen, 999). Surprisingly, lile aenion is direced o examining of he link beween exchange raes and fundamenals wih respec o srucural changes in cases where coinegraion does no hold. Sock and Wason (996) show ha univariae and bivariae macroeconomic ime series are subjec o subsanial insabiliies which resul in poor forecasing performance. Bacchea and Wincoop (2009) argue ha large and frequen variaions in he relaionship Mark (995) is he firs auhor who focuses on more han one exchange raes simulaneously. He includes he Canadian dollar, he Deuschmark, he Japanese yen and he Swiss franc expressed in US dollar. Chinn and Meese (995) do include he pound serling in US dollars as well as he US dollar and he Deuschmark in Japanese yen bu no he Swiss franc. 2 MacDonald and Taylor (994) invesigae he pound serling-us dollar exchange rae. 2
beween he exchange rae and macro fundamenals naurally develop when srucural parameers in he economy are unknown and subjec o changes. Goldberg and Frydman (996a, 200) provide evidence ha some periods exis in which he moneary model is valid and some oher periods in which his is no he case. Thus, he insabiliy of he moneary model in he daa generaing process migh serve as an explanaion for he findings of Cheung e al. (2005) which sugges ha model specificaions ha work well in one period do no necessarily work well in anoher period. 3 In he recen pas, models capable of aking differen regimes ino accoun have been applied o he moneary approach. Sarno, Valene and Wohar (2004) use a Markov regimeswiching model in order o invesigae he response of exchange raes o deviaions from fundamenal values in differen regimes. Sarno and Valene (2008) demonsrae ha exchange rae models ha opimally use he informaion in he fundamenals change ofen and his implies frequen shifs in he parameers". De Grauwe and Vanseenkise (2007) invesigae paricularly he adjusmen of he nominal exchange wih respec o changes in he fundamenals under differen inflaion regimes. Taylor and Peel (2000), Taylor, Peel and Sarno (200) and Kilian and Taylor (2003) make use of models ha allow for smooh ransiion beween wo saes, supporing he hypohesis ha exchange rae adjusmens owards equilibrium pahs is nonlinear. To be more specific, fundamenals become imporan if he deviaion from an equilibrium rae is large. Frömmel, MacDonald and Menkhoff (2005a,b) es direcly for he significance of differen regimes in he exchange rae deerminaion equaion of he real ineres rae differenial model. The laer is one of he rare bu meriorious conribuions using a model in which he coefficiens in he exchange rae deerminaion process iself are allowed o change. However, since he auhors specify heir model in firs differences, hey do no invesigae a long-run relaionship in a sric sense. 4 All oher conribuions focus on deviaions of he exchange rae from a fundamenal value which assumes coinegraion wih implied resricions wihou modelling he long-run srucure separaely. However, boh menioned regime-swiching approaches have in common ha hey only allow for a fixed number of perseveraive, i.e. regularly recurring, regimes. In early works, Schinasi and Swamy (989) and Wolff (987) apply a ime-varying coefficien model (TVP) o moneary models. They are able o show ha heir models display quie beer 3 See also Bacchea and Wincoop (2009). Parameer insabiliy, i.e. an unsable relaionship beween exchange raes and macro fundamenals, is confirmed by formal economeric evidence delivered by Rossi (2005). 4 In order o obain a long-run perspecive, Frömmel, MacDonald and Menkhoff use annual changes saring from a monhly daa se. 3
forecasing properies han fixed coefficien models. Hence, aking ino accoun ime-varying parameers appears o be a worhwhile nex sep owards a valid empirical model of he exchange rae. Differen marke surveys sugges ha differen fundamenals are imporan during differen periods (Gehrig and Menkhoff, 2006). This paern can also be derived from he imperfec knowledge approach which is based on he awareness ha marke paricipans inermienly revise heir views on how fundamenals influence he exchange rae (Frydman and Goldberg, 996b, 2007). Hence, i is reasonable o assume ha a srong and significan relaionship beween exchange raes and fundamenals exiss during sub-periods and ha is naure ends o change considerably over ime. From his poin of view, a fundamenal value of he exchange rae exiss in he sense ha a par of he exchange rae is driven by fundamenals. For his reason, a posiive analysis should be applied insead of a normaive one. Taking hese consideraions as a saring poin, we es for he significance of a couple of differen hypoheses in his dynamic conex. Firs, we check wheher here has been a sable long-run equilibrium relaionship among fundamenals and he US dollar exchange rae vis-à-vis he DM/euro since he breakdown of Breon Woods I. Second, we es wheher he regimes are no perseveraive (DEF) implying ha across differen regimes eiher he coefficien values for he same fundamenals or he significance differ(s). Third, we check empirically wheher here is a leas one regime ino which no fundamenals ener. Fourh, we es wheher he deviaion from he sepwise relaionship acs as an error-correcion erm. The remainder of his paper is organized as follows: Secion wo inroduces he concep of regime-sensiive coinegraion and gives a shor overview of he models we consider laer on. The economeric mehodology is described in Secion hree. We sar wih a muliple srucural change model developed by Bai and Perron (998, 2003) which we apply o he reduced form of srucural exchange rae models. Hypohesis can be rejeced if a leas one srucural change is found. As a nex sep, we use he esimaed breakpoins o generae indicaor funcions based on which we esimae he srucural model in order o obain esimaes for he differen regimes. For his purpose, we apply he fully-modified OLS esimaor by Phillips and Hansen (990) which is able o deal wih non-saionary variables as regressors and regressands. The resuls are hen evaluaed wih respec o he second and hird hypohesis in Secion four. Finally, we consruc an error-correcion erm and regress he 4
change of he exchange rae on his error-correcion erm in order o invesigae wheher he exchange rae adjuss o deviaions from a fundamenal equilibrium relaionship. 2. Moneary models of he exchange rae 2. Theories Afer he breakdown of Breon Woods I, exchange rae models were developed which see exchange raes as asse prices (Dornbusch, 976a; Frenkel, 976; Kouri, 976). All models of his kind have in common ha hey rely on a sable money demand funcion of he form M P r = L( Y, i) wih M he money supply, p he price level and L he money demand depending on real income (Y r ) and ineres raes (i). A basic assumpion of he sandard moneary model is ha he purchasing power pariy (PPP) holds. In he log-linearized form, he exchange rae can be expressed as he difference in price levels which is equal o he difference beween domesic and foreign money supply less real money demand based on money marke equaions, so ha he exchange rae is deermined as follows: s = α + ( β m β y + β i = α + β β m β y + β i f f f f f f 2 3 ) ( 2 3 ) f f f f f f m β m β2 y + β2 y + β3i β3 i In he lieraure, his model is widely known as he Frenkel and Bilson (FB) model. 5 In he original moneary model α is zero and β = β f due o he srucure of he money = demand funcion. Equaion (2) can be rewrien under he resricion ha he (semi-) elasiciies of he ineres raes are equal. This yields: f f f f f s = α + β m β m β y + β y + β ( i i ). (3) 2 2 3 f If he uncovered ineres rae pariy (UIP) holds, ( i i ) can be replaced by he expeced change in he exchange rae ( E ( s+ ) s ). Wih an expecaion generaing mechanism based upon PPP, he differences in ineres raes can hen be replaced by he differences in expeced raes of inflaion. 6 Since i is known ha he exchange rae ofen deviaes from he PPP he adjusmen owards he PPP value can be aken ino accoun in addiion o he expecaions concerning he expeced raes of. () (2) 5 β,2,3 are elasiciies and α is a consan erm. m and y are he logarihms of money supply and real income. The ineres raes are expressed as percenage. 6 This formulaion is equivalen o a money demand funcion in which he expeced raes of inflaion ener as opporuniy coss. 5
inflaion E ( s φ + π π. 7 The real ineres rae model (RID) by Frankel + s ) = ( s s) f (979) arises if he expecaion formaion process is combined wih he UIP and is solved for he expeced change in he exchange rae (equaion (4). f f f f f f s = α + β m β m β y + β y β i i ) + β ( π π ). (4) 2 2 3 ( 4 The negaive sign of he ineres rae differenial implies ha an increase in he differenial is associaed wih an appreciaion of he domesic currency. Wih he help of equaion (4) a similar process can be explained as in he overshooing case of Dornbusch (976a). In Dornbusch (976a) he exchange rae is negaively correlaed wih he ineres rae differenial bu wihou feedback on inflaion expecaions, i.e. β 4 is zero. Equaion (4) allows he exchange rae o deviae from PPP in he shor-run, i.e. i reacs negaively on ineres raes, bu sill posiively on inflaion rae expecaions. Following Frankel (979) o he word, f β and β mus be equal o one. 8 Since a disincion mus be made beween he Dornbusch model and he Frankel model we refer o he RID model when alking abou equaion (4). A weakness of he radiional moneary model is ha he real exchange rae is assumed o be consan in he long-run. In order o ake accoun of real shocks, Hooper and Moron (982) inroduce changes of he equilibrium real exchange rae ino he radiional moneary model (HM model). In addiion o nominal impac facors, he real side of he economy is inroduced by aking accoun of innovaions in he curren accoun. The equilibrium real exchange rae depends on he desire of domesic and foreign agens o accumulae (or decumulae) ne foreign asses in he long run. Since he desire o accumulae (or o decumulae) ne foreign asses is refleced by he equilibrium curren accoun surplus, he equilibrium real exchange rae is linked o he equilibrium ne foreign asse posiion and he equilibrium curren accoun posiion. An unexpeced rise in he curren accoun means ha oo many ne foreign asses are accumulaed which in urn reduces he demand for foreign capial and causes he domesic currency o appreciae nominally. Thus, unexpeced (posiive) shocks o he equilibrium ne foreign asse posiion resul in a nominal appreciaion. Hooper and Moron (982) proxy he ne foreign asses wih he cumulaed curren accoun. Thus, 7 φ denoes he adjusmen speed owards he equilibrium value s. π is he expeced rae of inflaion. 8 Neverheless, Driskill und Sheffrin (98) show ha overshooing requires β > 0 and f 0 6 β. <
equaion (4) can be exended by he cumulaed rade balances as a proxy for he curren accoun balance (eq. (5)). 9 f f f f f f f f s = α + β m β m β y + β y β i i ) + β ( π π ) β CTB + β CTB. (5) 2 2 3 ( 4 5 5 The Hooper and Moron model is usually applied by esimaing equaion (5) wih cumulaed overall domesic and foreign rade balance. Wihou a loss in generaliy he cumulaed overall rade balances can be replaced by he rade balances wih he same meaning because he equilibrium change in he ne foreign asse posiion is he equilibrium rade balance. In addiion o he real exchange rae moive, Hooper and Moron (982) also use he overall rade balances as an indicaor for he risk premium which arise from governmen deb, an insufficien holding of inernaional reserve and foreign indebedness. A fall in he ne foreign asse posiion (in paricular if i is negaive) increases he risk premium from which an increase in he exchange rae follows. Hence, he risk premium sensiively reacs o a worsening of a negaive ne foreign asse posiion. In a bilaeral case i is sraighforward o use he bilaeral cumulaed rade balance (BCTB) insead of he overall cumulaed rade balances (equaion (6)). f f f f f f 0 s = α + βm β m β 2 y + β 2 y β 3 ( i i ) + β 4 ( π π ) β 5BCTB. (6) Since i is expeced ha he PPP holds for raded goods raher han for a mixure of raded and non-raded goods as implicily assumed by using he overall price index, he prices of raded goods can be aken ino accoun (Dornbusch, 976b). If he overall price index, which is deermined by he money marke, consiss of prices of boh raded and non-raded goods and if he PPP is only valid for raded goods, he moneary approach yields an exchange rae deerminaion equaion of he form s = α + β T f f P f β3 ( i i ) + β4( π π ) + β6 β NT 6 P ft f f f f m β m β2 y + β2 y fnt P The proporion of raded o non-raded goods mirrors he real exchange rae. A rise in he price of radeables relaive o he price of non-radeables les he nominal exchange rae increase because he domesic good is subsiued by he foreign good. In he flex price model β 4 is equal o zero and he exchange rae reacs posiively o he ineres rae differenial (Wolff, 987). P. (7) 9 Since daa on he curren accoun are no available a a monhly frequency, i is adequae o proxy he curren accoun by he rade balance. 0 However, noe ha using he cumulaed bilaeral rade balance as a proxy for ne foreign asses covers only a par of he curren accoun. Besides he ransfers, income and rade in services are excluded. Since reurns on capial dominae he income variable he laer depends predominanly on reurns such as ineres raes which are included. Since rade in services was a minor issue over large pars of he sample i is reasonable o exclude i in our sudy. T denoes radeables and NT non radeables. 7
In applied moneary models, equaion (2) is ypically esimaed based upon a reduced form in which i is assumed ha he elasiciies for an economic variable are idenical in boh f f f counries. Hence, he resricions β = β, β 2 = β 2 and β 3 = β 3 apply (Meese and Rogoff, 983). However, any analysis in which he coefficiens are resriced o be equal for each variable ypically resuls in biased coefficiens (Haynes and Sone, 98). If he srucure of he economy is no known a priori, resriced coefficiens do no help in explaining he exchange rae. While he radiional moneary model assumes ha domesic and foreign asses are perfec subsiues he assumpion is relaxed by highlighing he role of risk as described by Hooper and Moron (982). A model ha explicily akes accoun risk premia ino accoun is he porfolio balance model (Branson, 977). If a risk premium becomes more imporan, i is preferable o use he porfolio balance approach. In he following we make use of a hybrid model which caches effecs ha can be found in boh moneary and porfolio models (Frankel, 983). As a consequence, we remove he resricions of parameer equaliy of he ineres rae differenial and he inflaion rae differenial in equaions (4), (5), (6), and (7). Thus, we sar our analysis as unresricive as possible and bear in mind dynamics semming from boh he porfolio balance approach and he moneary approach. Finally, we have four differen models which all rely on he baseline specificaion of he unresriced RID model exchange rae deerminaion equaion in equaion (4). 2.2 Long-run analysis wih ime-varying coefficiens Wolff (987) menions hree reasons why a ime-varying coefficien model should be superior o fix coefficiens models. Firs of all, he money demand funcion is subjec o insabiliies which cause he coefficiens in he exchange rae deerminaion equaion of a reduced model o change (Levenakis, 987). Anoher reason is he famous Lucas criique: coefficiens change if an anicipaed change in he policy regime occurs. The hird argumen is relaed o he long-run real exchange rae. The moneary model assumes ha purchasing power pariy holds in he long run from which follows ha he long-run real exchange rae is sable. Innovaions o he real exchange rae from he real side of he economy can lead o changes in he coefficiens. Because we explicily accoun for changes in he real exchange rae he laer issue deserves less aenion in our analysis wih respec o he choice of he esimaion echnique. A reason for choosing ime-varying coefficien models can also be derived from differen heories. In iner-emporal new open economy macroeconomic (NOEM) models (Obsfeld and Rogoff, 995), money demand does no depend on income, bu on real 8
consumpion. If we proxy real consumpion by real income, a change in he average rae of consumpion resuls in a change in he elasiciy of income in he exchange rae equaion. Thus, if consumpion shares do vary which is, for insance, rue for he US he exchange rae deerminaion equaion is hus also ime-varying. As argued by Wilson (979), an anicipaed policy change, i.e. an expansionary moneary policy, can generae dynamics which are differen from ha semming from unanicipaed changes. In Wilson (979) he overshooing dynamics are slighly differen from hose in Dornbusch (976a). A very imporan resul is ha an appreciaion period of he domesic currency coincides wih he increase in money supply while in he Dornbusch model a boos in money supply coincides wih a depreciaion. If anicipaed and unanicipaed shocks alernae, fixed coefficien models are inadequae because hey canno cach boh effecs simulaneously. According o he resuls gained by Sarno, Valene and Wohar (2004) or de Grauwe and Vanseenkise (2007), he adjusmen of exchange raes owards he long-run equilibrium relaionship does no appear o be ime-invarian. However, we expec ha adjusmen differs from period o period, a leas over a long span of daa. An adjusmen owards he long-run equilibrium relaionship can occur because he exchange rae predominanly reacs on he fundamenals or because, vice versa, he fundamenals reac o changes in exchange raes. In he laer case, i is possible ha he exchange rae does no adjus in sub-periods. Consequenly, he adjusmen coefficien has he poenial o differ beween sub-periods. Siklos and Granger (997) develop a framework well-suied o analyze hese issues in he necessary deail. They poin ou ha a coinegraion relaionship can be subjec o srucural changes and argue ha he common sochasic rends are only presen in specific periods. In his respec hey inroduce he concep of regime-sensiive coinegraion, or swich on swich off coinegraion. This concep of regime-sensiive coinegraion can be combined wih a ime-varying coefficien approach as follows. Le differen processes where X, 2 X and Y be Y 2 2 = S + β S + φ X X y β Z + ε, (8) = S + φ Z + ε x 2 = S + φ Z + ε 2 2 2 x2 2, and (9). (0) 9
and The variables S and Z are boh I() bu do no share a common sochasic rend. y ε are boh i.i.d. error processes which follow a normal disribuion wih zero mean. k Furhermore, β can be a ime-varying coinegraion parameer, i.e. x ε wih β = β +... + β wih k = [,2 ] () k k k k m k m j j k = ( T < < T ), wih j =,..., m. (2) In equaion (2) we do no allow for any overlap of he ime periods overlap and he coinegraion parameer is permied o be absen during sub-periods. From his i follows ha one of he wo common sochasic rends can vanish in equaion (8). Imposing coinegraion on X, 2 X and Y requires ha a linear combinaion of X, 2 X and Y wih coinegraion vecor of (, β, β 2 )' is saionary. Hence, he linear combinaion is: Y β X β X 2 2 = β S + β S 2 2 = ( φ β φ β φ ) Z 2 2 2 2 + φ Z β S β φ Z β ε + ε x 2 y β S 2 2 y β φ Z 2 2 2 2 x2 + ε β ε β ε. x β ε 2 x2 (3) Equaion (3) is a coinegraion relaionship if φ β φ β φ is zero and he 2 2 2 2 sochasic rend Z vanishes so ha coinegraion is swiched on. Similarly o equaions () and (2), a ime-varying represenaion of φ and k φ can be achieved. For his reason, hese parameers can be presen or absen in sub-periods. This resul is independen of he number of common sochasic rends involved in he sysem. If he condiion is no valid, coinegraion is swiched off. The combinaion of equaions (8), (9), (0), (), and (2) shows ha he sysem is driven by wo common sochasic rends which can be absen in subsequen periods. If one of he sochasic rends in equaion (8) is currenly absen, he corresponding k X variable does no ener he coinegraion vecor and he coinegraion vecor only conains wo elemens. Coinegraion is coninuously presen over he whole period of observaion while merely he composiion of he coinegraion vecor is changing. If a sysem has a leas one coninuous common sochasic rend, Y coninuously coinegraes wih k X only under he condiion ha φ β φ β φ is zero. Wih 2 2 2 2 ec = ε β ε β ε, he error-correcion erm herefore urns ou o be y x 2 x2 0
ec = Y β X β X 2 2, (4) for which he error-correcion presenaion resuls as follows Δ Y = α ( Y β X β X ) + η 2 2, (5) where η is a i.i.d. variable which follows a normal disribuion wih zero mean. In addiion o a ime-varying coinegraion vecor, we allow he causaliy beween he variables o change during he period of observaion. This means ha he dimension of he vecor which conains he adjusmen coefficiens can be reduced during sub-periods. Assuming ha he adjusmen of he k X is sill presen, as long as coinegraion prevails, α in equaion (5) does no only change is magniude, i can also be zero if Y does no adjus a all o he longrun relaionship. In a long-run relaionship analysis we hus are poenially confroned simulaneously wih swich on and off coinegraion, a changing coinegraion vecor and he adjusmen process. The difficuly wih our esimaions hen is o cope wih poenial overlaps of hese phenomena. Hence, our approach akes accoun of differen regimes. Hence, i is able o disinguish beween cases in which he coinegraion relaionship is swiched on and hose in which differen adjusmens are presen. For a mulivariae case we consider he erm Y = μ + β X + ε (6) wih k X = [ X,..., X ] for n =,..., K, (7) where K represens he maximum number of explanaory variables. 2 The marix has he dimension ( K ) and β he dimension ( K ) following models under closer scruiny: X. In our empirical analysis, we pu he Model one: Y f f f f ( m y i π m y i π ) ' = [ s ], X = (8) 2 μ is a regime-dependen consan erm. The variable ε represens an error erm.
Model wo: Y f f f f ( m y i π m y i π ) ' = [ s ], X = BCTB (9) Model hree: Y f f f f f ( m y i m y i ΔCTB ΔCTB ) ' = [ s ], X = π π (20) Model four: Y T ft [ f f f f p p ], s = X m y i π m y i π (2) NT p p = fnt ' 3. Modeling srucural changes and esimaing coinegraing relaions - mehodological issues 3. Tesing for muliple srucural changes In general, wo frameworks for ess for srucural change can be disinguished. The firs framework consiss of generalized flucuaion ess fi a model o he daa and derive an empirical process ha capures he flucuaions eiher in he residuals or in parameer esimaes. If he generaed process exceeds he boundaries of he limiing process, which can be derived from he funcional cenral limiing heorem, he null of parameer consancy has o be rejeced, meaning a srucural change occurs a he corresponding poin in ime (Zeiless e al, 2003). The classical and he OLS based CUSUM es and he flucuaion es of Nyblom (989) are well-known examples of hese mehods. These srucural change ess are predominanly designed for saionary variables. In he case of a coinegraion analysis an eigenvalue flucuaion es developed by Johansen and Hansen (999) which heavily relies upon Nyblom can be applied. While hese procedures have he advanage of no assuming a paricular paern of deviaion from he null hypohesis hey can eiher only idenify a single break or show general insabiliy. The second framework o es for srucural changes is o compare he OLS residuals from regressions for differen subsamples. This can be done, for example, by applying he F- saisics or he Chow es. In his paper, we adop an exension of he laer case developed by Bai and Perron (998, 2003). Their basic idea is o choose breakpoins such ha he sum of squared residuals for all observaions is minimized. 2
regimes As a saring poin, consider a muliple linear regression wih m breakpoins and m+ ' ' y = x κ + z δ + u, = T j +,..., T ), j ( j (22) for j=, m+ wih he convenion ha T = 0 0 and T T m = +. y is he dependen variable, x and z denominae he regressors and κ and δ are he coefficien vecors. Noe ha only δ varies over ime while κ is consan. Wih a sample of T he firs sep is o calculae he corresponding values for all possible ( T +) 2 T segmens. 3 The esimaed breakpoins T...T m by definiion represen he linear combinaion of hese segmens which achieve a minimum of he sum of squared residuals (Bai and Perron, 2003). Formally: Tˆ ˆ,..., Tm ) = arg mint,..., (,..., ). T ST T T m m (23) ( Bai and Perron (2003) develop a dynamic programming algorihm which compares all possible combinaions of he segmens. Their mehodology allows esing for muliple srucural breaks under differen condiions. 4 Wihin our framework, he locaion of he breakpoins is also obained by calculaing he sum of squared residuals. To selec he dimension of he model we apply he Bayesian Informaion Crierium (BIC) which according o Bai and Perron (200) works well in mos cases when breaks are presen. Afer calculaing he ess for all possible breakpoins he sequence ( T,..., ˆ ) T m ˆ is seleced as he configuraion a which he BIC achieves is minimum. Carrioni-Silvesre and Sanso (2006) show ha his approach yields a consisen esimae of he break fracion. The breakpoins obained in his fashion are a local minimum of he sum of squared residuals given he number of breakpoins bu no necessary a global minimum. 3 Bai and Perron (998) noe ha for pracical purposes less han T(T+) segmens are permissible, for example if a minimum disance beween each break is imposed. In he framework of his paper, breaks are allowed o occur every 2 monhs. 4 One possibiliy is o es he null of no change agains he hypohesis of a fixed number of breaks m=k using F- ess based on he sum of squared residuals under boh hypoheses. For an unknown number of breaks, one way is o allow a maximum number of breaks. In his case one can apply he so called double maximum es. The number of breakpoins is hen seleced by comparing he F-values described above for he differen numbers of breakpoins and selec he configuraion wih he highes F-value respecively he minimum of he sum of he squared residuals. Anoher possibiliy is o es sequenially for an addiional break using he l vs. l+ break ess. For deails see Bai and Perron (998, 2003). 3
I is imporan o noe ha he procedure of Bai and Perron has originally been developed for he case of saionary variables (I(0)). Neverheless, i can as well be applied o non-saionary variables which are inegraed of order one (I()). For insance, Siklos and Granger (997) use his mehodology o idenify srucural breaks in he ineres pariy equaion beween he Unied Saes and Canada in he conex of regime-sensiive coinegraion. In addiion, Zumaquero and Urrea (2002) poin ou ha he break esimaor is consisen also in he non-saionary case. Using disaggregaed price indexes for seven counries, hey es for srucural breaks in he coefficiens of coinegraing relaions which represen absolue and relaive purchasing power pariy. They also examine insabiliies in he adjusmen behaviour of price raios and exchange raes. Finally, Perron and Kejriwal (2008) demonsrae ha he resuls of Bai and Perron (998) in general coninue o hold even wih I(0) and I() variables in he regression. 5 This is also rue if one allows for endogenous I() regressors. 6 The use of informaion crieria as he BIC is also correc in boh cases. To check our resuls for robusness, we also apply he CUSUM es combined wih Andrews and Ploberg (994) in a similar way as Goldberg and Frydman (200) o deec possible breakpoins. However, wih no considerable differences arising from he resuls, we proceed using he breakpoins obained by he Bai and Perron mehodology. 3.2 Esimaing coinegraing relaions wih single equaions Afer idenifying he breakpoins we now urn o he issue of correc esimaion. As Bai and Perron s mehodology is designed for single equaions, we canno consider mulivariae sysem esimaors as proposed by Johansen (988) or Sock and Wason (988). Besides he radiional approach of Engle and Granger (988), several modified single esimaors have been developed. Examples are he fully modified esimaor by Phillips and Hansen (99) and he approach of Engle and Yoo (99). 7 Even in he case of a muli-dimensional coinegraion space, single equaion approaches can be used o achieve asympoically efficien esimaes of single coinegraing relaionships. For our purposes, he fully modified (FM) esimaor is he mos suiable mehod. In conras o radiional single equaion formulas i considers endogenous regressors (Phillips, 99). Phillips and Hansen (990) show ha he FM-OLS esimaor is hyperconsisen for a uni roo in single equaions auoregression. Phillips (995) proves ha his procedure is 5 This is only rue if, as in our case, he inercep is allowed o change across segmens. 6 For he case wihou uni roos, Perron and Yamamoo (2008) show ha he esimaion of he break daes via OLS is preferable o an IV procedure in he presence of endogenous regressors. 7 For a review of he differen esimaion mehods of esimaing coinegraing relaionships see Hargreaves (994), Phillips and Lorean (99) and Capporale and Piis (999). 4
reliable in he case of full rank or coinegraed I() regressors 8 as well as wih I(0) regressors. Hargreaves (994) runs a Mone Carlo simulaion and poins ou ha single esimaors, in general, are robus if more han one coinegraing relaion exiss, wih he FM-OLS esimaor doing bes. He concludes ha he FM-OLS esimaor should be preferred, even in advance o mulivariae mehods, if one wans o examine one coinegraing vecor and is unsure abou he coinegraing dimensionaliy. This is of paricular ineres for he aim of his paper as we are primarily ineresed in he long-run relaionship beween exchange raes and fundamenals and do no wan o pay oo much regard o oher coinegraing relaionships which migh arise beween he repored fundamenals. Caporale and Piis (999) claim ha he FM-OLS esimaor and he Johansen esimaor perform bes in finie samples. 9 The roo idea of his concep is o esimae coinegraing relaions direcly by correcing radiional OLS wih regard o endogeneiy and serial correlaion (Phillips, 995). Le z denominae an n-vecor where y denoes an r dimensional I() process while ( n r) = (( n r) + ( n r regressors. follows: ) 2 X is an dimensional vecor of coinegraed or possibly saionary u represens an n-vecor saionary ime series. Boh vecors can be pariioned as z y = x x 2 u u = u u 2 3 (24) The daa generaing process of y is represened by he following coinegraed relaion y = β x + u (25) The vecors of he regressors are specified as follows Δ x = u2 (26) x u3 2 = (27) The esimaor correcions can be applied wihou pre-esing he regressors for uni roos as boh correcions can be conduced by reaing all componens of x as non-saionary. For he non-saionary componens, his ransformaion reduces asympoically o he ideal 8 Noe ha he direcion of coinegraion does no need o be known. Regressors conaining a deerminisic rend are also allowed. 9 Furhermore, also Phillips and Hansen (990), Hargreaves (994) and Cappucio and Lubian (200) repor good finie sample properies of he FM-OLS esimaor. 5
correcion while he differenced saionary componens vanish asympoically. Such a correcion does no have any effec on he sub-vecors of x where serial correlaion or endogeneiy are no presen. 20 A furher advanage is ha we do no have o accoun for coinegraion beween he x regressors wihin his mehodology (Phillips, 995). To imply he correcions, we firs consider he long-run covariance marix Ω which can be decomposed ino a conemporaneous variance and he sums of auo-covariances (Hargreaves, 994). Ω = E( u u ' ) + k = 2 E( u 0 u ) + ' k k = 2 E( u u k ' 0 ) (28) ' Ω = +λ + λ (29) We define Δ as Δ = +λ (30) Esimaion of hese covariance parameers can be achieved by using he pre-whiened kernel esimaor suggesed by Andrews and Monahan (992). 2 The endogeneiy correcion hen has he form y * = y Ω ˆ xω ˆ 0 xxδx (3) The above correcion is employed o accoun for endogeneiies in he regressors x 0 linked wih any coinegraion beween x 0 and y. The second correcion akes ino accoun he effecs of serial covariances in he shocks he hisory of u and any serial covariance beween u 0 and u. The bias effec arises from he persisence of shocks due o he uni roos in x. The induced one-sided long-run covariance marices carry hese effecs in an OLS regression (Phillips, 995). They can be defined as 20 Wihou serial correlaion or endogeneiy he FM-OLS esimaor is idenical o he OLS esimaor. 2 Oher sudies adop he esimaor of Newey and Wes (987) which is robus o serial correlaion and heeroskedasiciy. For deails see Cappuccio and Lubian (2003). 6
Δˆ 0x = Ωˆ 00 Ωˆ 0x Ωˆ xx Ωˆ x0 (32) The correcion is hen given by Δˆ * x = Δˆ x Ωˆ xω ˆ xxδ ˆ 0 0 0 xx (33) Combining boh correcions he formula for he fully modified esimaor is 22 ˆ * *' * ' β = ( Y X TΔ )( X X ) 0x (34) 3.3 Regime shifs in Coinegraion models To apply he FM-OLS esimaor in a model wih srucural changes we proceed in a similar way as Hansen (2003) does in he Johansen framework by allowing he parameers o change heir values a he breakpoins. 23 We rewrie equaion ( 22 ) wih τ () as a consan y = ) + ' ' τ ( ) + xκ( ) + zδ j ( u (35) The piecewise consan ime-varying parameers are given by δ j + ( ) = δ... δ m m (36) κ j + ( ) = κ... κ m m (37) τ = τ +... τ (38) ( ) m m where he indicaor funcion for each subsample is defined as follows (Hansen, 2003) m j j = ( T + < < T ), J=,.m (39) ˆ β = Y' X ( X ' X ) 22 The radiional OLS esimaor is given by 23 We corroboraed our resuls wih a relaed approach inroduced by Gregory and Hansen (996). They model he changes in he inercep and he slope coefficiens relaive o he firs subperiod as a benchmark, running from 0 o T. The base model ' ' ' ' is hen wrien as y = τ + τ ( ) + xκ + xκ( ) + zδ + zδ j ( ) + u. 7
wih he convenion ha T 0 = 0 and T m = T. Defining dummies according o he indicaor funcion ensures ha we are able o obain esimaes for each period. 4. Daa and esimaed Models 4. Daa Our sample conains monhly daa running from January 975 unil December 2007. We use he aggregae M for money supply. Real income is proxied by he real producion index. As suggesed by Wolff (987) he producer price index serves as a proxy for radeable goods while he baske of non-radeables is refleced by he consumer price index (CPI). Furhermore, we use he overall rade balance as an approximaion of he cumulaed curren accoun. As seen in he HM model, he equilibrium flow deermines he equilibrium sock. Since he bilaeral rade balance can be expressed in wo currencies, i is no quie clear which denominaion currency should be used. In he case of our analysis a separae coinegraion analysis (no repored) shows ha he US dollar denominaed balance adjuss o he euro denominaed one. Thus, we choose he euro configuraion. For he shor-erm ineres raes we use money marke raes wih a mauriy of hree monhs. Exchange raes, money supply and real income are expressed in logarihms. All series are seasonally adjused and are aken from Inernaional Financial Saisics of he Inernaional Moneary Fund. In srong conras o oher sudies invesigaing he euro exchange rae, we rely on he Deuschmark and he fundamenals of Germany before he inroducion of he euro. The reason is ha we are ineresed in marke raes which could be conrased by using weighed ECU-Daa. In a sense, he Deuschmark has been a predecessor of he euro as i had a similar imporance on he foreign exchange marke. One reason was he big influence of he German Bundesbank (Fraianni and von Hagen (990). We herefore use a ime series which conains he German values unil December 998 and, from hen on, he values of he euro area. Consequenly, he Deuschmark / US dollar exchange rae is convered by he official Deuschmark / euro exchange rae in order o obain a level adjusmen. As a consequence, we also adjus he German fundamenals in levels o allow for a smooh ransiion o he euro area daa. Since we deal wih srucural break models in he empirical secion, we do no see any problems wih our proceeding. The reason is ha if a break due o daa adjusmen were imporan, he Bai-Perron es would signify a break around January 999. 8
4.2 Preliminary ess for uni roos and saionariy Alhough he FM-OLS esimaor and he Bai-Perron mehodology are able o handle a combinaion of I(0) and I() regressors, esing he daa for uni roos is necessary as a firs sep. Wih he exchange rae being an I() variable, he concep of coinegraion only makes sense if he fundamenals can also be reaed as I() processes. By definiion, a coinegraing relaionship can only exis beween variables which are inegraed of he same order (Engle and Granger, 987). Neiher can a saionary variable force a non-saionary variable o adjus, nor is a saionary relaionship beween I() and I(2) variables possible. Furhermore, inferences in a model wih I(2) variables are far more complicaed from a saisical poin of view. To es for uni roos, we apply he Phillips-Perron es, he KPSS es and he DF-GLS es. In he firs insance we es for saionariy in he levels. Differences are aken and esed again if a uni roo remains, i.e. if he corresponding variables are inegraed of order wo. If boh hypoheses are rejeced we conclude ha he variable is I(). In he case of he cumulaed overall rade he resuls of he ess sugges ha he balance is inegraed of order wo. Therefore, we decide o work wih differences for he US and he euro area series. This can be done wihou changing he underlying economic heory. The resuls of he ess are presened in Table. According o he resuls, all variables can be considered as being inegraed of order one alhough, in some cases, he evidence is mixed. Table abou here The KPSS es rejecs he hypohesis of saionariy for he change in he money supply of he Unied Saes, he change in he bilaeral rade balance and he wice differenced rade balance of he euro area. Furhermore, he hypohesis of a uni roo is rejeced for he change in he rade balance of he euro area according o he DF-GLS and he Phillips-Perron es. However, since he oher ess indicae conrary resuls for hese series we rea hem as I(). 4.3 Empirical resuls 4.3. Esimaion of he breakpoins The breakpoins we are able o idenify by applying he Bai and Perron mehodology are presened in Table 2. Obviously, breaks occur quie frequenly. Hence, we conclude ha here is no sable and unique long-run equilibrium relaionship among fundamenals and exchange raes since he breakdown of Breon Woods I. Anoher resul is ha, despie a couple of differences, also some significan similariies beween he various configuraions emerge. For 9
insance, he number of breakpoins always lies beween eigh and en even hough we allow for a shif every welve monhs. Furhermore, he daes of breakpoins for he differen models are locaed closely ogeher. An encouraging resul is ha, in many cases, major economic or poliical developmens are able o deliver good explanaions for insabiliies. Table 2 abou here The breaks in 976 and 977 (row 2 of Table 2) can clearly be addressed wih an eye on he macroeconomic urbulences arising from he oil price shocks and worldwide recession. Furhermore, insabiliies ofen occur wihin he epoch of he so-called pseudo-monearism policy of he FED wihin 979 and 982 (Timberlake, 993) or a he end of he rise of he US dollar during he mid 980s. From his poin of view, we feel legiimized o explain he breaks of model and 3 in a exbook-syle fashion by he beginning of he moneary experimen (row 3 of Table 2). In addiion, he regular inervenions by he reasury were sopped as had been announced in April 98 alhough unil 985 infrequen inervenions occurred (c.f. Desler and Henning, 989). In order o suppor he real economy, he federal funds rae sared o fall in mid 98. This dae coincides wih breakpoins in each model (row 4 of Table 2). The nex breakpoin locaed around Ocober 988 (row 7 of Table 2).can also well be raced back o a specific sance of moneary policy. In 988, he moneary policy sance on boh sides of he Alanic, i.e. of boh he US Fed and Germany s Bundesbank, became more resricive. Besides he usual moneary policy suspecs, he elecion of George Bush senior and he G-7 summi in Berlin 24 offer furher popular explanaions. For each model, breakpoins are idenified in February 992, shorly afer he German reunificaion (row 8 of Table 2). The following insabiliy in 992 and 993 (row 9 of Table 2) is usually aribued o he crisis of he European Moneary Sysem. A his ime, also significan changes in US and German moneary policies have o be aken ino accoun. Wihin a comparaively sable period unil he end of he 990s, he only insabiliy, in 997 (row of Table 2), is said o be caused by he Asian currency crisis and/or he worsening of he US rade balance which had sared in 996. Aferwards, breaks are repored by he Bai and Perron procedure for model one, wo and hree in 2000 (row 3 of Table 2) and for each model in November 2004 (row 4 of Table 2). In mid-2000, he American economy sared o slow down wih he American sock marke crashing. Ineresingly, he las break coincides exacly wih an even which saw he shor-erm ineres raes of he euro 24 In conras o previous meeings, he paricipans of he Berlin meeing did no publically claim ha flucuaions in he dollar were unwaned. 20
area declining below he level of US ineres raes. Of course, as far as he daing of breakpoins and heir economic inerpreaion are concerned, we preferred o follow quie sandard inerpreaions. Moreover, one should also no forge ha many oher imporan developmens are no refleced by breaks. However, in all hese cases he ineres rae differenial seems o play an imporan role. As becomes obvious afer a visual inspecion of Figure, many breaks correspond o and are hus poenially riggered by changes in he rend of he ineres rae differenial or o changes in is sign. Figure abou here 4.3.2 Inerpreaion of he ime varying coefficiens Moving one sep furher, we proceed by esimaing he coinegraion vecor via FM-OLS using he obained break daes. Table 3-6 conain he resuls for he specified models. Since configuraion is embedded in he oher hree configuraions, we predominanly draw on he resuls of configuraions 2 (Table 4), 3 (Table 5) and 4 (Table 6) and use configuraion (Table 3) jus for comparison. In order o ake accoun of he differen model specificaions proposed by Hooper and Moron (982), we draw on model 2 and model 3 o disinguish beween he bilaeral ne foreign asse posiion and he overall ne foreign (nfa) asse posiions of each counry (in our case, he changes in he nfas). A comparison of model 2 and 3 wih 4 helps us in separaing real effecs as he laer case doesn accoun for changes in he rade balances Table 3 abou here Models, 2 and 3 are broadly consisen wih he real ineres rae model (Equaion 4) in he firs sub-period afer our period of observaion sars (Row of Tables 3-5). From his poin of view, our empirical findings clearly corroborae he findings in he lieraure concerning he early period afer he breakdown of Breon Woods I (for an early overview see, for insance, Isard, 987). Only in he case of model 4 does he German inflaion expecaions variable ener he regression equaion wih an incorrec sign of is esimaed coefficien (Row, column 5 of Table 6). While he overall change in he overall ne foreign asse posiion (nfa) of he euro area/ of Germany in model 4 is no significan, he same variable urns ou o be significan a he % level in he US case (Row, columns 0 and of Table 6). Is negaive sign indicaes ha risk consideraions seem o be imporan. During his period, a worsening of he US rade accoun is linked o a depreciaion of he US dollar. I is imporan o noe ha he US money supply seems o be srongly linked o he exchange 2
rae. During his period boh variables appear o share common rends. (Row, column 6 of Tables 3, 5 and 6). Table 4 abou here From 977:05 ill 979:2 many coefficiens of model 3 show signs which are no consisen wih sandard heory (Row 2 of Table 5). The esimaed coefficiens of boh he German money supply and he German inflaion expecaions urn ou o be highly significan wih a negaive sign (row 2, column 2 and 5 of Table 5). When eiher he relaive price of radeables (row 2, column 0 and of Table 6). or he bilaeral nfa (row 2, column 0 of Table 4). is aken ino accoun in model 2 and 4, heir coefficiens display he correc sign and he significance of he money supply and he inflaion raes disappears. One can hink of several reasons for ha paern. On he one hand, he sub-periods of model 2 and 4 are similar in his example bu differen from model 3. On he oher hand, he second oil price shock ook place exacly in his period. I becomes obvious ha real shocks have an impac on he exchange rae and le he impac of nominal facors shocks vanish. Table 5 abou here The paern of he esimaion resuls for he sample period from 979:2 ill 98:06 in model 3 are again broadly consisen wih he heory (row 3 of Table 5). Despie he fac ha he above menioned episode of he moneary experimen iniiaed by he US Fed falls in his period, i becomes obvious ha he coefficiens for US money supply and inflaion raes are in line wih he heory, i.e. impacs of US moneary variables deermine he exchange rae. This is paricularly rue for model and 3 (row 3, columns 6 and 9 of Tables 3 and 5). The only deviaion from he real ineres rae model (equaion 4) is ha he German shor-erm ineres raes ener wih a posiive sign which indicaes ha he opporuniy coss of holding money are imporan in he shor run. Table 6 abou here Beween 98 and he end of 984 (Model ) respecively he beginning of 985 (Model 4) he esimaed coefficiens of US money supply and he US real income variable show signs which are no consisen wih sandard heory. (row 3, columns 2 and 3 of Tables 3 and 6). Following he broad picure conveyed in Figure 2 which displays he ime series of he macroeconomic indicaors in he Unied Saes from 973 unil 2007, we aribue his paern o he deepening recession in he US economy. However, if relaive prices of radeables are included, his yields signs of he esimaed coefficiens which are broadly consisen wih he underlying heory of secion 2. (row 3, columns 0 and of Table 6). 22