Structural changes, common stochastic trends, and unit roots in panel data

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1 Structural changes, common stochastc trends, and unt roots n panel data Jushan Ba Department of Economcs New York Unversty Josep Lluís Carron--Slvestre Department of Econometrcs, Statstcs and Spansh Economy Unversty of Barcelona Frst draft: July 2th, 22 Revsed: December th, 23 Abstract In ths paper we propose a new test statstc that consders multple structural breaks to analyse the non-statonarty of a panel data set. he methodology s based on the common factor analyss n an attempt to allow for some sort of dependence across the ndvduals. hus allowng for multple structural breaks n the Panel Analyss of Non-statonarty n Idosyncratc and Common components (PANIC) methodology ncreases the degree of heterogenety when assessng the stochastc propertes of the panel data set. Keywords: multple structural breaks, common factors, panel data unt root test, prncpal components JEL codes: C2, C22, C3, C5 Introducton Nowadays, the ncreasng applcaton of the panel data technques to the determnaton of tme seres stochastc propertes has led to the development of a wde range of new proposals n the econometrc lterature. he short tme perod s coverage that o er most of the avalable macroeconomc tme seres may be thought as the man reason behnd ths explodng phenomenon. hs lack of nformaton, n terms of tme observatons, mples a loss n the power of unt root, statonarty and contegraton tests. he combnaton of the nformaton n the tme and cross-secton dmensons to compose a panel data set of ndvduals,.e. countres or regons, onto whch perform the analyss of the stochastc propertes has revealed as a promsng way to ncrease the power of these tests.

2 hus, a gan n power s expected when performng a statstcal nference unt root, statonarty or contegraton test usng a panel data set made up of ndvduals that share, at rst, some smlartes. Bretung and Meyer (994), Im, Pesaran and Shn (997), Maddala and Wu (999) and Levn, Ln and Chu (22), on the unt root tests, and Pedron (995) and Phllps and Moon (999), on the contegraton analyss, are some of the most relevant papers. Comprehensve surveys of the eld can be found n Banerjee (999), Baltag (2) and Baltag and Kao (2). Although the determnstc component should not be of nterest when analysng the order of ntegraton of the tme seres, ts msspec caton can drve to msleadng conclusons. hus, a statonary tme seres that evolves around a breakng-trend model mght be characterzed as a non-statonary process f the order of ntegraton analyss fals to consder the structural breaks see Perron (989) for the unvarate tme seres framework and Carron--Slvestre, del Barro and López-Bazo (2) for the panel data framework. Our proposal focus on the presence of multple structural breaks a ectng the panel data set, so that takng nto account the presence of these structural breaks overcomes the nterferences that can cause the msspec caton error n the stochastc propertes of the panel. In ths paper we analyse the presence of multple structural breaks when testng for the unt root hypothess n a panel data framework. Some of the recent proposals n the panel data based unt root and statonarty tests have addressed ths queston by developng sutable tests see Im, Lee and eslau (22) for the LM test and Carron--Slvestre et al. (2) for the DF wth one structural break, and Carron--Slvestre et al. (22) for the KPSS tests wth multple structural breaks. However, our approach overcomes the crtcsm that has rased the assumpton of cross-secton ndependence n whch most of the panel data based tests rely, and models the cross-secton dependence n terms of the common factors as n Ba and Ng (2, 24). Bre y speakng, the dea s to establsh a dstncton between comovements and dosyncratc shocks that may be a ectng the ndvdual tme seres. Flterng out the comovements wll reduce the nose n the system, so that, the analyss wll focus on those shocks that are spec c for each ndvdual. Moreover, note that the cross-secton ndependence s more lkely to be ful lled when usng these dosyncratc shocks than when usng the raw data. he rest of the paper s organzed as follows. Secton 2 descrbes the model and the two determnstc spec catons that are consdered along the paper. hese models arse because of the d erent e ects that the structural breaks may cause on the determnstc part of the model. Secton 4 presents d erent pooled tests, whle n Secton?? we analyse the nte sample performance. Fnally, Secton 6 concludes. All proofs are presented n the Appendx. 2

3 2 Panel unt root test wth multple structural breaks Let us de ne the panel data model gven by: X ;t = D ;t + F t ¼ + e ;t ; () (I L) Ft = C (L) u t ; (2) ( ½ L) e ;t = H (L) " ;t ; (3) t =;::: ;, =;::: ;N,whereC (L) = P j= C jl j and H (L) = P j= H ;jl j. D ;t denotes the determnstc part of the model, F t s a (l )-vector that accounts for the common factors that are present n the panel and e ;t s the dosyncratc dsturbance term. Our analyss s based on the same set of assumptons n Ba and Ng (24). Let M<beagenerc postve number, not dependng on and N: Assumpton A: () for non-random ¼, k¼ k M; for random ¼, E k¼ k 4 M, () P N N = ¼ ¼ p!,a(l l) postve matrx. P Assumpton B: () u t» d (; u ), E ku t k 4 M, and () Var( Ft)= j= C j u Cj >, () P j= j kc jk <M; and (v) C () has rank l, l l. Assumpton C: () for each, " ;t» d (; " ), E j" ;t j 8 M, P j= j jh ;jj < M,! 2 = H () 2 ¾ 2 > ; ()E (" ;t" j;t )= ;j wth P N = j ;jj M for all j; () E p P N N = [" ;s" ;t E (" ;s " ;t )] 4 M, for every (t; s). Assumpton D: he errors " ;t, u t, and the loadngs ¼ are three mutually ndependent groups. Assumpton E: E kf k M, and for every =;::: ;N, E je ; j M. Assumpton A ensures that the factor loadngs are dent able. Assumpton B establshes the condtons on the short and long-run varance of F t.e. postve de nte short-run varance and long-run varance that can be of reduced rank n order to accomodate lnear combnatons of I () factors to be statonay. Assumpton C() allows for some weak seral correlaton n ( ½ L) e ;t,whereas C() and C() allow for weak cross-secton correlaton. Fnally, Assumpton E de nes the ntal condton on e ;t. hs model expresses the stochastc process X ;t as the sum of up to three d erent components, so that we can focus on each of these components to characterze X ;t n terms of ts stochastc propertes. Note that the non-statonarty of X ;t can be due to the non-statonarty of ether F t or e ;t,sothatwehave two potental sources of non-statonarty wth d erent economc nterpretatons. hus, the matrx F t collects the common e ects that are present across the cross-secton dmenson and, therefore, the non-statonarty of F t wll mean that all ndvduals n the panel are common non-statonary. hese e ects a ect the ndvduals wth d erent magntude (¼ ). However, even f X ;t s drven by a common non-statonary component (F t ), the dosyncratc e ect may be 3

4 e ;t» I (). hs wll mean that the stochastc shocks that only a ect each ndvdual are statonary. Hence, the non-statonarty analyss can be performed through the applcaton of unt root tests on F t and e ;t. Regardng the determnstc component, the spec caton that s adopted n themodelsqutegeneraltoallowforthepresenceofmultplestructuralbreaks. Spec cally, we formulate: Xm Xm D ;t = ¹ + t + µ ;k DU ;k;t + ;k D ;k;t; (4) k= that s, we allow for m structural breaks a ectng the mean of the tme seres. he dummy varables are de ned as DU ;k;t =and D ;k;t ³t = b;k for t>b;k and elsewhere, where b;k denotes the k-th date of the break for the -th ndvdual, k =;::: ;m, m. In fact, equaton (4) nests two d erent spec catons dependng on the e ect of the structural breaks on the determnstc components. On the one hand, we can ntroduce the constrant = ;k =, 8; k, n (4) to analyse the stochastc propertes of panel data sets formed by non-trended varables for nstance, the PPP hypothess ought to be tested usng ths spec caton. Hereafter, the constraned model s denoted as Model. Formally speakng, Model mples the followng determnstc spec caton: Xm D ;t = ¹ + µ ;k DU ;k;t ; k= whch ncludes ndvdual e ects and ndvdual shftng e ects. On the other hand, we wll denote the unconstraned model gven by (4) as Model 2, a spec- caton that s sutable for trended varables that may be a ected by structural breaks that shft both the ndvdual and the spec c tme trend for nstance, the analyss of the unt root hypothess n GDP should be based on ths spec- caton. Notce that both models assume that the structural breaks are dosyncratc for the ndvduals, snce () they can be postoned at d erent dates for each ndvdual, () they may have d erent magntude and () each ndvdual may have d erent number of structural breaks. herefore, our spec caton takes nto account a hgh degree of ndvdual s heterogenety. Once the model have been de ned n a general way, now we are gong to address the unt root null hypothess testng through the consderaton of two stuatons: rst, we assume that there are no common factors, ¼ =8 n () and, second, we allow for thepresenceofsuchcommonfactors,¼ 6=n (), =;::: ;N. For ease of exposton, at rst we take the date of the breaks as known. Once the lmt dstrbutons are derved, we ntroduce the dscusson about the procedures that can be appled n order to estmate them. k= 4

5 2. Indvduals are assumed to be ndependent across From a theoretcal pont of vew, t s of nterest to consder the smpl ed stuaton n whch ¼ =8 n () and fe ;t g s a stochastc process ndependent across =;::: ;N. In order to test the null hypothess that X ;t» I (), 8, =;::: ;N, we suggest to compute the square of the mod ed Sargan- Bhargava (MSB) test statstc de ned n Stock (999): MSB ( ) = 2 P ~ t= X;t 2 ; (5) where X ~ ;t = X ;t ~D ;t and ~¾ 2 s the long-run varance of ~X ;t. We have made explct the dependency of the test on the structural breaks through the consderaton of n the notaton, where =( ; ;::: ; ;m ), ;k = b;k =, k =;::: ;m, s the so-called vector of break fracton parameters. he lmt dstrbuton of (5) for the two d erent models consdered n the paper s gven n the followng heorem. heorem Let X ;t, =;::: ;N, t =;::: ;, be the stochastc process generated by () wth ¼ =8 and ½ =n (3). As ; b;k!n a way that ;k = b;k = remans constant, 8; k; =;::: ;N, k =;::: ;m,then the test n (5) converges to: () Model : MSB ( ) ) P m + k= ( ;k ;k ) 2 R V ¹ ;k (b)2 db (2) Model 2: MSB ( ) ) P m + k= ( ;k ;k ) 2 R V ;k (b)2 db where ) denotes weak convergence of the assocated measure of probablty, V ¹ ;k (b) =W ;k (b) R W ;k (s) ds; and V ;k (b) =W ;k (b) (4 6b) R W ;k (s) ds ( 6+2b) R sw ;k (s) ds, wthw ;k (b) the standard Brownan moton, and ; =and ;m+ =. heorem shows that the lmt dstrbuton of the MSB ( ) test s functon of Brownan motons and two nusance parameters.e. the break fracton parameters ( ) and the number of structural breaks (m ). Moreover and as shown n the Appendx, when there s only one structural break, m =,the lmt dstrbuton of the test s symmetrc around =:5. Fnally, note that for m =the lmt dstrbutons n heorem concde wth the ones gven n Stock (999). Besdes, although the stuaton n whch N =can be understood as a specal case, ths s of great nterest provded that t generalses the proposal n Perron (997) and Lumsdane and Papell (997) through the consderaton of multple structural breaks n the non-statonarty analyss. hus, our can be appled to test the null hypothess of unt root on a sngle tme seres allowng for the presence of multple structural breaks both under the null and alternatve hypotheses. As mentoned above, heorem ndcates that the lmt dstrbuton of the MSB test depends both on the number of structural breaks (m ) and ther locaton ( ). hs gves rse to two possble stuatons. Frst, practtoners should be ~¾ 2 5

6 wllng to assume that the number and dates of the structural breaks are known. For nstance, the German reun caton and the Euro currency s brth are two events for whch the exogenous nature of the structural breaks can be assumed. However, ths stuaton s rarely found n practce, so that the computaton of the MSB wll requre the applcaton of a consstent estmaton procedure to determne the number of structural breaks and the respectve vector of break fracton parameters. hs de nes the second stuaton of nterest. Let us now focus on the rst stuaton n whch both the number and the poston of the structural breaks are known. he MSB test can be computed and compared to the crtcal values drawn from the lmt dstrbutons n heorem. However, we beleve that the avalablty of the assossated p-value could be more nformatve when performng the statstcal nference. Provded that the MSB test has a non-standard lmt dstrbuton, the p-values have to be approxmated by smulatons. MacKnnon (994), Adda and Gonzalo (996), Hansen (997), and Ba and Ng (23) computed asymptotc p-values for test statstc wth non-standard dstrbuton. Here we follow MacKnnon (994) and estmate a set of response surfaces to approxmate the p-values of the MSB test. However, we generalse the prevous proposals and estmate response surfaces for the p- values that take nto account the sample sze. he estmaton s made assumng a probt model for the p-value (p ) as a functon of powers ³ of the quantle (q ), the sample sze and the break fracton parameters, log p p = g (q ;; ). We have essayed d erent functonal forms usng the Newey-West robust covarance estmator to analyse the ndvdual sgn cance of the parameters. In concrete, for the stuaton n whch m =the response surface s gven by: µ p log = p X j= ³ µ ³ j + ³ j q + ³ 2j q =2 + ³ 3j q =3 + ³ 4j q =4 j + u ; where, for each sample sze ( ),, quantles, =;::: ;, has been computed from the emprcal dstrbuton to estmate the model. We have conducted a Monte Carlo experment to obtan the emprcal dstrbuton of the MSB test for ={3, 35, 4, 45, 5, 55, 6, 65, 7, 75, 8, 85, 9,, 25, 5, 75, 2, 225, 25, 3, 35, 4, 45, 5, 2} usng 5, replcatons. he p-values response surfaces are collected n Panel A of able. Smlar response surfaces are presented n Panel A of able 2 for m =. Note that (6) does not produce a drect estmate of p. he estmate of p s obtaned from ^p = exp f^g (q ;; )g +expf^g (q ;; )g : Let us now focus on the procedures that are based on the endogenous determnaton of the breakng ponts. he proposal descrbed n Ba and Perron (998) s very convenent for the spec caton n Model 2, provded that both the number and dates of the breaks can be consstently estmated under the null hypothess takng the rst d erence of y t. herefore, the problem reduces to the (6) 6

7 able : Response surfaces for the p-values estmaton for m = Panel A Panel B Model Model 2 Model Model2 ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ Panel A corresponds to heorem and Panel B corresponds to heorem ³ 2. he functonal form s gven by log p p = P j= ³³ j + ³ j q + ³ 2j q =2 + ³ 3j q =3 + ³ 4j q =4 j + u. he R ¹ 2 of all these estmatons were.99. he ncluded parameters were sgn cant at the 5% level - we used the Newey-West robust estmator to compute the s.e. dentfycaton of level shfts on y t, a statonary varable, on whch the dynamc optmzaton algorthm n Ba and Perron (998) can be appled. Notwthstandng, for the Model we have to follow a d erent approach gven that takng the rst d erence of y t wll mply datng mpulse outlers addtve outlers (AO) and ths stuaton s not covered n Ba and Perron (998). he standard way to deal wth AO outlers requres the estmaton of a fully parametrsed ARMA model on whch the outler detecton analyss s performed usng a t statstc n an teratve fashon see say (986) and Chen and Lu (993), among others. hs teratve approach was followed n Franses and Haldrup (994) to allow for AO outlers n the ADF test. However, two man drawbacks can be hghlghted. Frst, t requres to control the dynamc structure.e. estmaton of a fully parametrsed ARMA model and, second, the t statstc that s used to detect the presence of outlers reles on the dstrbutonal assumptons about the error term. Instead, we could estmate the shft dates usng the proposals n Perron and Vogelsang (992) and Vogelsang (998). Bre y speakng, Perron and Vogelsang (992) date the breakng ponts n the addtve spec caton through the mnmsaton of the sgn cance test of the dummy parameters. On the other hand, Vogelsang (998) uses the sup PS test whch does not rely on the dynamc of the system and, hence, seral-correlaton parameters does not have to be est- 7

8 able 2: Response surfaces for the p-values estmaton for m = Panel A Panel B Model Model2 Model 2 ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^' ^' ^' ^' ^' ^' ^' ^' ^# ^# ^# ^# Panel A corresponds to heorem and Panel B corresponds to heorem ³ 2. he functonal form s gven by log p p = P j= ³³ j + ³ j q + ³ 2j q =2 + ³ 3j q =3 + ³ 4j q =4 j + P j= 'j + ' 2j 2 + ' j 3j 3 + P 3 j= + u. he R ¹ 2 of all these estmatons were.99. he ncluded parameters were sgn cant at the 5% level - we used the Newey-West robust estmator to compute the s.e. 8

9 mated. However, these proposals do not provde a good approxmaton. On the one hand, Perron and Vogelsang (992) show that the date of the break s not dent ed under the null alternatve of unt root. On the other hand, the test n Vogelsang (998) s not consstent when y t» I () snce t has the same lmtng dstrbuton under the null and the alternatve hypothess. herefore, ths test should not be used to estmate the locaton of the level shft. o overcome these lmtatons we propose the use of the procedure de ned n Carron--Slvestre (23), whch conssts on the dent caton of AO s n the rst d erenced tme seres wthout havng to specfy a fully parametrsed model as requred n the exstng proposals. Fnally, for further purposes t would be useful to derve the mean and varance of the lmt dstrbuton of MSB for Models and 2. Spec cally, these two moments are used to de ne one of the pooled tests n Secton 4. hey are presented n the followng Proposton. Proposton Let MSB ( ) =~¾ 2 2 P ~ t= X;t 2 be the test statstc wth lmt dstrbuton gven n heorem. Moreover, let» = E (MSB ( )) and & 2 = V (MSB ( )) be the mean and varance of MSB ( ) respectvely. hen, as ; b;k! n a way that ;k = b;k = remans constant, 8; k; = ;::: ;N, k =;::: ;m, () Model :» = 6 & 2 = (2) Model 2:» = 5 where ; =and ;m+ =. & 2 = 63 P m + P m + 45 P m + P m+ k= ( ;k ;k ) 2 and k= ( ;k ;k ) 4 ; k= ( ;k ;k ) 2 and k= ( ;k ;k ) 4 ; Note that these moments are functon of the break fracton parameters. Besdes, when there are no structural breaks they concde wth the mean and the varance of the lmt dstrbuton n Stock (999). hese results agree wth the lmt dstrbutons n heorem. 2.2 Allowng for common factors Let us now weaken the framework that has been consdered n the prevous secton takng nto account the presence of common factors n the panel data. Obvously, the man d culty comes from the fact that the factors and the dosyncratc components are unobserved so that, the rst step of the analyss les n gettng a consstent estmate of both components. Followng Ba and Ng (2, 24), n order to estmate these unobserved common factors we apply the prncpal components technque to the d erenced-detrended model whch, expressed n matrx notaton, s gven by: M X = M F¼ + M e x = f¼ + z ; (7) 9

10 where X =( X ;2 ; X ;3 ;::: ; X ; ) and e =( e ;2 ; e ;3 ;::: ; e ; ) are two (( ) )-vectors for the -th ndvdual, F =[ F F 2 ::: F l ] s a (( ) l)-matrx beng F j =( F j;2 ; F j;3 ;::: ; F j; ), j =;::: ;l, a (( ) )-vector, and ¼ =(¼ ; ;::: ;¼ ;l ) s the (l )-vector of loadng parameters for the -th ndvdual, =;::: h ³ ;N. On the other hand, we de ne M = I a (a a ) a,wtha ;t = D b; ;::: ;D ³b;m for Model ³ t t, beng D b;k =for t = b;k + and elsewhere, k =;::: ;m,and h ³ t a ;t = D b; ³b;m t ;::: ;D ;DU ;;t;::: ;DU ;m;t for Model 2. M s t the usual dempotent matrx of projecton nto the space spanned by a ;t.he estmated factors ^f ;t ;::: ; ^fl;t are the l egenvectors that corresponds to the l largest egenvalues of the ( ) matrx xx,bengx =[x ;::: ;x N ]. he matrx of estmated weghts, ^ =(^¼ ;::: ;^¼ N ),sgvenby^ =x ^f t. As a result, we can obtan an estmate of z from ^z = x ^f ^¼, that, after computng ts cumulated sum, produces a consstent estmaton of the dosyncratc dsturbance term, ~e ;t = P t j= ^z ;j = P t j= (M ^e ) j. Now, the null hypothess of unt root n the dosyncratc stochastc element,.e. e ;t» I (), can be tested through the computaton of the MSB test usng ~e ;t : MSB ( ) = 2 P t= ~e2 ;t ~¾ 2 ; (8) where ~¾ 2 s an estmaton of the long-run varance of f ~e ;t g. he followng heorem gves the asymptotc dstrbuton of (8). heorem 2 Let fx ;t g N; =;t= the stochastc process generated by () wth ¼ 6= 8. If ½ =n (3), and ; b;k!n a way that ;k = b;k = remans constant, 8; k; =;::: ;N, k =;::: ;m, then the test n (8) converges to: () Model : MSB ( ) ) R W 2 (r) dr (2) Model 2: MSB ( ) ) P m + k= ( ;k ;k ) 2 R V ;k 2 (b) db; where W (r) s the standard Brownan moton, V ;k (b) =W ;k (b) bw ;k () s a Brownan brdge, and ; =and ;m + =. heorem 2 shows that the lmtng dstrbuton of the MSB test for Model does not depend on the presence of the structural breaks, snce the e ect of the mpulse dummy s asymptotcally neglgble. hs result s also found n Im et al. (22) for the LM panel data based unt root test. However, ths s not true for the model that allow for structural breaks a ectng the tme trend. hus, the asymptotc dstrbuton of the test for Model 2 depends on the set of nusance parameters de ned by the break fracton parameters. Moreover, the asymptotc dstrbuton of the MSB test for m =s symmetrc around =:5 for Model 2, a feature that has also been hghlghted n the prevous secton. he response surfaces for the p-values estmaton are collected n Panel

11 B of ables and 2 for m =and m =respectvely. he mean and varance of the lmt dstrbuton of MSB for Models and 2 are presented n the followng Proposton. Proposton 2 Let MSB ( ) =~¾ 2 2 P t= ~e2 ;t be the test statstc wth lmt dstrbuton gven n heorem 2. Moreover, let» = E (MSB ( )) and & 2 = V (MSB ( )) be the mean and varance of MSB ( ) respectvely. hen, as ; b;k! n a way that ;k = b;k = remans constant, 8; k; = ;::: ;N, k =;::: ;m, () Model :» = 2 and &2 = 3 ; (2) Model 2:» = P m + 6 k= ( ;k ;k ) 2 and & 2 = P m + 45 k= ( ;k ;k ) 4 ; where ; =and ;m+ =. See Levn and Ln (992) for the proof of statement and the Appendx for the proof of the statement 2 of Proposton 2. 3 A smpl ed test statstc In ths Secton we propose a smpl ed test that explots the fact that the lmtng dstrbutons n heorems and 2 are wegthed sums of ndependent functonals of Brownan motons. We follow Busett and Harvey (2) and compute the MSB test as a weghted sum of partal sum processes so that we get rd of the break fracton parameters n the lmt dstrbutons. hs smpl caton reduces the amount of computaton e ort that has to be made to provde practtoners wth sutable sets of p-values for large m. However, ths approach s prmarly addressed for panels wth large provded that the approxmaton s for the lmt dstrbuton. Frst of all, let us focus on the stuaton where there are not common factors, that s, ¼ =8, =;::: ;N. he weghted MSB test, MSB ( ), sgvenby: MSB ( ) = P m+ k= µ ³ 2 b;k b;k P b;k t=b;k ~¾ 2 ~X 2 ;t ; (9) =;::: ;N,wthb; =and b;m + =. Now the computaton of the test dstngushes among m + subperods whch are rescaled by the square of the correspondng number of observatons. he lmt dstrbuton of the MSB ( ) test for the models wthout common factors s presented n the followng Corollary. Corollary Let X ;t, =;::: ;N, t =;::: ;, be the stochastc process generated by () wth ¼ =8 and ½ =8 n (3). As ; b;k!n a way

12 that ;k = b;k = remans constant, 8; k; =;::: ;N, k =;::: ;m,then the test n (9) converges to: () Model : MSB ( ) ) P m + R k= V ¹ ;k (b)2 db (2) Model 2: MSB ( ) ) P m + R k= V ;k (b)2 db where ) denotes weak convergence of the assocated measure of probablty, V ¹ ;k (b) =W ;k (b) R W ;k (s) ds; and V ;k (b) =W ;k (b) (4 6b) R W ;k (s) ds ( 6+2b) R sw ;k (s) ds, wthw ;k (b) the standard Brownan moton. he proof follows from heorem and, hence, s omtted. Smlar developments can be made for the spec caton n Model 2 wth common factors. For ths model, the MSB ( ) test should be computed as: µ P ³ 2 m+ MSB k= b;k b;k P b;k ~e 2 ( ) t=b;k ;t = () wth the lmtng dstrbuton gven n the followng Corollary. Corollary 2 Let fx ;t g N; =;t= the stochastc process generated by () wth ¼ 6= 8. If½ =8 n (3), and ; b;k!n a way that ;k = b;k = remans constant, 8; k; =;::: ;N, k =;::: ;m, then the test n () converges to: MSB ( ) ) mx + k= Z ~¾ 2 V 2 ;k (b) db; where V ;k (b) =W ;k (b) bw ;k () s a Brownan brdge. he proof follows from heorem 2 and, hence, s omtted. Note that the de nton of the weghted MSB test makes free the lmt dstrbuton of the break fracton parameters, although t stll depends on the number of structural breaks n fact, they belong to the famly of Cramér-von Mses dstrbutons wth (m +)-degrees of freedom. he asymptotc p-values of the lmt dstrbutons n Corollares and 2 can be computed from the response surfaces n able 3 Panel A for the lmt dstrbutons n Corollary and Panel B for the one n Corollary 2. hey are computed usng the methodology descrbed above usng up to m =5structural breaks wth =2; to approach the steps and 5, replcatons. It can be shown that the response surfaces n able 3 provdes a good approxmaton of the crtcal values for the Cramér-von Msses dstrbuton computed n Canova and Hansen (995) and Nyblom and Harvey (2). For nstance, for the Cramér-von Msses dstrbuton wth two degrees of freedom de ned by demeaned Brownan motons, these authors set the 95% quantle as see the second row of able n Canova and Hansen (995). Usng ths quantle (^q =:749) wth m =n the response surface for the Model 2 Panel B of able3 weobtan^p =:

13 able 3: Response surfaces for the p-values estmaton for the smpl ed test statstcs Panel A Panel B Model Model 2 Model 2 ^³ 3.87 ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ ^³ 3..7 ^³ ^³ ^³ Panel A corresponds to Corollary and Panel B corresponds to Corollary ³ 2. he functonal form s gven by log p p = P 3 j= ³³ j + ³ j q + ³ 2j q =2 + ³ 3j q =3 + ³ 4j q =4 m j + u. he R ¹ 2 of all these estmatons were.99. he ncluded parameters were sgn cant at the 5% level - we used the Newey-West robust estmator to compute the s.e. 3

14 he performance of the smpl ed test n nte samples mght not show good propertes. he statements n Corollares and 2 are vald as!,whch prevent the use of the P-value functons that have been estmated above n nte samples. he value of for whch the asymptotc results are of applance s somethng to be addressed n the Monte Carlo analyss, but we should menton n advance that the smpl ed test shows an emprcal sze dstorton even for =3. hus, we would lke to make avalable a test statstc that can be appled n nte samples, allowng for multple structural breaks, and for whch the computaton of sutable p-values (or crtcal values) would not represent a hgh cost. he pont here s the computaton of these nte sample p-values. Note that the lmtng dstrbutons n Corollares and 2 do not depend on the break fracton parameters, but just on the number of breaks. hs s because as ; b;k!n a way that ;k = b;k = remans constant, 8; k; = ;::: ;N, k =;::: ;m, then the lmtng dstrbutons can be expressed as the sum of m ndependent functonals of Brownan motons. When applyng ths strategy to the nte sample framework we nd that t s mpossble to get rd of the number of observatons that are nvolved n each regme. hus, we should compute the nte moments usng nte values for. One possble soluton consst on the use of an approxmate nte sample dstrbuton. hus, we can de ne by approx = =(m +)the nte sample sze for the -th ndvdual and approxmate the nte sample dstrbuton usng approx. hs smpl caton s specally appealng provded that ths nte sample dstrbuton wll converge to the lmtng dstrbuton as n Corollares and 2!. able?? presents the estmates for the P-value functons that can be used to obtan the correspondng nte sample p-values for up to m =5structural breaks. 4 Poolng the ndvdual tests he results contaned n Propostons and 2 de ne the rst way of poolng the ndvdual nformaton, whch gves rse to the followng test statstc: Z = p N MSB( ) ¹» ¹&» N (; ) ; where MSB( ) = N P N = MSB ( ), wth ¹» = N P N =» and ¹& 2 = N P N = &2 computed usng the statements n Propostons and 2. he standard normal dstrbuton s obtaned from the applcaton of the Lndberg- Lévy Central Lmt heorem (CL). As mentoned n Ba and Ng (2), ths way of poolng can drve to unsatsfactory results, spec cally when the asymptotc dstrbuton of the ndvdual tests s skewed, as ths s the case. Instead, they suggest to follow the proposal n Maddala and Wu (999) and Cho (2) that pool the p-values assocated to the ndvdual tests - henceforth, we denote these p-values as p, =;::: ;N. Under the assumpton of cross-secton ndependence, 2lnp» Â 2 2, a results that was used n Maddala and Wu (999) to 4

15 de ne the Fsher-type test statstc: P = 2 NX ln p» Â 2 2N: = Notce that ths statement does not requre N!to be sats ed, so ths test statstc s of applance for panels wth small cross-secton dmenson. Besdes, Cho (2) proposes the followng test when N!: P m = 2 P N = ln p 2N p» N (; ) ; 4N where the standard normal lmt dstrbuton s obtaned from the applcaton of the Lndberg-Lévy CL. As a result, the P m test s sutable for those panels wth large N. hs spec caton was chosen n Ba and Ng (24) to test the null hypothess of non-statonary panel usng the DF test. Whle the man advantage of the p-values poolng strategy comes from the fact that the de nton of the test can be adapted to the cross-secton dmenson, ts man drawback reles on the avalablty of the p-values. hey are provded by the response surfaces estmated n Secton 2. 5 Fnte sample performance We analyse the performance of the panel data unt root test n two d erent stuatons. Frst we consder the case n whch there are no common factors, that s, we study the propertes of the test assumng that the ndvduals are crosssecton ndependent. After that, we wll focus on those panels where the crosssecton dependence s drven by the presence of up to three common factors. In all these smulatons we assume that the date of the breaks are known. hree values for the number of ndvduals N = f2; 4; g have been consdered, wthasampleszeequalto =. he number of replcatons s r =5;. he DGP s gven by equatons () to (3) wth ¹» U [; ],» U [:2; :5], µ ;k» U [ ; 3] and ;k» U [:3; :9], whereu [ ] denotes the Unform dstrbuton. We have allowed one structural break randomly postoned accordng to» U [:5; :85]. Under the null hypothess e ;t» I () have been generated as a random walk wthout drft de ned by the cumulated sum of d N (; ) processes. he common factors are de ned followng the AR() model: F j = F j + ¾ F u t ; = f:5; ; g, j =;::: ;l,wththe where = f:5; :8; :9; :95g and ¾ 2 F factor loadngs gven by ¼ j» N (; ). he smulatons have spec ed l =and l =3common factors. he number of common factors are xed usng the panel BIC nformaton crteron n Ba and Ng (22) wth l max =6as the maxmum number of factors. able 4 reports the sample sze of the three d erent statstcs when there are no common factors. he test based on the standarsaton present a sze 5

16 able 4: Emprcal sze. Known breaks and no common factors Model Smpl ed test N Z P m P Z P m P Model 2 Smpl ed test N Z P m P Z P m P dstorton that ncreses wth the number of ndvduals. hs s n accordance wth Ba and Ng (24), where t s mentoned that poolng n ths way can lead to unsatsfactory results specally when the asymptotc dstrbuton of the ndvdual tests s skewed, as ths s the case. On the contrary, the tests based on the combnaton of the ndvdual p-values show have an emprcal sze close to the nomnal one. Note that ths s also true for the smpl ed test, whch ndcates the usefulness of our proposal n appled research. he pcture changes when we analyse the panel data set that allows for common factors. For Model all three test statstcs show good performance n terms of emprcal sze. he excepton s the P test, whch n some stuatons presents emprcal sze dstortons that lead to under reject the null hypothess see able 5. For Model 2 the P m test s the one wth the most stable emprcal sze, provded that the Z and P tests under reject the null hypothess. hs s also true for all the verson of the smpl ed tests see able 6. 6 Conclusons In ths paper, we have proposed new procedures for testng non-statonarty of panel data n the presence of multple structural breaks and dynamc common factors. In the absence of common factors, the lmtng dstrbutons are shown to be weghted sum of ndependent and dentcally dstrbuted Brownan motons (demeaned or detrended). hese results are of specal nterest for the sngle tme seres analyss.e. panels wth N =ndvdual provded that they extend the proposals n Perron (997) and Lumsdane and Papell (997), among others, and allow to test the unt root hypothess wth multple structural changes. When dynamc factors are present, the PANIC approach of Ba and Ng (24) s used to estmate the model. he lmtng dstrbutons of the test statstcs are nvarant to mean breaks. For breaks n the lnear trend, the lmtng dstrbutnos are 6

17 able 5: Emprcal sze for model and N =4. Known breaks wth common factors r = r =3 ¾ 2 F Z P m P Z P m P shown to be weghted sum of d Brownan brdges. We further ntroduced a smpl ed test statstc, and showed that the lmtng dstrbuton s nvarant to both mean and trend breaks. Pooled test statstc s also studed. Response surfaces for p-values of all test statstcs are computed. 7

18 able 6: Emprcal sze for model 2 and N =4. Known breaks wth common factors r = Smpl ed tests ¾ 2 F Z P m P Z P m P r =3 Smpl ed tests ¾ 2 F Z P m P Z P m P

19 7 Appendx: Proof of heorem 7. Proof of statement () he model that s consdered n ths statement s the one for non-trended varables where = ;k =, 8; k n (4). In addton, the constrant n ¼ = 8 s mposed n order to avod the presence of common factors that drve the behavour of the ndvdual tme seres. From ths spec caton, the estmated OLS resduals of the model are obtaned from ~e = M e, wth M = I a (a a ) a. Note that for m structural changes the determnstc part of the model gven n (4) can be expressed n terms of orthogonal regressors de nng a block dagonal matrx. ³³ he elements n the dagonal are gven by vectors k =(;::: ;) of dmenson b;k b;k, k =;::: ;m +,wthb; =and b;m =.hus, + the cross-product matrx of regressors a a s gven by 2 a a = 6 4 b; b;2 b;... b;m b;m b;m Usng the fact that b;k = ;k and de nng the (m m )-dagonal rescalng matrx P = dag =2 ;::: ; =2, P a a P can be expressed as P a a P = dag ( ; ; ( ;2 ; ) ;::: ; ( ;m ;m ) ; ( ;m )). On the other hand, under the null hypothess that e» I (), P a e ) ³ R ¾ ; R W (s) ds; ¾ ;2 R W ; (s) ds;::: ; ¾ ;m W R (s) ds; ¾ ;m W (s) ds. ;m hs means that for t b; for b; <t b;2 =2 ~e ;t ) ¾ W (r) ¾ =2 ~e ;t ) ¾ W (r) ¾ ; ( ;2 ; ) Z ; Z ;2 3 ; 7 5 W (s) ds; <r< ; ; ; W (s) ds; ; <r< ;2 ; and so on, so that, t can be establshed for b;k <t b;k =2 ~e ;t ) ¾ W (r) ¾ ( ;k ;k ) Z ;k ;k W (s) ds; ;k <r< ;k ; k =;::: ;m +,wth ; =and ;m + =. 9

20 he goal s to show that lmt dstrbuton of the test can be expressed as a sum of a set of ndependent ntegrals of detrended Brownan motons. o do so we rescale the Brownan motons so that we ensure that the ndex of thebrownanmotonneachsubsamplebelongsto[; ]. hus, notce that n the rst subsample <r< ; can be rescaled as = ; <r= ; < ; = ;,sothatb =(r= ; ) 2 [; ]. In general, for ;k <r< ;k t can be de ned ( ;k ;k ) = ( ;k ;k ) < (r ;k ) = ( ;k ;k ) < ( ;k ;k ) = ( ;k ;k ),sothatb =(r ;k ) = ( ;k ;k ) 2 [; ] Lee (996), Lee and Strazcch (2) and Bartley, Lee and Strazcch (22) use smlar developments when dervng the lmt dstrbuton of the KPSS test wth one structural break. herefore, for <b< we have µ Z =2 ~e ;t ) ¾ q( ;k ;k ) W ;k (b) W ;k (s) ds q = ¾ ( ;k ;k )V ¹ ;k (b) ; where V ¹ ;k (b) denotes the demeaned Brownan moton. hus, the lmt dstrbuton of the MSB ( ) test s gven by MSB ( ) ) 2; Z V ¹ ; (b)2 db + +( ;k ;k ) 2 Z + +( ;m ) 2 Z V ¹ ;m + (b)2 db; V ¹ ;k (b)2 db wth V ¹ ;k (b) =W ;k (b) R W ;k (s) ds, k =;::: ;m +, ndependent demeaned Brownan motons and provded that ~¾ 2! ¾ 2 - see Stock (999). Notce that for m =the lmt dstrbuton of the test s gven by MSB ( ) ) 2 Z V ¹ ; (b)2 db +( ) 2 Z V ¹ ;2 (b)2 db () whch t s shown to be symmetrc around = :5, provded that we can nterchange and ( ) n () and obtan the same asymptotc dstrbuton. Fnally, note that the lmt dstrbuton of MSB ( ) s the weghted sum of (m +)ndependent Cramér-von Mses dstrbutons -see Harvey h (2). he R expectatons of these Cramér-von Mses dstrbutons are E V ;k 2 (b) db = h R =6 where the varance are V V ;k 2 (b) db ==45, 8k =;::: ;m + see Levn and Ln (992). herefore, E [MSB ( )]=(=6) P m + k= ( ;k ;k ) 2 and V [MSB ( )]=(=45) P m + k= ( ;k ;k ) Proof of statement (2) hs statement presents the lmt dstrbuton of the test for trended varables. As before, we assume ¼ =8. Followng the steps on the prevous proof, notce 2

21 that for m structural changes the determnstc part of the model gven n (4) can be expressed n terms of orthogonal regressors de nng a block dagonal matrx. Now, the elements n the dagonal are gven by vectors k =(;::: ;) and ³ ³ ³³ t k = ; 2;::: ; b;k b;k both of dmenson b;k b;k, k = ;::: ;m +,wthb; =and b;m + =. hus, the cross-product matrx of regressors a a s block dagonal matrx, wth the³ k-thblockgvenbythe(2 2)- matrx a ;k a ;k wth elements a ;k a ;k [; ] = b;k b;k, a ;k a ;k [; 2] = ³ ³³ a ;k a ;k [2; ] = =2 b;k b;k b;k b;k + and a ;k a ;k [2; 2] = ³ ³³ ³ ³ =6 b;k b;k b;k b;k + 2 b;k b;k +. If we de ne the rescalng dagonal matrx P k = dag =2 ; 3=2,then =3( ;k ;k ) 3 =2( ;k ;k ) 2 P k a ;k a ;kp k! =2( ;k ;k ) 2 ( ;k ;k ) ; where ³ =(=2) ( ;k ;k ) 4. Hence, t can be shown that (P a a P ) = P dag a ; a ;P ;::: ; P a a ;m ;m P. On the other hand, P a e s a (2m )-vector de ned by stackng the m (2 )-vectors gven by 2 6 3=2 P 3 b;k t=b;k 4 + e " R ;k # ;t 5=2 P 7 ¾ W (s) ds ;k b;k t=b;k + te 5 ) R ¾ ;k ; ;t sw (s) ds ;k k =;::: ;m +. herefore, for b;k <t b;k t can be establshed that =2 ~e ;t ) ¾ W (r) ¾ h³³=3( ;k 3 ;k ) Z ;k =2( ;k ;k ) 2 r W (s) ds ;k ³ Z ;k =2( ;k ;k ) 2 ( ;k ;k ) r sw (s) ds ;k ;k <r< ;k ;k=;::: ;m +,wth ; =and ;m+ =. Rescalng the ndex of the Brownan motons n a way that b k =(r ;k ) = ( ;k ;k ) so that <b k <, s straghtforward to see that Z =2 ~e ;t ) ¾ q( ;k ;k ) W (b k ) (4 6b k ) W (s k ) ds k Z ( 6+2b k ) s k W (s k ) ds k q = ¾ ( ;k ;k )V ;k (b) ; # ; 2

22 where V ;k (b) denotes the detrended Brownan moton. Notce that V ;k (b) s equvalent to the detrended Brownan moton nvolved n the lmt dstrbuton of the MSB test n Stock (999) heorem. hus, the lmt dstrbuton of the MSB ( ) test MSB ( ) ) 2; Z V ; (b) 2 db + +( ;k ;k ) 2 Z + +( ;m ) 2 Z V ;m + (b) 2 db; V ;k (b) 2 db wth V ;k (b) =W ;k (b) (4 6b) R W ;k (s) ds ( 6+2b) R sw ;k (s) ds, k = ;::: ;m +, ndependent detrended Brownan motons and provded that ~¾ 2! ¾ 2 see Stock (999). Notce that, as before, for m =the lmt dstrbuton of the test s symmetrc around =:5. he mean and the varance of the lmt dstrbuton s gven by can be computed from the moment generatng functon n anaka (996), whch for our test s gven by m (') = " 3 p 2' sn p 2' +2 p ' cos p 2' 2 p ' p ' 5 # =2 : he rst dervatve of the moment generatng functon evaluated at ' =wll provde us the rst moment of the lmt dstrbuton: dm (')» = lm '! d' = 5 ; whereas the varance s obtaned from & 2 = lm '! d 2 m (') d' 2 µ 2 = 5 63 : herefore, E [MSB ( )]=(=5) P m + k= ( ;k ;k ) 2 and V [MSB ( )] = (=63) P m + k= ( ;k ;k ) 4. 8 Appendx: Proof of heorem 2 8. Proof of statement () Statement () n heorem s concerned wth Model, that s, the model for non-trended varables where = ;k =, 8; k n (4). he estmaton of the (d erenced and detrended) model produces the followng result: x ;t = ^f t^¼ +^z ;t : (2) 22

23 Subtractng (2) from (7) we obtan ^z ;t = z ;t + f t ¼ ^f t^¼ : Followng Ba and Ng (23), we can express the model as ^z ;t = z ;t + f t HH ¼ ^f t H ¼ + ^f t H ¼ ^f t^¼ = z ;t + ³f t H t ^f H ¼ ^f t ^¼ H ¼ = z ;t + v t H ¼ ^f t d ; (3) where v t = ³f t H t ^f and d = ^¼ H ¼. Let us de ne the partal sum process usng the estmated resduals as ~e ;t = P t s=2 ^z ;s = P t s=2 (M ^e ) s.by Lemmas 3 and C n Ba and Ng (23), =2 P t s=2 v sh ¼ = op () and =2 P t s=2 ^f s d = op (), sothat =2 ~e ;t = =2 tx (M e ) s + o p () : s=2 hepartalsumprocesscanbeexpressedntermsofthepopulatonresduals as: wth =2 ~e ;t = =2 tx (M e ) s = s=2 tx (M e ) s + o p () ; (4) s=2 tx ( e ;s (P e ) s ) ; s=2 where P = a (a a ) a. Note that P t s= e ;s = e ;t e ; and P e s a vector of zeros except for the ³ + b;k -th postons, k =;::: ;m. he cumulated process s equal to: ~e ;t =(e ;t e ; ) ³e ; b; + e ; b; DU ;;t ::: ³e ; b;m + e ; DU b;m ;m;t: If we assume that e ;t» I (): =2 ~e ;t = =2 (e ;t e ; ) =2 ³ e ; b; + e ; b; DU ;;t ::: =2 ³ e ; b;m + e ; b;m DU ;m;t + o p () ) ¾ W (r) ¾ W () ¾ dw ( ) du ::: ¾ dw ( m ) du m = ¾ W (r) ¾ dw ( ) du ::: ¾ dw ( m ) du m ; (5) 23

24 where W (r) denotes the standard Brownan moton and du k =for r> k and elsewhere, wth k = b;k =, k =;::: ;m. he lmt expresson of =2 ~e ;t gven by (5) nvolves two d erent knd of elements: () the Brownan moton, W (r), and () the d erence of Brownan motons, dw ( k), k =;::: ;m. Followng Perron (997), the e ect of these d erences can be understood as neglgble compared to W (r), so that, we can consder that =2 ~e ;t ) ¾ W (r). herefore, the test statstc converges to: MSB ( ) ) Z W 2 (r) dr; provded that ~¾ 2! ¾ 2. Notce that after consderng the neglgble e ect of the dw ( k) terms, k =;::: ;m, the asymptotc dstrbuton of the test does not depend on the break fracton parameters k, thats,thetestsnvaranttothe presence of structural breaks a ectng the mean of the tme seres. 8.2 Proof of statement (2) Let us now focus on the spec caton gven by Model 2, that s, the model for trended regressors where 6= ;k 6=, 8; k n (4). As n the prevous proof, the computaton of the partal sum process can be done from (3). However, we have to assess that =2 P t ^f s=2 s d = op (). Note that =2 P t ^f s=2 s d =2 P t s=2 ^f s kd k. From Ba and Ng (23), kd k = o p (), and =2 tx tx ³ ^f s = =2 ^fs Hf s + Hf s s=2 s=2 tx = =2 v s + H =2 s=2 tx = o p () + H =2 f s : o determne the order n probablty of =2 P t s=2 f s we rewrte the matrx of determnstc elements a n a ( (2m +))quas block dagonal matrx: a = DU D b; DU 2 D b;2 DU 3 ::: DU m D b;m DU m + ; s=2 tx s=2 f s where DU k =for b;k <t b;k and elsewhere, k =;::: ;m +,wth b; =and b;m + =. Now the elements that de nes the a matrx can be grouped n two d erent sets, the rst one ³ compounded by the DU k regressors and the second one composed by the D b;k regressors, k =;::: ;m +. Note the orthogonalty property that characterze the elements of each set. 24

25 Moreover, ths transformaton makes the P matrx to be block dagonal. Wthout loose of generalty and n order to smplfy cumbersome algebrac manpulatons, we derve the order n probablty of =2 P t s=2 f s assumng that m =, although the results are vald when m. hus,whenm =the P matrx s gven by: P = DU DU + ( + ) + D b D b + + D b DU DU2 D b + DU2 DU 2 : (6) he e ect of P on F can be analysed by parts. When multplyng the rst element of (6) by F produces: ( DU DU F = ³F F b t b : t>b he computaton of the partal sum process nvolves: =2 tx s=2 8 DU DU < F s s = : ³ 3=2 F b F t t b ³ =2 F b F t>b ; whch s O p (). he same result s found for the product nvolvng the fth element of (6). he second element of (6) gves: ( + ) + D ( b ( + ) D b F = + F b + t = b + t 6= b + ; so that the partal sum process s =2 tx s=2 ( + ) ³ D b D + F b s = s ( t b ( + ) =2 + F b + t>b ; wth O p () as order n probablty. For the thrd element we have + D ½ b DU 2 F F F = b + t = b + t 6= b + ; so that =2 tx s=2 8 D + b DU 2 < s F s = : t b µ F F + b ( + ) =2 t> b ; 25

26 whch s also O p (). he fourth element s + DU2 D ½ b F = t b + F b + t>b : hus, =2 tx s=2 ³ DU 2 D ( b + F s = s t b F b +(t b) ( + ) =2 t> b ; whch s O p (). Fnally, the fth element ( + DU2 DU 2 F = + ³ t b F F b + t>b ; wth cumulated sum 8 tx =2 DU 2 DU 2 < + s F s = : s=2 µ F F b + t b (t b) ( + ) =2 t> b ; whch s also O p ().herefore, all the partal sum processes nvolvng P F are O p (), a result that can be straghtforwardly extended to those stuatons that allow for multple breaks. Consequently, tx tx =2 ^f s d =2 ^f s kd k s=2 s=2 O p () o p () ; whch means that =2 P t ^f s=2 s d = op (). As n the prevous proof, the partal sum process of the estmated resduals s gven by (4). Now, the cumulatve process are gven by the prevous expressons but replacng F by e. he rst element of the partal sum process, whch nvolves the rst set of step dummy varables, converges to =2 P t s=2 DU DU e s ;s ) (r= ) W ( ) for t b and =2 P t s=2 DU DU e s ;s ) W ( ) for t>b. he second element produces =2 ( + ) + e ;b + ) dw ( ), an element that vansh asymptotcally. he thrd element s o p (), whereas the fourth element, whch nvolves the second set of step dummy varables, + e ;b t + b ) r dw ( ), another element that vansh asymptotcally. Fnally, the fth element s =2 P t s=2 + DU 2 DU 2 e s ;s ) r (W () W ( )) for t>b and elsewhere. 26

27 herefore, =2 ~e ;t ) ¾ W (r) ¾ (r= ) W ( ) for r and =2 ~e ;t ) r ¾ W (r) ¾ W ( ) ¾ (W () W ( )) for r>, andthemsb ( ) test statstc converges to: 2 P t= ~e2 ;t ~¾ 2 = 2 P b t= ~e2 ;t Z ~¾ P t= b + ~e2 ;t ~¾ 2 ) Z + [W (r) (r= ) W ( )] 2 dr W (r) W ( ) r 2 (W () W ( )) dr; provded that ~¾ 2! ¾ 2. However, the lmt dstrbuton of MSB ( ) can be expressed as the sum of two ndependent ntegrals. Let us de ne b = r= ; so that <b<. Usng the propertes of the Brownan motons the lmt dstrbuton can be wrtten n terms of b as W (r) (r= ) W ( ) = p W ; (b) b p W ; () = p (W ; (b) bw ; ()), sothat Z [W (r) (r= ) W ( )] 2 dr = 2 Z Z = 2 [W ; (b) bw ; ()] 2 db V; 2 (b) db; where V ; (b) denotes the demeaned Brownan moton. For the second ntegral, let us now de ne b = (r ) = ( ), so that < b <. Now, the lmt dstrbuton can be reexpressed n terms of b as W (r) W ( ) r (W () W ( )) = W ;2 (b) bw ;2 (), whch mples that Z W (r) W ( ) r 2 Z (W () W ( )) dr =( ) 2 V;2 2 (b) db; where V ;2 (b) denotes the demeaned Brownan moton. herefore, the asymptotc dstrbuton of the test when m =s gven by MSB ( ) ) 2 Z V 2 ; (b) db +( ) 2 Z V 2 ;2 (b) db; (7) where V ; (b) and V ;2 (b) are two ndependent Brownan brdges. Note also the symmetry of the asymptotc dstrbuton around =:5. As shown above, we can nterchange and ( ) n (7) and obtan the same asymptotc dstrbuton. In general, for k =;::: ;m + we have ;k b <t ;k b and the partal sum processes converges to ~¾ =2 ~e ;t ) W (r) (r ;k ) = ( ;k ;k ) (W ( ;k ) W ( ;k )), wth ; = and ;m+ =. Let us now de- ne b = (r ;k ) = ( ;k ;k ) so that < b <. As before, the 27

28 lmt dstrbuton of the partal sum processes s gven by ~¾ =2 ~e ;t ) p ;k ;k (W ;k (b) bw ;k ()), and the test statstc MSB ( ) =~¾ 2 2 P P =~¾ 2 2 b; t= ~e2 ;t + +P b;k t=b;k + ~e2 ;t + + P t=b;m + ~e2 ;t wth lmt dstrbuton gven by: MSB ) 2; Z V 2 ; (b) db + +( ;k ;k ) 2 Z + +( ;m ) 2 Z V;m 2 + (b) db; V;k 2 (b) db where V ;k ( ), k =;::: ;m +, denotes the demeaned Brownan moton and provded that ~¾ 2! ¾ 2 see below the proof of the consstency of the nonparametrc long-run varance estmaton. he lmt dstrbuton of MSB ( ) s the weghted sum of (m +)ndependent Cramér-von Mses dstrbutons. he expectatons of these Cramér-von Mses dstrbutons are E =6 where the varance are V h R V 2 ;k (b) db = h R V 2 ;k (b) db ==45, 8k =;::: ;m + -see Levn and Ln (992). herefore, E [MSB ( )]=(=6) P m + k= ( ;k ;k ) 2 and V [MSB ( )]=(=45) P m + k= ( ;k ;k ) Proof of the consstency of the long-run varance estmaton Let us de ne the AR() regresson on the estmated dosyncratc resduals: ~e ;t = b ~e ;t + ;t ; (8) ³ whch under the null hypothess of unt root mples that ~b = O p (= ). From (8) we can express the error term as: ~ ;t = ~e ;t + ³ ~ b ~e ;t ; wherefrom(3)tfollowsthat ~ ;t = z ;t + v t H ¼ ^f t d + ³ ~b ~e ;t = z ;t + w ;t ; wth z ;t =(M e ) t and w ;t = v t H ¼ ^f t d + ³ ~ b ~e ;t. For arbtrary tme seres a t and b t de ne: " # dnw ab = X JX X j a t b t + K (j) (a t b t+j + a t+j b t ) ; t= j= t= t= ~e2 ;t 28

29 wth K (j) = j= (J +). hen NW d zz s the Newey-West estmator of the long-run varance of z = M e. In order to proof the consstency of ths estmator we need to show that dnw ~ ~ dnw zz = o p () : From ~ ;t = z ;t + w ;t we have that dnw ~ ~ = dnw zz +2dNW zw + dnw ww : We next show that f J!and J=± N!, ± N =mn[n;], then dnw zw = o p () and dnw ww = o p () : Frst, notce that à dnw zw 2à JX j!=2 à + K (j) 4 X z;t 2 j= t= X! =2 à z;t 2 t= X j!=2 w;t+j 2 t= Notce that P t= z2 ;t = O p (). On the other hand, X w;t 2 t= jw ;t j 2 4 kv t k 2 H ¼ 2 4 ^f t 2 kd k ! =2 à X j t= z 2 ;t+j ³ ~b 2 ~e 2 ;t ;!=2 à X j w;t!= : t= so that X jw ;t j 2 4 t= X t= kv t k 2 H ¼ 2 4 X ^f 2 ³ t kd k 2 +4 ~ b 2 t= X t= ^e 2 ;t : P From Lemmas (a) and (c) n Ba and Ng (23), t= kv tk 2 = O p ± 2 N and kd k 2 ³ = O p ± 2 N respectvely, and ~ b P 2 t= ~e2 ;t ³ = ~ b 2 P 2 t= ~e2 ;t = O P p (= ). herefore, t= jw ;tj 2 O p ± 2 N. hese ntermedate results enable us to establsh that dnw z w (J +)Op ± N! : Moreover, snce P t= jw ;tj 2 O p ± 2 N then dnw ww (J +)Op ± 2 N! : hus, we have shown that the long-run varance can be consstently estmated through the applcaton of the non-parametrc Newey-West estmaton procedure, that s, we propose to use ~¾ 2 = NW d ~ ~. 29