Additional Evidence of Long Run Purchasing Power Parity with Restricted Structural Change. David H. Papell and Ruxandra Prodan.

Transcription

1 Addiional Evidence of Long Run Purchasing Power Pariy wih Resriced Srucural Change David H. Papell and Ruxandra Prodan May 005 We invesigae wo alernaive versions of Purchasing Power Pariy (PPP): reversion o a consan mean in he spiri of Cassel and reversion o a consan rend in he spiri of Balassa and Samuelson, using long-span real exchange rae daa for indusrialized counries. We develop uni roo ess ha boh accoun for srucural change and mainain a long run mean or rend. Wih convenional ess, previous research finds evidence of some varian of PPP for 9 of he 6 counries. Wih he uni roo ess in he presence of resriced srucural change, we find evidence of some varian of PPP for 5 addiional counries. We are graeful o paricipans in seminars a Berkeley, Davis, Emory, Michigan, Washingon, he Federal Reserve Bank of San Francisco, Texas Camp Economerics VIII, and he 005 Inernaional Economics and Finance Sociey Meeings, as well as o an anonymous referee, for helpful commens and discussions. Papell hanks he Naional Science Foundaion for financial suppor. Correspondence o: David Papell, el. (73) , dpapell@mail.uh.edu, Deparmen of Economics, Universiy of Houson, Houson, TX Ruxandra Prodan, el. (05) , rprodan@cba.ua.edu, Deparmen of Economics, Universiy of Alabama, Tuscaloosa, AL

2 . Inroducion Purchasing Power Pariy (PPP) is one of he oldes and mos sudied opics in inernaional economics. As ariculaed by Cassel (98), he absolue version of PPP posulaes ha he relaive prices (in differen currencies and locaions) of a common baske of goods will be equalized when quoed in he same currency. The relaive version of PPP, emphasizing arbirage across ime raher han across space, is ha he exchange rae will adjus o offse inflaion differenials beween counries. While Cassel undersood he possibiliy ha he exchange rae migh ransiorily diverge from PPP, he viewed he deviaions as minor. Modern versions of PPP, recognizing he imporance of slow speeds of adjusmen, define PPP as reversion of he real exchange rae o a consan mean. Drawing on ideas from Ricardo and Harrod, Balassa (964) and Samuelson (964) drew aenion o he fac ha divergen inernaional produciviy levels could, via heir effec on wages and home goods prices, lead o permanen deviaions from Cassel s absolue version of PPP. They linked PPP, exchange raes and inercounry real-income comparisons, arguing ha he absolue version of PPP is flawed as a heory of exchange raes. Assuming ha PPP holds for raded goods, heir argumen is based on he fac ha produciviy differenials beween counries deermine he domesic relaive prices of non-radables, leading in he long run o rend deviaions from PPP. 3 Obsfeld (993) uses hese ideas o develop a model in which real exchange raes conain a pronounced deerminisic rend. The ension beween hese wo approaches is eviden in recen sudies of PPP. Modern research on PPP, aking ino accoun he possibiliy of slow speeds of reversion, finds evidence of PPP if he uni roo null can be rejeced in favor of a saionary alernaive for real exchange raes. The quesion remains, however, wheher he saionary alernaive should be level saionariy, reversion o a consan mean, or rend saionariy, reversion o a consan rend. These issues are of cenral imporance in sudying longspan real exchange raes. In he conex of sudying real exchange raes during he pos-973 floaing exchange rae regime, produciviy differenials and he resulan possibiliy of rending assumes lesser imporance. Purchasing Power Pariy wih long-span real exchange raes has been exensively invesigaed. Abuaf and Jorion (990) and Lohian and Taylor (996) find evidence of long-run PPP by rejecing uni roos in favor of level saionary real exchange raes using Augmened-Dickey-Fuller (ADF) ess. Taylor Since daa on price indexes for differen counries do no measure a common baske of goods, empirical work on PPP usually ess relaive, raher han absolue, PPP. He idenified hree groups of disurbances: acual and expeced inflaion or deflaion, new hindrances o rade and shifs in inernaional movemens of capial. Bu even if he disurbances are recognized, heir quaniaive effec on deviaions from PPP is seen as confined wihin raher narrow limis. 3 There are several sudies which poin ou he limiaions of his heory: Asea and Mendoza (994) found lile evidence o suppor he proposiion ha deviaions from PPP reflec differences in he relaive prices of nonradables, Canzoneri (999) a al. found less favorable evidence on purchasing power pariy in raded goods, and Fizgerald (003) argues ha he classic relaionship beween produciviy and relaive price levels is modified by adding he erms-of-rade effecs o he model.

3 (00) develops a long-span nominal exchange rae and price level daa se for 7 indusrialized counries, producing 6 real exchange raes wih he Unied Saes dollar as he numeraire currency. He finds ha he uni roo null hypohesis can be rejeced a he 5% level in favor of eiher level or rend saionariy for ou of 6 real exchange raes using he Ellio, Rohenberg and Sock (996) generalized-leas-squares version of he Dickey-Fuller (DF-GLS) es wih allowance for a deerminisic rend. 4 Lopez, Murray, and Papell (005) argue ha Taylor s conclusion is sensiive o his use of subopimal lag selecion in uni roo ess. Using sandard lag selecion mehods, hey find ha ADF and DF- GLS ess produce he same resul: he uni roo null hypohesis can be rejeced a he 5% level in favor of eiher level or rend saionariy for only 9 ou of 6 real exchange raes of indusrialized counries wih he Unied Saes dollar as he numeraire currency. In his paper we analyze he 7 counries for which he uni roo null canno be rejeced, proposing a new mehodology in order o re-evaluae he Purchasing Power Pariy hypohesis. Much research has been conduced on esing for uni roos in he presence of a one-ime change in he mean and/or he rend of economic ime series. This is imporan in ess for PPP because, if here is a one-ime change in he mean, long-run PPP does no hold. The siuaion becomes more complicaed when, as in Lumsdaine and Papell (997), muliple srucural changes are allowed. Suppose ha he real exchange rae is subjec o wo changes in he mean. If he changes are offseing, he series reurns o a consan mean and long-run PPP holds. If he changes are no offseing, eiher because hey ac in he same direcion or because hey ac in opposie direcions bu are of differen magniude, he series does no reurn o a consan mean and long-run PPP does no hold. Dornbusch and Vogelsang (99) argue ha a qualified version of purchasing power pariy can sill be claimed in he presence of a one-ime shif in he mean level of he real exchange rae ha is deermined exogenously. They inerpre heir findings as supporing he Balassa-Samuelson model. Hegwood and Papell (998) formalize and generalize his idea, allowing for muliple srucural changes ha are deermined endogenously. They argue ha real exchange raes are level saionary, bu around a mean which is subjec o srucural change, and show ha reversion o a changing mean is much faser han reversion o a fixed mean. 5 Tess of he uni roo hypohesis in he presence of a one-ime change in he inercep have been developed for boh non-rending daa, as in Perron and Vogelsang (99), and rending daa, as in Perron (997) and Vogelsang and Perron (998). Subsequenly we es he uni roo hypohesis in real exchange raes allowing for a possible shif in he inercep of he rend funcion, considered o occur a an unknown 4 Taylor also repors resuls wih four Lain American counries and, for all counries, wih a world baske as numeraire. 5 Hegwood and Papell (998) use exchange raes for five counries wih he US dollar for 900 o 990, plus he wo exchange raes used by Lohian and Taylor (996).

4 ime. As an exension, we propose ess ha exend he previous models o incorporae wo endogenous break poins. In order o avoid confusion beween differing conceps of PPP, we call Purchasing Power Pariy (PPP) he rejecion of he uni roo null hypohesis in favor of an alernaive hypohesis of level saionariy in a model ha does no incorporae a ime rend. We call Trend Purchasing Power Pariy (TPPP) he rejecion of he uni roo null in favor of a rend saionary alernaive in a model ha incorporaes a ime rend. Following his erminology, we call Qualified Purchasing Power Pariy (QPPP) he rejecion of he uni roo hypohesis in favor of an alernaive hypohesis of regime-wise level saionariy (level saionariy afer allowing for one or wo changes in he inercep). We call Trend Qualified Purchasing Power Pariy (TQPPP) he rejecion of he uni roo hypohesis in favor of an alernaive hypohesis of regime-wise rend saionariy (rend saionariy afer allowing for one or wo changes in he inercep). We firs repor he resuls of ess ha do no impose resricions on he breaks. Allowing for one or wo srucural changes, we find evidence of eiher QPPP or TQPPP for 4 counries (a he 5% level). 6 These resuls are no necessarily a sep forward owards PPP or TPPP. They do no impose eiher he alernaive of a consan mean or he alernaive of a consan rend. In order o es for PPP while allowing for srucural change, we develop uni roo ess ha resric he coefficiens on he dummy variables ha depic he breaks o produce a consan mean or rend in he long run. We herefore accoun for srucural change bu sill mainain he long-run PPP or TPPP hypohesis. We call he model for nonrending daa PPP resriced srucural change and he model for rending daa Trend PPP resriced srucural change. I is imporan o undersand ha, in conras wih he ess ha allow for QPPP and TQPPP alernaives, rejecion of he uni roo null in favor of he resriced srucural change alernaives provides evidence of PPP or TPPP. 7 We rejec he uni roo hypohesis in favor of PPP or rend PPP resriced srucural change for 5 counries (a he 5% level). Combining hese ess wih he previous evidence from ADF ess for PPP and TPPP, Canada and he Neherlands are he only counries for which here is no evidence of any varian of PPP. In he case of Canada, where he uni roo null canno be rejeced in favor of any of our alernaive hypoheses, one possibiliy for he lack of evidence of PPP is because of he large depreciaion of he Canadian dollar agains he US dollar a he end of he sample. For he Neherlands, where he uni roo null can only be rejeced in favor of QPPP wih one break, here is a real appreciaion of he guilder agains he US dollar in he early 970s following he discovery of large naural gas deposis ha has no been reversed. 6 We use he erm counry as shorhand for real exchange rae wih he Unied Saes. 7 A relaed es was developed by Papell (00) o accoun for he large appreciaion and depreciaion of he dollar in he 980s. Using panel mehods for pos-973 daa and imposing a PPP resriced broken rend consrain he provides srong evidence of PPP. 3

5 In order o inerpre our findings, we conduc simulaions o evaluae he power of he ess under various alernaive hypoheses. The simulaions produce several ineresing resuls. Firs, he power of he ADF ess wihou srucural change is miniscule when he daa is generaed by a process ha incorporaes srucural change, even if he process is consisen wih PPP or TPPP (he breaks are of equal and opposie sign). 8 This implies ha he previous rejecions using ADF ess do provide evidence of PPP or TPPP wihou srucural change. Second, he ess wih resriced srucural change have very high power when he process has breaks of equal and opposie sign, moderae power when he process has no breaks, and very low power when he process has breaks ha are no consisen wih PPP or TPPP. This implies ha he addiional rejecions from he ess wih resriced srucural change consiue evidence of PPP and TPPP beyond wha can be obained wih he ADF ess. Rogoff s celebraed (996) Purchasing Power Pariy Puzzle involves he combinaion of slow speeds of convergence o PPP and high shor-erm volailiy of real exchange raes. Our paper can be considered o be a prelude o he PPP puzzle because evidence of PPP is necessary before one can sensibly measure he speed of convergence o PPP. Using ess ha do no allow for srucural change, Taylor (00) found evidence of PPP or TPPP a he 5 percen level for ou of 6 indusrial counries and Lopez, Murray, and Papell (005) found evidence of PPP or TPPP for only 9 ou of 6 counries. Wih he addiion of ess ha boh allow for srucural change and impose pariy resricions, evidence of PPP can be found for 0 counries and evidence of TPPP can be found for 4 addiional counries, for a oal of 4 ou of 6 counries.. Tesing for PPP and TPPP wih Srucural Change The purpose of he paper is o analyze long-run purchasing power pariy among indusrialized counries. We use annual nominal exchange raes and price indices. The laer are measured as consumer price deflaors or GDP deflaors, depending on heir availabiliy. The daa was obained from Taylor (00) and updaed by he auhors (using Inernaional Financial Saisics daa). I consiss of 07 o 9 years of real exchange raes for 6 indusrialized counries wih he Unied Saes dollar as he numeraire currency, saring beween 870 and 89 and ending in 998. Under purchasing power pariy (PPP), he real exchange rae displays long-run mean reversion. The real dollar exchange rae is calculaed as follows: q = e + p * p, () where q is he logarihm of he real exchange rae, e is he logarihm of he nominal exchange rae (he dollar price of he foreign currency) and levels, respecively. p and p * are he logarihms of he US and he foreign price 8 This resul varies wih he break size; he power of ADF es becomes higher if he size of he break is very small. 4

6 Previous research has provided evidence of PPP and TPPP using convenional uni roo ess. Using he Ellio, Rohenberg and Sock (996) generalized-leas-squares version of he Dickey-Fuller (DF-GLS) es, Taylor (00) rejecs he uni roo null in favor of eiher PPP or TPPP for ou of 6 indusrialized counries wih he U.S. dollar as numeraire a he 5% level and 4 addiional counries a he 0% level. Using ADF ess, he finds evidence of eiher PPP or TPPP for 9 ou of 6 counries a he 5% level and for 3 addiional counries a he 0% level. Lopez, Murray, and Papell (005) argue ha he reason for Taylor s srong rejecions lies in he lag selecion procedures used, which end o produce shor lag lenghs. As shown by Ng and Perron (995, 00), echniques ha produce shor lag lenghs have low power for ADF ess and are badly sized for DF-GLS ess. Using ADF ess wih general-o-specific lag selecion, hey find evidence of eiher PPP or TPPP by rejecing he uni roo null for 9 counries a he 5% level and addiional counries a he 0% level. Since ADF ess ha do no incorporae a ime rend have very low power o rejec uni roos wih rending daa, we inerpre hese resuls as providing evidence of PPP for he 8 counries for which he uni roo null is rejeced in favor of he PPP alernaive, Belgium, Germany, Finland, France, Ialy, Norway, Spain, and Sweden, and of TPPP for one counry, Ausralia, for which he uni roo null is rejeced in favor of he TPPP, bu no he PPP, alernaive. 9 We proceed o analyze he 7 counries, Canada, Denmark, Japan, Neherlands, Porugal, Swizerland, Unied Kingdom and Unied Saes, where Lopez, Murray, and Papell (005) could no rejec he uni roo null in favor of eiher level or rend saionariy. 0 Firs, we es for uni roos while allowing for srucural change, bu do no impose PPP or TPPP. Second, we es for uni roos in he presence of PPP or TPPP resriced srucural change... Tess for a uni roo in he presence of srucural change As Campbell and Perron (99) emphasized, nonrejecion of he uni roo hypohesis may be due o he misspecificaion of he deerminisic componens included as regressors. Uni roo ess ha ignore srucural change could fail o provide evidence of PPP when i acually holds ouside of he srucural shif. We invesigae he uni roo hypohesis in real exchange raes, bu no PPP or TPPP, by using previously developed ess for a uni roo in he presence of one break and by developing ess for a uni roo wih wo breaks. The inuiion ha moivaes he ess is o rea he breaks as being deermined ouside he daa generaing process. 9 They repor he same number of rejecions using DF-GLS ess wih MAIC lag selecion. 0 Given he resuls in Lohian and Taylor (996), i may be surprising ha we do no rejec he uni roo null for he Unied Kingdom. Hegwood and Papell (998), however, show ha, wih general-o-specific lag selecion, he uni roo null is no rejeced in favor of level saionariy, bu is rejeced in favor of saionariy around a one-ime change in he mean, for he Lohian and Taylor dollar/serling daa. While here is no heoreical reason o resric aenion o one or wo breaks, pracical consideraions involving compuing ime for simulaions and calculaing criical values precluded considering addiional breaks. 5

7 Tess for a uni roo in a non-rending ime series characerized by a single srucural change in is level are developed by Perron and Vogelsang (99). The possible changes are considered o occur a an unknown ime. We consider an Addiive Oulier ype (AO) model o model changes ha occur insananeously. Wih long-span real exchange raes, mos of he observaions from nominal fixed exchange rae regimes, where devaluaions and reevaluaions, especially following failed aemps o defend currencies, can lead o discree jumps (inercep changes). The AO model is esimaed using a wo-sep process. For a value of he break poin Tb, wih.0t<tb <.90T (where T is he sample size), he deerminisic par of he series is removed using he following regression: q = µ + γdu + ~ z, () where DU = if > Tb and 0 oherwise. The 0% rimming is used o avoid finding spurious breaks a he beginning and end of he sample. The uni roo es is hen performed using he -saisic for α = 0 in he regression: where k k i D( Tb) i + α~ z + ci ~ z i + i= 0 i= ~ z = ω ε, (3) D ( Tb) = if = Tb + and 0 oherwise. The inclusion of k + dummy variables is needed o ensure ha saisic on α is invarian o he value of runcaion lag parameer k. The recursive procedure of selecing he runcaion lag parameer k sars wih k max = 8 and i is repeaed unil he las lag is significan (use a criical value of.645). The break dae, Tb is chosen o minimize he -saisic on α. Saisics are compued for all break daes, aking ino accoun he rimming. The chosen break is ha for which he maximum evidence agains he uni roo null, in he form of he mos negaive -saisic on α, is obained. Tess for a uni roo in a rending ime series characerized by a single srucural change are developed by Perron (997) and Vogelsang and Perron (998). Including a ime rend, we follow he procedure described above and perform uni roo ess ha allow shifs in he inercep a an unknown ime. As previously, he AO model is esimaed using a wo-sep process. The deerminisic par of he series is removed using he following regression: q = µ + β + γdu + ~ z, (4) where 0% rimming is used o avoid finding spurious breaks. The uni roo es is hen performed using he saisic for α = 0 in he regression described by Equaion (3). The uni roo hypohesis is esed as in he previous case. 6

8 Criical values were compued using Mone Carlo mehods. We generae a uni roo series (wihou srucural change) wih 9 observaions (he maximum size of he sample), using an AR () model wih iidn(0,) innovaions. The AO model is esimaed as described above, wih he es saisic being he -saisic on α in Equaion (3). The criical values for he finie sample disribuions are aken from he sored vecor of 5000 replicaed saisics. 3 The resuls of he ess for a uni roo in he presence of one srucural change are repored in Table. We find evidence of Qualified Purchasing Power Pariy (QPPP) by rejecing he uni roo null for 3 ou of 7 counries a he 5% level, Denmark, Neherlands and Swizerland and for one addiional counry a he 0% level. Incorporaing ime rends, we find evidence of Trend Qualified Purchasing Power Pariy (TQPPP) by rejecing he uni roo null, for counry a he 5% level, Swizerland, and for 3 addiional counries a he 0% level. Combining he wo ess, we find evidence of eiher QPPP or TQPPP by rejecing he uni roo null for 3 counries a he 5% level. We proceed o exend he AO model of Perron and Vogelsang (99), for non-rending daa, o incorporae wo endogenous break poins. 4 Following he previous esing procedures, he AO model is esimaed using a wo-sep process. For values of he break poins Tb and Tb wih.0t< Tb i <.90T (where T is he sample size and i =,), he deerminisic par of he series is removed using he following regression: q = µ + γ DU + γ DU ~ z (5) where DU = if > Tb, 0 oherwise and + performed using he -saisic for α = 0 in he regression: where D( Tb ) = if = i DU = if > Tb, 0 oherwise. The uni roo es is hen k k k i D( Tb ) i + ωi D( Tb ) i + α~ z + ci ~ z i + i= 0 i= 0 i= ~ z = ω ε, (6) Tb i + and 0 oherwise. Saisics are compued for all possible combinaions of break daes, aking in accoun he rimming and no allowing breaks o occur in consecuive years. Finally, we exend he AO model of Vogelsang and Perron (998), for rending daa, o allow for wo breaks. We follow he procedure described above and perform uni roo ess ha allow shifs in he inercep a an unknown ime. The deerminisic par of he series is removed using he following regression: q = µ + β + γ DU + γ DU ~ z (7) + There are hree possible models in case of rending daa: inercep shif, inercep and slope shif and a slope shif. We did calculaions for all bu we didn find more rejecions or oher counries han in he model ha allows only for changes in he inercep. 3 We experimened by compuing daa specific criical values for several counries, and he resuls were unaffeced. 4 Lumsdaine and Papell (997) develop a uni roo es ha allows for wo endogenously deermined break poins in he conex of an innovaional oulier (IO) model for rending daa, where he srucural change is assumed o occur gradually. This assumpion is more appropriae for macroeconomic aggregaes han for real exchange raes. 7

9 The uni roo es is hen performed using he -saisic for α = 0 in he regression described by Equaion (6). The uni roo hypohesis is esed as in he previous case. The resuls of he ess for a uni roo in he presence of wo srucural changes are repored in Table. We find evidence of Qualified Purchasing Power Pariy (QPPP) by rejecing he uni roo null for 4 ou of 7 counries a he 5% level, Denmark, Porugal, Swizerland, and he Unied Kingdom, and for one addiional counry a he 0% level. Incorporaing ime rends, we find evidence of Trend Qualified Purchasing Power Pariy (TQPPP) by rejecing he uni roo null, for 4 counries a he 5% level, Denmark, Porugal, Swizerland, and he Unied Kingdom, and for one addiional counry a he 0% level. Using boh ess, we find evidence of eiher QPPP or TQPPP by rejecing he uni roo null for 4 counries a he 5% level. We conclude by combining he resuls of he ess wih one and wo breaks. There is one counry, he Neherlands, for which evidence of QPPP is found wih one, bu no wo, breaks. We herefore rejec he uni roo null in favor of eiher he QPPP or he TQPPP alernaive for 5 counries a he 5% level... Tess for PPP in he presence of resriced srucural change Using an AO model and allowing for one or wo inercep changes we found evidence of QPPP (or TQPP) for 5 counries. Neiher of hese ess, however, invesigaes he possibiliy ha a series may experience boh srucural change and reversion o he mean (or rend). In order o es for PPP (or TPPP) while allowing for srucural change, we develop uni roo ess ha resric he coefficiens on he dummy variables ha depic he breaks o produce a long-run consan mean or rend. We esimae AO models ha mainain he long run PPP hypohesis by removing he deerminisic par of he series using he following regression: q = µ + γ DU + γ DU ~ z, (8) + subjec o he resricion: γ + γ 0 (9) = where 0% rimming is used o avoid finding spurious breaks. The resricion in Equaion (9) is ha he coefficiens on he breaks are of equal and opposie sign. This imposes he PPP hypohesis because he mean following he second break is resriced o equal he mean prior o he firs break. In order o make he uni roo es invarian o ~γ and ~γ, we also impose he following resricion o Equaion (6): ω + ω 0 (0) = The uni roo es is hen performed using he saisic for α = 0 in he regression described by Equaion (6). The uni roo hypohesis is esed as in he previous case. Nex, we esimae he AO model which includes a ime rend, described by Equaion (7), subjec o he same resricion (9). This imposes he TPPP hypohesis because he rend following he second 8

10 break is resriced o equal he rend prior o he firs break. We follow he same procedure as before o choose he breaks and es he uni roo hypohesis. Criical values for he resriced models are calculaed using he same mehod as in he case of he AO model wih wo srucural breaks. Because of he resricions, heir values are lower han in he case of wo unresriced breaks. The resuls of he ess wih resriced srucural change are repored in Table. We find evidence of PPP Resriced Srucural Change by rejecing he uni roo null for ou of 7 counries a he 5% level, Porugal he Unied Kingdom. Incorporaing ime rends, we find evidence of Trend PPP Resriced Srucural Change by rejecing he uni roo null for 5 counries a he 5% level, Denmark, Japan, Porugal, Swizerland, and he Unied Kingdom. Combining he wo ess, we find evidence of eiher PPP or TPPP Resriced Srucural Change by rejecing he uni roo null for 5 counries a he 5% level. Canada and Neherlands are he counries for which we do no find evidence of eiher PPP or TPPP Resriced Srucural Change. In addiion, we find evidence of QPPP for he Neherlands. Combined wih previous evidence of Lopez, Murray, and Papell (005), among he 6 indusrialized counries, Canada is he only counry for which here is no evidence of some varian of saionariy. 3. Power of he univariae uni roo ess wih srucural change By developing and implemening ess for a uni roo in he presence of resriced srucural change, we add 5 more rejecions o uni roos in real exchange raes beyond hose found by using convenional ADF or DF-GLS ess. In order o deermine wheher hese rejecions consiue evidence of PPP, TPPP, PPP resriced srucural change, or rend PPP resriced srucural change, we sudy he power of he newly developed ess and proceed o invesigae heir finie sample performance. 3. Consrucion of he power simulaions The simulaion experimens address he following issues: (a) comparison of properies of he various ess (ADF, AO model allowing for wo srucural changes and AO resriced model), (b) power of AO models as a funcion of he magniude of he break. Wihin he Mone Carlo experimens we consider he following wo daa generaing processes: q q = + β) + αq µ ( + ε () ( β) + αq + γ DU + γ = µ + DU + ε () The power of he uni roo ess can be invesigaed by consrucing experimens wih arificial daa under a rue alernaive hypohesis where he real exchange rae is saionary (or rend saionary) wihou srucural change (), wih wo equal changes in he inercep of he same sign (), and wih 9

11 wo equal changes in he inercep of opposie sign (). The firs and hird experimens are consisen wih PPP or TPPP, he second is no. We perform uni roo ess on hese consruced series, abulaing how ofen he uni roo null is (correcly) rejeced. The wo differen ses of daa are generaed based on differen assumpions: we specify α = 0.8 and 0.9 for firs generaed sample () and α = 0.7 for he second generaed sample (). This reflecs he range of values of α repored in Table. In he case of generaed daa including wo srucural changes we consider cases where he breaks are equal and have eiher he same sign or he opposie sign. We accoun for breaks wih a magniude of 0., 0.3 and 0.5, which covers mos of he cases found in our daa. The iming of he breaks is se a he /3 and /3 of he sample. In all of he cases he sample size is T = 9 wih 0% rimming and 000 replicaions are used wih e = iidn(0,). We also use he finie sample criical values previously calculaed and repor resuls for ess of nominal size of 0%, 5% and %. 3. Simulaion resuls To address hese issues i is useful o sar wih saionary daa generaing processes () ha do no conain srucural change under he alernaive hypohesis. In his secion, we repor power resuls for ess wih a nominal size of 5% (deailed resuls for differen nominal sizes and values of α are presened in Table 3). Firs, as he errors become more persisen, he power decreases: The ADF es has good power when he daa generaing process is non-rending wih α = 0.8 bu lower power when α = 0.9. This is in accord wih previous research, considering he span and persisence of he daa. As expeced, ess for QPPP ha incorporae wo srucural changes have less power han he ADF es. Tess for PPP resriced srucural change have generally lower power han he ADF es (excep he case wih lower persisence) bu higher power han QPPP es. Table 3 also repors resuls for rending daa generaing processes. Applying he previous ess, including a ime rend, he resuls are fairly similar. The power of he ess, however, is lower wih rending han wih non-rending daa in mos cases. The resuls wih saionary daa generaing processes () ha allow for wo changes in he inercep are repored in Table 4. We sar by looking a cases where he breaks occur in he same direcion (inconsisen wih PPP). The es for QPPP wih wo srucural changes has very good power o rejec he uni roo null for all break sizes, alhough he simulaion resuls show evidence of nonmonoonic power: he power firs decreases and hen increases as he size of he break rises. 5 Because he daa generaing process wih wo breaks ha have he same sign is boh regimewise saionary and inconsisen wih PPP, here is an ambiguiy regarding how o inerpre power. If 5 The issue of non-monoonic power in models wih mean shifs or rend shifs is discussed by Vogelsang (997, 999). 0

12 ADF and PPP resriced srucural change ess have good power o rejec he uni roo null, hey will correcly provide evidence of saionariy bu incorrecly provide evidence of PPP. Since we are concerned wih PPP, no saionariy, low power becomes a desirable propery. Boh he ADF ess and he PPP resriced srucural change ess have exremely low power wih medium (0.3) and large (0.5) breaks, bu higher power in he case of very small breaks (0.). Thus, a leas for medium and large breaks, we are very unlikely o incorrecly find evidence of PPP. The resul for very small breaks is no surprising. As he size of he breaks decreases, he limi of a regime-wise saionary process will be a saionary process wihou breaks. Nex we consider a process which is consisen wih PPP (he breaks are equal and of opposie sign). The ADF es has low power for medium or large breaks. The QPPP es wih wo srucural changes has good power in all cases. The PPP resriced srucural change es has very high power when he process has breaks of equal and opposie sign, regardless of he magniude of he break. The simulaion resuls show evidence of non-monoonic power for boh he QPPP and he PPP resriced srucural change ess. As above, he power firs decreases and hen increases as he size of he break rises. Generaing rend saionary daa which allows for wo changes in he inercep and applying he previous ess, including a ime rend, we obain some similar and some fairly differen resuls han in he case of non-rending daa (Table 4). In he case of breaks ha occur in he same direcion, he ADF es and he TPPP resriced es have a fairly good power for all he break sizes because he ime rend adjuss o compensae for he srucural changes. The TPPP resriced srucural change es has he highes power when he daa generaing process includes wo srucural changes ha are equal and of opposie sign. The ADF es, in conras, has much lower power on processes ha are consisen wih TPPP (breaks ha are equal and of opposie sign). 3.3 Inerpreaion of he empirical resuls We proceed o inerpre our empirical resuls in he conex of he findings from he simulaions. Recall ha previous research rejeced he uni roo null a he 5% level for 9 ou of 6 counries wih he ADF es. Because he simulaion evidence shows ha he ADF ess have no power or very low power when he daa conains any varian of srucural change, his provides evidence of PPP or TPPP for hese counries. Applying our new es, which boh allows for srucural change and imposes pariy resricions, we add 5 more rejecions o he previous resuls: Denmark, Japan, Porugal, Swizerland and he Unied Kingdom. The combinaion of rejecion of he uni roo null wih ess ha incorporae resriced srucural change and failure o rejec he uni roo null wih ess ha do no incorporae resriced srucural change provides evidence of PPP (TPPP) resriced srucural change for hese counries.

13 The simulaion resuls also allow us o discriminae beween evidence of PPP and TPPP resriced srucural change. For Denmark, Japan and Swizerland he uni roo null is rejeced in favor of he resriced srucural change alernaive when he ess include a ime rend, bu no rejeced when he ess do no include a ime rend. This provides clear evidence of TPPP resriced srucural change. For Porugal and he Unied Kingdom, he uni roo null is rejeced in favor of he resriced srucural change alernaive in boh cases. Due o he fac ha ess for PPP resriced srucural change have very low power when he daa is acually generaed by a model wih TPPP resriced srucural change, we inerpre hese findings as evidence of PPP resriced srucural change. 6 As a furher check on our findings of PPP and TPPP resriced srucural change, we compare he break daes and coefficiens from he resriced srucural change models (Table ) o hose from he unresriced wo srucural change models (Table ). For Porugal and he Unied Kingdom, he comparison reinforces he findings of PPP resriced srucural change. For Denmark, Japan and Swizerland i reinforces he findings of TPPP resriced srucural change. The coefficiens on he breaks in he unresriced models are of opposie sign, and he break daes do no change dramaically wih he imposiion of he resricions. These resuls are illusraed in Figure. The speed of reversion o PPP has become an acive research opic. Using he values of α in Table, we calculae, for our preferred specificaions, he half-lives of PPP (or TPPP) deviaions, he ime ha i akes for a shock o he real exchange rae o reurn halfway o is long run PPP (or TPPP) resriced mean (or rend). The half-lives are all under wo years, ranging from.80 years for he Unied Kingdom o.97 years for Japan. The shor half-lives should no be surprising because, by measuring he reurn o he resriced mean or rend, we have aken ou he effecs of wo shocks ha are of sufficien imporance o cause he non-rejecion of he uni roo hypohesis in ess ha do no incorporae srucural change. 7 On he oher hand, we did no find any varian of PPP in Canada and he Neherlands. In he case of Canada one possibiliy for he lack of evidence of PPP is he very large depreciaion of he Canadian dollar agains he US dollar in he end of he sample. The Neherlands experiences only one srucural change, which is no consisen wih eiher one of he PPP alernaives. This is he classic example of he Duch disease, he large real appreciaion of he guilder following he discovery of large naural gas deposis in he Norh Sea. Wha caused he srucural changes o he real exchange raes? Wih he excepion of he Neherlands, hese changes eiher represen emporary (alhough long-lived) movemens away from (T)PPP or movemens ha resore (T)PPP. In addiion, he iniial movemen away from PPP is, in all cases, a real depreciaion agains he dollar, followed by a real appreciaion o resore PPP. Three of he 6 We conduced, bu do no repor, simulaions ha illusrae his resul, which is in accord wih he findings of Wes (987) on ADF ess ha do no incorporae srucural change. 7 We did no calculae he half-lives from he impulse response funcion or correc for median bias, as in Murray and Papell (00) because he half-lives are so shor ha hese correcions would no be paricularly imporan.

14 counries where we found evidence of PPP or TPPP resriced srucural change experienced a depreciaion of heir nominal exchange raes agains he dollar following he end of World War I (Porugal) or World War II (Denmark and he Unied Kingdom). For Swizerland, while he nominal exchange rae remained fixed, U.S. prices rose faser han Swiss prices, causing a real depreciaion saring in 943. For Japan, he onse of he deviaion from PPP was caused by a nominal depreciaion in 930. The offseing shifs ha resored PPP or TPPP are cenered around he collapse of he Breon Woods sysem of fixed exchange raes, which riggered a nominal appreciaion agains he dollar for Denmark, Japan and Swizerland, causing an appreciaion of heir real exchange raes. For he Unied Kingdom, he appreciaion of he pound seems o be caused by high inflaion relaive o he Unied Saes. These movemens in he nominal exchange raes and prices (raios o US prices) are illusraed in Figure. Following he esablishmen of flexible nominal exchange raes, hese counries experienced a reurn o he original level (or rend) of heir real exchange raes. 4. Conclusion Does long-run purchasing power pariy hold beween he Unied Saes and oher indusrialized counries? If here is reversion in he long run, is i o a consan mean, as in he spiri of he version of PPP developed by Cassel, or is i o a consan rend, as in he spiri of he version of PPP heory developed by Balassa and Samuelson. In order o make he disincion clear, we differeniae wo conceps: Purchasing Power Pariy (PPP) and Trend Purchasing Power Pariy (TPPP). Lopez, Murray, and Papell (005) have previously found evidence of a varian of PPP for 9 ou 6 indusrialized counries. In his sudy we focus on he remaining 7 counries. We firs invesigae he hypohesis ha he failure o rejec uni roos in some of he real exchange raes can be explained by he presence of srucural change. As a firs sep, we es for uni roos in he presence of one or wo changes in he inercep or shifs in he rend funcion. We find evidence of eiher Qualified PPP or Trend Qualified PPP for 4 ou of 7 counries. These ess, however, do no provide evidence of eiher PPP or TPPP. We hen consider he possibiliy ha a series may experience boh srucural change and reversion o is mean (or rend). We develop uni roo ess ha resric he coefficiens on he dummy variables ha depic he breaks o produce a consan mean or rend in he long run. These resricions ensure ha he rejecion of he uni roo in favor of he PPP or TPPP resriced srucural change is evidence of long-run (rend) purchasing power pariy. Wih hese new resriced ess, we add 5 counries o he previous PPP or TPPP evidence. Canada and he Neherlands are he only counries where we do no find evidence of any varian of PPP. The simulaion experimens reinforce our empirical resuls. Firs, we find evidence ha ADF ess have very low power o rejec he uni roo hypohesis in processes ha incorporae srucural 3

15 change, including QPPP, TQPPP, and boh PPP and TPPP resriced srucural change. Rejecions using ADF ess herefore provide srong evidence of PPP or TPPP wihou srucural change. Second, our resriced es has very good power when he process incorporaes srucural change ha is consisen wih he PPP or TPPP hypohesis, bu low or moderae power in oher cases. Taking in accoun he previous simulaion resuls, we conclude ha he resriced ess provide srong evidence of PPP resriced srucural change for Porugal and he Unied Kingdom and TPPP resriced srucural change for Denmark, Japan and Swizerland. This resul is reinforced by comparing break daes and coefficiens from unresriced and resriced srucural change models. Mos of he srucural changes are associaed wih movemens in nominal exchange raes. This paper posed wo quesions: Is here evidence of long-run PPP or TPPP among indusrialized counries and, if so, which varian does he evidence suppor? Combining previous resuls of convenional ess wih our new resriced ess we find evidence of PPP and/or TPPP for 4 of he 6 counries. By including counries which experience srucural change ha is consisen wih long-run PPP or TPPP, we increase he evidence by 5 counries compared wih convenional uni roo ess. Using a combinaion of economeric and simulaion evidence, we conclude ha PPP is suppored for 0 counries and TPPP is suppored for 4 counries. 4

16 References Abuaf, Niso and Philippe Jorion (990), Purchasing Power Pariy in he Long Run, The Journal of Finance, 45, Asea, Parick K. and E. G. Mendoza (994), The Balassa-Samuelson Model: A General Equilibrium Appraisal, Review of Inernaional Economics,, Balassa, Bela (964) The Purchasing Pariy Power Docrine: a Reappraisal, The Journal of Poliical Economy, 7, Campbell, John and Pierre Perron (99), Pifalls and Opporuniies: Wha Macroeconomiss Should Know abou Uni Roos, NBER Macroeconomics Annual, 4-0 Cassel, Gusav (98), Abnormal deviaions in inernaional exchanges, The Economic Journal, 8, Canzoneri, Mahew B., Rober E. Cumby and Behzad Diba (999), Relaive Labor Produciviy and he Real Exchange Rae in he Long Run: Evidence for a Panel of OECD Counries, Journal of Inernaional Economics, 47, Dornbusch, Rudiger and Timohy Vogelsang (99), Real Exchange Raes and Purchasing Power Pariy, Trade Theory and Economic Reform: Norh, Souh and Eas, Essays in Honnor of Bela Balassa, Cambridge, MA, Basil Blackwell. Ellio, Graham, Thomas J. Rohenberg and James H. Sock (996), Efficien Tess for an Auoregressive Uni Roo, Economerica, 64, Fizgerald, Doireann (003), Terms-of-Trade Effecs, Inerdependence and Cross-Counry Differences in Price Levels, working paper, Universiy of California, Sana Cruz. Hegwood, Naalie D. and David H. Papell (998), Quasi Purchasing Power Pariy, Inernaional Journal of Finance and Economics, 3, Lumsdaine, Robin L. and David H. Papell (997), Muliple Trend Breaks and he Uni Roo Hypohesis, The Review of Economics and Saisics, LXXIX, -8. Lopez, Claude, Chrisian J. Murray and David H. Papell (005), Sae of he Ar Uni Roo Tess and Purchasing Power Pariy, Journal of Money, Credi, and Banking, 37, Lohian, James R. and Mark P. Taylor (996), Real Exchange Rae Behavior: The Recen Floa from he Perspecive of he Pas Two Cenuries, Journal of Poliical Economy, 04, Murray, Chrisian J. and David H. Papell (00), The Purchasing Power Pariy Persisence Paradigm, Journal of Inernaional Economics, 56, -9. Ng, Serena and Pierre Perron, (995), Uni Roo Tess in ARMA Models wih Daa Dependen Mehods for he Selecion of he Truncaion Lag, Journal of he American Saisical Associaion, 90, Ng, Serena and Pierre Perron (00), Lag Lengh Selecion and he Consrucion of Uni Roo Tess wih Good Size and Power, Economerica, 69,

17 Obsfeld, Maurice (993), Model Trending Real Exchange Raes, Cener for Inernaional and Developmen Economic Research, working paper no. C93-0. Papell, David H. (00), The Grea Appreciaion, he Grea Depreciaion and he Purchasing Power Pariy Hypohesis, Journal of Inernaional Economics, 57, 5-8. Perron, Pierre (997), Furher Evidence on Breaking Trend Funcions in Macroeconomic Variables, Journal of Economerics, 80, Perron, Pierre and Timohy J. Voselgang (99), Nonsaionariy and Level Shifs wih an Applicaion o Purchasing Power Pariy, Journal of Business and Economic Saisics, 0, Rogoff, Kenneh (996), The Purchasing Power Pariy Puzzle, Journal of Economic Lieraure, 34, Samuelson, Paul A. (964) Theoreical Noes on Trade Problems, The Review of Economics and Saisics, 46, Taylor, Alan (00), A Cenury of Purchasing-Power Pariy, Review of Economics and Saisics, 84, Vogelsang, Timohy J. and Pierre Perron (998), Addiional Tess for a Uni Roo Allowing for a Break in he Trend Funcion a an Unknown Time, Inernaional Economic Review, 39, Vogelsang, Timohy J. (997), Sources of nonmonoonic power when esing for a shif in he rend of a dynamic ime series, Cener for Analyic Economics, Working Paper #97-4, Cornell Universiy. Vogelsang, Timohy J. (999), Sources of nonmonoonic power when esing for a shif in mean of a dynamic ime series, Journal of Economerics, 88, Wes, Kenneh (987), A Noe on he Power of Leas Squares Tess for a Uni Roo, Economics Leers, 4,

18 Table. Uni roo ess including one and wo srucural changes Real exchange rae α Break γ k α QPPP es including one srucural change Canada Denmark ** Japan Neherlands ** Porugal * Swizerland ** Unied Kingdom TQPPP es including one srucural change Canada Denmark * Japan * Neherlands Porugal Swizerland ** Unied Kingdom * Real exchange rae α Break γ Break γ k α QPPP es including wo srucural changes Canada Denmark ** Japan Neherlands * Porugal *** Swizerland ** Unied Kingdom *** TQPPP es including wo srucural changes Canada Denmark ** Japan * Neherlands Porugal *** Swizerland ** Unied Kingdom *** *, **, *** denoe significance a he 0%, 5% and % level of significance, respecively. The criical values for α are: -4.0 (0%), (5%) and (%) (QPPP es including one srucural change) -4.7 (0%), -5.0 (5%) and -5.6 (%) (TQPPP es including one srucural change) -5.4 (0%), -5.5 (5%) and (%) (QPPP es including wo srucural changes) (0%), (5%) and (%) (TQPPP es including wo srucural changes) 7

19 Table. Resriced srucural change ess Real exchange rae α Break Break *, **, *** denoe significance a he 0%, 5% and % level of significance, respecively. The criical values for α are: -4.7 (0%), (5%) and (%) (PPP resriced srucural change) -5.3 (0%), -5.59(5%) and -6. (%) (TPPP resriced srucural change) γ ( γ = - γ ) k α PPP resriced srucural change Canada Denmark Japan Neherlands Porugal *** Swizerland Unied Kingdom *** TPPP resriced srucural change Canada Denmark ** Japan ** Neherlands Porugal ** Swizerland *** Unied Kingdom *** 8

20 Table 3. Power agains no srucural change a) Non-rending daa ADF es % (-3.57) 5% (-.95) 0% (-.64) QPPP wo srucural change es % 5% 0% (-6.06) (-5.5) (-5.4) PPP resriced srucural change es % 5% 0% (-5.67) (-5.04) (-4.7). Saionary generaed daa: q = µ + αq + ε α = α = b) Trending daa ADF es including a ime rend % 5% 0% (-4.7) (-3.57) (-3.3) TQPPP wo srucural change es % 5% 0% (-6.45) (-5.96) (-5.69) TPPP resriced srucural change es % 5% 0% (-6.) (-5.59) (-5.3). Trend-saionary generaed daa: q = µ + β + αq + ε α = α =

21 Table 4. Power agains wo srucural changes non-rending daa generaing process ADF es α = 0.7 % (-3.57) 5% (-.95) 0% (-.64) QPPP wo srucural change es % 5% 0% (-6.06) (-5.5) (-5.4) PPP resriced srucural change es % 5% 0% (-5.67) (-5.04) (-4.7) 3. Saionary generaed daa wih wo breaks in he inercep: q = µ + αq + γ DU + γ DU + ε a) Coefficiens on he breaks are equal and have he same sign γ =., γ = = 0.3, γ = = 0.5, γ = γ γ b) Coefficiens on he breaks are equal and have opposie signs γ =., γ = = 0.3, γ = = 0.5, γ = γ γ ADF es % (-4.7) 5% (-3.57) 0% (-3.3) TQPPP wo srucural change es % 5% 0% (-6.45) (-5.96) (-5.69) TPPP resriced srucural change es % 5% 0% (-6.) (-5.59) (-5.3) 4. Trend-saionary generaed daa wih wo breaks in he inercep: q = µ + β + αq + γ DU + γ DU + ε I. α = 0. 7 a) Coefficiens on he breaks are equal and have he same sign γ =., γ = = 0.3, γ = = 0.5, γ = γ γ b) Coefficiens on he breaks are equal and have opposie signs γ =., γ = = 0.3, γ = = 0.5, γ = γ γ

22 Figure. Evidence of PPP or TPPP resriced srucural change A. PPP resriced srucural change 0.8 UK-US real exchange rae -4.6 Porugal-US real exchange rae RER UNRESTRICTED RESTRICTED RER UNRESTRICTED RESTRICTED B. TPPP resriced srucural change Japan-US real exchange rae RER UNRESTRICTED RESTRICTED Denmark-US real exchange rae RER UNRESTRICTED RESTRICTED Swizerland-US real exchange rae RER UNRESTRICTED RESTRICTED

23 Figure. Nominal exchange rae and price (raio o US) movemens for counries where we found evidence of PPP or TPPP resriced srucural change (in logs) A. PPP resriced srucural change.0.5 UK-US NER prices Porugal-US NER prices B. TPPP resriced srucural change 0 - Japan-US NER prices NER prices Denmark-US Swizerland-US NER prices