State of the Art Unit Root Tests and Purchasing Power Parity

Transcription

1 Sae of he Ar Uni Roo Tess and Purchasing Power Pariy Claude Lopez a Chrisian J. Murray b David H. Papell b a Universiy of Cincinnai, b Universiy of Houson April 2004 Absrac Alhough he uesion of wheher Purchasing Power Pariy (PPP) holds in he long run has been exensively sudied, he answer is sill conroversial. Some of he sronges evidence is provided by Taylor (2002), who concludes ha long-run PPP held over he wenieh cenury. We argue ha his conclusion is uie sensiive o he use of subopimal lag selecion in uni roo ess. Using superior lag selecion mehods, we find ha long run PPP held for he real exchange raes of only 9 ou of he 6 indusrialized counries in Taylor s sample wih he U.S. dollar as he base currency. Correspondence: Claude Lopez, Deparmen of Economics, Universiy of Cincinnai el: (53) , fax: (53) claude.lopez@uc.edu Chris Murray, Deparmen of Economics, Universiy of Houson el: (73) , fax: (73) , cmurray@mail.uh.edu David Papell, Deparmen of Economics, Universiy of Houson el: (73) , fax: (73) , dpapell@mail.uh.edu

2 . Inroducion The uesion of wheher Purchasing Power Pariy (PPP) holds in he long run has been he subjec of voluminous research. Some of he sronges evidence of long-run PPP is provided by Taylor (2002). An imporan conribuion of Taylor s wor is o consruc real exchange rae daa for over 00 years for 20 counries. For he firs ime, i is possible o invesigae PPP using long-horizon daa wih approximaely he same se of advanced counries as is commonly used in sudies wih pos-973 floaing real exchange rae daa. Afer performing he Augmened Dicey-Fuller (ADF) es and he more powerful DF-GLS es of Ellio, Rohenberg, and Soc (996), Taylor concludes ha PPP has held in he long run over he wenieh cenury for my sample of weny counries. The evidence is sufficien o lead him o wrie: If PPP holds in he long run, i is no longer producive o devoe furher aenion o he saionariy uesion. In his paper, we ae issue wih Taylor s conclusion ha PPP has held over he wenieh cenury. More precisely, we show ha srong evidence of PPP can be found for only 9 ou of 6 indusrialized counries wih he U.S. dollar as he base currency. The reason for he difference lies in he seemingly arcane opic of lag selecion for augmening uni roo regressions in he presence of serial correlaion. Taylor employs a mehod ha selecs very shor lag lenghs, mosly zero. Research by Hall (994) and Ng and Perron (995, 200) shows ha he use of oo shor lag lenghs lowers power for ADF ess and maes DF-GLS ess oversized. They recommend a general-o-specific procedure for ADF ess and a modified Aaie informaion crierion for DF-GLS ess. These procedures have become sandard pracice for conducing uni roo ess. We perform uni roo ess using sandard lag selecion echniues. Wheher we use he ADF or he DF-GLS es, he resul is he same. The uni roo null hypohesis can be rejeced in favor of eiher level or rend saionariy for 9 ou of 6 counries a he 5 percen level. We do no see how finding evidence of PPP for one more han half of he counies jusifies describing he wenieh cenury as A Cenury of Purchasing Power Pariy. The res of his paper is organized as follows. Secion 2 discusses he daa, uni roo ess, and issues regarding lag selecion. Secion 3 performs an exercise which demonsraes ha any desired conclusion abou he exisence of long run PPP can be

3 reached if one chooses he lag o rig he resuls. Secion 4 summarizes and offers concluding remars. 2. The Daa and Uni Roo Tess Taylor (2002) collecs nominal exchange rae and price level daa hrough 996 for 20 counries, each for over 00 years, yielding 9 US dollar denominaed real exchange raes. The price levels are consumer price deflaors or, if no available, GDP deflaors. We exend Taylor s daa hrough 998, and omi Argenina, Brazil, and Mexico, in order o focus solely on developed counries. This leaves us wih 6 dollar denominaed real exchange raes: Ausralia, Belgium, Canada, Denmar, Finland, France, Germany, Ialy, Japan, he Neherlands, Norway, Porugal, Spain, Sweden, Swizerland, and he Unied Kingdom. Taylor conducs boh ADF and DF-GLS ess, wih and wihou deerminisic ime rends. where The ADF es, wihou a deerminisic ime rend, runs he following regression: + ψ i = c + α i + u () is he naural logarihm of he real exchange rae. This regression includes lagged firs differences o accoun for serial correlaion. The more powerful DF-GLS es runs he following auxiliary regression: where µ µ = + α ψ + u (2) µ is he GLS demeaned real exchange rae. Tha is, ( ~ 2 ) ~ = z ~ z~ i µ i µ ~ βz =, where z =, β, ~ (,( α ),...,( α )), ~ z = (,( α),...,( α) ), α = + c /T, and c = 7. = 2 T T Taylor uses a Lagrange Muliplier (LM) crierion o choose he number of lags in he uni roo regressions, and in mos cases he seleced lag is zero. For he real exchange raes ha we consider in his paper, Taylor finds ha wih he ADF es, 6 of 6 uni roo ess are rejeced a he 5% level, while wih he DF-GLS es, 0 of he 6 uni roo ess are rejeced a he 5% level. The increased rejecion rae wih he DF-GLS es is The difference beween our resuls and Taylor s is no caused by exending he daa from 996 o 998. Taylor also ess for PPP using a world base as he base currency. 2

4 aribued o he higher power of he es. Much research has been devoed o he opic of lag selecion in uni roo ess, and o our nowledge, he LM crierion has no been sudied in his conex. 2 Also, here are well documened problems wih uni roo ess when he chosen lag is oo small. Specifically, as shown by Hall (994) and Ng and Perron (995), he ADF es suffers from low power when he lag lengh is oo small. Also, Ng and Perron (200) demonsrae ha he DF-GLS es suffers from size disorions when he lag is oo small. Since he former problems leads o oo few rejecions, and he laer problems leads o oo many, lag selecion alone may be responsible for he difference in rejecions rae ha Taylor finds beween ADF and DF- GLS ess. There exis lag selecion procedures whose properies are well undersood, and which offer a beer combinaion of size and power. Specifically, he exising lieraure suggess using he general-o-specific (GS) procedure of Hall (994) for ADF ess and he Modified Aaie informaion crierion (MAIC) of Ng and Perron (200) for DF-GLS ess. Accordingly, we repor ADF and DF-GLS ess in Table and Table 2 respecively. Every series ends in 998 bu no all have he same saring dae. The sample period for each series is indicaed in Tables and 2. For compleeness, we consider LM, GS, and MAIC lag selecion for each es, alhough we will focus on GS for ADF es, and MAIC for DF-GLS es. Wih GS lag selecion, we find 8 of 6 rejecions a he 5% level using ADF ess; 2 more han Taylor finds for he same 6 counries wih daa ending in 996. Wih MAIC lag selecion, we find 9 of 6 rejecions a he 5% level wih DF-GLS es, less han Taylor wih he same daa ending in 996. Thus, using proper lag selecion procedures, he difference in rejecion raes beween he ADF and DF-GLS es has almos been eliminaed. 3 Taylor also repors rejecions of he uni roo null based on ADF and DF-GLS ess wih he inclusion of a deerminisic ime rend. While rejecion of he uni roo null in his case is no evidence of PPP in he sric sense of a mean revering real exchange rae, 2 See Hall (994), Ng and Perron (995), and Ng and Perron (200). 3 The 8 rejecions in Table are no a proper subse of he 9 rejecions in Table 3. In oal, here are counries ha rejec he uni roo null in Tables and/or 3. We do no repor rejecions since his esing sraegy would lead our es o be oversized. 3

5 one can ascribe a Balassa-Samuelson inerpreaion o a saionary real exchange rae around a ime rend. 4 A sensible sraegy when esing for uni roos in persisen daa, when some of he series are clearly rending, is o employ a wo-sep procedure. Firs, es he uni roo null agains he alernaive of level saionariy. If he null is rejeced, conclude ha he series is level saionary, or in our case, ha long run PPP holds. If he null canno be rejeced agains level saionariy, es he uni roo null agains he alernaive of rend saionariy. If he null is rejeced, conclude ha he series is rend saionary. If he null canno be rejeced, conclude ha he series is I(), and ha long run PPP fails o hold. This esing sraegy, which Taylor essenially follows, has inuiive appeal when esing for uni roos in real exchange rae daa, since some series are clearly rending. Suppose ha he rue daa generaing process is level saionariy. In his case he uni roo es described by euaion () or (2) is appropriae. A ime rend is an exraneous regressor, and is inclusion reduces power. However, if he rue daa generaing process were rend saionariy, failing o include a ime rend also resuls in a reducion in power of he es. In addiion, his loss or power from excluding a ime rend when i should be presen is more severe han he reducion in power associaed wih including a ime rend when i is exraneous; see Wes (987). Finally, if he series is I(), hen boh seps of his procedure should lead o a failure o rejec he uni roo null. Using his wo-sep procedure, Taylor finds 3 addiional 5% rejecions for he ADF es, and addiional 5% rejecion for he DF-GLS es. He cies hese exra rejecions from including a ime rend as an imporan ingredien o his conclusion ha PPP holds for over a cenury. However, i appears ha i is he use of sub-opimal lag selecion which leads Taylor o find so many rejecions in he presence of linear ime rend, no he (rend) saionariy of he daa. When one uses proper lag selecion mehods, here is only one addiional uni roo rejecion for he ADF es and none for he DF-GLS es. We repor he resuls of boh ess, again using all 3 mehods of lag secion for compleeness. The ADF uni roo es wih a ime rend runs he following regression: + ψ i = c + b + α i + u, 4 See Papell and Prodan (2003) for furher discussion. 4

6 he resuls of which are repored in Table 3. For he ADF es wih GS lag selecion he 0% rejecion for he U.S./Ausralia exchange rae is srenghened o a 5% rejecion. Table 4 repors he resuls of he DF-GLS es, also allowing for a ime rend under he alernaive hypohesis, which runs he following auxiliary regression: where τ τ = α + ψ + u τ is he GLS derended real exchange rae. For he DF-GLS es wih MAIC lag selecion, here are no addiional uni roo rejecions. Thus, using sae of he ar uni roo ess and lag selecion, 9 of 6 counries are consisen wih PPP, while he remaining 7 are consisen wih he failure of PPP o hold in he long run. I is ineresing o noe ha he number of rejecions a he 5 percen level is he same for he ADF and he DF-GLS es use of he more powerful DF-GLS es does no mae any difference in his case. 3. Lag Selecion for he Insane We perform he following exercise o provide furher evidence ha lag selecion plays an imporan role in deermining he exisence, or lac hereof, of long run PPP. Suppose he lag lengh were chosen o minimize he p-value of he uni roo saisic. Tha is, one considers all possible values of, and chooses he value which maximizes he evidence agains he uni roo null. Euivalenly, his procedure maximizes he evidence ha long run PPP holds. Admiedly, choosing in his manner would lead o uni roo ess wih absurd size and power properies, and we cerainly do no advocae using his mehod of lag selecion in pracice. We employ i here o mae he poin ha lag selecion for uni roo ess is no merely a heoreical curiosiy bu can have dramaic implicaions for empirical findings. The resuls from his exercise are repored in Table 5. We conduc ADF and DF-GLS ess, boh wih and wihou ime rends, by choosing he lag lengh o maximize he evidence he long run PPP holds. 5 We focus only on he resuls of uni roo ess wihou ime rends, as here are no subsanive differences in rejecion freuencies of he uni i τ i 5 We only repor he decision o rejec or no a he 5% level. P-values of he uni roo -saisic and chosen lag lenghs are omied for breviy. 5

7 roo null by including a ime rend. 6 We find 0/6 and 3/6 findings of long run PPP for he ADF and DF-GLS respecively. These resuls are sronger han wha we find in Secion 2, where we employ sandard lag selecion procedures. Wih he DF-GLS es in paricular, his mehod of lag selecion would provide compelling evidence ha long run PPP holds for mos of he real exchange raes in Taylor s (2002) daa se. Conversely, we nex consider a lag selecion procedure which maximizes he p-value of he uni roo saisic. In conras o our previous exercise, we consider all possible values of, and choose he value which provides he greaes evidence agains long run PPP. These resuls are repored in Table 6, again for ADF and DF-GLS ess, wih and wihou ime rends. The resuls are even more sriing in his case. We find a failure o rejec he null ha long run PPP does no hold for 5/6 and 4/6 cases for he ADF and DF-GLS ess respecively. These resuls would sugges ha long run PPP did no hold over he 20 h cenury beween he U.S. and any oher counry in our sample excep for Finland and possibly Ialy. The message from his exercise is clear. Differen lag selecion mehods lead o drasically differen conclusions regarding he exisence of long run PPP. By choosing he lag lengh o rig he oucome, basically any desired conclusion can be reached. Therefore, i is imporan o employ lag selecion procedures which lead o uni roo ess wih good size and power properies, so ha he oucomes of uni roo ess are credible. 4. Conclusion Taylor (2002) consrucs long-horizon US dollar denominaed real exchange rae daa for 6 developed and 3 developing counries from nominal exchange raes and consumer price deflaors. This daa allows researchers, for he firs ime, o sudy long-horizon real exchange raes for developed counries wih comparable coverage and consrucion o he daa ha are commonly used o invesigae pos-973 real exchange raes. He concludes ha, using more powerful DF-GLS ess, long-run PPP has held over he wenieh cenury. We find his conclusion oo srong. When one performs uni roo ess wih superior mehods of lag selecion, he uni roo null hypohesis is rejeced in favor of long run PPP 6 Again, rejecing he uni roo null when a ime rend is included is no euivalen o finding long PPP in he sric sense of a mean revering real exchange rae. 6

8 a he 5% level for only 9 of 6 real exchange raes. To us, his evidence is no sufficien o conclude ha long run PPP has held over he las cenury or ha devoing furher aenion o he issue of saionariy would be unproducive. 7

9 References Ellio, G., Rohenberg, T., and Soc, J.H., 996, Efficien Tess for an Auoregressive Uni Roo, Economerica 64: Hall, A., 994, Tesing for a Uni Roo in Time Series wih Prees Daa-Based Model Selecion, Journal of Business and Economic Saisics 2: Ng, S. and P. Perron, 995, Uni Roo Tes in ARMA Models wih Daa Dependen Mehods for he Selecion of he Truncaion Lag, Journal of he American Saisical Associaion 90: Ng. S. and P. Perron, 200 Lag Lengh Selecion and he Consrucion of Uni Roo Tess wih Good Size and Power. Economerica 69: Papell, D.H., and R. Prodan, 2003, Long Run Purchasing Power Pariy: Cassel or Balassa-Samuelson? woring paper, Universiy of Houson. Taylor, A.M., 2002, A Cenury of Purchasing Power Pariy, The Review of Economics and Saisics 84: Wes, K.D., 987, A Noe on he Power of Leas Suares Tess for a Uni Roo, Economics Leers 24:

10 Table. ADF Uni Roo Tess Wihou Time Trends + ψ i i = c + α + u Counry Sample ADF LM LM ADF GS GS ADF MAIC MAIC Ausralia * Belgium ** 0-4.5*** -2.92** 3 Canada Denmar Finland *** *** -4.58*** 0 France *** -3.55*** Germany ** -2.95** Ialy ** *** ** 0 Japan Neherlands * Norway * *** Porugal * Spain ** Sweden ** *** -3.09** 2 Swizerland UK ** 0-2.6* ** 0 *, **, and *** denoe significance a he 0%, 5%, and % level respecively. 9

11 Table 2. ADF Uni Roo Tess Wih Time Trends + ψ i = c + b + α i + u Counry Sample ADF LM LM ADF GS GS ADF MAIC MAIC Ausralia * ** -3.22* 0 Belgium ** 0-5.*** -3.90** 0 Canada Denmar Finland *** *** -4.69*** 0 France *** -4.6*** Germany * -3.32* Ialy * *** * 0 Japan Neherlands * Norway ** Porugal Spain * Sweden * *** -3.40* 0 Swizerland ** UK * * 0 *, **, and *** denoe significance a he 0%, 5%, and % level respecively. 0

12 Table 3. DF-GLS Uni Roo Tess wihou Time Trends µ µ = α + ψ + u i µ i Counry Sample DF-GLS LM DF-GLS LM GS DF-GLS GS MAIC MAIC Ausralia ** *** -2.29** 0 Belgium *** *** -2.38** 3 Canada Denmar ** Finland *** *** -4.49*** 0 France ** Germany ** -2.52** -.97** 2 Ialy *** *** *** 0 Japan Neherlands * 0-2.5** -2.20** 2 Norway * ** Porugal * Spain ** ** ** 3 Sweden ** ** ** 2 Swizerland UK *** ** ** 4 *, **, and *** denoe significance a he 0%, 5%, and % level respecively.

13 Table 4. DF-GLS Uni Roo Tess wih Time Trends τ τ = α + ψ + u i τ i Counry Sample DF-GLS LM DF-GLS LM GS DF-GLS GS MAIC MAIC Ausralia * 0-3.7** -2.77* 0 Belgium *** 0-5.4*** -3.93*** 0 Canada Denmar * Finland *** *** -4.72*** 0 France *** -4.04*** Germany ** -3.35** -2.78* 2 Ialy ** *** ** 0 Japan Neherlands ** Norway *** Porugal Spain ** Sweden ** *** -3.44** 0 Swizerland *** -2.78* * 2 UK * * 0 *, **, and *** denoe significance a he 0%, 5%, and % level respecively 2

14 Table 5. Choosing o Maximize he Evidence ha PPP Holds Oucome of Uni Roo Tes a he 5% Significance Level ADF Tes DF-GLS Tes Counry w/o Time Trend w/ Time Trend w/o Time Trend w/ Time Trend Ausralia FTR Rejec Rejec Rejec Belgium Rejec Rejec Rejec Rejec Canada FTR FTR FTR FTR Denmar FTR FTR Rejec Rejec Finland Rejec Rejec Rejec Rejec France Rejec Rejec Rejec Rejec Germany Rejec FTR Rejec Rejec Ialy Rejec Rejec Rejec Rejec Japan FTR FTR FTR FTR Neherlands FTR FTR Rejec Rejec Norway Rejec Rejec Rejec Rejec Porugal Rejec FTR Rejec Rejec Spain Rejec FTR Rejec Rejec Sweden Rejec Rejec Rejec Rejec Swizerland FTR Rejec FTR Rejec UK Rejec Rejec Rejec Rejec Long Run PPP 0/6 9/6 3/6 4/6 The oucome of he es, rejecion or failure o rejec, is deermined using 5% criical values. 3

15 Table 6. Choosing o Minimize he Evidence ha PPP Holds Oucome of Uni Roo Tes a he 5% Significance Level ADF Tes DF-GLS Tes Counry w/o Time Trend w/ Time Trend w/o Time Trend w/ Time Trend Ausralia FTR FTR FTR FTR Belgium FTR FTR FTR Rejec Canada FTR FTR FTR FTR Denmar FTR FTR FTR FTR Finland Rejec FTR Rejec FTR France FTR FTR FTR FTR Germany FTR FTR FTR FTR Ialy FTR FTR Rejec FTR Japan FTR FTR FTR FTR Neherlands FTR FTR FTR FTR Norway FTR FTR FTR FTR Porugal FTR FTR FTR FTR Spain FTR FTR FTR FTR Sweden FTR FTR FTR Rejec Swizerland FTR FTR FTR FTR UK FTR FTR FTR FTR No Long Run PPP 5/6 6/6 4/6 4/6 The oucome of he es, rejecion or failure o rejec, is deermined using 5% criical values. 4