esting Weak Cross-Sectional Deendence in Large Panels M. Hashem Pesaran University of Southern California, and rinity College, Cambridge January, 3 Abstract his aer considers testing the hyothesis that errors in a anel data model are weakly cross sectionally deendent, using the exonent of cross-sectional deendence, introduced recently in Bailey, Kaetanios and Pesaran (). It is shown that the imlicit null of the CD test deends on the relative exansion rates of and. When = O ( ), for some < ; then the imlicit null of the CD test is given by < ( )=4, which gives < =4, when and tend to in nity at the same rate such that = ; with being a nite ositive constant. It is argued that in the case of large anels, the null of weak deendence is more aroriate than the null of indeendence which could be quite restrictive for large anels. Using Monte Carlo exeriments, it is shown that the CD test has the correct size for values of in the range [ ; =4], for all combinations of and, and irresective of whether the anel contains lagged values of the deendent variables, so long as there are no major asymmetries in the error distribution. Keywords: Exonent of cross-sectional deendence, Diagnostic tests, Panel data models, Dynamic heterogenous anels. JEL-Classi cation: C, C3, C33. his aer comlements an earlier unublished aer entitled "General Diagnostic ests for Cross Section Deendence in Panels", which was distributed in 4 as the Working Paer o. 435 in Cambridge Working Paers in Economics, Faculty of Economics, University of Cambridge. I am grateful to atalia Bailey and Majid Al-Sadoon for roviding me with excellent research assistance, and for carrying out the Monte Carlo simulations. I would also like to thank two anonymous referees as well as Alex Chudik, George Kaetanios, Ron Smith, akashi Yamagata, and Aman Ullah for helful comments. Financial suort from the ESRC Grant ES/I366/ is gratefully acknowledged.
Introduction his aer is concerned with tests of error deendence in the case of large linear regression anels where (the cross section dimension) is large. In the case of anels where is small (say or less) and the time dimension of the anel ( ) is su ciently large the cross correlations of the errors can be modelled (and tested statistically) using the seemingly unrelated regression equation (SURE) framework originally develoed by Zellner (96). In such anels where is xed as, traditional time series techniques, including log-likelihood ratio tests, can be alied. A simle examle of such a test is the Lagrange multilier (LM) test of Breusch and Pagan (98) which is based on the average of the squared air-wise correlation coe cients of the residuals. However, in cases where is large standard techniques will not be alicable and other aroaches must be considered. In the literature on satial statistics the extent of cross-sectional deendence is measured with resect to a given connection or satial matrix that characterizes the attern of satial deendence according to a re-seci ed set of rules. For examle, the (i; j) elements of a connection matrix, w ij, could be set equal to if the i th and j th regions are joined, and zero otherwise. See Moran (948) and further elaborations by Cli and Ord (973, 98). More recent accounts and references can be found in Anselin (988, ), and Haining (3, Ch. 7). his aroach, aart from being deendent on the choice of the satial matrix, is not aroriate in many economic alications where sace is not a natural metric and economic and socioolitical factors could be more aroriate. In the absence of ordering, tests of cross-sectional indeendence in the case of large anels have been considered in Frees (995), Pesaran (4), Pesaran, Ullah and Yamagata (8), Sara dis, Yamagata, Robertson (9), and Baltagi, Feng and Kao (). Recent surveys are rovided by Moscone and osetti (9), and Sara dis and Wansbeek (). he null hyothesis of these tests is the cross-sectional indeendence of the errors in the anel regressions, and the tests are based on air-wise correlation coe cients of the residuals, ^ ij, for the (i; j) units, comuted assuming homogeneous or heterogeneous sloes. he original LM test of Breusch and Pagan (98), and its modi ed version for large anels by Pesaran, Ullah and Yamagata (8), are based on ^ ij, and test the hyothesis that all airwise error covariances, Cov (u it ; u jt ), are equal to zero for i 6= j. In contrast, we show that the imlicit null of the CD test, roosed in Pesaran (4), which is based on ^ ij, is weak crosssectional deendence discussed in Chudik, Pesaran and osetti (), and further develoed in Bailey, Kaetanios and Pesaran (, BKP). More seci cally, we show that the imlicit null of the CD test deends on the relative exansion rates of and. In general, if = O ( ) for some in the range (; ], then the imlicit null of the CD test is given by < ( )=4; where is the exonent of cross-sectional deendence de ned by = [=(( ))] P P j=i+ ij = O( ), with ij denoting the oulation correlation coe cient of u it and u jt. measures the degree of cross-sectional deendence amongst the errors and attains its highest value of unity if the number of non-zero air-wise error correlations tend to in nity at the same rate as. his is related to the rate at which the largest eigenvalue of the correlation matrix of the errors, max (R), exands with, where R = ( ij ). = corresonds to the strong cross-sectional deendence case where max (R) rises at the same rate as. (see, for examle, Chamberlain, 983). he value of = corresonds to the oosite extreme where max (R) is xed in. Intermediate values of For emirical alications where economic distance such as trade atterns are used in modelling of satial correlations see Conley and oa () and Pesaran, Schuermann, and Weiner (4).
in the range (; ) corresond to cases where max (R) changes with but at a slower rate. here is also a corresondence between the rate of exansion of max (R) and the number of non-zero factor loadings in a factor reresentation of the errors which is discussed in the literature, although the focus has been on the extreme values of = and =. (see, for examle, Chudik, Pesaran and osetti ()). BKP show that is identi ed and can be estimated consistently if = <. his aer comlements BKP by showing that the null hyothesis that lies in the range [; =) can be tested using the CD statistic if is close to zero ( almost xed as ), but in the case where = ( and at the same rate) then the imlicit null of the CD test is given by < =4. he null of weak cross-sectional deendence also seems more aroriate than the null of crosssectional indeendence in the case of large anel data models where only ervasive cross deendence is of concern. For examle, in ortfolio analysis full diversi cation of idiosyncratic errors is achieved if the errors are weakly correlated, and cross-sectional error indeendence is not required. In estimation of anels only strong cross-sectional error deendence can ose real roblems, and in most alications weak cross-sectional error deendence does not ose serious estimation and inferential roblems. he small samle roerties of the CD test for di erent values of and samle sizes are investigated by means of a number of Monte Carlo exeriments. It is shown that the CD test has the correct size for values of in the range [; =4], for all combinations of and, and irresective of whether the anel contains lagged values of the deendent variables, so long as there are not major asymmetries in the error distributions. his is in contrast to the LM based tests (such as the one roosed by Pesaran, Ullah and Yamagata, 8) that require the regressors to be strictly exogenous. In line with the theoretical results, the CD test tends to over-reject if is large relative to and is within the interval (=4; =]. he CD test also has satisfactory ower for all values of > = and rises with so long as > =. he rest of the aer is organized as follows. he anel data model and the LM tests of error cross-sectional indeendence are introduced in Section. he concet of weak cross-sectional deendence is introduced and discussed in Section 3. he use of CD statistic for testing weak cross-sectional deendence is discussed in Section 4, where the asymtotic distribution of the test is rigorously established under the null of indeendence. he distribution of CD statistic under the more general null of weak deendence is considered in Section 5, and the conditions under which it tends to (; ) are derived. he alication of the test to heterogeneous dynamic anels is discussed in Section 6. Small samle evidence on the erformance of the test is rovided in Section 7. Section 8 concludes. Panel Data Models and the LM ye ests of Cross- Sectional Error Indeendence Consider the following anel data model y it = ix it + u it, for i = ; ; :::; ; t = ; ; :::; ; () where i indexes the cross section dimension and t the time series dimension, x it is a (k + ) vector of observed time-varying regressors (individual-seci c as well as common regressors). An individual-seci c intercet can be included by setting the rst element of x it to unity. he coe cients, i, are de ned on a comact set and allowed to vary across i. For each i, u it s
IID(; i ), for all t, although they could be cross-sectionally correlated. he deendence of u it across i could arise in a number of di erent ways. It could be due to satial deendence, omitted unobserved common comonents, or idiosyncractic air-wise deendence of u it and u jt (i 6= j) with no articular attern of satial or common comonents. he regressors could contain lagged values of y it, be either stationary (or integrated of order zero, I()) or have unit roots (or integrated of order, I()). But in the derivations below we assume x it s I(), and distinguish between the static and dynamic cases where the regressors are strictly exogenous and when they are weakly exogenous, seci cally when x it = (; y i;t ; :::; y i;t ). he testing rocedure is alicable to xed and random e ects models as well as to the more general heterogeneous sloe or random coe cient seci cations.. LM ye ests In the SURE context with xed and, Breusch and Pagan (98) roosed a Lagrange multilier (LM) statistic for testing the null of zero cross equation error correlations which is articularly simle to comute and does not require the system estimation of the SURE model. he test is based on the following LM statistic CD lm = j=i+ where ^ ij is the samle estimate of the air-wise correlation of the residuals. Seci cally, ^ ij; ^ ij = ^ ji = P P t= e ite jt = P = ; () t= e it t= e jt and e it is the Ordinary Least Squares (OLS) estimate of u it de ned by e it = y it ^ ix it ; (3) with ^ i being the OLS estimator of i comuted using the regression of y it on x it for each i; searately. he LM test is valid for relatively small and su ciently large. In this setting Breusch and Pagan show that under the null hyothesis of no cross-sectional deendence, seci ed by Cov (u it ; u jt ) = ; for all t, i 6= j; (4) CD lm is asymtotically distributed as chi-squared with ( )= degrees of freedom. As it stands this test is not alicable when. However, noting that under H, ^ a ij s with ^ ij, i = ; ; ::;, j = i + ; ; :::;, being asymtotically indeendent, the following scaled version of CD lm can be considered for testing the hyothesis of cross deendence even for and large: s CD lm = ( ^ ij ): (5) ( ) j=i+ he assumtion that u it s are serially uncorrelated is not restrictive and can be accommodated by including a su cient number of lagged values of y it amongst the regressors. 3
a It is now easily seen that under H with rst followed by we would have CD lm s (; ). However, this test is likely to exhibit substantial size distortions for large and small, a situation that can frequently arise in emirical alications. his is rimarily due to the fact that for a nite, E( ^ ij ) will not be correctly centered at zero, and with large the incorrect centering of the LM statistic is likely to be accentuated, resulting in size distortions that tend to get worse with. A bias corrected version of CD lm is roosed in Pesaran, Ullah and Yamagata (8) under the assumtions that the regressors are strongly exogenous and the errors are normally distributed. In what follows we roose a test of weak cross-sectional deendence, which we argue to be more aroriate for large anels, where mere incidence of isolated deendencies are of little consequence for estimation or inference. 3 Weak Error Cross-Sectional Deendence As noted in the introduction when is large it is often more aroriate to consider the extent of error cross-sectional deendence rather than the extreme null hyothesis of error indeendence that underlies the LM tye tests. his is in line, for examle, with the assumtion of aroximate factor models discussed in Chamberlain (983) in the context of caital asset ricing models. o this end we consider the following factor model for the errors u it = i ( if t + " it ) ; (6) where f t = (f t ; f t ; :::; f mt ) is the m vector of unobserved common factors (m being xed) with E(f t ) =, and Cov(f t ) = I m, i = ( i ; i ; :::; im ) is the associated vector of factor loadings, " it are idiosyncratic errors that are cross-sectionally and serially indeendent with mean zero and a unit variance, namely " it s IID(; ), and i is a scaler that controls the variance of u it. he degree of cross-sectional deendence of the errors, u it, is governed by the rate at which the average airwise error correlation coe cient, = [=( )] P P j=i+ ij, tends to zero in, where ij = Corr(u it ; u jt ). In the case of the above factor model we have, V ar(u it ) = i = i ( + i i); ij = i j, for i 6= j, where i = i + i i : (7) hen it is easily seen that = P i i ; (8) where = P i. Consider now the e ects of the j th factor, f jt, on the i th error, u it, as measured by ij, and suose that these factor loadings take non-zero values for M j out of the cross-section units under consideration. hen following BKP, the degree of cross-sectional deendence due to the j th factor can be measured by j = ln(m j )= ln(), and the overall degree of cross-sectional deendence of the errors by = max j ( j ). BKP refer to as the exonent of cross-sectional deendence. can take any value in the range to, with indicating the highest degree of cross-sectional deendence. Considering that i i = O(m) where m is xed as, the exonent of crosssectional deendence of the errors can be equivalently de ned in terms of the scaled factor loadings, 4
i = ( i ; i ; :::; im ). Without loss of generality, suose that only the rst M j elements of ij over i are non-zero, and note that 3 j; = M j @ ij + ij A = M M j j @M j ij A = j j = O( j ); where j = O( j M P Mj j ij i=m j+ ), and using (8) we have 6=, for a nite M j and as M j. Similarly, P = O( ): ij = In what follows we develo a test of the null hyothesis that < =. he case where > = is covered in BKP. he values of in the range [; =) corresond to di erent degrees of weak cross-sectional deendence, as comared to values of in the range (=; ] that relate to di erent degrees of strong cross-sectional deendence. 4 A est of Weak Cross-Sectional Deendence Given that is de ned by the contraction rate of, we base the test of weak cross-sectional error deendence on its samle estimate, given by b = ( ) j=i+ ^ ij ; (9) where ^ ij is already de ned by (). he CD test of Pesaran (4) is in fact a scaled version of b which can be written as = ( ) CD = b : () In what follows we consider the distribution of the CD statistic under three di erent null hyotheses. o establish comarability and some of the basic results we begin with CD statistic under hyothesis of cross-sectional indeendence de ned by H : i = ; for all i: () We then consider the asymtotic distribution of the CD statistic as and, such that = O( ), for <, and show that the imlicit null of the CD test is given by H w : < ( )=4: () As argued earlier, such a null is much less restrictive for large anels than the air-wise error indeendence assumtion that underlies the LM tye tests which are based on ^ ij. Initially, we derive the asymtotic distribution of the CD test in the case of the standard anel data model, () subject to the following assumtions: 3 he main results in the aer remain valid even if P i=m j + ij = O(). maintain the assumtion that P i=m j + ij =. But for exositional simlicity we 5
Assumtion : he factor model, (6), holds. he idiosyncratic errors, " it, are serially and cross-sectionally indeendent with mean zero and a unit variance, and are symmetrically distributed around for all i and t. f t s IID(; I m ), f t and " i;t are distributed indeendently for all t and t, and < i < K <. he factor loadings, i, are indeendently distributed across i: Assumtion a: he regressors, x it, are strictly exogenous such that E (" it j j ) =, for all i; j and t; (3) where i = (x i ; x i ; :::; x i ). Also i i is a ositive de nite matrix for any xed, i i ii ; as, with ii being a ositive de nite matrix, and i j ij = O(). Assumtion b: he following standardized errors it; = " it =( " im i " i ) = ; for i = ; ; :::; ; t = ; ; :::; (4) where " i = (" i ; " i ; :::; " i ) ; are cross-sectionally and serially indeendent and have fourth-order moments, namely E 4 it; < K <. (5) It is also assumed that the regressors are suitably bounded such that h it; = x it i i i i ; (6) have fourth-order moments, namely E h 4 it; < K <, where i = ( i; ; i; ; :::; i; ), and M i = I i ( i i) i. Assumtion 3: > k + 4 and the OLS residuals, e it, de ned by (3), are not all zero. Assumtion 4: he factor loadings, i, de ned by (6) satisfy the -summability condition i = O( ): (7) Remark: Assumtion is standard in the literature on multi-factor models, excet for the assumtion that the idiosyncratic errors are symmetrically distributed. his assumtion is needed to ensure that the standardized errors, it;, also have mean zero. he rst art of Assumtion b is met if su i E(" 6 it ) < K <, and > k + 4. his result can be established using Lemmas in Lieberman (994). For future use it is also worth noting that E it; = E " it " i M = + O : (8) i" i In fact when " it are normally distributed we have the following exact result E it; = k 3 ; which follows since ( k )" it =" i M i" i has an F distribution with and k degrees of freedom. In the case where " it are non-guassian, the moment conditions in (8) follow from standard alication of results in Lieberman (994), assuming the regressors are suitably bounded. 6
heorem Consider the regression model, (), and suose that Assumtions -3 hold, and the idiosyncratic errors, " it, are symmetrically distributed around, then under H : i =, and for all > and > k + we have E ^ ij = ; for all i 6= j; (9) E ^ ij^ is =, for all i 6= j 6= s; () E (CD) = ; () a (k + ) V ar(cd) = + ( k ) ( k ) ; () a = P P j=i+ r (A ia j ) < k + ; (3) ( ) where A i = i ( i i) i ; ^ ij and CD are de ned by () and (), resectively. 4 Proof:. First note that the air-wise correlation coe cients can be written as where it are the scaled residuals de ned by ^ ij = it jt ; (4) t= e it it = ( e i e ; (5) = i) e it is the OLS residuals from the individual-seci c regressions, de ned by (3), and e i = (e i ; e i ; :::; e i ). Also under H, e i = i M i " i, where " i = (" i ; " i; :::; " i ). herefore, conditional on x it, the scaled residuals, it, are odd functions of the disturbances, " it, and under Assumtion we have Hence, unconditionally we also have E ( it j i ) = ; for all i and t. E ( it ) =, for all i and t. Using this result in (4) now yields (recall that under H indeendent), E ^ ij = ; which in turn establishes that (using ()) the errors, " it ; are cross-sectionally E(CD) =, 4 Similar results can also be obtained for xed or random e ects models. It su ces if the OLS residuals used in the comutation of ^ ij are relaced with associated residuals from xed or random e ects seci cations. But the CD test based on the individual-seci c OLS residuals are robust to sloe and error-variance heterogeneity whilst the xed or random e ects residuals are not. 7
for any, and all > k +. Under H and Assumtions -3, ^ ij and ^ is are cross-sectionally uncorrelated for i; j and s, such that i 6= j 6= s. More seci cally E ^ ij^ is = = t= t = t= t = E it jt it st E ( it it ) E jt E (st ) = ; for i 6= j 6= s: Also since the regressors are assumed to be strictly exogenous, we further have 5 V ar ^ij = E ^ ij = r(mi M j )=( k ) : Using this result in () we have V ar(cd) = = ( )( k ) @ j=i+ [ (k + )] ( k ) + ( )( k ) [ (k + ) + r (A i A j )] A @ j=i+ r (A i A j ) A : Hence where a (k + ) V ar(cd) = + ( k ) ( k ) = + O ; a = P P j=i+ r (A ia j ) : ( ) But r (A i A j ) < r(a i ) r(a j ) = = k +, and we must also have a < k +. his comletes the roof of the theorem. he above results also suggest the following modi ed version of CD, CD gcd = h i + a (k+) = ; (6) ( k ) ( k ) which is distributed exactly with a zero mean and a unit variance. In cases where k is relatively large, and the regressors, x it ; are cross-sectionally weakly correlated, the term involving a in the exression for the variance of CD will be small and both statistics are likely to erform very similarly, and the CD test is recommended on grounds of its simlicity. o kee the analysis simle, and without of loss generality, in what follows we shall focus on the CD test. 5 For a roof see Aendix A. in Pesaran, Ullah, and Yamagata, (8,. 3-4). 8
4. he distribution of the CD test under H Consider now the distribution of the CD test. As shown in (), the elements in the double summation that forms the CD statistic are uncorrelated but they need not be indeendently distributed when is nite. herefore, when is nite the standard central limit theorems can not be exloited in order to derive the distribution of the CD statistic. 6 o resolve the roblem we rst re-write the CD statistic (de ned by ()) as s CD = @ ^ij A ; (7) ( ) j=i+ and recall that ^ ij = P t= it jt ; where it is de ned by (5). ow under H : i =, using standard results from regression analysis, we have h it; = it = it + = h it; ; (8) x it i i i i; ; (9) where it; = " it =(" i M i" i = ) =, and i = ( i ; i ; :::; i ). o simlify the notations we abstract from the deendence of it; on. It will also rove helful to recall that under Assumtions, a and b, E( it; ) =, Cov( it; ; jt; ) =, for all i 6= j, and for each i; V ar( it; ) = i = + O( ). Before deriving the asymtotic distribution of CD we resent some reliminary results in the following lemma and rovide the roofs in the Aendix. Lemma : Consider w t; = = it;, and h t; = = h it;, (3) where it; and h it; are de ned by (4) and (6), and suose that Assumtions and hold. hen E (w t; ) = ; E(wt; ) = + O ; (3) E (h t; ) = ; E h t; = x i i it x it + O = O (); (3) Cov (h t; ; w t ; ) = O( = ), Cov (w t; ; w t ; ) = O( ), for t 6= t ; (33) w t; = O (), h t; = O (), and w t; h t; : (34) t= 6 his corrects the statement made in error in Pesaran (4). 9
With these results in mind, we write CD as s CD = ( ) j=i+ t= it jt ; (35) and note that and hence j=i+ it jt = 4 s CD = ( ) t= 4 it it 3 5 ; (36) P 3 P it it 5 : (37) However, using (8), and = it = = it + ( ) = h it; (38) + = it it = it + ( ) ( ) = = = h it; Consider now the terms involving h it;, and note that h it; h it; + (39) = it : ; t= ( ) = h it; = h t; = O = (4) t= Further, t= = it t= h it; = O = : (4) ( ) = h it; = w t h t; ; (4) where w t, and h t; are de ned by (3). herefore, using (4), (4) and (4) in (38) and (39) and then substituting the results back in (37) we have CD = Z + o (); (43) t=
where s Z = ( ) t= 4 P 3 P it it 5 (44) and o () indicates terms that tend to zero in robability as and, in any order. o derive the distribution of Z ; recall that w t; = = P it, and write Z as s Z = ( ) (U V ) ; (45) where and V = U = t= P P t= it wt; E(wt; ) ; E( it) P P t= = it ; where it = it E( it), and E(w t; ) = + O. Under our assumtions, it are crosssectionally indeendently distributed with mean and a nite variance V ar ( it ) = E( 4 it) 4 i, such that su i V ar ( it ) < K <. Furthermore, it are serially uncorrelated as (see (33)). Hence, it readily follows that E(V ) = ; V ar (V ) = t= V ar ( it ) < su V ar ( it ) = O : (46) Consider now U, and recall that w t; is asymtotically temorally uncorrelated with E(w t; ) = ; and E(w t; ) = + O. Hence, i E(U ) = ; V ar(u ) = t= V ar wt; = t= h E wt; 4 E w t; i : (47) But, noting that it; are cross-sectionally indeendent, we have E wt; 4 = j= r= s= E it jt rt st " = 3 E # it + = 3 E(w t; ) + E 4 it E 4 it ; (48)
and substituting (48) into (47) we obtain V ar(u ) = t= E w t; + t= E 4 it : ow using (46) in (44), and then in (43), we have CD = U + O = + o(): Also, since E(w t; ) = + O U = t=, we have wt; E(wt; ) = t= wt; + O : But for any t and as ; w t; = = P it d (; ); and therefore wt; d t (); where t (), for t = ; ; :::; are indeendent chi-square variates with degree of freedom. his in turn imlies that as, wt; ; for t = ; ; :::;, are indeendent random variates with mean zero and a unit variance. Hence, U d (; ); as and, noting also that the term O = vanishes with, considering that = = O( += ) and <. o summarize: heorem Consider the anel data model (), and suose that Assumtions to 4 hold. hen under the null hyothesis of cross-sectional error indeendence,(), the CD statistic de ned by () has the limiting (; ) distribution as and, in any order. It is also clear that since the mean of CD is exactly equal to zero for all xed > k + and ; the test is likely to have good small samle roerties (for both and small), a conjecture which seems to be suorted by extensive Monte Carlo exeriments to be reorted in Section 7. We now show that the CD test is in fact alicable even if the errors are weakly correlated, namely the imlicit null of the CD test is weak cross-sectional error deendence, rather than indeendence. We argue that when is large the null hyothesis that all airs of errors are indeendently distributed is rather restrictive. What is needed is a less restrictive null that ostulate a su ciently large number of airs of errors are indeendently distributed. he analysis below formalizes what is meant by "a su ciently large number". 5 Asymtotic Distribution of the CD est Under Weak Cross- Sectional Error Deendence In this section we consider the asymtotic distribution of the CD statistic under the null of weak cross-sectional deendence, H w de ned by (). o this end we assume that for each i ( P t= " itf t = O =, P t= x itft = O =, P t= f tft = I m + O = (49) :
We also make the following standard assumtions about the regressors 7 i j = ij + O ( = ), i " i = O ( = ); for all i and j; (5) where ii is a ositive de nite matrix. Consider now the CD test statistic de ned by () and note that under H w, the vector of the OLS residuals is given by e i = i (M i " i + M i F i ) ; where F = (f ; f ; :::; f ) ; and as before M i = I i ( i i) i. In this case the distribution of ^ ij is quite comlicated and deends on the magnitude of the factor loadings and the cross correlation atterns of the regressors and the unobserved factors. It does not, however, deend on the error variances, i. Under Hw ; it de ned by (5), can be written as or more comactly where it = i f t + " it ( " i M i" i + " i M if i + i F M i F i ) = x it ( i i) i (F i + " i ) ( " i M i" i + " i M if i + i F M i F i ) = ; it = ~ it + = ~ hit; + ~g it; (5) ~ it = " it ; h ~ it; = x i i it i ~ i i ; and i = " im i " i + " im i F i + if M i F i = ; t= ~g it; = i f t x it ( i i) i F i : (5) Using (5) in (37), we now have s P 6 CD = ~ it + =~ P ~it h it; + ~g it; + =~ 3 h it; + ~g it; 7 4 5 : ( ) (53) Following the derivations in the revious section, it is ossible to show that under Assumtions -4, (49), and (5), the null of weak cross-sectional deendence given by (), then the CD statistics tends to (; ) if i E( ~ it), (54) 7 hese assumtions allow for the inclusion of lagged deendent variables amongst the regressors and can be relaxed further to take account of non-stationary I() regressors. 3
t= P ~g it; ; and t= o establish these results, we rst note that under Assumtions (49), and (5) Using this result we have E( ~ it) = i = + i i + O ( = ): " E it i ~g it; : (55) + i = i i i + i : i But under (7), P i i + i = O( ); and P i E(~ it), if <. Consider now the two exressions in (55), and note that t= P ~g it; = = t= + P ~ if t ~ i t= F F P x it ( i i) i F~ i (56) ~ i x it ( i i ) if~ i (57) ~ i (F i ) ( i i ) ( if) ~ i where ~ i = ~ i = i. But under (49) and (7) and setting = O ( ), we have F ~ F i ~ i = O += ; and Similarly, P ~ i = = = F P F ~ i t=, as, if + = <, or if < ( )=4. x it ( i i ) if~ i t= j= j= ~ if i ( i i ) x it x jt j j j F~ j ~ i (F i ) ( i i ) ( i j ) j j j F ~ j ~ if A i ~ if A i ; 4
where A i = i ( i i) i. But (using the norm kak = r(a A)) ~ if A i ~ i kf A i k ~ = i~ i [ r(f A i F)] = ; and F A i F = F i i i if O P ~ i~ i = = O ( ) ; and for = O( ), 8 t= x it ( i i ) if~ i = = O (), by Assumtion (49). Hence P ~ if A i = ~ if A i hus, the second term of (57) vanishes if < ( + )=4, which is satis ed if < ( easily established that the third term in (56) will also vanish if < ( + )=4. Finally, consider the second exression in (55) and note that = = = t= t= t= ~g it; h ~ if t x it ( i i ) if~ i i ; ~ i f t ft ~ i + ~ if i ( i i) x it x it ( i i) i F~ i ~ if F~ i ~ if A i F~ i : ~ if t x it ( i i) i F~ i ~ if A i = O = : )=4. It is also P P Using similar lines of reasoning as above, it is easily established that t= ~g it;, if < ( )=, which is satis ed if < ( )=4, considering that. he above results are summarized in the following theorem: heorem 3 Consider the anel data model (), and suose that Assumtions to 4, (49), and (5) hold. Suose further that and, such that =, where lies in the range (; ] and is a nite ositive non-zero constant. hen the CD statistic de ned by () has the limiting (; ) distribution as and, so long as, the exonent of cross- sectional deendence of the errors, u it, is less than ( )=4. In the case where and tend to in nity at the same rate the CD statistic tends to (; ) if < =4. he CD test is consistent for all values of > =, with the ower of the test rising in and. his theorem rovides a full characterization of the distribution of the CD test under di erent degrees of cross-sectional deendence, ranging from indeendence, weak deendence, to strong deendence. he theorem also establishes the asymtotic ower distribution of the CD test and shows that the test has su cient ower when > =, with the ower rising in. 8 ote that since i > then the order of P i and P ~ i will be the same. ; 5
6 Cross Section Deendence in Heterogeneous Dynamic Panels he analyses of the revious sections readily extend to models with lagged deendent variables. As an examle consider the following rst-order dynamic anel data model y it = a i + i y i;t + i u it, i = ; ; :::; ; t = ; ; :::; ; (58) where a i = i ( i ), y i = i + c i u i, and for each i the errors, u it, t = ; ; :::; are serially uncorrelated with a zero mean and a unit variance but could be cross-sectionally correlated. he above seci cation is quite general and allows the underlying AR() rocesses to be stationary for some individuals and have a unit root for some other individuals in the anel. In the stationary case E(y it ) = i, and if the rocess has started a long time in the ast we would have c i = i i =. In the unit root case where i =, c i could still di er across i deending on the number of eriods that the i th unit root rocess has been in oeration before the initial observation, y i. Given the comlicated nature of the dynamics and the mix of stationary and unit root rocesses that could revail in a given anel, testing for cross-sectional deendence is likely to be comlicated and in general might require and to be large. As it is well known the OLS estimates of a i and i for the individual series, as well as the xed and random e ects anel estimates used under sloe homogeneity ( i = ) are biased when is small. he bias could be substantial for values of i near unity. evertheless, as it turns out in the case of ure autoregressive anels (without exogenous regressors) the CD test is still valid for all values of i including those close to unity. he main reason lies in the fact that desite the small samle bias of the arameter estimates, the OLS or xed e ects residuals have exactly mean zero even for a xed, so long as u it, t = ; ; :::; are symmetrically distributed. o see this we rst write the individual AR() rocesses, (58), in matrix notation as i (y i i + ) = D i u i ; (59) where yi = (y i; y i ; :::; y i ), u i = (u i; u i ; :::; u i ), + is a ( + ) vector of ones, D i is a ( + ) ( + ) diagonal matrix with its rst element equal to c i and the remaining elements equal to i, and i i i = : B..... C @ A i he OLS estimates of individual intercets and sloes can now be written as ^ i = u i H i G M G H i u i u i H i G M G H i u ; i ^a i = i ( i ) + G H i u i G H i u i ^ i ; 6
where M = I ( ), H i = i D i, G = ( ; I ), G = (I ; ), and is a vector of zeros. Using these results we now have the following exression for the OLS residuals, e it = y it ^a i ^i y i;t, for t = ; ; :::; ; e it = ^i i (y i;t i ) + i u it G H i u i + G H i u i ^ i : Using (59) we also note that y i;t i = s t H i u i, where s t is a ( + ) selection vector with zero elements excet for its t th element which is unity. herefore, e it ; and hence it = e i e = i eit will be an odd function of u i, and we have E( it) =, t = ; ; ::;, under the assumtion that u i has a symmetric distribution. hus, under the null hyothesis that u it and u jt are cross-sectionally indeendent we have E(^ ij ) =, and the CD test continues to hold for ure dynamic heterogeneous anel data models. Under weak cross-sectionally deendent errors it is easily seen that the conditions (54) and (55) are satis ed under () as and. Finally, the CD test will be robust to structural breaks so long as the unconditional mean of the rocess remains unchanged, namely if E(y it ) = i ; for all t. For roofs and further discussions see Pesaran (4). 9 7 Small Samle Evidence In investigating the small samle roerties of the CD test we consider two basic anel data regression models, a static model with a single exogenous regressor, and a dynamic second-order autoregressive seci cation. Both models allow for heterogeneity of sloes and error variances and include two unobserved factors for modelling di erent degrees of cross-sectional deendence in the errors, as measured by the maximal cross-sectional exonents of the unobserved factors. he observations for the static anel are generated as where i s IID(; ); y it = i + i x it + u it ; for i = ; ; :::; ; t = ; ; :::; ; x it = ix x it + it ; i = ; ; ::: for t = ; ; :::; ; it s IID(; ); and x i = ( x) = i ; for i = ; ; :::. We do not exect the small samle roerties of the CD test to deend on the nature of the regressors, and throughout the exeriments we set ix = :9. We allow for heterogeneous sloes by generating them as i s IID(; ), for i = ; ; :::;. he errors, u it, are generated as a serially uncorrelated multi-factor rocess: u it = i f t + i f t + " it ; with " it s IID(; i" ), i" s IID ()=, for i = ; ; :::;. he factors are generated as f jt s IID(; ), for j = and. he factor loadings are generated as: ji = v ji ; for i = ; ; :::; M j and j = ; ; ji = i Mj ; for i = M j + ; M j + ; :::; and j = ; 9 In the more general case where the anel data model contains lagged deendent variables as well as exogenous regressors, the symmetry of error distribution does not seem to be su cient for the symmetry of the residuals, and the roblem requires further investigation. 7
where M j = [ j ] for j = ; ; v ji s IIDU( vj :5; vj + :5). We set vj = for j = ;. We set =, since our reliminary analysis suggested that the results are not much a ected by the choice of, although one would exect that the erformance of the CD test to deteriorate if values of close to unity are considered. In such cases larger samle sizes () are needed. Here by setting =, we are also able to consider the baseline case where the errors are cross-sectionally indeendent, which corresonds to setting the exonent of cross-sectional deendence to zero, = if =. But if 6= one does not obtain error cross-sectional indeendence by setting =. We considered a one-factor as well as a two-factor seci cation. In the one-factor case we set = (; :; :; :5; :35; :5; :65; :75; :85; :9; ): In the two-factor case, and = max( ; ). More seci cally, we set (; ); (:; ); (:; :); (:5; :5); (:35; :5); (:5; :5); ( ; ) = (:65; :5); (:75; :5); (:85; :5); (:9; :5); (:; :5) ; so that in the case of the two-factor model we also have = max( ; ) = (; :; :; :5; :35; :5; :65; :75; :85; :9; ): he dynamic anel data model was generated as a second-order autoregressive rocess with heterogeneous sloes: y it = ( i i ) i + i y i;t + i y i;t + u it : i and u it were generated exactly as in the case of the static seci cation. he autoregressive coe cients, i and i, were generated as i s IIDU(; :4);and i = :, for all i, and xed across relications. All exeriments were carried out for = ; 5; ; 5; 5 and = ; 5;, to evaluate the alicability of the CD test to anels where is much larger than. he number of relications was set to ;. he results are summarized in ables and for the static and dynamic seci cations, resectively. he tables give the rejection frequencies of the CD test for di erent values of, samle sizes, and. he left anels of the tables refer to the one-factor error models and the right anels to the two-factor case. For all values of and the rejection frequencies are around 5% (the nominal size of the CD test) when < =4 and start to rise signi cantly as aroaches and exceed the :5 threshold, and attains its maximum of unity for :75. hese ndings hold equally for static and dynamic models. However, at = =4, there is some evidence of over rejection (7% as comared to 5%) when is small relative to, namely for = and =. he Monte Carlo evidence matches the asymtotic theory remarkably well, and suggests that the test can be used fruitfully as a relude to the estimation and inference concerning the values of in the range [:7; ] which are tyically identi ed with strong factor deendence. See also Bailey, Kaetanios and Pesaran (). 8 Concluding Remarks his aer rovides a rigorous roof of the validity of the CD test roosed in Pesaran (4), and further establishes that the CD test is best viewed as a test of weak cross-sectional deendence. 8
he null hyothesis of the CD test is shown to be < ( )=4, where is the exonent of cross-sectional deendence introduced in Bailey, Kaetanios and Pesaran (), and measures the degree to which exands relative to, as de ned by = O( ), for values of <. It is shown that the CD test is articularly owerful against > =; and its ower rises with and in. As a test of weak cross-sectional deendence, the CD test continues to be valid under fairly general conditions even when is small and large. he test can be alied to balanced and unbalanced anels and is shown to have a standard normal distribution assuming that the errors are symmetrically distributed. he Monte Carlo evidence reorted in the aer shows that the CD statistic rovides a simle and owerful test of weak cross-sectional deendence in the case of static as well as dynamic anels. As a ossible area of further research it would be interesting to investigate if the test of crosssectional indeendence roosed in Hsiao, Pesaran and Pick () for non-linear anel data models can also be viewed as a test of weak-cross-sectional deendence, and in articular determine the range of values of for which the test has ower. 9
Aendix: Proof of the Lemma Using (6) and (4) we rst note that (conditional on i ) E(h it; ) =, V ar(h it; ) = i x it i i x it < K <, for all i, (6) Cov(h it; ; h jt; ) = O( ), for all i and j; Cov(h it; ; jt; ) =, for all i 6= j, Cov(h it; ; it; ) = E h it; it; = = i x i i it x it = O ; (6) and by (8), we have i = + O : Furthermore, since conditional on i, it and h it; are cross-sectionally indeendent we also have V ar(h t; ) = V ar (h it; ) = x it i i x it + O( ) = O(); (6) V ar (w t; ) = V ar it; = + O( ) = O(); (63) and similarly E (w t; h t; ) = Cov (h t; ; w t ; ) = E (h t; w t ; ) = Cov (w t; ; w t ; ) = E (w t; w t ; ) = E ( it it ) = ow let q t; = w t; h t;, and consider the limiting roerties of E h it; it; = O = ; (64) E (h it; it ) = O = ; for t 6= t ; "it " it E " i M = O( ); for t 6= t : i" i = q = q t; : t= First, using (64) we note that E (q t; ) = O =, and E (q ) = O =. Also Cov (q t; ; q t ; ) = E (w t; h t; w t ; h t ; ) E (w t; h t; ) E (w t ; h t ; ) = E (w t; h t; ) E (w t ; h t ; ) + E (w t; w t ; ) E (h t; h t ; ) +E (w t; h t ; ) E (h t; w t ; ) E (w t; h t; ) E (w t ; h t ; ) = E (w t; w t ; ) E (h t; h t ; ) + E (w t; h t ; ) E (h t; w t ; ), for t 6= t : = O( );
and E qt; = E wt; h t; = j= r= s= E it; jt; h rt; h st; = E it; h it; + " E # " it; r But by assumtion (see (8)), E it; h it; E 4 it; )E(h 4 it; E h it; # : < K <, and using (6) and (63) we have E qt; = O(). Recalling also that E (q ) = O = it follows that V ar(q t; ) = O(). Using this result in conjunction with Cov (q t; ; q t ; ) = O( ), established we have V ar (q ) = t= V ar(q t; ) + t= t = Cov (q t; ; q t ; ) = O( ); herefore, recalling that E (q ) = O =, we nally have q ; (65) as and tend to in nity (in any order). We also note that, = = it = h it; = = q t; = O = : (66)
References [] Anselin, L. (988), Satial Econometrics: Methods and Models, Dorddrecht: Kluwer Academic Publishers. [] Anselin, L. (), Satial Econometrics, in B. Baltagi (ed.), A Comanion to heoretical Econometrics, Blackwell, Oxford. [3] Bailey,., G. Kaetanios, and Pesaran, M.H. (), "Exonent of Cross-sectional Deendence: Estimation and Inference", University of Cambridge Working Paers in Economics 6, Faculty of Economics, University of Cambridge. [4] Baltagi, B. Q. Feng, and Kao, C. (), "esting for Shericity in a Fixed E ects Panel Data Model. he Econometrics Journal 4, 5-47. [5] Breusch,.S., and Pagan, A.R. (98), he Lagrange Multilier est and its Alication to Model Seci cations in Econometrics, Review of Economic Studies, 47, 39-53. [6] Chamberlain, G. (983), "Funds, factors and diversi cation in Arbitrage ricing theory", Econometrica 5, 35-33. [7] Chudik, A., Pesaran, M. H. and osetti, E. (), "Weak and strong cross-section deendence and estimation of large anels", he Econometrics Journal, 4, C45 C9. [8] Cli, A. and Ord, J.K. (973), Satial Aurocorrection, London: Pion. [9] Cli, A and Ord, J.K. (98), Satial Processes: Models and Alications, London: Pion. [] Conley,.G. and oa, G. (), Socio-economic Distance and Satial Patterns in Unemloyment, Journal of Alied Econometrics 7, 33-37. [] Frees, E. W. (995). Assessing cross-sectional correlation in anel data. Journal of Econometrics 69, 393 44. [] Haining, R.P. (3), Satial data Analysis: heory and Practice, Cambridge University Press, Cambridge. [3] Hsiao, C., M.H. Pesaran, Pick, A. (), "Diagnostic ests of Cross-section Indeendence for Limited Deendent Variable Panel Data Models", Oxford Bulletin of Economics and Statsitics, forthcoming. [4] Lieberman, O. (994), "A Lalace aroximation to the moments of a ratio of quadratic forms", Biometrika 8, 68-69. [5] Moscone, F. and osetti, E. (9), "A Review and Comarisons of ests of Cross-Section Indeendence in Panels", Journal of Economic Surveys, 3, 58 56. [6] Moran, P.A.P. (948), he Interretation of Statistical Mas, Biometrika, 35, 55-6. [7] Pesaran, M.H. (4), "General Diagnostic ests for Cross Section Deendence in Panels", CESifo Working Paer Series o. 9 ; IZA Discussion Paer o. 4. Available at SSR: htt://ssrn.com/abstract=5754.
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able : Rejection frequencies of the CD test at 5% signi cance level for static heterogeneous anels with one exogenous regressor One factor wo factors \= 5 \= 5..6.49.6..56.57.54..6.49.6..56.57.54..6.49.6..56.57.54.5.8.63.9.5.7.7.8.35.8.63.9.35.8.86.97.5.63.48.664.5.36.364.598.65.83.996..65.78.973..75.988...75.98.999..85....85....9....9............ 5.59.49.55. 5.63.5.55..59.49.55..63.5.55..69.54.67..74.6.56.5.69.54.67.5.74.6.56.35.83.74.96.35.85.78.86.5.383.546.756.5.97.576.76.65.97.997..65.883.995..75....75....85....85....9....9.............6.56.6..48.48.49..6.56.6..48.48.49..64.6.66..54.54.5.5.69.7.79.5.54.6.6.35.94.5.78.35.78.7.35.5.36.646.886.5.3.647.774.65.955...65.96...75....75....85....85....9....9............ 5.59.49.5. 5.55.5.5..59.49.5..55.5.5..6.5.53..6.58.56.5.6.5.53.5.59.59.55.35.85.84.9.35.75.79.95.5.37.54.86.5.3.464.84.65.994...65.996...75....75....85....85....9....9............ 5.54.5.5. 5.6.6.58..54.5.5..6.6.58..56.53.47..6.55.58.5.56.53.5.5.64.64.63.35.74.8.87.35.8.9.9.5.4.69.849.5.35.6.87.65....65....75....75....85....85....9....9........... is maximal cross-sectional exonent of the errors u it in the anel data model y it = i + i x it + u it, u it = i f t + i f t + i"" it, i = ; :::;, t = ; :::;. = max( j), where j corresonds to the rate at which P ij rises with (O ( j )), for j = ; factors.
able : Rejection frequencies of the CD test at 5% signi cance level for AR() heterogeneous anels One factor wo factors \= 5 \= 5..58.5.5..5.47.56..58.5.5..5.47.56..58.5.5..5.47.56.5.7.7.76.5.59.7.88.35.7.7.76.35.64.78.5.5.3.37.577.5.88.345.583.65.89.993.999.65.758.979.999.75.977...75.97...85.999...85.999...9....9............ 5.55.45.49. 5.47.5.55..55.45.49..47.5.55..59.5.57..49.54.6.5.59.5.57.5.49.54.6.35.8.7.87.35.6.76.9.5.6.599.77.5.7.49.75.65.85.997..65.758.996..75.997...75.995...85....85....9....9.............59.46.5..54.5.54..59.46.5..54.5.54..65.5.5..59.54.53.5.67.5.6.5.68.67.66.35.98.89.39.35.94.3.55.5.39.53.86.5.64.6.867.65.957...65.866...75....75....85....85....9....9............ 5.6.47.44. 5.56.48.5..6.47.44..56.48.5..57.53.47..59.55.55.5.57.53.47.5.59.5.55.35.76.8.88.35.7.77.89.5.3.577.88.5.54.55.85.65.996...65.99...75....75....85....85....9....9............ 5.5.47.53. 5.49.5.5..5.47.53..49.5.5..5.46.56..49.5.53.5.58.47.6.5.5.5.55.35.66.77.99.35.64.69.83.5.38.57.83.5.33.533.789.65.998...65.998...75....75....85....85....9....9........... is maximal cross-sectional exonent of the errors u it in the anel data model y it = ( i i) i + iy i;t + iy i;t + u it, u it = i f t + i f t + i"" it, i = ; :::;, t = ; :::;. = max( j), where j corresonds to the rate at which P ij rises with (O ( j )), for j = ; factors.