Is the Bayesian Approach Necessarily Better than the Classical Approach in Unit-Root Test?

Transcription

1 Universiy of Connecicu Economics Working Papers Deparmen of Economics January 2002 Is he Bayesian Approach Necessarily Beer han he Classical Approach in Uni-Roo Tes? Francis W. Ahking Universiy of Connecicu Follow his and addiional works a: hp://digialcommons.uconn.edu/econ_wpapers Recommended Ciaion Ahking, Francis W., "Is he Bayesian Approach Necessarily Beer han he Classical Approach in Uni-Roo Tes?" (2002). Economics Working Papers hp://digialcommons.uconn.edu/econ_wpapers/200218

2 Deparmen of Economics Working Paper Series Is he Bayesian Approach Necessarily Beer han he Classical Approach in Uni-Roo Tes? Francis W. Ahking Universiy of Connecicu Working Paper February Mansfield Road, Uni 1063 Sorrs, CT Phone: (860) Fax: (860) hp://

3 Absrac Some researchers, for example, Koop (1992), and Sims (1988), advocaed for Bayesian alernaives o uni-roo esing over he classical approach using he augmened Dickey-Fuller es (ADF). This paper sudies wha Koop (1992) called he Objecive Bayesian approach o uni-roo esing. We apply he objecive Bayesian uni-roo es o a sudy of long-run purchasing power pariy (PPP) for he pos- Breon Woods era. While he classical approach using he ADF es canno rejec he uni-roo hypohesis, he Bayesian approach, on he oher hand, suggess ha he uni-roo hypohesis is no srongly suppored by he sample daa. Raher, he rend-saionary hypohesis receives he highes poserior probabiliy in all cases excep for he Japanese yen/german mark real exchange rae where he saionary hypohesis receives he highes poserior probabiliy. In wo Mone Carlo simulaions, however, we find ha he objecive Bayesian es have relaively low power in disinguishing beween plausible alernaives, making i difficul o draw any conclusions concerning long-run PPP. We conclude ha, a leas for he objecive Bayesian es, he Bayesian approach is no necessary beer han he classical ADF approach. Journal of Economic Lieraure Classificaion: C22, F31

4 Is he Bayesian Approach Necessarily Beer han he Classical Approach in Uni-Roo Tes? 1. Inroducion The quesion of how bes o characerize he growh componen of macroeconomic ime series has been holy debaed since he publicaion of he sudy by Nelson and Plosser (1982). The mos widely held view before 1982 was ha mos macroeconomic ime series could be characerized as rend-saionary (TS) series, i.e., he series consiss of a deerminisic rend and saionary flucuaions around ha rend. Nelson and Plosser s sudy called ino quesion ha widely held view by demonsraing ha hireen ou of foureen U.S. macroeconomic ime series ha hey considered conained a uni roo, and hus were difference-saionary (DS) series, i.e., hey required differencing o induce saionariy. Alhough he debae is far from seled, 1 he focus of he debae has shifed o he appropriae saisical echniques o use in esing beween TS and DS series. In paricular, several economiss have srongly advocaed Bayesian alernaives o uni-roo esing. We sudy one of he Bayesian alernaives in his paper by applying i o anoher conroversial area in Economics, ha of long run purchasing power pariy (PPP) in he pos-breon Woods era. Thus, we hope o shed some ligh on hese wo conroversial areas of Economics. We presen a general discussion of he Bayesian approach o uni-roo esing in he nex secion. In secion 3, we review some of he curren lieraure on esing for long-run PPP. In secion 4, we discuss our Bayesian uni-roo es which has been called objecive Bayesian es by Koop (1992). We presen our empirical resuls in secion 5. We sudy he properies of he objecive Bayesian uni-roo es using Mone Carlo simulaions in secion 6. Finally, our summary and conclusions are in secion The Bayesian Approach o Uni-Roo Tesing I should be noed a he onse ha he classical (e.g., he regression approach) and he Bayesian approaches o economerics are complemenary raher han compeing. There are, however, major differences beween he wo. We will sar by reviewing some of he major differences in his secion. In he classical approach, he daa are reaed as random, and he parameers o be esimaed as non-random. The Bayesian approach reas unknown quaniies, such as he parameers in a model, as random variables and he daa as non-random. In he classical approach, hypoheses are eiher rejeced or failed o be rejeced a a pre-chosen level of significance, say 5% or 10%. The Bayesian approach is based on Bayes Theorem, which may be saed as: f ( y θ ) g( θ ) g( θ y) =, (1) f ( y) 1

5 where θ is a vecor of parameers, y is a vecor of sample observaions, f (y) is he densiy funcion for y, f ( y θ ) is he join densiy funcion, i is algebraically idenical o he likelihood funcion for θ, and conains all he sample informaion abou θ. The funcion g(θ ) is he prior densiy funcion for θ ; i summarizes he nonsample informaion abou θ. Finally, g( θ y) is he poserior densiy funcion for θ, i summarizes all he informaion abou θ afer y has been observed. The modeling objecive of he Bayesian approach is no o rejec or fail o rejec a hypohesis, bu o deermine how probable a hypohesis is relaive o oher compeing hypoheses. There are several ways of comparing hypoheses using Bayesian mehods. The mos common mehod is o calculae poserior odds raios for various compeing hypoheses based on prior and sample informaion. This gives he researcher he odds in favor of one hypohesis relaive o oher compeing hypoheses. For example, he poserior odds raio for comparing he null hypohesis ( H o an alernaive hypohesis ( ) can be wrien as 1 ) H 2 P( H1) P( H1 y) K 12 =, (2) P( H ) P( H y) 2 2 where P H ) / P( ) is called he prior odds raio, and P( H y) = P( θ H ) L( θ y, H ) dθ, i = 1, 2, is ( 1 H 2 i i i i i i he poserior probabiliy ha H i, i = 1, 2, were rue given he sample daa y, and P θ i H ) and L( θ i y, H ) ( i i are he prior densiy and likelihood funcion, respecively, under each hypohesis. In uni-roo esing, he frequenly used classical approach is he augmened Dickey-Fuller (ADF, 1981) es, where a linear regression of he following ype is ypically esimaed by ordinary leas squares: y = δ 0 + ρ y k 1 + δ i y i + i= 1 ε, (3) whereδ is a consan, is he firs difference operaor, i.e., y = y y, and is a serially uncorrelaed 0 1 error process. The null hypohesis is H : ρˆ 1, agains he alernaive ha ˆρ < 1. Several sudies, e.g., Hakkio 0 = (1986), and DeJong, Nankervis, Savin, and Whieman (1992), have shown ha he ADF es has low power agains plausible alernaives, especially agains rend-saionary alernaive. Indeed, sudies using Bayesian approaches o uni-roo esing, e.g., DeJong and Whieman (1991a, 1991b), and Koop (1992), have generally found much weaker evidence in favor of uni-roo, and much sronger evidence in favor of rend-saionariy. This led researchers such 2 ε

6 as Koop (1992), DeJong and Whieman (1991), Sims (1988), and Sims and Uhlig (1991) o advocae forcefully for Bayesian alernaives over he more radiional classical approach such as he ADF ess in uni-roo esing. 2 These researchers poined ou several advanages of he Bayesian approach over he classical approach. For example, DeJong e al. (1992), Koop (1994), and ohers have noed ha given ha classical uni-roo ess have low power agains rend-saionary alernaive, a small finie daa se generaed from a uni-roo model may be virually indisinguishable from a small finie daa se generaed from a rend-saionary model wih srong persisence. Faced wih he rend-saionary model, classical esing procedures would fail o rejec he null hypohesis of a uni roo. The Bayesian approach, however, would reveal ha boh he uni roo and he rend-saionary hypoheses receive similar poserior probabiliies. Thus, he Bayesian approach provides a more reasonable summary of sample informaion han he classical approach. Anoher problem wih he classical uni-roo ess is he disconinuiy of he classical asympoic heory a ρˆ = 1 [see Sims (1988) especially for a good discussion of his poin]. For example, he sandard asympoic heory applies when ρ ˆ <1, and does no apply when ρˆ = 1, or ρˆ > 1, and Mone Carlo sudies are necessary o esablish he appropriae asympoic criical values. The Bayesian approach, on he oher hand, since i is based on he likelihood funcion, which is coninuous in ρ, does no have he same disconinuiy problem. Koop (1994) also poined ou ha, in he classical approach, frequenly researchers have o rely on he asympoic criical values, which could differ subsanially from he small sample criical values. The Bayesian approach, on he oher hand, since i is condiional on he observed sample, provides exac small sample resuls. Despie he apparen advanages of he Bayesian approach over he classical approach in uni-roo esing, only a relaively small bu growing number of sudies have appeared using he Bayesian approach. The reasons may be ha he Bayesian approach requires a likelihood funcion and he use of prior informaion. The percepion is ha oher han for he simples cases, he Bayesian approach is compuaionally burdensome, and he use of prior informaion is by far he mos conroversial. Frequenly in many applicaions, however, here is considerable agreemen among researchers. For example, in uni-roo esing, many researchers would agree ha he alernaive o ρ = 1is ha ρ is concenraed near one, as opposed o say ρ is near 2 or 0.1. Thus, a reasonable prior may be o allocae significan probabiliy in he region near and including one. An alernaive is o use noninformaive (or fla) priors. The one serious drawback, however, is ha hey are frequenly improper, i.e., hey do no inegrae o one. An improper prior poses problems for hypohesis esing bu no for esimaion and predicion, however. As 3

7 far as compuaional burden is concerned, boh Sims (1988) and Koop (1992) have shown relaively simple mehods o implemen a Bayesian alernaive o classical uni-roo ess. In summary, here is no consensus on he meris of he classical vs. he Bayesian approach o uni-roo esing. Bayesian researchers have pu forh a srong case o a leas consider he Bayesian alernaive o uni-roo esing. We suspec ha all researchers would agree ha he Bayesian approach provides a useful alernaive empirical approach. 3 We sudy wha Koop (1992) called he objecive Bayesian es in his paper and apply i o a sudy of long-run PPP using daa from he pos-breon Woods era. In he nex secion, we review briefly he conroversies surrounding ess for long-run PPP. 3. Tesing for Long-Run PPP The heory of PPP has a long radiion in inernaional economics. I is a cenral building block in he moneary models of exchange rae deerminaion, e.g., Frenkel (1978) and Dornbusch (1976). In is simples form, PPP can be expressed as a relaionship beween he nominal exchange rae, and relaive prices, or, in naural logarihm form: e = P P, (4) * where e is he naural logarihm of he nominal exchange rae, defined as he domesic currency price of one uni of P foreign currency, is he naural logarihm of an index of he domesic price level, and is he naural logarihm of an index of he foreign price level. Recen empirical ess of PPP have mainly focused on he long run given ha here are frequen large and persisence shor-run deviaions from PPP. Tess of long-run PPP based on equaion (4) have mosly used he echniques of coinegraion of Engle and Granger (1987), or Johansen (1988). 4 A weak es of long-run PPP is o deermine wheher or no a coinegraion vecor exiss. The exisence of a leas one coinegraion vecor would imply ha an equilibrium long-run relaionship exiss beween he nominal exchange rae and he relaive prices, so ha here is no permanen deviaion from PPP. In addiion o finding a coinegraion vecor, a sronger es of long-run PPP, however, would also require ha he wo properies implied by PPP -- (a) symmery beween domesic and foreign price levels, and (b) proporionaliy beween he relaive prices and he nominal exchange rae -- are also saisfied. Anoher approach o esing long-run PPP is o rewrie equaion (4) in is real exchange rae equivalen form by imposing boh he symmery and he proporionaliy properies. Tha is, in naural logarihm form: * P q = e + P * P, (5) 4

8 where q is he naural logarihm of he real exchange rae. If PPP holds coninuously, = 0 for all. This is unrealisic, however, given ha we observe frequen deviaions from PPP. The quesion herefore is wheher q deviaions from PPP are ransiory or permanen. If deviaions from PPP are ransiory, he ime series of q is a saionary series. On he oher hand, if deviaions from PPP are permanen, hen he ime series of q is nonsaionary and conains a uni-roo. Tess for uni-roos have mosly been carried ou using he ADF es using a regression such as Equaion (3). Long-run PPP requires ha ρˆ <1. If ρˆ = 1, here is a uni-roo in he series q, shocks o he real exchange rae are permanen and long-run PPP does no hold. Earlier empirical ess of long-run PPP using eiher equaion (4) or (5), or varians of hem, have produced mixed resuls, bu largely unfavorable o long-run PPP, especially when daa from he pos-breon Woods floaing period are used [see he survey by Rogoff (1996)]. Several explanaions have been offered for his failure of longrun PPP in he pos-breon Woods period. Firs, i is known ha when German mark is used as he base currency, researchers end o find more favorable resuls for long-run PPP han when he U.S. dollar is he base currency [see Papell (1997), and Papell and Theodoridis (1998) for recen examples]. This leads Lohian (1998) o conclude ha he frequen failures o find evidence in favor of long-run PPP in he pos-breon Woods floaing exchange rae period is no a generic problem o his period. Raher, i is confined o using he U.S. dollar as he base currency and is resriced o he early o mid-1980s ime period when firs here was a subsanial real appreciaion of he U.S. dollar for , and an almos equal offseing real depreciaion of he U.S. dollar for Second, since he empirical mehodologies generally used o es equaion (4) or (5) are known o have low power in small samples, i is no possible o disinguish wheher he failure o find coinegraion (using equaion 4), or he failure o rejec he uni-roo hypohesis (using equaion 5) is due o he low power of he ess employed or ha long-run PPP does no hold in he pos-breon Woods floaing period. One way o increase he power of he empirical ess is o use longer span of daa. For example, Diebold, Hused, and Rush (1991), 5 using daa going back o he gold sandard period, Lohian, and Taylor (1996), using daa daing back o he1790s and early 1800s, found evidence o suppor long-run PPP. On he oher hand, Engel and Kim (1999), using monhly daa daing back o 1885, found evidence of a permanen (i.e., a uni-roo) componen in he U.S./U.K. real exchange rae. In addiion, Rogoff (1996) and ohers have noed ha sudies ha used long spans of daa ypically mix fixed and floaing exchange raes daa, and he economic implicaions of mixing daa from he wo exchange rae regimes is unclear. Moreover, long spans of ime series daa may poenially conain serious 5

9 srucural breaks. Engel (1996) also argued ha hese sudies can have serious size biases, and may fail o rejec a sizable uni roo. Finally, hese sudies also do no shed much ligh on he quesion of wheher or no PPP is a valid hypohesis in he pos-breon Woods floaing period. Anoher way o increase he power of he coinegraion and uni-roo ess is o use panel daa. Recen examples using daa from he pos-breon Woods floaing exchange rae period include sudies by Jorion and Sweeney (1996), Papell (1997), Papell and Theodoridis (1998), Koedijk, Schoman, and Van Dijk (1998), and Heimonen (1999). These sudies all found evidence o suppor long-run PPP. On he oher hand, O Connell (1998) found lile evidence o suppor long-run PPP afer accouning for serial correlaion. Papell (1997) found ha evidence in favor of long-run PPP is dependen on he size of he panel and he counries included. Rogoff (1996) also noed ha wih panel sudies, he evidence of long-run PPP ends o be much more favorable when high inflaion counries are included. Finally, he inerpreaion of he panel sudies resuls is no always very obvious. For example, Karlsson and Löhgren (2000) using Mone Carlo simulaions, found ha for panels wih long spans of daa, he null hypohesis of uni roos can be erroneously rejeced even when only a small proporion of he series is saionary. For panels wih shor spans of daa, however, Karlsson and Löhgren found ha he null hypohesis of uni roos is frequenly no rejeced even when a large fracion of he series is saionary. Thus, hey concluded ha he rejecion or he non-rejecion of he null hypohesis of uni roos in panel uni-roo ess do no provide sufficien evidence o conclude ha all he series in he panels are saionary or ha hey all have a uni roo. An excepion, however, is he panel sudy by Sarno and Taylor (1998). Using a special applicaion of he Johansen (1988) Likelihood raio, where he null hypohesis of a uni roo is rejeced only when all he series are saionary, Sarno and Taylor concluded ha heir four real exchange raes are joinly saionary series. Thus, panel uni roo ess have produced encouraging bu inclusive resuls. Sill oher researchers, using differen empirical mehodologies, found mixed resuls wih daa from he pos-breon Woods period. For example, Cheung and Lai (1998), using more efficien uni-roo ess, found more encouraging resuls han when using he ADF ess. Culver and Papell (1999), using he ess proposed by Kwiakowski, Phillips, Schmid and Shin (1992) where saionariy is he null, raher han he alernaive hypohesis, found favorable evidence o suppor long-run PPP. However, a recen sudy by Caner and Kilian (1999) shown ha severe size disorion can resul wih he use of convenional asympoic criical values for ess of he null hypohesis of saionariy if he model under he null hypohesis is highly persisen. On he oher hand, using sizeadjused criical values can overcome he problem of size disorions, bu resul in low power of he ess for 6

10 economically plausible values of he firs-order auoregressive (i.e., AR (1)) parameer. Finally, Baum, Barkoulas, and Caglayan (1999), allowing for fracional differencing or srucural breaks, found no evidence o suppor longrun PPP. Our brief review suggess ha he recen empirical sudies have ended o be more supporive of long-run PPP han he earlier sudies. The resuls are no very robus, however. Moreover, even-hough here is a growing body of lieraure ha suppors long-run PPP in he pos-breon Woods period, consisen individual counry ime series evidence from he pos-breon Woods period coninues o be scarce. 4. Tesing Long-Run PPP wih he objecive Bayesian Approach We are aware of only hree sudies ha have used he Bayesian approach o es for long-run PPP. Whi (1992), using he Bayesian approach discussed in Sims (1988), found evidence o suppor long-run PPP using pos- Breon Woods monhly daa for five real exchange raes, and pos-world War II annual daa for six real exchange raes. Schoman and van Dijk (1991) exended Sims s (1988) resuls by inroducing a consan erm in he model and found mixed resuls using monhly daa for eigh real exchange raes from he pos-breon Woods period. They concluded ha for four of heir eigh real exchange rae series, he random-walk hypohesis received abou equal poserior probabiliies as he hypohesis ha he series are saionary auoregressive processes of order one (i.e., AR(1) processes). Ahking (1995), using a Bayesian approach discussed in Koop (1992), and using boh monhly and quarerly daa from Canada s floaing exchange rae period in he 1950s, found mixed resuls, bu generally no very supporive of long-run PPP. In his paper, we also use he Bayesian approach discussed in Koop (1992) o sudy long-run PPP using daa from he pos-breon Woods era. Since we are using only univariae ime series of he real exchange raes, we avoid he poenial problems associaed wih using panel daa. Furhermore, since our daa are from he pos-breon Woods period, we also avoid he criicisms of using long spans of daa ha mixed boh fixed and floaing exchange raes daa. Koop s (1992) Bayesian uni-roo es is based on he work of Zellner and Siow (1980). 6 There are several feaures of his approach ha make i aracive for economiss looking for an alernaive o he classical uni-roo ess. Firs, i is compuaionally simple. I does no require complicaed numerical inegraion, and hence advanced programming skills are no necessary. Second, i requires informaive priors, hus avoiding he problems associaed wih using non-informaive priors. Third, i does no require significan subjecive prior informaion, only ha all he compeing hypohesis have equal prior probabiliy, leading Koop (1992) o call his an objecive Bayesian es. 7

11 As we will argue below (see foonoe 7), however, even-hough we assign equal prior probabiliy o all he compeing hypoheses, here is a bias in favor of he uni-roo hypohesis. The objecive of he analysis is o find he linear model ha would bes describe he ime series of he real exchange rae, H1 H 2 H 3 : : : q = ( q1, q2,..., qt )'. We consider hree hypoheses: k q = α 0 + αiq i + αk ξ1 i= 1 k q = 2 i= 1, (6) α 0 + αiq i + ξ, ( α = 0 1 ) (7) k+ k k q = α 0 ( αi ) q 1 ( αi ) q 2 α k q ( k 1) + ξ3 i= 2 i= 3, (8) ( k i= 1 α i = 1, α k+ 1 = 0) where = a linear deerminisic ime rend; and ξ j, j =1, 2, 3 is a serially uncorrelaed error process wih zeromean and consan variance. Hypohesis 1 ( H 1 ) is he null model. I hypohesizes ha he real exchange rae is a rend-saionary AR(k) process. hypohesizes a saionary AR(k) process for he real exchange rae, while H hypohesizes an AR(k) process wih a uni roo. Noe ha boh and are special cases of H wih linear resricions (given in parenheses nex o he respecive equaions), imposed on he null model. The rend-saionary hypohesis is included because several researchers have found ha he sochasic processes of some of he real exchange raes canno be adequaely modeled wihou he inclusion of a linear deerminisic ime rend [see Cheung and Lai (1998), and Koedijk, Schoman, and Van Dijk (1998) for recen examples]. Since he ime series of he real exchange raes ypically show evidence of a rend, he rend-saionary hypohesis offers an alernaive o he uni-roo hypohesis as he source of he rend in he daa. The linear deerminisic ime rend in he real exchange rae is generally inerpreed as represening sysemaic differences in produciviy growh beween radable and non-radable goods in he wo counries [see also Cheung and Lai (1998) for a recen discussion]. Thus, he presence of a linear deerminisic ime rend in he real exchange rae ime series is generally no inerpreed as a violaion of long-run PPP. 7 H 2 3 H 2 H 3 1 8

12 odds raios: We compare he hree hypoheses, based on boh prior and sample informaion, by calculaing he poserior P( H1) P( H1 q) K1 j =, j = 2, 3. (9) P( H ) P( H q) j j On he assumpion ha all hree hypoheses have equal prior probabiliy, i.e., P H ) = P( H ) = P( ), equaion (9) reduces o 1 = j 8 ( 1 2 H 3 P( H1 q) K =, j 2, 3 j P( H q). (10) Tha is, he poserior odds raio gives he raio of he probabiliies of he wo hypoheses holding given he sample daa. Following Koop (1992), we calculae he poserior odds raio for esing a se of exac linear resricions wih a formula suggesed by Zellner and Siow (1980). The Zellner-Siow poserior odds raio is calculaed approximaely as 0.5 r / 2 ( π / Γ[( r + 1) / 2])( v / 2) K1 j, j = ( v 1) / 2 (1 + rf / v) 2, 3, (11) where Γ[ ] = he Gamma funcion, v = T n, T = he oal number of observaions, n = he number of regressors in he null model, r = he number of linear resricions esed, and F = he usual F-saisics for esing he se of linear resricions. Afer we obained he poserior odds raios, we can hen calculae he poserior probabiliy for each of he hree hypoheses using Equaion (10). We discuss our daa se and presen our empirical resuls in he nex secion. 5. Empirical Resuls The source of our daa is he OECD G-7 counries, supplied on a diskee. Our daa consis of monhly observaions from April 1973 o February 1999 for he G-7 counries, and are no seasonally adjused. The G-7 counries are he U.S., he U.K., Canada, Germany, Ialy, France, and Japan. In all cases, we use he consumer price index as our measure of he average price level. The only bilaeral nominal exchange rae available on he diskee uses he U.S. dollar as he base currency, i.e., foreign currency per U.S. dollar. We are, however, also ineresed in wheher he use of non-u.s. dollar based real exchange raes may produce differen resuls, as oher sudies have found. We herefore also compued real exchange raes based on he pound serling, he Canadian dollar, he 9

13 German mark, he Ialian lira, he France franc, and he Japanese yen. These non-u.s. dollar based exchange raes are compued as cross-raes. 9 We sar our empirical ess by firs presening in Table 1 our es for uni-roo using he ADF es. This is moivaed by wo facors. Firs, our resuls provide an updae on previous research hrough he beginning of 1999, and i is ineresing o find ou wheher an addiion of several more years of daa would have made a difference in he ADF ess for uni roo. Second, he ADF uni-roo es resuls will provide a comparison o our Bayesian approach o uni-roo esing. Noe also ha we provide uni-roo es resuls for fory-wo real exchange raes (six real exchange raes for each of he seven currencies). Of course, here are only weny one differen real exchange raes since he real exchange rae of counry A s currency per uni of counry B s currency is simply he inverse of he real exchange rae of counry B s currency per uni of counry A s currency. This is done so ha we can examine how he real exchange raes based on non-u.s. dollar would behave compared o he U.S. dollar based real exchange rae. The ADF regression acually esimaed is l q = β 0 + φq 1 + βi q i + βl ν, (12) i= 1 where = a linear deerminisic ime rend, and ν is a serially uncorrelaed error process wih zero mean and consan variance. The lag lengh for he lagged firs-differences is deermined by he shores lag lengh ha produces serially uncorrelaed residuals a he 10% significance level using he Ljung-Box Q-saisic compued over he firs hiry-six lags of he residuals. The lag lenghs deermined by his procedure are shown in column 2 of Table In column 3, we show he -saisic for he hypohesis H : φ 0. As can be seen, wih he excepion of he Japanese yen/german mark real exchange rae, he null hypohesis of a uni roo canno be rejeced a he 5% and he 10% significance levels using eiher he criical values from Fuller (1976) or he lag-adjused criical values for exac sample size from Cheung and Lai (1995). For he Japanese yen/german mark real exchange rae, he null hypohesis of a uni roo can be rejeced a he 10% level using eiher Fuller s or Cheung and Lai s criical values. This is consisen wih he resuls of Cheung and Lai (1998), bu hey also found ha he French franc/german mark real exchange rae could be characerized as a saionary process, a resul ha we do no obain. Thus, he addiion of a few more years of monhly daa appears o have no impac on he power of he ADF uni-roo es. Wha is more surprising, however, is he finding ha his is rue regardless of he base currency used, conrary o some earlier resuls. The German mark-based real exchange rae produces marginally beer resul, wih he rejecion of he uni =

14 roo hypohesis a he 10% significance level for he Japanese yen/german mark real exchange rae. This is a very marginal improvemen over he U.S. dollar-based real exchange raes, however. Moreover, using Pound serlingbased or Ialian lira-based real exchange raes produce equally dismal resuls as using U.S. dollar-based real exchange raes. Thus, our ADF resuls provide no suppor o Lohian s (1998) asserion ha he frequen failures o find favorable evidence of long-run PPP in earlier sudies for he pos-breon Woods period is confined o using U.S. dollar as he base currency. Column 4 shows he 2 χ es saisic for he join null hypohesis of a uni roo and he absence of a linear deerminisic ime rend, i.e., H : φ = = No surprisingly, his hypohesis is no rejeced in all cases 0 β l excep for he Japanese yen/german mark real exchange rae, which can be rejeced a he 10% significance level using he criical values from Dickey and Fuller (1981). Once again, his is rue for all base currencies considered. Thus, based on he ADF uni-roo es resuls presened in Table 1, he evidence agains long-run PPP appears o be quie overwhelming. We nex urn o our Bayesian es resuls presened in Table 2. In column 2 of Table 2 we show he auoregressive lag lengh, deermined using he same mehod as in he ADF ess. 11 The nex hree columns of Table 2 give he poserior probabiliies of he hree hypoheses. Wha we find sriking is ha, of he hree hypoheses, he rend-saionary hypohesis receives he highes poserior probabiliy in all cases excep for he Japanese yen/german mark real exchange rae. 12 In ha case, he saionary hypohesis receives he highes poserior probabiliy. The French franc/german mark real exchange rae also deserves menion because i is he only case where he rend-saionary and he saionary hypoheses receive approximaely he same poserior probabiliies. Ineresingly, hese are he same wo real exchange raes ha Cheung and Lai (1998) have found o be well characerized by saionary or rend-saionary processes using he convenional ADF ess. The uni roo hypohesis, on he oher hand, receives no significan poserior probabiliy. In sum, he Bayesian resuls srongly suppor he hypohesis ha he real exchange rae series are rend-saionary AR processes. Or, pu differenly, he resricions on he null model are no srongly suppored by he daa. Finally, we wish o also noe wo oher poins. Firs, as wih he ADF resuls, we find no compelling evidence o sugges ha non-u.s. dollar based real exchange raes produce more favorable evidence in suppor of long-run PPP han U.S. dollar based real exchange raes. In our Bayesian resuls, we see ha he French franc/german mark and he Japanese yen/german mark real exchange raes receive relaively high poserior probabiliy for he saionary hypohesis. In his respec, he use of he German mark as he base currency produces differen resuls. Second, he real exchange raes of he four European counries 11

15 only (ha is, he real exchange rae expressed as he currency of European counry A per uni of European counry B s currency), do no behave differenly from hose ha are based on he non-european currencies. Since he four European counries in our sample all paricipaed in he European Currency Uni (ECU), membership in he ECU does no appear o conribue o he behavior of he real exchange rae. 6. The power of he objecive Bayesian uni-roo es In his secion, we perform wo Mone Carlo experimens o sudy he power of he objecive Bayesian uni-roo es. This will allow us o assess he reliabiliy of our resuls in Secion 5, and also invesigae wheher he objecive Bayesian approach is beer han he classical ADF approach in uni-roo ess. In he firs experimen, we use a daa generaing model of he following form: y ρ y u, = 0 1 where ~ iid N(0, ), and ρ varies from 0.95 o 0.99, in incremen of Thus, he daa u generaing process is a rend-saionary AR(1) model. The consan, he ime-rend coefficien, and he variance of he error process are averages from esimaing he null model (Equaion 6) wih daa from he weny-one real exchange rae series. Thus, hey represen economically plausible parameers. The values of ρ ha we choose are also based on wha we consider o be plausible alernaives for monhly series. For example, Sims (1988) argued ha for monhly daa, i is reasonable o concenrae he prior odds on he inerval (0.98, 1) as opposed o, say (0.5, 1). Second, here appears o be a consensus among economiss ha found favorable evidence for long-run PPP in he pos-breon Woods era ha deviaions from long-run PPP have a half-life of abou hree o five years [see e.g., Abuaf and Jorion (1990), Rogoff (1996)]. 13 According o Caner and Kilian (1999), his corresponds o a ρ value of 0.98 and 0.99, respecively, for hree and five years, using monhly daa. In sum, we believe ha he parameer values ha we choose for our Mone Carlo experimen are economically plausible. The power of he objecive Bayesian uni-roo es in his experimen is defined as he proporion of imes ha he hypohesis ha corresponds o he daa generaing model receives he highes poserior probabiliy in repeaed samples. Some Bayesian researchers may objec o his definiion of power. I means, for example, ha if he rend-saionary hypohesis receives 40% of he poserior probabiliy and he uni-roo hypohesis receives 41% of he poserior probabiliy, all he credi will be given o he uni-roo hypohesis and none o he rend-saionary hypohesis, while Bayesian researchers would view he wo hypoheses as equally likely. We believe, however, ha if in fac i is no possible o disinguish beween a rend-saionary series from a uni-roo series, hen in repeaed 12

16 samples, he objecive Bayesian uni-roo es should allocae roughly he same proporion of imes ha he rendsaionary hypohesis and he uni-roo hypohesis receive he highes poserior probabiliy. We generae 411 observaions, and discard he firs 100 observaions o avoid he iniializaion problem. The remaining 311 observaions correspond o he sample size of our real exchange raes. The experimen is replicaed 5,000 imes for each ρ value. The resuls are repored in he upper panel of Table 3. The resuls reveal a surprising paern. For <ρ < 0. 98, he rend-saionary hypohesis receives he highes poserior probabiliy he larges proporion of he imes. The power of he es increases from ρ = 0.95 and peaks a ρ = A ρ = 0.99, however, he uni-roo hypohesis receives he highes poserior probabiliy he larges proporion of he imes, suggesing an exremely low power of he objecive Bayesian es a his value of ρ. Before drawing any conclusion, we urn nex o our second experimen. In our second experimen, we use a daa generaing model of he following form: y ρy + u, = 1 where has he same properies as he firs experimen. In his experimen, however, we vary ρ from 0.95 o u 1.00, in incremen of Thus, he daa generaing model is a saionary AR(1) model and a random-walk wih drif model. We use he same procedure o generae he daa and he experimen is also replicaed 5,000 imes for each ρ value. The resuls are repored in he lower panel of Table 3. Ineresingly, for <ρ< 0. 99, he resuls mirror he resuls repored in he upper panel of Table 3. This, perhaps, is no surprising since he effec of he ime rend is exremely small, hus making he wo daa generaing models almos he same. A ρ = 1. 00, he rend-saionary hypohesis receives he highes poserior probabiliy he larges proporion of he imes, followed by he saionary hypohesis, while he uni-roo hypohesis receives he highes poserior probabiliy less han 1% of he ime. The resuls in Table 3 are no very encouraging o he objecive Bayesian uni-roo es. Firs, for 0.95 <ρ< 0.98, i canno disinguish beween a highly persisen rend-saionary model from a highly persisen saionary AR model. There appears o be a bias in favor of he rend-saionary model. A ρ = 0.99 he uni-roo hypohesis is favored regardless of he daa generaing model used. When he daa generaing model is a randomwalk wih drif model, he objecive Bayesian uni-roo is biased in favor of he rend-saionary model. Thus, jus as he classical ADF es is criicized frequenly for is bias in favor of finding a uni-roo, i appears ha he 13

17 objecive Bayesian uni-roo es can also be criicized for is bias in favoring rend-saionariy. In pracice hen, a leas for he objecive Bayesian uni-roo es, here is no evidence ha i is a beer saisical approach han he classical ADF es in uni-roo esing. Given he Mone Carlo resuls, i is impossible o draw any conclusions regarding long-run PPP. According o Table 3, he daa generaing processes of he real exchange raes can come from a rend-saionary model, a saionary AR model, or a random-walk wih drif model. 7. Summary and Conclusions Researchers generally agree ha he Bayesian approach offers an alernaive and useful way o he classical approach o empirical modeling. In uni-roo esing, Sims (1988), Sims and Uhlig (1991), and Koop (1994) have presened a srong case for favoring he Bayesian approach over he classical ADF ess. Thus, i is surprising ha only a relaively small number of sudies have appeared ha uilize he Bayesian approach. In his paper, we use he Bayesian approach o uni-roo esing discussed in Koop (1992) o es for long-run PPP. Our resuls indicae ha when he ADF es is used for uni-roo esing, he hypohesis of a uni-roo is no rejeced excep for he Japanese yen/german mark real exchange rae. This is also rue for he join null hypohesis of a uni roo and he absence of a linear deerminisic ime rend. Thus, he evidence agains long-run PPP appears o be overwhelming. Our Bayesian resuls, on he oher hand, provide a sark conras o he ADF resuls. In all cases, he hypohesis of a uni roo does no receive significan poserior probabiliy. Raher, sample informaion appear o srongly suppor he hypohesis of rend-saionariy for all cases excep he Japanese yen/german mark real exchange rae where he sample informaion sugges a saionary ime series. The French franc/german mark real exchange rae is he oher case where he saionary hypohesis receives significan poserior probabiliy. Nex, we sudy he power of he objecive Bayesian es using wo Mone Carlo simulaions. The resuls are no very encouraging. In paricular, using economically plausible parameers for monhly daa of he real exchange raes for our daa generaing models, we find ha he objecive Bayesian es canno disinguish beween a rend-saionary AR model from a saionary AR model when he ime rend effec is relaively small, and he ime series is highly persisence. The bias is in favor of finding a rend-saionary model. When ρ = 0.99, he objecive Bayesian es is biased in favor of a uni-roo. On he oher hand, when he daa generaing model is a random-walk wih drif model, he es is biased in favor of he rend-saionary hypohesis. We find his o be raher disappoining since he objecive Bayesian es has several desirable feaures ha make i an aracive alernaive o he ADF es in uni-roo esing. 14

18 The Mone Carlo resuls sugges ha he objecive Bayesian es is no necessarily beer han he classical ADF es in uni-roo esing. Based on he Mone Carlo resuls, we canno draw any conclusions regarding long-run PPP. I should be emphasized, however, ha our conclusions apply only o he objecive Bayesian es and do no apply o Bayesian ess in general. Moreover, since our Mone Carlo simulaions are carried ou using parameer values which are relevan only o our real exchange raes series used in his sudy, Mone Carlo sudy of he objecive Bayesian es may very well come o differen conclusions when differen parameer values are used. Neverheless, we believe ha he claim ha he Bayesian approach o uni-roo esing is beer han he classical approach mus be esablished on a case by case basis. 15

19 References Abuaf, Niso, Jorion, Philippe, Purchasing power pariy in he long run. The Journal of Finance 45, Ahking, Francis W., Tesing long-run purchasing power pariy wih a Bayesian uni roo approach: he experience of Canada in he 1950s. Applied Economics 29, Cheung, Yin-Wong, Lai, Kon S., Lag order and criical values of he augmened Dickey-Fuller es. Journal of Business and Economic Saisics 13, Cheung, Yin-Wong, Lai, Kon S., Pariy reversion in real exchange raes during he pos-breon Woods period. Journal of Inernaional Money and Finance 17, Culver, Sarah, Papell, David H., Long run purchasing power pariy wih shor-run daa: evidence wih a null hypohesis of saionariy. Journal of Inernaional Money and Finance 18, DeJong, David N., Nankervis, John C., Savin, N. E., Whieman, Charles, Inegraion versus rend saionariy in ime series. Economerica 60, DeJong, David N., Whieman, Charles H., 1991a. Reconsidering Trends and random walks in macroeconomic ime series. Journal of Moneary Economics 28, DeJong, David N., Whieman, Charles H., 1991b. The case for rend-saionariy is sronger han we hough. Journal of Applied Economerics 6, Dickey, David A., Fuller, Wayne A., Likelihood raio saisics for auoregressive ime series wih a uni roo. Economerica 49, Diebold, Francis X., Hused, Seven, Rush, Mark, Real exchange raes under he gold sandard. Journal of Poliical Economy 99, Dornbusch, Rudiger, Expecaions and exchange rae dynamics. Journal of Poliical Economy 84, Engel, Charles, Long-run PPP may no hold afer all. Discussion Series #96-05, Insiue for Economic Research, Universiy of Washingon. Engle, Rober F., Granger, Clive W. J., Co-inegraion and error correcion: represenaion, esimaion, and esing 55, Frenkel, Jacob A., A moneary approach o he exchange rae: Docrinal aspecs and empirical evidence, in J. A. Frenkel and H. G. Johnson, eds., The economics of exchange rae (Addison-Wesley, Reading) Fuller, Wayne A., Inroducion o Saisical Time Series. John Wiley and Sons, New York. Hakkio, Craig S., Does he exchange rae follow a random walk? A Mone Carlo sudy of four ess for a random walk. Journal of Inernaional Money and Finance 5, Hamilon, James D., Time Series Analysis. Princeon Universiy Press, Princeon, New Jersey. Heimonen, Kari, Saionariy of he European real exchange raes evidence from panel daa. Applied Economics 31, Johansen, S., Saisical analysis of coinegraion vecors. Journal of Economic Dynamic and Conrol 12,

20 Jorion, Philippe, Sweeney, Richard, Mean reversion in real exchange raes: Evidence and implicaions for forecasing. Journal of Inernaional Money and Finance 15, Journal of Applied Economerics. Ocober-December, Karlsson, Sune, Löhgren, Mickael, On he power and inerpreaion of panel uni roo ess. Economic Leers 66, Kim, Chang-Jin, Engel, Charles, The long-run U.S./U.K. real exchange rae. Journal of Money, Credi and Banking 31, Par 1, Koedijk, Kees G., Schoman, Peer C., Van Dijk, Mahijs A., The re-emergence of PPP in he 1990s. Journal of Inernaional Money and Finance 17, Koop, Gary, Objecive Bayesian uni roo ess. Journal of Applied Economerics 7, Koop, Gary, Recen progress in applied Bayesian economerics. Journal of Economic Survey 8, Lohian, James R., Some new sylized facs of floaing exchange raes. Journal of Inernaional Money and Finance 17, Lohian, James R., Taylor, Mark P., The recen floa from he perspecive of he pas wo cenuries. Journal of Poliical Economy 104, Murray, Chrisian J., Nelson, Charles R., The uncerain rend in U.S. GDP. Journal of Moneary Economics 46, Murray, Chrisian J., Papell, David H., The purchasing power pariy persisence paradigm. Journal of Inernaional Economics 56, Nelson, Charles, Plosser, Charles, Trends and random walks in macroeconomics ime series: Some evidence and implicaions. Journal of Moneary Economics 10, O Connell, Paul G. J., The overvaluaion of purchasing power pariy. Journal of Inernaional Economics, 44, Papell, David H., Searching for saionariy: Purchasing power pariy under he curren floa. Journal of Inernaional Economics 43, Papell, David H., Theodoridis, Hrisos, Increasing evidence of purchasing power pariy over he curren floa. Journal of Inernaional Money and Finance 17, Rogoff, Kenneh, The purchasing power pariy puzzle. Journal of Economic Lieraure 34, Sarno, Lucio, Taylor, Mark P., Real exchange raes under he recen floa: unequivocal evidence of mean reversion. Economics Leers 60, Schoman, Peer, Van Dijk, Herman K., A Bayesian analysis of he uni roo in real exchange raes. Journal of Economerics 49, Sims, Chrisopher A., Bayesian skepicism on uni roo economerics. Journal of Economic Dynamics and Conrol 12, Sims, Chrisopher A., Uhlig, Harald, Undersanding uni rooers: A helicoper our. Economerica 59,

21 Whi, Joseph A., The long-run behavior of he real exchange rae. Journal of Money, Credi and Banking 24, Zellner, Arnold, Siow, A., Poserior odds for seleced regressions. In: Bernardo, J. M., DeGroo, M. H., Lindley, D. V., Smih, A. F. M. (eds), Bayesian Saisics, Universiy Press, Valencia,

22 Foonoes 1 For a recen conribuion o he debae, see Murray and Nelson (2000). 2 In addiion o he auhors cied, Hamilon (1994), pp also conains a good discussion of he advanages and disadvanages of using he classical approach vs. he Bayesian approach in uni-roo esing. 3 See he lively debae on he classical vs. he Bayesian mehods of uni-roo esing in he Ocober-December 1991 issue of he Journal of Applied Economerics. 4 This assumes ha he nominal exchange rae and he relaive prices are inegraed of he same order, i.e., he ime series of he nominal exchange rae and he relaive prices conain he same number of uni roos. Previous empirical sudies have found ha, generally, hey conain one uni roo each. 5 I should be noed ha in addiion o using long span of daa, Diebold, Hused and Rush (1991) also used fracional differencing in heir model. Thus, i is no possible o disinguish wheher heir finding of favorable evidence of long-run PPP is due o he long span of daa or o he use of auoregressive fracionally inegraed moving-average process (ARFIMA). 6 We prefer Koop s Bayesian approach over Sims s approach because Sims s approach is limied o a raher resricive AR(1) process wihou a consan erm. Schoman and van Dijk (1991) have shown ha he inclusion of a consan erm increases he compuaional burden remendously. Koop s approach, however, allows us o use a more flexible AR process ha is no only a higher order AR process, bu also allows for he inclusion of a consan and a linear deerminisic ime rend. Furhermore, Koop also performed Mone Carlo simulaions and concluded ha his es has reasonable size and power for plausible values. The size and power of Sims s Bayesian es are unknown. 7 Oher researchers, Papell (1997), and Culver and Papell (1999), for example, argued ha he presence of a linear deerminisic ime rend is inconsisen wih long-run PPP, however. 8 One could argue ha even-hough he hree compeing hypoheses have equal prior probabiliy, he uni-roo hypohesis is he mos favored, and he rend-saionary hypohesis is he leas favored. For example, in equaion (3), he prior probabiliy for he uni-roo hypohesis, i.e., ρ =1 is percen, while he saionary inerval ( 0 <ρ< 1) also receives percen prior probabiliy, disribued uniformly in ha inerval, resuling in each poin in ha inerval receiving exremely low prior probabiliy. For he rend-saionary hypohesis, however, he percen prior probabiliy, in addiion o being disribued on he saionary inerval, mus also be disribued on he infinie inerval of values ha he ime-rend coefficien may ake. 9 This assumes cross-rae equaliy excep for ransacion coss. This is probably a valid assumpion for he G-7 counries. Alernaively, as long as he measuremen error is a saionary process, our ess for uni-roo will no be affeced. 10 We have also ried oher mehods of lag-lengh deerminaion, e.g., Akaike s Informaion Creeria. While he lag lenghs deermined are differen, hey do no produce significanly differen resuls ha hose repored in Table 1, however. 11 Raher han using he same Bayesian approach o deermine he lag lenghs, we used a non-bayesian mehod. This reduces he number of alernaive hypoheses we need o consider and allow us o allocae he enire prior probabiliy o he hree hypoheses in quesion which are wha we are mos ineresed in. To check he robusness of our Bayesian resuls, we have also used a uniform four lags for all he real exchange rae series. The resuls are no very differen from hose repored in he paper. The addiional Bayesian resuls are available as Appendix A from he auhor. 12 Noe also ha i is he Japanese yen/german mark real exchange rae ha provides he lone excepion o he uniformly unfavorable long-run PPP resuls in Table 1. 19

23 13 A recen paper by Murray and Papell (2002) has shown ha he half-life esimaes are exremely unreliable, however. 20

24 Table 1: Univariae ADF Tes Resuls l (φ ) 2 χ Base Currency: U.S. Dollar Canada U.K Germany Ialy Japan France Base Currency: Canadian Dollar U.S.A U.K Germany Ialy Japan France Base Currency: Pound Serling U.S.A Canada Germany Ialy Japan France Base Currency: German Mark U.S.A Canada U.K Ialy Japan ** 5.514** France Base Currency: Ialian Lira U.S.A Canada U.K Germany Japan France Base Currency: Japanese Yen U.S.A Canada U.K Germany ** 5.514** Ialy France Base Currency: French Franc U.S.A Canada U.K Germany Ialy Japan Noe: ** denoe he rejecion of he null hypohesis a he 10% significance levels.

25 Table 2: Poserior Probabiliies k Trend Saionary Saionary Uni Roo Base Currency: U.S. Dollar Canada U.K Germany Ialy Japan France Base Currency: Canadian Dollar U.S.A U.K Germany Ialy Japan France Base Currency: Pound Serling U.S.A Canada Germany Ialy Japan France Base Currency: German Mark U.S.A Canada U.K Ialy Japan France Base Currency: Ialian Lira U.S.A Canada U.K Germany Japan France Base Currency: Japanese Yen U.S.A Canada U.K Germany Ialy France Base Currency: French Franc U.S.A Canada U.K Germany Ialy Japan Noe: The poserior probabiliies may no sum o one because of rounding.