Nonlinear IV unit root tests in panels with cross-sectional dependency

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1 Journal of Econometrcs 11 (22) Nonlnear IV unt root tests n panels wth cross-sectonal dependency Yoosoon Chang Department of Economcs-MS22, Rce Unversty, 61 Man Street, Houston, TX , USA Abstract We propose a unt root test for panels wth cross-sectonal dependency. We allow general dependency structure among the nnovatons that generate data for each of the cross-sectonal unts. Each unt may have derent sample sze, and therefore unbalanced panels are also permtted n our framework. Yet, the test s asymptotcally normal, and does not requre any tabulaton of the crtcal values. Our test s based on nonlnear IV estmaton of the usual augmented Dckey Fuller type regresson for each cross-sectonal unt, usng as nstruments nonlnear transformatons of the lagged levels. The actual test statstc s smply dened as a standardzed sum of ndvdual IV t-ratos. We show n the paper that such a standardzed sum of ndvdual IV t-ratos has lmt normal dstrbuton as long as the panels have large ndvdual tme seres observatons and are asymptotcally balanced n a very weak sense. We may have the number of cross-sectonal unts arbtrarly small or large. In partcular, the usual sequental asymptotcs, upon whch most of the avalable asymptotc theores for panel unt root models heavly rely, are not requred. Fnte sample performance of our test s examned va a set of smulatons, and compared wth those of other commonly used panel unt root tests. Our test generally performs better than the exstng tests n terms of both nte sample szes and powers. We apply our nonlnear IV method to test for the purchasng power party hypothess n panels. c 22 Elsever Scence B.V. All rghts reserved. JEL classcaton: C1; C15; C32; C33 Keywords: Panels wth cross-sectonal dependency; Unt root tests; Nonlnear nstruments; Average IV t-rato statstcs 1. Introducton It s now wdely perceved that the panel unt root test s mportant. The test helps us to answer some of the mportant economc questons lke growth convergence and Correspondng author. Tel.: ; fax: E-mal address: yoosoon@rce.edu (Yoosoon Chang) /2/$ - see front matter c 22 Elsever Scence B.V. All rghts reserved. PII: S (2)95-7

2 262 Yoosoon Chang / Journal of Econometrcs 11 (22) dvergence, and purchasng power party (PPP), among many others. Moreover, t also provdes a means to mprove the power of the unt root test, whch s known to often yeld very low dscrmnatory power f performed on ndvdual tme seres. A number of unt root tests for panel data are now avalable n the lterature. Examples nclude the tests proposed by Levn et al. (1997), Im et al. (1997), Maddala and Wu (1999), Cho (21) and Chang (2). The reader s referred to Banerjee (1999) for some detaled dscussons on the exstng panel unt root tests and other related ssues. Rather unsatsfactorly, however, most exstng panel unt root tests assume crosssectonal ndependence, whch s qute restrctve gven the nature of economc panel data. Such tests are, of course, lkely to yeld based results f appled to the panels wth cross-sectonal dependency. Maddala and Wu (1999) conduct a set of smulatons to evaluate the performances of the commonly used panel unt root tests that are developed under the cross-sectonal ndependence when n fact the panel s spatally dependent. They, n partcular, show that the panel unt root tests based on ndependence across cross-sectonal unts, such as those consdered n Levn et al. (1997) and Im et al. (1997), perform poorly for cross-sectonally correlated panels. The cross-sectonal dependency s very hard to deal wth n nonstatonary panels. In the presence of cross-sectonal dependency, the usual Wald type unt root tests based upon the OLS and GLS system estmators have lmt dstrbutons that are dependent n a very complcated way upon varous nusance parameters denng correlatons across ndvdual unts. There does not exst any smple way to elmnate the nusance parameters n such systems. Ths was shown n Chang (2). None of the exstng tests, except for Chang (2) whch reles on the bootstrap method, successfully overcomes the nusance parameter problem n panels wth cross-sectonal dependence. In ths paper, we take the IV approach to solve the nusance parameter problem for the unt root test n panels wth cross-sectonal dependency. Our approach here s based upon nonlnear IV estmaton of the autoregressve coecent. We rst estmate the AR coecent from the usual augmented Dckey Fuller (ADF) regresson for each cross-sectonal unt usng the nstruments generated by an ntegrable transformaton of the gven tme seres. We then construct the t-rato statstc for testng the unt root based on the nonlnear IV estmator for the AR coecent. We show for each cross-sectonal unt that such nonlnear IV t-rato statstc for testng the unt root has lmtng standard normal dstrbuton under the null hypothess, just as n the statonary alternatve cases. The asymptotc normalty under the null ndeed establshes contnuty of the lmt theory for the t-statstc over the entre parameter space coverng both null and alternatve hypotheses. Ths clearly makes a drastc contrast wth the lmt theory of the standard t-statstc based on the ordnary least-squares estmator. More mportantly, we show that the ndvdual IV t-rato statstcs are asymptotcally ndependent even across dependent cross-sectonal unts. The cross-sectonal ndependence of the ndvdual IV t-rato statstcs follows readly from the asymptotc orthogonalty for the nonlnear transformatons of ntegrated processes by an ntegrable functon, whch s establshed n Chang et al. (21). We are therefore led to consder the average of these ndependent ndvdual IV t-rato statstcs as a statstc for testng jont unt root null hypothess for the entre panel. The actual test statstc s smply dened as a standardzed sum of the ndvdual IV t-ratos. We show n the paper that

3 Yoosoon Chang / Journal of Econometrcs 11 (22) such a normalzed sum of the ndvdual IV t-ratos has standard normal lmt dstrbuton as long as T mn and Tmax 1=4 log T max =T 3=4 mn, where T mn and T max denote, respectvely, the mnmum and maxmum numbers of the tme seres observatons T s for the cross-sectonal unts =1;:::;N. The usual sequental asymptotcs, upon whch most of the avalable asymptotc theores for panel unt root models heavly rely, are therefore not requred. We may thus allow the number of cross-sectonal unts to be arbtrarly small or large. 1 Our test s applcable for all panels that have large numbers of ndvdual tme seres observatons and are asymptotcally balanced n the sense mentoned above. Fnte sample performance of our average IV t-rato statstc, whch we call S N statstc, s examned va a set of smulatons, and compared wth that of the commonly used average statstc t-bar by Im et al. (1997). Our test generally performs better than the t-bar test n terms of both nte sample szes and powers. The smulatons conducted ndeed corroborate the standard normal lmt theory we provde here. The nte sample szes of S N are computed usng the standard normal crtcal values, and shown to very well approxmate the nomnal szes. Ths s qute contrary to the well-known nte sample sze dstortons of the t-bar test, see Maddala and Wu (1999) for example. The dscrmnatory powers of S N are yet notceably hgher than the t-bar test. We also apply our nonlnear IV method to test for the PPP hypothess usng the data sets from Papell (1997) and Oh (1996). Our test S N supports strongly the PPP relatonshps, contrary to most of the prevous emprcal ndngs whch are usually nconclusve. The rest of the paper s organzed as follows. Secton 2 ntroduces the model, assumptons and background theory. Secton 3 presents the nonlnear IV estmaton of the augmented autoregresson and derves the lmt theory for the nonlnear IV t-rato statstcs for each cross-sectonal unt. In Secton 4, we ntroduce a nonlnear IV panel unt root test and establsh ts lmt theory. It s, n partcular, shown that the test s asymptotcally standard normal. Secton 5 extends our nonlnear IV methodology to models wth determnstc components such as constant and lnear tme trend. In Secton 6, we conduct smulatons to nvestgate nte sample performance of the average IV t-rato statstc. Secton 7 provdes emprcal llustratons for testng the PPP usng our nonlnear panel IV unt root test. Secton 8 concludes, and mathematcal proofs are provded n the Appendx. 2. Model, assumptons and background theory We consder a panel model generated as the followng rst-order autoregressve regresson: y t = y ;t 1 + u t ; =1;:::;N; ;:::;T : (1) As usual, the ndex denotes ndvdual cross-sectonal unts, such as ndvduals, households, ndustres or countres, and the ndex t denotes tme perods. The number of tme seres observatons T for each ndvdual may der across cross-sectonal unts. 1 The asymptotcs developed here are T -asymptotcs. Throughout the paper, we assume that N s xed.

4 264 Yoosoon Chang / Journal of Econometrcs 11 (22) Hence, unbalanced panels are allowed n our model. We are nterested n testng the unt root null hypothess, = 1 for all y t gven as n (1), aganst the alternatve, 1 for some y t ;=1;:::;N. Thus, the null mples that all y t s have unt roots, and s rejected f any one of y t s s statonary wth 1. The rejecton of the null therefore does not mply that the entre panel s statonary. The ntal values (y 1 ;:::;y N )of(y 1t ;:::;y Nt ) do not aect our subsequent asymptotc analyss as long as they are stochastcally bounded, and therefore we set them at zero for expostonal brevty. It s assumed that the error term u t n model (1) s gven by an AR(p ) process speced as (L)u t = t ; (2) where L s the usual lag operator and p (z)=1 ;k z k for =1;:::;N. Note that we let (z) and p (whch s assumed to be xed) vary across, thereby allowng heterogenety n ndvdual seral correlaton structures. We assume: Assumpton 2.1. For =1;:::;N; (z) for all z 6 1. Under Assumpton 2.1, the AR(p ) process u t s nvertble, and has a movngaverage representaton u t = (L) t ; where (z)= (z) 1 and s gven by (z)= ;k z k : k= We allow for the cross-sectonal dependency through the cross-correlaton of the nnovatons t ;=1;:::;N, that generate the errors u t s. To dene the cross-sectonal dependency more explctly, we dene ( t ) T by t =( 1t ;:::; Nt ) (3) and denote by the Eucldean norm: for a vector x =(x ); x 2 = x2, and for a matrx A =(a j ); A = ;j a2 j. The data generatng process for the nnovatons ( t )s assumed to satsfy the followng assumpton. Assumpton 2.2. ( t )sand(;) sequence of random varables wth E t for some 4; and ts dstrbuton s absolutely contnuous wth respect to Lebesgue measure and has characterstc functon such that lm r () =; for some r. Assumpton 2.2 s strong, but s stll satsed by a wde class of data generatng processes ncludng all nvertble Gaussan ARMA models. Note that here the errors are

5 Yoosoon Chang / Journal of Econometrcs 11 (22) assumed to be d across tme perods. However, they are allowed to be cross-sectonally dependent. The techncal assumpton on the characterstc functon s requred for our subsequent asymptotcs on nonlnear functons of ntegrated processes as used n Park and Phllps (1999, 21). Dene a stochastc processes U T for t as [Tr] U T (r)=t 1=2 t on [; 1], where [s] denotes the largest nteger not exceedng s. The process U T (r) takes values n D[; 1] N, where D[; 1] s the space of cadlag functons on [; 1]. Under Assumpton 2.2, an nvarance prncple holds for U T, vz., U T d U as T, where U s an N -dmensonal vector Brownan moton wth covarance matrx. It s also convenent to dene B T (r) from u t =(u 1t ;:::;u Nt ), smlarly as U T (r). Then we have B T d B, where B =(B 1 ;:::;B N ) and B = (1)U. Ths s shown n Phllps and Solo (1992). Our theory reles heavly on the local tme of Brownan moton as n Park and Phllps (1999, 21), Chang and Park (1999) and Chang et al. (21). The reader s referred to these papers and the references cted there for the concept of local tme and ts use n the asymptotcs for nonlnear models wth ntegrated tme seres. To dene local tmes that appear n our lmt theory more precsely, we rst wrte the lmt vector Brownan moton gven n (4) explctly as U(r)=(U 1 (r);:::;u N (r)). We denote by L the (scaled) local tme of U, for =1;:::;N, whch s dened by 1 t L (t; s) = lm 1{ U (r) s j} dr: j 2j The local tme L s therefore the tme that the Brownan moton U spends n the neghborhood of s, upto tme t, measured n chronologcal unts. 2 Then we may have an mportant relatonshp t G(U (r)) dr = (4) G(s)L (t; s)ds (5) whch we refer to as the occupaton tmes formula. In addton to the Brownan motons U =(U 1 ;:::;U N ), we need to ntroduce another set of standard Brownan motons W =(W 1 ;:::;W N ). Throughout the paper, the Brownan moton W wll be assumed to be standard vector Brownan moton that s ndependent of U. We now ntroduce the class of regularly ntegrable transformatons n R, whch plays an mportant role n the subsequent development of our theory. 2 Usually, the local tme s dened n unts of quadratc varaton tme. Therefore, the local tme L U of U dened n the usual manner s gven by L U = 2L n terms of our local tme L, where 2 s the varance of U.

6 266 Yoosoon Chang / Journal of Econometrcs 11 (22) Denton 2.3. A transformaton G on R s sad to be regularly ntegrable f G s a bounded ntegrable functon such that for some constants c and k 6=( 2) wth 4 gven n Assumpton 2.2; G(x) G(y) 6 c x y k on each pece A of ts support A = m =1 A R. The regularly ntegrable transformatons are roughly ntegrable functons that are reasonably smooth on each pece of ther supports. The requred smoothness depends on the moment condton of the nnovaton sequence ( t ). Let be the maxmum order of the exstng moments. If 8, any pecewse Lpschtz contnuous functon s allowed. For the ndcator functon on a compact nterval to be regularly ntegrable, on the other hand, t s sucent to have 4. The denton of the regularly ntegrable functon n Denton 2.3 s dentcal to the one ntroduced n Park and Phllps (1999, 21). The asymptotc behavors of the nonlnear functons of an ntegrated tme seres are analyzed by Park and Phllps (1999, 21). For (y t ) generated as n (1), they provde, n partcular, the asymptotc theores for the sample moments gven by T G(y t) and T G(y ;t 1) t, whch are referred to n ther paper as the mean and covarance asymptotcs, respectvely, for varous types of functon G. Our subsequent theory s based upon the mean and covarance asymptotcs for G regularly ntegrable. The condtons n Assumpton 2.2 are requred to obtan the relevant asymptotcs. They are stronger than those requred for the usual unt root asymptotcs, because we need the convergence and nvarance of the sample local tme, as well as those of the sample Brownan moton, for the asymptotcs of ntegrable transformatons of ntegrated tme seres. We now obtan the Beverdge Nelson representatons for u t and y t. Let (1)=1 p ;k. Then t s ndeed easy to get u t = 1 (1) t + p = (1) t +(ũ ;t 1 ũ t ); p j=k ;j (u ;t k u ;t k+1 ) (1) where (1)=1= (1) and ũ t = p ;ku ;t k+1, wth ;k = (1) p j=k ;j. Under our condton n Assumpton 2.1, (ũ t ) s well dened both n a.s. and L r sense [see Brockwell and Davs (1991, Proposton 3.1.1)]. Under the unt root hypothess =1, we may now wrte y t = t u k = (1) t +(ũ ũ t ); (6) where t = t k, for all =1;:::;N. Consequently, y t behaves asymptotcally as the constant (1) multple of t. Note that ũ t s stochastcally of smaller order of magntude than t, and therefore wll not contrbute to our lmt theory.

7 Yoosoon Chang / Journal of Econometrcs 11 (22) Usng the speccaton of the regresson error u t gven n (2), we wrte model (1) as p y t = y ;t 1 + ;k u ;t k + t : Snce y t = u t under the unt root null hypothess, the above regresson may be wrtten as p y t = y ;t 1 + ;k y ;t k + t (7) on whch our unt root test wll be based. 3. IVestmaton and lmt theory In ths secton, we consder the IV estmaton of the augmented autoregresson (7). To deal wth the cross-sectonal dependency, we use the nstrument generated by a nonlnear functon F as F(y ;t 1 ) for the lagged level y ;t 1. For the lagged derences x t=(y ;t 1 ;:::;y ;t p ), we use the varables themselves as the nstruments. Hence for the entre regressors (y ;t 1 ;x t), we use the nstruments gven by (F(y ;t 1 );x t) =(F(y ;t 1 ); y ;t 1 ;:::;y ;t p ) : (8) The transformaton F wll be called the nstrument generatng functon (IGF) throughout the paper. We assume that Assumpton 3.1. Let F be regularly ntegrable and satsfy xf(x)dx. Roughly speakng, the condton gven n Assumpton 3.1 requres that the nstrument F(y ;t 1 ) s correlated wth the regressor y ;t 1. It s shown n Phllps et al. (1999, Theorem 3.2(a)) that IV estmators become nconsstent when the nstrument s generated by a regularly ntegrable functon F such that xf(x)dx=. In ths case, the IGF F s orthogonal to the regresson functon, whch s the dentty n ths case, n the Hlbert space L 2 (R) of square ntegrable functons. In the standard statonary regresson, an nstrument s nvald and the resultng IV estmator becomes nconsstent f, n partcular, t s uncorrelated wth the regressor. Such an nstrument falure also arses n our nonstatonary regresson wth an ntegrated regressor when the IGF s orthogonal to the regresson functon. Examples of the regularly ntegrable IGFs satsfyng Assumpton 3.1 nclude 1{ x 6 K}, any ndcator functon on a compact nterval dened by a truncaton parameter K, and ts varates such as sgn(x)1{ x K} and x1{ x K}. Also ncluded are functons of the type xe x. The IV estmator, for example, constructed from the ndcator functon F(x)= 1{ x 6 K} s smply the trmmed OLS estmator,

8 268 Yoosoon Chang / Journal of Econometrcs 11 (22) e., the OLS estmator whch uses only the observatons takng values n the nterval [ K; K]. Dene y = y ;p+1. y ;T ; y = y ;p. y ;T 1 ; X = x ;p +1. x ;T ; = ;p+1 where x t =(y ;t 1 ;:::;y ;t p ). Then the augmented autoregresson (7) can be wrtten n matrx form as y = y + X + = Y + ; (9) where =( ;1 ;:::; ;p ) ;Y =(y ;X ), and =( ; ). For the augmented autoregresson (9), we consder the estmator ˆ of gven by ( ) ( ˆ F(y ) y ˆ = =(W Y ) 1 W F(y ) ) 1 ( X F(y ) ) y y = ; (1) ˆ X y X X X y where W =(F(y );X ) wth F(y )=(F(y ;p );:::;F(y ;T 1)). The estmator ˆ s thus dened to be the IV estmator usng the nstruments W. The IV estmator ˆ for the AR coecent corresponds to the rst element of ˆ gven n (1). Under the null, we have ˆ 1=B 1 T A T ; (11) where A T = F(y ) F(y ) X (X X ) 1 X T T = F(y ;t 1 ) t F(y ;t 1 )x t ( T B T = F(y ) y F(y ) X (X X ) 1 X y T T = F(y ;t 1 )y ;t 1 F(y ;t 1 )x t and the varance of A T s gven by 2 EC T x t x t ( T ) 1 T x t x t under Assumpton 2.2, where C T = F(y ) F(y ) F(y ) X (X X ) 1 X F(y ) T T = F(y ;t 1 ) 2 F(y ;t 1 )x t ( T x t x t ) 1 T x t t ) 1 T x t y ;t 1 x t F(y ;t 1 ):. ;T ;

9 Yoosoon Chang / Journal of Econometrcs 11 (22) For testng the unt root hypothess H : = 1 for each =1;:::;N, we construct the t-rato statstc from the nonlnear IV estmator ˆ dened n (11). More speccally, we construct such an IV t-rato statstc for testng for a unt root n (1) or (7) as Z = ˆ 1 s(ˆ ) ; (12) where s(ˆ ) s the standard error of the IV estmator ˆ gven by s(ˆ ) 2 =ˆ 2 B 2 T C T : The ˆ 2 s the usual varance estmator gven by T 1 resdual from the augmented regresson (7), vz., p ˆ t = y t ˆ y ;t 1 ˆ ;k y ;t k = y t ˆ y ;t 1 x t ˆ : (13) T ˆ2 t, where ˆ t s the tted It s natural n our context to use the IV estmate ( ˆ ; ˆ ) gven n (1) to get the tted resdual ˆ t. However, we may obvously use any other estmator of ( ; ) as long as t yelds a consstent estmate for the resdual error varance. To derve the lmt null dstrbuton of the IV t-rato statstc Z ntroduced n (12), we need to obtan the asymptotcs for varous sample product moments appearng n A T ;B T and C T. For ths, we need to ntroduce a set of ndependent standard Brownan motons W 1 ;:::;W N ;x whch are ndependent of the Brownan motons U 1 ;:::;U N. The lmt theores are presented n the followng lemma. Lemma 3.2. Under Assumptons 2.1; 2.2 and 3.1; we have (a) T 1=4 T (b) T 1=2 (c) T 3=4 F(y ;t 1) t d ( (1) L (1; ) F(s)2 ds) 1=2 W (1); T F(y ;t 1) 2 d (1) L (1; ) F(s)2 ds; T F(y ;t 1)y ;t k p ; for k =1;:::;p jontly as T ; where (1)=1 p ;k. The results n Lemma 3.2 are smple extensons of the results n parts (c), () and (e) of Lemma 5 n Chang and Park (1999). For the detaled dscusson on the asymptotcs here, the reader s referred to Park and Phllps (1999, 21) and Chang et al. (21). For the regularly ntegrable IGF F, the covarance asymptotcs yelds a mxed normal lmtng dstrbuton wth a mxng varate dependng upon the local tme L of the lmt Brownan moton U, as well as the ntegral of the square of the transformaton functon F. It s very useful to note that T 1=4 T 1=2 T T 1 4 F(y ;t 1 ) t d T F( T B T )du ; 1 F(y ;t 1 ) 2 d T F( T B T ) 2

10 27 Yoosoon Chang / Journal of Econometrcs 11 (22) from whch we may easly deduce the results n parts (a) and (b) of Lemma 3.2 usng elementary martngale theory as n Park and Phllps (1999, 21) and Chang et al. (21). The lmt null dstrbuton of the IV t-rato statstc Z dened n (12) now follows readly from the results n Lemma 3.2. Theorem 3.3. Under Assumpton 2.1; 2.2 and 3.1; we have Z d W (1) N(; 1) as T for all =1;:::;N. The lmtng dstrbuton of the IV t-rato Z for testng = 1 s standard normal f a regularly ntegrable functon s used as an IGF. Moreover, the lmt standard normal dstrbutons, W (1) s, are ndependent across cross-sectonal unts =1;:::;N,aswe show n the next secton. Our lmt theory here s thus fundamentally derent from the usual unt root asymptotcs. Ths s due to the local tme asymptotcs and mxed normalty of the results n Lemma 3.2. The nonlnearty of the IV s essental for our Gaussan lmt theory. The result n Theorem 3.3, and thereby any of the subsequent result, s not applcable for the usual lnear unt root tests such as those by Phllps (1987) and Phllps and Perron (1988). We now consder the lmt behavor of our IV t-rato statstc under the alternatve of statonarty to dscuss the consstency of the test. Note that under the alternatve,.e., = 1, our IV t-rato Z gven n (12) can be expressed as T ( 1) Z = Z ( )+ ; (14) T s(ˆ ) where s(ˆ ) s dened n (13) and Z ( )= ˆ (15) s(ˆ ) whch s the IV t-rato statstc for testng = 1. Under the alternatve, we may expect that Z ( ) d N(; 1) f the usual mxng condtons for (y t ) are assumed to hold. Moreover, f we let B = plm T 1 T B T and C = plm T 1 T C T exst under sutable mxng condtons for (y t ), then the second term n the rght-hand sde of Eq. (14) dverges to at the rate of T. Ths s because T ( 1) and T s(ˆ ) p ; where 2 = 2 B 2 C. Hence, the IV t-rato Z dverges at the T -rate under the alternatve of statonarty, just as n the case of the usual OLS-based t-type unt root tests such as the augmented Dckey Fuller test. We also note that the IV t-ratos constructed wth regularly ntegrable IGF are normally dstrbuted asymptotcally, for all 6 1. The contnuty of the dstrbuton across the values of of the t-rato Z ( ) dened n (15) also allows us to construct the condence ntervals for from the IV estmator. As we have notced above, Z ( ) d N(; 1) for all values of ncludng unty, when the IGF F s a regularly

11 Yoosoon Chang / Journal of Econometrcs 11 (22) ntegrable functon. We may therefore construct 1(1 )% asymptotc condence nterval for as [ˆ z =2 s(ˆ ); ˆ + z =2 s(ˆ )] usng the IV estmator generated by any ntegrable functon F, where z =2 s the (1 =2)-percentle from the standard normal dstrbuton. Ths s one mportant advantage of usng the nonlnear IV method. The OLS-based standard t-rato has non-gaussan asymptotc null dstrbuton, called the Dckey Fuller dstrbuton. It s asymmetrc and skewed to the left, as tabulated n Fuller (1996). Therefore, the condence nterval whch s vald for all 6 1 cannot be constructed from the OLS based t-rato. 4. Panel nonlnear IVunt root test The test statstc we propose here to test for the unt root hypothess n a panel s bascally an average of t-rato statstcs for testng the unty of the AR coecent computed ndvdually from each cross-sectonal unt. More speccally, we test for the jont unt root null hypothess H : = 1 for all =1;:::;N usng an average statstc based on the ndvdual t-statstcs for testng = 1 n (7) constructed from the nonlnear IV estmator ˆ dened n (11). The average IV t-rato statstc s thus dened as S N = 1 N N =1 Z ; where Z s the ndvdual nonlnear IV t-rato statstc, dened n (12), for testng = 1 for the th cross-sectonal unt. For the average statstc S N, we allow each of the cross-sectonal unts =1;:::;N to have a derent sample sze T, and therefore unbalanced panels are permtted n our framework. Our test s based on nonlnear IV estmaton of the usual ADF type regresson for each cross-sectonal unt, usng as nstruments nonlnear transformatons of the lagged levels y ;t 1 s. In order to derve the lmt theory for the statstc S N, we rst nvestgate how the ndvdual IV t-rato statstcs Z s nteract n the lmt. We have T 1=4 T 1=4 j T T j 1 4 F(y ;t 1 ) t d T F( T B T )du ; 1 4 F(y j;t 1 ) jt d Tj F( T j B jtj )du j ; whch become asymptotcally ndependent f ther quadratc covaraton 1 4 j T T j F( T B T (r))f( T j B jtj (r)) dr (16)

12 272 Yoosoon Chang / Journal of Econometrcs 11 (22) converges a.s. to zero, where j denotes the covarance between U and U j. Ths was shown n Chang et al. (21). Below we ntroduce a sucent condton and establsh ther asymptotc ndependence subsequently. Let T mn and T max, respectvely, be the mnmum and the maxmum of T s for =1;:::;N. Assumpton 4.1. Assume T mn Then we have and T 1=4 max log T max T 3=4 mn : Lemma 4.2. Under Assumptons 2.1; 2.2; 3.1 and 4.1; the followng holds: 1 4 T T j F( T B T (r))f( T j B jtj (r)) dr p (17) and the results n Lemma 3.2 hold jontly for all =1;:::;N wth ndependent W s across =1;:::;N. The result n Lemma 4.2 s new, and shows that the product of the nonlnear nstruments F(y ;t 1 ) and F(y j;t 1 ) from derent cross-sectonal unts and j are asymptotcally uncorrelated, even when the varables y ;t 1 and y j;t 1 generatng the nstruments are correlated. Ths mples that the ndvdual IV t-rato statstcs Z and Z j constructed from the nonlnear IV s F(y ;t 1 ) and F(y j;t 1 ) are asymptotcally ndependent. Ths asymptotc orthogonalty plays a crucal role n developng lmt theory for our panel unt root test S N dened above, as can be seen below. The lmt theory for S N follows mmedately from Theorem 3.3 and Lemma 4.2, and s provded n Theorem 4.3. We have S N d N(; 1) under Assumptons 2.1; 2.2; 3.1 and 4.1. The lmt theory s derved usng T -asymptotcs only. It holds as long as all T s go to nnty and T s are asymptotcally balanced n a very weak sense, as we specfy n Assumpton 4.1. It should be noted that the usual sequental asymptotcs s not used here. 3 The factor N 1=2 n the denton of the test statstc S N n (16) s used just as a normalzaton factor, snce S N s based on the sum of N ndependent random varables. Therefore, the dmenson of the cross-sectonal unts N may take any value, small as well as large. The above result mples that we can do smple nference based on the standard normal dstrbuton even for unbalanced panels wth general cross-sectonal dependences. 3 The usual sequental asymptotcs s carred out by rst passng T to nnty wth N xed, and subsequently let N go to nnty, usually under cross-sectonal ndependence.

13 Yoosoon Chang / Journal of Econometrcs 11 (22) The normal lmt theory s also obtaned for the exstng panel unt root tests, but the theory holds only under cross-sectonal ndependence, and obtaned only through sequental asymptotcs. For example, the pooled OLS test by Levn et al. (1997) and the groupmean t-bar statstc by Im et al. (1997) have normal asymptotcs. However, they all presume cross-sectonal ndependence and ther normal lmt theores are obtaned through sequental asymptotcs. 4 The ndependence assumpton was crucal for ther tests to have normal lmtng dstrbutons, snce the ndvdual t-statstcs contrbutng to the average become ndependent only when the nnovatons t generatng the ndvdual unts are ndependent. Moreover, the sequental asymptotcs s an essental tool to derve ther results, and they do not provde jont asymptotcs. Here we acheve the asymptotc ndependence of ndvdual t-statstcs by establshng asymptotc orthogonaltes of the nonlnear nstruments used n the constructon of the ndvdual IV t-rato statstcs wthout havng to mpose ndependence across cross-sectonal unts, or relyng on sequental asymptotcs. 5. Nonlnear IVestmaton for models wth determnstc trends The models wth determnstc components can be analyzed smlarly usng properly demeaned or detrended data. A proper demeanng or detrendng scheme requred here must be able to successfully remove the nonzero mean and tme trend, whle mantanng the martngale property of the errors and ultmately the Gaussan lmt theory for our nonlnear IV unt root tests. We now ntroduce our demeanng and detrendng schemes. If the tme seres (z t ) wth a nonzero mean s gven by z t = + y t ; (18) where the stochastc component (y t ) s generated as n (1), then we may test for the presence of a unt root n (y t ) from the augmented regresson (7) dened wth the demeaned seres y t and y ;t 1 of z t and z ;t 1, vz., p y t = y ;t 1 + ;k y ;t k + e t; (19) where y t = z t 1 t 1 z k ; (2) t 1 y ;t 1 = z ;t 1 1 t 1 z k ; (21) t 1 y ;t k =z ;t k; k =1;:::;p (22) and (e t ) are regresson errors. 4 They also consder the case N=T k, where k s a xed constant, but the relevant asymptotcs s not rgorously developed. Moreover, they ntroduce common tme eects to ther model, and thereby allow for lmted cross-sectonal dependency.

14 274 Yoosoon Chang / Journal of Econometrcs 11 (22) The term (1=(t 1)) t 1 z k s the least-squares estmator of obtaned from the prelmnary regresson z k = + y k for k =1;:::;t 1: We note that the parameter s estmated from model (18) usng the observatons upto tme t 1. That we use the data upto the perod t 1 only, nstead of usng the full sample, for the estmaton of the constant leads to the demeanng based on the partal sum of the data up to t 1 as gven n (2) and (21), whch we call adaptve demeanng. 5 Note from (2) that even for the tth observaton z t,weuse (t 1)-adaptve demeanng to mantan the martngale property. No further demeanng s needed for the lagged derences z ;t k ;=1;:::;p, snce the derencng has already removed the mean. We may then construct the nonlnear IV t-rato statstc Z based on the nonlnear IV estmator for from regresson (19), just as n (12). Wth the adaptve demeanng the predctablty of our nonlnear nstrument F(y ;t 1 ) s retaned, and consequently our prevous results contnue to apply, ncludng the normal dstrbuton theory for the IV t-rato statstc. We may also test for a unt root n the models wth more general determnstc tme trends. As n the cases wth the models wth nonzero means, we may derve nonlnear IV unt root test Z n the same manner. More explctly, consder the tme seres wth a lnear tme trend z t = + t + y t ; (23) where (y t ) s generated as n (1). Smlarly, we may test for the unt root n (y t ) from regresson (7) dened wth the properly detrended seres yt ;y ;t 1 and y t k of z t ;z ;t 1 and z ;t k ;;:::;p as p yt = y;t 1 + ;k y;t k + e t ; (24) where yt = z t + 2 t 1 6 t 1 z k kz k 1 z T ; (25) t 1 t(t 1) T y;t 1 = z ;t t 1 6 t 1 z k kz k ; (26) t 1 t(t 1) y ;t k =z ;t k 1 T z T ; k =1;:::;p (27) and (e t ) are regresson errors. The varables z t and z ;t 1 are detrended usng the least-squares estmators of the drft and trend coecents, and, from model (23) usng agan the observatons 5 Ths method was formerly used by So and Shn (1999) to demean postvely correlated statonary AR processes. They found that the method reduces the bases of the parameter estmators.

15 upto tme t 1 only, vz., Yoosoon Chang / Journal of Econometrcs 11 (22) z k = + k + y k for k =1;:::;t 1: The term z T =T appearng n the dentons of yt and y;t k gven n (25) and (27) s the grand sample mean of z t ; 1 T T z k. The term s used n (25) to elmnate the remanng drft term of z t +(2=(t 1)) t 1 z k (6=t(t 1)) t 1 kz k, and n (27) to remove the nonzero mean of z ;t k, for k =1;:::;p. The adaptve detrendng of the data as gven n (25) and (26) above preserves the predctablty of our nstrument F(y;t 1 ). The nonlnear IV t-rato statstc Z s then dened as n (12) from the nonlnear IV estmator for from the regresson (24). 6 We may now derve the lmt theores for the statstcs Z and Z n the smlar manner as we dd to establsh the lmt theory gven n Theorem 3.3. In order to dene the lmt dstrbutons properly, we rst ntroduce some notatons. Dene adaptvely demeaned Brownan moton by r U (r)=u (r) 1 U (s)ds r for =1;:::;N, and denote ts local tme by L scaled as for L. Smlarly, we also dene adaptvely detrended Brownan moton as U (r)=u (r)+ 2 r U (s)ds 6 r r r 2 su (s)ds and analogously denote ts local tme by L for =1;:::;N.IfweletU () = and U () =, then both processes have well-dened contnuous versons on [; ), as shown n the proof of Corollary 5.1. The lmt theores gven n Lemma 3.2 extend easly to the models wth nonzero means and determnstc trends f we replace the lagged level y ;t 1 wth the lagged detrended seres y ;t 1 and y ;t 1 dened, respectvely, n (21) and (26). They are ndeed gven smlarly wth the local tmes L and L of the adaptvely demeaned and detrended Brownan motons U and U n the place of the local tme L of the Brownan moton U. Then the lmt theores for the nonlnear IV t-rato statstcs Z and Z for the models wth nonzero means and determnstc trends follow mmedately, and are gven n Corollary 5.1. Under Assumpton 2.1; 2.2 and 3.1; we have Z ;Z d N(; 1) as T for all =1;:::;N. 6 The adaptve demeanng and detrendng, n partcular, make our statstcs ndependent of the startng values. Note that the nonlnear nstruments F(y ;t 1 ) and F(y ;t 1 ) are now generated, respectvely, from the adaptvely demeaned and detrended seres, (y ;t 1 ) and (y ;t 1 ), for t =2;:::;T, and that they are forced to start at the orgn,.e., F(y 1 )=F() and F(y 1 )=F() a.s.

16 276 Yoosoon Chang / Journal of Econometrcs 11 (22) The standard normal lmt theory of the nonlnear IV t-rato statstcs contnues to hold for the models wth determnstc components. 6. Smulatons We conduct a set of smulatons to nvestgate the nte sample performance of the average IV t-statstc S N based on ntegrable IGFs for testng the unt root null hypothess H : = 1 for all =1;:::;N aganst the statonarty alternatve H 1 : 1 for some. In partcular, we explore how close are the nte sample szes of the test S N n relaton to the correspondng nomnal test szes, usng the crtcal values from ts lmt N(; 1) dstrbuton, and compare ts szes and powers wth those of the commonly used average statstc t-bar proposed by Im et al. (1997). For the smulatons, we consder the tme seres (z t ) wth a drft gven by model (18) wth (y t ) generated as n (1) and (u t ) as AR(1) processes, vz., u t = u ;t 1 + t : (28) The nnovatons t =( 1t ;:::; Nt ) that generate u t =(u 1t ;:::;u Nt ) are drawn from an N-dmensonal multvarate normal dstrbuton wth mean zero and covarance matrx. The AR coecents, s, used n the generaton of the errors (u t ) are drawn randomly from the unform dstrbuton,.e., Unform[:2; :4]. The parameter values for the (N N ) covarance matrx =( j ) are also randomly drawn, but wth partcular attenton. To ensure that s a symmetrc postve dente matrx and to avod the near sngularty problem, we generate followng the steps outlned n Chang (2). The steps are presented here for convenence: (1) Generate an (N N ) matrx M from Unform[,1]. (2) Construct from M an orthogonal matrx H = M(M M) 1=2. (3) Generate a set of N egenvalues, 1 ;:::; N. Let 1 = r and N = 1 and draw 2 ;:::; N 1 from Unform[r; 1]. (4) Form a dagonal matrx wth ( 1 ;:::; N ) on the dagonal. (5) Construct the covarance matrx as a spectral representaton = HH. The covarance matrx constructed n ths way wll surely be symmetrc and nonsngular wth egenvalues takng values from r to 1. We set the maxmum egenvalue at 1 snce the scale does not matter. The rato of the mnmum egenvalue to the maxmum s, therefore, determned by the same parameter r. We now have some control over the sze of the mnmum egenvalue and the rato of the mnmum to the maxmum egenvalues through the choce of r. The covarance matrx becomes sngular as r tends to zero, and becomes sphercal as r approaches to 1. For the smulatons, we set r at :1. For the estmaton of the model (7) for =1;:::;N, we consder the IV estmator ˆ dened n (1) whch uses the nstrument (F(y ;t 1 ); y ;t 1 ;:::;y ;t p ). The

17 Yoosoon Chang / Journal of Econometrcs 11 (22) Fg. 1. IGF F(x)=xe x. nstrument used for the lagged level y ;t 1 s generated by the ntegrable IGF F(y ;t 1 )=y ;t 1 e c y; t 1 ; where the factor c s nversely proportonal to the sample standard error of y t = u t and T 1=2. That s, T c = KT 1=2 s 1 (y t ) wth s 2 (y t )=T 1 (y t ) 2 ; where K s a constant. The value of K s xed at 3 for all =1;:::;N, and for all combnatons of N and T consdered here. 7 We note that the factor c n the denton of the nstrument generatng functon F s data-dependent through the sample standard error of the derence of the data y t. Hence, the value of c wll be determned for each cross-sectonal unt =1;:::;N. The shape of the ntegrable IV generatng functon F s gven n Fg. 1. Our asymptotcs requres the factor c to be constant. For practcal applcatons, however, we found t desrable to make c dependent upon T as suggested n the prevous equaton. Wth the choce of larger (smaller) value of c, we may have better sze (power) at the cost of power (sze). Ths s well expected. Notce that the larger the value of c, the more ntegrable the IGF F becomes. Our asymptotcs thus better predcts nte sample behavor of the test. On the other hand, the test loses nte sample powers as the factor c gets larger. As s well known from the standard regresson 7 The test S N constructed from the IGF F wth a lager value of K tends to have smaller rejecton probabltes unformly over all the choces of N and T. The IGF dened wth K = 3 seems to work best overally, and thus chosen for our smulatons. For the cases where the tme dmenson s small T = 25, the average nonlnear IV test S N slghtly over-rejects the null. In such cases, one mght use a lttle larger value of K to correct the upward sze dstorton.

18 278 Yoosoon Chang / Journal of Econometrcs 11 (22) theory, the optmal IGF s gven by the dentty F(x)=x, whch reduces our nonlnear IV estmator to the OLS estmator. As the IGF F tends to be more ntegrable, the resultng nonlnear IV estmator becomes less ecent, whch may lead to the power loss n our test. To test the unt root hypothess, we set = 1 for all =1;:::;N, and nvestgate the nte sample szes n relaton to the correspondng nomnal test szes. To examne the rejecton probabltes under the alternatve, we generate s randomly from Unform[.8,1]. The model s thus heterogeneous under the alternatve. The nte sample performance of the average nonlnear IV t-rato statstc S N s compared wth that of the t-bar statstc by Im et al. (1997), whch s based on the average of the ndvdual t-tests computed from the sample ADF regressons (7) wth mean and varance modcatons. More explctly, the t-bar statstc s dened as N ( t N N 1 N =1 t-bar = E(t )) N ; 1 N =1 var(t ) where t s the t-statstc for testng = 1 for the th sample ADF regresson (7), and t N = N 1 N =1 t. The values of the expectaton and varance, E(t ) and var(t ), for each ndvdual t depend on T and the lag-order p, and computed va smulatons from ndependent normal samples. The number of tme seres observaton T for each =1;:::;N s requred to be the same. 8 The panels wth the cross-sectonal dmensons N =5; 15; 25; 5; 1 and the tme seres dmensons T =25; 5; 1 are consdered for the 1%, 5% and 1% sze tests. 9 Snce we are usng randomly drawn parameter values, we smulate 2 tmes and report the ranges of the nte sample performances of the average nonlnear IV t-rato statstc S N and the t-bar test. Each smulaton run s carred out wth 1, smulaton teratons. Tables 1, 2 and 3 report, respectvely, the nte sample szes, the nte sample rejecton probabltes and the sze adjusted nte sample powers of the two tests. For each statstc, we report the mnmum, mean, medan and maxmum of the rejecton probabltes under the null and the alternatve hypotheses. As can be seen from Table 1, the nte sample szes of the test S N are qute close to the correspondng nomnal szes. The szes are calculated usng the crtcal values from the standard normal dstrbuton, and therefore the smulaton results corroborate the asymptotc normal theory for S N. The lmt theory seems to provde reasonably good approxmatons even when the number of tme seres observaton s relatvely small,.e., when T = 25, for all of the cross-sectonal dmensons consdered. On the other hand, the t-bar statstc exhbts notceable sze dstortons, as reported, for 8 Table 2 n Im et al. (1997) tabulates the values of E(t ) and var(t ) for T = 5; 1; 15; 2; 25; 3; 4; 5; 6; 7; 1 and for p =;:::;8. 9 For smplcty we use the same T for all cross-sectonal unts n our smulatons. However, our theory does permt heterogenety n the number T of tme seres observatons. It s also true that the t-bar test can be practcally mplemented for unbalanced panels, though Im et al. (1997) do not explctly allow for heterogeneous T s n ther theoretcal developments.

19 Yoosoon Chang / Journal of Econometrcs 11 (22) Table 1 Fnte sample szes N T Tests 1% Test 5% Test 1% Test Mn Mean Med Max Mn Mean Med Max Mn Mean Med Max 5 25 t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N nstance, n the prevous smulaton work by Maddala and Wu (1999). The drecton of the sze dstortons are upward n all cases for all 1%, 5% and 1% tests. The t-bar statstc suers from severe sze dstortons especally when the number of cross-sectonal unts s large relatve to the number of tme seres observatons. For example, when N = 1 and T = 25, the average nte sample szes of the t-bar statstcs for 1%, 5% and 1% tests are, respectvely, 15%, 36% and 49%. The sze dstortons become less serous as the tme dmenson gets large; however, they are stll qute notceable. The test S N s more powerful than the t-bar statstc for all 1%, 5% and 1% tests and for all N and T combnatons consdered, as can be seen clearly from the results on the nte sample rejecton probabltes and the sze adjusted powers,

20 28 Yoosoon Chang / Journal of Econometrcs 11 (22) Table 2 Fnte sample rejecton probabltes N T Tests 1% Test 5% Test 1% Test Mn Mean Med Max Mn Mean Med Max Mn Mean Med Max 5 25 t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N reported, respectvely, n Tables 2 and 3. The dscrmnatory power of S N s notceably much hgher than that of the t-bar statstc for the cases wth smaller T and N. For the 1% tests wth the combnatons (N; T) ={(15; 25); (25; 25); (5; 5)}, the power of the test S N s more than twce as large as that of the t-bar statstc. The S N stll performs much better than the t-bar even when T s large, f the cross-sectonal dmenson s small. The performance of the t-bar statstc mproves as both N and T ncrease, though the mprovement s more notceable wth the growth n T. The derences n the nte sample powers of S N and t-bar vansh as both N and T ncrease.

21 Yoosoon Chang / Journal of Econometrcs 11 (22) Table 3 Fnte sample powers N T Tests 1% Test 5% Test 1% Test Mn Mean Med Max Mn Mean Med Max Mn Mean Med Max 5 25 t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N t-bar S N Emprcal llustratons In ths secton, we apply the newly developed panel unt root test S N to test whether the PPP hypothess holds. The PPP hypothess has been tested by many researchers usng varous unt root tests, both n panel as well as n unvarate models. Examples nclude MacDonald (1996), Frankel and Rose (1996), Oh (1996), Papell (1997), O Connell (1998), just to name a few. There have been, however, conctng evdence, and the ssue does not seem to be completely settled. We consder the data used n Papell (1997), whch conssts of the real exchange rates for 2 countres computed from the IMF s Internatonal Fnancal Statstcs (IFS)

22 282 Yoosoon Chang / Journal of Econometrcs 11 (22) Table 4 PPP tests for IFS data t-bar T =5 T = 1 T =5 T = 1 AR 2 1:589 b 1:183 2:872 a 2:554 a Order 4 7:18 a 4:525 a 5:969 a 5:119 a BIC 4 2:74 a 3:49 a 3:719 a 4:695 a Max order 8 1:127 2:646 a :249 3:937 a Note: The superscrpts a, b and c denote, respectvely, the statstcal sgncance at 1%, 5% and 1% levels. S N tape, coverng the perod 1973:1 1998:4. 1 We also consder the data from the Penn World Table (PWT) analyzed n Oh (1996). 11 The emprcal results are summarzed n Tables 4 and 5, respectvely, for the results obtaned from the data from Papell (1997) and Oh (1996). We allow the models to have heterogeneous dynamc structures,.e., the models may have derent AR orders for ndvdual cross-sectonal unts. For each cross-sectonal unt the AR order s selected usng the BIC crteron wth the maxmum number of lags 4 or 8 for the quarterly IFS data, and wth 2 or 4 for the annual PWT data. To see how senstve are the test results wth respect to the speccatons of ndvdual dynamcs, we also look at the panels wth homogeneous dynamcs, where we do not allow the AR order to vary across the ndvdual unts and x the AR order for all cross-sectonal unts at 2 or 4 for the IFS data and at 1 or 2 for the PWT data. For the analyss of the PWT data, we looked at four derent groups of countres. For each groupof countres, the numbers of the tme seres observatons are derent, varyng from 3 to The IFS data have total 14 tme seres observatons. To examne the dependency on the sample sze also for the test results from the IFS data set, we consdered two sub-samples of szes 5 and 1. The sub-samples are obtaned by retanng the most recent observatons. For both data from Papell (1997) and Oh (1996), our test strongly rejects the unt root hypothess, whch s used n the emprcal studes as an ndrect evdence for the PPP relatonshp. As seen from Tables 4 and 5, our test rejects the presence of the 1 The quarterly data used n Papell (1997) covers the perod 1973:1 1994:3, but the data used here s extended to 1998:4. The countres consdered nclude Austra, Belgum, Denmark, Fnland, France, Germany, Italy, Japan, Netherlands, Norway, Span, Sweden, Swtzerland, Unted Kngdom, Ireland, Australa, Greece, New Zealand, Portugal, and Canada. The real exchange rate r t for the th country s computed usng the US dollar as the numerare currency, and calculated as r t = log(e t p t=p t ), where e t ; p t and p t denote, respectvely, the nomnal spot exchange rate for the th country, the US CPI, and the CPI for the th country. 11 The data used n Oh (1996) are yearly observatons from the Penn World Table, Mark 5.5. The data are collected for 111 countres for the perod , and extended to a longer perod for a group of 51 countres. For the longer sample, the data are analyzed for two sub-samples, the 22 OECD countres and G6 countres (Canada, France, Germany, Italy, Japan and Unted Kngdom). 12 For the groupof 111 countres, there are 3 annual observatons. But for the groupof 51 countres (ncludng ts sub-samples of 22 OECD countres and G6 countres), there are 41 tme seres observatons.