A Bias-Adjusted LM Test of Error Cross Section Independence

Transcription

1 A Bas-Adjusted LM Test of Error Cross Secton Independence M. Hashem Pesaran Cambrdge Unversty & USC Takash Yamagata Cambrdge Unversty May 6 Aman Ullah Unversty of Calforna, Rversde Abstract Ths paper proposes bas-adjusted normal approxmaton versons of Lagrange multpler (NLM) test of error cross secton ndependence of Breusch and Pagan (198) n the case of panel models wth strctly exogenous regressors and normal errors. The exact mean and varance of the Lagrange multpler (LM) test statstc are provded for the purpose of the bas-adjustments, and t s shown that the proposed tests have a standard normal dstrbuton for the fxed tme seres dmenson (T ) as the cross secton dmenson (N) tends to nfnty. Importantly, the proposed bas-adjusted NLM tests are consstent even when the Pesaran s (4) CD test s nconsstent. Also alternatve bas-adjusted NLM tests, whch are consstent under local error cross secton ndependence of any fxed order p, are proposed. The fnte sample behavor of the proposed tests are nvestgated and compared to the LM, NLM, and CD tests. It s shown that the bas-adjusted NLM tests successfully control the sze, mantanng satsfactory power n panel wth exogenous regressors and normal errors, even when cross secton mean of the factor loadngs s close to zero, where the CD test has lttle power. However, t s also shown that the bas-adjusted NLM tests are not as robust as the CD test to non-normal errors and/or n the presence of weakly exogenous regressors. JEL-Classfcaton: C1, C13, C33 Keywords: Cross Secton Dependence, Spatal Dependence, LM test, Panel Model, Bas-adjusted Test Pesaran and Yamagata gratefully acknowledge the fnancal supports from the ESRC (Grant No. RES ). Ullah gratefully acknowledges the fnancal supports from the Academc Senate, UCR. Faculty of Economcs, Unversty of Cambrdge, Cambrdge, CB3 9DD, UK. mhp1@cam.ac.uk. Department of Economcs, Unversty of Calforna, Rversde, CA, 951, USA. aman.ullah@ucr.edu Faculty of Economcs, Unversty of Cambrdge, Cambrdge, CB3 9DD, UK. ty8@econ.cam.ac.uk 1

2 1 Introducton Anumberofdfferent approaches already exst for testng cross secton ndependence n panel data models. An early contrbuton s due to Moran (1948) who provdes a test of spatal ndependence n the context of a pure cross secton model. Further developments of Moran s test are revewed n Anseln (1988, 1). Ths approach depends on the choce of the spatal matrx, and may not be approprate for many panels n economcs and fnance where space s not a natural metrc for modellng of cross secton dependence. An alternatve procedure would be to use the Lagrange multpler (LM) correlaton test of Breusch and Pagan (198), whch does not requre a pror specfcaton of a spatal matrx. The LM test s based on the average of the squared par-wse correlaton coeffcents of the resduals and s applcable n the case of panel data models where the cross secton dmenson (N) s small relatve to the tme dmenson (T ), and where Zellner s (196) seemngly unrelated regresson equaton (SURE) method can be used. Pesaran (4) examnes the normal approxmaton verson of the LM test (denoted by NLM) where the mean and varance of the test ndcator s approxmated up to O T 1.The NLM test s shown to exhbt substantal sze dstortons for N large and T small, a stuaton that can frequently arse n emprcal applcatons. Ths s prmarly due to the fact that for T fxed, the mean approxmaton of the LM statstc wll not be correct, and wth N large the ncorrect centerng of the test statstc s lkely to be accentuated, resultng n sze dstortons that tend to get worse wth N. Frees (1995) has proposed a verson of the Breusch and Pagan LM test, RAV E, based on squared par-wse Spearman rank correlaton coeffcents whch s applcable to panel data models where N s large relatve to T. However, Frees only provdes the dstrbuton of the test n the case of models wth only one regressor (ntercept), and ts generalzaton for models wth addtonal explanatory varables s not known. For example, the mean of RAV E appearng n Corollary 1 and Theorem of Frees (1995) may not be vald for the models wth explanatory varables. 1 Frees (1995) also proposes tests based on average par-wse sample correlatons of the seres across the dfferent cross secton unts. Hs R AV E test statstc s based on Spearman rank correlatons, and hs C AV E test statstc s based on Pearson rank correlatons. The latter s closely related the CD test also consdered n Pesaran (4). Pesaran (4) shows that unlke the LM test statstc, the CD statstc has exactly mean zero for fxed values of T and N, under a wde class of panel data models, ncludng heterogeneous dynamc models subject to multple breaks n ther slope coeffcents and error varances, so long as the uncondtonal means of y t and x t are tme-nvarant and ther nnovatons are symmetrcally dstrbuted. However, the CD test has an mportant drawback; namely t wll lack power n certan stuatons where the populaton average par-wse correlatons s zero, although the underlyng ndvdual populaton par-wse correlatons are non-zero. Ths could arse, for example, where under the alternatve hypothess cross dependence can be characterzed as a factor model wth mean zero factor loadngs. See Pesaran (4, p.14). In ths paper we propose bas-adjusted versons of the NLM tests, whch use the exact mean (and varance) of the LM statstc n the case of panel data models wth strctly exogenous regressors and normal errors. The adjustments are obtaned usng the results n Ullah (4), so that the centerng of the LM statstc s correct for fxed T and N. We consder two versons of bas-adjusted LM tests: the mean-bas-adjusted NLM test (denoted by NLM ), and mean-varance-bas-adjusted NLM test (denoted by NLM ). The NLM test s expected to be out-performed by the NLM test n small samples, however, ts smpler computaton s an advantage. Importantly, t wll be shown that these bas-adjusted tests are consstent even when 1 In fact usng Monte Carlo experments we found that the uncorrected verson of the R AV E test tends to behave smlarly to the uncorrected verson of the Breusch and Pagan LM test when N s large for models wth explanatory varables. These results are avalable from the authors upon request.

3 the cross secton mean of the factor loadngs s near zero, under whch Pesaran s CD test s not consstent. In cases where the cross secton unts can be ordered a pror, as wth spatal observatons, the proposed LM tests mght not be suffcently powerful as they do not explot the spatal nformaton. To deal wth ths problem we also propose a generalzaton of the NLM and NLM tests, whch capture the spatal patterns. We call them NLM(p) and NLM(p) tests, that correspond to the CD(p) test proposed n Pesaran (4). The fnte sample behavor of the bas-adjusted tests s nvestgated by means of Monte Carlo experments, and compared to that of the (non-bas-adjusted) LM and NLM tests, as well as to the CD test. It wll be shown that the bas-adjusted NLM tests successfully control the sze, mantanng reasonable power n panels wth exogenous regressors and normal errors, even when cross secton mean of the factor loadngs s close to zero, where the CD test has lttle power. Also ther spatal versons perform smlarly n the case of spatal cross secton dependence. However, t s shown that the bas-adjusted NLM tests are not as robust as the CD test to non-normal errors and/or n the presence of weakly exogenous regressors. The plan of the paper s as follows. Secton presents the panel data model and the exstng tests of cross secton ndependence, and formulate the bas-adjusted tests. Secton 3 reports the results of the Monte Carlo experments. Secton 4 provdes some concludng remarks. Model and Tests Consder the followng panel data model y t = β x t + u t,for =1,,...,N; t =1,,...,T, (1) where ndexes the cross secton dmenson and t thetmeseresdmenson,x t s a k 1 vector of strctly exogenous regressors wth unty on ts frst row. The coeffcents, β,aredefned on a compact set and allowed to vary across. Foreach, u t IID(, σ u ), for all t, although they could be cross-sectonally correlated. We frst provde an over-vew of the alternatve approaches advanced n the lterature to test the cross secton ndependence of the errors..1 Breusch and Pagan s Test of Cross Secton Independence In the SURE context wth N fxed and as T, Breusch and Pagan (198) proposed a Lagrange multpler (LM) statstc for testng the null of zero cross equaton error correlatons whch s partcularly smple to compute and does not requre the system estmaton of the SURE model. The test s based on the followng LM statstc LM = T N 1 X NX =1 j=+1 ˆρ j, () where ˆρ j sthesampleestmateofthepar-wsecorrelatonoftheresduals.specfcally, ˆρ j =ˆρ j = ³ PT t=1 e t P T t=1 e te jt 1/ ³, (3) PT t=1 jt 1/ e and e t s the Ordnary Least Squares (OLS) estmate of u t defned by e t = y t ˆβ x t, (4) wth ˆβ beng the estmates of β computed usng the OLS regresson of y t on x t for each, separately. Ths LM test s generally applcable and does not requre a partcular orderng of 3

4 the cross secton unts. However, t s vald for N relatvely small and T suffcently large. In ths settng Breusch and Pagan show that under the null hypothess specfed by Cov (u t,u jt )=, for all t, 6= j, (5) the LM statstc s asymptotcally dstrbuted as ch-squared wth N(N 1)/ degrees of freedom. As t stands ths test s not applcable when N. However, notng that under the null hypothess, as T T ˆρ j d χ 1, wth ˆρ j, =1,,..,N 1, j = +1,,...,N, beng asymptotcally uncorrelated, the followng scaled verson of the LM statstc can be consdered even for N and T large: s N 1 1 X NX NLM = (T ˆρ j 1). (6) N(N 1) =1 j=+1 Under H wth T frst and then N we have: NLM d N(, 1). However, a test based on ths result s lkely to exhbt substantal sze dstortons for N large and T small, a stuaton that can frequently arse n emprcal applcatons. Ths s prmarly due to the fact that for a fnte T, E(T ˆρ j 1) wll not be correctly centered at zero, and wth N large the ncorrect centerng of the LM statstc s lkely to be accentuated, resultng n sze dstortons that tend to get worse wth N. Recently Ullah (4) provdes unfed technques to obtan the exact and approxmate moments of econometrc estmators and test statstcs. We make use of ths approach to correct for the small sample bas of the LM statstc.. Fnte Sample Adjustments To obtan the bas-adjusted NLM tests we make the followng assumptons: Assumpton 1: Foreach, the dsturbances, u t, are serally ndependent wth the mean and the varance, < σ <. Assumpton : Under the null hypothess defned by H : u t = σ ε t, where ε t IIDN(, 1) for all and t. Assumpton 3: The regressors, x t, are strctly exogenous such that E(u t X )=, for all and t where X =(x 1,...,x T ) s a T k matrx, and X X s a postve defnte matrx. Assumpton 4: T>kand the OLS resduals, e t, n (4), are not all zero. Now we ntroduce the followng dempotent matrx of rank T k, M = I T H ; H = X (X X ) 1 X, (7) such that Tr(M )=T k, wherei T s an dentty matrx of order T. Smlarly M j = I T H j s the same as M wth X replaced by X j,andtr(m j )=T k. Then we can state the followng theorem. Theorem 1 : Consder the panel data model (1), and suppose that Assumptons 1-4 hold. Then the exact mean and varance of (T k)ˆρ j are, respectvely, gven by See also Frees (1995, p.395). μ Tj = E (T k)ˆρ 1 j = T k Tr(M M j ) (8) 4

5 and where υ Tj = Var h (T k)ˆρ j =[Tr(M M j )] a 1T +Tr (M M j ) a T, (9) a 1T = a T 1 (T k), a T =3 (T k 8) (T k +)+4 (T k +)(T k ) (T k 4). (1) Proof s gven n Appendx A.. Frstly, by usng (8), the mean-bas-adjusted NLM test statstc s defned by NLM = s 1 N(N 1) N 1 X NX =1 j=+1 (T k)ˆρ j μ Tj. (11) Note that the mean of NLM statstc s exactly zero for all T and N, and t s unlkely that the ncrease n N enhances the sze dstorton of the test. However, the varance of ths test statstc s stll subject to small sample bas. Therefore, under Assumptons 1-4, wth T frst, then N, we would have (under H ) NLM d N(, 1). Next, usng (8) and (9), the mean-varance-bas-adjusted NLM test statstc s defned as s N 1 NLM X NX (T k)ˆρ j μ Tj =. (1) N(N 1) υ Tj =1 j=+1 Under Assumptons 1-4, for all T,asN we would have (under H ) NLM d N(, 1). Clearly, the NLM test s more lkely to exhbt sze dstortons as compared to the NLM test. However, t has the advantage of beng relatvely smple to compute..3 Pesaran s (4) CD Test and ts Potental Inconsstency Pesaran (4) proposed a cross secton ndependence test, s N 1 T X NX CD = N(N 1) =1 j=+1 ˆρ j, (13) and t was shown that under H,forT suffcently large, as N, CD d N(, 1). 3 Unlke the NLM test statstc, the above statstc has exactly mean zero for fxed values of T and N, under a wde class of panel data models, ncludng heterogeneous dynamc models subject to multple breaks n ther slope coeffcents and error varances, so long as the uncondtonal means of y t and x t are tme-nvarant and ther nnovatons are symmetrcally dstrbuted. We also note that the NLM and NLM test statstcs have exact means zero for fxed values of T and N as well, so long as Assumptons 1 to 4 hold. However, as ponted out n Pesaran (4), under a partcular case the CD test would be nconsstent. To see ths, we frst specfy the error structure of the model (1) under the alternatve as H 1 : u t = γ f t + ε t, (14) 3 The CD test requres less restrctve verson of Assumpton, where ε t are symmetrcally..d. dstrbuted around zero wth unt varance. 5

6 where γ IID, σ γ, wth < σ γ <, are factor loadngs, f t IID(, 1) are unobserved common effects, and we assume E ft 4 = μf4 wth < μ f4 <, ε t IIDN(, 1), and E (ε t f s )=for all, t, and s. Under H 1 Cov (u t,u jt )=E (γ ) E γ j, and the CD test statstc s centred at f E (γ )=,evenwhenγ 6=for some and f t 6=. Therefore, under the alternatves wth E(γ )=the power of the CD test would not ncrease wth N. But, the power of the LM type tests nvolves the terms Cov u t,u jt = E γ E γ j μf4 (15) whch contnue to dffer from zero even when E(γ )=. Hence, the power of LM type tests wll ncrease wth N even under alternatves wth E(γ )=. These results wll contnue to hold under mult-factor alternatves..4 Tests for Local Cross Secton Independence As shown by Pesaran (4), the power of the CD test s adversely affected when the dependence under the alternatve hypothess s spatal (local). The spatal dependence of the errors can be modelled usng the spatal weght matrx, W =(w j ), whch s appled to a partcular orderng of the cross secton unts. It s often convenent to order the cross secton unts by ther topologcal poston, so that the p th order neghbors of the th cross secton unt can be defned as the + p and the p cross secton unts. Observng ths, under the alternatve hypothess of a p th order local dependence, Pesaran (4) proposes a p th order generalzaton of the CD test defned by s à px T CD(p) = p (N p 1) N s X s=1 =1! ˆρ,+s. (16) In a smlar manner, we may propose p th order NLM, NLM,andNLM tests defned by s 1 px N s X NLM(p) = T ˆρ p (N p 1),+s 1, (17) where s NLM(p) 1 px = p (N p 1) s=1 s NLM(p) px = p (N p 1) s=1 =1 N s X =1 N s X s=1 =1 (T k)ˆρ,+s μ T,+s, (18) (T k)ˆρ,+s μ T,+s υ T,+s, (19) μ T,+s = Tr(M M +s ) h, υ T,+s =[Tr(M M +s )] a 1T +Tr (M M +s ) a T. T k 3 Fnte Sample Behavor of the Tests of Cross Secton Independence In ths secton we nvestgate the fnte sample behavor of alternatve tests of cross secton ndependence by means of Monte Carlo experments. We shall focus on our proposed basadjusted NLM tests, NLM and NLM,defned by (11) and (1), respectvely, and compare ther performance to the nave LM and NLM tests defned by () and (6), respectvely, and 6

7 the CD test defned by (13). Intally we consder the experments n panels wth exogenous regressors. In vew of the valdty of CD test n the varous cases of dynamc models, we also consder experments n the cases of statonary and unt root dynamc panels wth and wthout parameter heterogenety and structural breaks, under whch the bas-adjusted NLM tests may not be vald. Fnally, we also provde small sample evdence on alternatve tests of cross secton ndependence aganst spatal alternatves. 3.1 Expermental Desgns Intally, we consder the data generatng process (DGP) specfed as y t = α + kx x`t β + u t,=1,,...,n; t =1,,...,T, (1) `= where α IIDN (1, 1), β` IIDN (1,.4). The covarates are generated as x`t =.6x`t 1 + v`t, =1,,...,N; t = 5,...,,...,T; ` =, 3,...,k () wth x`, 51 =where v`t IIDN(, τ ` / 1.6 ), τ ` IIDχ (6) /6. The dsturbances are generated as u t = c (γ,k) (γ f t + σ ε t ),=1,,...,N; t =1,,...,T, where f t IIDN (, 1), and σ IIDχ () /. The dosyncratc errors, ε t, are generated under two dfferent schemes, () normal errors, IIDN (, 1), and () ch-squared errors, IID χ (1) 1 /. The latter s to check the robustness of the tests to non-normal errors. The values of α, x`t, σ are drawn for each =1,,...,N,andthenfxed across replcatons. Under the null hypothess we have γ =for all, and under the alternatves we consder () γ IIDU[.1,.3], () γ IIDN (,.1), where under (), the CD test s nconsstent, as shown above. In order to examne the effects of changng the number of regressors, k =, 4, 6 are consdered. Meanwhle the same average populaton explanatory power of each cross secton regresson and the same degree of error cross secton correlaton are to be mantaned for all k. To ths end, c (γ,k) s set c (γ,k) = 1.4(k 1) for γ = (k 1), for γ IIDU[.1,.3], (k 1), for γ IIDN (,.1), so that R =.5 across experments, where R = E(σ u )/Var(y t) wth σ u = Var(u t) and Var(y t )=(k 1)E(β ` )+E(σ u ). For examnng the power of the frst order cross secton ndependence tests, the DGP defned by (1) for k =but wth spatally correlated errors are consdered: u t = λ (.5u 1,t +.5u +1,t )+σ ε t, (3) wth end ponts set at u 1t = u t + ε 1t and u Nt = u N 1t + ε Nt,whereσ IIDχ () /, ε t IIDN(, 1). For ths DGP, the fnte sample performance of the spatal verson of the tests, defned by (16), (17), (18) and (19), are examned n the case of p =1, and for the values of λ =,.1 and.1. 7

8 In the case of dynamc models, followng Pesaran (4), fve specfcatons are consdered. The frst s the heterogeneous frst order autoregressve (AR(1)) panel data model: y t = μ (1 β )+β y,t 1 + u t, (4) u t = γ f t + σ t ε t,=1,,...,n; t = 5, 49,...,T, wth y, 51 =. The dosyncratc errors, ε t, are generated under two dfferent schemes as above, () normal errors, IIDN(, 1), and () ch-squared errors, IID χ (1) 1 /. Here we focus on the heterogeneous slope experments where β IIDU[, 1). Thefxed effects, μ, are drawn as ε + η,wthη IIDN(1, ), thus allowng for the possblty of correlatons between fxed effects and the ntal values, y. γ, σ t = σ,andf t are generated n the same manner as specfed for the DGP wth exogenous regressors. The parameters η, β and σ t are fxed across replcatons. For examnng the emprcal sze of the tests n the case of structural break(s), two specfcatons are consdered. The frst dynamc DGP s subject to sngle break, specfed as (4) except μ IIDN (1, 1), β t = β t =.6 for t = 5,...,T/, β t =.8 for t = T/+1,...,T; σ t = σ t = 1.5 for t = 5,...,T/, σ t =1for t = T/+1,...,T,andε t IIDN (, 1). The second dynamc DGP s subject to multple breaks, specfed as (4) except β t =.5 for t = 5,..., and all, β t IIDU[, 1) for t =1,...,T, =1,...,N; σ t IIDχ ()/ for t = 5,...,T, =1,...,N. For both desgns, the frst 5 observatons are dscarded. Fnally, the DGP subject to unt root, whch s specfed as (4) except β t = β =1for all and t, σ t IIDχ ()/, are consdered. The test statstcs are computed usng the OLS resduals from the ndvdual regressons. For all experments the combnatons of sample szes N = 1,, 3, 5, 1, and T =, 3, 5, 1 are consdered. The nomnal sze of the tests s set at the 5% sgnfcance level. All experments are based on replcatons. 3. Monte Carlo Outcomes Table 1 reports the sze of the tests for the DGP wth dfferent number of exogenous regressors (k =, 4, 6) and normal errors. As shown n Pesaran (4), the CD test has the correct sze, and the LM and NLM tests severely over-reject the null partcularly for N T.Incontrastthe adjusted versons of the NLM test, partcularly the NLM verson defned by (1), successfully controls the sze for all combnatons of N and T, except when both k and N are large and T small. However, the NLM verson of the test defned by (11), tends to under-reject for small T, and such a tendency s accentuated as k s ncreased. In the case of γ IIDU[.1,.3], whose results are reported n Table, the bas-adjusted NLM tests and the CD test seem to have reasonable power. In the case of γ IIDN (,.1), whose results are reported n Table 3, as theory predcts the CD test has lttle power. The power of CD test ncreases wth T very slowly, but t does not ncrease wth N for gven T. Ths s because the sample average of factor loadngs for fnte N can be dfferent from zero far enough for the test to reject the null, for some replcatons, and the precson of ths happenng ncreases as T rses. On the other hand, the NLM and NLM tests mantan reasonable power under the same desgn. Overall, the powers of both the CD and bas-adjusted NLM tests ncrease faster wth N than T,andt seems that the number of regressors does not affect the power of these tests much (for the same average explanatory power and error cross secton dependence). The results for the case wth IID χ (1) 1 / errors are gven n Table 4, and show that the bas-adjusted NLM tests are generally not as robust to non-normal errors as the CD test. They tend to over-reject (moderately) for all combnatons of N and T. Table 5 summarzes the results of the spatal frst order tests. Interestngly, the szes of the NLM(1) test are now closer to ther nomnal levels, except when N s much larger than T.The NLM(1) test defned by (19), successfully controls the sze, and the NLM(1) test defned by 8

9 (18), tends to slghtly under-reject. The CD(1) test has the correct sze, as shown n Pesaran (4). All of the bas-adjusted NLM(1) tests and CD(1) test seem to have reasonable power under the alternatves defned by λ = ±.1. Tables 6 to 1 provde the results for the varous dynamc DGPs. For all experments, the CD test has the correct sze and the LM and NLM tests severely over-reject the null when N T. Ths s to be expected, as dscussed above, snce for small T relatve to N, themean approxmaton of ˆρ j 1 wll not be correct, and wth N large the ncorrect centerng of the test ndcator s lkely to be accentuated, resultng n sze dstortons that tend to get worse wth N. Unlke n the case of DGP wth exogenous regressors and normal errors, n the case of heterogenous dynamc AR(1) specfcatons wth IIDN (, 1) errors (Table 6), the bas-adjusted NLM tests tend to over-reject when N s much larger than T. Wth respect to the power, as was n the case of DGP wth exogenous regressors, the CD test has lttle power n the case of γ j IIDN (,.1). The results for IID χ (1) 1 / errors (Table 7) s smlar to those n Table 4. For the DGP wth a sngle structural break (Table 8), the bas-adjusted NLM tests reject the null too often. For example, when N =, the estmated sze of the NLM test s 1% for all T. In the case of multple structural breaks (Table 9), the bas-adjusted NLM tests tend to over-reject, especally for N T. Fnally, n the case of models wth unt roots (Table 1) the bas-adjusted NLM tests also tend to over-reject, wth the extent of over-rejecton ncreasng wth N. 4 Concludng Remarks Ths paper has proposed bas-adjusted normal approxmaton verson of the LM test (NLM) of cross secton ndependence. For the bas-adjustment, we derved the exact mean and varance of the test ndcator of the LM statstc n the case of the model wth strctly exogenous regressors and normal errors, based on the work n Ullah (4), so that the centerng of the LM statstc s correct for fxed T and large N. Importantly, the proposed bas-adjusted NLM tests are consstent even when the Pesaran s (4) CD test s nconsstent. Small sample evdence based on Monte Carlo experments suggests that the bas-adjusted NLM tests successfully control the sze, mantanng reasonable power n panels wth exogenous regressors and normal errors, even when cross secton mean of the factor loadngs s close to zero, where the CD test has lttle power. Also ther spatal versons perform smlarly n the case of spatal cross secton dependence. However, t s shown that the bas-adjusted NLM tests are not as robust as the CD test to non-normal errors and/or n the presence of weakly exogenous regressors. Clearly, t would be worth dervng the mean and varance of the LM test statstc n the case of dynamc models, and n the case where the errors are non-normal. Ths wll be the subject of future research. 9

10 A Appendx A.1 Evaluaton of the Frst Two Dervatves of E (W r ) Let us consder a quadratc form W = u Mu, wherethet 1 vector u N(μ, I T ) and M s an dempotent matrx of rank m T. Then W s dstrbuted as a non-central ch-squared dstrbuton wth the non-centralty parameter θ = μ Mμ/. When μ =, henceθ =, W s dstrbuted as a central ch-square dstrbuton. In what follows we evaluate de W r and dd E W r,whered = μ + /μ and r =1,,... Now d E W r = dd E W r = μ + μ E W r = μ E W r + μ + μ E W r + μ = E W r + μμ E W r + + θ E W r Mμμ M, μ E W r θ E W r θ E W r μ M, μμ M + Mμμ + M whereweuse μ E W r = θ E W r Mμ, (A.) μμ E W r = μ θ E W r μ M = θ E W r Mμμ M + θ E W r M. Frst we note from Ullah (4, p.193) that E W r = 1 Γ m r + r e θ Γ θ m +,whenθ6= (A.3)! Further, for s =1,,..., = = 1 Γ m r r Γ 1 =,whenθ6=. m (m ) (m 4)... (m r) (A.1) s θ s EW r = ( 1) s 1 Γ(r + s) e θ Γ m r + r Γ (r) Γ θ,whenθ6= (A.4) m +1s +! = ( 1) s 1 Γ(r + s) Γ m r r Γ (r) Γ,whenθ =. m + s Substtutng r =1, and s =1, n (A.3) and (A.4), and usng these results n (A.1) we get d E W r and dd E W r for θ 6=.Whenμ =, henceθ =,weobtan d E W r =, dd E W r = E W r + θ E W r M, (A.5) where E W r /θ s gven by (A.4) for s =1and θ =. = A. Proof of Theorem 1 From (3) and (4) ˆρ j = (u M M j u j ) (u Mu) u j Mjuj (A.6) = u M M ju ju jm jm u (u M u ) u j M ju j = u A j u u M u wth A j = M M j u j u jm j M u j M ju j, where M j and M are as defnedn(7),u N(, I T ), u j N(, I T ). Takng the expectatons on both sdes of (A.6) we can wrte E ˆρ j = Euj E ˆρ j u j, [A.1]

11 n whch E ˆρ u j u j = E A j u u M (A.7) u = E u A j u W 1 = d A j d E W 1 = Tr Aj d d E W 1 = Tr(A j ) E W 1 + μ A jμ E W 1 + E W 1 θ μ M A j μ + μ A jm μ + Tr(A j M ) + E W 1 μ θ M A j M μ, where we use (A.1), W = u M u, d = μ + /μ, the thrd equalty follows by usng the results n Ullah (4, (.8)), and θ = μ M μ /. Now But for μ = E ˆρ j E ˆρ j = E W 1 + E W 1 θ + θ u j M j M M j u j E u j M ju j + μ E (A j) μ E W 1 u μ j M j M M j u j E (A j ) μ + E u j M ju j E W 1 μ E (A j) μ. = E W 1 u j M j M M j u j E u j M + E W 1 u j M j M M j u j E ju j θ u j M ju j = E W 1 + E W 1 u j M jm M ju j E. θ u j Mjuj (A.8) (A.9) Now, wrtng B j = M j M M j,w j = u jm j u j, θ j = μ j M jμ j / and usng (A.7), u j B ju j E u j M = Tr(B j ) E W 1 j + μ jb j μ j E W 1 j ju j + E W 1 j μ θ jm jb jμ j + Tr(B jm j) j + E W 1 μ j M jb j M j μ j, θ j whch can be wrtten for μ j = as u j B ju j E = E W 1 u j Mjuj j + E W 1 j Tr(B j ), Tr(B j )=Tr(M M j ). θ j (A.1) (A.11) Substtutng (A.11) n (A.9) we get E ˆρ j = E W 1 + E W 1 E W 1 j θ 1 = m Tr(M M j ) m (m ) 1 = m Tr(MMj), and E mˆρ 1 j = m Tr(MMj), where m = T k and we use the results (A.3) and (A.4) for r =1and s =1. Next we consder ˆρ 4 j = u A j u / u M u, and takng the expectatons on both sdes of (A.14) we get E ˆρ 4 j = Euj E 4 ˆρ j u j, n whch usng Ullah (4, (.8)) E ˆρ 4 u j u j = E A ju W + E W 1 j Tr(M M j) θ j (A.1) (A.13) (A.14) (A.15) [A.]

12 = d A jd E W = d A j d d A j d E W = d A j d Tr Aj d d E W = Tr A jd d c(μ ), where c(μ )=Tr A j d d E W c(μ ) = Tr(A j) E W + μ A jμ E W + θ, and usng (A.7) (see also Ullah (4, Chapter )), E W μ A jμ. + E W θ In order to evaluate the term n the last equaltes of (A.15) we note that d c (μ )= μ + c (μ ), d d c (μ )= μ + μ μ c(μ )+ μ A jμ + Tr(A j) c (μ ) μ μ = μ μ c (μ )+μ c (μ μ )+c(μ )+ c (μ μ ) μ + c (μ μ μ ). Usng (A.18) n (A.15) we then get Tr A j d d c(μ ) = μ A j μ c (μ )+Tr(A j ) c (μ ) + Tr A jμ μ c (μ ) + Tr A j c (μ μ ) μ +Tr A j c (μ μ μ ), n whch, usng (A.16), we can verfy that (A.16) (A.17) (A.18) (A.19) where and where c(μ ) = Tr(A j ) μ + θ E W M μ θ +A j μ EW + E W M μ μ A j μ + Tr(A j ) + E W θ +(A jμ ) θ (4A jμ )+ 3 E W θ 3 E W θ E W M μ μ A jμ = a 1 (θ)(a j μ )+a (θ) μ A j μ M μ + a 3 (θ)tr(a j ) M μ, a 1 (θ) = E W + E W + θ a (θ) = a 3 (θ) = E W + θ θ + E W θ θ E W E W, θ + 3 E W, θ 3 E W, M μ μ A jμ (A.) c (μ μ μ ) = A ja 1 (θ)+m μ μ A ja (θ) (A.1) + A j μ μ M + μ A jμ M a (θ) + μ A jμ Mμ μ M a 4 (θ) +Tr(A j ) M a 3 (θ)+m μμ M Tr(A j ) a 5 (θ), a 4(θ) = θ a 5(θ) = θ E W 3 + E W θ θ 3 E W 4 + E W. θ 4 E W, [A.3]

13 In order to evaluate (A.19) for μ = we frst note that c () = Tr(A j ) E W + E W, θ c (μ μ ) =, c (μ μ μ ) = A j E W + E W + θ θ +Tr(A j ) M E W + θ θ Therefore, for μ =, we can smplfy Tr A j d d c(μ ) = [Tr(A j )] +Tr A j = 3[Tr(A j )] E W + E W + E W θ E W E W E W θ + θ. + θ E W, E W (A.) (A.3) because [Tr(A j)] = Tr Aj. Further, usng (A.3), (A.3) and (A.4) n (A.15) we get, for μ =, E ˆρ 4 j = 3E [Tr(A j)] a 1m (A.4) = 3 [Tr(M M j)] +Tr (M M j) a 1m, where E [Tr(A j )] = u j M jm M ju j E u j M ju j = [Tr(M M j )] +Tr (M M j ) a 1m, and Next, usng (A.1) and (A.4), we get V ˆρ j a 1m = (m 8)(m +)+4 (m +)m (m ) (m 4). = E ˆρ 4 j E ˆρ j (A.5) = [Tr(M M j)] 3a 1m 1 +6Tr (M M j) a m 1m, 4 whch gves where V mˆρ j =[Tr(M M j )] b m 1 +b m m Tr (M M j ), (m 8)(m +)+4 b m =3 =3m a 1m. (m +)(m ) (m 4) [A.4]

14 References Anseln, L., (1988). Spatal Econometrcs: Methods and Models. Dorddrecht: Kluwer Academc Publshers. Anseln, L., (1). Spatal econometrcs, In: B. Baltag (ed.) Econometrcs, Oxford: Blackwell. A companon to Theoretcal Breusch, T.S., Pagan, A.R., (198). The Lagrange multpler test and ts applcatons to model specfcaton tests n econometrcs. Revew of Economc Studes 47, Frees, E.W., (1995). Assessng cross-sectonal correlaton n panel data. Journal of Econometrcs 69, Moran, P.A.P., (1948). The nterpretaton of statstcal maps. Bometrka 35, Pesaran, M.H., (4). General dagnostc tests for cross secton dependence n panels. CESfo Workng Papers No.133. Ullah, A., (4). Fnte Sample Econometrcs. New York: Oxford Unversty Press. Zellner, A. (196). An effcency method of estmatng seemngly unrelated regresson equatons and tests for aggregaton bas. Journal of the Amercan Statstcal Assocaton 57, [R.1]

15 Table 1 Sze of Cross Secton Independence Tests wth Exogenous Regressors, γ = γ = Normal Errors wth Dfferent Number of Regressors (k) k = k =4 k =6 (T,N) LM NLM NLM* NLM** CD Notes: Data are generated as y t = α + k `= x`tβ` + u t, u t = c (γ,k) (γ f t + σ ε t ), =1,,...,N, t =1,,...,T,whereα IIDN (1, 1), wthx`t =.6x`t 1 + v`t, ` =, 3,..., k, = 1,,..., N, t = 5, 49,...,T, x`, 51 =, where v`t IIDN(, τ `/ 1.6 ), τ ` IIDχ (6) /6. β` IIDN(1,.4), f t IIDN(, 1), σ IIDχ () /, and ε t IIDN (, 1). α, x`t, σ are fxed across replcatons. c (γ,k) s chosen so that R = E(σ u)/var(y t ) =.5 wth σ u = Var(u t ) and Var(y t) =(k 1)E(β `) +E(σ u). LM s Breusch-Pagan (198) LM test, NLM s normal approxmaton verson of LM test, NLM and NLM are mean-adjusted and mean-varance-adjusted LM tests whch are proposed, respectvely, CD s Pesaran s (4) CD test, LM test s based on χ N(N 1)/ dstrbuton. NLM, NLM,NLM and CDtestsarebasedontwo-sdedN(, 1) test. All tests are conducted at 5% nomnal level. All experments are based on, replcatons. [T.1]

16 Table Power of Cross Secton Independence Tests wth Exogenous Regressors, γ IIDU[.1,.3] Normal Errors wth Dfferent Number of Regressors (k) k = k =4 k =6 (T,N) LM NLM NLM* NLM** CD Notes: The desgn s the same as that of Table 1 except γ IIDU[.1,.3]. [T.]

17 Table 3 Power of Cross Secton Independence Tests wth Exogenous Regressors, γ IIDN(,.1) Normal Errors wth Dfferent Number of Regressors (k) k = k =4 k =6 (T,N) LM NLM NLM* NLM** CD Notes: The desgn s the same as that of Table 1 except γ IIDN(,.1). [T.3]

18 Table 4 Sze of Cross Secton Independence Tests wth Exogenous Regressors, γ = γ = Non-normal Errors wth Dfferent Number of Regressors (k) k = k =4 k =6 (T,N) LM NLM NLM* NLM** CD Notes: The desgn s the same as that of Table 1 except ε t IID χ (1) 1 /. [T.4]

19 Table 5 Sze and Power of Frst Order Cross Secton Independence Tests wth Exogenous Regressors, Spatally Correlated Errors λ = λ =.1 λ =.1 (T,N) NLM(1) NLM(1)* NLM(1)** CD(1) Notes: The desgn s the same as that of Table 1 for k =, except errors are spatally correlated such that u t = λ (.5u 1,t +.5u +1,t )+σ ε t,wthendpontssetat u 1t = u t + ε 1t and u Nt = u N 1t + ε Nt. [T.5]

20 Table 6 Sze and Power of Cross Secton Independence Tests, Heterogeneous AR(1) Specfcaton wth Normal Errors γ = γ = γ IIDU[.1,.3] γ IIDN(,.1) (T,N) LM NLM NLM* NLM** CD Notes: See notes to Table1. The DGP s specfed as y t = μ (1 β )+β y,t 1 + u t,u t = γ f t + σ t ε t, =1,,..., N; t = 5,..., T,whereβ IIDU[, 1), μ ε + η, η IIDN (1, ), f IIDN (, 1), σ t = σ IIDχ () /, andε t IIDN (, 1). η, β and σ t are fxed across replcatons. y, 51 =and the frst 5 observatons are dscarded. [T.6]

21 Table 7 Table 8 Sze of Cross Secton Independence Tests Sze of Cross Secton Independence Tests Heterogeneous AR(1) wth Non-normal Errors DGP Subject to a Sngle Break γ = γ = γ = γ = (T,N) (T,N) LM LM NLM NLM NLM* NLM* NLM** NLM** CD CD Notes: See the notes to Table 6. The desgn s the same Notes: The DGP s specfed as y t μ = β t (y t 1 μ )+u t, as that of Table 6 except ε t IID χ (1) 1 /. u t = γ f t + σ t ε t,whereμ IIDN (1, 1); β t = β t =.6 and σ t = σ t = 1.5 for t = 5,...,T/; β t =.8 and σ t =1 for t = T/+1,..., T ; ε t IIDN (, 1). y, 51 =and the frst 5 observatons are dscarded. See also the notes to Table 6. [T.7]

22 Table9 Table1 Sze of Cross Secton Independence Tests Sze of Cross Secton Independence Tests DGP Subject to Multple Structural Break DGP Subject to Unt Root γ = γ = γ = γ = (T,N) (T,N) LM LM NLM NLM NLM* NLM* NLM** NLM** CD CD Notes: The desgn s the same as that of Table 8, Notes: The DGP s the same as that of Table 1 except β t =.5 for t = 5,..., and β t IIDU[, 1) except β t = β =1for all and t. See also the notes for t =1,..., T, =1,..., N; σ t IIDχ ()/ for to Table 6. t = 5,..., T, =1,...,N. See also the notes to Table 6. [T.8]