Non-linear adjustment to purchasing power parity: an analysis using Fourier approximations

Transcription

1 Non-linear adjusmen o purchasing power pariy: an analysis using Fourier approximaions Juan A. Jiménez Marín M. Dolores Robles Fernández juanangel@ccee.ucm.es mdrobles@ccee.ucm.es. Corresponding auhor. Absrac This paper esimaes he dynamics of adjusmen o long run purchasing power pariy (PPP) using daa for 8 mayor bilaeral US dollar exchange raes, over he pos-breon Woods period, in a non-linear framework. We use new uni roo and coinegraion ess ha do no assume a specific non-linear adjusmen process. Using a firs-order Fourier approximaion, we find evidence of non-linear mean reversion in deviaions from boh absolue and relaive PPP. This firs-order Fourier approximaion allows us o capure many feaures of he nonlinear decay deeced in he daa. Our resuls are consisen wih heoreical argumens on inernaional goods markes arbirage under ransacion coss as well as wih an emerging srand of empirical lieraure. In his sense, his paper conribues owards forming a consensus on he presence of nonzero ransacion coss across a broad range of counries. Keywords: Uni-roo es, Coinegraion es, Fourier approximaion, nonlinear model, exchange raes, purchasing power pariy. JEL classificaion: F, C, C We are graeful o A. Novales for helpful commens. We acknowledge he financial suppor of he Miniserio de Ciencia y Tecnología, Spain, hrough Projec BEC-965. M. D. Robles also acknowledge he financial suppor of Juna de Comunidades de Casilla la Mancha and FEDER hrough Projec PAI-5-7. J.A. Jiménez is visiing scholar of he Economics deparmen in George Washingon Universiy and acknowledges financial suppor from Fundación Caja Madrid and Univesidad Compluense de Madrid Deparameno de Fundamenos del Análisis Económico II, Universidad Compluense, Campus de Somosaguas, 8, Pozuelo de Alarcón, Madrid, Spain. Tel: Fax: + 996

2 . INTRODUCTION There is a large and growing lieraure indicaing ha radiional linear uni roos ess do no work well when here are imporan non-lineariies in he daa. This problem appears frequenly in he analysis of he heory of Purchasing Power Pariy (PPP), which saes ha he real exchange rae should equal one, or a leas have a endency o reurn quickly o one when ha long-run raio is disurbed for some reason. Someimes his version of PPP is called absolue PPP. Relaive PPP is he weaker saemen ha changes in naional price levels are always equal or a leas, end o equaliy over sufficienly long periods. The raionaliy behind he PPP is a simple arbirage hypohesis: If wo idenical goods are raded a differen prices in differen counries, a profiable arbirage opporuniy arises and he arbirageurs can buy he good cheaply in one locaion and sell i a a higher price in he oher. In he absence of arbirage coss, his process leads o convergence of he deviaions from PPP owards zero. Exchange-rae-adjused prices are equalized across counries, leaving no room for profiable arbirage opporuniies. Under his version of he PPP heory, one would expec saionariy in real exchange rae dynamics. The pariy condiion ress on he assumpion of perfec iner-counry commodiy arbirage and i is a cenral building block of many heoreical and empirical models of exchange rae deerminaion. Due o facors like ransacion coss, axaion, subsidies, acual or hreaened rade resricions, he exisence of non raded goods, imperfec compeiion, foreign exchange marke inervenion, and he differen composiion of marke baskes and price indices across counries, one may expec PPP o be valid only in he long run. Then, prices and nominal exchange rae should show a seady long-erm relaionship, and herefore one would expec coinegraion beween hem. Empirical sudies over long periods of ime have suppored long-run PPP (Diebol e al., 99, Taylor 996, Michael e al., 997). However, resuls are mixed when he recen floaing-rae period is examined. Using sandard uni roos ess Corbae and Ouliaris (988), Meese and Rogoff (988), Edison and Fisher (99) and Grilli and Kaminsky (99) canno rejec he uni roo null hypohesis for real exchange raes in he managed-floa regime. In conras, Perdroni (997), Frankel and Rose (996), Lohian (997), Oh (996), Wu (996) and Papell and Theodoridis (998) find srong evidence of mean reversion in real exchange raes by implemening panel daa varians of sandard uni-roo ess. However, O Connel (998) srongly dispues hese mean-reversion findings in real exchange raes as hey fail o conrol for cross-secional dependence in he daa. Addiional evidence agains reversion o

3 PPP based on a panel has been repored in Engel e al. (997). Papell (997) and Liu and Maddala (996) also find ha evidence of mean reversion in panels of real exchange raes is very sensiive o he group of counries considered. Recenly, an alernaive explanaion bases he persisen deviaions from pariy on he presence of marke fricions ha preclude commodiy rade. Dumas (99), Uppal (99), Sercu e al. (995) develop equilibrium models of real exchange rae deerminaion which ake ino accoun ransacion coss and show ha adjusmen of real exchange rae owards PPP is necessarily a nonlinear process. Marke fricions in inernaional rade inroduce a neural range, or band of inacion, wihin which deviaions from PPP are lef uncorreced, as hey are no large enough o cover ransacion coss. In his dynamic equilibrium framework, deviaions from PPP follow a nonlinear sochasic process ha is mean revering. Obsfeld and Taylor (997), and O Connell and Wei () provide some empirical evidence of he effec of ransacion coss in his conex. In all hese sudies, he nonlinear naure of he adjusmen process is invesigaed in erms of a TAR model (Tong, 99). The TAR model allows for a ransacion coss band wihin which no adjusmen in deviaions from he PPP akes place so ha deviaions may exhibi a uni roo behavior while ouside he band, as goods arbirage becomes profiable, he process swiches abruply o become saionary. Oher auhors, as Michael e al. (997), Baum e al. (), employ he exponenial smooh ransiion auoregression (ESTAR) framework o analyze he dynamic behavior of deviaions form PPP, finding evidence of nonlinear adjusmen. Overall, one way o circumven he assumpion of a linear dynamic process is o posi a paricular model of non linear adjusmen. However, if here is lile a priori informaion concerning he acual form of adjusmen, he esimaed model is likely o suffer from a misspecificaion error. Moreover, i is ofen difficul o discriminae among alernaive nonlinear models using sandard diagnosic ools. In his paper, we es non-linear price adjusmen mechanisms under he curren floa, employing firs order Fourier series o approximae he non-linear adjusmen process. Under his specificaion, i is no necessary o pre-specify he naure of he adjusmen process. This feaure makes his procedure more suiable han oher alernaives such as TAR or ESTAR. As Enders and Ludlow () remarks, if here is reversion bu no linear or hreshold reversion he Fourier approximaion migh bes characerize he adjusmen process, since he funcional form of he alernaive hypohesis need no be specified.

4 The paper is organized as follows: Secion presens a brief descripion of absolue and relaive PPP. Secion analyzes he daa esing PPP using linear ess. Secion describes a procedure ha can capure he presence of non-linear adjusmen and presens he main resuls. Finally, Secion 5 concludes.. BRIEF DESCRIPTION OF PPP The key heoreical concep underlying our analysis is he Purchasing Power Pariy. In is simples formulaion, PPP posis equaliy beween he price level in one counry and he exchange rae adjused price level in he oher. I herefore reas he real exchange rae he nominal exchange rae divided by he raio of he wo counries price levels as a consan. How close his descripion is o acual experience depends on how he real exchange rae behaves. A common sraegy o es for PPP hypohesis over he pas decade and a half consiss on analyzing he presence of a uni roo in he real exchange rae. To undersand he relaionship beween real exchange raes and PPP, consider he following expression for he real exchange rae: q = s p + p (), where q is he log of he real exchange rae, s is he log of he corresponding nominal exchange rae measured in unis of domesic currency per uni of foreign currency, and p and p, are he logarihms of foreign and domesic prices, respecively. The absolue PPP hypohesis saes ha q will equal a consan, call q. In his case, we can rewrie () as: s = q + p p (), Under floaing exchange raes, () is a relaionship among price levels and he nominal exchange rae. We can hus hink of he above equaion as defining a coinegraion relaionship beween hese hree variables. To see his more clearly, consider he following sochasic equaion analogue o (): Sricly speaking, he absolue PPP hypohesis saes ha he exchange rae beween he currencies of wo counries should equal he raio of he price levels of he wo counries, specifically: S = P / P, where S, is he nominal exchange rae measured in unis of domesic currency per uni of foreign currency, P is he price level in he domesic counry, and P is he price level in he foreign counry. The measures of consumer prices published by naional saisical agencies are ypically repored as indexes relaive o a base year (say, 995=). Thus, hey only measure he rae of change of he price level from he base year, no is absolue level. For his reason, i is difficul o es absolue PPP in a sric sense.

5 s = + p + p + e () β β β, where β i are coefficiens and e is an error erm. If we impose he condiion β = y β =-, i is possible o apply a simple uni roo es o he real exchange rae iself o es for absolue PPP. Tha hypohesis is very resricive and recen sudies on he validiy of PPP have focused in a weaker heory: he relaive PPP hypohesis, which saes a long-run relaionship beween nominal exchange rae and prices. Relaive PPP hypohesis requires ha e in () be a saionary process, ha is, ( s, p, p ) should form a coinegraed sysem (Engle and Granger, 987).. DATA AND PRELIMINAR PPP ANALYSIS In his paper, boh he relaive and he absolue PPP hypohesis are esed using he Consumer Prices Index (CPI) as a proxy for price levels of each counry s oupu. The nominal exchange raes are end-of-monh bilaeral US dollar exchange raes. In all cases, he USA is considered he foreign counry. EcoWin provides us wih all series exraced from he Inernaional Moneary Fund s Inernaional Financial Saisics daabase. The sample included daa for 9 indusrialized counries: he USA, Ausria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ialy, Japan, Mexico, he Neherlands, Norway, Porugal, Spain, Sweden, Swizerland, and he Unied Kingdom. The daabase spans he period January 97 o February 5 for Canada, Denmark, Japan, Norway, Sweden, Swizerland and he Unied Kingdom. For Mexico, he sample sars in January 98 and finishes in February 5, for Greece, i sars in April 98 and finishes in December, and for he Neherlands he sample spans from January 98 o December 998. For Ausria, Belgium, Finland, France, Germany, Ialy, Porugal, and Spain he sample spans from January 97 o December 998 (he Euro currency is The Neherlands fi heir currency o he German Mark up o January 98. April 98 is he dae of he enry of Greece ino he European Economic Communiy. Afer he counry s public finances wen bankrup in 98, Mexico began a profound ransformaion. The foreign deb crisis of 98-8 had a severe and irreversible effec on public finances and he Sae s economic and poliical hinking. Unil ha dae, Mexico s economy and poliics were heavily subsidized and proeced from compeiion. CPI daa for he unified Germany are only available from January 99.

6 creaed in January 999). Figure shows he sandardized real exchange raes for he eigheen counries. The mos relevan poin here is ha each real exchange rae has exhibied large and persisen deviaions around is mean. Neverheless, during his period he real exchange raes considered seem o rever o an unknown reference value. [Inser Figure ] We examine he evidence on he absolue PPP hypohesis esing for saionariy of he real exchange raes using he Augmened Dickey Fuller (ADF) and he more efficien Dickey Fuller Generalized Leas Squares (DF-GLS), proposed by Ellio, Rohenberg and Sock (ERS, 996).,5 Table presens boh ess wih and wihou a ime rend in he es regression. 6 The lag lengh for he differenced dependen variable on he righ hand side of he es regression is deermined using he Akaike informaion crierion (AIC) in boh ess. The ess produce lile suppor for absolue PPP version. ADF es never rejecs he null of nonsaionariy, while he DF-GLS only does i for four counries. For Germany and Norway he DF-GLS es rejecs he null hypohesis of a uni roo a he 5% significance level. For Greece and Ialy, his es rejecs he null hypohesis a he % significance level. Cheung e al. () es for a uni roo on he real exchange rae for five counries: France, Germany, Ialy, Japan, and he Unied Kingdom. They find saionariy on real exchange rae for all cases, excep Japan. For France, Germany and Ialy heir sample spans he same period of ime han ours. In conras o Cheung e al. (), we do no find saionariy of he real exchange rae for France (in his case we deec a differen lag in he DF-GLS es). [Inser Table ] DF-GLS es has subsanially improved power when an unknown mean or rend is presen. As Ellio e al. (996) proves, he modified es works well in small samples. 5 All individual series (nominal exchange raes and prices) have also been esed for he presence of uni roo using he ADF and DF-GLS ess. Consisen wih he lieraure, he null hypohesis of a single uni roo canno be rejeced in any case. To save space, hese resuls are no repored here, bu are available upon reques. 6 Some researchers, for example, Cheung and Lai (998), and Koedijk e al. (998), have found ha he sochasic processes of some of he real exchange raes canno be adequaely modeled wihou he inclusion of a linear deerminisic ime rend. The linear deerminisic ime rend is generally inerpreed as represening sysemaic differences in produciviy growh beween radable and non-radable goods in he wo counries. On he oher hand, oher researchers, for example, Papell and Theodoridis (998), and Amara and Papell () consider a linear ime rend in he real exchange rae as inconsisen wih long-run PPP. 5

7 We also es relaive PPP using Engle and Granger (987) mehodology. Firsly, we esimae expression () and hen we es for saionariy of he residual using he ADF saisic. Figure shows he esimaed sandardized disequilibrium series for all counries. [Inser Figure ] Addiionally, we use he coinegraion DF-GLS es developed by Perron and Rodríguez (). They propose a residual based es for coinegraion when residuals are consruced using GLS derended o quasi-differenced daa o each variable of he sysem. 7 As Table repors, he Engle and Granger (987) es rejecs he null of non-coinegraion only in one case, he Neherlands, and he DF-GLS in merely wo cases (Denmark and he Neherlands). These resuls fall shor of being supporive of long-run PPP. [Inser Table ] In summary, he evidence in favor of mean reversion in US dollar-based PPP deviaions series is quie weak (we rejec he absolue PPP excep for counries and only for cases we can no rejec he relaive PPP) for he sample period sudied here. As menioned in previous secions, several recen sudies propose an alernaive explanaion ha bases he persisence of managed-floa deviaions from pariy on he presence of marke fricions ha impede commodiy rade. Several models ake ino accoun ransacion coss and show ha adjusmen o he equilibrium is necessarily a non-linear sochasic process. However, his non-linear, mean-revering, process is hard o capure using linear uni roo ess.. NON-LINEAR ADJUSTMENT To es for saionariy of he deviaions from PPP, we apply he Enders and Ludlow () es (EL-es). This procedure allows us o sudy he non-linear mean-revering behavior of PPP wihou having o specify he kind of nonlinear adjusmen process. Enders and Ludlow () sugges he following modificaion of he Augmened Dickey-Fuller (ADF) es: 7 Perron and Rodríguez () analyze residual based ess for coinegraion. Among oher cases, hey consider he sandard ADF and derive heir asympoic disribuion assuming a general quasi-differencing parameer c and abulae is criical values. Their simulaions reveal an imporan power gain from using GLS derended daa, especially if he quasi-difference parameer is se as suggesed by Ellio e al. (996). 6

8 p y = α() y + δ y + ε, () i i i= where{ y } is he process of ineres (he real exchange rae in our case), and ε is he sochasic error erm. No specificaion of he funcional form of α() is required, since his can be approximaed by a sufficienly long Fourier series. However, in order o keep he problem racable, we consider only a Fourier approximaion using a single frequency as Enders and Ludlow () sugges: π () sen k π k α = a + a + b cos, (5) T T where k is an ineger number in he inerval [, T/]. If y denoes he real exchange rae, we can use () o es he absolue PPP hypohesis. To analyze he relaive PPP hypohesis, he EL-es can be easily generalized o es for coinegraion in he Engle and Granger (987) framework. Le he { e ˆ } sequence denoe deviaions from long-run equilibrium. The relaionship among exchange raes and prices are esimaed by OLS in model () as: = ˆ + ˆ + ˆ (6) eˆ s ( β βp β p ), where s, p and p have been defined above. The non-linear adjusmen process is: where α() is defined in (5). p eˆ = α() eˆ + δ eˆ + ε, (7) i i i= Insead of searching for a specific nonlinear adjusmen, he problem is reduced o finding he mos suiable values of a, a, b and k using equaions () [or (7)] and (5). 8 The sufficien and necessary condiion for a non-explosive adjusmen process, or, in oher words, for a mean-revering behavior is: a < + r r for r = a + b. (8) An imporan feaure of he EL-es is ha i allows for differen paerns of meanrevering behavior. In paricular, he es allows for nonlinear auoregressive decay wih 8 The ADF es is obained for he special case of a = b =. 7

9 possible periods of explosive and/or oscillaory behavior. In paricular, for a > and r < he series revers o an aracor if a <+r /. There are wo main differen ypes of decay:. If a +r < here is overall reversion ha can be monoone when a > r and oscillaory when a < r.. If a +r > here will be k periods when α() exceeds uniy. This implies ha while he overall process ulimaely revers o he aracor (direcly when a > r and wih oscillaions when a < r), he sequence exhibis periods of explosive behavior. 9 This feaure makes his es especially relevan in he analysis of he heory of long-run PPP. As we remarked above, inernaional rade is affeced by facors like ransacion coss, axaion, subsidies, rade resricions or official inervenion in he exchange markes ha could cause asymmeric movemens in exchange raes. These effecs make deviaions from PPP a mean-revering non-linear sochasic process, because exchange raes and domesic and foreign prices adjus o pas disequilibria as a funcion of muliple and complex facors. Someimes, he real exchange rae can be saionary bu fricions in inernaional rade inroduce some periods of inacion, wihin which he mispricing is lef uncorreced. Also, he properies of he es make his approach more suiable han oher alernaives such as TAR or ESTAR. Wih his procedure, we do no need o specify he funcional form of he alernaive hypohesis. As Enders and Ludlow () remark, if here is mean reversion bu he decay is no linear or i does no presens hreshold reversion hen he Fourier approximaion migh bes characerize he adjusmen process. We apply he EL-es o boh absolue and relaive PPP. For absolue PPP, we run a regression of he real exchange rae over a consan and a linear rend and compue he e ˆ series. Then, he following regression is esimaed for all he ineger values of k in he inerval o T/, o selec he mos suiable k frequency: p π k π k eˆ ˆ ˆ = c+ a sen + b cos e δi e i ε, T T + + (9) i= 9 Noe ha (5) has he equivalen represenaion: () a r cos π k ( d) If a +r > and since cos k ( d ) T α = + +, where r = a + b and d = arcsin ( a / r). T π + can equal uniy, here will k periods when α() exceed he uniy, here being explosive periods. Oscillaions appear when α() <, ha being he case when a < r and cos k ( d ) T π + =. Addiionally, we apply he es on he demeaned real exchange rae. To save space, we only presen he demeaned and derended case, bu resuls are available upon reques. 8

10 where c = a - and p- is he number of lags needed o compleely eliminae residual auocorrelaion. In he case of relaive PPP, $e denoe he residuals of he regression of he deviaions from PPP defined in (6) on a linear rend. We choose he value of k ha minimizes he sum of he squared residuals. This value is denoed k, and he coefficiens linked o such frequency by c, a and b. The period, T / k, indicaes he lengh of ime required for he process o repea a full cycle. Therefore, he bigger is k, he smaller is he period. In order o es for saionariy of deviaions of PPP, we compue hree saisics from he esimaion of (9): F_all, F_rig, and τ-saisic. The es procedure has he following srucure. Firs, we compue he F_all saisics, his is an F saisic for he null hypohesis c = a = b =. This saisic is used o es wheher he series in quesion is a random walk. If he F_all null hypohesis is rejeced, he series in quesion exhibis mean-reversion, which may be, linear, non-linear or boh. Second, if he F_all null hypohesis is no rejeced hen we compue he F_rig saisic for he null hypohesis a = b =, which imply r =. If his hypohesis is rejeced, here is a non-linear mean-revering behaviour in he daa. Finally, if he F_rig null hypohesis is no rejeced hen we compue he τ-saisic for he null hypohesis c =. Rejecing he hypohesis c = is a necessary condiion for mean-reversion only when r =. If his null hypohesis is also no rejeced hen we rejecs he PPP hypohesis. The power of he hree ess depends on he values of r and c. As r increases, he power of F_all, F_rig and τ-saisic increases. Besides, for small values of c and r, he power of F_rig is slighly higher han F-all and he τ-saisic. Addiionally, when he series rever o he mean, we discuss he differen ypes of decays: monoone decay, decay wih explosive periods, oscillaions, and oscillaions an explosive periods. The ype of reversion depend on he values of c and r. Esimaion is carried ou in differenced daa o allow for comparison wih he Engle and Granger (987) es for he linear case. We also analyze he demeaned relaive PPP. The resuls are almos he same ha in he derended case. For his reason, o conserve space, hese resuls are no repored here, bu are available upon reques. Enders and Ludlow () obain he criical values for hese hree saisics. However, if he F_rig es correcly indicaes r >, rejecing he null of he c es in favor of he alernaive hypohesis c <, is sufficien o guaranee reversion. 9

11 .. Resuls for absolue PPP EL-es resuls for absolue PPP are shown in Table. They provide sronger evidence on mean reversion in real exchange raes han linear uni roo es. In paricular, F_all rejecs he null of a nonsaionary behavior for Ausria, Belgium and Swizerland. Addiionally, he F_rig deecs nonlinear adjusmen processes for seven counries (Belgium, Canada, Denmark, France, Norway, Porugal and Swizerland). Also, he values of he saisic are very high for Ausria and Japan indicaing evidence on non-lineariy. [Inser Table ] As Table shows, i is no possible o rejec he null c = in any case. This saisic has a low power when a is near.9, as is our case. This finding is consisen wih he resuls of he linear uni-roo ess shown in Table. In he oher hand, he value of a fulfils he reversion condiion in (8) for all he counries. Neverheless, in he case of Ialy, Mexico, Porugal, Spain and Sweden he value of a is over.99. The esimaed values of k oscillae beween and 7. The minimum value is achieved for Canada, Spain and he Unied Kingdom, and he maximum for Norway. In erms of he paern of decay deeced, in all cases a >r and a +r >, indicaing he exisence of periods of explosive behavior in real exchange raes. These finding indicae he presence of periods of decay and periods of non-saionariy, his being compaible wih he exisence of marke fricions and asymmeries ha make arbirage profiable only under some condiions. Summarizing, using a nonlinear analysis ha exends he radiional ADF es, we find sronger evidence in favor of absolue PPP han in he linear one. We find evidence in favor of mean-reversion in seven real exchange raes whereas he sandard ADF es does no rejec he non-saionariy null in any counry. In he linear case, we find some evidence in favor of absolue PPP using DF-GLS es for Germany, Greece, Ialy and Norway whereas wih he EL-es we find ha evidence for Ausria, Belgium, Canada, Denmark, France, Norway, Porugal and Swizerland... Resuls for relaive PPP When we analyze he deviaions from PPP in he coinegraion framework, he evidence on non-linear mean revering is higher. As can be seen in Table, he F_all and F_rig saisics provides sronger evidence on mean reversion in he deviaions from relaive

12 PPP han in he linear coinegraion framework. In paricular, F_all rejecs he null of a nonsaionary behavior for Greece, Mexico, he Neherlands and Swizerland a 5%. Also, he values of his saisic are close o he % criical value for Norway and he Unied Kingdom. The F_rig deecs nonlinear decay for six counries (Greece, Ialy, Mexico, Norway Swizerland and he Unied Kingdom). Finally, he null c = is rejeced a he 5% significance level for Germany, Greece and he Neherlands. [Inser Table ] The esimaed values of a fulfill he coinegraion condiion for all he counries, wih lower values han in he absolue PPP case. Canada achieves he higher value (a =.986), and Germany he lower one (a =.7765). For all counries, excep for Germany, a > r and a +r >. This resul indicaes ha he deviaions from PPP ulimaely decay o an aracor bu he sequence exhibis periods of explosive performance. In he German case, here is a monoone adjusmen process. Also in his case, using he nonlinear coinegraion analysis we find sronger evidence in favor of relaive PPP han under he linear framework. We find evidence in favor of meanreversion in long-run relaionship residuals for eigh counries, whereas he Engle and Granger procedure only rejec he null of non-coinegraion for he Neherlands and Denmark... A non-linear error correcion model Summarizing, we find evidence of nonlinear mean revering behavior in deviaions from PPP (absolue, relaive or boh) in welve of he eigheen analyzed counries. These resuls conras wih hose we found previously in he linear analysis, wih evidence on a linear convergence o long run equilibrium in six counries. As expeced, for hose six counries we also find saionariy in deviaions from PPP wih he Fourier approximaion, bu wih a non-linear componen. These resuls sugges sronger evidence in favor of he long run PPP han ha found by anoher auhors. For example, Baum e al. () analyze he same panel of counries ha we do, wih he excepion of Mexico, using a shorer sample. They find nonlineariies only in seven counries wihin an ESTAR framework. In our case, we find a nonlinear adjusmen o PPP for he same counries, excep Japan and Finland. Addiionally, we find nonlinear adjusmens in oher seven counries.

13 Since in he majoriy of cases he analyzed exchange raes and prices are coinegraed and he adjusmen owards he long-erm relaionship is non-linear, we can esimae an error correcion model exhibiing Fourier-decay. This model allows us o disinguish beween shor and long erm Granger causaliy. Long-erm causaliy (Granger, 986) is he resul of including all variables lagged one period in he ECM. This causaliy will always occur a leas in one direcion since, according o Engle-Granger represenaion heorem, if wo variables are coinegraed, a leas one of hem mus respond o deviaions from he long run equilibrium relaionship. The specified VAR-ECM exhibiing Fourier-decay is in our case: s m α α α s p = + e + p + p T ε () p sen π k ˆ m α α α T i Φ ε i= m α α α cos π k p ε where Φi, i =, K, p are x coefficien marices. This model allows us o spli he adjusmen process o pas disequilibria in hree differen erms: a linear one, capured by he α i, i =,, coefficiens, and wo non-linear erms, capured by α i sen π k and i π k T α cos, i =,,. T [Inser able 5] In Table 5 we presen esimaion resuls for non-linear error correcion model (equaion ()) for he eigheen counries analyzed. The firs hree columns show he esimaed coefficiens of he long-run relaionship in expression (). The remaining columns show esimaed adjusmen coefficiens, which represen he adjusmen o pas disequilibria in he exchange rae equaion (cols 5 o 7), in he domesic price equaion (cols 8 o ) and in he foreign price equaion (cols o ). The exchange rae responds o pas disequilibria in all counries, suggesing ha he exchange raes adjus for deviaions from relaive PPP. In mos cases, he response of exchange raes is made up of a linear componen and a non-linear one, since no only α bu also α or α, are individually significan. 5 The evidence on price adjusmen o deviaions from PPP is much weaker. Finally, analyzing resuls for domesic price equaions we only find significan response o deviaions from relaive PPP for six counries: a linear response in France, Greece, Ausria, and nonlinear responses in Ausria, Mexico, he Neherlands and Norway. 5 We approximae he disribuion of hese significaion ess by a normal sandard disribuion.

14 The response of he foreign price is linear for Belgium, Denmark, Germany, Mexico and Spain, and nonlinear for France and he Neherlands. These resuls sugges ha exchange raes ake mos of he burden of adjusmens o long-run PPP, in mos cases hrough a non-linear correcion mechanism. Mos likely, his reflecs some degree of sickiness in consumer prices in all counries. We also analyze linear Granger causaliy (or shor erm linear causaliy) in order o sudy he emporal flow of informaion in he sysem. Using he sandard definiion, a variable is said o cause anoher, if he inroducion of he lags of he causal variable in he model of he caused one improves he forecas of he caused variable. 6 Resuls are shown in Table 6. [Inser able 6] We find lile evidence of bi-direcional linear causaliy beween exchange raes and prices. Lagged domesic prices changes are only significan in he equaion of exchange raes for France, Germany, Porugal and Sweden. Lagged foreign prices changes are only significan for France, Japan, Norway and Swizerland. On he oher hand, lagged exchange rae changes are significan in he case of Ialy, Mexico, he Neherlands, Norway and Swizerland in he equaion of domesic prices and only for Belgium, Denmark and Porugal in he case of foreign prices. No significan paern seems o emerge from hese observaions. Remarkably, we find shor-erm bilaeral feedback beween domesic and foreign prices changes, he lags of each one of hem being joinly significan in he equaion of he oher for almos all he counries in he sample. This resul may reflec he presence of exogenous shocks such as shocks in raw maerials prices like peroleum, simulaneously affecing domesic and foreign prices. 5. CONCLUSIONS Using nonlinear ess, his paper finds evidence of nonlinear mean-revering behavior in real exchange raes and in deviaions from long run PPP, which are consisen wih he purchasing power pariy hypohesis, adjused for marke fricions such as ransacion coss. We use a sample of monhly CPI indexes and nominal exchange raes for 97-, for a broad se of US rading parners. We find ha deviaions from PPP may go rough shor 6 To es he exisence of shor-erm causaliy, we sar from he esimaed VAR-ECM model (expression ()) and carry ou a join significance es of he lags of he causal variable in he equaion of he caused one.

15 periods of explosive behavior, wih overall mean revering performance. These resuls reinforce he insigh of previous sudies regarding he possible presence of non-linear, bu saionary adjusmen processes o long-run PPP. We find evidence of a nonlinear mechanism o correc for deviaions from long-run PPP. Our findings sugges ha he adjusmen o long-run equilibrium comes mainly from he exchange rae marke. In he shor erm, we find a bi-direcional flow of informaion beween domesic and foreign prices, possibly due o experiencing common shocks, while we rarely find Granger causaliy beween price changes and nominal exchange rae changes.

16 REFERENCES. Amara, J., D. Papell (). Tesing for purchasing power pariy using saionary covariaes, working paper, Deparmen of Economics, Universiy of Houson. Baum, C. F., J. T. Barkoulas and M. Caglayan (). Non-linear adjusmen o purchasing power pariy in he pos-breon Woods era. Journal of Inernaional Money and Finance,, Cheung, Y. W., K. S Lai and M. Bergman, (). Dissecing he PPP puzzle: The unconvenional roles of nominal exchange rae and price adjusmens, Journal of Inernaional Economics 6, 5-5. Cheung, Y. W., K. S Lai and S. Kon, (998). Pariy reversion in real exchange raes during he pos-breon Woods period, Journal of Inernaional Money and Finance, 8, Corbae, D. and S. Ouliaris (988). Coinegraion and ess of purchasing power pariy. Review of Economics and Saisics, 7, Davidson, R. and J. MacKinnon (99). Esimaion and Inference in Economerics. Oxford Universiy Press, Oxford. Diebold, F., S. Hused, and M. Rush (99). Real exchange rae under he gold sandard. Journal of Poliical Economy, 99, Dumas, B. (99). Dynamic equilibrium and he real exchange rae in a spaially separaed world. Review of Financial Sudies, 5, 5-8. Frankel, J. and A. Rose (996). A panel projec on purchasing power pariy: mean reversion wihin and beween counries. Journal of Inernaional Economics,, 9-. Edison, H. J. and E. Fisher (99). A long-run view of he European moneary sysem. Journal of Inernaional Money and Finance,, 5-7. Ellio, G., T. J. Rohenberg and J. H. Sock (996). Efficien ess for an auoregressive uni roo. Economerica, 6, Enders, W. and J. Loudlow (). Non-linear decay: Tess for an aracor using a Fourier approximaion. Working Paper --, Universiy of Alabama. Engel, M. K. Hendrickson and J. H. Rogers (997). Inra-naional, inra-coninenal and inra-planeary PPP. Inernaional Finance Discussion Papers, 589, Washingon DC: Board of Governors of he Federal Reserve Sysem. 5

17 Engle, R. and C. Granger (987). Co-inegraion and error correcion: represenaion, esimaion, and esing. Economerica, 55, Granger, C. (986). Developmens in he sudy of coinegraed variables. Oxford Bullein of Economics and Saisics, 8, -8. Grilli, V. and G. Kaminsky (99). Nominal exchange rae regimes and he real exchange rae: evidence for he Unied Saes and Grea Briain, Journal of Moneary Economics, 7, 9-. Koedijk, K., P. Schoman, and M. Van Dijk (998). The re-emergence of PPP in he 9 s, Journal of Inernaional Money and Finance, 7, 5-6. Liu, P. C., and G. S. Maddala (996). Do panel daa cross-counry regressions rescue purchasing power pariy (PPP) heory?. Working Paper, Deparmen of Economics, Ohio Sae Universiy. Lohian, J. (997). Muli-counry evidence on he behaviour of purchasing power pariy. Journal of Inernaional Money and Finance, 6, 9-5. MacKinnon, J. G. (996). Numerical disribuion funcions for uni roo and coinegraion ess. Journal of Applied Economerics,, Meese, R. and K. Rogoff (988). Was i real? The exchange rae ineres differenial relaion over he modern floaing exchange rae period. Journal of Finance,, 9-8. Michael, P., A. Nobay and D. Peel (997). Transacion coss and non-linear adjusmens in real exchange raes: an empirical invesigaion. Journal of Poliical Economy 5, O Connell, P. G. J. (998). The overvaluaion of purchasing power pariy. Journal of Inernaional Economics,, -9. O Connell, P. and S. Wei. () The bigger hey are he harder hey fall. Journal of Inernaional Economics, 56, -5. Obsfeld, M. and A. Taylor (997). Non-linear aspecs of goods-marke arbirage and adjusmen: Heckscher s commodiy poin revisied. Journal of Japanese and Inernaional Economics,, -79. Oh, K. Y. (996). Purchasing power pariy and uni roo ess using panel daa. Journal of Inernaional Money and Finance, 5, 5-8. Papell, D. H. (997). Searching for saionariy: purchasing power pariy under he curren floa. Journal of Inernaional Economics,, -. 6

18 Papell, D. H. and H. Theodoridis (998). Increasing evidence of purchasing power pariy over he curren floa. Journal of Inernaional Money and Finance, 7, -5. Pedroni, P. (997). Panel coinegraion: asympoic and finie sample properies of pooled ime series ess wih an applicaion o he PPP hypohesis (new resuls). Working paper, Deparmen of Economics, Indiana Universiy. Perron, P. and G. Rodríguez, (). Residual based es for coinegraion wih GLS derended daa. PhD disseraion, Universiy of Monreal. Sercu, P., R. Uppal, C. Van Hull, (995). The exchange rae in he presence of ransacion coss: implicaions for ess of purchasing power pariy. Journal of Finance, 5, 9 9. Taylor, A. (996). Inernaional capial mobiliy in hisory: purchasing-power pariy in he long run. Working Paper 57. Naional Bureau of Economic Research, Cambridge, MA. Tong, H. (99). Non-linear Time Series. Oxford Universiy Press, Oxford. Uppal, R. (99). A general equilibrium model of inernaional porfolio choice. Journal of Finance, 8, Wu, Y. (996). Are real exchange raes non-saionary? Evidence from a panel daa es. Journal of Money, Credi and Banking, 8,

19 Figure. Sandardized real exchange raes Ausria Belgium Canada Denmark Finland France Germany Greece Ia ly Japa n Mexico The Neherlands Norway Porugal Spain Sweden Swizerland Unied Kingdom Table. Uni roo es for he real exchange rae Obs ADF_C Lags ADF_C+T Lags GLS_C Lags GLS_C+T Lags Ausria Belgium Canada Denmark Finland France Germany Greece Ialy Japan Mexico The Neherlands Norway Porugal Spain Sweden Swizerland Unied Kingdom Noe: The ADF es is performed including a consan (ADF_C) and a consan and a ime rend in he regression (ADF_C+T). Asympoic criical values for coinegraion are aken from MacKinnon (996). Wih consan: -.87 (5 %), -.57 ( %). Consan and ime rend: -. (5 %), -. ( %). Elio e al. (996) propose a simple modificaion of he ADF_C (GLS_C) and he ADF_C+T (GLS_C+T) ess in which he daa are firs derended, so ha explanaory variables are aken ou of he daa prior o running he es regression. Asympoic criical values for GLS_C are hose of ADF -saisic when here is no consan: -.9 (5 %), -.6 ( %). Asympoic criical values for GLS_C+T are aken from Elio e al. (996), Table : -.89 (5 %), -.57 ( %). Saisical significance is indicaed by a single aserisk () for he % level, and a double aserisk () for he 5% level. 8

20 Figure. Sandardized deviaions from long run PPP (relaive PPP) Ausria Finland Ialy Norway Belgium - -. Canada France Germany Denmark Greece Japan Mexico Neherlands Porugal Spain Sweden Swizerland Unied Kingdom Table. Linear coinegraion es Obs ADF_C Lags ADF_C+T Lags GLS_C lags GLS_C+T Lags Ausralia Belgium Canada Denmark Finland France Germany Greece Ialy Japan Mexico The Neherlands Norway Porugal Spain Sweden Swizerland Unied Kingdom Noe: The ADF es is performed including a consan (ADF_C) and a consan and a ime rend in he regression (ADF_C+T). Asympoic criical values for coinegraion are aken from Davidson and MacKinnon (99). Wih consan: -. (5 %), -. ( %). Consan and ime rend: -.78 (5 %), -.5 ( %). Asympoic criical values for GLS_C and GLS_C+T are aken from Perron and Rodríguez (). Asympoic criical values for GLS_C are: -.76 (5 %), -.7 ( %). For GLS_C+T are: -.9 (5 %), -.7 ( %). Saisical significance is indicaed by a single aserisk () for he % level, and a double aserisk () for he 5% level. 9

21 Table Enders-Ludlow es for nonlinear uni roo in real exchange raes Obs Lags k c a b a +r / F_all F_rig Ausria (-.85) (-.8) (.6988) Belgium (-.7) (.67) (.976) Canada (-.) (.5) (.7) Denmark (-.7) (.5) (.97) Finland (-.7669) (-.69) (.97) France (-.697) (.656) (-.58) Germany (-.78) (-.6688) (-.986) Greece (-.67) (-.56) (.) Ialy (-.8) (.8679) (.687) Japan (-.77) (-.78) (.85) Mexico (-.76) (-.597) (-.8) The Neherlands (-.58) (.879) (.57) Norway (-.75) (-.85) (.76) Porugal (-.5657) (-.7885) (.7) Spain (-.9859) (.96) (.9) Sweden (-.95) (-.79) (.6) Swizerland (-.55) (.988) (.9989) Unied Kingdom (-.9867) (-.598) (-.5) Noe: The esimaed model is: ˆ sen p π π k e cos ˆ ˆ = c + a + b e + δ e + ε i i T T i= where e are he residual of he regression of he real exchange rae on δ ˆ +δ. τ-saisics in brackes. Criical values a 9% and 95% are respecively 7.6 and 8.5 for F_all saisic (H : c =a =b =), 7.7 and 8.7 for F_rig saisic (H : a =b =) and.8 and.5 for he τ-saisic (H : c =). Saisical significance is indicaed by a single aserisk () for he % level and double aserisk () for he 5% level.

22 Table. Enders-Ludlow es for nonlinear coinegraion Obs Lags k c a b a +r / F_all F_rig Ausria (-.786) (-.76) (.9) Belgium (-.99) (.767) (-.96) Canada (-.) (-.57) (-.65) Denmark (-.875) (.6) (.89) Finland (-.5) (.58) (.8) France (-.65) (.77) (.785) Germany (-.56) (-.8) (.685) Greece (-.57) (.57) (.976) Ialy (-.7586) (.88) (.8) Japan (-.58) (-.59) (-.98) Mexico (-.559) (-6.897) (.75) The Neherlands (-.) (.7) (-.) Norway (-.98) (.8) (.98) Porugal (-.) (.799) (.67) Spain (-.5) (-.9797) (-.96) Sweden (-.57) (.885) (.8597) Swizerland (-.) (.957) (.957) Unied Kingdom (-.66) (.565) (.58) Noe: The esimaed model is: ˆ sen p π π k e cos ˆ ˆ = c + a + b e + δ e + ε i i T T i= where e are he long-run regression derended residuals [model (6) in ex]. τ-saisics in brackes. Criical values a 9% ˆ and 95% are, respecively 8. and 9.6 for F_all saisic (H : c =a =b =), 7. and 8. for F_rig saisic (H : a =b =), and.8 and.5 for he τ-saisic (H : c =). Saisical significance is indicaed by a single aserisk () for he % level, and a double aserisk () for he 5% level.

23 Table 5. Vecor error correcion model wih Fourier Decay β β β Lags α α α α α α α α α Ausria (-.6) (-.5787) (.6) (.656) (.78) (.8) (-.97) (-.88) (-.58) Belgium (-.7) (.576) (-.79) (.99) (.89) (.8) (-.9) (-.867) (-.598) Canada (-.79) (-.8) (-.8577) (-.8) (-.8) (-.6588) (-.59) (.6699) (-.7) Denmark (-.7) (.86) (.9) (-.76) (-.7) (-.9) (-.67) (.888) (-.5) Finland (-.) (.686) (.68) (-.7) (-.987) (-.66) (.858) (-.98) (-.6) France (-.68) (.58) (.75) (.9) (.7659) (.6) (-.68) (.9) (.797) Germany (-.95) (-.89) (.75) (-.55) (-.98) (.575) (-.968) (-.87) (.7) Greece (-.55) (.) (.78) (.96) (.577) (-.668) (-.6) (.578) (-.599) Ialy (-.) (.765) (.568) (-.695) (-.) (.76) (-.68) (-.958) (-.7) Japan (-.65) (-.68) (-.757) (-.6) (-.8) (-.788) (-.765) (.7) (-.5) Mexico (.5) (-6.66) (.9) (-.7) (-.) (.6) (-.86) (-.59) (.7758) The Neherlands (-.87) (.68) (-.5) (-.9) (-.7) (-.7) (.579) (.6) (-.755) Norway (-.887) (.85) (.788) (-.97) (-.6) (.7) (.96) (-.69) (-.6) Porugal (-.85) (.5) (.6) (-.5) (-.6) (.) (-.7) (.6) (-.) Spain (-.6) (-.6) (-.7) (.9) (.8959) (-.6) (-.97) (-.86) (-.79) Sweden (-.956) (.65) (.97) (-.5699) (-.69) (.666) (-.86) (-.59) (.) Swizerland (-.) (.86) (.66) (.77) (.6855) (-.6) (-.6) (.5) (-.55) Unied Kingdom (-.65) (.69) (.5) (.57) (.7) (-.59) (-.599) (.7585) (-.) Noe: -saisics in brackes. Saisical significance is indicaed by a single aserisk () for he % level, a double aserisk () for he 5% level and a riple aserisk () for he % (H : α ij =). The long-run esimaed model is s = β + βp + βp + e and he non-linear vecor error correcion model is: s m α α α p s i ε π k p sen = m + α α α T eˆ + Φi p i + ε p i m π cos k = α p i α α ε T

24 Table 6. Shor erm linear Granger causaliy es s equaion p equaion p equaion DF p lags p lags s lags p lags s lags p lags Ausria (.5777) (.85) (.) (.96) (.9) (.) Belgium (.75) (.) (.6) (.58) (.599) (.85) Canada (.9758) (.675) (.7) (.) (.99) (.96) Denmark (.887) (.6) (.788) (.) (.6) (.7) Finland (.9889) (.557) (.57) (.) (.89) (.9) France (.) (.7) (.77) (.6) (.857) (.) Germany (.6) (.96) (.5899) (.566) (.) (.) Greece (.9) (.9) (.998) (.78) (.9) (.6) Ialy (.86) (.78) (.5) (.) (.58) (.88) Japan (.59) (.888) (.975) (.6) (.666) (.66) Mexico (.75) (.556) (.) (.) (.8) (.) The Neherlands (.996) (.97) (.85) (.) (.8967) (.) Norway (.668) (.866) (.85) (.9) (.98) (.66) Porugal (.) (.9) (.86) (.) (.9) (.6) Spain (.) (.97) (.6) (.) (.99) (.8) Sweden (.) (.5) (.75) (.) (.66) (.) Swizerland (.987) (.6) (.) (.7) (.876) (.78) Unied Kingdom (.88) (.6) (.97) (.7) (.7) (.) Noe: esimaed equaion is: s m α α α p s i ε π k p sen = m + α α α T eˆ + Φi p i + ε p i m π cos k = α p i α α ε T p-values in brackes. Saisical significance is indicaed by a single aserisk () for he % level, double aserisk () for he 5% level and a riple aserisk () for he %.