Structural Breaks and Unit Root Tests for Short. Panels

Transcription

1 Structural Breaks and Unt Root Tests for Short Panels Elas Tzavals* Department of Economcs Athens Unversty of Economcs & Busness Athens , Greece (emal: Ths verson December 008 Abstract In ths paper we suggest panel data unt root tests whch allow for a potental structural break n the ndvdual e ects and/or the trends of each seres of the panel, assumng that the tme-dmenson of the panel, T, s xed. The proposed test statstcs consder for the case that the break pont s known and for the case that t s unknown. Monte Carlo evdence suggests that they have sze whch s very close to the nomnal ve percent level and power whch s analogues to that of Harrs and Tzavals (1999) test statstcs, whch do not allow for a structural break. JEL class caton: C, C3 Keywords: Panel data; Unt roots; Structural Breaks; Sequental Unt Root tests; Central Lmt Theorem; economc convergence hypothess *Ths research was funded by the ESRC (UK) under grant R

2 1 Introducton The AR(1) model for short panels has been extensvely used n the lterature n studyng the dynamc behavour of many economc seres across d erent cross secton unts, when the tme dmenson of the panel s consdered as xed (small) [see Baltaz and Kao (000), and Arelano and Honore (00), nter ala]. Of partcular nterest s to use the model n examnng whether many economc seres contan a unt root n ther autoregressve component, across all the sectonal unts of the panel [see Levn, Ln and Chu (00), for a recent survey of these type of tests]. Recent applcatons of unt roots tests for panel data nclude: the examnaton of the economc growth convergence hypothess [see de la Fuente (1997), for a survey], the hypothess that stock prces and dvdends follow the random walk model [see Lo and MacKnlay (1995), nter ala] and, nally, the nvestgaton of the valdty of the purchasng power party hypothess [see Culver and Papell (1999), nter ala]. Motvated by recent studes suggestng that evdence of a unt root n the many economc seres may be attrbuted to a structural break n the determnstc components of the seres [see Perron (1999)], n ths paper we ntroduce short panel data unt roots test statstcs allowng for a potental structural break under the alternatve hypothess of statonary. 1. The paper proposes test statstcs for two cases: () when the break occurs n the ndvdual e ects and () when t occurs n the determnstc trends of the panel, or n both the determnstc trends and the ndvdual e ects. For both cases, the tmng of the structural break s assumed to be common for all the seres of the panel. Regardng the dmensonalty ssue of the panel, the proposed test statstcs are n lne wth Harrs and Tzavals (1999) tests. They assume that the tme dmenson (T ) of the panel s xed, whle the cross-secton dmenson (N) grows large. These 1 As rst ponted out by Perron (1989) for a sngle tme seres, gnorng a structural break n the determnstc component of a seres s expected to bas the unt root tests towards falsely acceptng the null hypothess of a unt root. For panel data, ths has been shown by Carron-Slvestre, Barro-Castro and Lopez-Bazo (001), who consdered the smpler, canoncal case of the AR (1) panel data model wth ndvdual e ects treatng the tme pont of the break as known.

3 tests are approprate for panels where T s smaller than N, as t s typcal n many mcro-economc (or nance) studes. The proposed test statstcs are based on the pooled least squares (LS) estmator of the autoregressve coe cent of the panel data model corrected for the nconsstency of the LS estmator [see Kvet (1995), and Harrs and Tzavals (1999)]. Ths nconsstency arses from two sources: the presence of ndvdual e ects and/or trends n the AR(1) auxlary panel data regresson, and the allowance for a potental structural break n the determnstc components of the panel. Employng the corrected for ts nconsstency LS estmator n drawng nference about unt roots n panel data models has the followng two man advantages: rst, t can lead to test statstcs whch are nvarant to the ntal condtons and/or the ndvdual e ects of the panel data model; second, t enables us to dentfy the autoregressve coe cent of the AR(1) panel data model under both the null and alternatve hypothess. In addton to the above, the only dstrbuton assumpton that s needed n dervng the lmtng dstrbuton of the test statstcs based on the corrected for ts nconsstency LS estmator s that the fourth-moment of the dsturbance terms should exst. The paper consders two categores of tests: () when the break pont s known and () when t s unknown. The second category of the tests may be proved very useful n cases where t s d cult to sngle out any major exogenous event that could have caused a common change n any of the determnstc components of the seres of the panel. When the tme-pont of the break s assumed to be known, we show that the lmtng dstrbuton of the proposed test statstcs for panel data models wth ndvdual e ects, and ndvdual e ects and/or trends s normal wth varance whch depends on the fracton the sample that the break occurs and the tme dmenson (T ) of the panel. For ths case, we employ the moments of the lmtng dstrbuton of the test statstc for Ths s n contrast to condtonal Maxmum Lkelhood based panel data unt root nferece procedures suggested n the lterature n order to crcuvmnet the problem panel data ntal observatons. These procedures requre the dsturbance terms of the panel to be normally dstrbuted. 3

4 the AR(1) model wth ndvdual e ects n order to analyse the consequences of gnorng a potental structural break n the determnstc component of the seres of the panel n drawng nference about unt roots. Our analyss shows that the tendency of the panel unt root tests to falsely accept the null hypothess may be attrbuted to the smaller varance of the lmtng dstrbuton of the test statstcs that gnore a potental break. When the break-pont s unknown, we suggest a sequental procedure to test for the null hypothess of unt roots for the above panel data models. Ths s based on the mnmum values of the one-sded test statstcs for a known date break whch are sequentally computed, for each pont of the sample. 3 The dstrbuton of the mnmum values of the test statstcs can be calculated by that of the mnmum values of a xed number of correlated normal varables. We derve the correlaton matrx of these varables and provde crtcal values of the dstrbuton of the mnmum test statstcs for both panel data models: wth ndvdual e ects, and ndvdual e ects and trends by smulaton methods. Ths s done for d erent values of T, after trmmng the ntal and nal parts of the seres of the panel, assumng that the dsturbance terms are normally dstrbuted. To examne the small-n sample performance of the tests, we run some Monte Carlo smulaton experments. The results of these experments ndcate that both testng procedures, consderng a known date break or not, have sze whch s very close to the 5% level and power whch ncreases both wth N and T, but grows faster wth T. Ths performance of the tests s analogous to that of Harrs and Tzavals tests, whch do not consder the case of a structural break. The paper s organzed as follows. The lmtng dstrbutons of the test statstcs for known or unknown break-pont are derved n Secton. In Secton 3, we present the results of the Monte Carlo study. In Secton 4 we present an applcaton of the sequental testng procedure n order to reexamne whether level of the real per capta ncome of each country s mean revertng to ts steady 3 Perron and Vogelsang (199), and Zvot and Andrews (199) adopted ths procedure n testng for a unt root n sngle tme seres. 4

5 state level, as the economc growth convergence hypothess asserts. Conclusons are summarzed n Secton 5. The test statstcs and ther lmtng dstrbuton.1 The date of the break pont s known Consder the followng rst-order autoregressve panel data model, denoted AR(1), y = y ; 1 + X () () + u, = 1; ; :::; N (1) where y = (y 1 ; :::; y T ) 0 s a (T X1) dmenson vector of tme seres observatons of the dependent varable of each cross-secton unt of the panel,, y ; 1 = (y 0 ; :::; y T 1 ) 0 s the vector y lagged one tme perod back, u = (u 1 ; :::; u T ) s a (T X1) dmenson vector of dsturbance terms and X () = e () ; e (1 ) ; () ; (1 ) s a (T X4) dmenson matrx whose columns vectors are approprately desgned so that to capture a common, across, structural break n the vector of coe cents () = (a () ; a (1 ) ; () ; (1 ) ) 0 of the determnstc components of the panel,.e. the ndvdual e ects and trends, at the tme pont T 0 =nt[t ], where (0; 1) denotes the break fracton of T and nt declares nteger number. Spec cally, the columns vectors of X () are de ned as follows: e () (1 ) t = 1 f t T 0 and 0 otherwse, and e t = 1 f t > T 0 and 0 otherwse; () = t f t T 0 and 0 otherwse, where t denotes the determnstc trend, whle (1 ) = t f t > T 0 and 0 otherwse. For model (1), the null hypothess s that = 1,.e. there s a unt root n the autoregressve component of the model, and that there s no break n the determnstc components of the panel. Under the alternatve hypothess, t s assumed that jj < 1 and there s a break n the ndvdual e ects.... The panel data auxlary regresson model gven by equaton (1) s appro- 5

6 prate to test the null hypothess that each seres of the panel follows a random walk wth drft aganst the alternatve hypothess that each seres s statonary around a broken trend. Note that under the null hypothess the drft parameters can also have a break,.e. a () 6= a (1 ) 6= 0. As we show n the appendx (see proof of Theorem 1), the ncluson of the determnstc trends n model (1) wth the same break pont as the ndvdual e ects renders the LS estmator of under the null hypothess nvarant to the ndvdual e ects, ndependently on whether they are subject to a structural break, or not. Fnally, note that the auxlary regresson panel data model (1) can nest the model where X () = e () (1 ; e ) and () = (a () (1 ) ; a ) 0. Ths model (1) s approprate for testng for the null hypothess that each seres of the panel follows a drftless random walk aganst the alternatve hypothess that each seres s statonary around a broken mean. Test statstcs of the null hypothess = 1 based on the above alternatve spec catons of the determnstc components of model (1),.e. X () = e () ; e (1 ) ; () ; (1 ) and X () = e () ; e (1 ), can be derved based on the pooled LS estmator of the autoregressve coe cent,. The lmtng dstrbutons of these test statstcs can be derved by notcng that the pooled LS estmator of, denoted ^, under the hypothess = 1 sats es where Q () = " N # 1 " X N # X ^ 1 = y; 0 1Q () y ; 1 y; 0 1Q () u, () =1 h I X () X ()0 X () 1 X ()0 =1 s the (T XT ) wthn transformaton matrx of the seres of the panel [Baltag (1995), nter ala]. Snce ^ s an nconsstent estmator of = 1, we need to correct the lmtng dstrbuton of ^ 1 for the nconsstency of the estmator n constructng test statstcs for the hypothess = 1 based on LS estmators. The nconsstency of ^ arses from the wthn transformaton of the data. Ths now allows for a potental break n the determnstc components of model (1). The lmtng dstrbuton of ^ 1 corrected for the nconsstency of ^ can be derved by makng the followng assumpton about the nature of the sequence 6

7 of the dsturbance terms {u t }. Assumpton 1: {u t } s a sequence of ndependently and dentcally dstrbuted (IID) random varables wth E(u t ) = 0, V ar(u t ) = u, and E(u 4 t ) = k u, 8 f1; ; :::; Ng and 8 t f1; ; :::; T g, where k < 1. Assumpton 1 enable us to derve the lmtng dstrbuton of the test statstc for the hypothess = 1 usng classcal (standard) asymptotc theory results, assumng that N goes to n nty and T s xed. The condton k < 1 of the assumpton mples that the fourth moment of the dsturbance terms, u t, exsts. Ths condton enables us to apply the law of large numbers (LLN) and the central lmt theorem (CLT) n drvng the lmtng dstrbuton of ^ 1 corrected for the nconsstency of ^, wthout makng any dstrbutonal assumptons about the sequence of the dsturbance terms. In so dong, note that we do not need to make any assumpton about the nature of the ntal observatons y o, and the ndvdual e ects a () and a (1 ) of the panel. Under the null hypothess, the test statstcs that we propose are nvarant to these nusance parameters. Ths s acheved by ncludng ndvdual e ects and trends n the auxlary panel data regresson model (1). 4 The next theorem presents the lmtng dstrbuton of unt root test statstcs for the alternatve spec catons of the determnstc components of panel data model (1), mentoned before. Theorem 1 Let the sequence fy t g be generated accordng to model (1), Assumpton 1 hold and the date of the structural break, T 0, be known. Then, under the null hypothess = 1, as N! 1 Z(; T ) p N(^ 1 B(; T )) L! N(0; C(k; u ; ; T )) (3) where 4 When X () = e () ; e (1 ), the suggested test statstc renders nvarant to the ntal condtons of the panel by ncludng only ndvdual e ects n the panel data auxlary regresson. 7

8 B(; T ) = p lm(^ 1) N!1 = tr[ 0 Q () ]ftr( 0 Q () )g 1, and 8 < T C(k; u; ; T ) = : k X j=1 9 = n jj + 4 utr(a () ) ; utr( 0 Q )o () ; a () where s a (T XT ) matrx de ned as r;c = 1, f r > c and 0 otherwse, A () [a j ] s a (T XT ) dmenson symmetrc matrx, de ned as A () = 1 (0 Q () + Q () ) B(; T )( 0 Q () ) and L! sgn es convergence n dstrbuton. The proof of the theorem s gven n the appendx. Below, we make some remarks whch hghlght some nterestng specal cases of the results of the theorem and dscuss how to mplement the test statstcs mpled by the theorem. Remark When X () = e () ; e (1 ), Theorem 1 gves a test statstc whch s approprate for the specal case where the panel model (1) under the null hypothess conssts of drftless random walks. Remark 3 When u t are NIID(0; u), k = 0. Then, the varance of lmtng n tr( dstrbuton of Z(; T ) s gven by tr(a )o () 0 Q () ). The results of Theorem 1 show that the nconsstency of the pooled LS estmator ^; gven by B(; T ), s a determnstc functon of the fracton of the structural break of the sample,, and the tme dmenson of the panel, T. Subtractng B(; T ) from ^ 1 leads to test statstcs of the hypothess = 1 for the cases of model (1) that X () = e () ; e (1 ) and X () = e () ; e (1 ) ; () ; (1 ). These statstcs are based on a correcton of the lmtng dstrbuton of ^ 1 for the nconsstency of the LS estmator ^. Gven consstent estmates of the nusance parameters of the varance functon C(k; u,; T ), k and u, the test 8

9 statstcs gven by the theorem approprately scaled by C(k; u; ; T ) can be readly used n practce to conduct nference about unt roots based on the tables of the standard normal dstrbuton. 5 Remark 3 shows that the test statstcs proposed by the theorem become nvarant to the nusance parameter u only under the normalty assumpton of the dsturbance terms.. The e ects of structural breaks on test for unt roots n dynamc panel data models The results of Theorem 1 can be used to analyse the consequences of gnorng a structural break n the determnstc components of the panel on drawng nference about unt roots based on dynamc, autoregressve panel data models wth determnstc components. The functonal form of Z(; T ) shows that there wll be two sources of bases of the panel unt root test statstcs when gnorng a potental break. The rst wll come from the term correctng for the nconsstency of the LS estmator,.e. B(; T ), whle the second from the varance of the lmtng dstrbuton of the test statstc, gven by C(k; u,; T ). Both of these terms depend on the fracton of the structural break of the sample,. The e ects of on the unt root tests can be rgorously studed by nvestgatng the senstvty of B(; T ) and C(k; u,; T ) to changes n the values of. To ths end, n the next corollary we derve analytc expressons of B(; T ) and C(k; u,; T ) for the specal, smpler case of model (1) where X () = e () ; e (1 ) assumng that u t are normally dstrbuted. Corollary 4 Let the sequence of the dsturbance terms,fu t g; be normally dstrbuted and the matrx of the determnstc components X () be spec ed as X () = e () ; e (1 ). Then, the lmtng dstrbuton of the test statstc gven by Theorem 1 s gven by z(; T ) p N(^ 1 b(; T )) L! N(0; c(; T )), (4) 5 Consstent estmates of k and u can be derved based on GMM estmates of the fourth and second moments of the rst d erences of the panel data y t. 9

10 where b(; T ) = 3(T )f 1 ()T + 0 ()g 1 ; and c(; T ) = 3 5 f 6()T ()T ()T ()T 3 + ()T + 1 ()T + 0 ()gf 1 ()T + 0 ()g 4, where the polynomal functons s and s are de ned n the appendx. The proof of the corollary s gven n the appendx. Some remarks on the results of the corollary are gven below. Remark 5 For = 0, the test statstc gven by Corollary leads to the test statstc derved by Harrs and Tzavals (1999) for the case that X () = e () ; e (1 ). Remark 6 For su cently large T, Corollary mples that z() T p N ^ () L! N 0; 3 6() 5 1 () 4 : (5) The test statstc gven by (5) s derved by takng lmts of b(; T ) and c(; T ) for T gong to n nty and scalng approprately by T. Fgures.1 and. graphcally present the values of the nconsstency correcton term b(; T ) and the varance c(; T ) of the test statstc z(; T ), gven by Corollary 4, wth respect to T (see horzontal axs). Ths s done for the followng set of values = f0:0; 0:5; 0:8g. To make nterestng comparsons wth the case that T s su cently large (see Remark 6), the values of b(; T ) and c(; T ) have been approprately scaled by T. The graphs lead to the followng conclusons. Frst, both the values of the nconsstency correcton term b(; T ) and the varance c(; T ) are smaller n 10

11 Fgure 1: T b(; T ) magntude when the statstc allows for a break. Ths s true ndependently on whether T s large (see horzontal lnes) or xed. These results mply that there wll be two counteractng e ects on the sze of the panel unt root test statstc z(; T ) whch does not allow for a structural break. The smaller magntude of b(; T ) wll tend to ncrease the sze of the test by shftng ts whole dstrbuton to the left of the emprcal dstrbuton of the test allowng for the break, whle the smaller varance wll tend to decrease the sze of the test by drawng n the left tal of ts dstrbuton. If the rst of these e ects, referred to as mean e ect, domnates the second, then the test wll be overszed. If the second e ect, referred to as varance e ect, s domnant, then the test wll be underszed, and thus wll have low power to reject the null hypothess aganst ts alternatve hypothess wth a potental break. The above analyss ndcates that the falure of the panel unt root statstcs to reject the null hypothess when gnorng a potental break n the ndvdual e ects of the panel [see Carron--Slvestre, Del Barro-Castro and Lopez-Bazo (001), for a smulaton study] may be attrbuted to a downwards based estmate of the varance of the test statstc whch gnores a potental break. 11

12 Fgure : T c(; T ) vs 36() 5 1() 4 (horzontal lnes) The second concluson whch can be drawn from the graphs s that a substantal number of tme seres observatons s requred n order to apply the large-t test statstc mpled by the corollary, denoted z(), nstead of the xed-t test statstc z(; T ), n the presence of a break, otherwse serous sze dstortons wll occur n the same way as n the case of gnorng a potental break, analysed above. Indeed, the graphs show that the number of the tme seres observatons of the panel whch are requred n order b(; T ) and c(; T ) to reach ther T asymptotes, derved for T gong to n nty, vares wth. Note that ths number reaches ts maxmum value when = 0:5,.e. the break pont s n the mddle of the tme dmenson of the panel. For ths case we need panels wth very hgh tme dmenson (e.g. T > 300) for the nconsstency correcton term b(; T ) and the varance of the lmtng dstrbuton of the statstc c(; T ) to reach ther asymptotc lmts, over T. Ths happens because, when = 0:5, the shft n the level of the mean of the seres of the panel exhbts ts longest persstence. The above analyss ndcates that large-t based panel data unt root test statstcs may requre a substantal number of tme seres observatons n order to be able to dstngush the persstency of the shft n the determnstc 1

13 component from that of the dsturbance terms..3 The date of the break pont s unknown The results of Theorem 1 are based on the assumpton that the break pont s known. In ths subsecton, we relax ths assumpton and propose test statstcs for model (1), wth X () = e () ; e (1 ) and X () = e () ; e (1 ) ; () ; (1 ) ; whch allow for a structural break at an unknown date. As n Perron and Vogelsang (199), and Zvot and Andrews (199), we vew the selecton of the break pont as the outcome of mnmzng the test statstcs gven by Theorem 1 over all possble break ponts of the sample, after trmmng the ntal and nal parts of each seres of the panel model (1). 6 That s, the mnmum values of the test statstcs Z(; T ), over all (0; 1); are chosen to gve the least favorable result for the null hypothess = 1. Let ^ nf denote the break pont at whch the mnmum value of Z(; T ), over all (0; 1); are obtaned. Then, the null hypothess wll be rejected f where c nf nf (0;1) Z(; T ) < c nf, denotes the sze a left-tal crtcal value from the dstrbuton of the mnmum values of Z(; T ), denoted enable us to calculate the dstrbuton of nf Z(; T ). The followng theorem (0;1) Z(; T ). nf (0;1) Theorem 7 Let Assumpton 1 hold and assume that the date of the break pont be unknown. Then, as N! 1, nf Z(; T ) d! nf N(0; R) (6) (0;1) (0;1) where R [r s ] s the correlaton matrx of the test statstcs Z(; T ), for all (0; 1); wth elements gven by r s 6 For trmmng the ntal and nal parts of the seres of the panel, note that T 0 T should range from 1 T to = T 1 for model (1) wthout the trends, whle for model M t ranges from T T to = T for model (1) wth the trends. For the later model, note that the ndvdual T e ects and the determnstc trends are not dent ed when = 1 T and = T 1 T. 13

14 k P T j=1 r s = a() jj a(s) jj + 4 utr(a () A (s) ) n k P o 1= n T j=1 a() jj + 4 utr(a () ) k P o 1=, T j=1 a(s) jj + 4 utr(a (s) ) for ; s (0; 1). Proof: The result of Theorem 3 follows mmedately from the result of Theorem 1 usng the contnuous mappng theorem. The functonal form of the correlaton coe cents r s of Z(; T ), for ; s (0; 1), can be derved usng E( () (s) ) = k P T j=1 a() jj a(s) jj + 4 utr(a () A (s) ), where (j) ned n the Appendx (see proof of Theorem 1). = u 0 A(j) u s de- Remark 8 When u t s NIID(0; u), then k = 0 and r s are gven by r s = tr(a () A (s) ) ftr(a () )g 1= ftr(a (s) )g 1=. The result of Theorem 7 shows that the crtcal values c nf of the lmtng dstrbuton of the statstcs nf Z(; T ) can be calculated by those of the (0;1) mnmum values of a xed number of correlated normal varables wth correlaton matrx R. In the case that u t are normally dstrbuted, Remark 8 shows that the crtcal values c nf become nvarant to the nusance parameter u. Table 1: Crtcal Values of nf N(0; R) (0;1) T A 1% :91 :95 :98 3:05 :9 :97 3:04 3:10 5% :15 :33 :37 :43 :31 :38 :43 :49 10% 1:83 :00 :04 :10 1:99 :07 :11 :16 Notes: Panel A of the table presents the crtcal values c nf for the specal case that model (1) contans only the ndvdual e ects n ts determnstc components,.e. X () = e () ; e (1 ), whle Panel B of the table presents the crtcal values for the full spec caton of the model, whch contans both the ndvdual e ects and trends,.e. X () = e () ; e (1 ) ; () ; (1 ). B 14

15 In Table 1, we present crtcal values of nf (0;1) N(0; R) at 1%, 5% and 10% sgn cance levels, and for d erent values of T, assumng that u t s NIID(0; u). 7 These are calculated from Monte Carlo experments as follows. For each replcaton, we generated a vector of observatons from a multvarate normal dstrbuton of dmenson T mnus the trmmng ponts of the sample wth correlaton matrx R, de ned n Remark 4. We then sorted the vector of observatons n order and we selected the mnmum value. The crtcal values reported n the table correspond to 1%, 5% and 10% percentle of the sorted vector of the replcated mnmum values. Not surprsngly, these values are well below the left-tal crtcal values of the normal dstrbuton, at the correspondng sgn cance levels, and devate more from them as T ncreases. 3 Smulaton results In ths secton we explore the nte sample performance of the test statstcs suggested n the prevous secton by conductng 5000 Monte Carlo experments. Ths s done for d erent combnatons of N and T. In each experment we assume that the dsturbance terms, u t, are generated as u t NIID(0; 1). 8 Tables (a)-(b) report the nomnal sze at a level of sgn cance 5% and the sze-adjusted power of the test statstcs whch consder the break pont as known. In partcular, Table (a) reports the results of the test statstc for the specal case that model (1) contans only the ndvdual e ects n ts determnstc part,.e. X () = e () ; e (1 ), whle Table (b) for the full spec caton of the model (1), where X () = e () ; e (1 ) ; () ; (1 ). To calculate the power of the test statstcs, we generated the panel data accordng to the followng two alternatve hypotheses: = f0:95; 0:90g. The ntal observatons of the panel, y 0, are set equal to zero for both of the above spec catons of model (1), as the test statstcs are nvarant to y 0. 7 A Rats programme calculatng the crtcal values c nf s avalable upon request. 8 Note that we only consder cases of u t and a wth unt varance, as our test statstcs under the null hypothess are nvarant to the varance u when u t NIID, and to the ndvdual e ects, a. 15

16 In the case that X () = e () ; e (1 ) ; () ; (1 ), we assume that the structural break n the ndvdual e ects occurs both under the null and alternatve hypotheses. In partcular, we generated the determnstc components of the panel as: a () = 0:0 and a (1 ) and = (1 )a (1 ) (1 ) = 0:5+a, where a NIID(0; 1), and () = 0. Ths s done n order to nvestgate whether the test statstc has the power to dstngush between panels consstng of random walks wth broken drft parameters and panels consstng of statonary seres around broken trends and ndvdual e ects. The results of Tables (a)-(b) clearly ndcate that the test statstcs gven by Theorem 1 have a sze very close to the 5% level n nte samples. Ths s true even for very small values of N, such as N = 5. The power performance of the tests s analogous to that of the xed-t panel unt root tests of Harrs and Tzavals (1999), who consdered the case of no structural break. The power ncreases as both N and T ncreases, and grows faster wth T rather than N. Consstent also wth the sngle tme seres tests, the power performance reduces when X () = e () ; e (1 ) ; () ; (1 ),.e. the panel ncludes both the ndvdual e ects and trends n ts determnstc part. For ths case, we found that one needs panels wth T > 5 to acheve good performance of the test statstc, as for the case that X () = e () ; e (1 ). Table 3 reports the results of the sze and the sze adjusted powers of the sequental test statstcs whch treat the break pont as unknown. Panel A of the table presents the results for the case that X () = e () ; e (1 ) ;whle Panel B for the case that X () = e () ; e (1 ) ; () ; (1 ). The seres of the panel for ths smulaton experment were generated n the same way as wth the prevous one, whch treats the break pont as known. To calculate the power of the tests we assumed that under the alternatve, there s a break at each possble tmepont of the sample, sequentally searched for a break. To calculate the sze of the tests we used the crtcal values of the tests at 5% sgn cance level reported n Table 1. The results of the table clearly show that the performance of the test sta- 16

17 tstcs for the case that the break pont s unknown s the same wth that of the known-break case. Ths s true for both test statstcs consdered,.e. for X () = e () ; e (1 ) and X () = e () ; e (1 ) ; () ; (1 ). These results support the vew that Harrs and Tzavals (1999) tests perform equally well for the case that they are adjusted for a structural break n any of the determnstc components of the seres of the panel. 4 Emprcal Applcaton As an emprcal applcaton of the tests, n ths secton we reexamne whether the level of the real per capta ncome of each country s mean revertng to ts steady state, as the economc convergence hypothess asserts [see Mankew, Romer and Wel (199), nter ala]. Ths mplcaton of the economc convergence hypothess s known as convergence. In partcular, we employ our sequental test statstcs for the spec caton of model (1) whch contans ndvdual e ects n ts determnstc part,.e. X () = e () ; e (1 ), n order nvestgate whether a unt root n per capta ncome, whch mples economc dvergence, can be rejected n favour of the alternatve hypothess of convergence. Ths spec - caton of the panel data model (1) s often used n practce [see Islam (1995), and Casell, Esquvel and Lefort (1996), nter ala] to test for the convergence hypothess because t has more power to reject the dvergence hypothess aganst convergence, compared wth the sngle tme seres based unt root tests [see Bernard and Durlauf (1994)] or the cross secton based tests suggested by Barro and Sala--Martn (1995). Ths happens because the panel data based tests for unt roots have better power to dstngush between the null hypothess of unt roots and ts alternatve of statonarty, as they can explot both cross secton and tme seres nformaton of the data. Note that the sequental panel data unt root test statstc whch we employ n ths paper n order to reexamne the convergence hypothess has also more power to reject the null hypothess of dvergence aganst ts alternatve of statonarty when there s a permanent shft n the level of the per capta ncome of each seres of the panel, across all 17

18 countres. We carry out our sequental unt root test e () ; e (1 nf (0;1) Z(; T ), when X() = ), for two d erent groups of countres: () the Non-ol countres (N = 89) and () the OECD countres (N = ). In order to mtgate the possble e ects of cross secton correlaton on the results of the tests, tme dummes have been ncluded n the auxlary regresson model (1), as suggested by O Connel (1998). To see f allowng for a structural break n the panel unt root tests makes any d erence n drawng nference about the economc convergence/dvergence hypothess, we have also conducted Harrs and Tzavals (1999) tests, whch do not allow for a break. We found that these tests can not reject the economc dvergence hypothess for both groups of countres consdered. 9 The values of the sequental statstc, over all ponts of the sample, are graphcally presented n Fgures 3(a)-(b); Fgure 3(a) presents the results of the statstc for the group of the Non-ol countres, whle Fgure 3(b) presents the results for the OECD countres. The graphs of the gures clearly ndcate that the null hypothess can be rejected only for the group of the OECD countres when consderng for a break. For ths group of countres, we found that the value of the test statstc s nf Z(; T ) = 4:10, whch s clearly smaller than (0;1) ts crtcal value at 5% sgn cance level, mpled by the crtcal values of Table 1. The break pont s found to occur n year 1978, just before the second olcrss n year For the group of the Non-ol countres, there s no evdence of convergence. Summng up, the results of ths secton support the vew that evdence of economc dvergence found by many emprcal economc growth studes even for groups of countres wth the same level of economc convergence may be attrbuted to the exstence of a structural break n the steady state of the per capta ncome, across countres. They also ndcate that not all the groups of 9 The values of Harrs and Tzavals test statstcs, denoted HT, are found to be: HT=4. for the group of the OECD countres and HT=8.99 for the group of the Nol-ol countres. 18

19 countres seems to convergence to ther steady state real per capta ncome. 5 Conclusons In ths paper we proposed panel data unt root testng procedures that allow for a structural break n the ndvdual e ects and/or trends of the panel, assumng that the tme dmenson of the panel s xed. The test statstcs allow for a break pont at ether a known or unknown date. When the break pont s consdered as known, we show that the test statstcs have normal lmtng dstrbutons whose varance depend on the fracton of the sample that the break occurs. When the break s consdered as unknown, we suggested a sequental testng procedure of the null hypothess of unt roots. Ths entals n computng the test statstcs for known break pont at each possble break pont of the sample, and then selectng the test statstcs wth the mnmum values to test the null hypothess. The mnmum values of the sequental test statstcs have a dstrbuton whose crtcal values can be tabulated by that of the mnmum values of a xed number of correlated normal varables, after trmmng for the ntal and nal tme-ponts of the sample. Crtcal values of these dstrbutons have been tabulated based on Monte Carlo smulatons. To evaluate the nte sample performance of the tests statstcs we run Monte Carlo smulatons. We found that both categores of the test statstcs, wth known and/or unknown break pont, have emprcal sze whch s very close to the 5% and power whch ncreases wth both dmensons of the panel, but faster wth the tme-dmenson. As an emprcal llustraton, we employed the test statstc for the panel data model wth ndvdual e ects n ts determnstc component under the alternatve hypothess n order to nvestgate whether evdence of economc dvergence across countres can be attrbuted to a possble structural break n the steady state of the per capta ncome of each country, whch s re ected n the ndvdual e ects of the panel. We found evdence of such a break n year 1978 for the group of the OECD countres. 19

20 A Appendx In ths appendx we present the proofs of the man theoretcal results of the paper. Proof of Theorem 1: To derve the lmtng dstrbuton of the test statstc, we proceed nto stages. We rst show that the pooled LS estmator, ^, s nconsstent, as N! 1. We then construct a normalsed statstc based on ^ corrected for ts nconsstency, and derve the lmtng dstrbuton under the null hypothess of ^, as N! 1. Decompose the vector y ; 1 under the null hypothess as y ; 1 = ey e () a () + 0 e (1 ) (1 ) a + 0 u, (7) where e s the (T X1) dmenson vector of untes and the matrx s de ned n the theorem. Premultplyng equaton (7) wth the matrx Q () yelds Q () y ; 1 = Q () 0 u, (8) snce Q () (e; 0 e () ; 0 e (1 ) ) = (0; 0; :::; 0). Note that equaton (8) also holds for the case that a () (1 ) = a = a under the null hypothess,.e. there s no break. Ths happens because Q () ( 0 e () a () + 0 e (1 ) (1 ) a ) = Q () 0 ea = 0. Substtutng (8) nto () and notcng that Q () s an ndempotent and symmetrc matrx yelds " N # " X N # 1 X ^ 1 = u 0 0 Q () u u 0 0 Q () u. (9) =1 =1 Takng probablty lmts of equaton (9) yelds 0

21 B(; T ) = p lm (^ 1) N!1 1 = E hu 0 0 Q () u E hu 0 0 Q () u h = tr 0 Q () n h 1 tr 0 Q o (), (10) by the LLN. Subtractng the term B(; T ) from (9) yelds = = ^ 1 B(; T ) ( X N h ) ( X N u 0 0 Q () u B(; T )(u 0 0 Q () u ) u 0 0 Q () u =1 ( N X =1 () =1 ) 1 ) ( X N 1 u 0 0 Q u) (), (11) =1 where () = u 0 0 Q () u B(; T )(u 0 0 Q () u ) s a random varable whch has zero mean by constructon and constant varance, denoted V ar( () ), 8. Usng standard results on quadratc forms, can be wrtten as = u 0 1 () = u Q () + Q () 0 Q () + Q () u B(; T )(u 0 0 Q () u ) B(; T )( 0 Q () ) = u 0 A () u, (1) u where A () = 1 0 Q () + Q () B(; T )( 0 Q () ) s a symmetrc matrx, snce 1 0 Q () + Q () and ( 0 M () ) are symmetrc matrces. Usng results on quadratc forms for symmetrc matrces, t can be seen that V ar( () ) s gven by 1

22 [see Anderson (1971)]. V ar( () ) = V ar[u 0 A () u ] TX = k + 4 utr j=1 a () jj A () (13) The result of the theorem can be proved by scalng (11) approprately and usng the followng asymptotc results, for N! 1: Ths yelds by the CLT, and 1 p N N X =1 () d! N(0; V ar( )) (14) p lm 1 N NX =1 h u 0 0 Q () u = utr 0 Q () (15) by the LLN, the Cramer-Wold lemma. These results hold under the condtons of Assumpton 1. Note that the condton k < 1 of the assumpton guarantees that V ar( () ) exsts. Proof of Corollary : To prove Corollary, rst notce that under normalty k = 0: To obtan the analytc results of the corollary, we wll expand the nconsstency correcton term b(; T ) and the varance V ar[ () ] by replacng h the matrx Q () wth M () 1 = I T e() e ()0 1 (1 )T e(1 ) e, (1 )0 gven for X () = e () ; e (1 ). Then, expandng b(; T ) yelds h b(; T ) = tr 0 M () n h o 1 tr 0 M () = 3(T ) 1 ()T + 0 () 1, (16) where 1 () = ( + 1), and 0 () =. The last result s derved by usng the followng two results tr 0 M () = T and tr 0 M () = 1 6 T (T ) 1 3 T.

23 To derve an analytc form for V ar[ () ], wrte V ar[ () ] = V ar u M () + M () u b(; T )Cov u b(; T ) V ar hu 0 0 M () u 0 M () + M () u ; u 0 0 M () u. (17) Expandng the component terms of V ar[ () ] n the above expresson yelds the followng results: V ar u M () + M () u = 1 4 utr 0 M () + M () 1 = 1 ( + 1)T + 1 T u, (18) V ar hu 0 0 M () u = 4 utr 0 M () = 1 45 ( )T ( + 1 )T u, (19) and Cov 1 = tr = u M () + M () u ; u 0 0 M () u 0 M () + M () 0 M () 1 3 ( + 1 )T Substtutng (18), (19) and (0) nto (17) yelds 4 u (0) 3

24 V ar[ () ] = V ar u M () + M () u + B LSDV (; T ) V ar hu 0 0 M () u B LSDV (; T )Cov u M () + M () u ; u 0 0 M () u = 4 u 6 ()T ()T ()T ()T 3 + ()T ()T + 0 ()] 1 ()T + 0 (), (1) where 6 () = , 5 () = , 4 () = , 3 () = 0, () = , 1 () = 16, and 0 () = 136. The results of the corollary follows mmedately by substtutng (16) and (1) nto the test statstc Z(; T ), gven by Theorem 1, and notcng that the p lm of the denomnator of ^ s gven by tr 0 M () = 1 ()T + 0 () (see equaton (16)). 4

25 References [1] Arelano E. and B. Honoré (00), Panel data models: Some recent developments, Handbook of Econometrcs, edted by Heckman J. and E. Leamer edtors, Vol 5, North Holland. [] Baltag B.H. (1995), Econometrc analyss of panel data, Wley, Chchester. [3] Baltag B.H. (000), Nonstatonary panels, contegraton n panels and dynamc panels: A survey n advances n econometrcs: nonstatonaty panels, panel contegraton and dynamc panels, edted by Baltag B.H, T.B. Fomby and R. C. Hll, 15, [4] Barro R.J. and X. Sala--Martn (1995), Economc growth, McCraw-Hll, New York. [5] Bernard A.B. and S.N. Durlauf (1995), Convergence of nternatonal output, Journal of Appled Econometrcs, 10, [6] Carron--Slvestre, T. Del Barro-Castro, and E. Lopez-Bazo (001), Level shfts n a panel data based unt root test. An applcaton to the rate of unemployment, mmeo, Department of Econometra, Unversty of Barcelona, Span. [7] Cassel F., G. Esquvel and F. Lefort (1996), Reopenng the convergence debate: a new look at cross-country growth emprcs, Journal of Economc Growth, 1, [8] Culver, S.E. and D.H. Papell (1999), Long-run power party wth short-run data: evdence wth a null hypothess of statonarty, Journal of Internatonal Money and Fnance, 18, [9] de la Fuente (1997), The emprcs of Growth and Convergence: A selectve revew, Journal of Economc Dynamcs and Control, 1,

26 [10] Harrs R.D.F. and E. Tzavals (1999), Inference for unt roots n dynamc panels where the tme dmenson s xed, Journal of Econometrcs, 91, [11] Islam N. (1995), Growth Emprcs: a panel data approach, Quarterly Journal of Economcs, 110, [1] Levn A., and C.F., Ln, and C-S. J, Chu (00), Unt root tests n panel data: asymptotc and nte-sample propertes, Journal of Econometrcs, 108, 1-4. [13] Lo A.W. and A.C. MacKnlay (1995), A non-random walk down Wall Street, Prnceton Unversty Press. [14] O Connell P.G.J (1998), The overvaluaton of purchasng power party, Journal of Internatonal Economcs, 44, [15] Perron P. (1989), The great crash, the ol prce shock, and the unt root hypothess, Econometrca, 57, [16] Perron P. (1990), Testng for a unt root n a tme seres wth a changng mean, Journal of Busness & Economc Statstcs, 8, [17] Perron P. and T., Vogelsang (199), Nonstatonarty and level shfts wth an applcaton to purchasng power party, Journal of Busness & Economc Statstcs, 10, [18] Zvot E. and D.W.K. Andrews (199), Further evdence on the great crash, the ol prce shock, and the unt-root hypothess, Journal of Busness & Economc Statstcs, 10,

27 Table (a): Monte Carlo Smulatons for known break pont N T = 0: SIZE: Power: = 0: = 0: :5 SIZE: Power: = 0: = 0: = 0:8 SIZE: Power: = 0: = 0: Notes: Ths table presents the results of the Monte Carlo smulatons for model (1) wth ndvdual e ects,.e. X () = e () ; e (1 ). 7

28 Table (b): Monte Carlo Smulatons for known break pont N T = 0: SIZE: 0:06 0:06 0:06 0:06 0:06 0:06 0:05 0:06 0:06 POWER: = 0:95 0:05 0:06 0:05 0:07 0:09 0:06 0:07 0:11 0:44 = 0:90 0:06 0:08 0:07 0:11 0:4 0:08 0:15 0:39 1:00 0:5 SIZE: 0:05 0:05 0:05 0:05 0:06 0:05 0:05 0:05 0:05 POWER: = 0:95 0:05 0:05 0:05 0:06 0:07 0:06 0:06 0:10 0: = 0:90 0:06 0:08 0:06 0:08 0:16 0:07 0:10 0:38 0:90 = 0:8 SIZE: 0:05 0:06 0:05 0:06 0:06 0:05 0:06 0:06 0:06 POWER: = 0:95 0:05 0:06 0:06 0:06 0:09 0:05 0:07 0:11 0:44 = 0:90 0:06 0:09 0:07 0:10 0:6 0:08 0:13 0:44 1:00 Notes:Ths table presents the results of the Monte Carlo smulatons for model (1) wth ndvdual e ects and trends,.e. X () = e () ; e (1 ) ; () ; (1 ) : 8

29 Table 3: Monte Carlo Smulatons for unknown break pont N T A SIZE: 0:08 0:09 0:07 0:08 0:07 0:06 0:07 0:07 0:09 POWER: = 0:95 0: 0:3 0:36 0:55 0:87 0:6 0:84 0:99 1:00 = 0:90 0:49 0:70 0:76 0:94 0:99 0:97 1:00 1:00 1:00 B SIZE: 0:06 0:06 0:05 0:06 0:07 0:05 0:06 0:07 0:06 POWER: = 0:95 0:05 0:06 0:05 0:06 0:09 0:06 0:07 0:09 0:49 = 0:90 0:06 0:09 0:07 0:11 0:7 0:08 0:14 0:7 1:00 Notes: Panel A of the table presents the results of the Monte Carlo smulatons for the sequental test statstcs for the cases that X () = e () ; e (1 ) (see Panel A) and X () = e () ; e (1 ) ; () ; (1 ) (see Panel B). 9

30 Fgure 3: Sequental test statstc for the Non-ol countres Sequental test statstc for the OECD countres. 30