More Powerful Unit Root Tests and the PPP Puzzle

Transcription

1 More Powerful Uni Roo Tess and he PPP Puzzle Claude Lopez a Chrisian J. Murray b David H. Papell b a Universiy of Cincinnai, b Universiy of Houson Ocober 005 Absrac Using median-unbiased esimaion, recen research has quesioned he validiy of Rogoff s remarkable consensus of 3-5 year half-lives of deviaions from PPP. These esimaes, however, are based on uni roo ess wih low power. We exend medianunbiased esimaion o he efficien univariae uni roo es of Ellio, Rohenberg, and Sock (996). We find ha median-unbiased esimaion based on he more powerful uni roo es has he poenial o ighen confidence inervals for half-lives. Using long horizon real exchange rae daa, we find ha he ypical lower bound of he confidence inervals for median-unbiased half-lives is above 3 years. Thus, while previous confidence inervals for half-lives are consisen wih virually anyhing, our igher confidence inervals now rule ou economic models wih nominal rigidiies as candidaes for explaining he observed behavior of real exchange raes. Therefore, while we obain more informaion using efficien uni roo ess on longer erm daa, his informaion moves us away from solving he PPP puzzle. 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 We hank Luz Kilian, Barbara Rossi, and seminar paricipans a Duke Universiy, Texas Economerics Camp 8, and he Souhern Economic Associaion meeings for helpful commens and discussions. Papell hanks he Naional Science Foundaion for financial suppor.

2 . Inroducion During he pas decade, a number of sudies using long-horizon daa have changed he focus of research on Purchasing Power Pariy (PPP) from he narrow quesion of wheher or no he real exchange rae conains a uni roo o he broader quesion of he persisence of deviaions from PPP. Abuaf and Jorion (990), Diebold, Hused and Rush (99), Glen (99), Cheung and Lai (994), and Lohian and Taylor (996) all reach he same conclusion: he hypohesis of a uni roo in real exchange raes can be rejeced and he half-life of he PPP deviaions varies beween 3 and 5 years. In his well-known survey, Rogoff (996) discusses he remarkable consensus of hese half-lives and coins he phrase purchasing power pariy puzzle o describe he difficuly in reconciling hese slow speeds of adjusmen wih he high shor-run volailiy of real exchange raes. The slow speed of adjusmen is problemaic for models wih nominal rigidiies which predic faser convergence o PPP of o year half-lives. Alhough he 3 o 5 year consensus has become he common saring poin in aemps o solve he PPP puzzle, he consensus iself is problemaic. The sudies cied above generally calculae leas squares poin esimaes of he half-lives from firs order auoregressive processes. Poin esimaes alone do no provide a complee measure of persisence. Cheung and Lai (000) supplemen poin esimaes wih convenional boosrap confidence inervals in order o measure he precision of he half-life esimaes. Their confidence inervals, however, are no valid under he uni roo null and, even if long run PPP holds, are biased downwards in small samples. In addiion, he leas squares esimaes of half-lives are biased downward, providing an inaccurae picure of he speed of adjusmen o PPP. Two recen papers address hese issues using classical esimaion echniques. 3 Murray and Papell (00) use he median-unbiased esimaion mehods of Andrews (993) and he approximaely median-unbiased mehods of Andrews and Chen (994) for Dickey-Fuller (DF) and Augmened Dickey-Fuller (ADF) regressions respecively. They calculae poin esimaes and confidence inervals for half-lives of PPP deviaions for Engel (000) raises he quesion of wheher hese rejecions are caused by size disorions. See Kilian (998), Hansen (999), Kilian (999), and Inoue and Kilian (00) for furher discussion of boosrapping auoregressive processes wih uni roos or near uni roos. 3 See Kilian and Zha (00) for a Bayesian perspecive.

3 long-horizon annual ( ) US dollar real exchange rae daa for 6 counries and pos-973 quarerly US dollar real exchange rae daa for 0 counries. Rossi (005) uses he ess of Ellio and Sock (00) and Hansen (999) o calculae bias-correced confidence inervals for half-lives of PPP deviaions for pos-973 quarerly US dollar real exchange rae daa for 7 counries. Despie he differences in mehodology, he resuls for he pos-973 daa in he wo papers are nearly idenical. The lower bounds of he 95% confidence inervals are mosly jus above one year, while he upper bounds are generally infinie. These resuls, however, do no help solve he PPP puzzle. While he lower bounds are consisen wih relaively fas convergence o PPP as prediced by models wih nominal rigidiies, he upper bounds are consisen wih a uni roo in real exchange raes and no convergence o PPP even in he very long run. 4 These resuls indicae ha univariae mehods are unlikely o be informaive abou he persisence of pos-973 real exchange raes. 5 Focusing on pos-973 raes, moreover, ignores mos of he available daa. While long-horizon daa mixes fixed and flexible nominal exchange rae regimes and, herefore, canno answer he quesion of wheher PPP would hold wih a cenury long flexible nominal exchange rae regime, i can poenially answer he quesion of wheher PPP has held over he las cenury. This poenial has been grealy faciliaed by he work of Taylor (00), who develops real exchange rae daa for over 00 years for 0 counries. An imporan conribuion of Taylor s work is ha, for he firs ime, i is possible o invesigae real exchange rae persisence using long-horizon daa wih approximaely he same se of advanced counries as is commonly used in sudies wih pos-973 daa. The purpose of his paper is o improve inference on he persisence of PPP deviaions in long-horizon real exchange raes. We exend he mehodology developed by Andrews (993) and Andrews and Chen (994) for Augmened Dickey-Fuller ess o he more powerful DF-GLS es of Ellio, Rohenberg, and Sock (996). We compue median- 4 Taylor (00) and Imbs e.al. (005) invesigae ime aggregaion and secoral heergeneiy bias, respecively, alhough heir resuls remain conroversial. We rea he measured real exchange rae as he objec of ineres, and hus do no consider hese poenial sources of bias. 5 Panel mehods have been used exensively o es for uni roos in pos-973 real exchange raes. Murray and Papell (005) and Mark, Choi and Sul (005) examine real exchange rae persisence wih panel mehods. Ellio and Pesaveno (005) use univariae uni roo ess wih saionary covariaes o invesigae PPP in pos 973 daa. These covariae augmened uni roo ess have no ye been exended o calculae unbiased half-life esimaes.

4 unbiased and approximaely median-unbiased poin esimaes and confidence inervals for half-lives of PPP deviaions for DF-GLS regressions. We use Taylor s (00) daa for 6 annual US dollar real exchange raes for developed counries wih over a cenury of daa for each counry. To our knowledge, his is he firs paper which correcs for median-bias in DF-GLS regressions. Andrews (993) shows how o calculae median-unbiased poin esimaes and confidence inervals for half-lives in DF regressions, and abulaes he bias for a range of parameer values and sample sizes. We conduc a similar abulaion for DF-GLS regressions. We find ha, while he esimaes from DF-GLS regressions are biased downwards, he exen of he bias is much less han in Dickey-Fuller regressions. In addiion, he confidence inervals for median-unbiased esimaors are igher for DF-GLS regressions han for ADF regressions. This demonsraes he poenial for sharper inference on he persisence of shocks o he real exchange rae han has been previously available. We proceed o calculae median-unbiased poin esimaes and confidence inervals for half-lives of PPP deviaions from he 6 long-horizon real exchange raes. The poin esimaes of he half-lives are considerably larger han would be expeced based on Rogoff s 3-5 year consensus. The median value (among he 6 raes) is.34 years for he DF-GLS regressions and 7.55 years for he ADF regressions. The upper bounds of he 95% confidence inervals are infinie in a number of cases. The median lower bounds of he 95% confidence inervals are 3.0 years for he DF-GLS regressions and 3.5 years for he ADF regressions. The major resul in boh Murray and Papell (00) and Rossi (005) is ha, for boh annual long-horizon and quarerly pos 973 real exchange raes, he confidence inervals of he half lives were so wide as o be consisen wih virually anyhing. We find a very differen resul here. Wih boh DF-GLS and ADF regressions, he median lower bounds of he 95% confidence inervals are over hree years. Since he half-lives ha would be prediced from models wih nominal rigidiies are generally o years, our resuls are clearly inconsisen wih he predicions from such models. Therefore, while we obain 3

5 greaer informaion abou he persisence of shocks o he real exchange rae, he PPP puzzle becomes even more problemaic.. Median-Unbiased Esimaion in DF-GLS Regressions. Murray and Papell (00) use he median-unbiased echniques of Andrews (993) and Andrews and Chen (994) o compue poin esimaes and confidence inervals for PPP half-lives. Since hese esimaes are based on ADF regressions, hey do no opimally exploi he sample informaion in erms of power. We propose an exension of he Andrews (993) and Andrews and Chen (994) mehodology o he DF-GLS regression. The objecive here is o obain igher confidence inervals han hose of Murray and Papell (00) o poenially shed more ligh on he PPP puzzle. The exension of median-unbiased esimaion o DF-GLS regressions is sraighforward. Andrews (993) exacly median-unbiased esimaor is based on DF regression q α + u, () = c + q whereas he Andrews and Chen (994) approximaely median-unbiased esimaor is based on he ADF regression k + ψ i i= q = c + α q q i + u, () where k lagged differences are included o accoun for serial correlaion. 6 Insead of working wih he daa in levels as in he ADF regressions, we simply work wih he GLS demeaned (or derended) daa in he auxiliary DF-GLS regression where k µ µ = q + i= q α ψ q + u, (3) µ q is he GLS demeaned real exchange rae. Tha is, ( ~ ) ~ = z ~ zq~ i µ i q µ ~ q βz =, where z =, β, q~ ( q,( q α q ),...,( q αq )), ~ z = (,( α),...,( α) ), = T T α = + c /T, and c = 7. Since deerminisic erms have been removed by GLS 6 The regression wih only a consan and a lagged dependen variable is Case in Andrews (993). Cases and 3 have no deerminisic regressors, and a consan and ime rend respecively. Since we are ineresed in he sric inerpreaion of PPP, for our purposes Case is appropriae. 4

6 demeaning, none are presen in he above regression. 7 When k = 0, as in Andrews (993), he median-unbiased esimaor is exac, and when k > 0, as in Andrews and Chen (994), he median-unbiased esimaor is approximae. 8. Exacly Median-Unbiased Esimaion We compue our exacly median-unbiased esimaor for equaion (3) wih k = 0 for he sample sizes considered by Andrews (993). We also repor 90% confidence inervals. Specifically, for each value of α, we generae 0 5 AR() processes wih iid Gaussian innovaions. To find he median-unbiased esimaor, we find he value of α such ha he median of he leas squares esimaor is equal o he leas squares esimae. For example, if he leas squares esimae of α is 0.95, and T + = 5, he medianunbiased esimae of α based on he DF-GLS regression is A similar exercise leads o he consrucion of confidence inervals. Our esimaor is repored in he firs row of Table, and Andrews esimaor, based on equaion (), is repored in he second row of Table. 9 The median-unbiased esimaor of α in he AR() case in only exac if he disribuion of he innovaions is correcly specified. If he errors are non Gaussian, which hey are likely o be in mos economic ime series, hen he above procedure will no produce exacly median-unbiased esimaes. However, Andrews (993) demonsraes ha he median-unbiased esimaor is quie robus o deparures from Normaliy. Specifically, if he error erms are skewed and kuroic, bu have finie variance, hen he approximaion error resuling from incorrecly assuming Gaussian errors is quie small. Two feaures of Table are imporan o highligh here. Firs, while median-bias is presen in he leas squares esimaor of α in DF-GLS regressions, i is no as severe as he bias in ADF regressions. This accords wih inuiion since bias worsens as he number of deerminisic regressors increases. The auxiliary DF-GLS regression conains 7 Again, since we are ineresed in he sric inerpreaion of PPP, we do no allow for deerminisic ime rends, alhough doing so is sraighforward. 8 See Andrews and Chen (994) and Murray and Papell (00) for furher deails concerning he compuaion of approximaely median-unbiased esimaors. 9 While our subsequen empirical applicaion repors 95% confidence inervals, we repor 90% confidence inervals in Table in order o direcly compare our esimaor o Andrews esimaor, for which he does no repor 95% confidence inervals. 5

7 no deerminisic erms, while he ADF regression conains a consan. Second, he confidence inervals from he DF-GLS regressions are igher han from he ADF regressions. Uniformly, he lower bounds of he confidence inervals for he medianunbiased esimaor of α are higher in he DF-GLS regressions han in he ADF regressions. Similarly, wih only a few excepions when T + = 40, he upper bounds from he DF-GLS regressions are higher han from he ADF regressions. Even hough boh he upper are lower bounds are higher, he confidence inervals are uniformly igher in he DF-GLS case. This derives from he greaer power of he DF-GLS es, and demonsraes he poenial o exrac more informaion on he persisence of shocks o real exchange raes han has been previously available.. Approximaely Median-Unbiased Esimaion When k > 0, even if he disribuion of he innovaions is correcly specified, he median-unbiased esimaor is no longer exac, bu approximae. In addiion, he half-life, which is based on he impulse response funcion, is a nonlinear ransformaion of an approximaely median-unbiased esimae, and is herefore biased. In his subsecion, we conduc a simulaion sudy of he half-life esimae o deermine how our proposed halflife esimaor performs relaive o ha of Andrews and Chen (994), in erms of bias and precision. We consider four values of α : 0.85, 0.90, 0.95, and. For each value of α, we generae muliple parameerizaions, eiher nd or 3 rd order auoregressions. The rue halflives of all he parameerizaions we consider range from 3.3 years o infiniy. For each process, we compue he approximaely median-unbiased esimae of he half-life, as well as he 95% confidence inerval, using our proposed mehodology, as well as ha of Andrews and Chen (994). In he simulaions, he value of k is se o is rue value a each ieraion, alhough in pracice i would have o be esimaed. The resuls are repored in Table. There are wo main feaures of Table worh noing. Firs, alhough no severe, our esimae of he half-life is downward biased for every daa generaing process we consider. In addiion, he bias we find in our esimaor is greaer han he bias of he Andrews and Chen (994) esimaor. While our esimaor is arguably ouperformed by 6

8 he Andrews and Chen esimaor in erms of poin esimaes of he half-life, i pains a more precise picure of he persisence of shocks o he real exchange rae. Our 95% confidence inervals are igher in every case. Our lower bounds of he half-life are always higher, and excep for he case where he rue half-life is infiniy, our upper bounds are always lower. The confidence inerval of he half-life is arguably more imporan han he poin esimae when one is rying o compare he persisence of shocks o he exchange rae wih he predicions from economic models. 0 Our proposed mehodology leads o noably igher confidence inervals han hose compued from he Andrews and Chen (994) mehodology, and demonsraes he abiliy o gain more informaion regarding he PPP puzzle when he median-unbiased esimaor is only approximae. 3. Empirical Resuls: The Persisence of Shocks o he Real Exchange Rae Taylor (00) collecs nominal exchange rae and price level daa hrough 996 for 0 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, Denmark, Finland, France, Germany, Ialy, Japan, he Neherlands, Norway, Porugal, Spain, Sweden, Swizerland, and he Unied Kingdom. The daa begin as early as 870, and exac saring daes for each real exchange rae are provided in Table 3. We compue median-unbiased esimaes of half-lives, and 95% confidence inervals, for our 6 dollar denominaed real exchange raes. The half-life is defined as he number of years required for a uni shock o dissipae by one-half, and is based direcly on he impulse response funcion for each real exchange rae. In Table 3, we repor half-life esimaes from DF-GLS regressions where he lag lengh has been chosen by he Modified Akaike informaion crierion (MAIC) of Ng and Perron (00). The medianunbiased esimaes are exac when k = 0, and are approximae when k > 0. 0 Using a differen mehodology, Rossi (005) only repors confidence inervals for half-lives. 7

9 The poin esimaes of he half-lives in Table 3 are larger han wha has been previously repored in he lieraure. The median poin esimae is.34 years, wih of he 6 half-lives lying ouside Rogoff s (996) 3-5 year inerval. This srenghens Murray and Papell s (00) conclusion ha he lieraure surveyed by Rogoff (996) does no accuraely represen he behavior of real exchange raes. Furhermore, he 95% confidence inervals pain a much differen picure of he persisence of deviaions from PPP, vis-à-vis models wih nominal rigidiies. The median confidence inerval for halflives of PPP deviaions is [3.0, ) years and, wih he excepion of he US/Finland real exchange rae, every lower bound is greaer han years. We would like o know wheher he larger poin esimaes and lower bounds of he confidence inervals ha we repor (compared wih previous work) are solely caused by differences in echniques, or if differences in he daa also play a role. The half-lives for long-horizon daa in Murray and Papell (00) are compued from ADF regressions using GS lag selecion, using daa for six counries from To remove one of hese differences, we have also compued median-unbiased half-lives from Taylor s (00) long erm daa se, based on ADF regressions wih GS lag selecion. These are repored in Table 4. The poin esimaes of he half-lives from ADF regressions in Table 4 are also larger han wha has been previously repored in he lieraure. The median poin esimae is 7.55 years, wih of he 6 half-lives lying ouside Rogoff s (996) 3-5 year inerval. As wih he DF-GLS regressions in Table 3, he 95% confidence inervals pain a much differen picure of he persisence of deviaions from PPP han models wih nominal rigidiies. The median confidence inerval for half-lives of PPP deviaions is [3.5, 30.] years and, again wih he excepion of he US/Finland real exchange rae, every lower bound is greaer han years. The differences in he poin esimaes and confidence inervals beween Table 4 and Murray and Papell (00) are no caused by he inclusion of addiional counries. For he six counries: Canada, France, Ialy, Japan, he Neherlands, and he Unied Kingdom, Since we have an even (6) number of real exchange raes, we repor he median as he average of he eighh and ninh larges values. For he upper bound of he 95% confidence inervals, his is he average of 7.86 and infiniy, which we repor as infiniy. 8

10 ha are included in boh sudies, he median poin esimae is 7.36 years and he median confidence inerval is [3.74, 37.68] years, compared o 3.98 years and [.69,.75] years in Murray and Papell (00). The wo mos imporan findings from Tables 3 and 4 are ha he median poin esimae is well above Rogoff s 3-5 year consensus and he median lower bound of he 95% confidence inerval is well above wha would be consisen wih models based on nominal rigidiies. These hold equally well for he six-counry subse. Anoher poenial explanaion for he differences wih previous work is he ime span of he daa. Taylor s (00) begins in 870 or 880 for mos counries, and ends in 996. We exend he end o 998. In Murray and Papell (00), we use daa from Lee (976) ha begins in 900 and ends (wih our augmenaion) in 996. In order o see wheher he larger half-lives are caused by he longer span of daa, we compued (bu do no repor) median-unbiased poin esimaes and confidence inervals, using ADF regressions, for he six-counry subse using Taylor s daa from The resuls are quie similar o he resuls for he same six-counry subse wih he full ime span of daa. Having ruled ou differences in echnique and differences in ime span as explanaions for he differences beween hese and previous resuls, we are lef wih differences in daa. Mos of he exisen work on persisence of PPP deviaions using long-horizon daa, including Abuaf and Jorion (990), Glen (99), Lohian and Taylor (996), and Murray and Papell (00), uses real exchange raes compued from nominal exchange raes and wholesale price indexes. Taylor (00) compues real exchange raes using consumer price indexes. Since he consumer price index conains a larger componen of non-raded goods han he wholesale price index, i is no surprising ha he persisence of PPP deviaions is larger. Since mos sudies of PPP using pos-973 floaing exchange raes use consumer price indexes, his is anoher dimension where Taylor s daa provides greaer comparabiliy beween long-horizon and pos-973 daa han was previously available. Given ha half-life confidence inervals based on he DF-GLS regression are shown in Table o be uniformly narrower han hose based on he ADF regression, i may seem puzzling ha he confidence inervals in Table 3 are no all igher han hose in Table 4. However, his is caused enirely by MAIC and GS choosing differen lag lenghs. In he 9

11 five cases where MAIC and GS choose he same lag lengh, he half-life confidence inervals based on our esimaor are igher han hose based on Andrews and Chen s (994) esimaor. We have focused on he differences in he confidence inervals beween he medianunbiased ADF and DF-GLS regressions. Boh ses of confidence inervals, however, are narrower han wha currenly exiss in he lieraure, and he message from Tables 3 and 4 is he same. Using he larges available daase, we are unable o reconcile he predicions of exchange rae models wih nominal rigidiies wih he behavior of real exchange raes. Therefore, while igher confidence inervals ranslae o more informaion abou he persisence of deviaions from PPP, his increase in informaion moves us away from solving he PPP puzzle. 4. Conclusion Rogoff s (996) remarkable consensus of 3-5 year half-lives of PPP deviaions was based on sudies using biased esimaes ha underesimae he magniude of he PPP puzzle. Subsequen work using daa for indusrialized counries from he pos-973 flexible exchange rae period has obained ambiguous conclusions. In Murray and Papell (00) and Rossi (005), he confidence inervals for half-lives are so wide ha hey are consisen wih virually anyhing. They range from a speed of reversion o PPP ha is prediced by models wih nominal rigidiies (half-lives beween and years) o he failure of PPP o hold in he long run (infinie half-lives). Murray and Papell (00) also examine long-horizon daa for six counries, and repor similar resuls. In his paper, we invesigae he purchasing power pariy puzzle for Taylor s longhorizon daa using more powerful echniques. We exend he median-unbiased esimaion mehodology developed by Andrews (993) and Andrews and Chen (994) o he efficien DF-GLS es of Ellio, Rohenberg, and Sock (996), and repor boh poin esimaes and confidence inervals. We show ha he more powerful DF-GLS es can produce igher confidence inervals han he more widely used ADF es. Rogoff (996) argues ha he combinaion of high shor-run real exchange rae volailiy and glacial speeds of mean reversion produce he PPP puzzle. Using he bes available daa and esimaion echniques, we find half-lives of PPP deviaions o be much 0

12 larger han his 3 5 year consensus. Anoher conribuion of our work is o augmen he informaion conveyed by poin esimaes wih confidence inervals. In our earlier work using long-horizon daa, as well as in work using pos-973 daa, median-unbiased confidence inervals for PPP deviaions were oo wide o be informaive. In his paper we see somehing much differen. Similar o previous work, he upper bounds of he confidence inervals are so high ha we canno rule ou he failure of PPP o hold in he long run. In conras o previous work, however, he lower bounds are also so high ha we can rule ou consisency wih models based on nominal rigidiies. While our quaniaive resuls are very differen from hose repored by Rogoff, our conclusions are in some respecs very similar. Using more powerful and more complee echniques wih beer daa moves us furher away from solving he PPP puzzle.

13 References Abuaf, N. and P. Jorion, 990, Purchasing power pariy in he long run, Journal of Finance 45, Andrews, D.W.K, 993, Exacly Median-Unbiased Esimaion of Firs Order Auoregressive/Uni Roo Models, Economerica 6: Andrews, D.W.K., and H.-Y. Chen, 994, Approximaely Median-Unbiased Esimaion of Auoregressive Models, Journal of Business and Economic Saisics : Cheung, Y.-W., and K.S. Lai, 994, Mean Reversion in Real Exchange Raes, Economics Leers 46, Cheung, Y.-W. and K.S. Lai, 000, On he Purchasing Power Pariy Puzzle, Journal of Inernaional Economics 5: Choi, C.Y., Mark, N., and D. Sul, 005, Unbiased Esimaion of he Half-Life o PPP Convergence in Panel Daa, forhcoming, Journal of Money, Credi and Banking. Diebold, F., Hused, S., and Rush, M. 99, Real Exchange Raes Under he Gold Sandard, Journal of Poliical Economy 99, 5-7. Ellio G, and Sock, J.H., Confidence Inervals for Auoregressive Coefficiens Near One, Journal of Economerics 03: Ellio, G. and Pesaveno, E., 005, On he Failure of PPP for Bilaeral Exchange Raes Afer 973, forhcoming, Journal of Money, Credi and Banking. Ellio, G., Rohenberg, T., and Sock, J.H., 996, Efficien Tess for an Auoregressive Uni Roo, Economerica 64: Engel, C.M., 000, Long-Run PPP May No Hold Afer All, Journal of Inernaional Economics 5: Frankel, J., 986, Inernaional Capial Mobiliy and Crowding Ou in he U.S. Economy: Imperfec Inegraion of Financial Markes or of Goods Markes? in R. Hafer, ed., How Open is he U.S. Economy?, Lexingon Books. Glen, J.H., 99, Real Exchange Raes in he Shor, Medium, and Long Run, Journal of Inernaional Economics 33, Hansen, B., 999, The Grid Boosrap and he Auoregressive Model, The Review of Economics and Saisics 8,

14 Imbs, J., Mumaz, H., Ravin, M. and H. Rey, 005, PPP Srikes Back: Aggregaion and he Real Exchange Rae, Quarerly Journal of Economics 0, -43. Inoue, A, and L. Kilian, 00, Boosrapping Auoregressive Processes wih Possible Uni Roos, Economerica 70: Kilian, L., 998, Small-Sample Confidence Inervals for Impulse Response Funcions, The Review of Economics and Saisics 80, Kilian, L., 999, Finie-Sample Properies of Percenile and Percenile- Boosrap Confidence Inervals for Impulse Responses, The Review of Economics and Saisics 8, Kilian, L., and T. Zha, 00, Quanifying he Uncerainy abou he Half-Life of Deviaions from PPP, Journal of Applied Economerics 7:07-5. Lohian, J., and M. Taylor, 996, Real Exchange Rae Behavior: The Recen Floa from he Perspecive of he Pas Two Cenuries. Journal of Poliical Economy 04: Murray, C.J., and D.H. Papell, 00, The Purchasing Power Pariy Persisence Paradigm, Journal of Inernaional Economics 56: -9. Murray, C.J., and D.H. Papell, 005, Do Panels Help Solve he PPP Puzzle?, Journal of Business and Economics Saisics 3: Ng. S. and P. Perron, 00 Lag Lengh Selecion and he Consrucion of Uni Roo Tess wih Good Size and Power. Economerica 69: Rogoff, K., 996, The Purchasing Power Pariy Puzzle, Journal of Economic Lieraure 34: Rossi, B., 005, Confidence Inervals for Half-Life Deviaions from Purchasing Power Pariy, Journal of Business and Economic Saisics 3: Taylor, A.M., 00, Poenial Pifalls for he Purchasing-Power-Pariy Puzzle? Sampling and Specificaion Biases in Mean-Reversion Tess of he Law of One Price, Economerica 69, Taylor, A.M., 00, A Cenury of Purchasing Power Pariy, The Review of Economics and Saisics 84,

15 Table. Exacly Median-Unbiased Esimaors DF-GLS Exacly Median-Unbiased Esimaor T+=40 T+=50 T+=60 T+=70 α/quanile α/quanile α/quanile α/quanile Andrews (993) OLS Exacly Median-Unbiased Esimaor T+=40 T+=50 T+=60 T+=70 α/quanile α/quanile α/quanile α/quanile

16 Table. Exacly Median-Unbiased Esimaors, Coninued DF-GLS Exacly Median-unbiased Esimaor T+=80 T+=90 T+=00 T+=5 α/quanile α/quanile α/quanile α/quanile Andrews (993) OLS Exacly Median-Unbiased Esimaor T+=80 T+=90 T+=00 T+=5 α/quanile α/quanile α/quanile α/quanile

17 Table. Exacly Median-Unbiased Esimaors, Coninued DF-GLS Exacly Median-Unbiased Esimaor T+=50 T+=00 α/quanile α/quanile Andrews (993) OLS Exacly Median-Unbiased Esimaor T+=50 T+=00 α/quanile α/quanile

18 Table. Relaive Performance of Approximaely Median-Unbiased Half-Life Esimaes Based on DF-GLS and ADF Regressions α = True Median-Unbiased Esimaes Half-Life DF-GLS 95% CI ADF 95% CI =.5, = [.0, ) [9., ) =.50, = [5.3, ) [.64, ) = 0.80, = [5.3, ) [3.97, ) = 0.60, = [4.03, ) [.74, ) α = 0.95 =.5, = [5.40, 36.64] 4.64 [5.8, ) =.50, = [6.90, 34.44] 3.49 [6.87, 68.34] = 0.80, = [3., 60.0].05 [.95, ) = 0.60, = [.49, 5.97] 0.5 [.4, ) α = 0.90 =.5, = [4., 4.50] 7.37 [4.0, 8.8] =.55, = 0. 85, [3.83,.8] 6.70 [3.74, 7.6] 3 = 0.0 = 0.60, = [.06,.77] 4.94 [.0, 58.8] α = 0.85 =.5, = [3.34, 8.4] 6. [3.35, 9.4] =.55, = 0. 85, [3.6, 7.35] 4.93 [3.64, 8.49] 3 = 0.5 = 0.60, = [0.86, 7.50] 3.4 [0.85, 0.96] 4

19 Table 3. Median-Unbiased Half-Lives in DF-GLS Regressions Counry Sample k MAIC α LS α MU 95% CI HL MU 95% CI Ausralia [0.85,.0] 8.3 [4.7, ) Belgium [0.77, 0.98] 3.73 [.37,.67] Canada [0.93,.0] 34.3 [9.55, ) Denmark [0.85,.0].4 [3.0, ) Finland [0.58, 0.84].0 [.7, 3.98] France [0.89,.0].89 [.56, ) Germany [0.89,.0] 4.7 [5.4, 7.86] Ialy [0.73, 0.94] 3.7 [.0,.0] Japan [0.99,.0] [0.50, ) Neherlands [0.87,.0] 0.6 [4.4, 34.9] Norway [0.9,.0]. [4., ) Porugal [0.88,.0] 7.68 [.87, ) Spain [0.85,.0] 9.36 [3.03, 35.07] Sweden [0.83, 0.99] 7.46 [.76,.4] Swizerland [0.95,.0] 7. [7.37, ) UK [0.78, 0.99] 3.9 [.86,.55] 5

20 Table 4. Median-Unbiased Half-Lives in ADF Regressions Counry Sample k GS α LS α 95% CI MU HL 95% CI MU Ausralia [0.84,.0] 8.8 [3.86, 36.00] Belgium [0.70, 0.90] 3.78 [.44, 6.67] Canada [0.88,.0] 6.98 [5.4, ) Denmark [0.88,.0] 4.96 [.9, ) Finland [0.46, 0.74]. [.57,.87] France [0.77, 0.96] 5.60 [.96, 3.89] Germany [0.86,.0] 0.44 [5.00, 7.00] Ialy [0.67, 0.86] 3.76 [.4, 5.69] Japan [0.97,.0] [9.0, ) Neherlands [0.85,.0] 9.3 [4.5, 4.4] Norway [0.8, 0.97] 6.78 [3.88, 4.6] Porugal [0.8,.0] 8.3 [.76, ) Spain [0.8, 0.98] 6.70 [3.54, 8.] Sweden [0.75, 0.95] 4.95 [.89, 0.4] Swizerland [0.9,.0] 7.64 [4.94, ) UK [0.76,.0] 4.0 [.9, 50.94] 6