PANEL UNIT ROOT TESTS UNDER CROSS-SECTIONAL DEPENDENCE: AN OVERVIEW

Transcription

1 Journal of Statstcs: Advances n Theory and Applcatons Volume, Number 2, 2009, Pages 7-58 PANEL UNIT ROOT TESTS UNDER CROSS-SECTIONAL DEPENDENCE: AN OVERVIEW LAURA BARBIERI Dpartmento d Scenze Economche e Socal Unverstà Cattolca del Sacro Cuore d Pacenza Va E. Parmense0 84, 2900 Pacenza Italy e-mal: laura.barber@uncatt.t Abstract The ncreasng avalablty of new datasets where the tme-seres dmenson and the cross-secton dmenson are of the same order of magntude asks for new technques for the analyss of ths pecular knd of data. In the panel unt root test framework, two generatons of tests have been developed: a frst generaton whose man lmt s the assumpton of cross-sectonal ndependence across unts; a second generaton of tests that rejects the cross-sectonal ndependence hypothess. Although wthn ths second generaton of tests dfferent approaches can be dstngushed on the bass of the way n whch the cross-sectonal dependence s modelled the one that has encountered the most attenton among researchers s the factor structure approach. Ths paper provdes an updated overvew of the man tests belongng to the second generaton, and underlnes the man ssues whch reman to be solved.. Introducton Over the past decades the avalablty of several new panel data sets where the number of tme seres observatons (T) and the number of 2000 Mathematcs Subject Classfcaton: 62-02, 62H5, 62M0. Keywords and phrases: non-statonary panel data, panel unt root tests, cross-secton dependence. Receved Aprl 7, Scentfc Advances Publshers

2 8 LAURA BARBIERI groups or ndvduals (N) are of the same order of magntude has ncreased the need for new econometrc technques whch are able to deal wth ths knd of data. Orgnally, the asymptotc propertes of regresson analyss n the panel context were derved under the hypothess of weak statonarty for each ndvdual tmeseres (for a survey, see Hsao [28]). Obvously, ths was not a very restrctve assumpton n the classcal panel data case, where the tme span s very short (usually 5 or 6 observatons), but t could be a very crtcal assumpton n the large panel data framework. In fact the presence of a unt root can sgnfcantly affect the asymptotc propertes of tme-seres regresson estmates and test statstcs and can lead to spurous results, f there are not contegratng relatons among ntegrated varables, (.e., Engle and Granger [2]). Workng n the large panel data context and developng approprate test statstcs requre the prelmnary soluton of the problem of carryng out an asymptotc analyss, as both N and T go to nfnty. Several theoretcally nterestng approaches have been developed consderng how the two ndexes go to nfnty (Phllps and Moon [46]); however, from a practcal pont of vew, sequental asymptotc results seem to be adequate n most cases. Ths procedure, consstng n lettng T go to nfnty frst, and then N go to nfnty second ( ( T, N hereafter), s easy to mplement, even though t can sometmes gve msleadng asymptotc results 2. The frst theoretcal works on the non-statonary panel data focused on testng for unt roots n unvarate panel, but snce the work of Quah [48] and Bretung and Meyer [9], the nterest on ths topc has consderably ncreased. ) seq The cross-sectonal unts may be households, frms, regons, countres and so forth. 2 Other alternatve approaches to the sequental lmts are the dagonal-path lmts, whch consst of mposng restrctons on the relatve rates at whch N and T go to nfnty, and the jont lmts whch allow both N and T to pass to nfnty smultaneously wthout placng specfc dagonal path restrctons on the dvergence.

3 PANEL UNIT ROOT TESTS UNDER 9 In general, the commonly used unt root tests as Augmented Dckey- Fuller (ADF) tests (Dckey and Fuller [9]) have non-standard lmtng dstrbutons that depends on whch determnstc components are ncluded n the regresson equaton 3. In fnte samples such tests based on a sngle tme-seres show lttle power n dstngushng the unt root from statonary alternatves, wth hghly persstent devatons from equlbrum. Ths problem seems to be partcularly severe n the small sample case. Moreover, standard unt root tests n the panel framework may show bad performances also because of the dfferent null hypotheses that are tested n ths case. When we consder the model yt = ρ yt + ut =, 2, K, N t =, 2, K, T n the sngle equaton case, we are nterested n testng ρ = 0 aganst the alternatve hypothess ρ < 0 and we apply a unt root for the frst tme-seres. Instead, n the panel data case, the hypothess whch we are nterested n s H 0 : ρ = 0 aganst H a : ρ < 0 for =, 2, K, N. Referrng to ths stuaton, two generatons of panel unt root tests have been developed. The frst generaton ncludes Levn, Ln and Chu s test [39]-LLC thereafter-, Im, Pesarvan and Shn [30]-IPS thereafter-, the Fsher-type test proposed frst by Maddala and Wu [42], later developed by Cho [4] and the Hadr test [24] for the null hypothess of statonarty The ADF test statstc converges to a functon of Brownan moton (Whte, [54]) under very general condtons (Sad and Dckey [49]). 4 In the remander of ths paper, and t are always assumed =, 2, K, N, t =, 2, K, T, where not dfferently specfed. 5 For a survey on ths frst generaton of tests, see Banerjee [6], Baltag and Kao [5]. Table A. n Appendx brefly summares n a tabular form the man characterstcs of ths frst generaton of tests.

4 20 LAURA BARBIERI The man lmt of these tests s that they are all constructed under the assumpton that the ndvdual tme-seres n the panel are crosssectonally ndependently dstrbuted. Ths condton s needed n order to satsfy the Lndberg-Levy central lmt theorem, and therefore obtan asymptotcally normal dstrbuted test statstcs. Nevertheless, ths s a crtcal statement because a large amount of lterature, (.e., Backus and Kehoe []) has provded evdence on the strong co-movements between economc varables, and t has been recognzed that the assumpton of ndependence across members of the panel s rather restrctve, partcularly n the context of cross-country regressons. Moreover, ths cross-sectonal correlaton may affect the fnte sample propertes of panel unt root test (O Connell [44]) 6. For nstance, the lmt dstrbuton of the usual Wald type unt root tests based upon ordnary least squares (OLS) and generalzed least squares (GLS) system estmators depend upon nusance parameters defnng correlatons across ndvdual unts. Varous attempts to elmnate the nusance parameters n such systems have been proposed 7 ; unfortunately, even when ths procedure could partly deal wth the problem, t s not approprate f par-wse cross-secton covarances of the error terms vary across the ndvdual seres. To overcome these dffcultes, a second generaton of tests rejectng the cross-sectonal ndependence hypothess has been proposed. Wthn ths generaton of tests, we can dstngush the contrbuton of Chang [2, 3], who proposed frst mposton of few or no restrctons on the resdual covarance matrx, and then the use of nonlnear nstrumental varable methods or bootstrap approaches to solve the nusance parameter problem due to cross-sectonal dependence or the more recent contrbuton of Cho and Chue [6] who propose subsamplng methods. 6 The assumpton of cross-sectonal ndependence can lead to severe sze dstortons and low power of the tests when ths hypothess s not satsfed (Banerjee et al. [8], Bretung and Das []). 7 For example, the cross-sectonally de-meanng of the seres before applcaton of the panel unt root test. Ths s what has been done by Im et al. [29], who consder a smple form of cross-sectonal correlaton usng tme-specfc effects.

5 PANEL UNIT ROOT TESTS UNDER 2 Nevertheless, the most popular approach n ths framework reles on the factor structure approach and ncludes contrbutons by Ba and Ng [3, 4], Phllps and Sul [47], Moon and Perron [43]), Cho [5] and Pesaran [45]. The attempt of the present work s to provde an overvew of the recent developments n panel unt root tests lterature and to underlne the man ssues whch reman to be solved. Ths s fundamental for the econometrc researcher, who wants to apply exstng tests or to develop new and better tests. The paper s organzed as follows. Secton II revews the second generaton of panel unt root tests. Partcular attenton s made to the ncreasngly popular approach that deals wth the problem of cross-sectonal dependence by means of a factor structure representaton. Other sutable alternatve approaches are dscussed before the end of the secton wth a comparson of the presented tests. Secton III brefly provdes a presentaton of some smple tests belongng to the second generaton whch verfy the null hypothess of statonarty. Fnally, some conclusons and an Appendx presentng n a tabular form the comparatve characterstcs of the dscussed tests close the paper. 2. The Second Generaton of Panel Unt Root Tests 8 As ntroduced prevously, the frst generaton of tests was constructed under the assumpton of cross-sectonal ndependence between ndvdual tme-seres n the panel. Nevertheless, ths s a very strong hypothess often dsproved by emprcal evdence. Ths s why, a new generaton of panel unt root tests that explctly consder the crosssecton correlaton between panel unts has been proposed n the lterature. To buld these knd of tests, a prelmnary ssue s the specfcaton of the cross-sectonal dependence. But snce ndvdual observatons n a cross-secton have no natural orderng ths specfcaton s not obvous. In 8 See Table A.2. n Appendx for a summary of the man characterstcs of ths second generaton of tests. The reader should note that ths area of research and the lnked lterature are stll under development, gven the dversty of the potental cross-sectonal correlatons.

6 22 LAURA BARBIERI order to deal wth ths problem, varous methods have been proposed and they can be organzed n two man streams: ) approaches modellng the cross-sectonal dependence n the form of a low dmensonal common factor model, (.e., Cho [5], Ba and Ng [3], Moon and Perron [43], Phllps and Sul [47], Pesaran [45]), and 2) other more general approaches, (.e., O Connell [44], Maddala and Wu [42], Taylor and Sarno [53], Chang [2, 3], Bretung and Das [0], Cho and Chue [6]), often consstng n mposng few restrctons on the resdual covarance matrx. These two streams are presented n the followng sub-sectons. 2.. The factor structure approach The factor structure approach assumes a common factor representaton n whch an observed seres s wrtten as a lnear combnaton of (unobserved) common and dosyncratc components. Ths model s estmated and condtoned out before constructng the panel unt root tests. The advantages of ths procedure are the possblty to model the cross-sectonal dependence by allowng the common factors to have dfferental effects on dfferent cross-secton unts as well as the reducton of the number of requred unobserved common factors. Among the tests adoptng a factor structure approach, we wll descrbe n the followng only those developed by Pesaran [45], Ba and Ng [2] and Moon and Perron [43] Pesaran [45] test Pesaran [45], presents a new and smple procedure for testng unt roots n dynamc panels subject to possbly cross-sectonally dependent as 9 Phllps and Sul [47] and Cho [5] propose test procedures that are very smlar to that of Moon and Perron [43]. Nevertheless, snce respect to the Moon and Perron approach the Phllps and Sul s one s more restrctve n term of the common components specfcaton, and that of Cho shows worst performance n term of sze and power, they are not dscussed here.

7 PANEL UNIT ROOT TESTS UNDER 23 well as serally correlated errors and proposes a cross-sectonally augmented verson of the IPS standardzed t -test (Im et al. [30]). In ths approach, the observatons y t are supposed to be generated accordng to a smple dynamc lnear heterogeneous panel data model y = ( ρ ) µ + ρ y + u, (2...) t t t where µ s a determnstc component, the ntal values y 0 are gven and the dsturbances follow a one-factor structure ut = (2...2) λ ft + εt n whch - the dosyncratc shocks, ε t are ndependently dstrbuted both across and t wth mean zero, varance moment; σ 2 and fnte fourth-order - the unobserved common factor f t s serally uncorrelated wth mean 2 zero, constant varance σ f and fnte fourth-order moment. Wthout loss of generalty, 2 σ f s set equal to one; - the varables ε t, ft and λ are assumed to be ndependently dstrbuted for all. The assumptons made about ε t and f t mply seral uncorrelaton for the u t 0. Equatons (2...) and (2...2) can be more convenently wrtten as y = ( ρ ) µ ( ρ ) y + λ f + ε, (2...3) t t t t beng y t = yt yt. Pesaran [45] consders the followng unt root hypothess 0 Ths assumpton as well as the assumpton that K = (there s only one common factor) could be relaxed. Pesaran [45] explctly consders ths stuaton n the follow of hs work.

8 24 LAURA BARBIERI H 0 : ρ = for all aganst the possbly heterogeneous alternatves H ρ : ρ < = for for =, K, N, = N, K, N, where the fracton of the statonary processes s such that N N, as N wth 0 <. Instead of basng the unt root tests on devatons from the estmated common factors, Pesaran [45] proposes a test based on standard unt root statstcs n a Cross-sectonally Augmented DF (CADF thereafter) regresson - that s a DF (or ADF) regresson whch s augmented wth the cross-secton averages of lagged levels and frst-dfferences of the ndvdual seres y t = a + b yt + c yt + d yt + et, (2...4) where e t s the regresson error and yt = N y j = jt N and y t = N N j = y jt are ncluded nto Equaton (2...4) as a proxy for the unobserved common factor f t 2. Let CADF be the ADF statstc for the -th cross-sectonal unt gven by the, t-rato of the OLS estmate bˆ of b n the CADF regresson (2...4). Ths s a natural extenson of the DF approach n order to deal wth resdual seral correlaton, where lagged changes of the seres are used to flter out the tme-seres dependence when T s suffcently large. 2 N Ths approxmaton s applcable f λ = N λ j j = and λ 0 for a fxed N and as N. Moreover, Equaton (2...4) s vald for serally uncorrelated u t. For a more general case, lagged values of y t, but also of yt need to be ncluded n the estmaton (Pesaran [45]).

9 PANEL UNIT ROOT TESTS UNDER 25 Indvdual CADF statstcs are used to develop a modfed verson of the IPS t-bar test denoted Cross-sectonally Augmented IPS (CIPS thereafter) that smultaneously take account of cross-secton dependence and resdual seral correlaton CIPS N = CADF. (2...5) N = The asymptotc null dstrbutons of the sngle CADF statstcs are smlar and do not depend on the factor loadngs. Unfortunately, CADF statstcs are correlated because of ther dependence on the common factor. Then, even f CIPS statstcs can be bult, t s not possble to apply standard central lmt theorems to them. Moreover, n contrast to the results obtaned by Im et al. [30] under cross-sectonal ndependence 3, the dstrbuton of the CIPS statstc s shown to be nonstandard even for large N. Pesaran also consders a truncated verson of CADF ( CADF ) to avod excessve nfluence of extreme outcomes that could arse for small T samples. The results for CADF and CIPS are vald also for CADF and the related CIPS and, even f t s not normal, the null asymptotc dstrbuton of CIPS statstc exsts and s free of nusance parameter. Pesaran proposes smulated crtcal values of CIPS for varous sample szes and three dfferent specfcatons of determnstc components, (.e., models wthout ntercept or trend, models wth ndvdual-specfc ntercepts and models wth ncdental lnear trends) and analyzes the small sample propertes of hs tests by Monte Carlo experments. In ths way, CADF tests-but mostly the truncated verson of the CIPS test-show satsfactory propertes for small N and T as well, even f the power of the tests crtcally depends on the sample szes and on whether the model contans lnear tme trends. 3 In ths case, a standardzed average of ndvdual ADF statstcs was normally dstrbuted for large N.

10 26 LAURA BARBIERI Followng Maddala and Wu [42] or Cho [4], Pesaran also proposes Fsher-type tests based on the sgnfcant levels of ndvdual CADF statstcs. In ths case as well, the statstcs do not have standard dstrbutons because of the prevous reasons. Pesaran [45] also extends hs approach to serally correlated resduals. For an AR ( p) error specfcaton, the relevant ndvdual p -th order cross-secton/tme- CADF statstcs can be computed from a seres augmented regresson p p y t = α + ρ yt + c yt + dj yt j + βj yt j + µ t. j= 0 j= 0 Fnally, note that both Pesaran s [45] CADF and CIPS tests are desgned for testng for unt roots when cross-sectonal dependence s due to a sngle common factor, but the CIPS test has better power propertes than the ndvdual CADF tests, and therefore t should be preferred Moon and Perron [43] tests Moon and Perron [43] represent the observaton seres y t as AR() processes wth fxed effects and assume, such as Pesaran [45], the presence of common factors n the error terms. They consder the followng dynamc panel model: y t = µ + xt, x t ρxt + ut, = (2..2.) u t = λ ft + et, where the observed y t varables ( =, 2, K, N; t =, 2, K, T ) are generated by a determnstc component µ and an autoregressve process x t, wth x 0 = 0 for all. In order to model the correlaton among the cross-sectonal unts, the error component u t s assumed to follow an approxmate factor model, where f t s a ( K ) vector of unobservable random factors, λ s the correspondng vector of non-random factor loadng for cross-secton and

11 PANEL UNIT ROOT TESTS UNDER 27 e t s an dosyncratc shock. The number of factors K s possbly unknown. As t s easy to note, n ths framework panel data are assumed to be generated by dosyncratc shocks and unobservable dynamc factors that are common to all the ndvdual unts but to whch each ndvdual reacts heterogeneously. Model (2..2.) can also be re-wrtten as y = ( ρ ) µ + ρ y + u. (2..2.2) t t t Comparng (2..2.2) and (2...), t s easy to note that Pesaran [45] and Moon and Perron [43] models are dentcal n the case, where a sngle common factor s present n the composte error term. For the error term u t n (2..2.) Moon and Perron make smlar assumptons to those of Pesaran [45], namely j = Γ ( L) ε, where Γ ( L) = γ j jl and ε ~.. d. ( 0, ) =0 t across - et t and t have a fnte eghth moment; furthermore, the possblty of contegratng relatonshps among the ntegrated dosyncratc shocks t Ej = e s = s s excluded; j - ft = Φ( L) ηt, where Φ ( L) = φ j jl s a K-dmensonal lag =0 polynomal and η t ~.. d. ( 0, I K ) so that u t s..d. across and t; furthermore, the covarance matrx of f t s (asymptotcally) postve defnte so that under the null hypothess the nonstatonary factors can be contegrated; - there exsts at least one common factor n the data and ther maxmum number K ( K K < ) s assumed to be known a pror. Also, the contrbuton of each factor to at least one of the y t s assumed p N to be sgnfcant by mposng λ λ > 0; = λ however, ths N assumpton does not mply that all cross-sectons respond to all factors so that some of the factor loadngs could be zero;

12 28 LAURA BARBIERI 2 e - short-run varance σ ( = γj ), = j 0 2 long-run varance ω 2 ( = e j= 0 j 2 ( γ ) ) l = j= 0 j j+ l as well as the one-sded long-run covarance ϕe ( = ( γ γ )) are well defned for all dosyncratc dsturbances 2 N 2 2 e t and have non-zero cross-sectonal averages σe = σ, ω = = e e N N 2 2 N 2 ω N and ϕ. = e e = N ϕ = e In referrng to the model (2..2.), the null hypothess of nterest s H 0 : ρ = for all =, K, N, (2..2.3) aganst the heterogeneous alternatve H a : ρ < for some. It s straghtforward to note from (2..2.) that, under the null T hypothess, two components nfluence y t : the ntegrated factors f = s T s = s and the ntegrated dosyncratc errors e. Moreover, whle both ntegrated or contegrated factors are allowed n the model, the possblty of contegratng relatons of the ntegrated dosyncratc errors s ruled out. When common factors are present n the panel, tests based on the assumpton of cross-sectonal ndependence among unts suffer from sze dstortons. To overcome ths dffculty, Moon and Perron [43] transform the model n order to elmnate the common components of the y t seres and apply the unt root test on defactored seres. The resultng test statstcs have a normal asymptotc dstrbutons as those of Im et al. [30] or Levn and Ln [37, 38]; moreover beng computed from defactored data, they are also cross-sectonal ndependent. More specfcally, to remove cross-sectonal dependence n (2..2.), Moon and Perron use a projecton onto the space orthogonal to the factor loadngs, (.e., the space generated by the columns of the matrx of factor s

13 PANEL UNIT ROOT TESTS UNDER 29 loadng Λ = ( λ, K, λ ) N ). Then, Λ s estmated 4 to construct the projecton matrx Q = I Λ ( Λ Λ ) Λ. Λ N 2 e Let Λˆ and Q be the estmator of the matrx Λ and of the Λˆ k correspondng projecton matrx; further, denote the estmates of shortrun and long-run varances, σ and ω, as ϕˆ and ω, respectvely 5, and let ϕˆ e and 2 e ˆ 2 ω e be ther cross-sectonal averages. The unt root test s mplemented from the defactored data obtaned as Y QΛ (beng Y the matrx of observed data). Specfcally, the unbased pooled estmator of ρ suggested by Moon and Perron [43] s ρ pool = e t ( Q ˆ Y ) NTϕˆ Λ e k, tr( Y Q ˆ Y ) tr Y Λk ˆ 2 e (2..2.4) where Y s the matrx of lagged observed data and tr ( ) the trace operator. pool From estmator ρ, two modfed t-statstcs based on pooled estmaton of the frst-order seral correlaton coeffcent of the data are suggested for the null hypothess (2..2.2) a t = T N ( ˆ + ρpool ), 2ˆ ϕ ωˆ 4 e 4 e ( a) 4 For ths purpose, Moon and Perron suggest to employ the prncpal component method used n Stock and Watson [5] and Ba and Ng [3]. 5 To ths am, Moon and Perron use kernel estmators (see Moon and Perron, [43]. Note that, once prevous estmates have been obtaned, the authors also dscuss how to consstently estmate the number of factors K.

14 30 LAURA BARBIERI b t T N ˆ 2 ( ˆ + ωe ρ ) tr( Y Y ), = pool 2 NT Q Λk ( b) ˆ 4 ϕ e + ρˆ pool beng the bas-corrected pooled autoregressve estmate of (2..2.4). Moon and Perron show that as N and T, wth N T 0, the statstcs ( a) and ( b) have a lmtng standard normal dstrbuton under the null hypothess. It s possble to note that, snce the Moon and Perron s model obtaned by removng the cross unt dependence s smlar to the LLC model wth common autoregressve root, under the cross-unt ndependence hypothess the convergence rate of the pooled estmator (corrected or not) of the autoregressve root s the same as the one obtaned n the LLC model. Fnally, Moon and Perron smulatons show that the tests are very powerful and have good sze when no estmaton of determnstc components s necessary, (.e., only a determnstc constant s ncluded n the model) for dfferent specfcatons and dfferent values of T and N. When such estmaton s necessary, the tests have no power beyond ther sze. Note that the Moon and Perron [43] tests usng defactored data allow for multple common factors. Therefore, ther use has to be recommended when cross-secton dependence s expected to be due to several common factors Ba and Ng [3] test Ba and Ng [3] propose a dfferent procedure to test for panel unt root allowng for cross-secton correlaton as well as contegraton. It does not treat cross-secton dependence as a dsturbance as the prevously presented tests dd: the nature of the co-movements of economc varables are themselves an object of nterest n the analyss. Ba and Ng consder a balanced panel wth N cross-secton unts wth T tme seres observatons and the followng model representaton: y = α + β t + λ F + e (2..3..a) t t t,

15 PANEL UNIT ROOT TESTS UNDER 3 F = F + (2..3..b) t t ft, e t = ρ (2..3..c) et + εt, wth j - ft = Φ( L) ηt, where Φ( L) = φ j jl s a K-dmensonal lag =0 polynomal such that rank ( Φ ( ) ) = k and η t ~.. d. ( 0, η ) wth fnte fourth-order moment. Consequently, F t are assumed to follow an AR() process contanng k K ndependent stochastc trends and K K statonary components 6. - the dosyncratc terms e t are also modelled as AR() processes and are allowed to be ether I ( 0) or I ( 0); furthermore, ε t = Γ ( L) ε t wth 2 ε ~.. d. ( 0, σ ) ε 7. t Instead of drectly testng the nonstatonarty of y t ( =, K, N ) 8, ths approach analyzes the common and ndvdual components separately; ths s why, t s referred as PANIC (Panel Analyss of 6 Ths means that Ft has a short-run covarance matrx of full rank; at the same tme, as rank ( Φ () ) = k, the long-run covarance matrx has reduced rank and contegraton relatons among the common factors are allowed. In such a context, smlar to Moon and Perron [43], (asymptotcally) redundant factors are excluded. 7 The cross-sectonal ndependence of the dosyncratc term s mposed only n order to valdate pooled testng. In such a context η s not necessarly a dagonal matrx and ths procedure seems to be more general than those by Moon and Perron [43], whch assumes uncorrelaton for the nnovatons of the common factors. 8 Note that a seres defned as the sum of two components wth dfferent dynamcs could have a very dfferent dynamc from those of ts consttuents. As a consequence, f one component s I() and the other one s I(0), t could be dffcult to establsh f a unt root exsts from the observaton of y t alone, especally, f ths seres contans a large statonary component. In fact, n ths stuaton, tests for the null hypothess of a unt root on yt can be overszed whle tests for the null hypothess of statonarty wll have no power (Schwert [50]).

16 32 LAURA BARBIERI Nonstatonarty n the Idosyncratc and Common Components). The am of ths procedure s to determne f nonstatonarty comes from a pervasve ( F t ) or an dosyncratc source ( E t ), and to construct vald pooled tests for panel data when the ndvduals are correlated. 9 In terms of model (2..3.), ths means determnng the number of nonstatonary factors k and testng whether ρ = wth =, K, N. Frst of all, n order to analyze the factors F t and the dosyncratc components e t that are both unobserved, Ba and Ng try to fnd consstent estmates of these seres preservng ther ntegraton features 20. They accomplsh ths goal by referrng to an approprate transformaton of y t; namely, they employ the frst dfferences,.e., yt = y t yt, f the only ntercept s ncluded n the model, (.e., y t = c d + λ F + e ) whle they apply the detrended y,.e., y = y y, t t T where yt = y, t 2 t T n the presence of a lnear trend = ( y t = c + βt + λ Ft + et ). In other words, Ba and Ng [3] suggest to proceed estmatng n a d frst step the common factors and dosyncratc errors n yt or y t by a smple prncpal component method; and then, n a second step, recumulatng these estmators, denoted as fˆ t and e ˆt, respectvely, n order to remove the effect of possble over-dfferencng t t t Fˆ t T T = ˆ, ˆ fs Et = e ˆs. (2..3.2) s= 2 s= 2 9 Note that the possblty that one or more common factors are ntegrated allows Ba and Ng test to consder the possble presence of cross-secton contegraton relatonshps. 20 Ths means that the common varatons must be obtaned wthout referrng to statonarty assumptons and/or contegraton restrctons.

17 PANEL UNIT ROOT TESTS UNDER 33 At ths tme, the null hypothess of a unt root s tested separately for the common factor Fˆ and for each dosyncratc component E. t - Common factors statonarty analyss In order to test the nonstatonarty of the common factors, when only a factor s detected 2, Ba and Ng [3] suggest to use an ADF test; when several common factors are detected, they employ a modfed verson of Stock and Watson [5] common trend test. In the former case, (.e., K = ), the unvarate augmented autoregresson whch they refer to s ˆ t Fˆ t = Dt p + θ ˆ ˆ 0Ft + θj Ft j + ξt, (2..3.3) j= where ξ t s the regresson error and D t s a polynomal functon contanng ether a constant α or a lnear trend α + β t. Now, let c ADF F ˆ and τ F ADF ˆ denote the t-statstc for θ 0 n the case, where D t contans only a constant and n the case, where t contans a lnear trend, respectvely. Ba and Ng [3] show that the lmtng dstrbutons of these statstcs correspond to the dstrbuton of the DF test for the constant only or the lnear trend case. When several common factors are detected ( K > ), ndvdually testng the factors for the presence of a unt root may overstate the number of common trends. Ths s why, n order to select k, (.e., the number of common ndependent stochastc trends n the common factors) Ba and Ng mplement an teratve procedure, smlar to the Johansen [33] trace test for contegraton. 0 = 2 The number of factors s estmated followng the Ba and Ng s [2] procedure. It s straghtforward to verfy that k = 0 corresponds to the case, where there are N contegratng vectors for N common factors,.e., all factors are I(0).

18 34 LAURA BARBIERI They consder demeaned or detrended factor estmates, dependng on whether the model (2..3.) contans the ntercept only, or also a tme trend. Specfcally, they refer to F ~ t whch s alternatvely defned as F ~ t = Fˆ ˆ t F t, wth ˆ T F ( ) ˆ t = T F, t = 2 t n the ntercept only case and as the resduals from a regresson of the lnear trend case. Fˆ t on a constant and a tme trend, n ~ Referrng to such defned F, the proposed test procedure can be descrbed as follow. Frstly, let m = K. t. If βˆ are the m egenvectors assocated to the m largest egenvalues of Xˆ 2 T ~ ~ T F t tft = 2 and Xˆ t s the matrx defned as ~ = β ˆ, t s possble to consder two dfferent statstcs: t F t (a) Let K ( j) = j ( J + ), j =, K, J; n ths case, the consdered τ statstc Q ( m), n the constant only case, or MQ c ( m), n the lnear M c c trend case, s defned as: T[ ν ˆ ( m) ], where νˆ ( m) s the smallest egenvalue of Φˆ c c T ( m ) = ( ˆ ˆ ˆ ˆ ) ( ˆ ˆ ) 2 XtXt + Xt Xt T + J t= 2 T c T ˆ ˆ Xt Xt t= 2 wth ˆ ( ) ˆ ˆ = K j ξ ξ, t j t ξˆ = T j t = 2 t beng the resduals from estmatng a frst order VAR n X ˆ t.

19 PANEL UNIT ROOT TESTS UNDER 35 (b) For p (the augmentaton for the autoregresson n (2..3.3)) fxed that does not depend on N or T, the consdered statstc Q ( m), n the M c f τ constant only case, or MQ f ( m), n the lnear trend case, s defned as T[ ν ˆ ( m) ], where νˆ ( m) s the smallest egenvalue of f Φˆ f T 2 t= f t t ( m ) = ( xˆ xˆ + xˆ xˆ ) xˆ xˆ, 2 t t t T t t= 2 where xˆ t = ˆ ( L) Xˆ t s obtaned by flterng Xˆ t by, that s, the polynomal coeffcents of an estmate Var ( p) n Xˆ,.e., ˆ ( L) = I L K ˆ p L ˆ p. 2. If the null hypothess H 0 : k = m s rejected, t s necessary to set m = m and return to Step. If the null hypothess s not rejected, we set k ˆ = m and we can stop. Then, f there are K > common factors, Ba and Ng consder two tests: the frst corrects for seral correlaton of arbtarary form by nonparametrcally estmatng the relevant nusance parameters. The second flters the factors under the assumpton that they have fnte order VAR representatons. Ths s why they have been called MQ c and MQ f, respectvely 22. It s obvous that the MQ f test s vald only when the common trends can be represented as fnte order AR ( p) processes whereas MQ c s more general. The lmtng dstrbutons of these tests are non-standard; Ba and Ng provde %, 5%, and 0% crtcal values for all four statstcs and varous m. t ˆ L m M c c, τ M c, τ f 22 Note that Q ( m) and Q ( m) statstcs are modfed verson of the Q c and Q f tests developed n Stock and Watson [5].

20 36 LAURA BARBIERI - Idosyncratc components statonarty analyss In order to test the non-statonarty of the dosyncratc components, a method based on meta-analyss s used 23. Specfcally, Ba and Ng mplement a methodology that conssts n poolng ndvdual ADF t- statstcs computed for each defactored Ê t n a model wth no determnstc term such as p E ˆ ˆ ˆ t = d0et + dj Et j + vt, (2..3.4) = where v t denotes a regresson error. Let ADF c τ () (f a constant s ncluded n the DGP) and ADF () Ê E ˆ (f a constant and a lnear trend are ncluded n the DGP) be the ndvdual t- statstcs to test the hypothess d 0. 0 = The lmtng dstrbuton of ADF c () Ê concdes wth the usual DF dstrbuton for the case of no constant and the 5% crtcal value s τ Instead, the asymptotc dstrbuton of ADF () E ˆ s proportonal to the recprocal of a Brownan brdge. Unfortunately, snce the crtcal values for ths dstrbuton are not tabulated, smulatons are requred. Thus, contrary to the other panel unt root tests prevously descrbed, these statstcs do not take the advantages of a lmtng standard normal dstrbuton. Ths happens because the panel nformaton has been used to consstently estmate e t, but not to analyze ts dynamc propertes. As Ba and Ng ponted out, PANIC procedure s characterzed by some sgnfcant features: frst, the tests on the factors do not depend on whether e t s I() or I(0), as well as the tests on the dosyncratc errors do not depend on whether F t s I() or I(0); second, the unt root tests for e s vald whether e jt, j, s I() or I(0), and n any event, such t knowledge s not necessary. 23 Ths procedure was orgnally ntroduced n Maddala and Wu [42] as well as n Cho [4].

21 PANEL UNIT ROOT TESTS UNDER 37 The ndependence of the lmtng dstrbuton of ADF c () Ê and τ ADF () E ˆ on the common factors makes possble for Ba and Ng [3] to propose a pooled Fsher-type test 24 as suggested n Maddala and Wu [42] or Cho [4]. The test statstc s gven by P o Eˆ N o 2 log p () ˆ 2N d t E = = N ( 0, ), (2..3.5) 4N where o P denotes Ê c Ê τ Eˆ P or P, dependng on the determnstc o specfcaton, and P () E ˆ s the assocated p-value of the ADF test on the estmated resdual e ˆt. For N and T, ths statstc converges n dstrbuton to a standard normal dstrbuton, but only f ndependence among the error terms s assumed: n ths case, pooled testng s vald and t s possble to derve the statstc dstrbuton. 25 Smulatons show that Ba and Ng test 24 In prncple, also an IPS-type test usng a standardzed average of the above descrbed t-statstcs should be possble. 25 Ths seems a contradcton: the am of Ba and Ng [3] test was specfcally to take nto account ths ndvdual dependence. Nevertheless, note that Ba and Ng do not assume the cross-sectonal ndependence hypothess on the whole seres y t as Im et al. [30] or Maddala and Wu [42] do, but they only hypothesze the asymptotc ndependence among the ndvdual components e t. Under ths hypothess, the test statstcs based on the estmate components ê t are asymptotcally ndependent and the p-values pe ˆ are also ndependently dstrbuted accordng to an unform law on [ 0, ]. In ths way, the hypothess that all ndvdual components e t for =, K, N are I() s suffcent to assure that the test statstc c P E ˆ or τ P E ˆ sze panel case, see Cho [4]) s standard normally dstrbuted, for all panel szes N (for the large

22 38 LAURA BARBIERI has good fnte sample propertes n term of sze and a power even for small panel ( N = 40). In concluson, the PANIC approach has the mportant advantage of allowng to obtan consstent estmators for the common factors and dosyncratc components, whether they are statonarty or nonstatonarty 26. Furthermore, t solves the problem of the sze dstorton 27 and takes advantage of the cross-secton relatonshp nformaton (Banerjee and Zangher [7]) n order to obtan pooled tests that are more powerful than the unvarate ones Other approaches Among other sutable solutons for the problem of cross-secton correlaton, 28 the procedures proposed by Bretung and Das [0], Chang [2] and Cho and Chue [6] seem to be partcular nterestng and are llustrated n the next few pages Bretung and Das [0] test Bretung and Das [0] propose a test procedure based on OLS t- statstcs wth panel corrected standard error (PCSE) (Joansson [32]), that does not requre any Monte Carlo smulatons to be computed, and can be used dfferently from the GLS test procedure even n cases, where T s less than N. Specfcally, under a weak error dependence assumpton 29, they consder the autoregressve model y t = φyt + Γ yt + Γ2 yt 2 + K + Γp yt p + ut, (2.2..) 26 Note that the re-accumulaton of the unobserved component estmates removes the effect of possble overdfferencng when the factors or the errors are statonary. 27 See note See, for example, Drscoll and Kraay [20], Conley [7], O Connell [44], Maddala and Wu [42], Taylor and Sarno [53], Chang [3], Harvey and Bates [27]. 29 Specfcally, they assume that all the egenvalues of the error covarance matrx are bounded when N.

23 PANEL UNIT ROOT TESTS UNDER 39 where yt = [ yt, y2t, K, ynt ], yt = [ y. t, y2. t, K, yn. t ], Γj = dag( γ, K, ) for j =, K, p, and the error vector u = [ u u, u Nt ] j γ Nj t t, 2t K, s..d. wth zero mean, postve defnte varance-covarance matrx Ω and fnte fourth-order moment 30. In order to test the null hypothess of nterest H : φ 0, aganst 0 = the alternatve H : φ 0, they suggest usng the robust t-statstc 0 < t rob = T yt yt t= T ˆ yt Ωyt t=, (2.2..2) where y t = y Γ y Γ y K Γ t t 2 t 2 y p t p 3 and Ωˆ s a consstent estmator of Ω. Note that under the null hypothess, the pre-whtened seres y t s a vector random walk and ( ) = Ω. y t s whte nose wth E y t y t Beng the GLS estmator more effcent than the OLS one, they also consder a more powerful test based on the GLS estmator of φ and computed wth the pre-whtened seres tgls = T yt t= T yt t= Ωˆ Ωˆ yt yt, (2.2..3) that obvously could only be computed f Ωˆ s not sngular, and so when T > N. 30 In ther study, Bretung and Das [0] also consder smple adjustment procedure n order to nclude ndvdual specfc ntercepts and tme trends n the model. 3 Bretung and Das show that ther results are stll vald even f the matrces Γ j are replaced by consstent estmates.

24 40 LAURA BARBIERI Bretung and Das [0] show that the two statstc are asymptotcally standard normally dstrbuted as ( T N ). 32 The authors also, seq show by means of Monte Carlo smulatons that the robust OLS t- statstc performs well wth respect to sze and power, whereas the GLS ones may suffer from severe dstortons n small and moderate sample szes. Bretung and Das [] extend ther prevous study and analyze the behavour of a seres of panel unt root tests assumng a factor structure approach 33. Specfcally, they consder ther prevous tests computed on non pre-whtened y seres n three dfferent stuatons and show that: - when both common and dosyncratc components are nonstatonary, the robust OLS t-statstc has a non standard lmtng dstrbuton f Ω does not have bounded egenvalues but follows a factor structure; whereas, the GLS test, that apply a transformaton n order to remove the common factors, has a standard normal dstrbuton; - when the dosyncratc components are statonary whle the common components are I () (cross-unt contegraton case), both statstcs dverge as T, N ; Note that when the autoregressve representaton (2.2..) s not the same for all unts, an ndvdual specfc autoregresson must be ftted for each unt and the seres are prewhtened by usng y ˆt = yt γˆ y, t K γ ˆ p y,. t p In these cases, Bretung and Das [0] suggest to estmate the lag order p by usng a consstent nformaton crteron as the Schwarz one (Lütkepohl and Krätzg [4]) appled to the ndvdual seres yt, K, y ( =, K, N ). 33 In that case, the common factor structure can be ncorporate n the GLS statstc mposng some structure on the covarance matrx Ω,.e., Ω = ΛΛ + wth Λ matrx of factor loadngs and covarance matrx of the dosyncratc nnovatons. In order to estmate Ω consstent estmators of Λ and are requred and the prncpal component procedure suggested by Ba and Ng [3] or Moon and Perron [43] can be adopted. 34 Bretung and Das [] observe that n such a context also the prewhtenng approach may fal, f the short-run dynamcs are ndvdual specfc.

25 PANEL UNIT ROOT TESTS UNDER 4 - when the common components are statonary whle the dosyncratc components are I ( ) the OLS-based statstc s not applcable snce t tends to nfnty and so tends to ndcate statonary tme seres as T, N. On the other hand GLS test, removng the cross-secton dependence, s asymptotcally vald Chang [2] test Chang [2] proposes an alternatve non-lnear nstrumental varable (IV) approach n order to solve the nusance parameter problem afflctng the dstrbuton of the unt root tests n the presence of cross-sectonal correlaton. To do ths, Chang [2] tres to make the panel statstcs asymptotcally nvarant to cross-sectonal dependence: for each crosssecton unt, the author estmates the AR coeffcent from an usual ADF regresson usng the nstruments generated by an ntegrable transformaton of the lagged values of the endogenous varable. Then, for testng the unt root based on these N nonlnear IV estmators, Chang constructs N ndvdual t-statstcs that have lmtng standard normal dstrbuton under the null hypothess. Fnally, a cross-sectonal average of the ndvdual IV t-rato statstcs s consdered, as n the IPS approach. Specfcally, Chang consders a panel model generated by a frst-order autoregressve regresson 35 such as y t ρ yt + ut, = (2.2.2.) where, as usual, =, K, N denotes ndvdual cross-sectonal unts and t, K, T = denotes tme-seres observatons. Note that the total number T for each ndvdual may dffer across unts,.e., unbalanced panels are allowed. The ntal values ( y0, K, yn0 ) are set at zero for smplcty. The error term u t s gven by an AR ( p ) nvertble process λ ( L) = ε, ( ) u t t 35 The model wth determnstc components can be analyzed smlarly usng demeaned or detrended data.

26 42 LAURA BARBIERI p j j where λ ( L) = β L, wth L denotng the usual lag operator. = j 2 Cross-sectonal dependence of the nnovatons ~.. d. ( 0, σ ) generate the errors u t s s allowed. The null hypothess of nterest s H : for all y, 0 ρ = aganst the alternatve hypothess H : < for some y. a ρ t t ε that Thus, the null hypothess mples that all y t s have unt roots, and t s rejected, f one at least of the y t s s statonary wth ρ <. In ths way, rejecton of the null hypothess does not mply that the whole panel s statonary. Gvng the (2.2.2.) and ( ), t s possble to rewrte the model as t ε y t p = ρ yt + βjut j + εt j=, and, snce y = u t regresson becomes t under the null hypothess of unt root, the above y t p = ρ yt j= + βj yt j + εt. ( ) In order to deal wth the cross-sectonal dependence, Chang uses the nstrument generated by a non-lnear functon F ( ) of lagged values y t,.e., F ( yt ), whch s called the Instrument Generatng Functon (IGF). F () s a regularly ntegrable functon whch satsfes xf ( x) dx 0,.e., the nonlnear nstrument F ( ) s correlated wth the regressor y t. +

27 PANEL UNIT ROOT TESTS UNDER 43 For the lagged demeaned dfferences x = ( y, K, y ), the varables themselves are used as nstruments. Let X = ( ) xp + x,, T dfferences, y ( y, y ) ( ε,, ε ) t t t p K be the ( T, ) matrx of the lagged t = p, T ε = p + K T the vector of resduals. p K the vector of lagged values and The augmented regresson ( ) can be wrtten n matrx form as y = y ρ + X β + ε ( ) l t, where β = ( β,, β ). K Under the null hypothess, the nonlnear IV p estmator of the parameter ρ denoted as ρ, s gven by ρˆ [ F ( y ) y F ( y ) = X ( X X ) X y ] F ( y ) ε F ( y ) X ( X X ) l l and ts varance by l ˆ [ X ε ] l l l 2 σˆ ρ = 2 σˆ ε [ F ( y ) y F ( y ) X ( X X ) X y ] 2 l l l [ F ( y ) F ( y ) F ( y ) X ( X X ) X F ( y )], l l l l l 2 2 where σˆ ε = ( T ) εˆ t = t regresson ( ). T and εˆ t s the ftted resdual from augmented For testng the unt root hypothess H 0 : ρ = for all y t, Chang constructs a t-rato statstc from the nonlnear IV estmator ρ ˆ, denoted by Z, n ths way: Z ρˆ = N ( 0, ) for all =, K N, σˆ T ρˆ d where the asymptotcal convergence to a standard normal dstrbuton s assured, f a regularly ntegrable functon s used, as an IGF. Ths asymptotc Gaussan result s fundamentally dfferent from the usual unt root lmt theores and t s essentally due to the nonlnearty

28 44 LAURA BARBIERI of the IV. More mportantly, the lmt dstrbutons of ndvdual statstcs are cross-sectonally ndependent. Hence, these asymptotc orthogonaltes lead to propose a panel unt root test based on the crosssectonal average of these ndvdual ndependent statstcs. Chang proposes an average IV t-rato statstc, defned as Z SN = N N Z, = 2 where N s just as a normalzaton factor. It s straghtforward to show that S N has a lmtng standard normal dstrbuton 36 and, then t s possble to do smple nference even for unbalanced panels wth general cross-sectonal dependence. Moreover, Chang s lmt theory does not requre a large spatal dmenson; consequently N may take any value, large or small. Chang s [2] approach s very general and has good fnte sample propertes. The smulaton results show that the fnte sample szes of S N, calculated usng the standard normal crtcal values, are close to the nomnal ones. Moreover, the S N test seems to be more powerful than the IPS one and mproves sgnfcantly upon the t-bar test under crosssectonal dependency, especally, for panels wth smaller T and N. However, Im and Pesaran [3] showed that Chang s test s vald only, f N s fxed as T. Ther Monte Carlo smulatons show that Chang s test s consderably over-szed for moderate degrees of crosssecton dependence, even for relatvely small values of N. Furthermore, the non lnear transformaton nvolved n the Chang s procedure may compromse the power of the test and t s not clear how to choose the transformaton n an optmal way Cho and Chue [6] test Cho and Chue [6] propose subsamplng technques to deal wth cross-sectonal dependence. Ther procedure s very general and can be 36 It should be noted that the usual sequental asymptotc s not used here. The lmt theory s derved for T, whch n not followed by N.

29 PANEL UNIT ROOT TESTS UNDER 45 appled to possbly nonstatonary panel data. Moreover, t allows the regressors to be statonary, nonstatonary or a mxture of both types and t does not requre the estmaton of the cross-sectonal correlaton, so far reducng the rsk of sze dstorton due to msspecfcaton wth respect to a factor structure approach. It also allows for cross-sectonal contegraton wthout requrng knowledge of the contegraton coeffcents and ranks. Cho and Chue s [6] tests have fnte sample dstrbutons that do not depend on nusance parameters 37. Specfcally, n the fnte N, nfnte T case, they consder a panel unt root test ξ Nn computed on the whole dsposable sample and observe that ts lmtng dstrbuton s lkely to depend on nusance parameters for crosssectonally correlated or contegrated panels. Nevertheless, t s possble to approxmate ths lmtng dstrbuton by means of the subsamplng method. Gven a panel unt root test ξ Nbs computed wth smaller blocks of consecutvely observed tme seres { y s, K, y. s+ b startng from } N = s ( s T b ) and wth block sze b, the emprcal dstrbuton functon based on the computed values of the tests can be expressed as L ξ NTb T ( x) = + { ξnbs x}, T b + where { ξnbs x} =, f ξ Nbs x and { ξnbs x} = 0, f ξ Nbs > x. ξ Assumng b and b T 0 as T, LNTb( x) approxmates the lmtng dstrbuton unformly n x and can be appled n a panel unt root context. Specfcally, Cho and Chue subsample the LLC, IPS and Fsher-type tests for panel unt root and show that they are consstent wth reasonably fnte-sample propertes, mostly n the noncontegrated panel case. Nevertheless, as Cho and Chue [6] pont out subsamplng s not always the best method to use, snce t depends on the nature of the problem. Sometmes other methods, (.e., Chang [3]) may work better n fnte samples. b s= 37 Cho and Chue [6] procedure ncludes panel unt root and contegraton tests as specal cases.

30 46 LAURA BARBIERI 2.3. Comparson among the prevous presented tests From the prevous dscussons, t s straghtforward to note that Ba and Ng [3], Pesaran [45] and Moon and Perron [43] tests assume the same dynamc structure of the data and are computatonally smple to mplement (they only requre some tabulated crtcal values and the selecton of the number of common factor K). Specfcally, as Gengenbach et al. [22] pont out, the use of a factor model allows to represent correlaton or contegraton between panel unts n a sutable way and the assumpton of ndependence between common factors and error terms (requred for pooled testng) seems to be more unrestraned than the assumpton of ndependence between cross-sectons (made n the frst generaton of unt root tests). Nevertheless, Ba and Ng [3] approach s more general than the ones of Pesaran [45] and Moon and Perron [43]. In partcular, one can observe that - n Ba and Ng [3] the non-statonarty of the seres can be due to common or dosyncratc sources (so that the ntegraton order of the two factors can dffer), whle Pesaran [45] and Moon and Perron [43] assume common and dosyncratc stochastc trends under the null hypothess Pesaran [45] and Moon and Perron [43] do not allow for contegraton among the y t as well as between the observed data and the common factors but Ba and Ng [3] models nclude both possbltes. - Pesaran s [45] and Ba and Ng s [3] models may nclude ether a constant or a lnear trend; whereas Moon and Perron [43] test s proposed for the case n whch only a constant s present. Due to these features, whle the Ba and Ng [3] test s able to dstngush cases, where the observed non-statonarty depends only on a non-statonary common factor; n such crcumstances the other two tests tend to reject the non-statonarty of the seres. 38 Note that on the bass of these observatons, the null hypothess assumed by Pesaran [45] and Moon and Perron [43] s a specal case of the Ba and Ng s one [3].

31 PANEL UNIT ROOT TESTS UNDER 47 Gengenbach et al. [22] analyze the small sample behavour of the proposed tests and show that: - Moon and Perron [43] test s more powerful than the Pesaran [45] one, but the latter s smpler to compute; - n order to detect the non-statonarty of the dosyncratc components, the c E P ˆ test s more powerful than the ADF ; - n order to detect the non-statonarty of the common factor the c F ADF ˆ test shows good small sample propertes when N and T 50 and the seral correlaton n the common factor s not too persstent; - n a mult-factor settng, the c Eˆ c MQ c seems to overcome the c MQ f test but both tests fal to dstngush hgh statonary seral correlaton from non-statonarty n the common factors 39. Due to ths observatons, Gengenbach et al. [22] provde a procedure for testng the presence of unt root n panels wth dynamc factors. Frst, on the bass of ths approach, f the cross-sectonal dependence s suspected to be generated by a sngle common factor, the Pesaran s [45] CIPS test s used for testng for the non-statonarty n the data; on the contrary, f cross-secton dependence s suspected to be generated by mult common factors, a Moon and Perron [43] test s preferred. Also, the c P E ˆ and the c ADF F ˆ tests are used n order to test for the non-statonarty of the dosyncratc components and the common factors, respectvely. Now, the rejecton by the Ba and Ng [3] tests of the null hypothess of unt roots for the dosyncratc components but not for the common factor, and the rejecton by the Pesaran [45] and Moon and Perron [43] 39 Ths topc s well analysed also by Bretung and Das [].