Working Paper Testing weak cross-sectional dependence in large panels. CESifo working paper: Empirical and Theoretical Methods, No.

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1 econstor Der Oen-Access-Publkatonsserver der ZBW Lebnz-Informatonszentrum Wrtschaft he Oen Access Publcaton Server of the ZBW Lebnz Informaton Centre for Economcs Pesaran, M. Hashem Workng Paer estng weak cross-sectonal deendence n large anels CESfo workng aer: Emrcal and heoretcal Methods, o. 38 Provded n Cooeraton wth: Ifo Insttute Lebnz Insttute for Economc Research at the Unversty of Munch Suggested Ctaton: Pesaran, M. Hashem () : estng weak cross-sectonal deendence n large anels, CESfo workng aer: Emrcal and heoretcal Methods, o. 38 hs Verson s avalable at: htt://hdl.handle.net/49/57947 utzungsbedngungen: De ZBW räumt Ihnen als utzern/utzer das unentgeltlche, räumlch unbeschränkte und zetlch auf de Dauer des Schutzrechts beschränkte enfache Recht en, das ausgewählte Werk m Rahmen der unter htt:// nachzulesenden vollständgen utzungsbedngungen zu vervelfältgen, mt denen de utzern/der utzer sch durch de erste utzung enverstanden erklärt. erms of use: he ZBW grants you, the user, the non-exclusve rght to use the selected work free of charge, terrtorally unrestrcted and wthn the tme lmt of the term of the roerty rghts accordng to the terms secfed at htt:// By the frst use of the selected work the user agrees and declares to comly wth these terms of use. zbw Lebnz-Informatonszentrum Wrtschaft Lebnz Informaton Centre for Economcs

2 estng Weak Cross-Sectonal Deendence n Large Panels M. Hashem Pesaran CESIFO WORKIG PAPER O. 38 CAEGORY : EMPIRICAL AD HEOREICAL MEHODS APRIL An electronc verson of the aer may be downloaded from the SSR webste: from the RePEc webste: from the CESfo webste:

3 CESfo Workng Paer o. 38 estng Weak Cross-Sectonal Deendence n Large Panels Abstract hs aer consders testng the hyothess that errors n a anel data model are weakly Cross-sectonally deendent (CD), usng the exonent of cross-sectonal deendence ntroduced recently n Baley, Kaetanos and Pesaran (). It s shown that the mlct null of the CD test deends on the relatve exanson rates of and. It s argued that n the case of large anels, the null of weak deendence s more arorate than the null of ndeendence whch could be qute restrctve for large anels. Usng Monte Carlo exerments, t s shown that the CD test has the correct sze for values of the cross-sectonal exonent that le n the range [, /4], for all combnatons of and, and rresectve of whether the anel contans lagged values of the deendent varables, so long as there are no major asymmetres n the error dstrbuton. JEL-Code: C, C3, C33. Keywords: exonent of cross-sectonal deendence, dagnostc tests, anel data models, dynamc heterogenous anels. M. Hashem Pesaran Faculty of Economcs Unversty of Cambrdge mh@cam.ac.uk January hs aer comlements an earler unublshed aer enttled "General Dagnostc ests for Cross Secton Deendence n Panels", whch was dstrbuted n 4 as the Workng Paer o. 435 n Cambrdge Workng Paers n Economcs, Faculty of Economcs, Unversty of Cambrdge. I am grateful to atala Baley and Majd Al-Sadoon for rovdng me wth excellent research assstance, and for carryng out the Monte Carlo smulatons. I would also lke to thank George Kaetanos, Ron Smth, akash Yamagata, and Aman Ullah for helful comments and dscussons. Fnancal suort from the ESRC Grant ES/I366/ s gratefully acknowledged.

4 Introducton hs aer s concerned wth tests of error deendence n the case of large lnear regresson anels where (the cross secton dmenson) s large. In the case of anels where s small (say or less) and the tme dmenson of the anel ( ) s su cently large the cross correlatons of the errors can be modelled (and tested statstcally) usng the seemngly unrelated regresson equaton (SURE) framework orgnally develoed by Zellner (96). In such anels where s xed as, tradtonal tme seres technques, ncludng log-lkelhood rato tests, can be aled. A smle examle of such a test s the Lagrange multler (LM) test of Breusch and Pagan (98) whch s based on the average of the squared ar-wse correlaton coe cents of the resduals. However, n cases where s large standard technques wll not be alcable and other aroaches must be consdered. In the lterature on satal statstcs the extent of cross-sectonal deendence s measured wth resect to a gven connecton or satal matrx that characterzes the attern of satal deendence accordng to a re-sec ed set of rules. For examle, the (; j) elements of a connecton matrx, w j, could be set equal to f the th and j th regons are joned, and zero otherwse. See Moran (948) and further elaboratons by Cl and Ord (973, 98). More recent accounts and references can be found n Anseln (988, ), and Hanng (3, Ch. 7). hs aroach, aart from beng deendent on the choce of the satal matrx, s not arorate n many economc alcatons where sace s not a natural metrc and economc and socooltcal factors could be more arorate. In the absence of orderng, tests of cross-sectonal ndeendence n the case of large anels have been consdered n Frees (995), Pesaran (4), Pesaran, Ullah and Yamagata (8), Sara ds, Yamagata, Robertson (9), and Baltag, Feng and Kao (). Recent surveys are rovded by Moscone and osett (9), and Sara ds and Wansbeek (). he null hyothess of these tests s the cross-sectonal ndeendence of the errors n the anel regressons, and the tests are based on ar-wse correlaton coe cents of the resduals, ^ j, for the (; j) unts, comuted assumng homogeneous or heterogeneous sloes. he orgnal LM test of Breusch and Pagan (98), and ts mod ed verson for large anels by Pesaran, Ullah and Yamagata (8), are based on ^ j, and test the hyothess that all ar-wse error covarances, E (u t ; u jt ), are equal to zero for 6= j. In contrast, we show that the mlct null of the CD test, roosed n Pesaran (4), whch s based on ^ j, s weak cross-sectonal deendence dscussed n Chudk, Pesaran and osett (), and further develoed n Baley, Kaetanos and Pesaran (, BKP). More sec cally, we show that the mlct null of the CD test deends on the relatve exanson rates of and. In general, f = O ( ) for some n the range (; ], then the mlct null of the CD test s gven by < ( )=4; where s the exonent of cross-sectonal deendence de ned by = [=(( )] P P j=+ j = O( ), wth j denotng the oulaton correlaton coe cent of u t and u jt. BKP show that s dent ed and can be estmated consstently f = <. hs aer comlements BKP by showng that the null hyothess that les n the range [; =) can be tested usng the CD statstc f s close to zero ( almost xed as ), but n the case where = ( and at the same rate) then the mlct null of the CD test s gven by < =4. he null of weak cross-sectonal deendence also seems more arorate than the null of crosssectonal ndeendence n the case of large anel data models where only ervasve cross deendence For emrcal alcatons where economc dstance such as trade atterns are used n modellng of satal correlatons see Conley and oa () and Pesaran, Schuermann, and Wener (4).

5 s of concern. For examle, n ortfolo analyss full dvers caton of dosyncratc errors s acheved even f the errors are weakly correlated, and cross-sectonal error ndeendence s not requred. (e.g. Chamberlan, 983). In estmaton of anels only strong cross-sectonal error deendence can ose real roblems, and n most alcatons weak cross-sectonal error deendence does not ose serous estmaton and nferental roblems. he small samle roertes of the CD test for d erent values of and samle szes are nvestgated by means of a number of Monte Carlo exerments. It s shown that the CD test has the correct sze for values of n the range [; =4], for all combnatons of and, and rresectve of whether the anel contans lagged values of the deendent varables, so long as there are not major asymmetres n the error dstrbutons. hs s n contrast to the LM based tests (such as the one roosed by Pesaran, Ullah and Yamagata, 8) that requre the regressors to be strctly exogenous. In lne wth the theoretcal results, the CD test tends to over-reject f s large relatve to and s n the range (=4 =]. he CD test also has satsfactory ower for all values of > = and rses wth so long as > =. he rest of the aer s organzed as follows. he anel data model and the LM tests of error cross-sectonal ndeendence are ntroduced n Secton. he concet of weak cross-sectonal deendence s ntroduced and dscussed n Secton 3. he use of CD statstc for testng weak cross-sectonal deendence s dscussed n Secton 4, where the asymtotc dstrbuton of the test s rgorously establshed under the null of ndeendence. he dstrbuton of CD statstc under the more general null of weak deendence s consdered n Secton 5, and the condtons under whch t tends to (; ) are derved. he alcaton of the test to heterogeneous dynamc anels s dscussed n Secton 6. Small samle evdence on the erformance of the test s rovded n Secton 7. Secton 8 concludes. Panel Data Models and the LM ye ests of Cross- Sectonal Error Indeendence Consder the followng anel data model y t = x t + u t, for = ; ; :::; ; t = ; ; :::; ; () where ndexes the cross secton dmenson and t the tme seres dmenson, x t s a k vector of observed tme-varyng regressors (ndvdual-sec c as well as common regressors). An ndvdualsec c ntercet can be ncluded by settng the rst element of x t to unty. he coe cents,, are de ned on a comact set and allowed to vary across. For each, u t s IID(; ), for all t, although they could be cross-sectonally correlated. he deendence of u t across could arse n a number of d erent ways. It could be due to satal deendence, omtted unobserved common comonents, or dosyncractc ar-wse deendence of u t and u jt ( 6= j) wth no artcular attern of satal or common comonents. he regressors could contan lagged values of y t, be ether statonary (or ntegrated of order zero, I()) or have unt roots (or ntegrated of order, I()). But n the dervatons below we assume x t s I(), and dstngush between the statc and dynamc cases where the regressors are strctly exogenous and when they are weakly exogenous, sec cally when x t = (; y ;t ; :::; y ;t ). he testng rocedure s alcable to xed and random e ects models as well as to the more general heterogeneous sloe or random coe cent sec catons. he assumton that u t s are serally uncorrelated s not restrctve and can be accommodated by ncludng a su cent number of lagged values of y t amongst the regressors.

6 . LM ye ests In the SURE context wth xed and, Breusch and Pagan (98) roosed a Lagrange multler (LM) statstc for testng the null of zero cross equaton error correlatons whch s artcularly smle to comute and does not requre the system estmaton of the SURE model. he test s based on the followng LM statstc CD lm = j=+ where ^ j s the samle estmate of the ar-wse correlaton of the resduals. Sec cally, ^ j; ^ j = ^ j = P P t= e te jt = P = ; () t= e t t= e jt and e t s the Ordnary Least Squares (OLS) estmate of u t de ned by e t = y t ^ x t ; (3) wth ^ beng the OLS estmator of comuted usng the regresson of y t on x t for each ; searately. he LM test s vald for relatvely small and su cently large. In ths settng Breusch and Pagan show that under the null hyothess of no cross-sectonal deendence, sec ed by Cov (u t ; u jt ) = ; for all t, 6= j; (4) CD lm s asymtotcally dstrbuted as ch-squared wth ( )= degrees of freedom. As t stands ths test s not alcable when. However, notng that under H, ^ a j s wth ^ j, = ; ; ::;, j = + ; ; :::;, beng asymtotcally ndeendent, the followng scaled verson of CD lm can be consdered for testng the hyothess of cross deendence even for and large: s CD lm = ( ^ j ): (5) ( ) j=+ a It s now easly seen that under H wth rst followed by we would have CD lm s (; ). However, ths test s lkely to exhbt substantal sze dstortons for large and small, a stuaton that can frequently arse n emrcal alcatons. hs s rmarly due to the fact that for a nte, E( ^ j ) wll not be correctly centered at zero, and wth large the ncorrect centerng of the LM statstc s lkely to be accentuated, resultng n sze dstortons that tend to get worse wth. A bas corrected verson of CD lm s roosed n Pesaran, Ullah and Yamagata (8) under the assumtons that the regressors are strongly exogenous and the errors are normally dstrbuted. In what follows we roose a test of weak cross-sectonal deendence, whch we argue to be more arorate for large anels, where mere ncdence of solated deendences are of lttle consequence for estmaton or nference. 3

7 3 Weak Error Cross-Sectonal Deendence As noted n the ntroducton when s large t s often more arorate to consder the extent of error cross-sectonal deendence rather than the extreme null hyothess of error ndeendence that underles the LM tye tests. hs s n lne, for examle, wth the assumton of aroxmate factor models dscussed n Chamberlan (983) n the context of catal asset rcng models. o ths end we consder the followng factor model for the errors u t = ( f t + " t ) ; (6) where f t = (f t ; f t ; :::; f mt ) s the m vector of unobserved common factors (m beng xed) wth E(f t ) =, and Cov(f t ) = I m, = ( ; ; :::; m ) s the assocated vector of factor loadngs, and " t are dosyncratc errors that are cross-sectonally and serally ndeendent wth a unt varance, namely " t s IID(; ). he degree of cross-sectonal deendence of the errors, u t, s governed by the rate at whch the average ar-wse error correlaton coe cent, = [=( )] P P j=+ j, tends to zero n, where j = Corr(u t ; u jt ). In the case of the above factor model we have, V ar(u t ) = = ( + ); j = j, for 6= j, where = + : (7) hen t s easly seen that = P ; (8) where = P. Consder now the e ects of the j th factor, f jt, on the th error, u t, as measured by j, and suose that these factor loadngs take non-zero values for M j out of the cross-secton unts under consderaton. hen followng BKP, the degree of cross-sectonal deendence due to the j th factor can be measured by j = ln(m j )= ln(), and the overall degree of cross-sectonal deendence of the errors by = max j ( j ). s the exonent of that gves the maxmum number of errors, M = max j (M j ), that are ar-wse correlated. he remanng M unts wll only be artally correlated. BKP refer to as the exonent of cross-sectonal deendence. can take any value n the range to, wth ndcatng the hghest degree of cross-sectonal deendence. Consderng that = O(m) where m s xed as, the exonent of cross-sectonal deendence of the errors can be equvalently de ned n terms of the scaled factor loadngs, = ( ; ; :::; m ). Wthout loss of generalty, suose that only the rst M j elements of j over are non-zero, and note that 3 M j j; j + =M j+ j A = j M j j 3 he man results n the aer reman vald even f P =M j + j = O(). mantan the assumton that P =M j + j =. A = j j = O( j ); But for exostonal smlcty we 4

8 where j = O( j M P Mj j j ), and usng (8) we have 6=, for a nte M j and as M j. Smlarly, P = O( ): j j = In what follows we develo a test of the null hyothess that < =. he case where > = s covered n BKP. he values of n the range [; =) corresond to d erent degrees of weak cross-sectonal deendence, as comared to values of n the range (=; ] that relate to d erent degrees of strong cross-sectonal deendence. 4 A est of Weak Cross-Sectonal Deendence Gven that s de ned by the contracton rate of, we base the test of weak cross-sectonal error deendence on ts samle estmate, gven by b = ( ) j=+ ^ j ; (9) where ^ j s already de ned by (). he CD test of Pesaran (4) s n fact a scaled verson of b whch can be wrtten as = ( ) CD = b : () In what follows we consder the dstrbuton of the CD statstc under three d erent null hyotheses. o establsh comarablty and some of the basc results we begn wth CD statstc under hyothess of cross-sectonal ndeendence de ned by H : = ; for all : () We then consder the asymtotc dstrbuton of the CD statstc as and, such that = O( ), for <, and show that the mlct null of the CD test s gven by H w : < ( )=4: () As argued earler, such a null s much less restrctve for large anels than the ar-wse error ndeendence assumton that underles the LM tye tests whch are based on ^ j. Intally, we derve the asymtotc dstrbuton of the CD test n the case of the standard anel data model, () subject to the followng assumtons: Assumton : he factor model, (6), holds. he dosyncratc errors, " t, are IID(; ), are symmetrcally dstrbuted around for all and t, f t s IID(; I m ), f t and " ;t are dstrbuted ndeendently, < < K <. he factor loadngs,, are ndeendently dstrbuted across : Assumton : he regressors, x t, are strctly exogenous such that E (" t j ) =, for all and t; (3) where = (x ; x ; :::; x ), and E 4 t < K < ; (4) 5

9 where t = " t =( " M " ) =, M = I ( ), and s a ostve de nte matrx for any xed, ; as, wth beng a ostve de nte matrx. 4 Assumton 3: > k + and the OLS resduals, e t, de ned by (3), are not all zero. 5 Assumton 4: he factor loadngs,, de ned by (6) satsfy the -summablty condton = O( ): (5) heorem Consder the regresson model, (), and suose that Assumtons -3 hold, and the dosyncratc errors, " t, are symmetrcally dstrbuted around, then under H : =, and for all > and > k + we have E ^ j = ; for all 6= j; (6) E ^ j^ s =, for all 6= j 6= s; (7) E (CD) = ; (8) a (k + ) V ar(cd) = + ( k ) ( k ) ; (9) a = P P j=+ r (A A j ) < k + ; () ( ) where A = ( ) ; ^ j and CD are de ned by () and (), resectvely. 6 Proof:. Frst note that the ar-wse correlaton coe cents can be wrtten as where t are the scaled resduals de ned by ^ j = t jt ; () t= e t t = ( e e ; () = ) e t s the OLS resduals from the ndvdual-sec c regressons, de ned by (3), and e = (e ; e ; :::; e ). Also under H, e = M ", where " = (" ; " ; :::; " ). herefore, condtonal on x t, the scaled resduals, t, are odd functons of the dsturbances, " t, and under Assumton we have E ( t j ) = ; for all and t. 4 he fourth-order moment of t exsts f su E(" 6 t ) < K <, and > k + 4. hs result can be establshed usng Lemmas n Leberman (994). 5 he requrement > k + can be relaxed under sloe homogenety assumton, = where xed e ects resduals can be used n the constructon of the CD statstc nstead of e t. 6 Smlar results can also be obtaned for xed or random e ects models. It su ces f the OLS resduals used n the comutaton of ^ j are relaced wth assocated resduals from xed or random e ects sec catons. But the CD test based on the ndvdual-sec c OLS resduals are robust to sloe and error-varance heterogenety whlst the xed or random e ects resduals are not. 6

10 Hence, uncondtonally we also have E ( t ) =, for all and t. Usng ths result n () now yelds (recall that under H ndeendent), E ^ j = ; whch n turn establshes that (usng ()) the errors, " t ; are cross-sectonally E(CD) =, for any, and all > k +. Under H and Assumtons -3, ^ j and ^ s are cross-sectonally uncorrelated for ; j and s, such that 6= j 6= s. More sec cally E ^ j^ s = = t= t = t= t = E t jt t st E ( t t ) E jt E (st ) = ; for 6= j 6= s: Also snce the regressors are assumed to be strctly exogenous, we further have 7 V ar ^j = E ^ j = r(m M j )=( k ) : Usng ths result n () we have V ar(cd) = = ( )( k j=+ [ (k + )] ( k ) + ( )( k ) [ (k + ) + r (A A j )] j=+ r (A A j ) A : Hence where a (k + ) V ar(cd) = + ( k ) ( k ) = + O ; a = P P j=+ r (A A j ) : ( ) But r (A A j ) < r(a ) r(a j ) = = k +, and we must also have a < k +. hs comletes the roof of the theorem. he above results also suggest the followng mod ed verson of CD, CD gcd = h + a (k+) = ; (3) ( k ) ( k ) 7 I am grateful to Aman Ullah for drawng my attenton to ths result. Also recall that E ^ j =. 7

11 whch s dstrbuted exactly wth a zero mean and a unt varance. In cases where k s relatvely large, and the regressors, x t ; are cross-sectonally weakly correlated, the term nvolvng a n the exresson for the varance of CD wll be small and both statstcs are lkely to erform very smlarly, and the CD test s recommended on grounds of ts smlcty. o kee the analyss smle, and wthout of loss generalty, n what follows we shall focus on the CD test. 4. he dstrbuton of the CD test under H Consder now the dstrbuton of the CD test. As shown n (7), the elements n the double summaton that forms the CD statstc are uncorrelated but they need not be ndeendently dstrbuted when s nte. herefore, when s nte the standard central lmt theorems can not be exloted n order to derve the dstrbuton of the CD statstc. 8 o resolve the roblem we rst re-wrte the CD statstc (de ned by ()) as s CD ^j A ; (4) ( ) j=+ and recall that ^ j = P t= t jt ; where t s de ned by (). ow under H : =, usng standard results from regresson analyss, we have h t; = t = t + = h t; ; (5) x t ; (6) where t = " t =(" M " = ) =, and = ( ; ; :::; ). It wll also rove helful to note that under Assumtons and, E( t ) =, Cov( t ; jt ) =, for all 6= j, and for each ; V ar( t ) = = E( " t " M" ) < K <. Furthermore, we have Hence, for each t E(h t; ) =, V ar(h t; ) = x t x t < K <, (7) Cov(h t; ; h jt; ) =, and Cov(h t; ; t ) = O : (8) h t; = = h t; = O (); and w t = = Wth these relmnary results n mnd, we wrte CD as s CD = ( ) j=+ 8 hs corrects the statement made n error n Pesaran (4). t= t = O (). (9) t jt ; (3) 8

12 and note that and hence j=+ t jt = 4 s CD = ( ) t= 4 t t 3 5 ; (3) P 3 P t t 5 : (3) However, usng (5), = t = = t + ( ) = h t; + = t t = t + ( ) ( ) = = = h t; h t; + = h t; ; t Consder now the terms nvolvng h t;, and note that (usng (7), (8), and (9)) : t= ( ) = h t; = t= = h t; = O = Further, t= = t t= h t; = O = : ( ) = h t; = w t h t; ; where w t, and h t; are de ned by (9), and are bounded ndeendently dstrbuted random varables wth zero means, and w t are serally uncorrelated. Hence w t h t;, t= as and, n any order. Smlarly, = = t = h t; t= = = w t h t; = O = : 9

13 herefore, where s Z = ( ) CD = Z + o (); (33) t= 4 P 3 P t t 5 (34) and o () ndcates terms that tend to zero n robablty as and, n any order. o derve the dstrbuton of Z ; recall that w t = = P t, and wrte Z as s Z = ( ) (U V ) ; (35) where and V = U = t= P P t= t w t E(wt ) ; E( t) P P t= = t ; where t = t E( t), and E(wt ) = P E( t): Under our assumtons, t are cross-sectonally and temorally ndeendently dstrbuted wth mean and a nte varance V ar ( t ) = E( 4 t) 4, such that su V ar ( t ) < K <. Hence, t readly follows that E(V ) = ; V ar (V ) = t= V ar ( t ) < su V ar ( t ) = O : (36) Consder now U, and recall that w t s temorally ndeendent, wth E(w t) = ; and E(w t ) = P E( t) < su < K <. Hence E(U ) = ; V ar(u ) = t= V ar wt = t= h E wt 4 E w t : (37) But, notng that t are cross-sectonally ndeendent, E wt 4 = j= r= s= E t jt rt st " = 3 E # t + = 3 E(w t) + E 4 t E 4 t ; (38)

14 hence, substtutng (38) nto (37) we get V ar(u ) = t= ow usng (36) n (34), and then n (33), we have Also, notng that we obtan U = E w t + t= CD = U + O = + o(): E(w t ) = t= w t = = t : E( t) = + O ; w t E(wt ) = t= E 4 t : w t +; But for any t and as ; w t d (; ); and therefore wt d t (); where t (), for t = ; ; :::; are ndeendent ch-square varates wth degree of freedom. hs n turn mles that as, wt ; for t = ; ; :::;, are ndeendent random varates wth mean zero and a unt varance. Hence, U d (; ); as and, notng also that the term O = vanshes wth, consderng that = = O( += ) and <. herefore, the CD test s vald for and tendng to n nty n any order. It s also clear that snce the mean of CD s exactly equal to zero for all xed > k + and ; the test s lkely to have good small samle roertes (for both and small), a conjecture whch seems to be suorted by extensve Monte Carlo exerments to be reorted n Secton 7. 5 Asymtotc Dstrbuton of the CD est Under Weak Cross- Sectonal Error Deendence In ths secton we consder the asymtotc dstrbuton of the CD statstc under the null of weak cross-sectonal deendence, H w de ned by ( ). o ths end we assume that for each ( P t= " tf t = O =, P t= x tft = O =, P t= f tft = I m + O = (39) : We also make the followng standard assumtons about the regressors 9 j = j + O ( = ), " = O ( = ); (4) 9 hese assumtons allow for the ncluson of lagged deendent varables amongst the regressors and can be relaxed further to take account of non-statonary I() regressors.

15 where s a ostve de nte matrx. Consder now the CD test statstc de ned by () and note that under H w, the vector of the OLS resduals s gven by e = (M " + M F ) ; where F = (f ; f ; :::; f ) ; and as before M = I ( ). In ths case the dstrbuton of ^ j s qute comlcated and deends on the magntude of the factor loadngs and the cross correlaton atterns of the regressors and the unobserved factors. It does not, however, deend on the error varances,. Under Hw ; t de ned by (), can be wrtten as or more comactly where t = f t + " t ( " M " + " M F + F M F ) = x t ( ) (F + " ) ( " M " + " M F + F M F ) = ; t = ~ t + = ~ ht; + ~g t; (4) ~ t = " t ; h ~ t; = x t ~ ; and Usng (4) n (3), we now have = " M " + " M F + F M F = ; s CD = ( ) ~g t; = f t x t ( ) F : (4) t= 4 P ~ t + = ht; ~ +~g t; P ( ~ t + = ht; ~ +~g t; ) 3 5 : (43) Followng the dervatons n the revous secton, t s ossble to show that under Assumtons -4, (39), and (4), the null of weak cross-sectonal deendence gven by (), then the CD statstcs tends to (; ) f t= P ~g t; ; E( ~ t), (44) t= o establsh these results, we rst note that under Assumtons (39), and (4) = + + O ( = ): ~g t; : (45)

16 Usng ths result we have E( ~ t) = " E t + = + : But under (5), P + = O( ); and P E(~ t), f <. Consder now the other two exressons n (45), and note that t= P ~g t; = = t= + P ~ f t ~ t= F F P x t ( ) F~ (46) ~ x t ( ) F~ (47) ~ (F ) ( ) ( F) ~ where ~ = ~ =. But under (39) and (5) and settng = O ( ), we have F ~ F ~ = O += ; and Smlarly, P ~ F P F ~, as, f + = <, or f < ( )=4. = = = x t ( ) F~ t= t= j= j= ~ F ( ) x t x jt j j j F~ j ~ (F ) ( ) ( j ) j j j F ~ j ~ F A ~ F A ; where A = ( ). But (usng the norm kak = r(a A)) ~ F A ~ kf A k ~ = ~ [ r(f A F)] = ; 3

17 and F A F = F F O P ~ ~ = = O ( ) ; and for = O( ), = O (), by Assumton (39). Hence P ~ F A = t= x t ( ) F~ = ~ F A ~ F A = O = : hus, the second term of (47) vanshes f < ( + )=4, whch s sats ed f < ( easly establshed that the thrd term n (46) also vansh f < ( + )=4. Fnally, consder the second exresson n (45) and note that )=4. It s also = = = t= t= t= ~g t; h ~ f t x t ( ) F~ ; ~ f t ft ~ + ~ F ( ) x t x t ( ) F~ ~ F F~ ~ F A F~ : ~ f t x t ( ) F~ ; P P Usng smlar lnes of reasonng as above, t s easly establshed that t= ~g t;, f < ( )=, whch s sats ed f < ( )=4, consderng that. he above results are summarzed n the followng theorem: heorem Consder the anel data model (), and suose that Assumtons to 4, (39), and (4) hold. Suose further that and, such that =, where les n the range (; ] and s a nte ostve non-zero constant. hen the CD statstc de ned by () has the lmtng (; ) dstrbuton as and, so long as, the exonent of cross- sectonal deendence of the errors, u t, s less than ( )=4. In the case where and tend to n nty at the same rate the CD statstc tends to (; ) f < =4. he CD test s consstent for all values of > =, wth the ower of the test rsng n and. he ower roertes of the CD test follows drectly from the dervatons rovded above. ote that snce > then the order of P and P ~ wll be the same. 4

18 6 Cross Secton Deendence n Heterogeneous Dynamc Panels he analyses of the revous sectons readly extend to models wth lagged deendent varables. As an examle consder the followng rst-order dynamc anel data model y t = ( ) + y ;t + u t, = ; ; :::; ; t = ; ; :::; ; where y = + c " ; and for each the errors, u t, t = ; ; :::; are serally uncorrelated wth a zero mean and a unt varance but could be cross-sectonally correlated. he above sec caton s qute general and allows the underlyng AR() rocesses to be statonary for some ndvduals and have a unt root for some other ndvduals n the anel. In the statonary case, f the rocess has started a long tme n the ast we would have c = ( ) =. In the unt root case where =, c could stll d er across deendng on the number of erods that the th unt root rocess has been n oeraton before the ntal observaton, y. Gven the comlcated nature of the dynamcs and the mx of statonary and unt root rocesses that could reval n a gven anel, testng for cross-sectonal deendence s lkely to be comlcated and n general mght requre and to be large. As t s well known the OLS estmates of c and for the ndvdual seres, as well as the xed and random e ects anel estmates used under sloe homogenety ( = ) are based when s small. he bas could be substantal for values of near unty. evertheless, as t turns out n the case of ure autoregressve anels (wthout exogenous regressors) the CD test s stll vald for all values of ncludng those close to unty. he man reason les n the fact that deste the small samle bas of the arameter estmates, the OLS or xed e ects resduals have exactly mean zero even for a xed, so long as u t t = ; ; :::; are symmetrcally dstrbuted. o see ths we rst wrte the ndvdual AR() rocesses n matrx notatons as (y + ) = D u ; (48) where y = (y ; y ; :::; y ), u = (u ; u ; :::; u ), + s a ( + ) vector of ones, D s a ( + ) ( + ) dagonal matrx wth ts rst element equal to and the remanng elements equal to, and = : B..... A he OLS estmates of ndvdual ntercets and sloes can now be wrtten as ^ = u H G M G H u u H G M G H u ; ^ = ( ) + G H u G H u ^ : 5

19 where H = D, G = ( ; I ), G = (I ; ), and s a vector of zeros. Usng these results we now have the followng exresson for the OLS resduals, e t = y t ^ ^ y ;t, e t = ^ (y ;t ) + u t G H u + G H u ^ : Usng (48) we also note that y ;t = s t H u, where s t s a ( + ) selecton vector wth zero elements excet for ts t th element whch s unty. herefore, e t ; and hence t = e e = et wll be an odd functon of u, and we have E( t) =, t = ; ; ::;, under the assumton that u has a symmetrc dstrbuton. hus, under the null hyothess that u t and u jt are cross-sectonally ndeendent we have E(^ j ) =, and the CD test contnues to hold for ure dynamc heterogeneous anel data models. Under weak cross-sectonally deendent errors t s easly seen that the condtons (44) and (45) are sats ed under () as and. Fnally, the CD test wll be robust to structural breaks so long as the uncondtonal mean of the rocess remans unchanged, namely f E(y t ) = ; for all t. For roofs and further dscussons see Pesaran (4). 7 Small Samle Evdence In nvestgatng the small samle roertes of the CD test we consder two basc anel data regresson models, a statc model wth a sngle exogenous regressor, and a dynamc second-order autoregressve sec caton. Both models allow for heterogenety of sloes and error varances and nclude two unobserved factors for modellng d erent degrees of cross-sectonal deendence n the errors, as measured by the maxmal cross-sectonal exonents of the unobserved factors. he observatons for the statc anel are generated as where s IID(; ); y t = + x t + u t ; for = ; ; :::; ; t = ; ; :::; ; x t = x x t + t ; = ; ; ::: for t = ; ; :::; ; t s IID(; ); and x = ( x) = ; for = ; ; :::. We do not exect the small samle roertes of the CD test to deend on the nature of the regressors, and throughout the exerments we set x = :9. We allow for heterogeneous sloes by generatng them as s IID(; ), for = ; ; :::;. he errors, u t, are generated as a serally uncorrelated mult-factor rocess: u t = f t + f t + " t ; wth " t s IID(; " ), " s IID ()=, for = ; ; :::;. he factors are generated as f jt s IID(; ), for j = and. he factor loadngs are generated as: j = v j ; for = ; ; :::; M j and j = ; ; j = Mj ; for = M j + ; M j + ; :::; and j = ; In the more general case where the anel data model contans lagged deendent varables as well as exogenous regressors, the symmetry of error dstrbuton does not seem to be su cent for the symmetry of the resduals, and the roblem requres further nvestgatons. 6

20 where M j = [ j ] for j = ; ; v j s IIDU( vj :5; vj + :5). We set vj = for j = ;. We set =, snce our relmnary analyss suggested that the results are not much a ected by the choce of, although one would exect that the erformance of the CD test to deterorate f values of close to unty are consdered. In such cases larger samle szes () are needed. Here by settng =, we are also able to consder the baselne case where the errors are cross-sectonally ndeendent, whch corresonds to = f =. But f 6= one does not obtan error cross-sectonal ndeendence by settng =. We consdered a one-factor as well as a two-factor sec caton. In the one-factor case we set = (; :; :; :5; :35; :5; :65; :75; :85; :9; ): In the two-factor case, and = max( ; ). More sec cally, we set (; ); (:; ); (:; :); (:5; :5); (:35; :5); (:5; :5); ( ; ) = (:65; :5); (:75; :5); (:85; :5); (:9; :5); (:; :5) ; so that n the case of the two-factor model we also have = max( ; ) = (; :; :; :5; :35; :5; :65; :75; :85; :9; ): he dynamc anel data model was generated as a second-order autoregressve rocess wth heterogeneous sloes: y t = ( ) + y ;t + y ;t + u t : and u t were generated exactly as n the case of the statc sec caton. he autoregressve coe cents, and, were generated as s IIDU(; :4);and = :, for all, and xed across relcatons. All exerments were carred out for = ; 5; ; 5; 5 and = ; 5;, to evaluate the alcablty of the CD test to anels where s much larger than. he number of relcatons was set to. he results are summarzed n ables and for the statc and dynamc sec catons, resectvely. he tables gve the rejecton frequences of the CD test for d erent values of, samle szes and. he left anels of the tables refer to the one-factor error models and the rght anels to the two-factor case. For all values of and the rejecton frequences are around 5% (the nomnal sze of the CD test) when < =4 and start to rse sgn cantly as aroaches and exceed the :5 threshold, and attans ts maxmum of unty for :75. hese ndngs hold equally for statc and dynamc models. However, at = =4, there s some evdence of over rejecton (7% as comared to 5%) when s small relatve to, namely for = and =. he Monte Carlo evdence matches the asymtotc theory remarkably well, and suggests that the test can be used frutfully as a relude to the estmaton and nference concernng the values of n the range [:7; ] whch are tycally dent ed wth strong factor deendence. See also Baley, Kaetanos and Pesaran (). 8 Concludng Remarks hs aer rovdes a rgorous roof of the valdty of the CD test roosed n Pesaran (4), and further establshes that the CD test s best vewed as a test of weak cross-sectonal deendence. 7

21 he null hyothess of the CD test s shown to be < ( )=4, where s the exonent of cross-sectonal deendence ntroduced n Baley, Kaetanos and Pesaran (), and measures the degree to whch exands relatve to, as de ned by = O( ), for values of <. It s shown that the CD test s artcularly owerful aganst > =; and ts ower rses wth and n. As a test of weak cross-sectonal deendence, the CD test contnues to be vald under farly general condtons even when s small and large. he test can be aled to balanced and unbalanced anels and s shown to have a standard normal dstrbuton assumng that the errors are symmetrcally dstrbuted. he Monte Carlo evdence reorted n the aer shows that the CD statstc rovdes a smle and owerful test of weak cross-sectonal deendence n the case of statc as well as dynamc anels. As a ossble area of further research t would be nterestng to nvestgate f the test of crosssectonal ndeendence roosed n Hsao, Pesaran and Pck () for non-lnear anel data models can also be vewed as a test of weak-cross-sectonal deendence, and n artcular determne the range of values of for whch the test has ower. 8

22 References [] Anseln, L. (988), Satal Econometrcs: Methods and Models, Dorddrecht: Kluwer Academc Publshers. [] Anseln, L. (), Satal Econometrcs, n B. Baltag (ed.), A Comanon to heoretcal Econometrcs, Blackwell, Oxford. [3] Baley,., G. Kaetanos, and Pesaran, M.H. (), "Exonent of Cross-sectonal Deendence: Estmaton and Inference", Unversty of Cambrdge Workng Paers n Economcs 6, Faculty of Economcs, Unversty of Cambrdge. [4] Baltag, B. Q. Feng, and Kao, C. (), "estng for Shercty n a Fxed E ects Panel Data Model. he Econometrcs Journal 4, [5] Breusch,.S., and Pagan, A.R. (98), he Lagrange Multler est and ts Alcaton to Model Sec catons n Econometrcs, Revew of Economc Studes, 47, [6] Chamberlan, G. (983), "Funds, factors and dvers caton n Arbtrage rcng theory", Econometrca 5, [7] Chudk, A., Pesaran, M. H. and osett, E. (), "Weak and strong cross-secton deendence and estmaton of large anels", he Econometrcs Journal, 4, C45 C9. [8] Cl, A. and Ord, J.K. (973), Satal Aurocorrecton, London: Pon. [9] Cl, A and Ord, J.K. (98), Satal Processes: Models and Alcatons, London: Pon. [] Conley,.G. and oa, G. (), Soco-economc Dstance and Satal Patterns n Unemloyment, Journal of Aled Econometrcs 7, [] Frees, E. W. (995). Assessng cross-sectonal correlaton n anel data. Journal of Econometrcs 69, [] Hanng, R.P. (3), Satal data Analyss: heory and Practce, Cambrdge Unversty Press, Cambrdge. [3] Hsao, C., M.H. Pesaran, Pck, A. (), "Dagnostc ests of Cross-secton Indeendence for Lmted Deendent Varable Panel Data Models", Oxford Bulletn of Economcs and Statstcs, forthcomng. [4] Leberman, O. (994), "A Lalace aroxmaton to the moments of a rato of quadratc forms", Bometrka 8, [5] Moscone, F. and osett, E. (9), "A Revew and Comarsons of ests of Cross-Secton Indeendence n Panels", Journal of Economc Surveys, 3, [6] Moran, P.A.P. (948), he Interretaton of Statstcal Mas, Bometrka, 35, [7] Pesaran, M.H. (4), "General Dagnostc ests for Cross Secton Deendence n Panels", CESfo Workng Paer Seres o. 9 ; IZA Dscusson Paer o. 4. Avalable at SSR: htt://ssrn.com/abstract=

23 [8] Pesaran, M.H., Schuermann,., and Wener, S.M. (4), Modelng Regonal Interdeendences usng a Global Error-Correctng Macroeconomc Model, Journal of Busness Economcs and Statstcs (wth Dscussons and a Rejonder),, 9-8. [9] Pesaran, M.H., A. Ullah, and. Yamagata, (8), "A Bas-Adjusted LM est Of Error Cross Secton Indeendence", he Econometrcs Journal,, 5 7. [] Sara ds, V.,. Yamagata, D. Robertson (9), "A est of Cross Secton Deendence for a Lnear Dynamc Panel Model wth Regressors", Journal of Econometrcs, 48, [] Sara ds, V. and. Wansbeek, (), "Cross-sectonal Deendence n Panel Data Analyss", Econometrc Revews, forthcomng. [] Zellner, A. (96), An E cent Method for Estmatng Seemngly Unrelated Regressons and ests of Aggregaton Bas, Journal of Amercan Statstcal Assocaton, 58,

24 able : Rejecton frequences of the CD test at 5% sgn cance level for statc heterogeneous anels wth one exogenous regressor One factor wo factors \= 5 \= s maxmal cross-sectonal exonent of the errors u t n the anel data model y t = + x t + u t, u t = f t + f t + "" t, = ; :::;, t = ; :::;. = max( j), where j corresonds to the rate at whch P j rses wth (O ( j )), for j = ; factors.

25 able : Rejecton frequences of the CD test at 5% sgn cance level for AR() heterogeneous anels One factor wo factors \= 5 \= s maxmal cross-sectonal exonent of the errors u t n the anel data model y t = ( ) + y ;t + y ;t + u t, u t = f t + f t + "" t, = ; :::;, t = ; :::;. = max( j), where j corresonds to the rate at whch P j rses wth (O ( j )), for j = ; factors.