Testing Cross-Sectional Correlation in Large Panel Data Models with Serial Correlation

Transcription

1 Artcle Testng Cross-Sectonal Correlaton n Large Panel Data Models wth Seral Correlaton Bad H Baltag 1, Chhwa Kao and Bn Peng 3, * 1 Department of conomcs & Center for Polcy Research, 46 ggers Hall, Syracuse Unversty, Syracuse, Y , USA; bbaltag@maxwellsyredu Department of conomcs, 365 Farfeld Way, U-1063, Unversty of Connectcut, Storrs, CT , USA; chh-hwakao@uconnedu 3 Department of Fnance, 53 School of conomcs, Huazhong Unversty of Scence and Technology, Wuhan , Chna * Correspondence: bpeng01@husteducn; Tel: Academc dtors: In Cho and Ryo Oku Receved: 3 July 016; Accepted: 19 October 016; Publshed: 4 ovember 016 Abstract: Ths paper consders the problem of testng cross-sectonal correlaton n large panel data models wth serally-correlated errors It fnds that exstng tests for cross-sectonal correlaton encounter sze dstortons wth seral correlaton n the errors To control the sze, ths paper proposes a modfcaton of Pesaran s Cross-sectonal Dependence CD test to account for seral correlaton of an unknown form n the error term We derve the lmtng dstrbuton of ths test as, T The test s dstrbuton free and allows for unknown forms of seral correlaton n the errors Monte Carlo smulatons show that the test has good sze and power for large panels when seral correlaton n the errors s present Keywords: cross-sectonal correlaton test; seral correlaton; large panel data model JL Classfcaton: C13; C33 1 Introducton Ths paper studes testng for cross-sectonal correlaton n panel data when seral correlaton s also present n the dsturbances It does that for the case of strctly-exogenous regressors 1 Cross-sectonal correlaton could be due to unknown common shocks, spatal effects or nteractons wthn socal networks Ignorng cross-sectonal correlaton n panels can have serous consequences In tme seres wth seral correlaton, exstng cross-sectonal correlaton leads to effcency loss for least squares and nvaldates nference In some cases, t results n nconsstent estmaton; see Lee [1] and Andrews [] Testng the cross-sectonal correlaton of panel resduals s therefore mportant One could test for a specfc form of correlaton n the error lke spatal correlaton; see Anseln and Bera [3] for cross-sectonal data and Baltag et al [4] for panel data, to menton a few Alternatvely, one could test for correlaton wthout mposng any structure on the form of correlaton among the dsturbances The null hypothess, n that case, s testng the dagonalty of the covarance or correlaton matrx of the -dmensonal dsturbance vector u t u 1t,, u t, whch s usually assumed to be ndependent over tme, for t 1,, T When s fxed and T s large, the tradtonal multvarate statstcs technques, ncludng log-lkelhood rato and Lagrange multpler tests, are applcable; see, for example, Breusch and Pagan [5], who propose a Lagrange 1 The ncluson of predetermned varables, whch s the weakly-exogenous case, alters the results conometrcs 016, 4, 44; do:103390/econometrcs wwwmdpcom/journal/econometrcs

2 conometrcs 016, 4, 44 of 4 Multpler LM test, whch s based on the average of the squared par-wse correlaton coeffcents of the least squares resduals However, as becomes large because of the growng avalablty of the comprehensve databases n macro and fnance, ths so-called hgh dmensonal phenomenon brngs challenges to classcal statstcal nference As shown n the Random Matrx Theory RMT lterature, the sample covarance and correlaton matrces are ll-condtoned snce ther egenvectors are not consstent wth ther populaton counterparts; see Johnstone [6] and Jang [7] ew approaches have been consdered n the statstcs lterature for the testng the dagonalty of the sample covarance or correlaton matrces; see Ledot and Wolf [8], Schott [9] and Chen et al [10], to menton a few The above tests for raw data cannot be used drectly to test cross-sectonal correlaton n panel data regressons snce the dsturbances are not observable ose caused by substtutng resduals for the actual dsturbances may accumulate due to large dmensons, and ths n turn may lead to based nference The bas for cross-sectonal correlaton tests n large panels depends on the model specfcaton, the estmaton method and the sample szes and T, among other thngs For example, Pesaran et al [11] consder an LM test and correct ts bas n a large heterogeneous panel data model; Baltag et al [1] extend Schott s test [9] to a fxed effects panel data model and correct the bas caused by estmatng the dsturbances wth fxed effects resduals n a homogeneous panel data model Followng Ledot and Wolf [8], Baltag et al [13] propose a bas-adjusted test for testng the null of sphercty n the fxed effects homogeneous panel data model However, ths method does not test cross-sectonal correlaton drectly Rejecton of the null could be due to cross-sectonal correlaton or heteroscedastcty or both A general test for cross-sectonal correlaton was proposed by Pesaran [14] Hs test statstc s based on the average of par-wse correlaton coeffcents, defned as CD P CD, Cross-sectonal Dependence The test s exactly centered at zero under the null and does not need bas correcton Pesaran [15] extends hs test statstc to test the null of weak cross-sectonal correlaton and derves ts asymptotc dstrbuton usng jont lmts Ths test s robust to many model specfcatons and has many applcatons Recent surveys for cross-sectonal correlaton or dependence tests n large panels are provded by Moscone and Tosett [16], Sarafds and Wansbeek [17] and Chudk and Pesaran [18] The asymptotcs and bas-correcton of exstng tests for cross-sectonal correlaton n large panels are carred out under some, albet restrctve, assumptons For nstance, the errors are normally dstrbuted; /T c 0, as, T, and so on One fundamental restrcton s that the errors are ndependent over tme In fact, the presence of seral correlaton n panel data applcatons s lkely to be the rule rather than the excepton, especally for macro applcatons and when T s large Ignorng seral correlaton does not affect the consstency of estmates, but t leads to ncorrect nference In RMT, when u 1, u,, u T are ndependent across t 1,, T, and s large, the Lmtng Spectral Dstrbuton LSD of the correspondng sample covarance matrx s the Marchenko-Pastur M-P law; see Ba and Slversten [19] xstng correlaton among these dsturbances may cause a devaton of the LSD from the M-P law Indeed, Ba and Zhou [0] show that the LSD of the sample covarance matrx wth correlatons n columns s dfferent from the M-P law Gao et al [1] show smlar results for the sample correlaton matrx Therefore, the cross-sectonal correlaton tests, whch heavly depend on the assumpton of ndependence over tme, could lead to msleadng nference f there s a seral correlaton n the dsturbances To better understand the effects of potental seral correlaton on the exstng tests of cross-sectonal correlaton, let us assume that the T 1 ndependent random vectors u u 1,, u T, for 1,, are observable The correlaton coeffcents ρ j of any u and u j j are defned by u u j/ u u j Ther means are zero vectors If all of the elements of each u are ndependent and dentcally sphercally dstrbuted, Murhead [] shows that ρ j 1/T When s fxed, the summaton of all dstnct 1/ terms of ρ j wll be small, as T In Secton 3, we show that f all of the elements of each u follow a multple Movng Average model of order one MA1 wth parameter θ, then ρ j [ 1/T + θ /T + Tθ ] As, the extra term θ /T + Tθ can

3 conometrcs 016, 4, 44 3 of 4 accumulate and lead to extra bas for the exstng LM type tests n panels Although CD P s centered at zero, t may stll encounter sze dstortons because seral correlaton s gnored Ths paper proposes a modfcaton of Pesaran s CD test of cross-sectonal correlaton when the error terms are serally correlated n large panel data models Frst, usng results from RMT, we study the frst two moments of the test statstc and propose an unbased and consstent estmate of the varance wth unknown seral correlaton under the null Second, we derve the lmtng dstrbuton of the test under the asymptotc framework wth, T smultaneously n any order wthout any dstrbuton assumpton We also dscuss ts local power propertes under a mult-factor alternatve Monte Carlo smulatons are conducted to study the performance of our test statstc n fnte samples The results confrm our theoretcal fndngs The plan for the paper s as follows The next secton ntroduces the model and notaton, exstng LM type tests and the Cross-sectonal Dependence CD test It then presents our assumptons and the proposed modfed Pesaran s CD test statstc Secton 3 derves the asymptotcs of ths test statstc Secton 4 reports the results of the Monte Carlo experments Secton 5 provdes some concludng remarks All of the mathematcal proofs are provded n the Appendx Throughout the paper, we adopt the followng notaton For a squared matrx B, trb s the trace of B; B trb B 1/ denotes the Frobenus norm of a matrx or the ucldean norm of a d vector B; denotes convergence n dstrbuton; and denotes convergence n probablty We use, T to denote the jont convergence of and T when and T pass to nfnty smultaneously K s a generc postve number not dependng on nor T Model and Tests Consder the followng heterogeneous panel data model p y t β x t + u t, for 1,, ; t 1,, T, 1 where and t ndex the cross-secton dmenson and tme dmenson, respectvely; y t s the dependent varable, and x t s a k 1 vector of exogenous regressors The ndvdual coeffcents β are defned on a compact set and allowed to vary across The null hypothess of no cross-sectonal correlaton s H 0 : covu t, u jt 0, for all t, j, or equvalently as H 0 : ρ j 0, for j, where ρ j s the par-wse correlaton coeffcents of the dsturbances defned by ρ j t1 T u tu jt 1/ 1/ t1 T u t t1 T u jt Under the alternatve, there exsts at least one ρ j 0, for some j For the panel regresson model 1, the resduals are unobservable In ths case, the test statstc s based on the resdual-based correlaton coeffcents ˆρ j Specfcally, ˆρ j t1 T e te jt 1/ 1/, 3 t1 T e t t1 T e jt where e t s the Ordnary Least Squares OLS resduals usng T observatons for each 1,, These OLS resduals are gven by e t y t x t ˆβ, 4

4 conometrcs 016, 4, 44 4 of 4 wth ˆβ beng the OLS estmates of β from 1 for 1,, Let M I T P X, where P X X X X 1 X, and X s a T k matrx of regressors wth the t-th row beng the 1 k vector x t We also defne u u 1,, u T, e e 1,, e T and v e / e, for 1,, The OLS resduals can be rewrtten n vector form as e M u, and the resdual-based par-wse correlaton coeffcents can be rewrtten as ˆρ j v v j, for any 1 j 1 LM and CD Tests For fxed and T, Breusch and Pagan [5] propose an LM test to test the null of no cross-sectonal correlaton n wthout mposng any structure on ths correlaton It s gven by 1 LM BP T 1 j+1 ˆρ j 5 LM BP s asymptotcally dstrbuted as a Ch-squared dstrbuton wth 1/ degrees of freedom under the null However, for a typcal mcro-panel dataset, s larger than T; and the Breusch-Pagan LM test statstc s not vald under ths large, small T setup In fact, Pesaran [14] proposes a scaled verson of ths LM test as follows LM P j+1 T ˆρ j 1 6 Pesaran [14] shows that LM P s dstrbuted as 0, 1 wth T frst, then under the null However, T ˆρ j 1 s not correctly centered at zero wth fxed T and large Hence, Pesaran et al [11] propose a bas-adjusted verson of ths LM test, denoted by LM PUY They show that the exact mean and varance of T k ˆρ j are gven by and ] νtj [T var k ˆρ j [ where a 1T a T 1, and a T k T 3 [ ] µ Tj T k ˆρ j 1 T k tr [ M M j ], 7 LM PUY { tr [ M M j ]} a 1T + tr { [M M j ] } a T, 8 T k 8T k+ T k+t k T k 4 1 ] LMPUY s gven by T k ˆρ j µ Tj ν Tj 9 Pesaran et al [11] show that LM PUY s asymptotcally dstrbuted as 0, 1 under the null and the normalty assumpton of the dsturbances as T followed by Alternatvely, Pesaran [14] proposes a test based on the average of par-wse correlaton coeffcents rather than ther squares The test statstc s gven by CD P T j+1 ˆρ j 10 Pesaran [15] shows that ths test s asymptotcally dstrbuted as 0, 1 wth, T He also extends ths to test the null of weak cross-sectonal correlaton Assumptons and the Modfed CD Test Statstc So far, all of the methods surveyed above for testng cross-sectonal correlaton n panel data models assume that the dsturbances are ndependent over tme Ignorng seral correlaton usually

5 conometrcs 016, 4, 44 5 of 4 results n effcency loss and based nference In fact, we show n Secton 3 that the exstence of seral correlaton leads to extra bas n the LM-type tests For the CD P test n 10, t s stll centered at zero wth seral correlaton, but ts varance s affected by seral correlaton As a result, we also expect sze dstortons n CD P To correct for ths, we consder a modfcaton of ths test statstc that accounts for an unknown form of seral correlaton n the dsturbances Frst, we ntroduce the assumptons needed: Assumpton 1 Defne ξ ξ 0, ξ 1,, ξ T and ε ε 0, ε 1,, ε T We also assume that ξ σ ε, for 1,,, where ε s a random vector wth mean vector zero and covarance matrx I T Let ε t denote the t-th entry of ε, for any 1,, ε t has a unformly bounded fourth moment, and there exsts a fnte constant, such that ε 4 t 3 + Followng Ba and Zhou [0], the dsturbances u t u 1t, u t,, u t are generated by u t d s ξ t s, for t 1,, T, 11 s0 where ξ t ξ 1t, ξ t,, ξ t, for t 0, 1,, T, are IID random vectors across tme, and {d s } s0 s a sequence of numbers satsfyng s0 d s < K < Assumpton 1 allows the error term u t to be correlated over tme The condton s0 d s < K < excludes long memory-type strong dependence We need bounded moment condtons to ensure large, T asymptotcs for panel data models wth seral correlaton The condtons n Assumpton 1 are qute relaxable; they are satsfed by many parametrc weak dependence processes, such as statonary and nvertble fnte-order Auto-Regressve and Movng Average ARMA models Under Assumpton 1, the covarance matrx of each u s Σ σ Σ, where Σ s a T T symmetrc postve defnte matrx The random vector u can be wrtten as u σ Γε, where ΓΓ Σ The generc covarance matrx Σ of each u captures the seral correlaton Ba and Zhou [0] use ths representaton and show that 1/TtrΣ κ s bounded for any fxed postve nteger κ More specfcally, consderng a multple Movng Average model of order one MA1 u t ξ t + θξ t 1, t 1,, T, 1 where θ < 1 and u t, ξ t, u and ξ are defned n Assumpton 1 For ths case, Σ MA δ lr T T, where δ lr 1 + θ, l r; θ, l r 1; 0, l r > 1 One can also verfy that for 11, we have the followng generc representaton, Σ ϖ lr T T, where ϖ lr We use ths representaton throughout the paper for convenence Assumpton The regressors, x t, are strctly exogenous, such that and X X s a postve defnte matrx 13 d s d l r +s 14 s0 u t X 0, for all 1,, and t 1,, T, 15 Assumpton 3 T > k and the OLS resduals, e t, defned by 4, are not all zeros wth probablty approachng one Assumptons and 3 are standard for model 1; see Pesaran [14] and Pesaran et al [11] We mpose the assumpton that the regressors are strctly exogenous We do not mpose any restrctons

6 conometrcs 016, 4, 44 6 of 4 on the dstrbuton of the errors or the relatve convergence speed of, T Ths framework s qute relaxable whle LM-type tests usually mpose the normalty assumpton and restrctons on the relatve speed of and T, namely /T c 0, Under these assumptons, the OLS estmates for model 1 are consstent, but neffcent We focus on the term used n Pesaran s CD test [14] 1/ T n 1 ˆρ j 16 In the next secton, we derve the frst two moments of ths test statstc and later derve ts lmtng dstrbuton under ths general unknown form of seral correlaton over tme 3 Asymptotcs 31 Asymptotc Dstrbuton under the ull In ths secton, we study the asymptotcs of the test statstc T n defned n 16 To derve ts lmtng dstrbuton, we frst consder ts frst two moments Theorem 1 Under Assumptons 1 3 and the null gven n, T n 0 17 and γ var T n 1 ˆρ j 1 tr M j ΣM j M ΣM tr M Σ tr M j Σ, 18 where M I T X X X 1 X, and Σ s defned by 14 Theorem 1 shows that the mean of the test statstc s zero Its varance depends on Σ, whch s a generc form contanng seral correlaton In fact, as shown n the proof of Theorem 1 see the Appendx B, tr M j ΣM j M ΣM / [ tr M Σ tr M j Σ ] In the specal case where the error terms are ndependent over tme, Σ I T, and reduces to tr M j M /T k, whch yelds the results gven n quaton 7 for the LM PUY test ˆρ j statstc wth no seral correlaton However, wth seral correlaton n the errors, an extra bas term s ntroduced n LM PUY snce tr M j ΣM j M ΣM tr M Σ tr M j Σ tr Mj M T k 0, f Σ I T More specfcally, let us assume that u, 1,,, are observable, then ρ j tr Σ /tr Σ For the MA1 process defned by 1, tr Σ /tr Σ 1/T + θ /T + Tθ and tr Σ /tr Σ 1/T, for θ 0 The extra term θ /T + Tθ accumulates n the LM test statstc and leads to extra bas as As dscussed above, we expect that LM PUY to have serous sze dstortons when seral correlaton s present n the dsturbances Unlke LM-type tests, the test statstc T n s centered at zero; t does not need bas adjustment ote that f u t are ndependent over tme, our model reduces to that of Pesaran [14] Let γ0 be the varance of T n wthout seral correlaton; t can be wrtten as γ 0 1 [ ˆρ j T k T k + trp ] X P Xj T k, 19

7 conometrcs 016, 4, 44 7 of 4 where P X X X X 1 X and P Xj X j X j X j 1 X j The above result s the exact varance for T n wthout seral correlaton; t s derved by Pesaran [15] A modfed verson of CD P s also gven by Pesaran [15] usng ths exact varance From Theorem 1, γ s dfferent from γ 0 f Σ I T Hence, we also expect CD P to have sze dstortons when seral correlaton s present n the dsturbances ext, we consder the lmtng dstrbuton of the proposed test The result s gven n the followng theorem Theorem Under Assumptons 1 3 and the null n, as, T, we have γ 1 T n d 0, 1 0 Theorem shows that approprately standardzed γ 1 T n s asymptotcally dstrbuted as a standard normal It s vald for and T tendng to nfnty jontly n any order However, we do not observe Σ n a panel data regresson model; and an estmate of the varance γ s needed for practcal applcatons Followng Chen and Qn [3], an unbased and consstent estmator of γ under the null s obtaned usng the cross-valdaton approach proposed n the followng theorem: Theorem 3 Let ˆγ 1 1 v v j v,j v j v v,j, where v,j 1 v τ 1 τ,j Under Assumptons 1 3 and the null n, ˆγ γ As, T, ˆγ p γ 1 Defne CD R ˆγ 1 T n As, T, CD R d 0, 1 Theorem 3 shows that ˆγ s a good approxmaton for the varance, and we do not need to specfy the structure of Σ In other words, the test statstc allows the error terms of model 1 to be dependent over tme Furthermore, CD R s a modfed verson of CD P, so they are lkely to perform very smlarly wth respect to many model specfcatons see Pesaran [14] 3 Local Power Propertes We now consder the power analyss of the test aturally, the power propertes depend on the specfcatons of the alternatves One general alternatve specfcaton that allows for global cross-sectonal correlaton n panels s the unobserved mult-factor model Under ths alternatve, the new error terms are defned by u u + σ Fλ σ Γε + Fλ, 3 where F f 1, f,, f T denotes the T r common factor matrx and λ s the r factor loadng vector Under the null hypothess, λ 0, for all We now consder the followng Ptman-type local alternatve 1 H a : λ T 1/4 1/ δ, for some, 4 where δ s a non-random and non-zero r 1 vector, whch does not depend on or T To smplfy the analyss, we add the followng assumpton: We only consder the case that the number of non-zero factor loadng vectors s or of order, whch means the model has strong error cross-sectonal correlaton For the weak error cross-sectonal correlaton case, we conjecture that t s smlar to Pesaran [15]

8 conometrcs 016, 4, 44 8 of 4 Assumpton 4 1 f t IID0, I r ; f t are ndependent of ε t, x t, for all and t; 3 for each, T 1/ T t1 u t f t O p 1; T 1/ T t1 x t f t O p 1 and T 1 T t1 f t f t I r + O p T 1/ ; 4 T 1/ X X j O p 1 and T 1/ X u O p 1, for all and j The followng theorem gves the power propertes under the local alternatve 4 Theorem 4 Under Assumptons 1 4 and local alternatve 4, as, T, where ψ plm,t γ 1 1 γ 1 T n 1/ d ψ, 1, 5 T 1/ 1 δ δ j tr 1/ M Σtr 1/ M j Σ 0 From Theorem 4, the test has nontrval power aganst the local alternatve that contracts to the null at the rate of T 1/4 1/ Hence, Theorem 4 establshes the consstency of the proposed test at the rate of T under the alternatve, as long as ψ 0 4 Monte Carlo Smulatons Ths secton conducts Monte Carlo smulatons to examne the emprcal sze and power of the proposed test CD R defned n n heterogeneous panel data regresson models We also look at the performance of LM PUY and CD P defned by 9 and 10, respectvely, for comparson purposes We consder four scenaros: 1 the errors are ndependent over tme, wth no seral correlaton; the errors follow a movng average model of order one MA1 over tme; 3 the errors follow an Auto-Regressve model of order one AR1 over tme; 4 the errors follow an Auto-Regressve and Movng Average of order 1,1 ARMA1, 1 over tme Fnally, we provde small sample evdence on the power performance of the modfed CD R test aganst a factor and spatal auto-regressve model of order one alternatves, whch are popular n economcs for modelng cross-sectonal correlaton 41 xpermental Desgn Followng Pesaran et al [11], our experments use the followng data-generatng process y t α + β x t + u t, 1,, ; t 1,, T, 6 x t ηx t 1 + υ t, 7 where α IID1,1; β IID1,004 x t s a strctly exogenous regressor, and we set η 06 and υ t IIDφ /1 06 wth φ IIDχ 6/6, for 1,, The error terms of 6 are generated usng the followng four data generatng processes 1 IID : u t ξ t ; 8 MA1 : u t ξ t + θξ t 1 ; 9 3 AR1 : u t ρu t 1 + ξ t ; 30 4 ARMA1,1 : u t ρu t 1 + ξ t + θξ t 1, 31 where ξ t σ ε t ; σ IIDχ / and ε t IID0,1 We further set θ 08 and ρ 06 To check the robustness of the tests to non-normal dstrbutons, ε t are generated from a ormal0,1 and a Ch-squared dstrbuton χ / 1 To examne the emprcal power of the tests, we consder two dfferent cross-sectonal correlaton alternatves: factor and spatal models The factor model s generated by u t λ f t + u t, for 1,, ; t 1,, T, 3

9 conometrcs 016, 4, 44 9 of 4 where f t IID0,1 and λ IIDU[01,03]; In ths case, u t replaces u t n 6 for the power studes u t s generated by the four scenaros defned by 8 31, respectvely For the spatal model, we consder a frst-order spatal auto-correlaton model SAR1, u t δ 05u,t + 05u +1,t + ut, 33 where δ 04 and u t are defned by 8 31, respectvely The experments are conducted for 10,0,30,50,100,00 and T 10,0,30,50,100 For each par of, T, we run 000 replcatons To obtan the emprcal sze, we conduct the proposed test CD R and CD P at the two-sded 5% nomnal sgnfcance level and LM PUY at the postve one-sded 5% nomnal sgnfcance level 4 Smulaton Results Table 1 reports the emprcal sze of CD P, LM PUY and CD R for normal and ch-squared dstrbuted errors The error terms are assumed to be ndependent over tme The results show that all of the tests have the correct sze wth dfferent, T combnatons under both normal and ch-squared scenaros Those are consstent wth the theoretcal fndngs The only exceptons are for small or T equal to 10, especally for LM PUY Table reports the emprcal sze of the three tests wth MA1 error terms defned by 9 The results show that CD R has the correct sze for all, T, but CD P has sze dstortons for dfferent, T combnatons because the dsturbances are MA1 over tme For example, under the normalty scenaro, the sze of CD P s 935% for 10 and T 0; t becomes 111% when T grows to 100 LM PUY suffers serous sze dstortons, because of the extra bas caused by gnorng seral correlaton From Table, the emprcal sze of LM PUY s 100% as or T becomes larger than 30 Tables 3 and 4 report the emprcal sze of the tests wth AR1 and ARMA1,1 errors under the two dstrbutons: normal and ch-squared scenaros ote that CD R s over-szed n Table 4 for the ch-squared case when T 10 However, t has the correct sze as T gets larger than 0 In contrast, LM PUY has serous sze ssues, rejectng 100% of the tme, and CD P s over szed by as much as 5% Overall, n comparson wth CD P and LM PUY, the proposed test CD R controls for sze dstortons when seral correlaton n the dsturbances s present and s not much affected when seral correlaton s not present Tests CD R CD P LM PUY,T Table 1 Sze of tests wth IID errors over tme ormal Ch-Squared otes: Ths table reports the sze of CD P, LM PUY and CD R wth u t ξ t, where ξ t σ ε t ; σ IIDχ / ε t IID0, 1 and are generated from normal and Ch-squared dstrbutons The tests are conducted at the 5% nomnal sgnfcance level

10 conometrcs 016, 4, of 4 Tests CD R CD P LM PUY,T Table Sze of tests wth MA1 errors ormal Ch-Squared otes: Ths table reports the sze of CD P, LM PUY and CD R wth u t ξ t + θξ t 1, where ξ t σ ε t ; σ IIDχ / ε t IID0, 1 and are generated from normal and Ch-squared dstrbutons The tests are conducted at the 5% nomnal sgnfcance level Tests CD R CD P LM PUY,T Table 3 Sze of tests wth AR1 errors ormal Ch-Squared otes: Ths table reports the sze of CD P, LM PUY and CD R wth u t ρu t 1 + ξ t, where ξ t σ ε t ; σ IIDχ / ε t IID0, 1 and are generated from normal and Ch-squared dstrbutons The tests are conducted at the 5% nomnal sgnfcance level

11 conometrcs 016, 4, of 4 Tests CD R CD P LM PUY,T Table 4 Sze of tests wth ARMA1,1 errors ormal Ch-Squared otes: Ths table reports the sze of CD P, LM PUY and CD R wth u t ρu t 1 + ξ t + θξ t 1, where ξ t σ ε t ; σ IIDχ / ε t IID0, 1 and are generated from normal and Ch-squared dstrbutons The tests are conducted at the 5% nomnal sgnfcance level Table 5 summarzes the sze-adjusted power of CD R wth MA1, AR1 and ARMA1,1 errors under the factor model alternatve Results show that CD R performs reasonably well under the two dstrbuton scenaros especally for and T > 10 Table 6 confrms the power propertes of CD R for MA1, AR1 and ARMA1,1 errors under the SAR1 alternatve, especally for large and T DGP MA1 AR1 ARMA1, 1,T Table 5 Sze adjusted power of CD R : factor model ormal Ch-Squared otes: Ths table computes the sze adjusted power for CD R wth a factor model that allows for cross-sectonal correlaton n the errors: ut λ f t + u t u t are generated by MA1, AR1 and ARMA 1,1 defned by 9 31 ξ t σ ε t ; σ IIDχ / ε t IID0, 1 and are generated from normal and Ch-squared dstrbutons

12 conometrcs 016, 4, 44 1 of 4 DGP MA1 AR1 ARMA1, 1,T Table 6 Sze adjusted power of CD R : SAR1 model ormal Ch-Squared otes: Ths table computes the sze adjusted power for CD R wth a SAR1 model that allows for cross-sectonal correlaton n the error: ut δ05u,t + 05u +1,t + u t wth δ 04 u t are generated by MA1, AR1 and ARMA 1,1 defned by 9 31 ξ t σ ε t ; σ IIDχ / ε t IID0, 1 and are generated from normal and Ch-squared dstrbutons 5 Conclusons In ths paper, we fnd that n the large heterogeneous panel data model, LM PUY exhbts serous sze bas when there s seral correlaton n the dsturbances Whle CD P s centered at zero, t stll encounters sze dstortons caused by gnorng seral correlaton We modfy Pesaran s CD P test to account for seral correlaton of an unknown form n the error term and call t CD R Ths paper has several novel aspects: frst, an unbased and consstent estmate of the varance under the assumptons and the null of no cross-secton correlaton s proposed wthout knowng the form of seral correlaton over tme Second, the lmtng dstrbuton of the test s derved as, T n any order Thrd, t s dstrbuton free Smulatons show that the proposed test CD R successfully controls for sze dstortons wth seral correlaton n the error term It also has reasonable power under the alternatves of a factor model and a spatal auto-correlaton SAR1 model for dfferent seral correlaton specfcatons Author Contrbutons: All authors contrbuted equally to the paper Conflcts of Interest: The authors declare no conflct of nterest Appendx Ths Appendx ncludes proofs of the man results n the text The Appendx ncludes two parts: Part A ncludes some useful lemmas, whch are frequently used n the proofs of the theorems; Part B gves the proofs of all of the theorems ncluded n the paper Let us ntroduce some notaton before proceedng: For two matrces B b j and C c j, we defne B C b j c j denotes summaton over mutually-dfferent ndces, eg, summaton over { 1,, j 1, j : 1,, j 1, j are mutually dfferent} 1,,j 1,j means

13 conometrcs 016, 4, of 4 Appendx A Some Useful Lemmas Lemma A1 Let F and G be non-stochastc symmetrc and postve defnte matrces Defne r u Fu u Gu Under Assumptons 1, we have a r k [ ε Fε k] [ε Gε ] k ; b ε Fε trf ; c ε Fε trf + tr F + trf F; d trf F tr F The Proof of part a s gven by Leberman [4], and the proofs of b d are from Proposton 1 of Chen et al [10]; hence, we omt the proof here Lemma A Defne B j M j ΣM j, for any j, respectvely Under Assumptons 1 3 and the null n, we have a ˆρ j trb B j ; trb trb j b ˆρ 4 j trb B j +tr B B j ; tr B tr B j + trb c For any j 1 j, ˆρ j 1 ˆρ B j1 +tr B B j1 1/+ trb B j +tr B B j 1/ j trb j1 trb j tr B Proof Recall that the par-wse correlaton coeffcents s defned as ˆρ j v v j T v t v jt, t1 where v are the scaled resdual vectors defned by v e e e e 1/ s the OLS resdual vector from the ndvdual-specfc least squares regresson, and t s gven by e M u M σ Γε, wth M I T P X I T X X X 1 X, where M s dempotent Consder part a, ˆρ j v v j e e j e 1/ e e j e j 1/ e A j e e e, where A j e je j e j e j Then [ ] [ ] e ˆρ j ˆρ j ε j A j e e e ε j

14 conometrcs 016, 4, of 4 Snce e M σ Γε, and usng parts a and b of Lemma A1, we have Together wth the above results, we have: Consder part b, e A je e e ε j [ tr Γ M A j M Γ ] ε j Γ M j M ΓΓ M M j Γε j ε j Γ M j Γε j tr Γ M j M ΓΓ M M j Γ tr Γ M j Γ tr M j ΣM j M ΣM tr M j Σ ˆρ j tr M j ΣM j M ΣM tr M Σ tr M j Σ tr B B j tr B tr B j trγ M A j M Γ trγ Moreover, M Γ [ ] e [ ρ 4 j ρ 4 j v j A j e ε e e v j Γ ] M A j M Γε tr Γ v j M Γ [ tr Γ M A j M Γ + tr Γ M A j M Γ + tr Γ M A j M Γ Γ M A j M Γ ] Usng part a of Lemma A1, we have tr B [ tr Γ M A j M Γ ] ε j Γ M j M ΓΓ M M j Γε j ε j Γ M j Γε j Usng part c of Lemma A1, we also have ε j Γ M j M ΓΓ M M j Γε j ε j Γ M j M ΓΓ M M j Γε j trγ M j M ΓΓ M M j Γ + tr Γ M j M ΓΓ M M j Γ + tr Γ M j M ΓΓ M M j Γ Γ M j M ΓΓ M M j Γ [ ] ε j Γ M j Γε j tr B B j + tr B B j + tr Γ M j M ΓΓ M M j Γ Γ M j M ΓΓ M M j Γ + tr B B j + tr B B j Wth the fact that ε j Γ M j Γε j trb j, we obtan [ ext, we consder tr Γ M A j M Γ ] [ tr Γ M A j M Γ ] + tr B B j + tr B B j tr B j [ tr Γ M A j M Γ ] ε j Γ M j M ΓΓ M M j Γε j ε j Γ M j Γε j + tr B B j + tr B B j tr B j

15 conometrcs 016, 4, of 4 Hence, Consder part c; snce ote that v B j1 v v B j v nequalty and Hence: ˆρ 4 j tr B B j + tr B B j tr B tr B j ˆρ j1 ˆρ j ˆρ j1 ˆρ j v ˆρ j1 v ˆρ j v v B j 1 v v B j v tr B j1 trbj [ v B j 1 v ] 1/ [ v B j v ] 1/ by usng the Cauchy Schwarz v B ε j 1 v Γ M M j1 ΓΓ M j1 M Γε ε + tr B B j1 + tr B B j1 Γ M Γε tr B + tr ˆρ B B j1 ˆρ j1 + tr 1/ B B j1 + tr B B j + tr 1/ B B j j tr B j1 trbj tr B Lemma A3 Under Assumptons 1 3 and the null n, for any fxed postve number k, we have a 1 T tr Σ k O1; b 1 T trbk O1; c 1 T trb 1 B B k O1, for 1 k Proof Part a s drectly from Ba and Zhou [0]; hence, we omt t here ext, we consder part b Snce I T P X s dempotent, for any 1,, ; hence, tr B k tr [ I T P X ΣIT P X ] k [IT ] k tr P X Σ By usng the nequalty that for any postve defnte matrces A and B see Bushell and Trustrum [5] tr AB k tr A k B k, we have IT tr B k tr P X Σ k tr Σ k Usng part a, then 1 B T tr k 1 T tr Σ k O1 For part c, snce for each B l, l 1,,k, t s postve sem-defnte We also have B l Σ, l 1,,k By usng the facts that for any matrces A, B, wth A B and C postve defnte, trac trbc, we conclude that Part c holds 1 T trb 1 B B k 1 T tr Σ k O1

16 conometrcs 016, 4, of 4 Appendx B Proof of the Theorems Appendx B1 Proof of Theorem 1 Proof Snce e X 0 and ε, 1,,, are ndependent, t s easy to show that ˆρ j 0, whch further mples T n 0 ext, we consder the varance of T n var 1 ˆρ j 1 ˆρ j To calculate the above term, we have three cases to dscuss: 1 1,, j 1, j are mutually dfferent ˆρ 1 j 1 ˆρ j 0 1, j 1 j By usng Lemma A, we have ˆρ j j j 1 trb B j trb trb j ˆρ 1 j 1 ˆρ j 3 1, 1 j 1 j Snce v 1, v j1, v 1 and v j are ndependent, we have ˆρ 1 j 1 ˆρ 1 j v 1 v j1 v 1 v j 0 Hence, the above results gve us the varance of T n, whch s and Theorem 1 s proven γ var T n Appendx B Proof of Theorem 1 1 1,j tr B B j tr B tr B j, tr M j ΣM j M ΣM tr M Σ tr M j Σ Proof To prove ths theorem, we need to employ the Martngale central lmt theorem Bllngsley [6] For that purpose, we defne F 0 {φ, Ω}, F as the σ-feld generated by {ε 1, ε,, ε } for 1 Let r denote the condtonal expectaton gven fltraton F r [ 0 ] Wrte L n 1 D, wth D,1 0 More specfcally, For every, we can further show that 1 D, 1 1/ D, F, 0 v v j Hence, D, [ 1 ] s a martngale dfference sequence wth respect to F, 1 Let δ D F, By applyng the Martngale central lmt theorem, t s suffcent to show that, as, T, 1 δ p 1 var T n and 1 D 4, var T n 0

17 conometrcs 016, 4, of 4 Lemmas B1 and B prove the above condtons Hence, we can apply the Martngale central lmt theorem, and as, T, we have γ 1 T n d 0,1 Lemma B1 Under Assumptons 1 3 and the null, as, T, where δ [ D F, ] 1 δ var T n p 1, Proof To prove Lemma B1, we frst show that 1 δ vart n Then, we wll show that as, T, var /var T n 0 It s easy to show that 1 δ δ 1 1 { ]} [D F, var T n ext, we only need to show that the second condton s satsfed We frst consder the magntude of vart n From Lemma A3, we know that tr B j B tr B tr B j OT 1, whch mples var T n OT ow, consder var 1 δ Let Q j v j, then: δ [ D F, ] 1 v Q jq j v F, 1 1 ε Γ M Q j Q j M Γε ε M Γ F ΓM ε, Q j M ΓΓ M Q j tr B Therefore, we need to show the magntude of var 1 Q j M ΓΓ M Q j Rewrte Q j M ΓΓ M Q j j 1 1 j 1 v j 1 B v j and: j 1 1 j 1 v j 1 M ΓΓ M v j ext, we consder j 1 1 j 1 v j 1 B v j j 1 1 j 1 v j 1 B v j v j B v j j 1 1 j 1 j 3 1 [ ε j Γ ] M j B M j Γε j ε j Γ M j Γε j j 4 1 v j1 B v j v j3 B v j4 To calculate the magntude order of the above term, we have three cases to dscuss: tr B j B tr B j

18 conometrcs 016, 4, of 4 1 j 1 j j 3 j 4 j j 1 j j 3 j 4 ε v j B j Γ M j B M j Γε j ε v j ε j Γ M j Γε j j Γ M j B M j Γε j [ ] ε j Γ M j Γε j tr B j B + tr Bj B + tr Bj B B j B tr B j 3 + tr Bj B tr B j v j1 B v j1 v j3 B v j3 v j1 B v j1 v j3 B v j3 tr B j1 B tr tr Bj3 B B j1 tr B j3 3 j 1 j 3 j j 4 [ ] v j1 B v j v j1 B v j v j1 B v j v j B v j1 v j tr Γ M j1 B M j Γε j ε j Γ M j B M j1 Γ tr M j1 Σ ε j Γ M j Γε j tr B j B B j1 B tr B j1 tr Bj Hence, [ ] varq j ΓM Γ Q j Q j ΓM Γ Q j Q j ΓM Γ Q j Bj1 B tr Bj B + j 1 1 tr j 1,j j 1 j 1 1 j 1,j j 1 j 1 1 j 1,j j 1 tr B j1 tr Bj tr Bj B B j1 B tr tr B j B B j1 tr Bj tr B j Bj B B j1 B tr tr + + B j1 tr Bj tr B j B tr B j tr B j B tr B j It further leads to var δ var δ 1 j 1 1 j 1,j j 1 1 tr B j B tr B tr B j tr Bj B B j1 B tr B tr B j1 tr Bj By usng Lemma A3, we have var δ 1 [ 1 K O T 3 + O 1 ] T

19 conometrcs 016, 4, of 4 As, T, var 1 δ /var T n 0 Lemma B1 s proven Lemma B Under Assumptons 1 3 and the null, as, T, 1 D 4, var T n 0 Proof Rewrte [ ] { [ ]} D 4, D 4, F, v Q jq j v F, tr Γ M Q j Q j M Γ + trγ M Q j Q j M Γ + tr Γ M Q j Q j M Γ Γ M Q j Q j M Γ tr B By usng the results from Lemma B1, we have [ ] tr Γ M Q j Q j M Γ Q j B Q j + tr j 1 1 j 3 1,j 3 j 1 j 1 1 j 1,j j 1 Bj1 B tr Bj3 B tr B j1 tr Bj3 tr Bj B B j1 B tr B j1 tr Bj tr B j B tr B j Snce and thus 1 trγ M Q j Q j M Γ tr Γ M Q j Q j M Γ tr Γ M Q j Q j M Γ Γ M Q j Q j M Γ tr Γ M Q j Q j M Γ, D 4, + + K 1 K 1 K 1 K T O 1 j 1 1 j 1,j j 1 1 tr B j B tr B tr B j 1 j 1 1 j 1,j j 1 1 T tr Bj1 B tr Bj B tr B tr B j1 tr Bj3 tr Bj B B j1 B tr B tr B j1 tr Bj Hence, 1 D4, var T n Appendx B3 Proof of Theorem 3 0, as, T Lemma B s proven Proof We want to show ˆγ γ and ˆγ γ o p 1 ote that

20 conometrcs 016, 4, 44 0 of 4 ˆγ v,j,j a 1 + a + a 3 + a 4, say v j v,j v j v v,j v v j v v j v j v,j v v,j v j v + v v,j v j v,j It s easy to show that the frst term a 1 γ, and a 0,,3,4 Therefore, we prove the frst part By usng Lemma A3 and Theorem 1, we have γ OT 1 Hence, to prove ˆγ γ o p 1, we only need to show that vara 1 o p T and a o p γ, for,3,4 Let us consder vara 1 vara 1 a 1 γ ˆρ j 1 1 j 1 1 j ρ 1 j 1 ρ j 4 tr B j B tr B tr B j 1 tr B j B tr B tr B j ow, we only consder the term 1 j 1 1 j 1 ρ 1 j 1 ρ j There are three cases for ths term, and Lemma A s used frequently: 1 1,, j 1 and j are mutually dfferent 1, j 1 j and 1 j 1 3 1, 1 j 1 j ρ 1 j1 ρ j tr B 1 B j1 tr B B j tr B 1 tr B1 tr B tr B O p 1 T ρ 4 j tr B B j + tr B B j 1 tr B tr O p B j T + tr ρ B B j1 ρ j1 + tr 1/ B B j1 + tr B B j + tr 1/ B B j j tr B j1 trbj tr B 1 O p T From the above results, we have 1 vara 1 O p T

21 conometrcs 016, 4, 44 1 of 4 p Hence a 1 γ 1 Consder the second term a, whch s equal to 1 v v jv j v τ The frst term,j,τ of v v jv j τ v s,j,τ v v j 1 v j1 v τ v v j v j v τ,j 1,j,τ,j 1,j,τ O trm j M τ ΣM τ M ΣM M j1 ΣM j1 M j Σ 4 T 3, tr B τ tr B j trbj1 tr B by usng Lemmas A and A3 By usng part c of Lemma A3, the second term of v v jv j τ v s,j,τ v v jv j v τ O p 3 T,j,τ Hence, a O p 1 T 3/ + O p 3/ T 1, whch further mples a o p γ Snce a a 3, a 3 o p γ Consder a 4 ; t can be dvded nto two terms 1 1 v v τv j v τ and,j,τ 1 1 v v τ 1 v j v τ,j,τ 1,τ It s easy to show that the former term s O p 1 a, then t s op γ We only need to consder the latter term v v τ 1 v j v τ,j,τ 1,τ,j,τ 1,τ v v τ 1 v j v τ [ v v ] τ 1 v j v τ O 4 T,,j,τ 1,τ by usng Lemma A A3 Hence, the latter term s O p T 1 The above results together lead to a 4 o p γ The frst part of Theorem 3 holds; the second part of Theorem 3 s drectly derved by usng Theorem and the frst part of Theorem 3 Appendx B4 Proof of Theorem 4 Proof The OLS resduals under the local alternatve are defned by M u σ M Γε + M Fλ, thus 1/ T n 1 1/ 1 M Γε + M Fλ M j Γε j + M j Fλ j M Γε + M Fλ M j Γε j + M j Fλ j M Γε + M Fλ M j Γε j + M j Fλ j ε Γ M Γε + ε Γ M Fλ + λ 1/ 1/ F M Fλ ε j Γ M j Γε j + ε j Γ M j Fλ j + λ j F M j Fλ j Consder the denomnator ote that ε Γ M Γε trσm + [trσm ] + tr ΣM ΣM O p T, whch lead to ε Γ M Γε OT Consder the term ε Γ M Fλ Snce ε Γ M Fλ ε Γ Fλ ε Γ X X X 1 X Fλ

22 conometrcs 016, 4, 44 of 4 From Assumpton 4, we have X F O p T 1/, ε Γ F O p T 1/ and ε Γ X O p T 1/, whch lead to ε Γ Fλ O p T 1/4 1/ and ε Γ X X X 1 X Fλ O p T 1/4 1/ Hence, ε Γ M Fλ o p ε Γ M Γε Smlarly, by usng Assumpton 4, we also have λ F M Fλ o p ε Γ M Γε From the above results, we further have It results n 1/ T n 1 where T n1 T n3 ε Γ M Γε + ε Γ M Fλ + λ F M Fλ 1 + o p 1ε Γ M Γε T n1 + T n + T n3 + T n4, 1 1 1/ ε Γ M M j Γε j + ε Γ M M j Fλ j + λ F M M j Γε j + λ F M M j Fλ j 1 + op 1ε Γ M Γε 1/ 1 + o p 1ε j Γ M j Γε j 1/ ε Γ M M j Γε j D ; T j n 1/ λ F M M j Γε j D and T j n / 1/ λ F M M j Fλ j D j D j 1 + o1ε Γ M Γε 1/ 1 + o1ε j Γ M j Γε j 1/ From Theorem, γ 1 T n1 d 0, 1 From Theorem 1, T n1 O p T 1/ Consder T n We observe that T n 0 and T n 1 1 Consder the term ε Γ M M j Fλ j ε Γ M M j Fλ j D j ε Γ M M j Fλ j + D j j 1 1 j j 1 ε Γ M M j Fλ j D j ; ε Γ M M j1 Fλ j1 λ j F M j M Γε D j1 D j wth ε Γ M M j Fλ j ε Γ Fλ j ε Γ X X X 1 X Fλ j ε Γ X j X j X j 1 X j Fλ j + ε Γ X X X 1 X X jx j X j 1 X j Fλ j Usng Assumpton 4 and under the local alternatve, we frst have ε Γ Fλ j O p T 1/4 1/ ; we then have ε ΓX X X 1 X Fλ j O p T 1/4 1/ snce ε ε ΓX X X 1 X F ΓX X 1 X X F O p 1; T T T we last have ε Γ X X X 1 X X jx j X j 1 X j Fλ j O p T 1/4 1/ Hence, ε Γ M M j Fλ j O p T 1/4 1/ Together wth the fact that D j O p T, the frst term of T n s of order O p T 3/ 1 Smlar to the proof of above, ε Γ M M j1 Fλ j1 λ j F M j M Γε O p T 1/ 1 ; wth the facts that D j1 O p T and D j O p T; the second term of T n s of order O p T 3/ Thus, T n O p T 3/4 o p T n1 Smlarly, T n3 o p T n1 Consder T n4 ote that λ F M M j Fλ j λ F Fλ j λ F X X X 1 X Fλ j λ F X j X j X j 1 X j Fλ j + λ F X X X 1 X X jx j X j 1 X j Fλ j

23 conometrcs 016, 4, 44 3 of 4 From Assumpton 4, we know that λ F Fλ j λ TI r + O p T 1/ λ j p Tλ λ j Snce F X X X 1 X F F X X 1 X X F O p 1, T T T λ F X X X 1 X Fλ j o p λ F Fλ j Smlarly, we can also show that the thrd and the fourth terms are of smaller order of the frst term Hence, λ F M M j Fλ j 1 + o p 1λ F Fλ j 1/ ote that λ F Fλ j Tλ λ j 0, and 1 λ F Fλ j D j O p T 1/ ; hence, γ 1 p T n4 O p 1 One can also show that D j tr 1/ M Σtr 1/ M j Σ 1/ Let ψ plm,t γ 1 1 ; from all of the above results, as, T, γ 1 T n1 ψ T 1/ 1 δ δ j tr 1/ M Σtr 1/ M j Σ d 0, 1 References 1 Lee, L Consstency and ffcency of Least Squares stmaton for Mxed Regressve, Spatal Autoregressve Models conom Theory 00, 18, 5 77 Andrews, DWK Cross-Secton Regresson wth Common Shocks conometrca 005, 73, Anseln, L; Bera, AK Spatal Dependence n Lnear Regresson Models wth an Introducton to Spatal conometrcs In Handbook of Appled conomc Statstcs; Ullah, A, Gles, D, ds; Marcel Dekker: ew York, Y, USA, 1998; pp Baltag, BH; Song, SH; Koh, W Testng Panel Data Regresson Models wth Spatal rror Correlaton J conom 003, 117, Breusch, TS; Pagan, AR The Lagrange Multpler Test and Its Applcaton to Model Specfcatons n conometrcs Rev con Stud 1980, 47, Johnstone, IM On the Dstrbuton of the Largest genvalue n Prncpal Components Analyss Ann Stat 001, 9, Jang, TF The Lmtng Dstrbutons of genvalues of Sample Correlaton Matrces Sankhyā 004, 66, Ledot, O; Wolf, M Some Hypothess Tests for the Covarance Matrx When the Dmenson s Large Compared to the Sample Sze Ann Stat 00, 41, Schott, JR Testng for Complete Independence n Hgh Dmensons Bometrka 005, 9, Chen, SX; Zhang, LX; Zhong, PS Tests for Hgh Dmensonal Covarance Matrces J Am Stat Assoc 010, 105, Pesaran, MH; Ullah, A; Yamagata, T A Bas-Adjusted LM Test of rror Cross-Secton Independence conom J 008, 11, Baltag, BH; Feng, Q; Kao, C A Lagrange Multpler Test for Cross-Sectonal Dependence n a Fxed ffects Panel Data Model J conom 01, 170, Baltag, BH; Feng, Q; Kao, C Testng for Sphercty n a Fxed ffects Panel Data Model conom J 011, 14, Pesaran, MH General Dagnostc Test for Cross Secton Dependence n Panels CSfo Workng Paper Seres o 19, IZA Dscusson Paper o 140 Avalable onlne: accessed on May Pesaran, MH Testng Weak Cross-Sectonal Dependence n Large Panels conom Rev 015, 34, Moscone, F; Tosett, A Revew and Comparsons of Tests of Cross-Secton Dependence n Panels J con Surv 009, 3, Sarafds, V; Wansbeek, T Cross-Sectonal Dependence n Panel Data Analyss conom Rev 01, 31,

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