ABias-AdjustedLMTestofErrorCrossSectionIndependence

Transcription

1 ABas-AdjustedLMTestofErrorCrossSectonIndependence M.HashemPesaran Cambrdge Unversty& USC TakashYamagata Cambrdge Unversty AmanUllah Unversty of Calforna, Rversde Frst Verson May 006, Ths Verson July 007 Abstract Ths paper proposes a bas-adjusted verson of Breusch and Pagan (1980 Lagrange multpler(lm test statstc of error cross secton ndependence, n the case of panel models wth strctly exogenous regressors and normal errors. The exact mean and varance of the test ndcator of the Lagrange multpler(lm test statstc are provded for the purpose of the bas-adjustments. It s shown that the centerng of the LM statstc s correct for fxed T and N. Importantly, the proposed bas-adjusted LM test s consstent even when the Pesaran s(004 CD test s nconsstent. Also an alternatve bas-adjusted LM test, whch s consstent under local error cross secton dependence of any fxed order p, s proposed. The fnte sample behavor of the proposed tests are nvestgated and compared to that of thelmandcdtests. Itsshownthatthebas-adjustedLMtestssuccessfullycontrolthe sze, mantanng satsfactory power n panel wth exogenous regressors and normal errors. However,tsalsoshownthatthebas-adjustedLMtestsnotasrobustastheCDtestto 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 WewouldlketothankthepartcpantsofsemnaratUSCfortherhelpfulcomments. Wearepartcularly grateful to two anonymous referees for ther suggestons and useful comments. 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. FacultyofEconomcs,UnverstyofCambrdge,Cambrdge,CB39DD,UK.mhp1@cam.ac.uk. DepartmentofEconomcs,UnverstyofCalforna,Rversde,CA,951,USA.aman.ullah@ucr.edu CFAP,JudgeBusnessSchool,UnverstyofCambrdge,Cambrdge,CB1AG,UK.ty8@cam.ac.uk 1

2 1 Introducton A number of dfferent 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, 001. Ths approach depends on the choce of the spatal matrx,andmaynotbeapproprateformanypanelsneconomcsandfnancewherespaces 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 (1980, whch doesnotrequreapror specfcatonofaspatalmatrx. TheLMtestsbasedontheaverage 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 beused. ButtheLMtestexhbtsubstantal szedstorton when N slargerelatvetot, a stuaton frequently encountered n emprcal applcatons. (Pesaran, 004. Frees (1995 has proposed a verson of the Breusch and Pagan LM test, RAVE, based on squared par-wse Spearman rank correlaton coeffcents whch s applcable to panel data modelswheren slargerelatvetot. However,Freesonlyprovdesthedstrbutonofthetestn the case of models wth only one regressor(ntercept, and ts generalzaton for models wth addtonalexplanatoryvarabless notknown. Forexample, themeanofrave appearngn Corollary1andTheoremofFrees(1995maynotbevaldforthemodelswthexplanatory varables. 1 Frees (1995 also proposes tests based on averagepar-wse sample correlatons of theseresacrossthedfferentcrosssectonunts. HsR AVE teststatstcsbasedonspearman rankcorrelatons,andhsc AVE teststatstcsbasedonpearsonrankcorrelatons. Thelatter s closely related the CD test also consdered n Pesaran(004. Pesaran(004 shows that unlke the LM test statstc, the CD statstc has exactly mean zeroforfxedvaluesoft andn,underawdeclassofpaneldatamodels,ncludngheterogeneous dynamc models subject to multple breaks n ther slope coeffcents and error varances, solongastheuncondtonalmeansofy t andx t aretme-nvarantandthernnovatonsare 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(pesaran, 004, p.14. Inthspaperweproposebas-adjustedversonsoftheLMtests,whchusetheexactmean 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(004, sothatthecenterngofthelmstatstcscorrectforfxedt andn. Importantly,twllbe shown that the bas-adjusted test s consstent even when 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, theproposedlmtestsmghtnotbesuffcentlypowerfulastheydonotexplotthespatalnformaton. To deal wth ths problem, followng to Pesaran(004, we also propose a generalzaton of the bas-adjusted LM tests, whch capture the spatal patterns. The fnte sample behavor of the bas-adjusted LM test s nvestgated by means of Monte Carlo experments, and compared to that of the (non-bas-adjusted LM test, as well as to the CD test. It wll be shown that the bas-adjusted LM test successfully controls the sze, mantanng reasonable power n panels wth exogenous regressors and normal errors, even when 1 In fact usng Monte Carlo experments we found that the uncorrected verson of the R AVE test tends to behavesmlarlytotheuncorrectedversonofthebreuschandpaganlm testwhenn slargeformodelswth explanatory varables. These results are avalable from the authors upon request.

3 crosssectonmeanofthefactorloadngssclosetozero,wherethecdtesthaslttlepower. Also ther spatal versons perform smlarly n the case of spatal cross secton dependence. However, tsshownthatthebas-adjustedlmtestsnotasrobustasthecdtesttonon-normalerrors and/or n the presence of weakly exogenous regressors. Theplanofthepapersasfollows. SectonpresentsthepaneldatamodelandtheBreusch and Pagan LM test and formulate the bas-adjusted LM tests. Secton 3 dscuses Pesaran s (004 CD test and ts potental nconsstency. Secton 4 reports the results of the Monte Carlo experments. Secton 5 provdes some concludng remarks. ModelandTests Consder the followng panel data model y t =β x t+u t,for=1,,...,n;t=1,,...,t, (1 wherendexesthecrosssectondmensonandtthetmeseresdmenson,x t sak 1vector ofstrctlyexogenousregressorswthuntyontsfrstrow. Thecoeffcents,β,aredefnedon acompactsetandallowedtovaryacross. Foreach,u t IIDN(0,σ u,forallt,although they could be cross-sectonally correlated. We frst revew the Breusch and Pagan(1980 test..1 Breusch and Pagan s Test of Cross Secton Independence In the SURE context wth N fxed and as T, Breusch and Pagan (1980 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. ThetestsbasedonthefollowngLMstatstc LM =T N 1 N =1 j=+1 ˆρ j, ( whereˆρ j sthesampleestmateofthepar-wsecorrelatonoftheresduals. Specfcally, ˆρ j =ˆρ j = ( T t=1 e t T t=1 e te jt 1/ ( T 1/, (3 t=1 jt e ande t stheordnaryleastsquares(olsestmateofu t defnedby e t =y t ˆβ x t, (4 wth ˆβ bengtheestmatesofβ computedusngtheolsregressonofy t onx t foreach, separately. Ths LM test s generally applcable and does not requre a partcular orderng of thecross sectonunts. Inths settng whereu t IIDN(0,σ u tcanbeshown thatunder the null hypothess H 0 : Cov(u t,u jt =0, foralltand j, (5 the LM statstc s asymptotcally dstrbuted as ch-squared wth N(N 1/ degrees of freedom. However,theLMtestslkelytoexhbtsubstantalszedstortonsforN largeandt small,a stuaton that can frequently arse n emprcal applcatons. Recently Ullah (004 provdes unfed technques to obtan the exact and approxmate moments of econometrc estmators and test statstcs. We make use of ths approach to correct forthesmallsamplebasofthelmstatstc. 3

4 . Fnte Sample Adjustments To obtan the bas-adjusted LM test we make the followng assumptons: Assumpton1: Foreach,thedsturbances,u t,areserallyndependentwththemean0 andthevarance,0<σ <. Assumpton : Under thenullhypothess defnedby H 0 :u t =σ ε t, ε t IIDN(0,1 forallandt. Assumpton3: Theregressors,x t,arestrctlyexogenoussuchthate(u t X =0,forall andt,wherex =(x 1,...,x T,and X X sapostvedefntematrx. NowwentroducethefollowngdempotentmatrxofrankT k, M =I T H ; H =X (X X 1 X, (6 suchthattr(m =T k,wherei T sandenttymatrxofordert. SmlarlyM j =I T H j s thesameasm wthx replacedbyx j,andtr(m j =T k. Thenwecanstatethefollowng theorem. Theorem 1 : Consder the panel data model (1, and suppose that Assumptons 1-3 hold. Thentheexactmeanandvaranceof(T kˆρ j are,respectvely,gvenby and where µ Tj =E (T kˆρ 1 j = T k TrE(M M j (7 { υ Tj =Var (T kˆρ j ={TrE(M M j } a 1T +Tr E (M M j } a T, (8 a 1T =a T 1 (T k, a (T k 8(T k++4 T =3. (9 (T k+(t k (T k 4 Proof s gven n Appendx A.. Usng(7 and(8, the bas-adjusted LM test statstc s defned as LM adj = N(N 1 N 1 N =1 j=+1 (T kˆρ j µ Tj υ Tj. (10 UnderAssumptons1-3,andassumngthatH 0 defnedby(5holds,frstt thenn we have LM adj d N(0,1. (11 Remark ForafxedT (>k+8,tseaslyseenthatlm adj hasexactlymeanzero,unlke thelmtest. However,evenunderthenormaltyassumpton,thecovarancebetween T kˆρ j and T kˆρ k forj k doesnotdsappearunlesst. Therefore,for(11,thesequental asymptotc,wheret frstfollowedbyn,srequred..3 Tests for Local Cross Secton Independence The power of the LM test s adversely affected when the dependence under the alternatve hypothesssspatal. Thespataldependenceoftheerrorscanbemodelledusngthespatal weghtmatrx,w=(w j,whchsappledtoapartcularorderngofthecrosssectonunts. It s often convenent to order the cross secton unts by ther topologcal poston, so that the p th orderneghborsofthe th crosssectonuntcanbedefnedasthe+pandthe pcross secton unts. SeePesaran(004. 4

5 Forsuchspatalalternatvesthefollowngp th orderbas-adjustedlmtestcanbeused p N s (T kˆρ,+s µ T,+s LM(p adj =, (1 p(n p 1 υ T,+s s=1 =1 where µ T,+s = TrE(M M +s {, υ T,+s ={TrE(M M +s } a 1T +Tr E (M M +s } a T. T k 3 The CD Test and ts Potental Inconsstency TodealwththelargeN basofthelmtest,pesaran(004suggestsusngthecdstatstc defned by N 1 T N CD= ˆρ N(N 1 j. (13 =1 j=+1 HeshowsthatunderH 0 andfort suffcentlylarge,cd d N(0,1asN. 3 Unlkethe LMteststatstc,theabovestatstchasexactlymeanzeroforfxedvaluesofT andn,under a wde class of panel data models, ncludng heterogeneous dynamc models subject to multple breaksntherslopecoeffcentsanderrorvarances,solongastheuncondtonalmeansofy t andx t aretme-nvarantandthernnovatonsaresymmetrcallydstrbuted. However,aspontedoutnPesaran(004,theCDtestneednotbeconsstentforalternatves of nterest. To see ths, consder the followng alternatves H 1 :u t =γ f t +ε t, (14 whereγ IID ( 0,σ γ,0<σ γ <,andthefactorsf t IID(0,1areunobservedcommon effects. AlsoassumethatE ( ft 4 =µf4 wth0<µ f4 <,ε t IIDN(0,1,andE(ε t f s =0 forall,t,ands.underh 1 Cov(u t,u jt =E(γ E ( γ j, andthecdteststatstcscentredat0fe(γ =0,evenwhenγ 0forsomeandf t 0. Therefore,underthealternatveswthE(γ =0thepoweroftheCDtestwouldnotncrease wthn. But,thepoweroftheLMtypetestsnvolvestheterms Cov ( u t,u jt =E ( γ E ( γ j µf4 (15 whchcontnuetodfferfromzeroevenwhene(γ =0. Hence,thepowerofLMtypetestswll ncrease wth N even under alternatves wth E(γ =0. These results wll contnue to hold under mult-factor alternatves. Underthealternatvehypothessofap th orderlocaldependence,pesaran(004proposes ap th ordergeneralzatonofthecdtestdefnedby ( p T CD(p= p(n p 1 N s s=1 =1 ˆρ,+s. (16 3 The CD test s based on a less restrctve verson of Assumpton, and only requres the errors, ε t, to be symmetrcally..d. dstrbuted around zero wth a unt varance. 5

6 4 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 LM test, LM adj, whch s defned by (10, and compare ts performance to that of the LM test and the CD test, whch are defned by ( and (13, respectvely. Intally we consder the experments n panels wth exogenous regressors. We shall also nvestgate the small sample propertes of the proposed test n the case of dynamc panels wth and wthout parameter heterogenety and structural breaks. Recall that the bas-adjusted LM test need not bevaldfordynamcpanels,andtwouldbenterestngtoseehowtperformsascomparedto the CD test. Fnally, we provde small sample evdence on the performance of the tests aganst spatal alternatves. 4.1 Expermental Desgns Intally, we consder the data generatng process(dgp specfed as y t =α + k x lt β +u t, =1,,...,N;t=1,,...,T, (1 l= whereα IIDN(1,1,β l IIDN(1,0.04. Thecovaratesaregeneratedas x lt =0.6x lt 1 +v lt,=1,,...,n;t= 50,...,0,...,T;l=,3,...,k ( wthx l, 51 =0wherev lt IIDN(0,τ l /( 1 0.6, τ l IIDχ (6/6. Thedsturbances are generated as u t =c (γ,k (γ f t +σ ε t, =1,,...,N;t=1,,...,T, where f t IIDN(0,1, and σ IIDχ (/. The dosyncratc errors, ε t, are generated under two dfferent schemes, ( normal errors, IIDN(0, 1, and ( ch-squared errors, IID ( χ (1 1 /. Thelatterstochecktherobustnessoftheteststonon-normalerrors. Thevaluesofα,x lt,σ aredrawnforeach=1,,...,n,andthenfxedacrossreplcatons. Underthenullhypothesswehaveγ =0forall,andunderthealternatvesweconsder (γ IIDU0.1,0.3, (γ IIDN(0,0.1, whereunder(,thecdtestsnconsstent,asshownabove. Inordertoexamnetheeffects ofchangngthenumberofregressors, k=,4,6areconsdered. Meanwhlethesameaverage populaton explanatory power of each cross secton regresson and the same degree of error cross sectoncorrelatonaretobemantanedforallk. Tothsend,c (γ,k sset c (γ,k = 1.04(k 1 forγ = (k 1, forγ IIDU0.1,0.3, (k 1, forγ IIDN(0,0.1, so that R = 0.5 across experments, where R = E(σ u /Var(y t wth σ u = Var(u t and Var(y t =(k 1E(β l +E(σ u. For examnng the power of the frst order cross secton ndependence tests, the DGP defned by(1fork=butwthspatallycorrelatederrorsareconsdered: u t =λ(0.5u 1,t +0.5u +1,t +σ ε t, (3 6

7 wth end ponts set at u 1t = u t +ε 1t and u Nt = u N 1t +ε Nt, where σ IIDχ (/, ε t IIDN(0,1. For ths DGP, the fnte sample performance of the spatal verson of the tests,defnedby(16and(1,areexamnednthecaseofp=1,andforthevaluesofλ=0, 0.1 and 0.1. In the case of dynamc models, followng Pesaran(004, 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= 50, 49,...,T, wth y, 51 = 0. The dosyncratc errors, ε t, are generated under two dfferent schemes as above, ( normalerrors, IIDN(0,1, and ( ch-squared errors, IID ( χ (1 1 /. Here wefocusontheheterogeneousslopeexpermentswhereβ IIDU0,1. Thefxedeffects,µ, are drawn as ε 0 +η, wth η IIDN(1,, thus allowng for the possblty of correlatons betweenfxedeffectsandthentalvalues,y 0. γ,σ t =σ,andf t aregeneratednthesame mannerasspecfedforthedgpwthexogenousregressors. Theparametersη,β andσ 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 exceptµ IIDN(1,1, β t =β t =0.6fort= 50,...,T/,β t =0.8 for t=t/+1,...,t; σ t = σ t = 1.5 for t = 50,...,T/, σ t = 1 for t = T/+1,...,T, and ε t IIDN(0,1. The second dynamc DGP s subject to multple breaks, specfed as (4 except β t =0.5 for t = 50,...,0 and all, β t IIDU0,1 for t = 1,...,T, = 1,...,N; σ t IIDχ (/ for t= 50,...,T,=1,...,N. Forbothdesgns,thefrst50observatonsaredscarded. Fnally,theDGPsubjecttountroot,whchsspecfedas(4exceptβ t =β=1forall andt,σ t IIDχ (/,sconsdered. The test statstcs are computed usng the OLS resduals from the ndvdual regressons. For all experments the combnatons of sample szes N = 10,0,30,50,100,00 and T = 0,30,50,100areconsdered. Thenomnalszeofthetestsssetatthe5%sgnfcancelevel. All experments are based on 000 replcatons. 4. Monte Carlo Outcomes Table1reportstheszeofthetestsfortheDGPwthdfferentnumberofexogenousregressors (k=,4,6andnormalerrors. AsshownnPesaran(004,theCDtesthasthecorrectsze, and the LM test severely over-rejects the null partcularly for N T. In contrast the basadjustedlmtestsuccessfullycontrolstheszeforallcombnatonsofn andt, exceptwhen both k and N are large and T small. In the case of γ IIDU0.1,0.3, whose results are reportedntable,thebas-adjustedlmtestandthecdtestseemtohavereasonablepower. Inthecaseofγ IIDN(0,0.1,whoseresultsarereportednTable3,astheorypredctsthe CDtesthaslttlepower. ThepoweroftheCDtestncreaseswthT veryslowly, buttdoes notncreasewthn forgvent. Thssbecausethesampleaverageoffactorloadngsforfnte N cannotbedfferentfromzerofarenoughforthetesttorejectthenull,forsomereplcatons, and the precson of ths happenng ncreases as T rses. On the other hand, the bas-adjusted LM test mantans reasonable power under the same desgn. Overall, the powers of both the CDandthebas-adjustedLMtestncreasefasterwthN thant,andtseemsthatthenumber ofregressorsdoesnotaffectthepowerofthesetestsmuch 4. The results for the case wth IID ( χ (1 1 / errors are gven n Table 4, and show thatthebas-adjustedlmtestsgenerallynotasrobusttonon-normalerrorsasthecdtest. Ittendstoover-reject(moderatelyforallcombnatonsofN andt. 4 Itsworthemphaszngthatwekeepthepopulatongoodnessofftofthemodelfxedfordfferentnumber of exogenous regressors n our Monte Carlo experments. 7

8 Table 5 summarzes the results of the spatal frst order tests. The bas-adjusted LM test, LM(1 adj,whchdefnedby(1,successfullycontrolsthesze. TheCD(1testhasthecorrect sze, whch s consstent to the results n Pesaran(004. Both the bas-adjusted LM test and thecd(1testseemtohavereasonablepowerunderthealternatves,whereλ=±0.1. Tables6to10provdetheresultsforthevarousdynamcDGPs. Forallexperments,the CDtesthasthecorrectszeandtheLMtestseverelyover-rejectsthenullwhenN T. Ths stobeexpected,asdscussedabove,snceforsmallt relatveton,themeanapproxmaton ofˆρ j 1wllnotbecorrect,andwthN largethencorrectcenterngofthetestndcators 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(0, 1 errors(table 6, the bas-adjusted LM test tends to over-reject when N s much larger than T. Wth respect to the power, as wasnthecaseofdgpwthexogenousregressors,thecdtesthaslttlepowernthecaseof γ j IIDN (0,0.1. TheresultsforIID ( χ (1 1 / errors(table7aresmlartothose n Table4. Forthe DGPwth asnglestructural break(table 8, thebas-adjusted LM test rejectsthenulltoooften. Forexample,whenN =00,theestmatedszeoftheLM adj tests 100%forallT. Inthecaseofmultplestructuralbreaks(Table9,thebas-adjustedLMtest tendstoover-reject,especallyforn T. Fnally,nthecaseofmodelswthuntroots(Table 10 the bas-adjusted LM tests tends to over-reject, wth the extent of over-rejecton ncreasng wth N. 5 Concludng Remarks Ths paper has proposed a bas-adjusted LM test of cross secton ndependence. For the basadjustment,wedervedtheexactmeanandvaranceofthetestndcatorofthelmstatstcn the case of the model wth strctly exogenous regressors and normal errors, based on the work n Ullah (004, so that the centerng of the LM statstc s correct for fxed T and large N. Importantly, the proposed bas-adjusted LM test s consstent even when the Pesaran s(004 CD test s nconsstent. Small sample evdence based on Monte Carlo experments suggests that the bas-adjusted LM test successfully controls 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 basadjustedlmtestsarenotasrobustasthecdtesttonon-normalerrorsand/ornthepresence of weakly exogenous regressors. Clearly,twouldbeworthdervngthemeanandvaranceoftheLMteststatstcnthecase ofdynamcmodels,andnthecasewheretheerrorsarenon-normal.thswllbethesubject of our future research. 8

9 A Appendx A.1 EvaluatonoftheFrstTwoDervatvesofE(W r Let us consder a quadratc form W =u Mu, where the T 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 = 0, hence θ = 0, W s dstrbuted as a central ch-square dstrbuton. In whatfollowsweevaluatede ( W r anddd E ( W r,whered=µ+ / µandr=1,,... Now whereweuse d E ( W r = dd E ( W r = µ E( W r = µ µ 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, θ E( W r Mµ, µ θ E( W r µ M FrstwenotefromUllah(1974,p.147thatform>r Further,fors=1,,..., =0 µ E( W r θ E( W r = θ E( W r θ E( W r µ M, (A.1 µµ M+Mµµ +M Mµµ M+ θ E( W r M. (A. E ( W r = 1 Γ ( m r+ re θ Γ ( θ m +,whenθ 0 (A.3! = 1 Γ ( m r r Γ ( 1 = m (m (m 4...(m r,whenθ=0. s θ sew r = ( 1 s 1 Γ(r+s e θ Γ ( m r+ r Γ(r Γ ( m +1s+θ,whenθ 0 (A.4! = ( 1 s 1 Γ(r+s r Γ(r =0 Γ ( m r Γ ( m +s,whenθ=0. 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 forθ 0. Whenµ=0,henceθ=0,weobtan d E ( W r =0, dd E ( W r =E ( W r + θ E( W r M, (A.5 where E ( W r / θsgvenby(a.4fors=1andθ=0. A. ProofofTheorem1 From(3and(4 ˆρ j = (u M M ju j (u M u ( u j M ju j (A.6 = u M M j u j u jm j M u (u Mu( u j Mjuj = u A ju u Mu wtha j = MMjuju jm jm u j Mjuj, wherem andm j areasdefnedn(6,u N(0,I T,u j N(0,I T. FrstweconsderthecasewhereX and X j arefxedsothatthem andm j arefxedmatrces. Takngtheteratveexpectatonsonbothsdesof(A.6 wecanwrte E (ˆρ j =Euj E (ˆρ j uj, A.1

10 n whch E (ˆρ j u j ( u = E A ju u j u Mu = E ( u A ju W 1 u j = ( d ( A jd E W 1 { =Tr Aj dd E ( W 1 } = Tr(A ( je W 1 ( + µ ( A jµ E W 1 + E ( W 1 θ µ MAjµ +µ AjMµ +Tr(AjM + θ E ( W 1 µ MAjMµ, (A.7 where we use (A.1, W = u M u, d =µ + / µ wth µ =X β, the thrd equalty follows by usng the resultsnullah(004,(.8,andθ =µ M µ /. Wenotethatalltheexpectatonsontherghthandequaltes n(a.7arewthrespecttou forcondtonalu j. Now E (ˆρ j = E ( W 1 + ( u j M jm M ju j E u + ( µ j Mjuj E(Ajµ ( E W 1 E ( W 1 ( u µ θ E(Ajµ +E j M jm M ju j u j Mjuj + θ E ( W 1 µ E(A jµ, (A.8 wherealltherghthandexpectatons,excepte(w 1,arewthrespecttou j. Butforµ =0 E (ˆρ j = E ( ( W 1 u j M jm M ju j E + E ( W 1 u j M jm M ju j u j Mjuj E( θ u j Mjuj = E ( W 1. + E ( W 1 θ E( u j M j M M j u j u j M ju j Now,wrtngB j =M jm M j,w j =u jm ju j,θ j =µ j Mjµ j /wthµ j =Xjβ j andusng(a.7, ( u j B j u j E u j M ju j = Tr(B ( je W 1 ( j + µ ( jb jµ j E W 1 j + E ( W 1 j µ θ jm jb jµ j +Tr(B jm j j + θ E ( W 1 µ j MjBjMjµ j, j (A.9 (A.10 whchcanbewrttenforµ j =0as ( u j B ju j E = u j Mjuj Substtutng(A.11 n(a.9 we get E (ˆρ j E ( W 1 j = E ( W 1 + E ( W 1 θ + E ( W 1 j Tr(B j,tr(b j=tr(m M j. (A.11 θ j E ( W 1 j + E ( W 1 j Tr(M M j (A.1 θ j = = 1 m Tr(M M j m(m 1 m Tr(MMj, and E ( mˆρ 1 j = m Tr(MMj, wherem=t kandweusetheresults(a.3and(a.4forr=1ands=1. Next we consder ˆρ 4 j =( u A ju / ( u M u, andtakngtheexpectatonsonbothsdesof(a.14weget (A.13 (A.14 E (ˆρ 4 j =Euj E (ˆρ 4 j u j, A.

11 n whch usng Ullah(004,(.8 E (ˆρ 4 j uj =E (u A ju W u j (A.15 = ( d A jd E ( W = ( d A j d ( d A j d E ( W = ( d A jd Tr Ajd d E ( W = Tr A jd d c(µ, wherec(µ =Tr A j d d E ( W,andusng(A.7(seealsoUllah(004,Chapter, c(µ = Tr(A ( je W ( + µ Ajµ + θ E ( W µ A jµ. E ( W Inordertoevaluatethetermnthelastequaltesof(A.15wenotethat ( d c(µ = µ + c(µ, d d c(µ = =µ µ c(µ +µ Usng(A.18n(A.15wethenget µ ( µ + µ µ + E ( W µ θ Ajµ +Tr(Aj (A.16 µ c(µ + µ c(µ c(µ +c(µ + c(µ µ µ + c(µ µ µ. Tr A j d d c(µ = ( µ A jµ c(µ +Tr(A j c(µ { + Tr A jµ nwhch,usng(a.16,wecanverfythat µ +Tr A j c(µ µ µ } c(µ +Tr A j c(µ µ µ, (A.17 (A.18 (A.19 where and c(µ = Tr(A j E ( W M µ µ θ +A jµ EW + E ( W ( M µ θ µ Ajµ (A.0 + E ( W M µ µ Ajµ +Tr(Aj θ + E ( W θ +(A jµ θ (4A jµ + 3 E ( W θ 3 E ( W ( M µ µ A jµ = a 1(θ(A jµ +a (θ ( µ A jµ Mµ +a 3(θTr(A jm 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, µ µ c(µ = A ja 1(θ+M µ µ Aja(θ + { A j µ µ M +µ A } jµ M a (θ + ( µ A jµ Mµ µ M a 4(θ +Tr(A jm a 3(θ+M µµ M Tr(A ja 5(θ, (A.1 A.3

12 where a 4 (θ = θ a 5(θ = θ E ( W 3 + E ( W θ Inordertoevaluate(A.19forµ =0wefrstnotethat c(0 = Tr(A j Therefore,forµ =0,wecansmplfy θ 3 E ( W E ( W 4 + E ( W. θ 4 c(µ µ = 0, µ µ c(µ = A j E ( W + E ( W + θ θ +Tr(A jm E ( W + θ θ Tr A j d d c(µ = Tr(A j +Tr ( A j = 3Tr(A j E ( W E ( W, + E ( W, (A. θ E ( W + E ( W E ( W E ( W θ + E ( W + θ θ. + θ E ( W, E ( W (A.3 becausetr(a j =Tr ( Aj. Further,usng(A.3,(A.3and(A.4n(A.15weget,forµ =0, E (ˆρ 4 j = 3E { Tr(A j } a 1m (A.4 = 3 { Tr(M M j +Tr (M M j } a 1m, wheretherghthandexpectatonswthrespecttou j andts E { (u Tr(A j } j M j M M j u j = E u j M ju j and Next, usng(a.1 and(a.4, we get V (ˆρ j a 1m= = { Tr(M M j +Tr (M M 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 V ( ( mˆρ j =Tr(MM j b 1 m +b mtr (M M j, (A.6 m where (m 8(m++4 b m=3 =3m a 1m. (m+(m (m 4 When X and X j arerandom then theresultsn (A.1 and (A.5 are the same by replacng Tr(M M j, Tr(M M j,tr(m M j wththerexpectedvalues. A.4

13 References Anseln, L.,(1988. Spatal Econometrcs: Methods and Models. Dorddrecht: Kluwer Academc Publshers. Anseln, L., (001. Spatal econometrcs, In: B. Baltag (ed. A companon to Theoretcal Econometrcs, Oxford: Blackwell. Breusch, T.S., Pagan, A.R.,(1980. 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.,(004. General dagnostc tests for cross secton dependence n panels. CESfo Workng Papers No.133. Ullah, A.,(1974. On the samplng dstrbuton of mproved estmators for coeffcents n lnear regresson. Journal of Econometrcs, Ullah, A.,(004. 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

14 Table 1 SzeofCrossSectonIndependence TestswthExogenousRegressors, γ =γ=0 Normal Errors wth Dfferent Number of Regressors(k k= k=4 k=6 (T,N LM LM adj CD Notes: Data are generated as y t = α + k l= x ltβ l +u t, u t = c (γ,k (γ f t+σ ε t, = 1,,...,N, t = 1,,...,T, where α IIDN(1,1, wth x lt = 0.6x lt 1 +v lt, l =,3,...,k, = 1,,...,N, t = 50, 49,...,T, x l, 51 = 0, where v lt IIDN(0,τ l/ ( 1 0.6, τ l IIDχ (6/6. β l IIDN(1,0.04, f t IIDN(0,1, σ IIDχ (/, and ε t IIDN(0,1. α, x lt, σ are fxed across replcatons. c (γ,k s chosen so that R = E(σ u/var(y t = 0.5 wth σ u = Var(u t and Var(y t =(k 1E(β l +E(σ u. LMstheBreusch-Pagan(1980LMtest,LM adj sthebas-adjustedlmtestandcdsthepesaran s(004cdtest. LMtestsbased on χ N(N 1/ dstrbuton. LM adj s based on postve-one-sded N(0,1test. CD test s based on two-sded N(0,1test. All tests are conducted at 5% nomnal level. All experments are based on,000 replcatons. T.1

15 Table Powerof CrossSectonIndependenceTestswthExogenousRegressors, γ IIDU0.1,0.3 Normal Errors wth Dfferent Number of Regressors(k k= k=4 k=6 (T,N LM LM adj CD Notes: ThedesgnsthesameasthatofTable1exceptγ IIDU0.1,0.3. T.

16 Table 3 Powerof CrossSectonIndependence TestswthExogenousRegressors, γ IIDN(0,0.1 Normal Errors wth Dfferent Number of Regressors(k k= k=4 k=6 (T,N LM LM adj CD Notes: ThedesgnsthesameasthatofTable1exceptγ IIDN(0,0.1. T.3

17 Table 4 SzeofCrossSectonIndependence TestswthExogenousRegressors, γ =γ=0 Non-normal Errors wth Dfferent Number of Regressors(k k= k=4 k=6 (T,N LM LM adj CD Notes: ThedesgnsthesameasthatofTable1exceptε t IID χ (1 1 /. T.4

18 Table 5 Sze and Power of Frst Order Cross Secton Independence Tests wth Exogenous Regressors, Spatally Correlated Errors λ=0 λ=0.1 λ= 0.1 (T,N LM(1 adj CD( Notes: The desgn s the same as that of Table 1 for k =, except errors are spatally correlated such that u t = λ(0.5u 1,t+0.5u +1,t+σ ε t, wth end ponts set at u 1t=u t+ε 1t andu Nt=u N 1t+ε Nt. T.5

19 Table 6 Sze and Power of Cross Secton Independence Tests, Heterogeneous AR(1 Specfcaton wth Normal Errors γ =γ=0 γ IIDU0.1,0.3 γ IIDN(0,0.1 (T,N LM LM adj CD Notes: SeenotestoTable1. TheDGPsspecfedasy t=µ (1 β +β y,t 1+u t,u t=γ f t+σ tε t,=1,,...,n;t= 50,...,T,whereβ IIDU0,1,µ ε 0+η, η IIDN(1,,f IIDN(0,1,σ t=σ IIDχ (/,andε t IIDN(0,1. η,β andσ t arefxedacrossreplcatons. y, 51=0andthefrst50observatonsare dscarded. T.6

20 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=0 γ =γ=0 (T,N (T,N LM LM LM adj LM adj CD CD Notes: SeethenotestoTable6. Thedesgnsthesame Notes: TheDGPsspecfedasy t µ =β t (y t 1 µ +u t, asthatoftable6exceptε t IID χ (1 1 /. u t =γ f t +σ t ε t,whereµ IIDN(1,1;β t =β t =0.6and σ t=σ t= 1.5fort= 50,...,T/;β t =0.8andσ t=1 fort=t/+1,...,t;ε t IIDN(0,1. y, 51=0and thefrst50observatonsaredscarded. SeealsothenotestoTable6. T.7

21 Table 9 Table 10 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=0 γ =γ=0 (T,N (T,N LM LM LM adj LM adj CD CD Notes: ThedesgnsthesameasthatofTable8, Notes: TheDGPsthesameasthatofTable10 exceptβ t =0.5fort= 50,...,0andβ t IIDU0,1 exceptβ t =β=1forallandt. Seealsothenotes fort=1,...,t,=1,...,n;σ t IIDχ (/for totable6. t= 50,...,T,=1,...,N. SeealsothenotestoTable6. T.8