Long Run Purchasing Power Parity: Cassel or Balassa-Samuelson?

Transcription

1 Long Run Purchasing Power Pariy: Cassel or Balassa-Samuelson? David H. Papell and Ruxandra Prodan Universiy of Houson November 003 We use long-horizon real exchange rae daa for 6 indusrialized counries o 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. We develop uni roo ess ha boh accoun for srucural change and mainain a long run mean or rend. Using convenional ess, we find evidence of some varian of PPP for 9 of he 6 counries. Wih he resriced ess, we find evidence for 5 addiional counries. The Cassel version of PPP is suppored for 0 counries and he Balassa-Samuelson version is suppored for 4 counries. The simulaion resuls show good size and power of he resriced ess. We are graeful o paricipans in seminars a Emory Universiy, he Universiy of Michigan, Texas Camp Economerics VIII and he 00 Souhern Economic Associaion Meeings for helpful commens and discussions. Papell hanks he Naional Science Foundaion for financial suppor. Correspondence o: Deparmen of Economics, Universiy of Houson, Houson, TX Ruxandra Prodan, el. (73) , fax (73) , rprodan@mail.uh.edu David Papell, el. (73) , fax (73) , dpapell@mail.uh.edu

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 good 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 long-horizon real exchange raes. In he 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 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-horizon real exchange raes has been invesigaed using boh approaches. In he Cassel radiion, 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. Noneheless, he convergence o PPP is very slow, wih half-lives of PPP deviaions beween 3 and 5 years. 4 Murray and Papell (00) and Lopez, Murray, and Papell (00), using median unbiased echniques o measure persisence and calculaing confidence inervals as well as poin esimaes, find even longer halflife esimaes. In he Balassa-Samuelson radiion, Taylor (00) creaes a long-horizon real exchange rae daa se for 7 indusrialized and 4 Lain American counries. He finds ha long-run PPP can be suppored in almos all cases using he Ellio, Rohenberg and Sock (996) generalized-leassquares version of he Dickey-Fuller (DF-GLS) es wih allowance for a deerminisic rend. Cuddingon and Liang (000) and Lohian and Taylor (000) invesigae he implicaions of allowing for rending in he Lohian and Taylor (996) daa. Engel (000) decomposes he real exchange rae ino wo processes, one saionary (relaive price of raded goods) and he oher non-saionary (relaive price of non-raded goods). He shows ha, due o size bias, he non-saionary componen of he real exchange rae may no be deeced in ess for long run PPP. Ng and Perron (00), following he same argumen as Engel, show ha even if in some cases esing for uni roo is difficul, improvemens can be made by minimizing Type I error and maximizing he power of he ess. 5 Saring wih Perron (989), much research has been conduced on esing for uni roos in he presence of a one-ime change in he mean, rend, or boh he mean and 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 4 The combinaion of slow speed of convergence o PPP and high shor-erm volailiy of real exchange raes is called he The Purchasing Power Pariy Puzzle by Rogoff (996). 5 In order o minimize size disorions hey propose a new informaion crieria, MAIC, o selec he auoregressive lag lengh

4 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. 6 The firs sep in our invesigaion is o use ess ha do no incorporae srucural change. 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. The daa is from Taylor (00), and consiss of 07 o 9 years of real exchange raes for 6 indusrialized counries wih he Unied Saes dollar as he numeraire currency. 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. Using convenional Augmened Dickey-Fuller (ADF) uni roo ess, we find evidence of PPP by rejecing he uni roo null (a he 5% level) for only 8 ou of 6 counries. 7 Incorporaing ime rends, we also find evidence of TPPP for 7 counries. Combining he wo ess, we find evidence of eiher PPP or TPPP for 9 ou of 6 counries. This is weaker evidence han found by Taylor (00). The reason for our weaker evidence is ha we use lag selecion mehods ha provide beer size and power properies for he ess, no ha our ADF ess have lower power han Taylor s DF-GLS ess. 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 ime. As an exension, we propose ess ha exend he previous models o incorporae wo endogenous break poins. 6 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). 7 We use he erm counry as shorhand for real exchange rae wih he Unied Saes. 3

5 Following our earlier 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 QPPP for 3 ou of 6 counries and of TQPP for ou of 6 counries (a he 5% level). In no case, however, do we find evidence of TQPPP for a counry for which we do no find evidence of QPPP and so, combining he wo ess, we find evidence of eiher QPPP or TQPPP for 3 counries. These resuls are no necessarily a sep forward owards PPP. They do no impose eiher he Cassel alernaive of a consan mean or he Balassa-Samuelson 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 non-rending 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 8 We rejec he uni roo hypohesis in favor of PPP resriced srucural change for 7 counries (a he 5% level). When he alernaive is rend PPP resriced srucural change, we rejec he uni roo null for counries. Combining hese ess wih he 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 9. In 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 8 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. 9 For Germany and Ausralia, we found evidence of PPP and TPPP, respecively, bu did no find evidence of PPP or TPPP resriced srucural change. 4

6 is a real appreciaion of he florin agains he US dollar in he early 970s ha has no been reversed. We have presened evidence ha, using ess ha do no allow for srucural change, PPP or TPPP can be found for only 9 ou of 6 counries. Wih ess ha boh allow for srucural change and impose pariy resricions, evidence of PPP or TPPP can be found for 4 ou of 6 counries. In order o evaluae hese resuls, we conduc simulaions o evaluae he power of he no break, wo breaks, and resriced wo breaks 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). 0 This implies ha he 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. As a furher check on our findings of PPP or TPPP resriced srucural change, we compare he break daes and coefficiens from he resriced o hose from he unresriced srucural change models. Pu ogeher, he resuls in his paper provide srong evidence of PPP or TPPP for 4 ou of 6 long-run real exchange raes.. Tesing for PPP and TPPP wih and wihou 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 was updaed by increasing he sample (using Inernaional Financial Saisics daa from 00). The daa covers a se of 6 indusrialized counries and US as he base counry, sars from 870 for he longes series, and ends in This resul varies wih he break size and he level of persisence of he series; he power of ADF es becomes higher if he size of he break is very small and/or he level of persisence is very low. The counries are Ausralia, Belgium, Canada, Denmark, Finland, France, Germany, Ialy, Japan, Neherlands, Norway, Porugal, Spain, Sweden, Swizerland, Unied Kingdom and Unied Saes. The number of years varies across counries, bu we have a complee cross secion of 7 counries beween 89 and

7 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 p and p * are he logarihms of he US and he foreign price levels, respecively. We proceed in hree seps: Firs, we es for PPP and TPPP wihou allowing for srucural change. Second, we es for uni roos while allowing for srucural change, bu do no impose PPP or TPPP. Third, we es for uni roos in he presence of PPP or TPPP resriced srucural change... Convenional ess for saionariy We firs check for he saionariy of he daa using he Augmened Dickey-Fuller (ADF) uni roo es, which regresses he firs difference of he logarihm of he real exchange rae on a consan, is lagged level and k lagged firs differences: q k + ci q i + ε i= = µ + αq () The number of lags is chosen by using he general-o specific recursive -saisic procedure suggesed by Campbell and Perron (99) and Ng and Perron (995). We se he maximum value of k equal o 8, and use a criical value of.645 from he asympoic normal disribuion o assess significance. The null hypohesis of a uni roo is rejeced in favor of he alernaive of level saionariy if α is significanly differen from 0. In his case, he rejecion of he uni roo null provides evidence of Purchasing Power Pariy (PPP). If a ime rend is included in Equaion (), he null hypohesis of a uni roo is rejeced in favor of he alernaive of rend saionariy if α is The choice of he maximum value of k is somewha arbirary. On he one hand, one would like a large value o have as unresriced a procedure as possible. On he oher hand, a large value of k max yields problems of mulicolliniariy in he daa and also subsanial loss of power. 6

8 significanly differen from 0. In his case, rejecion of he uni roo null provides evidence of Trend Purchasing Power Pariy (TPPP). The resuls of he ADF ess are repored in Table. We find evidence of PPP by rejecing he uni roo null for 8 ou of 6 counries a he 5% level, Belgium, Germany, Finland, France, Ialy, Norway, Spain, and Sweden, and for 3 addiional counries a he 0% level. Incorporaing ime rends, we find evidence of TPPP by rejecing he uni roo null for 7 counries a he 5% level, Ausralia, Belgium, Finland, France, Ialy, Norway, Sweden and for 3 addiional counries a he 0% level. Combining he wo ess, we 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, 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. 3 Taylor (00) repors sronger evidence of PPP. Using he Ellio, Rohenberg and Sock (996) DF-GLS es, he rejecs he uni roo null for 0 ou of 6 counries a he 5% level and for 3 addiional counries a he 0% level wih he U.S. dollar as he numeraire currency. He also finds sronger evidence of TPPP, rejecing he uni roo null for 8 ou of 6 counries a he 5% level and for 3 addiional counries a he 0% level. Combining he wo ess, he finds evidence of eiher PPP or TPPP for counries a he 5% level and 4 addiional counries a he 0% level. He also repors marginally sronger rejecions wih ADF ess han we find, rejecing he uni roo null in favor of eiher PPP or TPPP for 9 ou of 6 counries a he 5% level and for 3 addiional counries a he 0% level. 4 The reason for he differences in he resuls comes from differences in lag selecion procedures. Taylor uses a Lagrange Muliplier echnique, which ends 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 he echniques recommended by Ng and Perron o produce ess wih good size and power, general-o-specific 3 Wes (987) shows ha ADF ess ha do no include a ime rend have zero asympoic power if he series is rend saionary. Taylor (00) indicaes ha a leas for some series a deerminisic erm may be presen, he rend being sizeable:.5% per annum in he case of Japan, 0.74% per annum for Swizerland, bu small in all oher cases. 4 The comparisons wih Taylor s resuls are for indusrialized counries wih he U.S. dollar as numeraire. He also repors resuls wih four Lain American counries and, for all counries, wih a world baske as numeraire. While we add wo years o Taylor s daa, none of he comparisons are affeced by using his original sample. 7

9 lag selecion for ADF ess and MAIC lag selecion for DF-GLS ess, we rejec approximaely as much as Taylor using ADF ess and less ofen using DF-GLS ess. Lopez, Murray and Papell (00), using DF-GLS ess wih MAIC lag selecion, rejec he uni roo null in favor of he PPP alernaive for 9 ou of 6 counries a he 5% level. We conduced DF-GLS ess wih MAIC lag selecion and TPPP as he alernaive hypohesis, and could only rejec he uni roo null for 5 ou of 6 counries a he 5% level and for 4 addiional counries a he 0% level. Combining he wo ess, evidence of eiher PPP or TPPP using DF- GLS ess wih MAIC lag selecion can be found for 9 counries a he 5% level and addiional counries a he 0% level. Wih up o 9 years of daa, i should no be surprising ha he power gains from using DF-GLS ess raher han ADF ess do no make much difference in pracice. Wheher we use ADF ess wih general-o-specific lag selecion or DF-GLS ess wih MAIC lag selecion, evidence of eiher PPP or TPPP can be found for only 9 of he 6 counries a he 5% level.. 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. 5 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. Analyzing long-run real exchange raes, we mainly draw he real exchange raes from a nominal fixed exchange rae regime, where devaluaions and reevaluaions, especially following failed aemps o defend currencies, can lead o discree jumps (inercep changes). 5 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. 8

10 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 + DU + ~ z (3) where DU = if T>TB, 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: k i = 0 k i + α~ z + c i ~ z i + i = ~ z = ω D ( Tb ) ε i (4) where 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 kmax =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. 6 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 + DU + ~, z (5) 6 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. 9

11 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 (4). The uni roo hypohesis is esed as in he previous case. 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 (4). The criical values for he finie sample disribuions are aken from he sored vecor of 5000 replicaed saisics. 7 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 ou of 6 counries a he 5% level, Ausralia, Belgium, Denmark, Finland, France, Ialy, Neherlands, Norway, Spain, Sweden 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 7 counries a he 5% level, Belgium, Finland, France, Ialy, Norway, Sweden and Swizerland, and for 5 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 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. 8 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<tbi<.90t (where T is he sample size and i =,), he deerminisic par of he series is removed using he following regression: q = µ + ~ z DU + DU + (6) where DU = if T>TB, 0 oherwise and is hen performed using he saisic for α = 0 in he regression: DU = if T>TB, 0 oherwise. The uni roo es k k k ω i D( Tb) i + ω i D( Tb) i + α~ z + ci ~ z i + i= 0 i= 0 i= ~ z = ε (7) 7 We experimened by compuing daa specific criical values for several counries, and he resuls were unaffeced. 8 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. 0

12 where D(TBi) = if = TBi+ 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 + ~ z DU + DU + (8) The uni roo es is hen performed using he saisic for α = 0 in he regression described by Equaion (7). 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 3. We find evidence of Qualified Purchasing Power Pariy (QPPP) by rejecing he uni roo null for ou of 6 counries a he 5% level, Ausralia, Belgium, Denmark, Finland, France, Ialy, Norway, Porugal, Spain, Sweden, 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 counries a he 5% level, Belgium, Denmark, Finland, France, Ialy, Norway, Porugal, Spain, Sweden, 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 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 3 counries a he 5% level..3 Tess for PPP in he presence of resriced srucural change We firs presened evidence of PPP (or TPPP) for 9 ou of 6 counries using a convenional ADF es. Then, using an AO model and allowing for one or wo inercep changes we found evidence of QPPP (or TQPP) for 3 counries. Neiher of hese ess, however, invesigaes he possibiliy ha a series may experience boh srucural change and reversion o

13 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 + ~ z DU + DU + subjec o he resricion: + 0 (0) = (9) where 0% rimming is used o avoid finding spurious breaks. The resricion in Equaion (0) 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. The uni roo es is hen performed using he saisic for α = 0 in he regression described by Equaion (7). 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 (8), subjec o he same resricions (0). This imposes he TPPP hypohesis because he rend following he second 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 4. We find evidence of PPP Resriced Srucural Change by rejecing he uni roo null for 7 ou of 6 counries a he 5% level, Finland, France, Ialy, Norway, Porugal, Spain and he Unied Kingdom, and for wo addiional counries a he 0% level. Incorporaing ime rends, we find evidence of Trend PPP Resriced Srucural Change by rejecing he uni roo null for counries a he 5% level, Belgium, Denmark, Finland, France, Ialy, Japan, Norway, Porugal, Spain, Sweden, Swizerland, and he Unied Kingdom, and for one addiional counry a he 0% level. Combining he wo ess, we find evidence of eiher PPP or TPPP Resriced Srucural Change by rejecing he uni roo null for counries a he 5% level. Ausralia, Canada, Germany and he Neherlands are he counries for which we do no find evidence of eiher PPP or TPPP Resriced Srucural Change. Among hese counries, we previously found evidence of PPP for Germany and of TPPP for Ausralia by using ADF ess.

14 daa. 9 Wihin he Mone Carlo experimens we consider he following wo daa generaing Combining he ess, we find evidence of some varian of PPP for 4 of he 6 counries. In addiion, we find evidence of QPPP for he Neherlands. Canada is he only counry for which here is no evidence of some varian of saionariy. 3. Size and power of univariae uni roo ess wih srucural change The cenral empirical resul of he paper is ha uni roo null can be rejeced in favor of a varian of PPP for 4 ou of 6 counries, 5 more rejecions han from a convenional ADF es. 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 finie sample performance of he newly developed ess and proceed o invesigae heir size and power. 3. Consrucion of he size and power simulaions The simulaion experimens address he following issues: (a) comparison of various ess properies (ADF, AO model allowing for wo srucural changes and AO resriced model), (b) power of AO models as a funcion of he level of persisence and he magniude of he break, (c) power of he PPP resriced ess on rending daa and rend PPP resriced ess on non-rending processes: q q = + ( β) + αq µ + ε ( β) + αq + DU + = 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 () and, also, allowing for wo changes in he inercep (). 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 = 0.7, 0.8 and 0.9 for firs generaed sample () and α = 0.5, 0.6, 0.7 and 0.8 for he second 9 We do no repor simulaion resuls for AO models wih one break because hey are similar o hose wih wo breaks in he same direcion. 3

15 generaed sample (). This reflecs he range of values of α repored in Tables 4 (wih and wihou srucural changes). 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% and α = 0.7, 0.8 and 0.9 (deailed resuls for differen nominal sizes and values of α are presened in Table 5). Firs, as he errors become more persisen, he power decreases: The ADF es has good power when he daa generaing process is saionary wih α = 0.7 and 0.8 bu lower power when α = 0.9. This is in accord wih previous research, considering he span and persisence of he daa. 0 As expeced, ess for QPPP ha incorporae wo srucural changes have less power han he ADF es. Tess for PPP resriced srucural change have lower power han he ADF ess (excep in he case of low persisence) bu higher power han QPPP es. Generaing rend saionary daa processes and applying he previous ess, including a ime rend, we obained fairly similar resuls wih he previous ones. However, he power of he ess, in mos of he cases, is lower han in he above case. As in he case of non-rending daa, he power of rend-resriced ess exceeds he power of ADF ess wih low persisence. Consider now experimen () in which we generae saionary daa and allow for wo changes in he inercep (Table 6). 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, excep for he case wih high persisence (α = 0.8). For low levels of persisence 0 A coefficien of α of.8 corresponds o a half-life wih annual daa of 3. years. 4

16 (α = 0.5, 0.6) here is generally a small decrease in power as he breaks become larger. For higher levels of persisence (α = 0.7, 0.8) he simulaion resuls show evidence of nonmonoonic power. The power firs decreases and hen increases as he size of he break rises. Because he daa generaing process is boh regime-wise saionary and inconsisen wih PPP, here is an ambiguiy in he conceps of size and power when we evaluae he ADF ess and he PPP resriced srucural change ess. Tess wih good power o rejec he uni roo null will be oversized wih regard o he PPP hypohesis. Since we are primarily concerned wih PPP, we will focus on he laer. Boh he ADF ess and he PPP resriced srucural change ess generally have good size wih medium (0.3) and large (0.5) breaks, being even undersized for all bu very low levels of persisence. However, in he case of very small breaks (0.) he resuls show subsanial size disorion for all values of α. This 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 good power wih small breaks a all bu very high levels of persisence, moderae power for medium breaks and low levels of persisence (α= 0.5,α = 0.6), and low power for medium or large breaks wih high levels of persisence (α= 0.7,α = 0.8). The QPPP es wih wo srucural changes has good power in all cases, wih he power decreasing as he process becomes more persisen. 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, excep for he mos highly persisen processes (α = 0.8). The simulaion resuls show evidence of non-monoonic power for boh he QPPP and he PPP resriced srucural change ess a higher levels of persisence (α = 0.7, 0.8). 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 generaed daa (Table 7). In he case of breaks ha occur in he same direcion, he ADF es and he TPPP resriced es have good power for all he break sizes, excep for highly persisen daa, because he ime rend adjuss o compensae for he srucural changes. The TPPP resriced srucural change es has he highes power when Afer reaching a cerain size, any addiional increase in he magniude of he break will no modify he power of he es. The issue of non-monoonic power in models wih mean shifs or rend shifs is discussed by Vogelsang (997, 999). 5

17 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 PPP (breaks ha are equal and of opposie sign). Finally, we conduc simulaions o deermine wheher our ess can discriminae beween PPP and TPPP resriced srucural change. As repored in Table 8, we run uni roo ess ha incorporae PPP resriced srucural change on various ypes of rending daa. If he daa is rending wih no breaks, ess ha incorporae PPP resriced srucural change have essenially zero power. If he daa is rending wih breaks of equal and opposie sign (TPPP resriced srucural change), ess ha incorporae PPP resriced srucural change have low power in he case of low persisence (α= 0.5,α = 0.6) and virually zero power in he case of higher persisence (α= 0.7,α = 0.8). These resuls are in accord wih he resuls of Wes (987) on ADF ess ha do no incorporae srucural change. We do no repor he resuls of TPPP resriced srucural change ess on various ypes of non-rending daa because hey are idenical o he resuls repored earlier for rending daa. 3 If he daa is non-rending, eiher wih no breaks or wih breaks of equal and opposie sign (PPP resriced srucural change), ess ha incorporae TPPP resriced srucural change have fairly good power, alhough no as good as ess ha incorporae PPP resriced srucural change. This is also in accord wih previous work on ADF ess ha do no incorporae srucural change, where including an unnecessary ime rend as a regressor will decrease power. 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 we 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 3 If a ime rend is included as a regressor, i does no maer wheher he daa is generaed wih a zero or a non-zero rend. 6

18 uni roo null wih ess ha do no incorporae resriced srucural change provides evidence of PPP (TPPP) resriced srucural change for hese counries. 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. Based on our simulaion resuls, where 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. 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 4) o hose from he unresriced wo srucural change models (Table 3). 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. 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 (Figure ). There are numerous poliical and economic facors ha have he poenial o cause shocks o real exchange raes. We explore possible explanaions for some of he srucural changes found wih resriced models. Mos of he counries where we found evidence of PPP or TPPP resriced srucural change experienced a depreciaion of heir exchange rae agains he dollar associaed wih he wo World Wars (Porugal, he Unied Kingdom, Denmark and Swizerland). On he oher hand, he collapse of he Breon Woods sysem, in 97, riggered a posiive shock in he Unied Kingdom, Japan and Swizerland, causing a srong appreciaion of heir exchange raes agains he dollar. Following he esablishmen of flexible nominal exchange raes, hese counries experienced a reurn o he original level (or rend) of heir real exchange raes. 7

19 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 version of PPP developed by Cassel, or is i o a consan rend, as in 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). Using he longes span of available hisorical daa, we firs perform convenional Augmened Dickey-Fuller (ADF) uni roo ess which, wih over a cenury of daa, provide sufficien power o rejec he uni roo null in favor of a level (or rend) saionary alernaive. We find evidence of PPP for 8 counries, Belgium, Germany, Finland, France, Ialy, Norway, Spain, and Sweden, and evidence of TPPP for one addiional counry, Ausralia. We proceed o 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 a one ime change in he inercep or a one ime shif in he rend funcion. We find evidence of eiher Qualified PPP or Trend Qualified PPP for ou of 6 counries. Nex we exend he previous ess o include wo srucural changes. In his case we find evidence of eiher QPPP or TQPPP for ou of 6 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. There are hree conclusions ha we can draw: Firs, we find evidence ha ADF ess have very low power o rejec he uni roo hypohesis in processes ha incorporae srucural 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. On he oher hand, 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. Finally, ess ha incorporae 8

20 PPP resriced srucural change have very low power if he daa is generaed by a model wih TPPP resriced srucural change, which allows us o discriminae beween evidence of PPP and TPPP resriced srucural change. 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. 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? Using convenional and 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 approximaely 50% compared wih convenional uni roo ess. Using a combinaion of economeric and simulaion evidence, we conclude ha he Cassel version of PPP is suppored for 0 counries and he Balassa-Samuelson version of TPPP is suppored for 4 counries. 9

21 References Abuaf, Niso and Philippe Jorion (990), Purchasing Power Pariy in he Long Run, The Journal of Finance, 45, Asea, Parick K. and Max Corden (994), The Balassa-Samuelson Model: An Overview, Review of Inernaional Economics,, 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, Ben-David, Dan; Robin Lumsdaine and David H.Papell (003), Uni Roos, Poswar Slowdowns and Long Run Growh: Evidence from Two Srucural Breaks, Empirical Economics, 8, 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, Cheung,Yin Wong and Kon S. Lai (000), On he Purchasing Pariy Puzzle, Journal of Inernaional Economic, 5, Cuddingon, John T. and Hong Liang (000), "Purchasing Power Pariy over wo cenuries?", Journal of Inernaional Money and Finance, 9, Dornbusch, Rudiger (988), Purchasing Power Pariy, The New Palgrave Dicionary of Economics, Macmillan. 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, Engel, Charles (000), Long run PPP may no hold afer all, Journal of Inernaional Economics, 57, Fizgerald, Doireann (003), Terms-of-Trade Effecs, Inerdependence and Cross-Counry Differences in Price Levels, working paper, Harvard Universiy. 0

22 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, May 997, -8. Lopez, Claude, Chrisian J. Murray and David H. Papell (00), Sae of he Uni Roo Tess and he PPP Puzzle, working paper, Universiy of Houson. 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, Lohian, James R. and Mark P. Taylor (000), Purchasing Power Pariy Over Two Cenuries: Srenghening he Case for Real Exchange Rae Sabiliy: A Reply o Cuddingon and Liang, Journal of Inernaional Money and Finance, 9, Murray, Chrisian and David H. Papell (00), The Purchasing Power Pariy Persisence Paradigm, Journal of Inernaional Economics, February 00, -9. Ng, Serena and Pierre Perron (00), Lag Lengh Selecion and he Consrucion of Uni Roo Tess wih Good Size and Power, Economerica, 69, Ng, Serena and Pierre Perron (00), PPP May no Hold Afer All: a Furher Invesigaion, Annals of Economics and Finance, 3, O Connell, Paul G.J. (998), The Overvaluaion of Purchasing Power Pariy, Journal of Inernaional Economics, 44, -9. 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. Papell, David H. and Hrisos Theodoridis (00), The Choice of Numeraire Currency in Panel Tess of Purchasing Power Pariy, Journal of Money, Credi and Banking, Augus, 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. (994), Faces of Balassa-Samuelson Thiry Years Laer, Review of Inernaional Economics,, 0-6.