Robust estimation for structural spurious regressions and a Hausman-type cointegration test

Transcription

1 Joural of Ecoomerics 14 (8) Robus esimaio for srucural spurious regressios ad a Hausma-ype coiegraio es Chi-Youg Choi a, Lig Hu b, Masao Ogaki b, a Deparme of Ecoomics, Uiversiy of Texas a Arligo, USA b Deparme of Ecoomics, Ohio Sae Uiversiy, Columbus, OH , USA Received February 6; received i revised form 1 Jue 7; acceped Jue 7 Available olie 6 July 7 Absrac This paper aalyzes a approach o correcig spurious regressios ivolvig ui-roo osaioary variables by geeralized leas squares (GLS) usig asympoic heory. This aalysis leads o a ew robus esimaor ad a ew es for dyamic regressios. The robus esimaor is cosise for srucural parameers o jus whe he regressio error is saioary bu also whe i is ui-roo osaioary uder cerai codiios. We also develop a Hausma-ype es for he ull hypohesis of coiegraio for dyamic ordiary leas squares (OLS) esimaio. We demosrae our esimaio ad esig mehods i hree applicaios: (i) log-ru moey demad i he U.S., (ii) oupu covergece amog idusrial ad developig couries, ad (iii) purchasig power pariy (PPP) for raded ad o-raded goods. r 7 Elsevier B.V. All righs reserved. JEL classificaio: C1; C15 Keywords: Spurious regressio; GLS correcio mehod; Dyamic regressio; Tes for coiegraio 1. Iroducio I he ui-roo lieraure, a regressio is echically called a spurious regressio whe is sochasic error is ui-roo osaioary. This is because he sadard -es eds o be spuriously sigifica eve whe he regressor is saisically idepede of he regressad i ordiary leas squares (OLS). Moe Carlo simulaios have ofe bee used o show ha he spurious regressio pheomeo occurs wih regressios ivolvig ui-roo osaioary variables (see, e.g., Grager ad Newbold, 1974; Nelso ad Kag, 1981, 198). Phillips (1986, 1998) ad Durlauf ad Phillips (1988) amog ohers have sudied he asympoic properies of esimaors ad es saisics for regressio coefficies of hese spurious regressios. This paper aalyzes a approach o correc spurious regressios ivolvig ui-roo osaioary variables by geeralized leas squares (GLS) usig asympoic heory. This aalysis leads o a ew robus esimaor ad a ew es for dyamic regressios. The robus esimaor is cosise for srucural parameers o jus whe Correspodig auhor. Tel.: ; fax: address: ogaki.1@osu.edu (M. Ogaki) /$ - see fro maer r 7 Elsevier B.V. All righs reserved. doi:1.116/j.jecoom.7.6.

2 8 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) 7 51 he regressio error is saioary bu also whe i is ui-roo osaioary uder cerai codiios. We also develop a Hausma-ype es for he ull hypohesis of coiegraio for dyamic OLS esimaio. Ecoomic models ofe imply ha cerai variables are coiegraed. However, ess ofe fail o rejec he ull hypohesis of o coiegraio for hese variables. Oe possible explaaio of hese es resuls is ha he error is ui-roo osaioary because of a osaioary measureme error i oe variable or osaioary omied variables. I such cases, i is sill possible o cosisely esimae srucural variables uder cerai codiios. Whe he error is ui-roo osaioary bu srucural parameers ca be recovered, he regressio is called a srucural spurious regressio. As a example of a srucural spurious regressio, cosider a regressio o esimae he moey demad fucio whe moey is measured wih a osaioary error. Currecy held by domesic ecoomic ages for legiimae rasacios is very hard o measure, sice currecy is held by foreig resides ad is also used for black marke rasacios. Therefore moey may be measured wih a osaioary error. As show by Sock ad Waso (199) amog ohers, whe he moey demad fucio is sable i he log ru, we have a coiegraig regressio if all variables are measured wihou error. If he variables are measured wih saioary measureme errors, we sill have a coiegraig regressio. If moey is measured wih a osaioary measureme error, however, we have a spurious regressio. We ca sill recover srucural parameers uder cerai codiios. The crucial assumpio is ha he osaioary measureme error is o coiegraed wih he regressors. Aoher example of a srucural spurious regressio is a regressio of moey demad wih osaioary omied variables. Cosider he case i which moey demad is sable i he log ru ad a measure of shoe leaher coss of holdig moey is icluded as a argume. If a ecoomericia omis he measure of he shoe leaher coss from he moey demad regressio ad he measure is osaioary, he he regressio error is osaioary. The shoe leaher coss of holdig moey are relaed o he value of ime ad, herefore, o he real wage rae. Because he real wage rae is osaioary i sadard dyamic sochasic geeral equilibrium models wih a osaioary echological shock, he omied measure of he shoe leaher coss is likely o be osaioary. I his case, he moey demad regressio ha omis he measure is spurious, bu we ca sill recover srucural parameers uder cerai codiios. The crucial assumpio is ha he omied variable is o coiegraed wih he regressors. Our srucural spurious regressio approach is based o he GLS soluio of he spurious regressio problem aalyzed by Ogaki ad Choi (1), 1 who use a exac small sample aalysis based o he codiioal probabiliy versio of he Gauss Markov Theorem. We develop asympoic heory for wo esimaors moivaed by he GLS correcio: he GLS correced dyamic regressio esimaor ad he feasible GLS (FGLS) correced dyamic regressio esimaor. Because Ogaki ad Choi oly used a exac small sample aalysis, hey did o cosider he FGLS correced esimaor. We will show hese esimaors o be cosise ad asympoically ormally disribued i spurious regressios. Whe he error erm is i fac saioary ad hece he variables are coiegraed, he GLS correced esimaor is o efficie, bu he FGLS correced esimaor, like he OLS esimaor, is supercosise. Hece, FGLS esimaio is a robus procedure wih respec o he error specificaio. The FGLS correced esimaor is asympoically equivale o he GLS correced esimaor i spurious regressios, ad i is asympoically equivale o he OLS esimaor i coiegraig regressios. I some applicaios, i is hard o deermie wheher or o he error i he regressio is saioary or uiroo osaioary because es resuls are icoclusive. I such applicaios, he FGLS correced esimaor is aracive because i is cosise i boh siuaios as log as he mehod of he dyamic regressio removes he edogeeiy problem. This approach aurally moivaes a Hausma-ype es for he ull hypohesis of coiegraio agais he aleraive hypohesis of o coiegraio (or a spurious regressio) i he dyamic OLS framework. 1 Aoher approach would be o ake he firs differece o iduce saioariy ad he use isrumeal variables. This is he approach proposed by Lewbel ad Ng (5) for heir osaioary raslog demad sysem. Our approach explois he paricular form of edogeeiy assumed by may auhors i he coiegraio lieraure ad avoids he use of isrumeal variables. Our approach yields more efficie esimaors as log as he paricular form of edogeeiy is correcly specified. This is especially impora whe weak isrumes cause problems. This es ca also be called a Durbi Wu Hausma ype es as i is closely relaed o ideas ad ess i Durbi (1954) ad Wu (197) as well as a family of ess proposed by Hausma (1978).

3 We cosruc his es by oig ha while boh he dyamic OLS ad GLS correced dyamic regressio esimaors are cosise i coiegraio esimaio, he dyamic OLS esimaor is more efficie. O he oher had, whe he regressio is spurious oly he GLS correced dyamic regressio esimaor is cosise. Hece, we could do a coiegraio es based o he specificaio o he error. We show ha uder he ull hypohesis of coiegraio he es saisics have a w limi disribuio, while uder he aleraive hypohesis of a spurious regressio he es saisics diverge. I some applicaios he assumpio ha he spurious regressio is srucural uder he aleraive hypohesis is o very aracive. If he violaio of coiegraio arises for reasos oher ha osaioary measureme error or omied variables, i is hard o believe ha he resulig spurious regressio is srucural. For his reaso we relax he assumpio ha he spurious regressio is srucural ad show ha he Hausmaype coiegraio es saisic sill diverges uder he aleraive hypohesis. Dyamic OLS is used i may applicaios of coiegraio. However, few ess for coiegraio have bee developed for dyamic OLS, wih he excepio of Shi s (1994) es. As i Phillips ad Ouliaris (199), he popular augmeed Dickey Fuller (ADF) es for he ull hypohesis of o coiegraio was origially desiged o be applied o he residual from saic OLS raher ha he residual from dyamic OLS. Because he saic OLS ad dyamic OLS esimaes are ofe subsaially differe, i is desirable o have a es for coiegraio applicable o dyamic OLS. Aoher aspec of our Hausma-ype es is ha i is for he ull hypohesis of coiegraio. Ogaki ad Park (1998) argue ha i is desirable o es he ull hypohesis of coiegraio raher ha ha of o coiegraio i may applicaios where ecoomic models imply coiegraio. Usig Moe Carlo experimes, we compare he fiie sample performace of he Hausma-ype es wih he es proposed by Shi (1994), which is a locally bes ivaria es for he ull of zero variace of a radom walk compoe i he disurbaces. Accordig o he experime resuls, he Hausma-ype es is domia i boh size ad power up o he sample size of. Shi s es becomes more powerful whe he sample size icreases, bu oly a he cos of higher size disorio. I some applicaios, i is appropriae o cosider he possibiliy ha measureme error is Ið1Þ ad is o coiegraed wih he regressors. For hese applicaios, he ADF es is applicable uder he ull hypohesis of a srucural spurious regressio, as show by Hu (6). For such applicaios, we recommed ha boh he ADF es ad he Hausma-ype es be applied because i is o clear which ull hypohesis is more appropriae. We demosrae our esimaio ad esig mehods i hree applicaios: (i) log-ru moey demad i he U.S., (ii) oupu covergece amog idusrial ad developig couries, ad (iii) purchasig power pariy (PPP) for raded ad o-raded goods. I he firs applicaio, we focus o esimaig ukow srucural parameers, while i he las wo applicaios we purpor o esig for coiegraio wih he Hausma-ype coiegraio es where we relax he assumpio ha he spurious regressio uder he aleraive hypohesis is srucural. The res of he paper is orgaized as follows. Secio gives ecoomeric aalysis of he model, icludig asympoic heories ad fiie sample simulaio sudies. Secio preses models of osaioary measureme error ad osaioary omied variables as well as he empirical resuls of hree applicaios. Secio 4 coais cocludig remarks.. The ecoomeric model C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) Cosider he regressio model y ¼ b x þ Z, (1) where fx g is a m-vecor iegraed process geeraed by Dx i ¼ v i. Afer compleig he firs draf, i has come o our aeio ha he Hausma-ype es was origially proposed by Ferádez-Macho ad Mariel (1994) for he saic OLS coiegraig regressio wih sric exogeeiy ad wihou ay serial correlaio. The es has o bee popular probably because hese assumpios are hard o jusify i applicaios, ad because he es was o developed for dyamic regressios.

4 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) 7 51 The error erm i (1) is assumed o be Z ¼ Xm X k i¼1 j¼ k e ¼ re þ u. g i;j v i; j þ e, ðþ ðþ Assumpio 1. Assume ha v ¼ðv 1 ;...; v m Þ ad u are zero mea saioary processes wih Ejv i j a o1, Eju j a o1 for some a4, ad srog mixig wih size a=ða Þ. We also assume ha he dyamic regressio mehod removes he edogeeiy problem. Tha is, Eðu v s Þ¼ for all ; s. We call his he sric exogeeiy assumpio for he dyamic regressio. The codiios o v ad u esure he ivariace priciples: for r ½; 1Š, =P ½rŠ v! d VðrÞ ad =P ½rŠ u! d UðrÞ, where VðrÞ is a m-vecor Browia moio wih covariace P 1 j¼ Eðv v jþ ad UðrÞ is a Browia moio wih variace P 1 j¼ Eðu u j Þ. The fucioal ceral limi heorem holds for weaker assumpios ha assumed here (de Jog ad Davidso, ), bu he codiios assumed above are geeral eough o iclude may saioary Gaussia or o-gaussia ARMA processes ha are commoly assumed i empirical modelig. Le v ¼ðDx 1; k ;...; Dx 1; ;...; Dx 1;þk ;...; Dx m; k ;...; Dx m; ;...; Dx m;þk Þ, ad c ¼ðg 1; k ;...; g 1; ;...; g 1;k ;...; g m; k ;...; g m; ;...; g m;k Þ. We esimae he srucural parameer b i he regressio y ¼ b x þ c v þ e. (4) The iferece procedure abou b differs accordig o he differe assumpios o he error erm e i (). Whe jrjo1, e is saioary, ad hece regressio (4) is a coiegraio regressio wih serially correlaed error. Whe r ¼ 1, e is a ui-roo osaioary process ad he OLS regressio is spurious. Boh models are impora i empirical sudies i macroecoomics ad fiace. I he ex wo secios, we will sudy he asympoic properies of differe esimaio procedures uder hese wo assumpios. Uder he assumpio ha r ¼ 1, OLS is o cosise while boh he GLS correcio ad FGLS correcio will give cosise ad asympoically equivale pffiffi esimaors. Uder he assumpio ha jrjo1, he GLS correced esimaor is o efficie as i is coverge, bu he FGLS esimaor is coverge ad asympoically equivale o he OLS esimaor. Therefore, FGLS is robus wih respec o he error specificaios (r ¼ 1orjrjo1)..1. Regressios wih Ið1Þ error I his secio we cosider he siuaio whe he error erm is Ið1Þ, i.e., r ¼ 1 i (). The esimaio mehods we sudy are dyamic OLS, he GLS correcio, ad he FGLS correcio The dyamic OLS spurious esimaio We sar wih he dyamic OLS esimaio of regressio (4). Uder he assumpio of r ¼ 1, his regressio is spurious sice for ay value of b he error erm is always Ið1Þ. I Appedix A, we show ha he DOLS esimaor ^b dols has he followig limi disribuio: Z 1 Z 1 ð^b dols b Þ! d VðrÞVðrÞ dr VðrÞUðrÞ dr. (5) ^c i he esimaio is also icosise wih ^c c ¼ O p ð1þ. As remarked i Phillips (1986, 1989), i spurious regressios he oise is as srog as he sigal. Hece, uceraiy abou b persiss i he limiig disribuios.

5 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) GLS correced esimaio Whe r ¼ 1, we ca filer all variables i regressio (4) by akig he full firs differece ad use OLS o esimae Dy ¼ b Dx þ c Dv þ u ¼ y Dz þ u, (6) where y ¼ðb ; c Þ ad z ¼ðx ; v Þ. This procedure ca be viewed as GLS correced esimaio. 4 If we le y ~ dgls deoe he GLS correced esimaor, he we ca show ha pffiffiffi ð ydgls ~ y Þ! d Nð; OÞ, (7) where O ¼ Q LQ wih Q ¼ EðDz Dz Þ ad L beig he log-ru variace marix of Dz u. Thus b i a srucural spurious regressio ca be cosisely esimaed (joily wih c), ad he esimaors are asympoically ormal. I he special case whe m ¼ 1, fv 1 g ad fu g are i:i:d: sequeces ad Z ¼ e, (7) gives ha pffiffiffi ð ydgls ~ y Þ! d Nð; s u =s 1v Þ, where s u ad s 1v are he variaces of u ad v 1, respecively..1.. The FGLS esimaio To use GLS o esimae a regressio wih serial correlaio i empirical work, a Cochrae Orcu FGLS procedure is usually adoped. This procedure also works for spurious regressios as show by Phillips ad Hodgso (1994). They show ha he FGLS esimaor is asympoically equivale o ha i he differeced regressio whe he error is ui-roo osaioary. I he prese paper, we will show ha he FGLS correcio o he dyamic regressio provides a cosise ad robus esimaor for srucural spurious regressios. Le he residual from he OLS regressio (4) be deoed by ^e, ^e ¼ y ^b x ^c v. To coduc he Cochrae Orcu GLS esimaio, we firs ru a AR(1) regressio of ^e, ^e ¼ ^r ^e þ ^u. (8) I ca be show ha ð^r 1Þ ¼O p ð1þ. Coduc he followig Cochrae Orcu rasformaio of he daa: ~y ¼ y ^r y ; ~x ¼ x ^r x ; ~v ¼ v ^r v. (9) The cosider OLS esimaio of he regressio ~y ¼ b ~x þ c ~v þ error ¼ y ~z þ error, (1) where ~z ¼ð~x ; ~v Þ. The OLS esimaor of y i (1) is compued as " # " # ~y fgls ¼ X X ~z ~z ~z ~y. (11) The limiig disribuio of ~ y fgls ca be show o be he same as i (7). Iuiively, eve hough he dyamic OLS esimaor is icosise, he residual is ui-roo osaioary because o liear combiaio of y ad x is saioary. Therefore, r approaches uiy i he limi, ad ~z behaves asympoically equivalely o Dz. A deailed proof of resuls i his secio is give i Appedix A. 4 This is a coveioal GLS procedure whe u is i:i:d: Whe u is serially correlaed as i our approach, we ame his procedure GLS correced dyamic esimaio.

6 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) Regressios wih IðÞ error I his secio, we cosider he asympoic disribuios of he hree esimaors (he DOLS esimaor, he GLS correced esimaor, ad he FGLS correced esimaor) uder he assumpio of coiegraio, i.e., jrjo1 i()...1. The dyamic OLS esimaio Uder he assumpio of coiegraio, he DGP of y is y ¼ b x þ c v þ e ; e ¼ re þ u ; jrjo1. (1) Applyig he ivariace priciple, for r ½; 1Š, =P ½rŠ e! d EðrÞ, where EðrÞ is a Browia moio wih variace P 1 j¼ Eðe e j Þ. The limiig disribuio of he OLS esimaor of b, which is asympoically idepede of ^c, is kow o be Z 1 Z 1 ð^b dols b Þ! d VðrÞVðrÞ dr VðrÞ deðrþ. (1)... GLS correced esimaio We ow ake he full firs differece as we did i he spurious regressios, he regressio becomes Dy ¼ b Dx þ c Dv þ e e ¼ y Dz þ e e. (14) Noe ha his rasformaio leads o a loss i efficiecy sice he esimaor b ~ p dgls is ow ffiffiffi coverge raher ha coverge as he DOLS esimaor is. Wih some mior revisios o equaio (7), he limiig disribuio of he esimaor i his case ca be wrie as pffiffiffi ð ydgls ~ y Þ! d Nð; O Þ, (15) where O ¼ Q L Q. Q is agai defied as Q ¼ EðDz Dz Þ ad L is he log-ru variace marix of vecor Dz De. I he special case whe m ¼ 1, fv 1 g ad fu g are i:i:d: sequeces, Z ¼ e, ad O ¼ s e ð1 c eþ=s 1v, where c e is he firs-order auocorrelaio coefficie of fe g.... The FGLS esimaio Isead of akig he full firs differece, if we esimae he auoregressio coefficie i he error ad use his esimaor o filer all sequeces, we will obai a esimaor ha is asympoically equivale o he DOLS esimaor. Iuiively, i he case whe he error e ¼ u is serially ucorrelaed, he AR(1) coefficie ^r will coverge o zero, ad hece he rasformed regressio will be asympoically equivale o he origial regressio. If, o he oher had, he error is saioary ad serially correlaed, he he AR(1) coefficie will be less ha uiy, ad, as show i Phillips ad Park (1988), he GLS esimaor ad he OLS esimaor i a coiegraio regressio are asympoically equivale. If we coduc he Cochrae Orcu rasformaio (9) ad esimae b i he regressio ~y ¼ b ~ ~x þ ~c ~v þ error, (16) he Appedix B shows ha he limiig disribuio of b ~ is he same as he limi of he OLS esimaor give i (1)... FGLS: a robus esimaor wih respec o he order of errors From our discussios o he FGLS esimaor i Secios.1. ad.., we ca summarize he FGLS correced esimaor i he followig proposiio: Proposiio 1. Suppose Assumpio 1 holds. I spurious regressios, he FGLS correced esimaor is asympoically equivale o he GLS correced esimaor, ad is limi disribuio ca be wrie as pffiffiffi ð yfgls ~ y Þ! d Nð; OÞ.

7 I coiegraio regressios, he FGLS correced esimaor is asympoically equivale o he DOLS esimaor, ad is limi disribuio ca be wrie as Z 1 Z 1 ð^b fgls b Þ! d VðrÞVðrÞ dr VðrÞ deðrþ. So FGLS is o oly valid whe he regressio is spurious bu also asympoically efficie whe he regressio is coiegraio. Remarks. C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) If a cosa is added o (4), we ca show ha he GLS or FGLS correced esimaors are asympoically equivale o ha give i (7) uder he assumpio of spurious regressios.. If a red erm is added o (4) (his is he case i which he deermiisic coiegraio resricio is o saisfied i he ermiology of Ogaki ad Park, 1998), he he GLS correced esimaio leads o a sigular covariace marix for he esimaor whe r is less ha oe i absolue value. This is because a red erm i (4) leads o a cosa erm i he firs differeced regressio (14) ad because he log-ru variace of he firs differece of e muliplied by a cosa is zero. Therefore, our mehods do o apply o regressios wih ime reds.. Uder some codiios, he mehods proposed i his paper also apply o oher model cofiguraios, such as regressios where he regressors have drifs. These exesios will be sudied i fuure work..4. Fiie sample performace of he hree esimaors From he above aalysis, we show ha FGLS correced esimaio is a robus procedure wih respec o error specificaios. I his secio, we use simulaios o sudy is fiie sample performaces compared o he oher wo esimaors. I he simulaio we cosider he case whe x is a scalar variable ad geerae v ad u from wo idepede sadard ormal disribuios while leig e ¼ re þ u. The srucural parameer is se o b ¼, ad c v ¼ :5v. The umber of ieraios i each simulaio is 5, ad i each replicaio 1 þ observaios are geeraed, of which he firs 1 observaios are discarded. Table 1 shows he bias ad he mea square error (MSE) of all hree esimaors for r ¼ ; :95, ad 1. Whe r ¼, he regressio is coiegraio wih i:i:d: error. I is clear ha he DOLS esimaor is he bes oe whe ¼ 5. Whe reaches 1, however, he FGLS esimaor becomes almos as good as he DOLS esimaor. Whe r ¼ :95, he regressio is coiegraio wih serially correlaed error. I his case, he GLS ad FGLS esimaors are much beer ha he DOLS esimaor. Whe he sample size icreases, he FGLS esimaor Table 1 The bias ad square roo of he mea square error of hree esimaors r DOLS esimaor GLS correced esimaor FGLS correced esimaor Bias Square roo of MSE Bias Square roo of MSE Bias Square roo of MSE r ¼ ¼ ¼ ¼ r ¼ :95 ¼ ¼ ¼ r ¼ 1 ¼ ¼ ¼

8 4 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) dyamic OLS regressio esimaor rho =.98 rho =.99 rho = GLS correced dyamic regressio esimaor rho =.98 rho =.99 rho = FGLS correced dyamic regressio esimaor rho =.98 rho =.99 rho = Fig. 1. Compariso of hree esimaors whe ¼ 1 ad r! 1. becomes he bes oe. Fially, whe r ¼ 1, he regressio is spurious, ad, as expeced, he GLS correced esimaor performs bes. Fig. 1 plos he empirical disribuio of hese hree esimaors (mius he rue value) whe ¼ 1 ad as r approaches 1. The figures show ha he DOLS esimaor becomes flaer ad flaer as r! 1. The GLS esimaor remais largely he same for r close o uiy. The FGLS esimaor becomes a bi flaer whe r reaches 1, bu i sill shows a clear peak aroud zero. From he fiie sample performace, i ca be see ha he FGLS esimaor is almos as good as he DOLS esimaor i coiegraio ad sigificaly ouperforms he DOLS esimaor i spurious regressios. The GLS esimaor is he bes whe r approaches 1, bu i suffers from a sigifica loss i efficiecy whe r is small. So we may wa o ake he full differece oly whe we are very sure ha he error is ui roo osaioary. Oherwise, he FGLS esimaor is a good choice..5. Hausma specificaio es for coiegraio.5.1. The es saisic ad is asympoic properies I his secio, we cosruc a Hausma-ype coiegraio es based o he differece of wo esimaors: a OLS esimaor (^b dols Þ ad a GLS correced esimaor ( b ~ dgls ). This is equivale o comparig esimaors i a level regressio ad i a differeced regressio. The es is for he ull of coiegraig relaioships agais he aleraive of a spurious regressio: H : jrjo1 agais H A : r ¼ 1.

9 Our discussios so far sugges ha uder he ull of coiegraio, boh OLS ad GLS are cosise bu he OLS esimaor is more efficie. Uder he aleraive of a spurious regressio, however, oly he GLS correced esimaor is cosise. pffiffi Le ^V b deoe a cosise esimaor for he asympoic variace of ð bdgls ~ bþ. Uder our assumpios, i coverges o he correspodig submarix of O uder he ull hypohesis ad o he correspodig submarix of O uder he aleraive. For example, whe m ¼ 1, fv 1 g ad fu g are idepede i:i:d:, ad = 1, where ^w deoes he residuals from OLS esimaio of he Z ¼ e, ake ^V b ¼ 1 P ^w P Dx differeced regressio. Uder he ull of coiegraio, spurious regressio, ^V b! p s u =s 1v. We defie he Hausma-ype es saisic as: ^V b! p s e ð1 c eþ=s 1v. Uder he aleraive of h ¼ ð b ~ dgls ^b dols Þ ^V b ð~ b dgls ^b dols Þ. (17) Proposiio. Suppose Assumpio 1 holds. Uder he ull hypohesis of coiegraio, h! d w ðmþ. Uder he aleraive of spurious regressios, h ¼ O p ðþ. Proof. Uder he ull of coiegraio, pffiffi ð bdgls ~ ^b p dols Þ¼ ffiffi ð bdgls ~ p b Þ ffiffiffi ð^bdols b Þ p ¼ ffiffi ð bdgls ~ b Þþo p ð1þ! d Nð; V b Þ, where V b is he asympoic variace of ~ b dgls uder he assumpio of coiegraio. Therefore, if ^V b is a cosise esimaor for V b, h ¼ ð ~ b dgls ^b dols Þ ð ^V b Þ ð ~ b dgls ^b dols Þ! d w ðmþ. Uder he aleraive of spurious regressios, pffiffi ð bdgls ~ ^b p dols Þ¼ ffiffi ð bdgls ~ p b Þ ffiffiffi ð^bdols b Þ pffiffi ¼ O p ð1þþo p ð Þ pffiffi ¼ O p ð Þ. Hece, h ¼ O p ðþ uder he aleraive. We ca exed he es o allow edogeeiy uder he aleraive. Cosider he followig DGP: y ¼ b x þ c v þ fs þ e, e ¼ re þ u, where fs g saisfies he same codiios as u ad v bu is correlaed wih fv g. The saisic defied i (17) ca be applied o es he hypoheses: agais H : jrjo1 ad f ¼ H A : r ¼ 1adfa: C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) The asympoics of h uder he ull H are he same as ha uder H. Uder he aleraive H A, we show i Appedix C ha he DOLS esimaor has he same asympoic disribuio as ha uder H A ad h ¼ O p ðþ. Therefore, his Hausma-ype es is cosise for he ull hypohesis of coiegraio agais he aleraive of spurious regressios, regardless of wheher he exogeeiy assumpio holds uder he aleraive..5.. Fiie sample properies of he Hausma-ype coiegraio es Before applyig he Hausma-ype coiegraio es empirically, i will be isrucive o examie is fiie sample properies i compariso wih oher comparable ess uder he same ull hypohesis. To his ed, we

10 6 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) 7 51 Table Fiie sample performace of he Hausma-ype coiegraio es T Hausma-ype es Shi s es Power Size (5%) Power Size (5%) Noe: The Hausma-ype coiegraio eshis sipulaedii Secio.5. Noparameric esimaor of log ru variace used is based o he QS kerel wih he badwidh of ieger 8ðT=1Þ 1=4. coduc a small simulaio experime based o he followig dyamic regressio model: y ¼ g 1 Dx þ1 þ bx þ g Dx þ e, ð18þ e ¼ re þ u, ð19þ where g 1 ¼ :, b ¼, g ¼ :5, ad seig r ¼ :9 for he size performace ad r ¼ 1 for he power performace. We cosider sample sizes of f5; 1; ; ; 5g ha are commoly ecouered i empirical aalysis. I he simulaios, pseudo-radom umbers are geeraed usig he GAUSS (versio 6.) RNDNS procedures. Each simulaio ru is carried ou wih 5 replicaios. A each replicaio, 1 þ radom umbers are geeraed, of which he firs 1 observaios are discarded o avoid a sar-up effec. Table repors seleced fiie sample properies of he Hausma-ype coiegraio es ogeher wih a residual-based es uder he ull of coiegraio due o Shi (1994, Shi s es), who exeded he KPSS es i he paramerically correced coiegraig regressio. I he simulaios, he leghs of he lead ad lag erms for DOLS ad DGLS are chose by he BIC rule. 5 A oparameric esimaio mehod for log-ru variace esimaio is employed usig he QS kerel wih he badwidh of ieger ½8ð=1Þ 1=4 Š. The resuls i Table illusrae wo pois. Firs, he empirical size of he Hausma-ype es is close o he omial size, i paricular, whe he sample size is relaively large, whereas Shi s es suffers from a serious oversize problem. Secod, i erms of power, he Hausma-ype es domiaes Shi s es for moderae sample sizes ha are very likely o be ecouered empirically. Shi s es seems more powerful whe is relaively large, bu oly a he cos of severe size disorios. Overall, our simulaio resuls provide evidece i favor of he Hausma-ype es.. Empirical applicaios I his secio we apply he GLS-ype correcio mehods ad he Hausma-ype coiegraio es o aalyze hree macroecoomic issues: (i) log ru moey demad i he U.S., (ii) oupu covergece amog idusrial ad developig couries, ad (iii) PPP for raded ad o-raded goods. The mai purpose of he firs applicaio is o illusrae he spurious regressio approach o esimaig ukow srucural parameers. Ideificaio of he srucural parameers i his applicaio is based o osaioary measureme error or osaioary omied variables ha are explaied i he followig wo secios. The mai purpose of he oher wo applicaios is o apply he Hausma-ype coiegraio es. The aleraive hypohesis is o ake as srucural spurious regressios i he las wo applicaios..1. A model of osaioary measureme error Cosider a se of variables ha are coiegraed. Oe model of a srucural spurious regressio is based o he case i which oe of he variables is measured wih osaioary measureme error. Le y be he rue 5 I is a ieresig research opic o ivesigae he performace of various lag legh selecio rules, bu would be beyod he scope of his paper.

11 value of y, ad assume ha y ¼ b x þ c v þ e () is a dyamic coiegraig regressio ha saisfies he sric exogeeiy assumpio. 6 Le y be he measured value of y, ad assume ha he measureme error saisfies y y ¼ cm v þ e m, (1) where e m is Ið1Þ ad is expecaio codiioal o x s for all s is zero. Here, he crucial assumpio for ideificaio is ha he measureme error is o coiegraed wih x. The y ¼ b x þ c v þ e, () where c ¼ c þ c m, ad e is Ið1Þ ad saisfies he sric exogeeiy assumpio. 7.. A model of osaioary omied variables Aoher case ha leads o a srucural spurious regressio is a model of osaioary omied variables. y ¼ b x þ y x þ c1 v þ c v þ e, () where x is a vecor of Ið1Þ variables ad v is a vecor of leads ad lags of he firs differeces of x. We imagie ha he ecoomericia omis x from his regressio. We assume ha y x þ c v ¼ cm v þ e m, (4) where e m is Ið1Þ ad is expecaio codiioal o x s for all s is zero. Here, he crucial assumpio for ideificaio is ha he x is o coiegraed wih x. The y ¼ b x þ c v þ e, (5) where c ¼ c 1 þ c m, ad e is Ið1Þ ad saisfies he sric exogeeiy assumpio. This model is observaioally equivale o he model of osaioary measureme error wihi our sigle equaio approach. However, he assumpios made i boh cases are cocepually differe... U.S. moey demad The log-ru moey demad fucio has ofe bee esimaed uder a coiegraig resricio amog real balaces, real icome, ad he ieres rae. The resricio is legiimae if he moey demad fucio is sable i he log ru ad if all variables are measured wihou osaioary error. Ideed, Sock ad Waso (199) foud supporive evidece of sable log-ru M1 demad by esimaig coiegraig vecors. However, if eiher moey is measured wih a osaioary measureme error or osaioary omied variables exis, he we have a spurious regressio, ad he esimaio resuls based o a coiegraio regressio are quesioable. Firs, cosider he model of a osaioary measureme error described above. To be specific, we follow Sock ad Waso (199) ad assume ha he dyamic regressio error is saioary ad he sric exogeeiy assumpio holds for he dyamic regressio error whe moey is correcly measured. We he assume ha moey is measured wih a muliplicaive measureme error. We assume ha he log measureme error is ui-roo osaioary ad ha he residuals of he projecio of he log measureme error o he leads ad lags of he regressors i he dyamic regressio saisfy he sric exogeeiy assumpio. Give ha a large compoe of he measureme error is arguably currecy held by foreig resides ad black marke paricipas, he log measureme error is likely o be very persise. Therefore, he assumpio ha he log measureme error is ui-roo osaioary may be a leas a good approximaio. The assumpio ha he 6 Noe ha ay variable ca be chose as he regressad i a coiegraig regressio. Therefore, we choose he variable wih osaioary measureme error as he regressad. 7 Here, we assume ha he dimesios of c ad c m are he same wihou loss of geeraliy because we ca add zeros as elemes of c ad c m as eeded. C.-Y. Choi e al. / Joural of Ecoomerics 14 (8)

12 8 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) 7 51 measureme error is o coiegraed wih he regressors is plausible if he error is maily due o currecy held by foreig resides. Secod, cosider he model of osaioary omied variables. A possible omied variable is a measure of he shoe leaher cos ha represes rasacio coss. For example, i he lieraure of moey demad esimaio, he real wage rae has someimes bee used as a regressor for his reaso. Because he real wage rae is Ið1Þ i sadard dyamic sochasic geeral equilibrium models wih a Ið1Þ echological shock, he omied measure of he shoe leaher cos is osaioary. If he real wage rae is he omied variable, he assumpio ha i is o coiegraed wih he regressors ha iclude log icome is o very plausible. However, i is possible ha he rue omied variable ha represes he shoe leaher cos is o he real wage rae ad is o coiegraed wih log icome. We apply our GLS correcio mehod o esimae he log-ru icome ad ieres elasiciies of M1 demad durig he period of To his ed, he regressio equaios are se up wih he real moey balace (M=P) as regressad ad icome (y) ad ieres (i) as regressors. Followig Sock ad Waso (199), he aual ime series for M1 deflaed by he e aioal produc price deflaor is used for M=P, he real e aioal produc for y ad he six-moh commercial paper rae i perceages for i. M=P ad y are i logarihms. Three differe regressio equaios are cosidered depedig o he measures of ieres. We have ried he followig hree fucioal forms (equaio 1 has bee sudied by Sock ad Waso, 199): l M ¼ a þ b lðy P Þþgi þ e ðequaio 1Þ, l M ¼ a þ b lðy P Þþglði Þþe ðequaio Þ, l M P ¼ a þ b lðy Þþgl 1 þ i þ e ðequaio Þ. i I is worh oig ha he liquidiy rap is possible for he laer wo fucioal forms as emphasized by Bae ad de Jog (7). 9 Whe he daa coai periods wih very low omial ieres raes, he laer wo fucioal forms may be more appropriae. Table preses he poi esimaes for b (icome elasiciy of moey demad) ad g based o he hree esimaors uder scruiy: he dyamic OLS esimaor, he GLS correced dyamic regressio esimaor, ad he FGLS correced dyamic regressio esimaor. 1 Several feaures emerge from he able. Firs, all he esimaed coefficies have heoreically correc sigs: posiive sigs for icome elasiciies ad egaive sigs for g for he firs wo fucioal forms ad posiive sigs for g for he hird fucioal form. Secod, he GLS correced esimaes of he icome elasiciy are implausibly low for all hree fucioal forms for low values of k ad icrease o more plausible values ear oe as k icreases. 11 The fac ha he resuls become more plausible as k icreases suggess ha he edogeeiy correcio of dyamic regressios works i his applicaio for moderaely large values of k such as hree ad four. The resuls for lower values of k are cosise wih hose of he low icome elasiciy esimaes of firs differeced regressios ha were used i he lieraure before 198. Therefore, he esimaors i he old lieraure of firs differeced regressios before coiegraio became popular are likely o be dowward biased because of he edogeeiy problem. Third, all poi esimaes of he hree esimaors are very similar, ad he Hausma-ype es fails o rejec he ull hypohesis of coiegraio for large eough values of k. Hece, here is lile evidece agais coiegraio. However, i should be oed ha a small radom walk compoe is very hard o deec 8 Readers are referred o Appedix D for he empirical guidelies o he use of esimaio ad esig echiques developed i his paper. We hak Yougsoo Bae for providig he daa used i Bae ad de Jog (7) o us. This daa se exeds Sock ad Waso s daa up o 1997 whe he six-moh commercial rae was discoiued. 9 Bae ad de Jog poi ou ha oliear coiegraio mehods are eeded if we are o evaluae hese differe fucioal forms wih a commo se of assumpios. I is beyod he scope of he prese paper o develop spurious regressio mehods for oliear coiegraio models. 1 For he FGLS correced dyamic regressio esimaor, he serial correlaio coefficie of he error erm is esimaed before beig applied o he Cochrae Orcu rasformaio. This coefficie is assumed o be uiy i he GLS correced dyamic regressio esimaor which is equivale o regressig he firs differece of variables wihou a cosa erm. 11 Whe k is icreased beyod five (he maximum k i he able), poi esimaes for icome elasiciy esimaes sabilize aroud uiy.

13 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) Table Applicaio o log u U.S. moey demad Esimaor k Equaio 1 Equaio Equaio ^b ^g ^b ^g ^b ^g DOLS.891 (.8).84 (.5).86 (.61).7 (.69).858 (.61).16 (.71) 1.97 (.79).9 (.5).861 (.47).1 (.5).859 (.47). (.54).94 (.74).98 (.4).86 (.4).1 (.44).861 (.4).9 (.46).98 (.59).1 (.19).865 (.1).7 (.4).86 (.1).6 (.6) (.6).14 (.).861 (.9). (.).858 (.).9 (.4) (.56).18 (.18).86 (.9).9 (.).86 (.9).48 (.4) BIC [lag] [4] [5] [5] GLS-correced.45 (.8).14 (.4).415 (.79).86 (.1).415 (.79).89 (.) 1.69 (.11).9 (.9).664 (.19).191 (.4).664 (.19).199 (.44).799 (.17).47 (.1).815 (.1).41(.49).815 (.1).49 (.51).811 (.17).58 (.15).85 (.11).7 (.57).849 (.11).78 (.58) (.149).66 (.18).87 (.14).6 (.6).868 (.14).67 (.64) 5.9 (.16).7 (.19).98 (.149).84 (.65).94 (.148).91 (.67) BIC [lag] [] [] [] FGLS-correced AR(1).887 (.5).78 (.).86 (.44).88 (.75).861 (.158).96 (.116) (.54).41 (.9).86 (.6).64 (.).86 (.5).7 (.).84 (.54).6 (.11).86 (.1). (.).86 (.1).11 (.1).884 (.47).8 (.1).874 (.6).5 (.6).87 (.6).4 (.7) 4.91 (.5).89 (.1).874 (.7).7 (.7).871 (.7).5 (.8) (.46).1 (.1).879 (.7).4 (.7).876 (.7).49 (.8) BIC [lag] [] [] [] FGLS-correced AR().887 (.5).78 (.).86 (.44).88 (.75).861 (.158).96 (.116) (.5).4 (.9).85 (.8).61 (.).85 (.7).7 (.).847 (.48).68 (.11).861 (.8).7 (.8).859 (.8).16 (.9).99 (.41).9 (.11).877 (.4).8 (.5).875 (.5).6 (.6) 4.97 (.4).98 (.1).877 (.5). (.5).874 (.5).8 (.6) 5.96 (.9).15 (.11).878 (.1).4 (.).874 (.).5 (.) BIC [lag] [] [5] [5] HAUSMAN es ADF-BASED es.88 z.891 z.85 z Noe: l M ¼ a þ b lðy P Þþgi þ e ðequaio 1Þ, l M ¼ a þ b lðy P Þþglði Þþe ðequaio Þ, l M P ¼ a þ b lðy Þþg l 1 þ i þ e ðequaio Þ. i GLS-correced (FGLS-correced) deoes he GLS (FGLS) correced dyamic regressio esimaor. Figures i he parehesis represe sadard errors. k deoes he maximum legh of leads ad lags. I FGLS correced esimaio, he serial correlaio coefficie i he error erm is esimaed before beig applied o he Cochrae Orcu rasformaio, whereas i is assumed o be uiy i GLS correced esimaio which is aalogous o regressig he firs differece of variables wihou a cosa erm. Hausma es represes he Hausma-ype coiegraio es as sipulaed i Secio.5. The es saisic is cosruced as ð ^G dgls ~G dols ÞSð ^G dgls ~G dols Þ! d w ðþ where G ¼½b; gš ad S ¼ varð ~ b dgls Þ covð ~ b dgls ;~g dgls Þ covð ~ b dgls ;~g dgls Þ varð~g dgls Þ. The criical values of w ðþ are 4.61, 5.99 ad 9.1 for 1%, 5%, ad 1% sigificace levels. The criical values of he ADF-based ess are :88 ad :57 for 5% ad 1% sigificace levels. z represes ha he ull hypohesis ca be rejeced a 5%.

14 4 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) 7 51 wih ay es for coiegraio. Therefore, i is assurig o kow ha all hree esimaors are similar for large eough values of k ad ha he esimaes are robus wih respec o wheher he regressio error is IðÞ or Ið1Þ. We repor he value of k chose by he Bayesia iformaio crierio (BIC) rule hroughou our empirical applicaios i order o give some guidace i ierpreig resuls. 1 A deailed aalysis of how k should be chose is beyod he scope of his paper because his issue has o bee seled i he lieraure of dyamic coiegraig regressios. Table also repors he resuls whe he ADF es is applied o he OLS residuals. The resuls show evidece agais he ull hypohesis of srucural spurious regressios ad hereby corroborae he resuls from he Hausma-ype es uder he opposie ull hypohesis..4. Oupu covergece across aioal ecoomies I his secio, we apply he echiques o re-examie a log sadig issue i macroecoomics, he hypohesis of oupu covergece. For his applicaio ad he ex, our mai purpose is o o esimae ukow srucural parameers bu o es he ull hypohesis of coiegraio wih he Hausma-ype es. For his purpose, we do o eed he sric exogeeiy assumpio uder he aleraive hypohesis of o coiegraio (or a spurious regressio). As a key proposiio of he eoclassical growh model, he covergece hypohesis has bee popular i macroecoomics ad has araced cosiderable aeio i he empirical field, paricularly durig he las decade. Besides is impora policy implicaios, he covergece hypohesis has bee used as a crierio o discer bewee he wo mai growh heories, exogeous growh heory ad edogeous growh heory. Despie his aeio, i remais he subjec of coiuig debae maily because he empirical evidece supporig he hypohesis is mixed. Neverheless, he esablished lieraure based o popular ieraioal daa ses such as he Summers Heso (Summers ad Heso, 6) daa se suggess as a sylized fac oupu covergece amog idusrialized couries bu o amog developig couries ad o bewee idusrialized ad developig couries. Give ha a mea saioary sochasic process of oupu dispariies bewee wo ecoomies is ierpreed as supporive evidece of sochasic covergece, ui-roo or coiegraio esig procedures are ofe used by empirical researchers o evaluae he covergece hypohesis. I his vei, our echiques proposed here fi i he sudy of oupu covergece. We cosider four developig couries (Columbia, Ecuador, Egyp, ad Pakisa) alog wih four idusrial couries (Demark, New Zealad, Souh Africa, Swizerlad). The raw daa are exraced from he Pe World Tables of Summers ad Heso (6) ad cosis of aual real GDP per capia (RGDPCH) over he period of The followig wo regressio equaios are cosidered wih regard o he coiegraio relaio: y D ¼ a þ by I þ e, ð6þ y I ¼ a þ byi þ e, ð7þ where y D ad y I deoe log real GDP per capia for developig ad idusrial couries, respecively. Table 4 preses he resuls which exhibi a large variaio i esimaed coefficies. Recall ha our ieres i his applicaio lies i he coiegraio es based o he Hausma-ype es. As ca be see from Table 4, irrespecive of coury combiaios, he ull hypohesis of coiegraio ca be rejeced whe developig couries are regressed oo idusrial couries, idicaig ha here is lile evidece of oupu covergece bewee developig couries ad idusrial couries. The picure chages dramaically whe idusrial couries are regressed oo idusrial couries as i (7). Table 4 also repors ha he Hausma-ype es fails o rejec he ull of coiegraio i all cases cosidered. Our fidig is herefore cosise wih he oio of covergece clubs which is ake as a sylized fac i he growh lieraure (e.g. Durlauf ad Quah, 1999; Easerly, 1). 1 I fac, we are o sure wheher he BIC is he righ mehod. Give he sesiiviy of esimaio resuls, we repor he resuls from various k s i he moey demad applicaio. I he followig wo applicaios for he PPP ad oupu covergece, we jus repor he resuls based o BIC maily because he resuls are o very sesiive o k aroud he BIC choice.

15 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) Table 4 Applicaio o oupu covergece Regressad Regressor k DOLS GLS-correced FGLS-correced Hausma es Regressio 1 COL DEN [1].815 (.168).569 (.88).767 (.17) 5.77 z NZL [] 1.67 (.7).1 (.86) 1.54 (.49) z SWI [] 1. (.5).94 (.77) 1.5 (.6) z ZAF [] 1.6 (.8).68 (.15) 1.7 (.51).998 z ECU DEN [].789 (.41).94 (.164).819 (.).68 z NZL [] 1.8 (.711).6 (.159) 1.4 (.5).8 y SWI [] 1.7 (.4).58 (.16) 1.5 (.) 7.86 z ZAF [] 1.81 (.461).1 (.7) (.64) 8. z EGT DEN [] (.15).664 (.15) 1.1 (.4) z NZL [4].77 (.54) 1.51 (.47).54 (.4) 9.57 z SWI [] (.75).94 (.71) 1.8 (.64) z ZAF [] 1.79 (.685).55 (.6) 1.75 (.115) 1.17 z PAK DEN [] 1.18 (.181).97 (.1) 1.1 (.49) 5.94 z NZL [] (.49).1 (.15) 1.7 (.115) 41.9 z SWI [] 1.5 (.4).89 (.7).6 (.5) 4.48 z ZAF [] 1.81 (.48).578 (.18) 1.8 (.17) z Regressio DEN NZL [4] (.4) (.1) (.41).149 SWI [4] 1. (.4) 1.5 (.14) 1.85 (.5).11 ZAF [] 1.68 (.8) 1.5 (.8) (.7).14 NZL DEN [].69 (.).496 (.1).65 (.8).96 SWI [].785 (.88).455 (.1).794 (.19).81 ZAF [] 1.51 (.158).88 (.44) 1. (.146).1 SWI DEN [1].749 (.48).774 (.17).67 (.45).48 NZL [] 1.15 (.98).81 (.8) (.9) ZAF [] 1.7 (.1).69 (.175) 1.89 (.18).77 ZAF DEN [1].54 (.75).54 (.17).48 (.68).78 NZL [].87 (.19).667 (.14).76 (.119).8 SWI [1].75 (.16).66 (.114).647 (.114).89 Noe: See he oes i Table. Aual daa coverig 1951 are used for four developig couries (COL: Columbia; ECU: Ecuador; EGT: Egyp; PAK: Pakisa) ad four idusrial couries (DEN: Demark; NZL: New Zealad; SWI: Swizerlad; ZAF: Souh Africa). k deoes he legh of lead ad lag erms for DOLS ad DGLS chose by he BIC rule. regressio 1: lðy DEV Þ¼aþb lðy IND Þþe, regressio : lðy IND Þ¼aþb lðy IND Þþe..5. PPP for raded ad o-raded goods As a major buildig block for may models of exchage rae deermiaio, PPP has bee oe of he mos heavily sudied subjecs i ieraioal macroecoomics. Despie exesive research, he empirical evidece o PPP remais icoclusive, largely due o he ecoomeric challeges ivolved i deermiig is validiy. As is geerally agreed, mos real exchage raes show very slow covergece which makes esimaig log-ru relaioships difficul wih exisig saisical ools. The lieraure suggess a umber of poeial explaaios for he very slow adjusme of relaive prices: volailiy of he omial exchage-rae, marke fricios such as rade barriers ad rasporaio coss, imperfec compeiio i produc markes, ad he presece of oraded goods i he price baske. Accordig o he commodiy-arbirage view of PPP, he law of oe price holds oly for raded goods, ad he deparures from PPP are primarily aribued o he large weigh placed o o-raded goods i he CPI. This view has obaied suppor from may empirical sudies based o

16 4 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) 7 51 Table 5 Applicaio o PPP for raded ad o-raded goods Esimaor Traded goods No-raded goods FRA ITA JPN U.K. U.S. FRA ITA JPN U.K. U.S. DOLS (.1) (.165) (.6) (.1) (.198) (.165) (.41) (.99) (.78) (.44) BIC [] [] [] [] [] [] [] [5] [1] [] GLS-correced (.9) (.81) (.411) (.65) (.14) (.178) (.176) (.198) (.171) (.8) BIC [] [] [] [] [] [] [] [] [1] [] FGLS-correced (.9) (.149) (.96) (.16) (.4) (.15) (.1) (.1) (.6) (.16) BIC [1] [1] [] [1] [1] [] [1] [5] [1] [] Hausma es [1] z 1.6 z z z z Noe: Resuls are for f T ¼ a þ bp T þ e ad f N ¼ a þ bp N þ e usig Caada as a base coury. Figures i parehesis represe sadard errors. Eries iside square brackes represe he legh of leads ad lags chose by BIC. Hausma es represes he Hausma-ype coiegraio es as sipulaed i Secio.5. The es saisic is cosruced as ð^b dgls ~ b dols Þ =Varð ~ b dgls Þ! d w ð1þ. The criical values of w ð1þ are.71,.84 ad 6.6 for e, five, ad oe perce sigificace level. z represes ha he ull hypohesis of ^b dgls ¼ ~ b dols ca be rejeced a 5% sigificace level. disaggregaed price idices. They ed o provide ample evidece ha prices for o-raded goods are much more dispersed ha for heir raded couerpars ad cosequely o-raded goods exhibi far larger deviaios from PPP ha raded goods. Give ha geeral price idices ivolve a mix of boh raded ad o-raded goods, highly persise deviaios of o-raded goods from PPP ca lead o he lack of coclusive evidece o he log ru PPP relaioship. As i he previous applicaio, our mai purpose for his applicaio is o o esimae ukow srucural parameers bu o es he ull hypohesis of coiegraio wih he Hausma-ype es. Le p ad p deoe he logarihms of he cosumer price idices i he base coury ad foreig coury, respecively, ad s be he logarihm of he price of he foreig coury s currecy i erms of he base coury s currecy. Log-ru PPP requires ha a liear combiaio of hese hree variables be saioary. To be more specific, log-ru PPP is said o hold if f ¼ s þ p is coiegraed wih p such ha e IðÞ i f T ¼ a þ bp T þ e, f N ¼ a þ bp N þ e, where he superscrips T ad N deoe he price levels of raded goods ad o-raded goods, respecively. Followig he mehod of Sockma ad Tesar (1995), Kim (5) recely aalyzed he real exchage rae for oal cosumpio usig he geeral price deflaor ad he real exchage rae for raded ad o-raded goods usig implici deflaors for o-service cosumpio ad service cosumpio, respecively. 1 We use Kim s daa se o apply our echiques o he liear combiaio of secorally decomposed variables. Table 5 preses he resuls usig quarerly price ad exchage rae daa for six couries: Caada, Frace, Ialy, Japa, U.K., ad U.S. for he period of 1974 Q1 hrough 1998 Q4. Wih he Caadia dollar used as umeraire, Table 5 preses he esimaes for b which should be close o uiy accordig o log-ru PPP. For raded goods, esimaes are above uiy i mos cases, bu he variaio across esimaes does o seem subsaial, resulig i o-rejecio of he ull of coiegraio i all cases cosidered. By sharp coras, he Hausma-ype coiegraio es rejecs he ull hypohesis i every coury whe he price for o-raded goods is used. I is oeworhy ha here exiss a cosiderable differece bewee he GLS-correced esimaes for b ad heir DOLS ad FGLS couerpars which are far greaer ha uiy. Tha is, supporive evidece of 1 For deails, see he Appedix for he descripio of he daa. We hak J.B. Kim for sharig he daa se.

17 C.-Y. Choi e al. / Joural of Ecoomerics 14 (8) PPP is foud for raded goods bu o for o-raded goods, cogrue wih he geeral iuiio as well as he fidigs by oher sudies i he lieraure such as Kakkar ad Ogaki (1999) ad Kim (5) Cocludig remarks ad fuure work I his paper, we aalyzed a approach o correcig spurious regressios ivolvig ui-roo osaioary variables by geeralized leas squares usig asympoic heory. This aalysis leads o a ew robus esimaor ad a ew es for dyamic regressios. We cosidered wo esimaors o esimae srucural parameers i spurious regressios: he GLS correced dyamic regressio esimaor suggesed by Choi ad Ogaki (1) ad he FGLS correced dyamic regressio esimaor. A GLS correced dyamic regressio esimaor is a firs differeced versio of a dyamic OLS regressio esimaor. Asympoic heory shows ha, uder some regulariy codiios, he edogeeiy correcio of he dyamic regressio works for he firs differeced regressios for boh coiegraig ad spurious regressios. This resul is useful because i is o iuiively clear ha he edogeeiy correcio works i regressios wih saioary firs differeced variables eve hough i has bee used for coiegraig regressios. For he purpose of he esimaig srucural parameers whe he possibiliy of osaioary measureme error or osaioary omied variables cao be ruled ou, we recommed he FGLS correced dyamic regressio esimaors because hey are robus. They are cosise boh whe he error is IðÞ ad Ið1Þ. They are asympoically as efficie as dyamic OLS whe he error is IðÞ ad as efficie as GLS correced dyamic regressio whe he error is Ið1Þ. This feaure may be especially aracive whe he FGLS correced dyamic esimaor is exeded o a pael daa seig. Pas works o osaioary ime series paels assume ha all regressios i a pael are eiher coiegraig or spurious. However, whe he umber of crosssecioal observaios icreases, i is very likely ha we may observe a IðÞ=Ið1Þ mixed pael, i.e., he regressio errors are IðÞ i some regressios ad Ið1Þ i ohers. Oe example is ha we may rejec PPP i some couries while o rejecig i i ohers. To esimae he srucural parameer i a IðÞ=Ið1Þ mixed pael, we ca firs ru a FGLS correcio of each idividual equaio, so ha he pooled pael esimaor always akes he fases covergece rae. Hu (5) sudies his exesio. We also developed a Hausma-ype coiegraio es by comparig he dyamic OLS regressio esimaor ad he GLS correced dyamic regressio esimaor. As oed i he iroducio, his ask is impora o merely because few ess for coiegraio have bee developed for dyamic OLS, bu also because ess for he ull hypohesis of coiegraio are useful i may applicaios. For his es, he spurious regressio obaied uder he aleraive hypohesis does o have o be srucural. We demosraed our esimaio ad esig mehods i hree applicaios: (i) log-ru moey demad i he U.S., (ii) oupu covergece amog idusrial ad developig couries, ad (iii) PPP for raded ad o-raded goods. I he firs applicaio of esimaig he moey demad fucio, he resuls sugges ha he edogeeiy correcio of he dyamic regressio works wih a moderaely large umber of leads ad lags for he GLS correced dyamic regressio esimaor. The GLS correced dyamic regressio esimaes of he icome elasiciy of moey demad are very low wih low orders of leads ad lags, ad he icrease o more plausible values as he order of leads ad lags icreases. Dyamic OLS esimaes are close o he GLS correced dyamic regressio esimaes for a large eough order of leads ad lags, ad we fid lile evidece agais coiegraio wih he Hausma-ype coiegraio es. The FGLS correced dyamic regressio esimaes are very close o he GLS correced dyamic regressio esimaes ad he dyamic OLS esimaes for sufficiely large orders of leads ad lags. Hece, i he firs applicaio, he FGLS correced dyamic regressio esimaor works well i he sese ha i yields esimaes ha are close o hose of he esimaor ha seems o be correcly specified. This is cofirmed by our simulaio resuls i Secio, idicaig ha he 14 Egel (1999) fids lile evidece for log-ru PPP for raded goods wih his variace decomposiio mehod. However, i should be oed ha his mehod is desiged o sudy variaios of real exchage raes over relaively shorer periods. Park ad Ogaki (7) show ha his variace decomposiio has a uexpeced propery whe he variace of log-ru differeces is used ad provides lile iformaio abou log-ru variaios of real exchage raes.