Nonlinear mean-reversion in the real exchange rate: evidence from a panel of OECD countries

Transcription

1 Nonlinear mean-reversion in he real exchange rae: evidence from a panel of OECD counries Absrac Hassan Shirvani Universiy of S Thomas Naalya Delcoure Universiy of S Thomas We examine he presence of mean-reversion in he real exchange raes of a panel of 0 OECD using he sandard Dickey-Fuller es and smooh ransiion auoregression (STAR) framework. Our resuls provide srong suppor for he long run empirical validiy of he purchasing power pariy in mos all major indusrial counries. Our findings aribue he nonlinear saionariy of he real exchange raes o cerain nonlineariies in he shor run adjusmen pahs of hese raes. Such nonlineariies in he adjusmen process may arise by more aggressive marke reacion and governmen inervenion when is deparure from equilibrium is more drasic. Keywords: ime series, exchange raes, marke efficiency Nonlinear mean reversion, Page

2 Inroducion Since he pah-breaking work of Nelson and Plosser (98), i is widely believed ha mos macroeconomic and financial ime series are characerized by linear uni roo processes. As poined ou by Campbell and Perron (99), his finding has imporan implicaions for boh heoreical and empirical research. In paricular, he presence of uni roos indicaes ha he shor run deparures of he underlying variables from heir long run equilibrium values are fairly persisen and irreversible, a finding which is a odds wih many of he exising heories of economic and financial behavior, where such deparures are assumed o be shor-lived and selfcorrecing. Given he unfamiliar implicaions of uni roo processes, here has been a hreepronged scholarly response. Firs, having acceped he basic premise ha mos ime series daa do conain uni roos, some researchers have aemped o raionalize his fac by new heoreical models. Examples include he developmen of he real business cycle heory (Kydland and Presco, 98) o explain he presence of uni roos in real oupus, and he efficien markes hypohesis (Fama, 99) o jusify he random walk characer of sock prices. Second, anoher srand of research has aemped o refue he Nelson-Plosser findings of uni roos in major ime series by using more powerful economeric echniques, such as he Bayesian (Schoman and van Dijk, 99) and panel daa uni roo ess (Frankel and Rose, 996). Finally, oher researchers have ried o aribue he failure o rejec he null of uni roos in many macroeconomic and financial variables o he use of inadequae alernaive hypoheses. Insead of esing for uni roos agains he alernaive of saionariy around a linear rend, as is commonly done in sandard uni roo ess, i is recommended ha hese ess be conduced agains he alernaive of saionariy around a nonlinear rend. Examples of hese nonlinear alernaive hypoheses, ofen jusified by he presence of nonlineariies in he underlying adjusmen processes, include he use of hreshold auoregression (Balke and Fomby, 997), shifed inerceps and broken rends (Perron, 989; Zivo and Andrews, 99), auoregressions subjec o ceilings and floors (Pesaran and Poer, 997), asymmeric auoregression (Enders and Granger, 998), and smooh ransiion auoregression (Terasvira, 994; Kapeanios, Shin, and Snell, 00). Among he above nonlinear alernaive hypoheses, he smooh ransiion auoregression (STAR) has received considerable aenion in recen years (see van Dijk, Terasvira, and Franses, 00, for an excellen survey). Like many oher nonlinear approaches o ime series modeling, he STAR approach is based on he exisence of a number of disinc regimes, each becoming operaional a a differen ime in response o a differen se of circumsances. Unlike oher approaches, however, where he ransiion from one regime o anoher is usually sudden and abrup, he STAR model assumes hese ransiions o be gradual and smooh. In oher words, by allowing he exisence of middle ground regimes, he STAR model provides a poenially more flexible framework o capure he ime series behavior of many macroeconomic and financial ime series. Given he preceding discussion, his paper aemps o illusrae he use of he STAR framework in analyzing he ime series behavior of he real exchange rae in 0 OECD counries over he period. More specifically, he STAR model is used as he alernaive nonlinear saionariy hypohesis for esing he null hypohesis of uni roos in he real exchange raes. Given he imporance of he real exchange rae as a key macroeconomic variable, here already exiss a voluminous empirical lieraure on is ime series behavior (see Rogoff, 996, and Sarno and Taylor, 00, and Taylor, 006, for excellen surveys of he lieraure). The main purpose of mos of his lieraure is o esablish wheher he real exchange rae is characerized Nonlinear mean reversion, Page

3 by he presence of uni roos during he recen floa, a condiion unfavorable o he long run mean reversion of he real exchange raes and, hence, o he empirical validiy of he long run purchasing power pariy (PPP). In paricular, Roll (979), Adler and Lehmann (983), Taylor (988), Mark (990), and Fraser e al. (99), among ohers, have found evidence of uni roos in he real exchange rae for a large number of indusrial and developing counries. Since hese findings are inconsisen wih he long run PPP, oher researchers using a hos of oher approaches, including fracional inegraion, mulivariae coinegraion, panel daa uni roo ess, and longer sample periods, have uncovered evidence more favorable o he saionariy of he real exchange raes (Taylor and McMahon, 988; Kugler and Lenz, 993; Frankel and Rose, 996; Lohian and Taylor, 996; MacDonald, 996; Baum e al., 999; Flores, e al., 999; Cheung and Lai, 000; Taylor, 00). The laer aemps o salvage he long run PPP, however, suffer from a number of drawbacks. Mos imporanly, hey are all based on essenially linear processes ha ignore such nonlinear effecs as ransacions coss, variable speeds of adjusmen, regime shifs, and monopolisic pricing effecs (e.g., Dumas, 99; Ohanian and Sockman, 997; Chari e al. 000; Baum e al. 00; Sarno e al., 004). In addiion, by relying on longer sample periods, some of hese sudies include periods of fixed exchange raes which have been shown o bias he resuls in favor of he saionariy of he real exchange raes (Rogoff, 996; Engel, 000). Finally, incorporaing he foregoing nonlinear effecs ino he real exchange rae behavior, a number of recen sudies have relied on he STAR model o derive evidence in suppor of he long run PPP (Taylor, e al., 00; Chorareas e al., 00; Lahinen, 006; Sekioua and Karanasos, 006). This paper advances he evidence on he empirical validiy of he long run PPP by esing for he nonsaionariy of he real exchange rae agains a nonlinear (STAR) alernaive for 0 major OECD counries during he recen period of floa, Thus our sudy covers a larger number of major indusrial counries and for a longer span of ime han earlier sudies. In addiion, insead of using bilaeral exchange raes, as is common in mos earlier empirical work, we rely on he rade-weighed real exchange raes, as repored by he OECD. The res of he paper is organized as follows. Secion II discusses he economeric mehodology employed. Secion III presens he empirical resuls. Secion IV concludes. Mehodology As saed earlier, he main objecive of his paper is o es he null hypohesis of uni roos in he real exchange raes of a sample of OECD counries agains he alernaive of saionariy wihin a smooh ransiion auoregression (STAR) framework. A univariae STAR process in a mean-zero (i.e., derended) sochasic process y can be expressed as: y β y + γ y Θ( θ; y ) + ε,,...,, () = d = T where ε iid(0, σ ), β and γ are unknown parameers, represening wo alernaive auoregressive regimes, and Θ( θ; y d ) is he ransiion funcion, wih θ = speed of rendreversion, and d = delay parameer. In addiion, he ransiion funcion is assumed o ake he following exponenial form: Θ( θ ; y ) = exp( θy ), () d d where i is assumed ha θ > 0 and d>. Clearly, he ransiion funcion can adop any value beween zero and one. Combining () and (), we obain: Nonlinear mean reversion, Page 3

4 [ exp( θy )] ε, y = β y + γy d + (3) which can alernaively rewrien as: y = φ y + γy [ exp( θy d )] + ε, (4) where φ = β. Clearly, if φ = θ = 0, y will have a uni roo as one possible auoregressive regime, and if φ = 0 and θ > 0, y will follow a nonlinear bu saionary process as an alernaive regime, assuming ha -< γ <0. Furhermore, he delay parameer d is chosen o maximize he goodness of fi of (4) over {,,., d max }, where d max is deermined by using one of he usual lag selecion procedures. If, following Kapeanios, Shin, and Snell (00, henceforh, KSS), he condiion φ = 0 is imposed, (4) can be rewrien as: { exp( θy d } y = γ y ) + ε. (5) Now, he null hypohesis of a uni roo agains he alernaive of a nonlinear STAR saionariy can be expressed as: H 0 : θ = 0, (6) H : θ > 0. (7) Since, under he null hypohesis, he nuisance parameer γ canno be idenified (Davies, 987), he paper follows Luukkonen, Saikkonen, and Terasvira (988) and derives a es by approximaing (5) by a firs order Taylor expansion (wih lagged values of he firs differences of y added o whien he error process a la Dickey and Fuller, 979): ρ y = ρ y e (8) j= j j + δ y y d + Thus, he null hypohesis can be esed as a es of δ = 0, agains he alernaive of δ < 0, by using he following saisic: NL = δ / s. e.( δ ), (9) using he criical values abulaed by KSS. KSS also sugges an alernaive join F es of φ = δ = 0 in he following: ρ y = ρ y φ δ (0) j = j j + y + y y d + e, based on he criical values provided by Enders and Granger (998). Empirical Resuls This secion presens he empirical resuls of esing for he presence of uni roos in he real exchange raes of our sample counries, using he mehodology discussed in he preceding secion. The daa, which are aken from he OECD files of he RATS sofware package, are quarerly, calculaed as he logs of he real exchange raes, and cover he 970:-0: period. As a firs sep in he analysis of he ime series properies of he real exchange raes, his secion conducs he sandard Dickey-Fuller uni roo ess of hese raes agains he alernaive hypoheses ha hey are saionary around a consan (since a visual inspecion of he real exchange raes indicaed no clear rend for any of he counries in he sample, no ime rends were included in he Dickey-Fuller ess). As is well known, he implemenaion of he Dickey- Fuller es requires he whiening of he error erms associaed wih he auxiliary equaions of Nonlinear mean reversion, Page 4

5 hese ess by adding an appropriae number of lags of he firs differences of he underlying variables o hese equaions. To esablish he appropriae lag lengh for each of he sample counries, he Akaike informaion crierion (Akaike, 973) is used. The Dickey-Fuller uni roo es resuls are given in Table. As seen from he able, he null of a uni roo canno be rejeced for seveneen of he weny sample counries (excepions are Germany, Neherlands, and Norway), indicaing he absence of mean-reversion for he real exchange rae in an overwhelming majoriy of he OECD counries. This finding, aken a face value, offers unfavorable evidence abou he empirical validiy of he long run PPP for OECD counries. Having esablished he random walk behavior of he real exchange raes wihin he sandard Dicky-Fuller framework, his secion now proceeds o examine he ime series properies of hese raes wihin a STAR model. As saed in he preceding secion, he STAR model ess for he presence of uni roos in he real exchange raes agains he alernaive hypohesis ha hese raes are saionary wihin a smooh regime-swiching framework. As also seen from he previous secion, he implemenaion he STAR approach requires ess of significance of cerain esimaed coefficiens in he auxiliary equaions (8) and (0). Specifically, his involves a es of significance of δ in he following equaion: ρ y = ρ y e () j= j j + δ y y d + See Table (Appendix). Or, alernaively, an F es of join significance of φ and δ in he following equaion: y = ρ j= ρ y j φ δ j + y + y y d + e, where he numbers of he lags used in he above equaions are he same as hose previously seleced by he Akaike mehod for he Dickey-Fuller ess. The esimaion of he above equaions, however, also requires he selecion of an appropriae value for d, he delay parameer. To his end, and for each of he sample counries, each of he above equaions is firs esimaed for all values of < d < d max, where d max represens he opimal lag lengh previously seleced by he Akaike mehod. Nex, he value of d wih he bes fi, i.e., wih he lowes significan p- value, is seleced as he opimal delay parameer o be used in he esimaion of he corresponding counry equaions. These esimaed equaions are hen used o conduc he STAR significance ess. The resuls of hese ess are repored in Table. I can be seen from he able ha, based on boh ses of es resuls, he STAR approach rejecs he null of uni roos for sixeen of he weny counries in he sample in favor of saionariy wihin a nonlinear STAR framework (he only excepions being Ausralia, Greece, Ialy, and Porugal where nonlinear saionariy is rejeced by he es bu no he F es). These resuls indicae ha for almos all he counries in he sample, here is significan evidence ha he real exchange rae has a endency o rever o is long erm mean, wih any shor erm deparure from he long erm mean being ransiory and shor-lived. Thus, hese resuls are clearly consisen wih similar findings in he lieraure, referred o earlier in he paper, which documen he long run mean-reversion of he exchange raes in major indusrial counries. These resuls, however, are clearly a odds wih he sandard Dickey-Fuller es resuls, which, as we have seen earlier, end o suppor he random walk characer of he real exchange raes. See Table (Appendix). () Nonlinear mean reversion, Page 5

6 As a final word, here is a need o jusify he nonlinear saionariy of he real exchange raes for almos all he OECD counries. Since he auoregressive equaions (8) and (0) above are essenially shor run adjusmen pahs of he real exchange raes owards heir long run equilibrium values, he nonlinear saionariy of he real exchange raes can hus be inerpreed as he nonlinear adjusmen behavior of hese raes. More specifically, he STAR model assumes ha he speed of adjusmen is a funcion of he size of he deviaion of he acual real exchange raes from heir equilibrium values. This siuaion can arise mosly in connecion wih he prevailing balance of paymens reacion funcions, where he exchange raes, wheher hrough marke forces or governmen policies, are expeced o be riggered more srongly in response o a larger disequilibrium in he exchange raes. For small deviaions from he equilibrium values, marke reacions or policy responses are usually mued, as such small deparures may be deemed oo limied and ransiory o warran sronger response. However, for more dramaic deparures of he exchange raes from heir hisorical means, he exen of marke reacion and governmen inervenion may be more exensive, wih he adjusmen of he real exchange raes o heir equilibrium values aking place a a much faser rae. Conclusion This paper has shown ha based on he sandard Dickey-Fuller uni roo es, which is a es of saionariy around a consan, he real exchange raes of a large panel of OECD counries follow random walks and are, hus, largely unfavorable o he long run PPP. The paper, however, has also shown ha mos of hese exchange raes are saionary wihin a nonlinear STAR framework, which renders hem consisen wih he empirical validiy of he PPP in he long run. Finally, he paper has aribued he nonlinear saionariy of he real exchange raes o cerain nonlineariies in he shor run adjusmen pahs of hese raes. Such nonlineariies in he adjusmen process can arise by more aggressive marke reacion and governmen inervenion when is deparure from equilibrium is more drasic. References Adler, M. and B. Lehmann, 983, Deviaions from Purchasing Power Pariy in he Long Run, Journal of Finance, 38, Akaike, H., 973, Informaion Theory and an Exension of he Maximum Likelihood Principle, Second Inernaional Symposium on Informaion Theory: Akademiai Kiado, 67-8 Balke, N.S. and T.B. Fomby, 997, Threshold Auoregression, Inernaional Economic Review, 38, Baum C., J. Barkoulas, and M. Caglayan, 999, Long Memory of Srucural Breaks: Can Eiher Explain Nonsaionary Real Exchange Raes under he Curren Floa? Journal of Inernaional Financial Markes, Insiuions, and Money, 9 (4), Baum C., J. Barkoulas, and M. Caglayan, 00, Nonlinear Adjusmen o he Purchasing Power Pariy in he pos-breon Woods Era, Journal of Inernaional Money and Finance, 0 (3), Campbell, J. and P. Perron, 99, Pifalls and Opporuniies: Wha Macroeconomiss Should Know abou Uni Roos, Naional Bureau of Economic Research, Macroeconomics Conference, Cambridge, Mass. February. Nonlinear mean reversion, Page 6

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9 Appendix Counry Table Uni Roo Tes Resuls (Dickey and Fuller, 979) Counry Lags es Counry Lags es Ausralia Ialy Ausria Japan Belgium -.3 Neherlands -3.08* Canada Norway -3.5* Denmark Porugal Finland -.47 Spain -.5 France Sweden -.74 Germany 3-3.0* Turkey Greece UK Ireland -.65 US * Indicaes significan a he 5 percen level. d for and (F) ess saisic Table STAR Tes Resuls (Kapeanios, Shin, and Snell, 00) F Counry saisic d for and (F) ess saisic F saisic Ausralia 4(3) * Ialy () * Ausria 0(5) -5.73* 96.79* Japan 7(4) -3.60* 6.6* Belgium 0() -4.50* 38.50* Neherlands 9() -4.44* 00.9* Canada (0) -3.74* 6.09* Norway () -7.3* * Denmark 8() -5.96* 9.3* Porugal () * Finland 9(9) -3.9* 70.4* Spain (6) -4.* 58.99* France 0() -8.3* 4.9* Sweden () -7.90* * Germany () -5.09* 87.50* Turkey (5) -5.93* 45.69* Greece 9() * UK 5(4) -8.50* 54.5* Ireland 7(9) -3.78* 3.5* US () -.73* 43.94* *Indicaes significan a he 5 percen level. Nonlinear mean reversion, Page 9