Panel cointegration rank test with cross-section dependence

Transcription

1 Panel contegraton rank test wth cross-secton dependence Josep Lluís Carron--Slvestre Laura Surdeanu y AQR-IREA Research Group Department of Econometrcs, Statstcs and Spansh Economy Unversty of Barcelona June 2007 Abstract In ths paper we propose a test statstc to determne the contegraton rank of VAR processes n panel data allowng for cross-secton dependence among the tme seres n the panel data. The cross-secton dependence s accounted for through the spec caton of an approxmate common factor model, whch allows to cover stuatons where there s cross-contegraton,.e. contegraton among the cross-secton dmenson. The framework that s consdered n the paper embedes as a partcular case the stuaton where tme seres n the panel data set are cross-secton ndependent. Fnte sample szes and powers are smulated va a Monte Carlo experment. JEL Class caton: C12, C22 Keywords: Panel data contegraton, contegratng rank, cross-secton dependence, cross-contegraton Av. Dagonal, Barcelona. Tel: ; fax: ; e-mal: carron@ub.edu y Av. Dagonal, Barcelona. e-mal: lsurdesu7@docd2.ub.edu 1

2 1 Introducton Snce the poneerng work of Engel and Granger (1987) and Johansen (1988, 1995), the lterature on contegraton grew at a rapd pace. Even though ths topc has been covered extensvely n tme seres, the analyss of contegraton n panel data s n early stages of development. The two man approaches to the analyss of contegraton n panels are the resdual-based and the system-based approaches. Regardng the resdual-based approxmaton, we can nd n the econometrc lterature proposals that specfy ether the null hypothess of no contegraton see Pedron (1995, 1999, 2004) and Kao (1999), among others or the null hypothess of contegraton see McCoskey and Kao (1998) and Westerlund (2005). The system-based approach has reled on the panel extenson of Johansen s methodology. Thus, based on the vector autoregressve model (VAR) framework, Larsson, Lyhagen and Löthgren (2001) propose the panel Johansen s test analog to determne the rank of the contegratng space. The proposals mentoned above de ne the so-called rst generaton of panel contegraton tests, n whch the tme seres (ndvduals) that de ne the panel data set are assumed to be cross-secton ndependent. Unfortunately, ths assumpton s crucal for the lmtng dstrbutons that are obtaned n these papers and, n most cases, t s not sats ed from an emprcal pont of vew. Volaton of the cross-secton ndependence assumpton mples that Central Lmt Theorems (CLT) cannot be appled and, hence, the panel data based statstcs do not converge to the standard normal dstrbuton. Provded that n most cases the economc tme seres of d erent sectors, ctes, regons or countres are closely related, the use of these panel data statstcs to analyse the presence of contegraton can lead to msleadng conclusons. The challenge to overcome ths lmtaton has gven rse to the so-called second generaton of panel contegraton tests. Proposals that consder the presence of cross-secton dependence among the tme seres that de ne the panel data set nclude Ba and Carron--Slvestre (2005), Banerjee and Carron--Slvestre (2006), and Gengenbach, Palm and Urban (2006) for the sngle equaton framework, and Groen and Klenberger (2003) and Bretung (2005) for the vector error correcton (VECM) framework. For a more detaled lterature revew on panel contegraton see Bretung and Pesaran (2007). The am of ths paper s to solve some lmtatons of the exstng lterature on panel contegraton analyss. To ths end, we propose a test statstc to determne the contegratng rank n a panel system of equatons allowng for the presence of cross-secton dependence across the systems of varables n the panel set-up. We deal wth cross-secton dependence by means of approxmate common factor models as proposed n Ba and Ng (2004), Ba and Carron--Slvestre (2005), Banerjee and Carron--Slvestre (2006), and Gengenbach, Palm and Urban (2006), among others. The novelty of our approach s that t takes nto account the possblty that there mght be more than one contegratng relatonshp among the varables that de ne the system for each ndvdual, at the tme that controls for the presence of cross-secton dependence among the d erent systems n a parsmonous way through the use of common factors. To the best of our knowledge, ths has not been addressed n the lterature. Our proposal relates to other test statstcs that are avalable n the lterature to determne the number of stochastc trends n ndvdual systems. Thus, the statstc determnes the 2

3 number of stochastc trends usng the prncpal component approxmaton as n Stock and Watson (1988) and Ba and Ng (2004). Other statstcs that base on the degeneraton of the moment matrx that nvolves the tme seres n the system are the ones n Phllps and Oulars (1990), Shntan (2001), Harrs and Posktt (2004) and Ca and Shntan (2006). However, none of these approaches consders the case where the varables n the model are a ected by global stochastc trends, whch n our set-up are captured by the common factors. In order to nvestgate the small-sample propertes of the panel contegraton rank test that we propose, we conduct a Monte Carlo smulaton. We estmate two d erent models dependng on the form of the determnstc component. One model where the determnstc term conssts of only the constant and another where the determnstc term conssts of the constant and the lnear trend. The results of the smulaton study ndcate that, n general, panel data statstc performs better than the unvarate one. The remander of the paper s organzed as follows. Secton 2 presents the model, assumptons and the test statstc that s used to determne the contegratng rank. In addton, we dscuss the way n whch the ndvdual statstcs can be combned to specfy a panel data contegratng rank statstc. Secton 4 analyses the nte sample of our approach, both n an ndvdual-by-ndvdual framework and n a panel set-up, by means of Monte Carlo smulaton. Fnally, some concludng remarks are presented n Secton 5. The Appendx collects all the proofs. 2 Model and assumptons Let Y ;t be a (k 1) vector of stochastc process wth the data generatng process (DGP) de ned as: Y ;t = D ;t + u ;t (1) u ;t = F t + e ;t (2) (I L) F t = C (L) w t (3) (I L) e ;t = G (L) " ;t ; (4) where D ;t denotes the determnstc component, whch n ths paper can be ether D ;t = henceforth, ths spec caton s denoted as the only constant case or D ;t = + t hereafter, the lnear tme trend case t = 1; : : : ; T and = 1; : : : ; N. Note that the case of non-determnstc components D ;t = 0 s also covered n our framework as a partcular case of the only constant case. The component F t denotes a (q 1) vector of common factors and s a (k q) matrx of factor loadngs. Fnally, e ;t s a (k 1) vector that collects the dosyncratc stochastc component. It s worth mentonng that the presentaton of the model s done n a general way, so that the case where there are no common factors at all,.e. = 0 8, can be embeded n our framework further spec c comments on ths concern are gven below. Let M < 1 be a generc postve number, not dependng on T and N. Throught the paper, we use kak to denote the Eucldean norm tr (A 0 A) 1=2 of matrx A. The stochastc processes that partcpate on the de nton of the DGP are assumed to satsfy the followng assumptons: 3

4 Assumpton A: () for non-random, k k M; for random, E k k 4 M, () P 1 N N =1 0 p!, a (q q) postve de nte matrx. Assumpton B: () w t d (0; w ), E kw t k 4 M, and () V ar (F t ) = P 1 j=0 C j w Cj 0 > 0, () P 1 j=0 j kc jk < M; and (v) C (1) has rank q 1, 0 q 1 q. P Assumpton C: () for each, " ;t d (0; " ), E k" ;t k 4 M, () V ar (" ;t ) = 1 j=0 G ;j " G 0 ;j > 0, () P 1 j=0 j kg ;jk < M; and (v) G (1) has rank r. Assumpton D: The errors " ;t, w t, and the loadngs are three mutually ndependent groups. Assumpton E: E kf 0 k M, and for every = 1; : : : ; N, E ke ;0 k M. The de nton of the (k q) loadng matrx s gven by ;1;1 = ;1;2 ;2;1 so that we can mpose restrctons on how the factors a ect the elements of Y ;t n (1). Thus, some of the factors can only a ect one subset of the varables, say, the seres that de nes the contegratng space, but not the other varables, and the other way round. Therefore, stuatons where ;1;2 = 0 and/or ;2;1 = 0 are covered n ths set-up. Note that t s possble that both ;1;2 6= 0 and ;2;1 6= 0, whch s the general stuaton that s assumed henceforth. The unobservable common factors are estmated usng the prncpal component approach suggested n Ba and Ng (2002, 2004). Let us consder the general determnstc component gven by D ;t = + t. Takng the rst d erence of the model we have ;2;2 Y ;t = + F t + e ;t : (5) We can de ne the dempotent matrx M = I T 1 ( 0 ) 1 0, wth a (T 1) 1 vector of ones. Then, MY = MF 0 + Me : y = f 0 + z : Note that when the determnstc component s D ;t =, takng rst d erences removes the constant term, so that n ths case we can de ne M = I T 1 and the rest of our dscusson apples wthout sole mod caton. The common factors are extracted as the q egenvectors correspondng to the q largest egenvalues of the (T 1) (T 1) matrx yy 0, where y = [y 1 ; : : : ; y N ] s a (T 1) Nk matrx that s de ned usng the (T 1) k matrces y, = 1; : : : ; N. The matrx of estmated weghts, ^ = ^1 ; : : : ; ^ N, s gven by ^ = y 0 ^f. We can obtan an estmate of z from ^z = y ^f ^0, as n Ba and Ng (2004). Note that we can recover the common factors as ^F t = P t ^f j=2 j and the dosyncratc component as ^e ;t = P t j=2 ^z ;j. The determnaton of the contegratng rank can be devsed usng ^e ;t n the usual VECM representaton. The model gven by (1) and (2) can be wrtten as Y ;t = D ;t + F t + e ;t : (6) 4

5 Note that f we assume the smplest VAR model for the dosyncratc dsturbance term, we have e ;t = B e ;t 1 + " ;t ; and de nng B I p = = 0, we get e ;t = [I p (I p + 0 ) L] n (6), for D ;t = + t, gves 1 " ;t, whch substtuted Y ;t = D ;t + F t + [I p (I p + 0 ) L] 1 " ;t Y ;t = [I p (I p + 0 ) L] D ;t + [I p (I p + 0 ) L] F t + " ;t The model can be expressed as: = [ 0 + (I p + 0 ) ] 0 t + 0 Y ;t 1 + F t 0 F t 1 + " ;t : (Y ;t D ;t F t ) = 0 (Y ;t 1 D ;t 1 F t 1 ) + " ;t e ;t = 0 e ;t 1 + " ;t : (7) We can see that the contegratng rank can be obtaned from the analyss of the dosyncratc stochastc component. Instead of usng the Johansen LR statstc as n Larsson, Lyhagen and Löthgren (2001), n ths paper we propose to determne the rank wth a test statstc that s based on the multvarate verson of the square of the mod ed Sargan-Bhargava (MSB) statstc proposed n Stock (1999). The de nton of the testng procedure bulds upon the d erent rates of convergence of the elements on the Q^e^e matrx under the null hypothess. Wthout loss of generalty, let us assume that the rank of the contegratng space s 0 < r < k. We can de ne the orthogonal matrx A = [A 1 : A 2 ] wth A 1 a (k r) matrx and A 2 a (k m) matrx, m = k r, such that the rst r elements of the rotated vector e A ;t = A 0 e ;t = (A 0 1e ;t ) 0 ; (A 0 2e ;t ) 0 0 are I(0) and the other m elements are I(1). Accordngly, we de ne the partton of the long-run varance matrx as 11; = 12; : e A e A Furthermore, note that T 1 1 Q^e^e ^ 21; 22; T 1 Q^e A ^e A ^ 1 = T 1 1=2 ^ ^e A ^ea ^e A Q^e ^ea A ^e A ^e ^e = where () denotes the egenvalues of the matrx between parenthess. The determnaton of the number of stochastc trends n the system reles on the followng sequental testng procedure: 1. Frst, assume that the contegratng rank s zero,.e. set m = k. 2. Specfy the null hypothess that there are l = m common stochastc trends (H 0 : l = m) aganst the alternatve hypothess that there are l < m common stochastc trends (H 1 : l < m). 3. Estmate A 2 as the m egenvectors that corresponds wth the m largest egenvalues of T 1 Q^e^e. ^ 1=2, ^e A ^ea 5

6 4. De ne the unvarate MSB statstc as MSB j; (m) = T mn 1 A Q^e 2 ^e A 2 ^ 1 ^e A 2 ^e A 2 = mn T 1 1=2 ^ ^e A 2 ^e A A 2 Q^e 2 ^e A 2 ^ 1=2 ^e A 2 ^e A 2 = ^ 1 ; (8) where the subscrpt j = f; g refers to the determnstc component that s used n the model for the D ;t = determnstc spec caton and for the D ;t = + t one beng 1 < < m the egenvalues of T 1 A Q^e 2 ^e A 2 ^ 1 sorted n ascendng ^e A 2 ^e A 2 order, and mn () denotng the mnmum egenvalue operator. 5. Compare the value of the MSB j; (m) statstc wth the correspondng crtcal values from the left tal of the dstrbuton.e. the null hypothess s rejected f MSB j; (m) s smaller than the crtcal value. 6. If the null hypothess of H 0 : l = m common stochastc trends s rejected, specfy l = m 1 and return to step 2. The process contnues tll ether the null hypothess s not rejected or when l = 0 s acheved. The estmaton of A e 2 e A 2 can be obtaned n a parametrc way from the estmaton of the VECM model spec caton. Expressed n matrx notaton, we have e A 2 = e A 2 ; 1 + e A 2 e A 2 (I ;p (L)) = e A 2 ; 1 + " ;p (L) + " e A 2 = e A 2 ; 1 (I ;p (L)) 1 + " (I ;p (L)) 1 ; where p denotes the number of lags of e A 2 ; that are consdered. Followng Ng and Perron 1 0 1, (2001), we de ne = I ^;p (1) T 1^" 0 ^" I ^;p (1) where the lag V AR ^ ^e A 2 ^e A 2 order of the model p s estmated usng the mod ed nformaton crteron n Qu and Perron (2006) assumng that the contegratng rank s zero note that under the null hypothess we assume that there are m stochastc trends n the system de ned by the m varables of e A 2 ;t. The lmtng dstrbuton of the MSB j; (m) statstc, j = f; g, s establshed n the followng Theorem. Theorem 1 Let Y ;t, = 1; : : : ; N, t = 1; : : : ; T, be a (k 1) vector of stochastc processes wth the DGP gven by (1) to (4). Under the null hypothess that there are m = k r common stochastc trends, wth p! 1 and p 3 =T! 0 as T! 1, the MSB statstc gven n (8) converges to: R (a) For the only constant model: MSB ; (m) ) mn 1 W 0 (s) W (s) 0 ds R (a) For the lnear trend model: MSB ; (m) ) mn 1 V 0 (s) V (s) 0 ds ; 6

7 where ) denotes weak convergence, W (s) s an (m 1) vector of ndependent standard Brownan motons, and V (s) = W (s) sw (1) s an (m 1) vector of ndependent Brownan brdges. The proof of Theorem 1 s gven n the Appendx. It has to be stressed that our framework treats as a specal case the stuaton n whch the tme seres are assumed to be crosssecton ndependent. Thus, note that f we mpose = 0 8, then equaton (6) s gven by Y ;t = D ;t +e ;t. The estmaton of the parameters of the determnstc component can be done specfyng the model n rst d erences so that y = z, where y = MY and z = Me as before, M = I T 1 for the constant and M = I T 1 ( 0 ) 1 0 for the tme trend determnstc spec catons. Then, de nng ^e ;t = P t j=2 y ;j the computaton of the MSB statstc proceeds as above, gvng rse to test statstcs wth the same lmtng dstrbuton as the ones reported n Theorem 1. The crtcal values for the MSB j (m) statstc, j = f; g, are reported n Table 1 for d erent sample szes. These nte sample crtcal values are computed usng the autoregressve spectral densty estmator, where the lag order of the model p s ^ V AR ^e A 2 ^e A 2 estmated usng the mod ed nformaton crteron n Qu and Perron (2006) assumng that the contegratng rank s zero. Followng Ng and Perron (2001) and Qu and Perron (2006), we have spec ed the upper bound for p as nt 12 (T=100) 1=4. We need to gve further detals on the way n whch the maxmum lag order s spec ed... to be spec c, we need to nform about: The maxmum number of lags that were used to obtan the crtcal values for the d erent T : h 1. Do we used the rule p ;max = nt 12 (T=100) 1=4? If so, n whch cases? We need to report "asymptotc crtcal values" whch can be computed assumng that the dsturbance terms are d and usng T = 1000 We can report the estmated response surfaces to approxmate the p-values, or menton that they are avalable upon request The MSB statstc that s presented n the paper s consstent under the alternatve hypothess that there are less common stochastc trends than the ones spec ed under the null hypothess. The followng Theorem presents the rate at whch the MSB statstc dverges under the alternatve hypothess. Theorem 2 Let Y ;t, = 1; : : : ; N, t = 1; : : : ; T, be a (k 1) vector of stochastc processes wth the DGP gven by (1) to (4). Under the alternatve hypothess that there are l < m common stochastc trends we have that MSB j; (m) = O p (T 1 ), j = f; g. The proof s gven n the Appendx. The result n Theorem 2 shows that the MSB statstc dverges under the alternatve hypothess at a faster rate than the contegratng rank test statstcs proposed, for nstance, n Phllps and Oulars (1990), Shntan (2001), and Ca and Shntan (2006). Note that ths faster rate of convergence derves from the use of the autoregressve spectral densty estmator ^ V AR ^e A ^ea 7.

8 3 Panel data contegratng rank tests The ndvdual MSB statstcs can be pooled to de ne panel data statstcs, whch are expected to ncrease the performance of the statstcal nference when estmatng the contegratng rank. In ths secton we de ne up to four d erent panel data statstcs dependng on the way n whch the ndvdual nformaton s combned. The performance of these four alternatves s analyzed by smulaton n the next secton. The rst panel data MSB (PMSB) statstc s based on the standardzed mean of the ndvdual statstcs P MSB Z j (m) = p N(MSBj (m) E(MSB j (m))) p ; (9) V ar(msbj (m)) where MSB j (m) = N P 1 N =1 MSB j; (m), and E(MSB j (m)) and V ar(msb j (m)) are the mean and the varance of the MSB j (m) statstc computed from (8). The lmtng dstrbuton of the PMSB statstc s gven n the followng Theorem. Theorem 3 Let Y ;t, = 1; : : : ; N, t = 1; : : : ; T, be a (k 1) vector of stochastc processes wth the DGP gven by (1) to (4). Under the null hypothess that there are m = k r common stochastc trends, wth p! 1 and p 3 =T! 0 as T! 1, the PMSB statstc gven n (9) converges to: P MSBj Z (m) ) N(0; 1): As n Pedron (2004), n order to prove Theorem 3 we requre only the assumpton of nte second moments of the random varables characterzed as Brownan moton functonals R mn 1 R W 0 (s) W (s) 0 ds, mn 1 0, V 0 (s) V (s) ds 0 whch wll allow us to apply the Lndberg-Levy Central Lmt Theorem as N! 1. The mean and the varance of the MSB j (m) statstc, j = f; g, that have been computed by smulaton are presented n Table 2. It s possble to de ne panel data statstcs based on the combnaton of the ndvdual p- values. Maddala and Wu (1999) de nes the panel data Fsher-type statstc P MSBj F (m) = 2 P N =1 ln ' 2 2N, where ' denotes the p-value of the MSB j; (m) statstc, j = f; g. Although the P MSBj F (m) statstc s vald for nte N, Cho (2001) suggests to compute the followng tests when N! 1: P MSB C1 j (m) = P MSB C2 j (m) = 2 P N =1 ln ' 2N p 4N ) N (0; 1) 1 p N N X =1 1 (' ) ) N (0; 1) ; where () denotes the standard cumulatve dstrbuton functon. 4 Monte Carlo smulaton We now analyze the small sample performance of the MSB panel contegraton rank test for the two determnstc spec catons that are consdered n ths paper. The DGP that s used 8

9 n ths secton s based on the ones n Toda (1995) and Sakkonen and Lütkepohl (2000) and has the followng form: e1;;t e 2;;t Y ;t = D ;t + F t + e ;t (10) 0 e1;;t = 1 "1;;t + (11) 0 I k r e 2;;t 1 " 2;;t F j = F j 1 + F w t (12) Ir " ;t = d N 0; ; (13) I k r where = 1; : : : ; N, t = 1; : : : ; T and j = 1; : : : ; q. Note that when there s no contegraton among any of the tme seres (r = 0) equaton (11) reduces to e ;t = e ;t 1 + " ;t wth " ;t = d N (0; I k ). Besdes, when all tme seres are statonary (r = k) equaton (11) reduces to e ;t = e ;t 1 + " ;t. The parameter sets are de ned as follows. We consder a system de ned by k = 3 varables. For the determnstc component, we have consdered D ;t = + t, wth U [ 1; 1] and U [ 0:5; 0:5], where U denotes the unform dstrbuton. The dosyncratc contegratng rank s nvestgated usng = ai r wth a = f0:5; 0:8; 0:9; 1g. Furthermore, we de ne = b1 r(k r) where 1 r(k r) denotes a r(k r) matrx of ones, and b = f0; 0:4; 0:8g. Note that controls the correlaton among the " ;t dsturbance terms. As for the common factor component, we specfy N (1; 1), = f0:9; 0:95; 1g, 2 F = f0:5; 1; 10g and w t N (0; 1), j = 1; : : : ; q, wth q = f1; 3g common factors. The number of common factors s estmated usng the panel Bayesan nformaton crteron (BIC) n Ba and Ng (2002) usng q max = 6 as the maxmum number of common factors. The smulatons are performed n GAUSS usng the COINT 2.0 lbrary. The emprcal sze and power of the statstcs are obtaned usng 1000 replcatons wth the level of sgn cance set at the 5% level for all d erent combnatons of ndvduals N = f1; 20; 40g and number of tme seres observatons T = f50; 100; 200g. For concseness, we report only the results for N = 1 and for N = 20. The results for N = 40 are very smlar to those for N = 20 and they are avalable upon request. The smulatons have used ether the crtcal values or the mean and the varance n Tables 1 and Unt-by-unt analyss In ths secton we nvestgate the performance of the statstc when N = 1, provded that ths s the rst tme that the multvarate MSB statstc s used to estmate the number of common dosyncratc stochastc trends. Tables 3 to 6 present the results for the only constant and the tme trend cases. Frst, we can see that the results do not depend on the stochastc propertes of the common factor, provded that the performance of the MSB statstcs s smlar regardless of whether the common factor s I(0) or I(1), and regardless of the magntude of the dsturbance varance that partcpates n the generaton of the common factors ( 2 F ). As for the estmaton of the common factors, the procedure always detects the true number of common factors. The performance of the MSB statstc depends on how close s the autoregressve parameter a 9

10 to one. Thus, for a = 0:5 (Tables 3 and 5) we can see that the statstcal procedure that has been proposed n ths paper selects the correct number of stochastc trends n most of cases, especally for T > 50. As expected, the behavour of the procedure worsens for a = 0:8 (Tables 4 and 6), although t tends to select the correct number of stochastc trends as T ncreases. It s worth mentonng that these conclusons are reached regardless of the determnstc component that s used. 4.2 Panel data analyss The multvarate results for both the only constant and the tme trend cases when N = 20 are presented n Tables 7 to A.2. The de nton of the MSB panel data statstc helps to ncrease the ablty of the statstcal nference to select the correct number of stochastc trends. Ths mprovement s notceable for the smaller sample sze that we have consdered, where the use of the panel data based statstc reduces the tendency shown by the ndvdual MSB statstc to overestmate the number of common stochastc trends. Thus, f we compare the results n Tables 4 and A.2 we can see that the number of stochastc trends that s selected usng the ndvdual MSB test tends to be greater than the true one, whereas ths overestmaton feature s not so marked when usng the panel data statstc. Examnaton of Tables 6 and A.2 reveals a smlar pattern for the case of lnear trend. The performance of the statstc mproves as the sample sze ncreases. The power of the panel statstc s one or almost one, especally for T > 50. Another concluson that arses from the examnaton of the results s that the statstc s slghtly underszed regardless of the determnstc spec caton. As n the unvarate case, the procedure always detects the correct number of common factors. Another smlarty wth the ndvdual test statstc s that the multvarate statstc does not depend on the stochastc propertes of the common factor and the magntude of the dsturbance varance that partcpates n the generaton of the common factors. 5 Concluson In ths paper we propose a new test statstc to estmate the contegratng rank both n a untby-unt analyss and n a panel data framework. Our proposal covers the case of cross-secton dependence, whch s a relevant stuaton from both theoretcal and emprcal pont of vews. The set-up that s consdered n the paper allows to cover wld cross-secton dependence cases,.e. cases where the tme seres of one ndvdual systems are contegrated wth tmes seres of other ndvdual systems. Ths stuaton can be consdered as the multvarate extenson of the cross-contegraton concept de ned earler n the lterature. The performance of the proposal s nvestgated wth Monte Carlo smulatons. In general, the panel data based MSB statstc provdes wth better estmaton of the number of stochastc trends that are present n each ndvdual system. 10

11 A Mathematcal appendx A.1 Proof of Theorem 1 A.1.1 The only constant case Note that the estmated d erence of the dosyncratc stochastc term s: ^z ;t = z ;t + f t ^ ^ft : Followng Ba and Ng (2003), we can express the model as ^z ;t = z ;t + H 1 Hf t H 1 ^ft + H 1 ^ft ^ ^ft = z ;t + H 1 Hf t ^ft ^ H 1 ^ft = z ;t + H 1 v t d ^ft ; (14) where v t = Hf t ^ft and d = ^ H 1. Let us de ne the partal sum process usng the estmated resduals as ^e ;t = P t j=2 ^z ;j = P t 0, j=2 [M ^e ] j where []j denotes the le j of the matrx between brackets. By Lemma 3 and C1 n Ba and Ng (2004), T 1=2 P t s=j H 1 v j = op (1) and T 1=2 P t j=2 d ^f j = op (1), so that wth T 1=2^e ;t = T 1=2 tx [Me ] j = j=2 tx j=2 tx j=2 [Me ] j 0 + op (1) ; (15) e 0;j [P e ] j ; where here P s a matrx of zeros provded that for the only constant case we have M = I T 1. The cumulated process s equal to T 1=2^e ;t = T 1=2 e ;t T 1=2 e ;1 + o p (1). If we rotate the vector ^e ;t and de ne ^e A ;t = A 0^e ;t = (A 0 1^e ;t ) 0 ; (A 0 2^e ;t ) 0 0 we can see that T 1=2 A 0 1^e ;t = o p (1) provded that A 0 1^e ;t de nes the statonary relatonshps and T 1=2 A 0 2^e ;t = O p (1) gven that A 0 2^e ;t de nes the I(1) stochastc trends. Therefore, T 1=2 A 0^e ;t = T 1=2 A 0 1 (e ;t e ;1 ) 0 ; T 1=2 A 0 2 (e ;t e ;1 ) 0 0 ) 0 0 r; 1=2 22; (W (s) W (0)) r; 1=2 22; W (s) 0 0 where 0 r s an r vector of zeros, W (s) denotes a k r vector of ndependent standard Brownan motons, and W (0) = 0. Then, we can see that T 1 Q^e A ^e A A0 = T 2^e ^e A 0 0 ) 0 1=2 R 1 22; W 0 (s) W (s) 0 ds 1=2 22; ; 11

12 gven that T 2 A 0 1e 0 e A 1 = o p (1), T 2 A 0 1e 0 e A 2 = o p (1) and T 2 A 0 2e 0 e A 2 = O p (1). Therefore, usng these elements the lmtng dstrbuton of the multvarate MSB statstc s gven by: MSB ; (m) = mn T 1 Q^e^e ^ 1 ^e ^e p = mn T 1 Q^e A ^e A ^ 1 ^e A ^ea = mn T 1 ^ 1=2 ^e A ^ea Q^e A ^e A ^ 1=2 ^e A ^ea Z 1 ) mn W (s) W (s) 0 ds ; 0 provded that ^ ^e A ^e A! e A e A, where! p denotes convergence n probablty, and where W (s) denotes an m = k r vector of ndependent standard Brownan motons. A.1.2 The lnear tme trend case The proof for ths determnstc component follows the one for the constant case, but where the projecton matrx M = I T 1 ( 0 ) 1 0. As above, T 1=2^e ;t s gven by (15) where now ^e ;t = = tx [M e ] j = j=2 tx j=2 e 0 ;j tx e 0;j [P e ] j = j=2 t 1 T 1 TX e ;t : t=2 tx j=2 e 0 ;j h ( 0 ) 1 0 e As before, we de ne ^e A ;t = A 0^e ;t = (A 0 1^e ;t ) 0 ; (A 0 2^e ;t ) 0 0, wth T 1=2 A 0 1^e ;t = o p (1) and T 1=2 A 0 2^e ;t = O p (1) provded that A 0 2^e ;t de nes the I(1) stochastc trends. Then, whch mples that T 1=2 A 0^e ;t ) 0 0 r; 1=2 22; (W (s) sw (1)) 0 0 ; j T 1 Q^e A ^e A A0 = T 2^e ^e A 0 0 ) 0 1=2 R 1 22; V 0 (s) V (s) 0 ds 1=2 22; ; where V (s) = W (s) sw (1) s a vector of ndependent Brownan brdges. Therefore, MSB ; (m) ) mn Z 1 0 V (s) V (s) 0 ds ; gven that ^ ^e A ^e A p! e A e A. 12

13 A.2 Proof of Theorem 2 Let us consder the null hypothess that there are m = k r stochastc trends. From the proof of Theorem 1 we have that T 1 A 0 1e 0 e A 1 = O p (1), T 1 A 0 1e 0 e A 2 = O p (1) and T 2 A 0 2e 0 e A 2 = O p (1), so that T 2 A 0 1e 0 e A 1 = O p (T 1 ) and T 2 A 0 1e 0 e A 2 = O p (T 1 ). Consequently, under the alternatve hypothess that there are l < m stochastc trends the rank of the matrx T 1 Q^e A ^e A wll be l < m. Usng these elements, we can see that the cross-products nvolvng I(0) stochastc processes n T 1 Q^e A ^e A tend to zero at rate O p (T 1 ). Let us now focus on the estmate of the long-run covarance matrx. Note that under both the null and the alternatve hypotheses T 1^" 0 ^" = O (1), wth T 1^" 0 p ^"! ". Snce all roots of the determnant of I ^;p (L) le outsde the unt crcle nterval, we can de ne ^ 1, ;1 (L) = I ^;p (L) wth ;1 (L) = (I + ;1 L + ;2 L 2 + ) and where the sequence of matrx coe cents f ;s g 1 s=0 s absolutely summable. Then, ;1 (1) < 1 so V AR p that ^! 0 ^e A ^ea ;1 (1) " ;1 (1). Therefore, the long-run covarance matrx estmator converges to a postve de nte matrx under both the null and the alternatve hypotheses note that ths result can be seen as the generalzaton of the one n Perron and Ng (1998) and Stock (1999). Fnally, note that under the alternatve hypothess that there are l (< m) stochastc trends rank T 1 Q^e A ^e A ^ 1 ^e A ^ea = l, where the elements that cause rank de - cency tend to zero at rate O p (T 1 ). Ths proves the consstency of the MSB statstc under the alternatve hypothess. 13

14 References [1] Baltag, B. and Kao, C. (2000) Nonstatonary Panels, Contegraton n Panels and Dynamc Panels: A Survey, Advances n Econometrcs 15, [2] Bretung, J. (2005) A parametrc approach to the estmaton of contegraton vectors n panel data, Econometrc Revews 24, [3] Bretung, J. and Pesaran, M.H. (2007) Unt Roots and Contegraton n Panels, forthcomng n Matyas, L. and Sevestre, P., The Econometrcs of Panel Data (Thrd Edton), Kluwer Academc Publshers. [4] Ca, Y. and Shntan, M. (2006) On the alternatve long-run varance rato test for a unt root, Econometrc Theory, 22, [5] Cho, I. (2001) Unt root tests for panel data, Journal of Internatonal Money and Fnance 20, [6] Engle, R.F. and Granger C.W.J. (1987) Co-ntegraton and error correcton: representaton, estmaton and testng, Econometrca 55, [7] Gengenbach, C., Palm, F. C. and Urban, J. P. (2006) Contegraton testng n panels wth common factors, Oxford Bulletn of Economcs and Statstcs 68(Supplement), [8] Groen, J.J.J. and Klebergen, F. (2003) Lkelhood-based contegraton analyss n panels of vector error-correcton models, Journal of Busness & Economc Statstcs 21(2), [9] Hadr, K. (2000) Testng for statonarty n heterogeneous panel data, The Econometrcs Journal 3(2), [10] Harrs D. and Posktt, D. S. (2004) Determnaton of contegratng rank n partally nonstatonary processes va a generalsed von-neumann crteron, Econometrcs Journal, 7, [11] Johansen, S. (1988) Statstcal analyss of contegratng vectors, Journal of Economc Dynamcs and Control 12, [12] Johansen, S. (1991) Estmaton and hypothess testng of contegratng vectors n Gaussan vector autoregressve models, Econometrca 59, [13] Johansen, S. (1995) Lkelhood-Based Inference n Contegrated Vector Autoregressve Models, Oxford: Oxford Unversty Press. [14] Kao, C. (1997) Spurous regresson and resdual-based tests for contegraton n panel data, Journal of Econometrcs 90, [15] Larsson, R., Lyhagen, J. and Löthgren, M. (2001) Lkelhood-based contegraton tests n heterogeneous panels, Econometrcs Journal 4,

15 [16] Maddala, G. S. and Wu, S. (1999) A Comparatve Study of Unt Root Tests wth Panel Data and a New Smple Test, Oxford Bulletn of Economcs and Statstcs, Specal Issue, 61, pp [17] Mark, N. C. and Sul, D. (2003) Contegraton Vector Estmaton by Panel DOLS and Long-run Money Demand, Oxford Bulletn of Economcs and Statstcs 65(5), [18] McCoskey, S. and Kao, C. (1998) A resdual based test of the null of contegraton n panel data, Econometrc Revews 17, [19] Pedron, P. (2004) Panel contegraton: asymptotc and nte sample propertes of pooled tme seres tests wth an applcaton to the PPP hypothess, Econometrc Theory 3, [20] Perron, P. and Ng, S. (1998) Useful mod catons to some unt root tests wth dependent errors and ther local asymptotc propertes, Revew of Economc Studes, 63, [21] Phllps, P. C. B. and Oulars, S. (1990) Asymptotc propertes of resdual based tests for contegraton, Econometrca, 58, [22] Qu, Z. and Perron, P. (2006) A mod ed nformaton crteron for contegraton tests based on a VAR approxmaton. Mmeo, Department of Economcs, Boston Unversty. [23] Sakkonen, P. and Lutkepohl, H. (2000) Testng for the contegratng rank of a VAR process wth an ntercept, Econometrc Theory 16, [24] Shntan, M. (2001) A smple contegratng rank test wthout vector autoregresson, Journal of Econometrcs, 105, [25] Stock, J. H. (1999). A Class of Tests for Integraton and Contegraton, n Engle, R. F. and H. Whte (ed.), Contegraton, Causalty and Forecastng. A Festchrft n Honour of Clve W. F. Granger. Oxford Unversty Press. [26] Stock, J. H. and Watson, M. C. (1988) Testng for common trends, Journal of the Amercan Statstcal Assocaton, 83, [27] Toda, H.Y. (1994) Fnte sample propertes of lkelhood rato tests for contegratng ranks when lnear trends are present, Revew of Economcs and Statstcs 76(1), [28] Westerlund, J. (2005) A panel CUSUM test of the null of contegraton, Oxford Bulletn of Economcs and Statstcs 62,

16 Table 1: Crtcal values for the MSB and MSB statstcs MSB statstc T = 50 T = 100 T = 200 k 1% 5% 10% 1% 5% 10% 1% 5% 10% MSB statstc T = 50 T = 100 T = 200 k 1% 5% 10% 1% 5% 10% 1% 5% 10% k denotes the number of stochastc trends under the null hypothess. Smulatons are based on 10,000 replcatons. 16

17 Table 2: Smulated mean and varance of the MSB and MSB statstcs MSB statstc T = 50 T = 100 T = 200 k Mean Varance Mean Varance Mean Varance MSB statstc T = 50 T = 100 T = 200 k Mean Varance Mean Varance Mean Varance k denotes the number of stochastc trends under the null hypothess. Smulatons are based on 10,000 replcatons. 17

18 Table 3: Proportons for MSB wth a = 0:5 and N = 1 Panel A: r = 0 Panel B: r = 1 Panel C: r = 2 Panel D: r = T 2 F;j j Note: MSB s the statstc for the lnear tme trend model for N = 1 and r denotes the number of stochastc trends. Smulatons are based on 1; 000 replcatons usng the 5% crtcal values from Table 1. For P anel A, = dag (0:8; 0:5; 0:5), for P anel B, = dag (0:8; 0:5) and for P anel C, = dag (0:8). 18

19 Table 4: Proportons for MSB wth a = 0:8 and N = 1 Panel A: r = 0 Panel B: r = 1 Panel C: r = 2 Panel D: r = T 2 F;j j Note: MSB s the statstc for the lnear tme trend model for N = 1 and r denotes the number of stochastc trends. Smulatons are based on 1; 000 replcatons usng the 5% crtcal values from Table 1. For P anel A, = dag (0:8; 0:5; 0:5), for P anel B, = dag (0:8; 0:5) and for P anel C, = dag (0:8). 19

20 Table 5: Proportons for MSB wth a = 0:5 and N = 1 Panel A: r = 0 Panel B: r = 1 Panel C: r = 2 Panel D: r = T 2 F;j j Note: MSB s the statstc for the lnear tme trend model for N = 1 and r denotes the number of stochastc trends. Smulatons are based on 1; 000 replcatons usng the 5% crtcal values from Table 1. For P anel A, = dag (0:8; 0:5; 0:5), for P anel B, = dag (0:8; 0:5) and for P anel C, = dag (0:8). 20

21 Table 6: Proportons for MSB wth a = 0:8 and N = 1 Panel A: r = 0 Panel B: r = 1 Panel C: r = 2 Panel D: r = T 2 F;j j Note: MSB s the statstc for the lnear tme trend model for N = 1 and r denotes the number of stochastc trends. Smulatons are based on 1; 000 replcatons usng the 5% crtcal values from Table 1. For P anel A, = dag (0:8; 0:5; 0:5), for P anel B, = dag (0:8; 0:5) and for P anel C, = dag (0:8). 21

22 Table 7: Proportons for P MSB wth a = 0:5 and N = 20 Panel A: r = 0 Panel B: r = 1 Panel C: r = 2 Panel D: r = T 2 F;j j