The UIP: An Unbiased and Efficient Estimator

Transcription

1 The UIP: An Unbiased and Efficien Esimaor W. A. Razzak Absrac This paper examines he uncovered ineres rae pariy condiion a long and shor run horizons (low and high frequency). An asympoically unbiased and an efficien nonlinear dynamic leas square esimaor is used o esimae he long-run coefficiens. Condiional on coinegraion, shor-run relaionships over he cycles are hen examined. There is evidence in favor of UIP. [JEL # F3, F4, C3] The auhor is an advisor a he Economics Deparmen of he Reserve Bank of New Zealand. Conac address is Reserve Bank of New Zealand, P. O. Box 498, Wellingon, New Zealand, razzakw@rbnz.gov.nz. Views expressed in his paper do no necessarily reflec hose of he Reserve Bank. This work has benefied grealy from discussions and close work wih Francisco Nadal De Simone. I am graeful o Peer Phillips, Nelson Mark, Rober Flood, Andrew Rose, Yin Wong Cheung, Menzie Chinn, John McDermo and paricipans of New Zealand Economic Meeing in Chrischurch 997 and he Far Easern Meeing of he Economeric Sociey 999 for heir generous commens.

2 . Inroducion Obsfeld and Rogoff (996, p. 63) show a close graphical relaionship beween he rade-weighed dollar exchange rae and he real ineres rae differenial uncovered ineres rae pariy, UIP and ask why doesn he visual impression come hrough in empirical analysis? This voluminous lieraure s conclusion is generally no in favor of he UIP condiion. In heir answer, Obsfeld and Rogoff (996) appeal o Baxer s (994) argumen ha in order o find a link beween he exchange rae and he ineres rae differenials, one should filer he daa properly. A very high frequencies (i.e., cycles -5quarers) one should expec o find nohing bu noise relaed o he irregular componen of he ime series. Using he Baxer-King Band-Pass filer, she finds a correlaion beween he exchange raes and ineres rae differenials a boh rend and business cycle. There are wo oher separae papers ha deal wih long and shor run movemens in he exchange rae and ineres rae differenials. Nadal De Simone and Razzak (997) used he Phillips and Lorean (99) nonlinear dynamic leas squares and repored a very significan coinegraing relaionship (rend) beween nominal rade-weighed dollar exchange rae and he long-erm nominal ineres rae differenial. Recenly, Meredih and Chinn (998) used regressions in differences and showed evidence in favor of UIP (shor erm) when long-erm ineres raes are used insead of shor-erm raes. These wo papers indicae ha here migh be some evidence in favor of UIP a boh he rend and he cycle. The objecive of his paper is o examine nominal UIP condiion a boh he rend and he cycle. I use an asympoically efficien and unbiased esimaor (Phillips and Lorean, 99) o esimae he coefficiens ha govern he longrun relaionship beween bilaeral exchange raes and long-erm nominal ineres raes such as he 0-year governmen bond yield. A he cycle, I Examples are Frankel (979), McNown and Wallace (989), Baillie and Pecchenino (99), Johansen and Juselius (99), Edison and Pauls (993), McNown and Wallace (994), McCallum (994), Kim and Mo (995), Barolini and Bodnar (995), Meredih and Chinn (998) and Edison and Melick (999). The heoreical debae abou how ineres raes affec he exchange raes is found in Dornbusch (976), Frankel (979), Chrisiano e al (998), Mussa (976, 979), Frenkel (976) and Bilson (978, 979). For insance, see Messe and Rogoff (983) and Messe and Rogoff (988) and MacDonald and Nagayasu (999) research on real exchange raes and real ineres rae differenials.

3 examine he relaionship by assessing common serial correlaion afer removing common rends (Vahid and Engle, 993). I find evidence of correlaion beween he exchange rae and long-erm ineres rae differenials a boh he rend and he cycle. Secion wo consiss of six sub-secions. Sub-secion. defines he UIP condiion;. describes he daa;.3 discusses he Phillips-Lorean mehod;.4 examines he order of inegraion of he ime series and ess for coinegraion;.5 esimaes he long-run coefficiens using he Phillips-Lorean; and.6 examines he cyclical comovemens using he Vahid-Engle (993) mehod. Secion hree is a summary.. Empirical Evidence.. The Definiion and Empirical Form of UIP I briefly discuss he uncovered ineres rae pariy condiion. UIP is expressed by: e + s, + s, + s * = E e + b( i i ), () where e is he naural logarihm of he spo exchange rae, E is he mahemaical expecaions operaor condiional on he informaion se a ime, i is ime U.S. annualized nominal ineres rae or yield o mauriy on a pure discoun bond ha maures a ime s. Similarly i * is he foreign magniude. The size of he parameerb depends on he unis of he ineres raes. For example, if he ineres raes are expressed as decimals of he 0- s 40 year governmen bond yields, hen b is 0 (e.g., ); b could also be or 0.0. When he error erm in () is whie noise, he ineres rae differenial is he only explanaory variables because i reflecs all available informaion a ime. There are differen ways o empirically esimae UIP and none of hem escapes criicisms (see Edison, Melick, 999). 3 One sandard empirical version ha is ypically esed in he lieraure is: 3 The expeced exchange rae poses a problem for esimaion because i unobservable. Firs differencing he spo rae is no he righ way o deal wih expecaions. Making he expeced exchange rae a funcion of a se of macroeconomic fundamenals is perhaps a reasonable way of dealing wih problem. 4 = 4

4 3 e * = a + b( i + s i, + s ), + ω () In his regression, he expeced exchange rae E e + s is assumed o be a consan a, which is a very big assumpion. The error erm in () is assumed o be Niid and b is he slope coefficien. Leas squares esimaion of his equaion is problemaic because he exchange rae and probably ineres raes have uni roos. Economiss use differenced regressions o avoid spurious regressions. If he ineres raes are 0-year governmen bond yields, he exchange raes have o be he 0-year changes. To avoid all problems associaed wih uni roos regressions and differencing, essenially, a more elaborae dynamic regression in levels of equaion will be esimaed in his paper using he Phillips-Lorean (99) non-linear dynamic leas squares mehod. The esimaed coefficiens â and bˆ represen he coinegraion coefficiens. When ω is whie noise, and he parameer bˆ is no s differen from, he consan erm â is inerpreed as he average long-run 4 exchange rae.. The Daa Monhly daa covering he period January 980 o July 997 from he Inernaional Moneary Fund s Inernaional Financial Saisics (IFS) are used in his paper. The exchange rae is defined as he foreign currency price of he US dollar. The exchange raes are he DM-USD, GBP-USD, YEN- USD, and he CAD-USD a he end of each monh. The U.S. dollar-french Franc and he U.S. dollar-ialian Lire are excluded from he sample because France and Ialy mainained some significan capial conrol measures unil lae 980s which I hink hey may affec he ess of he UIP. The reason I sar he sample in 980 is o avoid periods of capial conrols. Briain abolished exchange conrols in 979 (Aris and Taylor, 989). Japan abolished is exchange conrol policies in 980 (Fukao, 990). The long-erm ineres rae is he 0-year governmen bond yield rae (IFS, line 6, where e.g., six percen is wrien 6.0 insead of 0.06). Figures o 4 plo he exchange raes on he LHS axis and he ineres rae differenials on he RHS axis. The ineres rae differenial is defined as he 0-year governmen bond yield of he Unied Saes minus ha of he foreign

5 4 counry (i.e., Germany, Briain, Japan, and Canada respecively). Visually, here seems o be a srong relaionship beween he DM-USD and he GBP- USD exchange raes and he corresponding ineres rae differenials. A weaker relaionship appears beween he CAD-USD exchange raes and he corresponding ineres rae differenials. The YEN-USD ineres rae differenial relaionship appears he weakes among all four pairs. If Obsfeld and Rogoff (996) are righ we should no uncover significan long and shor-run relaionships beween he DM-USD and he GBP-USD and he corresponding ineres rae differenials. Before esimaing he UIP condiion, I briefly discuss he heory of he Phillips-Lorean (99) non-linear dynamic leas squares (lags and leads regression)..3 The Phillips-Lorean (99) mehod I briefly explain he mehodology used in his paper o esimae he coefficiens ha govern he long-run relaionship beween he exchange rae and he ineres rae differenials. This mehod is based on he Triangular Represenaion Theorem. Consider he following riangular sysem of equaions: (3), (4) y = α + β y + u y = u where [ u u ] u = is a saionary vecor. Phillips and Lorean (99) consider esimaion and inference in he above sysem. Equaion (3) can be esimaed using single equaion mehods, and provided ha he equaion is appropriaely augmened he asympoic properies of he esimaor can be readily deermined. The asympoic (disribuional) properies hinge on he inerrelaionships ha exis beween u and u, which are assumed o be saionary. If he variance-covariance marix of u is block diagonal (so ha he pariioned elemens of he vecor do no co-vary) and u is also (i.i.d.) hen equaion (3) can be esimaed using leas squares. The esimaes will be normally disribued, a leas asympoically, and will be equivalen o

6 5 maximum likelihood esimaes of he parameers of he sysem. If he variance covariance marix of u is block diagonal bu he residuals are auocorrelaed (bu also block diagonal) he regression can be augmened wih lagged error correcion erms, couneracing he auocorrelaion of u. This soluion works because he regressors y are super-exogenous. They are independen of he pas and fuure hisory of u, i.e. hey are orhogonal o { u } +. = If u and u are correlaed one can alleviae he problems ha his causes by augmening he regression wih leads and lags of y. One can hink of his as projecing u agains leads and lags of u and hen applying he Frisch Waugh - Lovell heorem (see Davidson and MacKinnon, 993). The error from he projecion is uncorrelaed wih u, and he simulaneiy problem is hus deal wih. The inclusion of leads eliminaes feedback from u back o u and i is imporan for valid condiioning. The basic problem is ha, a priori, we do no know wheher u correlaes wih leads of u or vice versa. The inclusion of lag differences eliminaes he simulaneiy problem, while he auocorrelaion problem may need o be remedied by including lags of he error correcion erm oo. Phillips and Lorean hen esimae he following equaion using leas squares: K p y = α + β y + d k + ν k= i= p ( y k β y k ) + d i y i (5) Phillips and Lorean demonsrae ha he parameer esimaes from esimaing his single equaion are equivalen o maximum likelihood esimaion of he sysem and hence are efficien. Furhermore, he parameers are asympoically normally disribued. The disribuions of he parameers of he coinegraing vecor and corresponding saisics depend on he null hypoheses concerning he inerrelaionships. If he series are inegraed bu no coinegraed he regression is spurious, and he asympoic heory is found in Phillips (986). Barnhar, McNown and Wallace (999) used his echnique o es he forward rae unbiasedness hypohesis and demonsraed ha he esimaor is mos

7 6 unbiased among many differen mehods used in esing. The assumpions of uni roos and coinegraion need o be esed firs, which is he nex sep..4. Uni Roos and Coinegraion Before esimaion, ess for uni roos and coinegraion mus be carried ou. There seems o be an agreemen among economiss ha exchange raes, during he pos Breon-Woods era, may conain sochasic rends. The lieraure on uni roos and coinegraion is vas and i will no be reviewed here. Suffice o say ha here is a valid concern among economiss abou he appropriaeness of he ess for uni roos and heir power agains saionary alernaives. The choice of a paricular esing mehodology is no sraighforward. Ulimaely, one may no be able o deermine wheher a paricular ime series conains a uni roo or no. I seems ineviable, however, ha one mus make a choice. Because here is no consensus on he issues of uni roo in long-erm ineres rae differenials and coinegraion beween he exchange rae and he ineres rae differenials, i is perhaps useful o subjec he daa o various differen ess. This sraegy reduces he risk of being on he wrong side. A decision can hen be based on wheher he resuls of various ess converge or no. For example, when differen ess for uni roos move in one direcion, e.g., indicaing a uni roo, one can be a lile more confiden in he resuls han when he ess diverge. 4 In his paper, four mehods o es for uni roos are used. All of hem have he same null hypohesis of a uni roo and are available in mos saisical packages. These are he DF es (Dickey and Fuller, 979, 98), he ADF es (Said and Dickey, 984), he Z es (Phillips, 987 and Perron, 988), he DF-GLS J es (Ellio, Rohenberg and Sock, 996), and Perron (997) es. Resuls for uni roo ess are repored in able a and b. 5 The ess ha are considered in his paper seem o agree wih convenional wisdom ha he exchange raes have uni roos. 4 For he comparison o be meaningful, i is imporan ha he ess have he same null hypohesis. 5 The choice of he lag srucure always has been an issue. The objecive of he lags is o remove serial correlaion. Wih his objecive in mind, I look a differen crieria o choose he lags. For example, he lag order is se as he highes significan lag order using an approximae 95 percen confidence inerval from eiher he auocorrelaion funcion or he parial auocorrelaion funcion of he firs-differenced series. The maximum lag order is he square roo of he sample size. I also es backward using F ess, and look a

8 7 However, here are wo excepions regarding he ineres rae differenials. The ADF and he Z ess disagree wice regarding he ineres raes differenials. The ADF es rejeced he null hypohesis ha he YEN-USD 0-year ineres rae differenial is I (), while he Z es finds a uni roo in i. Then he ADF es could no rejec he null hypohesis of a uni roo in he CAD-USD 0-year ineres rae differenial, while he Z es rejeced i. The DF-GLS a poin-opimal invarian es, which has improved power when an unknown mean or rend is presen in he daa indicaes ha he null of a uni roo wih a consan and a linear rend canno be rejeced for any variable. Finally, all he ime series are esed using Perron es (997). 6 Table b shows he resuls for wo models: he innovaional oulier model (model ) allows only a change in he inercep under boh he null and he alernaive hypohesis, and he addiive oulier model (model ) allows a change in boh he inercep and he slope. According o he resuls of able b, he null hypohesis of uni roo could no be rejeced for he daa excep for he YEN- USD 0-year ineres rae differenial. Therefore, one can conclude ha he Perron es indicaes ha he YEN-USD 0-year ineres rae differenial does no have a uni roo when allowance is made for a change in he inercep and he slope. The hypohesis of he uni in he YEN-USD ineres rae differenials is rejeced by wo ess ou of four. I is herefore assumed o be I (0). The DM-USD, GBP-USD and he CAD-USD ineres rae differenials on he oher hand seem o diverge for prolonged periods of ime; hey are likely o have sochasic rends. Schwarz IC and AIC. Unnecessary lags are eliminaed. Every ime, serial correlaion is checked using he Barle - Kolmogrov - Smirnov es for whie noise. 6 The es allows for a shif in he inercep of he rend funcion and a shif in he slope when he dae of he possible change is no fixed a priori bu is deermined endogenously. Two mehods are used o deermine he break poin (T b ). Firs, we selec he breaking poin ha minimizes he -saisic for esing he null of uni roo ( αˆ ). Second, we selec he breaking poin ha minimizes he -saisic on he parameer associaed wih he change in he inercep (model ) ( θˆ ), or he change in slope (model ) ( γˆ ). The lag parameer is chosen following a general o specific recursive procedure so ha he coefficien on he las lag in an auoregression of order k is significan and ha he las coefficien in an auoregression of order k greaer ha is insignifican, up o a maximum order k max.

9 8 To es he null hypohesis ha he exchange raes are no coinegraed wih he ineres rae differenials, he same sraegy used o es for uni roo is followed; many differen esing mehods are used. And a search for a consensus is pursued. Noe ha if he ineres rae differenials are I (0) hen esing for no coinegraion wih he exchange raes is meaningless. For his reason, I will drop he YEN-USD from he coinegraion analysis ha follows. Firs, I use he Engle-Granger (987) and he Engle and Yoo (987) procedures. The Engle-Granger mehod is suiable for bi-variae sysems. The null hypohesis ha he residuals from he OLS regressions of he nominal exchange rae on a consan, or on a consan and a rend, and he ineres rae differenial is esed for uni roo. Typically, he ADF es is used as recommended by Engle and Granger s original paper. Second he Phillips- Perron-Phillips-Ouliaris es o es he same hypohesis is used. 7 Given he sample, boh ess seem o indicae ha here is no saisical evidence of coinegraion in all pairs of daa. 8 Third, I use he Johansen and Juselius (990) maximum likelihood mehod. Alhough his mehod is more appropriae for mulivariae cases, I use i o es for no-coinegraion beween nominal exchange raes and ineres rae differenials because many papers repor his es. Resuls are repored in able. 9 The sample is adequae (in erms of he number of observaions and he span), bu I also repor he correced he criical values for small sample using he Cheung and Lai (993) approach. 0 Based on heλ max and he race saisics 7 The same approach o selecing he lag srucure in he ess for uni roos in individual series is used here. 8 I am no aware of any evidence of coinegraion beween he nominal exchange rae and ineres rae differenials in he lieraure based on hese ess. Resuls are no repored, bu hey are available upon reques. The resul is consisen wih Meese and Rogoff (988), Edison and Pauls (993), and Kawai and Ohara (997) for he real exchange raes. 9 Gonzalo (994) compares five differen residual-based ess for coinegraion including he Engle-Granger es. Among hem, he recommends using he Johansen-Juselius (990) mehod. This es has been very popular in he lieraure, bu jus like any oher uni roo es, i is highly criicized for is lack of power in finie samples, and among oher problems is sensiiviy o he choice of he lag lengh. I sar wih a lag srucure similar o ha we adoped in he Engle-Granger es. The choice is based on he mulivariae serial correlaion ess recommended by Johansen and repored in CATS. 0 The lag srucure is deermined using he same mehodology explained earlier. The residuals of he models are carefully checked for whieness each ime using mulivariae Ljung-Box, LM and LM4 ess.

10 9 a he 95% level, he es indicaes ha he nominal exchange rae and he nominal ineres rae differenials are coinegraed. There is a single coinegraing vecor in each of he hree pairs. The esimaed long-run coefficiens afer imposing a rank of one and normalizing on he exchange rae are 0.8, 0.6 and 0.0 for he DM-USD he GBP-USD and he CAD- USD pairs respecively. Having found a coinegraing relaionship beween each of he pairs of exchange raes and ineres rae differenials, we move o using he Phillips- Lorean approach as a final check on he long-run comovemen ha we are ineresed in..5 Esimaion of he Long-run Coefficiens As I explained earlier, his single-equaion esimaion echnique of Phillips and Lorean is asympoically equivalen o a maximum likelihood on a full sysem of equaions under Gaussian condiions. This echnique provides esimaors ha are saisically efficien, and whose -raios can be used for inference in he usual way. Mos imporanly, his mehod akes ino accoun boh he serial correlaion of he errors and he endogeneiy of he regressors ha are presen when here is a coinegraion relaionship. Recenly, Barnhar, McNown and Wallace (999, p 88-89) provided evidence ha his procedure produces no bias in he esimaes of he parameers in forward premium equaion. I esimae wo regressions, he resriced and he unresriced given by equaion (6) and (7) respecively: k * * * = a + b( i i ) + i ( i i ) i + ρ[ e a b( i i ) i= k e ] + v e δ, (6) k k* * * * = a + b i + b i i i θ i i ρ e a [ bi bi i= k i= k ] ϕ + η, (7) I am mosly ineresed in he magniudes $a, b $ in he resriced regression and a, ˆb and ˆb in he unresriced regression, because hey are he parameer The Phillips Informaion Crieria (Phillips, 996) is used o deermine he opimal lag srucure along wih he coinegraion vecor. I find a hree lag-lead srucure o be sufficien o eliminae he serial correlaion. However, because I am conscious abou Phillips and Lorean warning of over-fiing, I sar reducing he number of leads by one. Every ime, I check he serial correlaion, he parameer esimaes, and heir significance. I find ha a srucure of hree lags and wo leads gives he same resuls as a hree lag-lead srucure wihou compromising he whieness of he residuals. A furher reducion in he leads inroduces some serial correlaion.

11 0 esimaes ha deermine he long-run relaionship beween he nominal exchange rae and he nominal 0-year bond yield differenial condiional on he dynamics. Unlike he regressions in differences repored in Meredih and Chinn (998), he Phillips-Lorean is in levels and i is condiional on a more elaborae dynamic. Resuls of he nonlinear leas square esimaion for boh pairs of currencies are repored in able 3. The parameers of he resriced regression are significan. I repor he -raios and he p-values. These resuls are consisen wih he Johansen-Juselius es resuls boh in magniudes and in qualiy. The various diagnosics of he residuals indicae ha he residuals are whie noise, serially uncorrelaed, and homoskedasic. The magniudes of $ b are almos idenical for he DM-USD, he GBP-USD and he CAD-USD. These are 0., 0.0 and 0.0 respecively. In he long run, an increase in he nominal ineres rae differenial appreciaes he home currency. When here is a significan slope relaionship beween he exchange rae and he ineres rae differenials like he ones we obained, he esimaes of he consan erms $a can be inerpreed as average long-run values of he nominal exchange rae. For example, for he DM-USD he value is 0.34 and he anilog is.40. So he equilibrium DM-USD exchange rae over he sample is.40. I is 0.69 for he GBP-USD and.5 for he CAD-USD. The sample means are.95, 0.6 and.7 respecively. These numbers indicae ha on average and during he period 980:-997:7, he US dollar exceeded is long-run equilibrium value agains he DM, bu undersho is equilibrium raes agains he Pound and he Canadian Dollar. The unresriced regression produces similar resuls. However, he resricions (7 resricions on he levels and he dynamics) only hold in he DM-USD case as indicaed by he F es. Also, I repor a chi-square es of he resricion ha he long-run coefficien on he Unied Saes ineres rae is equal in magniude (wih opposie sign) o he foreign ineres rae coefficien. We canno rejec he hypohesis ha he long-run coefficiens are equal in magniude in he case of DM-USD. Also, I provide a graphical presenaion of he long-run relaionships. For each pair, he long run is compued from he RHS of equaion (6), * f = aˆ + bˆ( i i ). Then I plo he exchange raes and f. The plos are shown in

12 figures 5, 6 and 7. The correlaion beween he exchange raes and f are visually clear in he DM-USD and he GBP-USD cases. The correlaion values beween he DM-USD exchange raes and he corresponding f is 0.80 (equivalen R is 0.64), bu smaller in he cases of he GBP-USD and he CAD-USD. Alernaively, one can also plo e aˆ agains ˆ * b( i i ) wihou any loss in informaion..6 Examining Cyclical Comovemens Finally, I use he Vahid-Engle (993) mehod o es for common cycles for all pairs of currencies ha are coinegraed wih he ineres rae differenials. The es is condiional on he presence of coinegraion. This mehod explois he serial correlaion common beween he firs-difference of he wo coinegraed variables o es for common-cycles. In essence i searches for comovemens among he saionary componens of he ime series jus like coinegraion searches for comovemens among he non-saionary or uni roo componens of he ime series. Precisely, his mehod invesigaes common serial correlaion among he firs differences of a se of coinegraed variables. Thus, i implies ha he remainders afer removing heir rends from heir levels are common cycles. Resuls are repored in able 4. In he DM-USD, we canno rejec he null hypohesis ha here is a leas one common cycle. The null hypohesis of a common cycle in he pair GBP-USD is rejeced alhough here is a common rend beween ha exchange rae and he GBP-USD ineres rae differenial. Also, here seems o be a significan relaionship beween he exchange rae CAD-USD and he ineres rae differenials a he cycle. I conclude ha he exchange raes conain uni roos. The ineres rae differenials beween he Unied Saes and Germany and Briain conain uni roos. The ineres rae differenials beween he Unied Saes and Canada may conain uni roos, bu i is unclear wheher he ineres rae differenials beween he Unied Saes and Japan is a uni roo process. Consequenly, I find he DM-USD, he GBP-USD and he CAD-USD exchange raes and he corresponding ineres rae differenials o be coinegraed. The evidence of coinegraion beween he CAD-USD exchange rae and ineres rae differenials is weaker han he oher wo. Furher, he UIP holds very well in he cases of he DM-USD in he sense ha he resricions ha he ineres rae differenials in he levels, firs difference, he lags and he leads have

13 coefficiens equal in size and opposie in sings could no be rejeced. Also, here is evidence of common cycles beween he DM-USD, and CAD-USD and he corresponding ineres rae differenials. 3. Summary Differen ess o es he null hypohesis ha he nominal exchange raes and he long-erm nominal ineres rae differenials are uni roo processes are used. Daa are monhly for he DM-USD, he GBP-USD, he YEN-USD, and he CAD-USD from January 980 o July 997. Previous findings ha he null hypohesis of a uni roo in exchange raes canno be rejeced are confirmed. However, i is sill unclear wheher all of he long-erm ineres rae differenials are uni roo processes. The 0-year bond raes differenials beween he U.S. and Germany and he Unied Saes, he U.S. and Briain and beween he U.S. and Canada conain a uni roo, bu i is more significan in he firs wo cases han in he case of he Unied Saes and Canada. I is unclear wheher he ineres rae differenials beween he Unied Saes and Japan conain uni roos. I is highly probable ha he relaionship is an I(0). Consequenly, I found ha he DM-USD and he GBP-USD exchange raes and he corresponding long-erm ineres rae differenials are coinegraed. There is weaker evidence ha he CAD-USD exchange rae is coinegraed wih he corresponding ineres rae differenials. Exacly like esing for uni roos, he finding of coinegraion is sensiive o he power of he es used. The Engle- Granger and Phillips-Perron-Phillips-Ouliaris ype ess canno rejec he null hypohesis ha he exchange raes and he ineres rae differenials are no coinegraed for all pairs of counries. However, coinegraion can be found using he Johansen-Juselius (990) and confirmed by he Phillips-Lorean (99) mehods. Using long-erm ineres rae differenials suggess ha here is srong saisical evidence in favor of UIP in he pairs DM-USD and GBP-USD, also beween he CAD-USD and he long-run ineres rae differenials, albei weaker. Furher. On he basis ha nominal exchange raes and long-erm nominal ineres rae differenials are coinegraed, I invesigae he presence of common cycles. Using he Vahid-Engle (993) mehod, saisical evidence of common cycles beween nominal exchange raes and ineres rae differenials in he cases of DM-USD and he CAD-USD, bu no in he case of GBP-USD.

14 3 References Aris, M. J. and M. P. Taylor, Abolishing Exchange Conrol: The UK Experience, Cener for Economic Policy Research, paper No. 94 (989). Baillie R. and R. Pecchenino, The Search for Equilibrium Relaionship in Inernaional Finance: The Case of he Moneary Model, Journal of Inernaional Money and Finance 0 (99), Barnhar, S. W., R. McNown and M. S. Wallace, Non-Informaive Tess of he Unbiased Forward Exchange Rae, Journal of Financial and Quaniaive Analysis, Vol. 34, No. (999), Barr, D. and J. Y. Campbell, Inflaion, Real Ineres Raes, and he Bond Marke: A Sudy of UK Nominal and Index-Linked Governmen Bond Prices, Journal of Moneary Economics 39 (997), Barolini, L. and G. Bodnar, Are Exchange Raes Excessively Volaile? And Wha Does Excess Volailiy Mean, Anyway? IMF Working Paper WP/95/85, (995). Baxer, M., The Real Exchange Raes and he Real Ineres Differenials: Have We Missed he Business-Cycle Relaionship? Journal of Moneary Economics 33 (994), Baxer, M. and R. G. King, Measuring Business Cycles. Approximae Band-Pass Filers for Economic Time Series, Working Paper N0. 50, Naional Bureau of Economic Research (995). Bilson, J.F.O., The Deusche Mark/Dollar Rae-A Moneary Analysis, in Karl Brunner and Allan Melzer, (eds.), Carnegie-Rocheser Conference Series on Public Policy, Vol. Amserdam: Norh Holland (979), Bilson, J.F.O., The Moneary approach o he Exchange Rae-Some Empirical Evidence, IMF Saff Papers (978), 0-3. Brunner K. and A. Melzer, Moneary Economics, Oxford, Unied Kingdom: Basil Blackwell (989).

15 4 Cebula, R. J., The Impac of Ne Inernaional Capial Inflows on Long Term Ineres Raes in France, Alanic Economic Journal 5 (997), Cheung, Y.W. and K. S. Lai, Finie Sample Sizes of Johansen Likelihood Raio Tes for Coinegraion, Oxford Bullein of Economics and Saisics 55 (993), Chrisiano, L. J., M. Eichenbaum, and C. L. Evans, Moneary Policy Shocks: Wha Have We Learned and o Wha End? Working Paper N0 6400, Naional Bureau of Economic Research (998). Dickey, D. and W. Fuller, The Likelihood Raio Saisic for Auoregressive Time Series wih a Uni Roo, Economerica 49 (98), Dickey, D. and W. A. Fuller, Disribuion of Esimaes for Auoregressive Time Series Wih Uni Roo, Journal of American Saisical Associaion 74 (979), Dornbusch, R., Expecaions and Exchange Rae Dynamics, Journal of Poliical Economy 84 (976), Edison, H. J. and W. R. Melick, Alernaive Approaches o Real Exchange Raes and Real Ineres Raes: Three Up and Three Down, Inernaional Journal of Finance and Economics 4 (999), 93-. Edison, H. J. and B. D. Pauls, A Re-Assessmen of he Relaionship Beween Real Exchange Raes and Real ineres Raes: , Journal of Moneary Economics 3 (993), Ellio, G., T. J. Rohenberg, and J. Sock, "Efficien Tess for an Auoregressive Uni Roo, Economerica 64 (996), pp Engel, C., Long-Run PPP May No Hold Afer All, Naional Bureau of Economic Research, Working Paper: 5646, July (996), 3. Engle, R.F. and C.W.J. Granger, Coinegraion and Error Correcion: Represenaion, Esimaion and Tesing, Economerica 55 (987), Engle, R. F. and B. S. Yoo, Forecasing and Tesing in Coinegraed Sysems, Journal of Economerics 35 (987),

16 5 Frankel, J., "Quanifying Inernaional Capial Mobiliy in he 980s", in B. D. Berheim and J. B. Shoven (eds.) Naional Saving and Economic Performance (99), The Universiy of Chicago Press. Frankel, J., On he Mark: A Theory of Floaing Exchange Raes Based on Real Ineres Differenials, American Economic Review (979), Frenkel, J., A Moneary Approach o he Exchange Rae: Docrinal Aspecs and Empirical Evidence, Scandinavian Journal of Economics 78 (976), Fukao, M., Japan s Foreign Exchange Conrols and he Balance of Paymens, Bank of Japan Moneary and Economic Analysis (990), Gonzalo, J., Five Alernaive Mehods of Esimaing Long-Run Equilibrium Relaionships, Journal of Economerics 60 (994), Johansen, S. and K. Juselius, Tesing Srucural Hypoheses in a Mulivariae Coinegraion Analysis of he PPP and UIP for he UK, Journal of Economerics 53 (99), -44. Johansen S. and K. Juselius, The Full Informaion Maximum Likelihood Procedure for Inference on Coinegraion--Wih Applicaion o he Demand for Money, Oxford Bullein of Economics and Saisics 5 (990), Juselius, K. and R. MacDonald, Inernaional Pariy Relaionship Beween Germany and he Unied Saes: A join Modeling Approach, Working Paper, Insiue of Economics, Universiy of Copenhagen (000). Kawai, M. and H. Ohara, Non Saionariy of Real Exchange Raes in he G7 Counries: Are They Coinegraed wih Real Variables, Journal of he Japanese and Inernaional Economies (997), Kim, B.J.C. and S. Mo, Coinegraion and he Long-Run Forecas of he Exchange Rae, Economics Leers 48 (995), MacDonald, R. and Nagayasu, The Long-Run Relaionship beween Real Exchange Rae and Real Ineres Rae Differenials: A Panel Sudy, IMF Working Paper No. 99/37.

17 6 McCallum, B., A Reconsideraion of he uncovered ineres pariy relaionship, Journal of Moneary Economics Vol. 33 No. (994), McNown, R. and M. Wallace, Coinegraion Tess of he Moneary Exchange Rae Model for Three High-Inflaion Economies, Journal of Money, Credi and Banking 6 (994), McNown, R. and M. Wallace, Naional Price Levels, Purchasing Power Pariy, and Coinegraion: A Tes of Four High Inflaion Economies, Journal of Inernaional Money and Finance 8 (989), Meese, R. and K. Rogoff, Was i real? The exchange rae-ineres rae differenial relaion over he modern floaing-rae period, Journal of Finance 43 (988), Meese, R. and K. Rogoff, Empirical Exchange Rae Models of he Sevenies: Do They Fi Ou-of-Sample? Journal of Inernaional Economics (983), Meredih, G. and M. D. Chinn, "Long-Horizon Uncovered Ineres Rae Pariy", working paper No 6797, Naional Bureau of Economic Research (998). Mussa, M., Exchange Raes in Theory and in Realiy, Essays in Inernaional Finance, No. 79 (990), Princeon Universiy. Mussa, M., Empirical Regulariies in he Behavior of he Exchange Raes and Theories of he Foreign exchange Marke, in Karl Brunner and Allan Melzer, (eds.), Policies for Employmen, Prices, and Exchange Raes. Carnegie-Rocheser Conference, Vol. II Amserdam, Norh Holland (979), Obsfeld, M. and K. Rogoff, Foundaions of Inernaional Macroeconomics, MIT Press, Cambridge, Ma (996). Perron, P., "Furher Evidence on Breaking Trend Funcions in Macroeconomic Variables," Journal of Economerics 80 (997), Perron, P., Trends and Random Walks in Macroeconomic Time Series, Journal of Economic Dynamics and Conrol (988), Phillips, P. C. B., Time Series Regressions wih a Uni Roo, Economerica 55 (987),

18 7 Phillips, P. C. B., Economeric Model Deerminaion, Economerica 64 (996), Phillips, P. C. B. and M. Lorean, Esimaing Long-Run Equilibria, The Review of Economic Sudies 58(3) 99, Said, S. and D. A. Dickey, Tesing for Uni Roos in Auoregressive-Moving Average Models of Unknown Order, Biomerika 7 (984), Vahid, F. and R. Engle, Common Trends and Common Cycles, Journal of Applied Economerics Vol. 8 (993), Viner, A., Inside Japanese Financial Markes", Dow Jones-Irwin, Homewood, Illinois, (988) - 7 -

19 8 Table a Tess for Uni Roos in Nominal Exchange Raes and Nominal Ineres Rae Differenials y consan + rend + ρ y + δ y + ε i i i= 980: o 997:7 k Saisics for ρ =0 y Lags ADF τ Phillips Perron z DF-GLS DM-USD GBP-USD YEN-USD CAN-USD i (US)- i (Germany) i (US)- i (Briain) i (US)- i (Japan) * i (US)- i (Canada) * -.89 The exchange rae is measured in naural logarihms. Ineres raes are he 0-year bond raes. Lags are he same across all ess. ADF is he ADF saisic for H 0 : uni roo. The 5% criical value is Phillips-Perron is he Phillips-Perron saisic for H 0 : uni roo. The 5% criical value is -.7. We do no repor he saisics for join hypohesis ess for ρ and he consan, and ρ and rend. DF-GLS is he Ellio, Rohenberg and Sock saisic for Ho: uni roo wih a linear rend. The 5 percen criical value is.89. The es is he -saisic for esing a = 0 in he regression o + a y a p y ε p and y y β o y = α y + o ˆ βˆ =

20 9 Table b Perron (997) Uni Roo Tes for Nominal Exchange Raes and Nominal Ineres Rae Differenials Model: y = u + θdu + β + δd( T ) + αy b k + ci y i + i= Y T b α α e DM-USD 986: GBP-USD 986: YEN-USD 985: CAN-USD 987: i (US)- i (Germany) 988: i (US)- i (Briain) 980: i (US)- i (Japan) 994: * i (US)- i (Canada) 986: Model : y = u + θdu + β + δdt + δd( T ) + αy b k + ci y i + i= Y T b α α DM-USD 985: GBP-USD 99: YEN-USD 985: CAN-USD 987: i (US)- i (Germany) 989: i (US)- i (Briain) 985: i (US)- i (Japan) 989: * i (US)- i (Canada) 995: e T b is he value ha minimizes he -saisic for esing α =. Tess for T b are chosen so as o minimize he -saisic on he parameer associaed wih model or model gave he same resuls. The 5 percen criical values for model and model are 4.0 and 5.08, respecively.

21 0 Table The Johansen-Juselius Maximum Likelihood Tes for Coinegraion 980: o 997:9 Eigen Values λ max Trace H 0 : r p-r λ max 95% Trace 95% lag DM-USD * 0.49 * Residuals L-B ~ χ (0.74) LM ~ χ.09 4 (0.70) LM4 ~ χ (0.99) GBP-USD * 4.87 * Residuals L-B ~ χ (0.90) LM ~ χ (0.3) LM4 ~ χ (0.45) CAD-USD * Residuals L-B ~ χ 0 (0.6) LM ~ χ (0.03) LM4 ~ χ (0.4) r is he number of coinegraed vecors. p is he number of variables. The 95% criical values correced for small samples using Cheung and Lai (993) are used o evaluae he resuls. The models include drif erms. L-B is a mulivariae Ljung-Box es based on esimaed auo crosscorrelaion of he firs T/4 lags. The LM LM4 are he Lagrange muliplier ess.

22 Table 3 The Phillips-Lorean Non -Linear Dynamic Leas Square Esimaor 980: o 997:7 Resriced Model k * * * = a + b( i i ) + δ i ( i i ) i + ρ[ e a b( i i ) i= k e ] + v Unresriced Model k k * * * = a + bi + bi + i i i + θ i i i + ρ[ e a bi bi i= k i= k e δ ] + η a DM-USD GBP-USD CAD-USD DM-USD GBP-USD CAD-USD 0.34 (8.35) (-7.07) 0.4 (3.65) 0.09 (0.49) -0. (-.0) 0.55 (.75) [0.000] [0.000] [0.000] [0.674] [0.690] [0.0058] b 0. (5.60) 0.0 (3.04) 0.0 (.50) [0.000] [0.350] ρ 0.94 (5.3) 0.93 (50.) 0.97 (79.8) 0.93 (5.8) 0.90 (33.6) 0.97 (68.8) [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] b (5.80) 0.09 (3.38) 0.09 (.50) [0.000] [0.000] [0.389] b (- -0. (-4.88) -0.0 (-.6).83) [0.000] [0.05] [0.0665] Se DW Breusch-Pagan 6.0 [0.6349] 9.53 [0.993] 7.6 [0.590] 7.88 [0.689] 3.4 [0.5700].79 [0.6947] ARCH () 0.35 [0.5846] 5.5 [0.4] 6.86 [0.866].00 [0.4456].86 [0.0390] 3.3 [ Barle s-kolmogrov Smirnov F * 0.5 * χ 3.77 [.058] 5.6 [.078] 0.77 [.377] - - Se: Sandard error of he esimaes. The -raios are in parenheses and P-values are in square brackes. Barle s-kolmogrov-smirnov 5% criical value is F ess he resricion on he long-run parameers and he dynamics. χ is o es he resricions on he long-run parameers only.

23 Squared Canonical Correlaion Table 4 The Vahid-Engle Tes for Common Cycles 980: o 997:9 k Cks (, ) ~ χ H 0 : s Degrees of Freedom P-value DM-USD s> * s> GBP-USD * s> * s> CAD-USD s> * s> s Cks (, ) = ( T k ) ln( λ i ), where T is he sample size, k is he lag lengh, and λ is he canonical correlaion. i= s is he smalles squared canonical correlaion beween he exchange rae and he ineres rae differenial. +, where r is he number of coinegraed vecors (i.e., ), and p is he dimension of he sysem The degrees of freedom are s spk sr sp (i.e., ). An aserisk means significan a he 5% level.

24 3 Figure : Exchange Rae & Ineres Rae Differenial USD-DM (80:-97:7).4 DM-USD i-i* DM-USD Jan-80 Nov-80 0 Sep-8 Jul-8 May-83 Mar-84 Jan-85 Nov-85 Sep-86 Jul-87 May-88 Mar-89 Jan-90 Figure : Exchange Rae & Ineres Rae Differenial USD-GBP (80:-97:7) GBP-USD i-i* Nov-90 Sep-9 Jul-9 May-93 Mar-94 Jan-95 Nov-95 Sep-96 Jul i-i* GBP-USD Jan-80 Nov-80 Sep-8 Jul-8 May-83 Mar-84 Jan-85 Nov-85 Sep-86 Jul-87 May-88 Mar-89 Jan-90 Nov-90 Sep-9 Jul-9 May-93 Mar-94 Jan-95 Nov-95 Sep-96 Jul i-i*

25 4 Figure 3: Exchange Rae & Ineres Rae Differenial US-YEN (80:-97:7) 5.8 YEN-USD i-i* YEN-USD i-i* Jan-80 Nov Sep-8 Jul-8 May-83 Mar-84 Jan-85 Nov-85 Sep-86 Jul-87 May-88 Mar-89 Jan-90 Nov-90 Sep-9 Jul-9 May-93 Mar-94 Jan-95 Nov-95 Figure 4: Exchange Rae & Ineres Rae Differenial USD-CAD (80:-97:7) CAD-USD i-i* Sep-96 Jul CAD-USD Jan-80 Nov-80 Sep-8 Jul-8 May-83 Mar-84 Jan-85 Nov-85 Sep-86 Jul-87 May-88 Mar-89 Jan-90 Nov-90 Sep-9 Jul-9 May-93 Mar-94 Jan-95 Nov-95 Sep-96 Jul i-i*

26 5 Figure 5: Nominal Exchange Rae & a+b(i-i*) DM-USD (80:-97:7) Jan-80 Dec Jan-80 Nov Jan-80 Nov-80 Nov-8 Oc-8 Sep-83 Aug-84 Jul-85 Jun-86 DM-USD Sep-8 Jul-8 May-83 Mar-84 Jan-85 Nov-85 Sep-86 Jul-87 GBP-USD a+b(i-i*) May-87 Apr-88 Mar-89 Feb-90 Jan-9 Dec-9 Nov-9 Oc-93 a+b(i-i*) May-88 Mar-89 Jan-90 Nov-90 Sep-9 Jul-9 May-93 Sep-94 Aug-95 Jul-96 Jun-97 Figure 6: Nominal Exchange Rae & a+b(i-i*) GBP-USD (80:-97:7) Figure 7: Nominal Exchange Rae & a+b(i-i*) CAD-USD (80:-97:7) CAD-USD a+b(i-i*) Sep-8 Jul-8 May-83 Mar-84 Jan-85 Nov-85 Sep-86 Jul-87 May-88 Mar-89 Jan-90 Nov-90 Sep-9 Mar-94 Jan-95 Nov-95 Sep-96 Jul-97 Jul-9 May-93 Mar-94 Jan-95 Nov-95 Sep-96 Jul

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