Philip Shively Lafayette College. Abstract

Transcription

1 Threshold saionary real exchange raes: a nonlinear, mulivariae approach Philip Shively Lafayee College Absrac Purchasing power pariy implies ha real exchange raes are saionary. However, he finding of nonsaionary real exchange raes has been difficul o dismiss. Using a nonlinear, hree regime srucural bivariae hreshold model, his paper finds evidence of hreshold saionary real exchange raes which is consisen wih purchasing power pariy adjused for marke fricions such as ransacion coss. Ciaion: Shively, Philip, (2003) "Threshold saionary real exchange raes: a nonlinear, mulivariae approach." Economics Bullein, Vol. 6, No. 2 pp Submied: April 28, Acceped: May 8, URL: hp:// 03F30003A.pdf

2 1. Inroducion Tess for saionary versus nonsaionary real exchange raes are equivalen o ess of purchasing power pariy, a fundamenal arbirage condiion in inernaional finance which mainains ha real exchange raes are saionary. Alhough few, if any, economiss believe ha real exchange raes are nonsaionary, he finding of nonsaionary real exchange raes has been difficul o dismiss. For example, see Adler and Lehmann (1983), Cumby and Obsfeld (1984), Edison (1987), Frenkel (1981), Krugman (1978) and Roll (1979) who apply regression (based) ess, Abuaf and Jorion (1990) and Meese and Rogoff (1988) who apply he Dickey - Fuller uni-roo es, and Corbae and Ouliaris (1988), Mark (1990), Pael (1990) and Taylor (1988) who apply coinegraion ess. As economeric ess have become more powerful and daa spanning longer ime periods are used, some recen sudies have found evidence ha real exchange raes are consisen wih a very long-run saionary process. For example, see Abuaf and Jorion (1990) and Lohian and Taylor (1996) who apply he Dickey - Fuller uni-roo es, Cheung and Lai (1993a) and Seigerwald (1996) who apply coinegraion ess, Cheung and Lai (1993b) who apply fracional inegraion echniques, and Glen (1992) and Grilli and Kaminsky (1991) who apply he variance raio es. Economeric ess of saionary versus nonsaionary real exchange raes are generally univariae ess of emporary versus permanen innovaions. I is possible o deermine he size of independen emporary and permanen real-exchange-rae innovaions using mulivariae, srucural models which exploi he join behavior of real exchange raes and addiional, relaed ime series. For example, Clarida and Gali (1994) and Evans and Lohian (1993) esimae rivariae srucural models which find imporan shor-run emporary realexchange-rae innovaions, bu ha permanen innovaions dominae real exchange raes. Lasrapes (1992) esimaes a bivariae srucural model and finds ha permanen innovaions dominae real exchange raes across all horizons. There is a growing consensus ha real exchange raes follow a nonlinear process. Theoreically, a nonlinear process for real exchange raes is suppored by models ha include 1

3 marke fricions such as he model in Dumas (1992) which includes ransacion coss. Empirically, a nonlinear process for real exchange raes is suppored by recen sudies such as Bleaney and Mizen (1996) who es for hreshold nonlineariy, Coakley and Fueres (2001) who es for nonparameric coinegraion, Serleis and Gogas (2000) who es for nonlinear chaoic dynamics, and Chen and Wu (2000), Michael, Nobay and Peel (1997), Saranis (1999), and Sarno (2000) who es for nonlineariy using various forms of smooh ransiion auoregressive models. This paper akes he nonlineariy and (near) nonsaionariy of real exchange raes as is saring poin. I esimaes a nonlinear, srucural bivariae hreshold model in order o deermine he size of independen emporary and permanen real-exchange-rae innovaions in hree real-exchange-rae regimes. The idea is ha here is a middle regime where real exchange raes are nonsaionary due o marke fricions such as ransacion coss, and wo ouer regimes where real exchange raes are saionary. This nonlinear, hreshold saionary process for real exchange raes is consisen wih purchasing power pariy adjused for marke fricions such as ransacion coss. The model explois he join behavior of real exchange raes and real ineres-rae differenials, which are relaed hrough uncovered ineres pariy adjused for differenial inflaion raes. The model idenifies independen emporary and permanen real-exchange-rae innovaions in each regime using covariance resricions and he Blanchard and Quah (1989) resricion on he long-run muliplier of one srucural innovaion. This nonlinear, hreeregime srucural bivariae hreshold model finds ha permanen innovaions dominae real exchange raes in he middle regime, bu ha emporary innovaions have a large impac on real exchange raes in he wo ouer regimes. This evidence of hreshold saionary real exchange raes is consisen wih purchasing power pariy adjused for marke fricions such as ransacion coss. 2. The Model Le s be he naural logarihm of he domesic price of a uni of foreign currency and le p be he naural logarihm of a price index. Then he naural logarihm of he ime 2

4 real exchange rae is q = s + p p, where an aserisk denoes a variable from he foreign counry. Le π = p p 1 be he inflaion rae and le i be a nominal ineres rae, hen he real ineres rae is ρ = i π and he ime real-ineres-rae differenial is ˆρ = (ρ ρ ). Le y = ( q, ˆρ ) where q = q q 1 is he percen change (firs difference) in he real exchange rae. Then he bivariae hreshold vecor auoregression (TVAR) under consideraion is p j y = c j + φ (j) i y i + ɛ (j) for τ j 1 < q τ j (1) i=1 where j = {1, 2, 3} indexes he hree real-exchange-rae regimes, c j p j > 0, φ (j) i is a (2 x 2) parameer marix, and ɛ (j) is a consan vecor, = (ɛ q j, ɛˆρ j ) is a mean zero vecor of innovaions wih covariance srucure Σ j. The hreshold variable used o pariion he hree real-exchange-rae regimes is q, and he hreshold values are = τ 0 < τ 1 < τ 2 < τ 3 = where τ 1 and τ 2 are he wo nonrivial hreshold value which pariion he hree real-exchangerae regimes. Threshold models are linear wihin a regime bu nonlinear across regimes. They can be esimaed in each regime using leas squares condiional on he hreshold variable, he number of regimes, and he lag lenghs. Based on a nonresricive se of assumpions, Tsay (1998, p ) shows ha leas squares esimaes are srongly consisen as he regime sample sizes go o infiniy. The TVAR model in Equaion (1) can be rewrien as Φ j (L)y = ɛ (j) for τ j 1 < q τ j (2) where Φ j (L) has he appropriae lags p j and he consan vecor c j is suppressed. The model assumes ha he percen real-exchange-rae changes and he real ineres-rae differenials are saionary. Therefore Equaion (2) can be invered and represened by he infinie-order moving-average model y = C j (L)ɛ (j) for τ j 1 < q τ j (3) where C j (L) = Φ j (L) 1 and he conemporaneous effec of ɛ (j) on y is he ideniy marix. 3

5 The srucural, bivariae hreshold model which idenifies he independen emporary and permanen real-exchange-rae innovaions in each regime is where η (j) = (η T j y = A j (L)η (j) for τ j 1 < q τ j (4), η P j ) is a mean zero vecor of orhogonalized innovaions wih a covariance srucure normalized o he ideniy marix. regime by comparing Equaions (3) and (4) and observing ha ɛ (j) C j (k)a j (0), where A j (0) is he conemporaneous effec of η (j) The srucural model is idenified in each = A j (0)η (j) and A j (k) = on y in regime j. Therefore he four elemens of A j (0) jus idenify he srucural model in each regime. Covariance resricions esablish hree of he four resricions necessary o idenify A j (0) since Σ j = A j (0)A j (0). The fourh resricion is he Blanchard and Quah (1989) idenifying resricion ha he long-run dynamic response of he level of he real exchange rae o he srucural innovaion η T j is zero, or k=1 a (j) 1,1(k) = 0. These four resricions jus idenify he four elemens of A j (0) in each regime. The las resricion idenifies η T j as he emporary realexchange-rae innovaion and η P j as he permanen real-exchange-rae innovaion. 3. Empirical Resuls Nominal exchange raes (line ae) for Germany, Japan and he Unied Kingdom (U.K.) relaive o he Unied Saes (U.S.) as he domesic currency; wholesale price indexes (line 63) for Germany, Japan, he U.K. and he U.S.; and money-marke ineres raes (line 60b) for Germany, Japan, he U.K. and he U.S. are from he Inernaional Moneary Fund s Inernaional Financial Saisics. The daa are monhly during he floaing-exchange-rae period from April 1973 hrough December 2000 excep for Germany which ends in December 1998 when he German mark was linked o he Euro. The TVAR model assumes ha he percen real-exchange-rae changes and he real ineres-rae differenials are saionary. The Dickey - Fuller uni-roo es finds ha all of he percen real-exchange-rae changes and real ineres-rae differenials decisively rejec he uni-roo null hypohesis a he 0.01 level in favor of he mean-saionary alernaive. 4

6 The TVAR model is esimaed using 4 VAR lags in each regime, a choice suppored by a likelihood raio es which finds ha he reduced form VAR residuals are highly consisen wih whie noise. Tables I, II and III presen real-exchange-rae forecas-error variance decomposiions along wih sandard errors which are boosrapped using 1,000 repeiions of he model. The forecas-error variance decomposiions are normalized such ha he variances of he wo srucural innovaions sum o 100%. Therefore he ables only lis he percenage of he variance due o emporary real-exchange-rae innovaions. 1 The wo hreshold values τ 1 and τ 2 are seleced in a grid search across he empirical range of each real exchange rae such ha he wo ouer regimes have eiher 25, 35, 45 or 55 observaions. Empirical resuls are presened for he wo hreshold values which have he larges percenage of emporary real-exchange-rae innovaions in he wo ouer regimes. The Germany - U.S. real exchange rae selecs hreshold values which place 25 observaions in he lower regime and 45 observaions in he upper regime. The Japan - U.S. real exchange rae selecs hreshold values which place 35 observaions in boh ouer regimes. And he U.K. - U.S. real exchange rae selecs hreshold values which place 55 observaions in boh ouer regimes. Summary of Resuls: When he regimes are no aken ino accoun, hen he forecaserror variance decomposiions in Tables I, II and III show ha permanen innovaions dominae all hree real exchange raes. This is consisen wih previous empirical evidence. When he regimes are aken ino accoun, hen he forecas-error variance decomposiions in Tables I, II and III show ha permanen innovaions dominae each real exchange rae in he middle regime, bu ha emporary innovaions have a large impac on all hree real exchange raes in he wo ouer regimes. Germany - Unied Saes Real Exchange Rae in Table I: In he lower regime wih 25 observaions, emporary innovaions have a large impac on he Germany - U.S. real exchange rae accouning for 70% of he iniial forecas-error variance. In he middle regime wih 234 observaions, permanen innovaions dominae he Germany - U.S. real 1 Forecas-error variance decomposiions are no presened for he real ineres-rae differenials since heir purpose here is o idenify independen emporary and permanen real-exchange-rae innovaions. 5

7 exchange rae accouning for over 98% of he forecas-error variance a all forecas horizons. In he upper regime wih 45 observaions, emporary innovaions have an exremely large and persisen impac on he Germany - U.S. real exchange rae accouning for over 88% of he forecas-error variance hrough he 8-monh forecas horizon and over 50% of he forecas-error variance hrough he 41-monh forecas horizon. Japan - Unied Saes Real Exchange Rae in Table II: In he lower regime wih 35 observaions, emporary innovaions have an exremely large impac on he Japan - U.S. real exchange rae accouning for 82% of he iniial variance. In he middle regime wih 258 observaions, permanen innovaions dominae he Japan - U.S. real exchange rae accouning for over 88% of he forecas-error variance a all forecas horizons. In he upper regime wih 35 observaions, emporary innovaions have an exremely large and persisen impac on he Japan - U.S. real exchange rae accouning for 99% of he iniial variance and over 50% of he forecas-error variance hrough he 29-monh forecas horizon. Unied Kingdom - Unied Saes Real Exchange Rae in Table III: In he lower regime wih 55 observaions, emporary innovaions have a large impac on he U.K. - U.S. real exchange rae accouning for 64% of he iniial forecas-error variance. In he middle regime wih 218 observaions, permanen innovaions dominae he U.K. - U.S. real exchange rae accouning for over 97% of he forecas-error variance a all forecas horizons. In he upper regime wih 55 observaions, emporary innovaions have a large and persisen impac on he U.K. - U.S. real exchange rae accouning for 68% of he iniial variance. 4. Conclusion This paper esimaes a nonlinear, hree-regime srucural bivariae hreshold model using Germany - U.S., Japan - U.S. and U.K. - U.S. daa, and finds evidence of hreshold saionary real exchange raes. This evidence is consisen wih purchasing power pariy adjused for marke fricions such as ransacion coss. 6

8 References Abuaf, N., and P. Jorion (1990) Purchasing Power Pariy in he Long Run Journal of Finance 45, Adler, M., and B. Lehmann (1983) Deviaions from Purchasing Power Pariy in he Long Run Journal of Finance 38, Blanchard, O., and D. Quah (1989) The Dynamic Effecs of Aggregae Demand and Supply Disurbances American Economic Review 79, Bleaney, M., and P. Mizen (1996) Are Real Exchange Raes Saionary? Evidence from Serling Bilaeral Raes Economic Noes 25, Chen, S., and J. Wu (2000) A Re-Examinaion of Purchasing Power Pariy in Japan and Taiwan Journal of Macroeconomics 22, Cheung, Y., and K. Lai (1993a) Long - Run Purchasing Power Pariy During he Recen Floa Journal of Inernaional Economics 34, Cheung, Y., and K. Lai (1993b) A Fracional Coinegraion Analysis of Purchasing Power Pariy Journal of Business and Economic Saisics 11, Clarida, R., and J. Gali (1994) Sources of Real Exchange-Rae Flucuaions: How Imporan Are Nominal Shocks? Carnegie-Rocheser Conference Series on Public Policy 41, Coakley, J., and A. Fueres (2001) Nonparameric Coinegraion Analysis of Real Exchange Raes Applied Financial Economics 11, 1-8. Corbae, D., and S. Ouliaris (1988) Coinegraion and Tess of Purchasing Power Pariy Review of Economics and Saisics 70, Cumby, R. and M. Obsfeld (1984) Inernaional Ineres Rae and Price Level Linkages Under Flexible Exchange Raes: A Review of Recen Evidence in Exchange Rae Theory and Pracice by J. Bilson and R. Marson, Eds., Universiy of Chicago Press: Chicago, Dumas, B. (1992) Dynamic Equilibrium and he Real Exchange Rae in a Spaially Separaed World Review of Financial Sudies 5, Edison, H. (1987) Purchasing Power Pariy in he Long Run: A Tes of he Dollar / Pound Exchange Rae, Journal of Money, Credi, and Banking 19, Evans, M., and J. Lohian (1993) The Response of Exchange Raes o Permanen and Transiory Shocks Under Floaing Exchange Raes Journal of Inernaional Money and Finance 12, Frenkel, J. (1981) Flexible Exchange Raes, Prices and he Role of News : Lessons from he 1970 s Journal of Poliical Economy 89, Glen, J. (1992) Real Exchange Raes in he Shor, Medium and Long Run Journal of Inernaional Economics 33,

9 Grilli, V., and G. Kaminsky (1991) Nominal Exchange Rae Regimes and he Real Exchange rae: Evidence from he Unied Saes and Grea Briain, Journal of Moneary Economics 27, Krugman, P. (1978) Purchasing Power Pariy and Exchange Raes: Anoher Look a he Evidence Journal of Inernaional Economics 8, Lasrapes, W. (1992) Sources of Flucuaions in Real and Nominal Exchange Raes Review of Economics and Saisics 74, Lohian, J., and M. Taylor (1996) Real Exchange Rae Behavior: The Recen Floa from he Perspecive of he Pas Two Cenuries Journal of Poliical Economy 104, Mark, N. (1990) Real and Nominal Exchange Raes in he Long Run: An Empirical Invesigaion Journal of Inernaional Economics 28, Meese, R., and K. Rogoff (1988) Was I Real? The Exchange Rae - Ineres Rae Differenial Relaion Over he Modern Floaing - Rae Period Journal of Finance 37, Michael, P., A. Nobay, and D. Peel (1997) Transacions Coss and Nonlinear Adjusmen in Real Exchange Raes: An Empirical Invesigaion Journal of Poliical Economy 105, Pael, J. (1990) Purchasing Power Pariy as a Long - Run Relaion Journal of Applied Economerics 5, Roll, R. (1979) Violaions of Purchasing Power Pariy and heir Implicaions for Efficien Inernaional Commodiy Markes in Inernaional Finance and Trade by M. Sarna and G. Szego, Eds., Ballinger: Cambridge, MA, Saranis, N. (1999) Modeling Non-Lineariies in Real Effecive Exchange Raes Journal of Inernaional Money and Finance 18, Sarno, L. (2000) Real Exchange Rae Behavior in he Middle Eas: A Re-Examinaion Economics Leers 66, Serleis, A., and P. Gogas (2000) Purchasing Power Pariy, Nonlineariy and Chaos Applied Financial Economics 10, Seigerwald, D. (1996) Purchasing Power Pariy, Uni Roos, and Dynamic Srucure Journal of Empirical Finance 2, Taylor, M. (1988) An Empirical Examinaion of Long - Run Purchasing Power Pariy Using Coinegraion Techniques Applied Economics 20, Tsay, R. (1998) Tesing and Modeling Mulivariae Threshold Models Journal of he American Saisical Associaion 93,

10 Table I Forecas-Error Variance Decomposiions: Germany - Unied Saes Real Exchange Rae Forecas No Lower Middle Upper Horizon Regimes Regime Regime Regime (Monhs) (304 obs) (25 obs) (234 obs) (45 obs) Percenage of he Variance Due o Temporary Innovaions: (12.2) (29.7) (10.3) (31.7) (12.0) (26.9) (10.2) (31.0) (11.9) (25.5) (10.0) (30.6) (11.4) (22.4) (9.6) (30.0) (10.9) (18.9) (9.1) (29.3) (10.4) (16.0) (8.6) (28.6) (9.0) (11.9) (7.2) (26.5) (7.8) (10.7) (6.1) (24.6) (4.8) (9.2) (3.5) (18.2) (3.3) (8.4) (2.4) (13.5) (2.4) (7.7) (1.7) (10.0) (1.9) (7.2) (1.4) (7.5) (1.5) (6.8) (1.1) (5.7) (1.3) (6.5) (0.9) (4.4) (1.1) (6.4) (0.8) (3.6) (0.9) (6.2) (0.7) (3.0) (0.8) (6.2) (0.6) (2.6) Sandard errors in parenheses are boosrapped using 1,000 repeiions of he model. The daa are monhly from April 1973 hrough December

11 Table II Forecas-Error Variance Decomposiions: Japan - Unied Saes Real Exchange Rae Forecas No Lower Middle Upper Horizon Regimes Regime Regime Regime (Monhs) (328 obs) (35 obs) (258 obs) (35 obs) Percenage of he Variance Due o Temporary Innovaions: (14.6) (30.5) (12.8) (29.7) (14.5) (28.5) (13.0) (29.8) (14.5) (27.4) (13.2) (30.3) (14.2) (23.4) (13.2) (29.3) (13.7) (20.5) (12.9) (29.2) (13.1) (18.3) (12.5) (28.5) (11.1) (13.8) (11.0) (28.0) (9.3) (11.1) (9.4) (28.0) (4.9) (6.9) (5.3) (24.6) (3.1) (5.3) (3.3) (20.6) (2.2) (4.3) (2.3) (17.0) (1.6) (3.7) (1.7) (14.0) (1.3) (3.3) (1.4) (11.4) (1.1) (3.0) (1.1) (9.1) (0.9) (2.8) (1.0) (7.3) (0.8) (2.6) (0.8) (6.0) (0.7) (2.5) (0.8) (5.1) Sandard errors in parenheses are boosrapped using 1,000 repeiions of he model. The daa are monhly from April 1973 hrough December

12 Table III Forecas-Error Variance Decomposiions: Unied Kingdom - Unied Saes Real Exchange Rae Forecas No Lower Middle Upper Horizon Regimes Regime Regime Regime (Monhs) (328 obs) (55 obs) (218 obs) (55 obs) Percenage of he Variance Due o Temporary Innovaions: (16.3) (28.8) (9.5) (28.3) (15.1) (28.2) (9.5) (26.8) (13.6) (23.3) (9.6) (22.7) (12.4) (22.3) (9.7) (22.3) (11.4) (19.4) (9.6) (21.4) (10.5) (16.1) (9.2) (20.9) (8.4) (11.7) (8.0) (19.0) (6.8) (9.3) (6.8) (17.3) (3.5) (5.3) (3.9) (12.5) (2.2) (3.7) (2.6) (9.6) (1.5) (2.8) (1.9) (7.7) (1.2) (2.3) (1.5) (6.4) (1.0) (1.9) (1.3) (5.4) (0.8) (1.6) (1.1) (4.6) (0.7) (1.5) (0.9) (4.0) (0.6) (1.3) (0.8) (3.5) (0.5) (1.2) (0.7) (3.0) Sandard errors in parenheses are boosrapped using 1,000 repeiions of he model. The daa are monhly from April 1973 hrough December