Directed Tests of No Cross-Sectional Correlation in Large-N Panel Data Models

Transcription

1 Directed ests of o Cross-Sectional Correlation in Large- Panel Data Models Matei Demetrescu Christian Albrechts-University of Kiel Ulrich Homm University of Bonn Revised version: April 9, 205 Abstract Cross-sectional dependence in panel data can significantly affect the inference about slope parameters. Existing procedures test however for cross-unit dependence per se. Based on the principle of information matrix equality tests of White 982, Econometrica 50, -25, we propose directed tests for no crosssectional dependence. he new tests essentially check whether there is cross-unit dependence that is momentous for the variance of OLS slope parameter estimators and thereby for standard inferential procedures. he tests rely on suitably weighted sample residual cross-covariances or cross-correlations, and are in this sense a generalization of Pesaran s 2004, CESifo Working Paper 229 test for cross-sectional error dependence. We derive the joint, asymptotics of the proposed tests, which, under the null, follow asymptotic chi-squared distributions without restrictive sphericity or distributional assumptions. Moreover, the relative rates at which and grow to infinity are only mildly restricted. We use Monte Carlo simulation to gauge the finite sample power of the directed tests and compare it to extant procedures. he performance of the proposed tests in terms of size and power is good, even when the number of cross-sections is decisively larger than the number of time observations. When using the outcome of the directed tests to decide whether to use panel-robust standard errors or not for inference on slope parameters, the slope parameter tests are not affected. hey are, however, affected when using alternative cross-dependence tests for such decision: the use of generic error cross-correlation tests may induce serious size distortions in slope parameter tests and lead to wrong conclusions in applied work. Key words: Cross-unit correlation; Information matrix equality; Joint asymptotics; Pre-test JEL classification: C2 Hypothesis esting, C23 Models with Panel Data he authors would like to thank the editor M. Hashem Pesaran, three anonymous referees, Jörg Breitung, Kai Carstensen and Jean-Marie Dufour for very helpful comments and suggestions, as well as Benjamin Hillmann for computational research assistance. Corresponding Author: Institute for Statistics and Econometrics, Christian-Albrechts- University of Kiel, Olshausenstr , D-248 Kiel, Germany. address: mdeme@stat-econ.uni-kiel.de.

2 Introduction Cross-sectional dependence in panel data can arise for various reasons, such as global shocks unaccounted for be they economic, political or technological in nature or unobserved shocks affecting only a subset of cross-sectional units. Such dependence can have dramatic effects on the asymptotic and finite-sample properties of the least squares estimator and standard inferential procedures in panel data models; see e.g. Andrews 2005 and also the more recent survey of Chudik and Pesaran 203. One way to deal with potential cross-unit error dependence is to use so-called panelrobust covariance matrix estimation techniques, for instance as proposed by Arellano 987, Beck and Katz 995 or Driscoll and Kraay 998. Another way is to employ a preliminary test for cross-sectional dependence and adjust the estimation and inference procedures according to the outcome of the pretest. he relevant issue for practitioners is then the reliability of the procedure as a whole, i.e. of the pretesting step and the inference on the slope parameters taken together. o test for error cross-unit dependence, Breusch and Pagan 980 proposed a Lagrange multiplier [LM] test. he test, relying on the squared pairwise correlations between the residuals of each unit, is appropriate for panels with large time dimension and small cross-sectional dimension, and is therefore more often employed in the seemingly unrelated regressions framework of Zellner 962. When is large relative to, however, it is well-known that the LM test is severely oversized; see for instance Pesaran 2004 and Pesaran, Ullah, and Yamagata While the LM test is essentially relying on squared Pearson correlation coefficients, Frees 995 suggests the use of Spearman rank correlation coefficients. Frees statistic also tends to exhibit size distortions when is large relative to, even if the distortions are less pronounced than in the case of the LM test. One way to amend the size distortions of the initial LM test is that of Pesaran et al hey compute the expected values and variances of the squared correlation coefficients under the assumption of normally distributed disturbances and adjust the LM test accordingly. he resulting adjusted LM test is asymptotically normal as followed by. he empirical size of the test is controlled for moderateto-large and small. he normality assumption does not appear to be restrictive, as is documented by simulation experiments with χ 2 - and t-distributed errors; see e.g. Pesaran et al and Baltagi et al. 20. Another correction has recently been suggested by Baltagi et al. 202, who compute the bias of the LM statistic based on fixed-effects pooled OLS residuals for, and / c, which leads to a bias-corrected version of the LM test. on-negligible size distortion can occur, however, when is very large relative to cf. Baltagi et al., 20 2

3 Pesaran 2004, 205 puts forward an alternative approach; the corresponding test employs residual Pearson correlation-coefficients without squaring them, on the grounds that the null of an average zero correlation is often more relevant for practitioners, say in portfolio management. he test has correct empirical size in finite samples even when is much larger than. But it has, by construction, low power when error correlation is present and the correlation coefficients roughly sum up to zero. his can be the case when the disturbances are generated from a factor model where the loadings average to zero. Pesaran 205 analyzes the implicit null hypothesis of Pesaran s 2004 test, which is given by weak dependence rather than by independence of errors between cross-sections. Finally, following John 972, Baltagi et al. 20 discuss a sphericity test which can be used to infer about cross-correlation when the data are cross-sectionally homoskedastic. he common feature of these three approaches is that one ultimately test for the existence of cross-unit correlation per se. Here, we derive tests for the null of no crosscorrelation based on the information matrix equality test principle of White 982. We may interpret our procedure as being directed against cross-sectional correlation that impacts the estimate of the covariance matrix of the slope parameter estimators. From this perspective, our tests check whether the simple OLS variance matrix estimate significantly differs from a cross-correlation robust estimate. As a consequence, our procedure leverages the residual cross-covariances with sample cross-section covariances of the explanatory variables. his, of course, restricts the set of alternatives against which the test has power; but it is precisely these detected alternatives that are relevant from the standpoint of covariance matrix estimation for slope parameter estimators and thus for standard inferential procedures. Concretely, we discuss in Section 2 several versions of the test, which differ in how cross-sectional variance heterogeneity or heteroskedasticity is taken into account. hey are all robust to cross-sectional heteroskedasticity under the null, and differences arise in the presence of cross-sectional heteroskedasticity only under the alternative. Moreover, they are equivalent under homoskedasticity. he baseline variant uses the residual cross-covariances directly, whereas the second weights them using the unitspecific residual variances. he third variant relies on cross-correlations rather than cross-covariances; when focussing on an intercept only, it turns out to be essentially Pesaran s 2004 test. While our procedures inherit the good power properties of Pesaran s 2004 test when the correlation coefficients all have the same sign, we expect power gains in other directions of the space of alternative hypotheses and provide Monte Carlo evidence in favour of this conjecture. Furthermore, we establish the asymptotic distribution for and going to infinity 3

4 jointly; a χ 2 limiting distribution of the test is obtained without relying on a specific distribution for the disturbances. For the test based on residual cross-covariances, the relative rates of and are not explicitly restricted to a certain path. For the other two variants, not restricting the shape of the distribution of the disturbances comes at the price of having to control the growth rates of the cross-sectional dimension relative to more strictly. Our Monte Carlo simulations in Section 3 show that the directed tests have correct empirical size for as little as 0 time periods 20 for the correlations-based test, even for much larger. he Monte Carlo analysis furthermore illustrates the severe shortcomings of several alternative procedures for testing for no cross-correlation when used as a pretest to decide between ordinary and panel-robust standard errors. In contrast, our directed test procedures work reliably. Let us introduce some notation before proceeding. We denote vectors by boldface symbols. Let denote the Euclidean vector norm and the corresponding induced r matrix norm. Further, r stands for both the L r vector norm, r, and for r the L r norm of a random variable or vector, E r. he Kronecker product of two matrices is denoted by, and diagd i denotes the diagonal matrix having d i, i =,...,, as diagonal elements. Finally, C is a generic constant whose value may differ from occurrence to occurrence. 2 esting for no cross-correlation 2. Information matrix equality testing o motivate our tests, let us first analyze the homogenous panel data model y i,t = α + x i,tβ + u i,t, i =,...,, t =,...,, where the zero-mean disturbances u i,t are assumed to be independent of the regressors x i,t. We work with a total number of K regressors including the intercept, so x i,t R K. We shall relax the assumptions on α and β later on. For deriving the likelihood function, we assume normality and independence of the disturbances, u i,t iid 0, σ 2. ormality is only required to justify the test statistics; we shall establish their limiting behavior under considerably weaker conditions, including e.g. cross-sectional heteroskedasticity; see Subsection 2.3. Let now y i denote the vector containing the observations for cross-section i, X i be the K regressor matrix {x i,t,k } with x i,t, =, and X the matrix stacking the individual regressor matrices X i. he vectors u i are the individualunit disturbance vectors, and the -dimensional vector u contains the stacked 4

5 individual-unit disturbances u i. Correspondingly, u t = u,t,..., u,t denotes the cross-section of errors at time t. Finally, the contemporaneous covariance matrix of the errors is given by E u t u t = Σ = {σ ij } i,j=,...,. Under the null, Σ = Σ 0 = σ 2 I, whereas under the alternative we use the parameterization Σ = σ 2 Ω with Ω positive definite and normalized such that tr Ω =. he null is recovered when Ω = I. hen, conditional on the regressors, the log-likelihood of model is given under the null of no cross-sectional correlation by l = C 2σ 2 i= y i,t α x i,tβ 2, t= from which the score and the Hessian can be derived, s = u i,t = σ 2 x i= t= i,t u i,t σ 2 X u H = x i,t = σ 2 x i,t x i,t x σ i,t 2 X X i= t= when treating for simplicity σ 2 as known. When not restricting Σ, the covariance matrix of the score is given by Covs X = σ 2 X Ω I X = σ 4 X Σ I X; under the null of no cross-sectional correlation and homoskedasticity, the information matrix equality holds, σ 4 X Σ I X = σ 2 X X, with Σ = Σ 0 = σ 2 I being correctly specified. Under the alternative, the equality does not hold, and a test statistic for sphericity is immediately obtained by plugging in an unrestricted estimate of Σ, say ˆΣ, and a restricted estimate of σ 2, say ˆσ 2. One rejects the null when the equality is significantly violated, i.e. when X ˆΣ I X ˆσ 2 X X = ˆσij ˆσ 2 i = j X ix j, 2 i= j= with the indicator function, is significantly different from zero. In this respect we obtain an alternative to the John test recently discussed by Baltagi et al. 20. Unlike the case of the John test, however, it is straightforward to test against cross-sectional correlation only. We simply need to check whether the 5

6 terms not involving σ ii of the difference in 2 are zero or not, i.e. we focus on whether ˆσij ˆσ 2 i = j X ix j = i= j= i j i ˆσ ij X ix j + X jx i j= is significantly different from zero. In fact, we only need to look at the lower triangular elements of the matrix on the r.h.s. due to its symmetry, so our directed tests rely on S = i ˆσ ij vechx ix j + X jx i, 3 j= where ˆσ ij are suitable estimators σ ij and vech is the half-vec operator. Having derived the statistic S on the basis of a homogenous panel data model, one possibility to compute error covariance estimates ˆσ ij is via pooled OLS residuals. Indeed, a number of cross-sectional dependence tests rely on pooled or pooled fixedeffects residuals e.g. Baltagi et al., 20, 202. Other procedures, such as the ones of Pesaran 2004 or Pesaran et al. 2008, rely however on unit-wise residuals. We too shall resort to unit-wise residuals in the following; the main reasons to do so are discussed in more detail in Subsection 2.2 which also addresses the relation of the directed test to the literature. We therefore let ˆσ ij = with M Xi = I X i X ix i X i. 2 K û iûj = K y im Xi M Xj y j In may be interesting to note that, analogously, a cross-sectional heteroskedasticity test can be based on i= ˆσ ii ˆσ 2 vechx ix i ; this is essentially a White heteroskedasticity test against cross-sectional heteroskedasticity. In order to decide on the significance, the statistic S yet has to be normalized. Lemma in the Appendix indicates that CovS = K 2 j= σ ii σ jj E trm Xi M Xj vech X ix j + X jx i vech X i X j + X jx i, provided that the disturbances are independent of the regressors and that the moments exist. Precise assumptions about the components of our model are provided and discussed in the following subsection. 2 Alternatively, one could use the QML estimator which is not corrected for degrees of freedom; the correction improves however the finite-sample properties of the test statistic while not affecting the asymptotics. 6

7 ˆV = A natural estimator for the covariance matrix of S is then given by K 2 i ˆσ iiˆσ jj tr M Xi M Xj vech X i X j + X jx i vech X i X j + X jx i, j= where, as above, ˆσ ij is a consistent estimator of σ ij based on û i. Standardizing S with its estimated covariance matrix leads to the following statistic for testing the null of no cross-sectional correlation: CD σ X = S ˆV S. he following subsection will analyze its relation to the test of Pesaran 2004 and argue that it delivers a directed test of cross-section independence: should S be significantly different from zero, one rejects the null of no cross-sectional correlation in favor of cross-correlation that affects inference on the slope parameters β. he statistic CDX σ was derived under the assumption of cross-unit homoskedasticity. While we prove in Section 2.3 that the statistic is robust to cross-unit heteroskedasticity, it may be useful to examine a variant which explicitly accounts for heteroskedasticity to begin with. Under the null hypothesis of no cross-unit correlation, we have E u t u t = Σ 0 = diagσ ii so the score becomes and the Hessian is given by s = X Σ 0 I u H = X Σ 0 I X. At the same time, the covariance matrix of the score is given under the alternative by Cov s X = X Σ 0 I Σ I Σ 0 I X. By having allowed for cross-unit heteroskedasticity, the diagonals of Cov s X and H are equal, so, by focussing again on the lower triangular elements and plugging in the corresponding estimates, we obtain S w = i ˆσ ij vechx ˆσ ix j + X jx i 4 iiˆσ jj j= 7

8 as basis for a no cross-correlation test. Analogously, its variance can be estimated by ˆV w = K 2 i tr M Xi M Xj vech X ˆσ i X j + X jx i vech X i X j + X jx i, iiˆσ jj j= leading to CD w X = S w ˆV w S w. his statistic can be seen, with a mild abuse of terminology, as a WLS variant of CD σ X, since S w = i ˆρ ij vech X i ˆσii X j + X j ˆσjj ˆσjj X i, ˆσii j= is, up to a negligible 3 term, the statistic 3 computed in the WLS-transformed model y i,t σ ii = α σ ii + x i,t σ ii β + u i,t σ ii. We compare the variants theoretically and in finite samples in Sections 2.3 and 3, but not before introducing a correlations-based version with quite a nice interpretation. 2.2 Discussion o put the new tests in relation with existing ones, let us now consider a third variant of the directed tests obtained by replacing the estimated covariances in S, cf. Equation 3, by estimated correlation coefficients: R = i ˆρ ij vech X ix j + X jx i, where ˆρij = j= û iûj. 5 û iûi 2 û jûj 2 In practice, CovR can be approximated using ˆV R = i vech X ix j + X jx i vech X i X j + X jx i, j= since Eˆρ 2 for large and iid sampling. he resulting statistic is denoted as CD ρ X = R ˆV R R. 3 Under the conditions of Proposition 2 further below. 8

9 Under homoskedasticity, the three variants CDX σ, CDw X and CDρ X can be checked to be asymptotically equivalent; see the proof of Proposition 2 in the appendix. Under crosssectional heteroskedasticity, however, we document power gains of the WLS version in Section 3 discussing the finite-sample properties of our tests. ote that, if the model only includes an intercept, the correlation-based form of our test statistic reduces to CD ρ X = 2 i 2 ˆρ ij which is the square of the test statistic proposed by Pesaran he directed tests j= may thus be seen as an extension of Pesaran s test idea. hen, to understand what the additional terms in our statistic stand for, let us now examine the standard fixed effect panel data model y i,t = α i + x i,tβ + u i,t, i =,...,. 6 By letting ȳ i and X i denote the variables after within-group demeaning, we obtain by stacking equations ȳ = Xβ + ū, with ȳ = ȳ,..., ȳ and otherwise obvious notation. Under serial independence of the disturbances we make this assumption explicit in the following subsection, the covariance matrix of the OLS estimator of β is given by Cov ˆβ X = X X X Σ I X X X. Under serial error dependence, clustered standard errors would have been the suitable choice; see e.g. Driscoll and Kraay, 998. Under the null hypothesis of no crosssectional correlation we have that σ ij = 0 for all i j, and Σ = Σ 0 = diag σ ij. Comparing the covariance matrix of the fixed-effects OLS estimator with the one obtained assuming no cross-unit correlation, we should have equality under the null, X Σ I X = X Σ 0 I X. 7 Checking whether the equality is not significantly violated in the sample i.e. comparing the usual heteroskedasticity-robust with the heteroskedasticity and cross-correlation robust, or panel-robust, covariance matrix estimator takes us to a test directed at detecting cross-unit correlation affecting inference on the slope parameters in the fixed-effects framework. We thus obtain essentially the same test statistic as before, say CD σ X, de- 9

10 rived from the information matrix equality principle with intercepts concentrated out. Moreover, this fixed-effects variant is nothing else than the complement of Pesaran s 2004 CD test: the CD test focusses on cross-correlations leveraged by intercepts only, while CD ρ X leverages with cross-products of mean-adjusted regressors which are by construction orthogonal to the intercepts. Also, since S essentially consists of weighted pairwise error cross-unit covariances, the implicit null of our test is closely related to that of the CD test as discussed by Pesaran 205. Actually, an adaptation of the arguments of Pesaran 205 to the setup of Proposition would show that the CD test and the directed tests have the same implicit null, i.e. an exponent of cross-sectional dependence 0 α < 2 ɛ/4 when = O ɛ. 4 In brief, the CD X tests include the initial CD test, and the CD X leverages the expectation of the product of regressors from different units are either zero or constant, so there is no qualitative difference in the kind of hypotheses that can be detected by CD and CD X. he simulations in Section 3 shows the behavior of the CD tests to be the same under some data generating processes exhibiting weak cross-dependence o conclude the discussion on the construction and interpretation of the new tests, let us examine the issue of which residuals to use for computing the error covariance estimates ˆσ in more detail. On the one hand, the interpretation as directed tests is based on a homogenous or at most a fixed-effects panel data model. On the other hand, the directed tests only fit the framework of Pesaran 2004 when using unit-wise residuals. Unit-wise residuals have the additional advantage of being consistent in panels with coefficient heterogeneity, whereas the pooled fixed-effects OLS residuals would contain a component due to heterogeneity. Under regressor cross-unit dependence, this component is cross-sectionally correlated, so, unless one is specifically interested in detecting such types of cross-dependence, heterogeneity turns out to be a nuisance when detecting error cross-correlation. o sum up, using residuals from individual regressions has the advantage of robustifying the cross-correlation testing procedure against parameter heterogeneity; the pooled residuals may be used too, should parameter heterogeneity not be of concern. 5 he following subsection derives the asymptotic limiting distribution of the three variants of our test and discusses the assumptions under which we work. 4 he implicit null is different when C, but the correlation-based versions require some restrictions on the relative rates at which and go to infinity which exclude proportionality. 5 he results provided in the following subsection build on unit-wise residuals. he proof of Proposition first shows that the estimation effect is negligible, and establishes then the limiting distribution of the statistic building on the true disturbances. With pooled residuals converging at a higher rate under the null of no cross-correlation, Proposition arguably holds for pooled residuals as well. 0

11 2.3 Limiting behavior Let us now state the conditions for the panel data generating process. he assumptions on the disturbances are standard in the panel data literature. Assumption. he disturbances u i,t are independent of the regressors x i,t,k for all i, i, t, t and k K, and satisfy. u i,t = ɛ i,t σii with lim inf i σ ii > 0, lim sup i σ ii < ; 2. ɛ i,t iid 0, over both i and t with E ɛ 8 i,t <. he first condition allows for cross-unit heteroskedasticity and excludes the possibility that some units dominate in the limit. he second is e.g. implied by the normality of the disturbances assumed by Pesaran et al or Baltagi et al. 202, but does not restrict the distribution of the disturbances beyond typical moment restrictions. In what concerns the regressors, we essentially require mild forms of uniformity of their properties as. Without such uniformity or analogous conditions, there may not be a limit as, jointly. Before stating the assumption, we introduce some auxiliary notation that helps dealing with the fact that some of the cross-product moments of the regressors from different units may converge to zero, while others may converge to a non-zero constant. In other words, different regressors from different units may, but do not have to, be uncorrelated. Let v ij = D vech X ix j + X jx i, where D is a /2KK + /2KK + diagonal matrix whose diagonal elements are either when E t= x i,t,k = 0 or E t= x i,t,kx j,t,k + t= x i,t,k x j,t,k = 0, or when the respective expectation is nonzero. We implicitly assume that the regressors only exhibit short-range dependence. Long-range dependence can easily be dealt with using a different standardization, but we omit the details here; integration or cointegration of the regressors is however excluded since it would lead to stochastic limiting behavior of the sample cross-product averages. Let us furthermore agree on the notation that the mth element of v ij, m =,..., /2KK +, is given by v ij,m = t= x i,t,kx j,t,k + t= x i,t,k x j,t,k or v ij,m = t= x i,t,kx j,t,k + t= x i,t,k x j,t,k for suitable k and k. Assumption 2. he regressors x i,t,k, k = 2,..., K, are stochastic and satisfy X. Pr lim, sup i X i i < = ; 2. lim, sup i ; t E xi,t,k 6 < k =,..., K; v ij 4 3. lim, sup i,j ; t E,m < m =,... /2K K +.

12 where is the matrix norm induced by the Euclidean vector norm and v ij,m the standardized cross-product of the regressors as defined above. Furthermore, the space spanned by the vectors E v ij has dimension /2K K +. he first condition stated in the assumption is standard for stochastic regressors and just formalizes the requirement that the moment matrix of the regressors is invertible in each unit of the panel. Depending on the distributional properties of the regressors, it may imply some restrictions on the relative rates at which and are allowed to diverge, but the restrictions are not obvious. Considering e.g. independent units, the probability to observe regressor moment matrices in the neighbourhood of singularity is the key quantity: the faster it vanishes in, the faster can grow. At the other end of the scale, should the regressors be common to all units i.e. extreme regressor cross-dependence, it suffices that the condition be fulfilled in, case which is wellunderstood from standard regression analysis and there is no relative rate restriction. he second and third conditions are typical moment restrictions: the second is similar to the moment condition on the disturbances, while the third focuses of the cross-product sample moments of the regressors in a given unit and requires a specific form of uniformity of their convergence. For instance, a factor model such as x i,t = Λ i f t + e i,t with Λ i a K L matrix of loadings and independence of the factors from the idiosyncratic errors generates them under suitable moment conditions on the components f t and e i,t and uniformity conditions on the loadings Λ i, as can easily be checked. Finally, no independence across the panel is required and the regressors may be allowed to be common as long as the dimensionality condition is fulfilled. he condition ensures the covariance matrix ˆV to be well-behaved; see Remark for how to deal with the situation where the condition is violated. he requirement that the regressors be stochastic for k = 2,..., K simplifies proofs and notation, but is not essential for the proofs. In fact one can treat deterministic regressors as stochastic ones, provided that the sequence of regressor values behaves, in terms of distributions, like realizations of a stochastic regressor obeying the assumption; see e.g. Amemiya 985, Chapter 4. he two assumptions allow us to establish a χ 2 limiting distribution for the three variants of the proposed directed test. Proposition. Under the above assumptions, we have as, that CD σ X d χ 2 2 KK+. Proof: see the Appendix. 2

13 Remark. he covariance matrix V is not always well-behaved, a leading example being the case where the regressors are common across units. here are two possibilities of dealing with a rank-deficitary matrix V. he first would be to simply exclude some redundant regressor cross-products. he second relies on the work of Andrews 987 and amounts to using a generalized inverse of an estimator of V, but one which ensures that the rank of ˆV converges to the true rank of V : the limiting distribution then remains chi-squared, but with rk V degrees of freedom. We provide simulation evidence that the use of the Moore-Penrose inverse in the extreme case of common regressors i.e. unity rank of V and ˆV, together with χ 2 critical values, works reliably. Remark 2. Assumption requires strict exogeneity of the disturbance terms. It can be seen from the proof of the proposition that relaxing this to weak exogeneity, say, is difficult, since the limiting null distribution relies on uncorrelatedness of the disturbances and cross-moments of the regressors. his prevents the application of the directed tests in dynamic panels, for instance. It is possible though to eliminate the regressors that are not strictly exogenous from the vector of cross-moments; the CD test of Pesaran 2004, which has been proved to work in dynamic panels under certain circumstances, can be seen as such an exogenized statistic. Remark 3. Halunga, Orme, and Yamagata 202 bootstrap the LM test of Breusch and Pagan 980 to obtain, besides an improved behavior for large, robustness to heteroskedasticity in the time dimension as well. Along these lines, an examination of the proof of Proposition reveals that an Eicker-White type covariance matrix estimator Eicker, 967; White, 980 is given by Ṽ = i k û j,t û i,t û k,t û l,t vech X ix j + X jx i vech X i X j + X jx i. t= j= k=2 l= Using Ṽ instead of ˆV thus robustifies CDX σ heteroskedasticity in the time dimension. against conditional as well as unconditional In what concerns the WLS and the correlation-based versions of our test, we need to seriously restrict the rate at which may grow to infinity. he finite-sample experiments in Section 3 suggest that the restriction is only binding for very small and large. he literature sometimes assumes symmetry of the disturbances to relax such rate restrictions; we do not find the assumption plausible in general and rather recommend the use of CD σ X when is dangerously small. Proposition 2. Under the additional assumption that 3 / 0, we have as, that CD ρ X = R ˆV R R d χ 2 2 KK+ 3

14 and Proof: see the Appendix. CD w X = S w ˆV w S w d χ 2 2 KK+. Remark 4. Results Similar to Propositions and 2 hold as well for the test variants building on X rather than on X, yet with /2 K K degrees of freedom. 3 Finite-sample behavior In this section, the empirical size and power of the tests for cross-section independence is assessed by means of Monte Carlo experiments. We first present the competing procedures to keep the paper self-contained. he simulation scenarios are described in Section 3.2, and we discuss the results in Section Alternative test procedures For completeness, we start with the LM test of Breusch and Pagan 980, although, because of its known severe size distortions when is large relative to cf. Pesaran et al., 2008, or Moscone and osetti, 2009, its use is not recommended for comparable with, or larger than,. he LM test of Breusch and Pagan 980 builds on the statistic i LM = ˆρ 2 ij, with ˆρ ij defined as in 5. Under the null hypothesis and for fixed and, LM approaches a standard normal distribution. However, for small, ˆρ 2 ij is not centered at 0. For large cross-sectional dimension in relation to the time dimension, this can lead to substantial overrejection. j= Frees test 995 rank correlation test is given by R 2 AV E = 2 i where ˆr ij is the Spearman rank correlation-coefficient between residual û i and û j. Frees 995 derives the limit distribution of Q = R 2 AV E for the case that the intercept is the only regressor in 6. he resulting limit, Q, is a weighted sum of two independent χ 2 random variables. Because of dependence on, critical values are cumbersome to compute. When is not small, Frees 995 suggests to use the j= ˆr 2 ij, 4

15 following approximately normally distributed statistic F RE = Q, where VarQ = VarQ he adjusted LM test: Pesaran et al compute the exact finite sample expectations and variance of ˆρ 2 ij imposing that, in addition to Pesaran s assumptions, the errors ɛ i,t are normally distributed. heir statistic is given by where 2 i LM adj = j= Kˆρ 2 ij µ ij ν ij, µ ij = k tr EM Xi M Xj, ν 2 ij = tr EM Xi M Xj 2 a + 2 tr E M Xi M Xj 2 a 2 with a = a 2 [ ] 2 K, and a K 8 K = 3. K + 2 K 2 K 4 As followed by, LM adj converges to a standard normal distribution. he test controls size much better, also when is large relative to. he bias-corrected LM test: Baltagi et al. 202 analyze the asymptotics of the LM test under joint, asymptotics where / c 0,. When using the pooled fixed-effects residuals, ũ = M Xu, they prove that the statistic i LM bc = ρ 2 ij 2 2 with ρ ij the correlation of ũ i and ũ j has a limiting standard normal distribution. 6 Like for the correction of Pesaran et al or the John test below, normality is required. he size control is again essentially better than that of the initial LM statistic; see Baltagi et al j= Pesaran s test: o work around the bias problem of the LM test for finite, 6 his contrasts with the previous tests, where the residuals are obtained from separate cross-section regressions. One may conjecture that, applying techniques from Baltagi et al. 20, the asymptotic results from Section 2.3 are still valid when fixed-effects residuals are employed, but we do not pursue the topic here. 5

16 Pesaran 2004 suggests to use ˆρ ij instead of ˆρ 2 ij and considers the statistic 2 i CD P = ˆρ ij. He shows that, under his assumptions, E[ˆρ ij ] = 0 and CD P approaches a standard normal distribution for,. Indeed, his test has correct empirical size also when is large relative to ; see Section 3. A standard critique of the test is that it lacks power against alternatives under which i j= ρ ij 0. he John test: A different approach is taken by Baltagi et al. 20. heir procedures tests for spherical disturbances. null hypothesis includes homoskedasticity. j= hat is, apart from independence, the Although this only allows for a limited comparison with tests of cross-section independence, the John test is included in the MC experiments, since Baltagi et al. 20 found a favorable performance relative to CD P and LM adj under homoskedasticity. he test statistic is given by J = Ω tr 2 Ω2 tr 2 2 2, 8 where Ω = K t= ũtũ t and ũ t = ũ,t,..., ũ,t contains the residuals for period t from a fixed-effects regression in model 6. A crucial assumption of Baltagi et al. 20 is that the errors are normally distributed. hen, under H 0, the statistic in 8 is asymptotically standard normal as, with / c [0,. 3.2 Simulation setup Similar to Pesaran et al and Moscone and osetti 2009, we use the following data generating process for i =,..., : y i,t = α i + β x,i,t + β 2 x 2,i,t + u i,t, where u i,t = γ i f t + σ i ɛ i,t, f t iid 0,, α i iid,, β = β 2 =. Several scenarios are considered for the regressors x l,i,t, for the cross-sectional variances σi 2, for the factor loadings γ i and for the variance σi 2 of the idiosyncratic error components. In addition to Pesaran et al and Moscone and osetti 2009 we simulate 6

17 regressors that, thanks to a factor structure, are correlated across cross-sections, x l,i,t = f x l,t γx l,i + ɛ x l,i,t where f x l,t iid 0, and ɛ x l,i,t iid 0, 0.. his makes the DGP a factoraugmented panel data model without correlation between the common components of the regressors and of the errors. It comes at no surprise, that, without such regressor cross-dependence, the CD X tests have little power, since the terms vech X ix j + X jx i in 3 basically contain empirical covariances between regressors in different sections. But the point of the CD X tests is to check whether standard inferential procedures, such as the OLS-based t-test on slope coefficients, are invalid when neglecting error cross-section correlation. 7 Since the regressor cross-correlation affects the behavior of the CD X family of tests under the alternative, we shall consider three cases for the regressor loadings γ x l,i. hus, γx l,i iid U 0.2, 0.2 with Ua, b standing for a uniform distribution on a, b, captures relatively weak regressor cross-dependence; γ x l,i iid U0.3, 0.7 stands for moderate regressor cross-dependence, and γ x l,i iid U3, 5 models strong regressor cross-dependence. We report here results for the case of moderate regressor cross-dependence: typically, weak and strong regressor cross-dependence lead to little differences so we omit them to save space. In some of the cases namely some of the power studies, the results were different, though not by a large margin; we comment briefly on the additional results in such cases and the tables are available upon request from the authors. he null scenario, S0, is designed to study size. We start with homoskedastic errors σ 2 i = and simulate the idiosyncratic error components ɛ i,t as standard normal. here is no error cross-correlation, S0: γ i = 0 for all i =,...,. he regressors are moderately cross-dependent as described above. he errors fulfill both the symmetry assumption of Pesaran 2004 and the normality assumption of Pesaran et al or Baltagi et al. 20 such that, considering the slope parameter homogeneity, all considered tests have correct asymptotic size; this will allow for meaningful power comparisons. o check the robustness of the results, the baseline scenario S0 is expanded by several cases. We build on the baseline scenario and mention for each additional case only the features that differ. First, we consider a case where the errors are heteroskedastic 7 Unreported Monte Carlo simulations show that the t-test for the null β = has correct empirical size in the absence of regressor cross-section correlation, even if cross-sectional error correlation is present. 7

18 with σ 2 i iidχ 2 in addition to being non-normal, ɛ i,t iidχ 2 / 2. Furthermore, building on normality and homoskedasticity again, we consider a case with slope heterogeneity where the regressor cross-dependence is expected to induce residual cross-correlation, a case where the regressors are common such that V has reduced rank and we may study the behavior of the test when employing a generalized matrix inverse and adjusted critical values to compute the test see Remark, and two cases of weak cross-correlation. he implementation details are as follows. For the slope heterogeneity case, β is kept constant and we generate β 2i iidu0.5, ; the variability matches roughly the one on the heterogenous simulation setup of Chudik and Pesaran 203, see Eq. 67; we did not follow the design of Chudik and Pesaran 203 since it included weakly exogenous regressors. For the common regressors case, we simply set x i,t,l = f x l,t and ɛ x i,t = 0, resort to the Moore-Penrose generalized inverse of ˆV, and employ χ 2 critical values. For the weak cross-dependence designs we followed Chudik and Pesaran 203 in setting the loadings γ i to zero for i > [ α ] + ; we work with α = 0.5 and α = 0.75 and γ i iid U0., 0.3 for i [ α ] like in the first power scenario below. We also simulated with cross-dependence index α = 0.25, yet the figures are virtually the same as those for α = 0.5 and we do not report them to save space. In the following scenarios we investigate the power of the tests. he baseline scenario exhibits positive cross-unit covariances, S: γ i iid U0., 0.3, with moderate regressor cross-dependence as described above. he errors are, like in the baseline Scenario 0, homoskedastic and Gaussian. While, in the three variations of scenario, all covariances σ ij are positive, they will approximately net out in scenario 2, i 2 j= ρ ij 0 thanks to the loadings γ i roughly averaging to zero. It has been argued cf. Pesaran et al., 2008 that, in the latter case, the CD P test lacks power, and we examine our tests for similar behavior: S2: γ i iid U 0.3, 0.3. he third considered scenario duplicates Scenario, S3: γ i iid U0., 0.3, up to the distribution of the errors, which are now generated as ɛ i,t iidχ 2 / 2 with constant unity variances, σ ii =. For S4, we replicate the baseline Scenario 2, S4: γ i iid U 0.3,

19 but allow for heteroskedasticity as in S3 and switch to heteroskedastic errors with σi 2 iidχ 2 to allow for an assessment of the advantages of the WLS variant CDX w, if any. Finally, Scenario 5 works with the common regressors DGP considered as a variation of scenario S0, with the addition of cross-error correlation following the baseline scenario S, i.e. the case of a positive average loadings of the error factors. he question studied is whether the rank correction to get size control affects or not the power properties of the CD tests and the effectiveness as a pretest. For each scenario and varying numbers of cross-sections and time periods we employ 5000 Monte Carlo replications. All tests are conducted at the 5% nominal level. We also report, for each considered power study, the size of the tests of the null β = using either the usual or the panel-robust standard errors see Section 2.2. he choice of the standard error to be used for the slope parameter test is done according to the outcome of each cross-correlation test considered: a rejection of the null of no cross-correlation prompts the use of panel-robust standard errors, instead of standard errors accounting for cross-sectional heteroskedasticity only. We do this because, in terms of plain rejection frequencies, the CD X tests benefit from multiplication with the elements of X ix j +X jx i such that large sample covariances of the regressors boost the power of the directed tests. Raw power may therefore not be the best comparison criterion. Rather, it is more informative to find out how cross-dependence tests affect the behavior of subsequent slope parameter tests. 3.3 Simulation results We begin with the discussion of the scenario S0 not exhibiting cross-unit error correlation. able gives the empirical rejection frequencies under the null of the compared tests. he LM test is clearly unreliable, even for the largest considered and the smallest. As expected from the literature, the Frees test behaves much nicer here, but is still oversized under error skewness and heteroskedasticity able 2, with rejection frequencies between 6% and 9%, typically around 7.5%. he adjusted and the bias-corrected LM tests perform even better 4% to 6%, in particular the LM b test; the figures worsen slightly in able 2, where sizes up to 8% emerge for larger under skewness and heteroskedasticity, where the John test rejects in 90% to 00% of the cases, which is explained by the departure from the Gaussianity assumption under which its asymptotic properties have been derived able with normal errors shows that the John test does hold size when the assumption is met. he behavior of the other tests is barely affected by the changed shape of the error distribution. he best size control is offered by Pesaran s test CD p, which is virtually at 5% throughout, and by our CDX σ test, which is only marginally more liberal. he CDρ X 9

20 LM F RE LM a LM b CD p J CD σ X CDρ X CDw X able : Size: homoskedasticity and normal idiosyncratic error components; for further details see the text LM F RE LM a LM b CD p J CD σ X CDρ X CDw X able 2: Size: skewed heteroskedastic idiosyncratic error components; for details see the text 20

21 and CDX w variants have good size control when is not small compared to, as predicted by Proposition 2; for = 0 one should rely on CDX σ, but the CD X tests are otherwise practically equivalent in terms of size control for all. able 3 gives the behavior under slope coefficient heterogeneity. Given the moderate regressors cross-dependence, neglected heterogeneity induces residual cross-correlation. his however appears to be quite weak, the tests relying on fixed-effects OLS estimation still hold size. As before, the correlations-based CD X tests have some difficulties for small. able 4 then gives size in the case where the use of a generalized inverse together with χ 2 critical values is required. We note that = 0 is too small for the asymptotics to deliver a good finite-sample approximation for the CD X tests, but otherwise size control is quite good considering the nonstandard asymptotic problem. An interesting finding is that the extreme cross-dependence influences the finite-sample behavior of the LM and Frees tests: since the distortions decrease with, the likely explanation is the difference between errors u i, t and the residuals û i,t, which cross-correlates due to the common regressors. he other tests behavior does not change compared to the baseline S0. For the last two variations of scenario S0, ables 5 and 6 illustrate the behavior under weak error cross-dependence. While for α = 0.5 the size is controlled by all tests that controlled size in the baseline scenario S0, we notice that, for α = 0.75, the CD tests tend to reject slightly more often as expected under the null. o sum up, only the LM a, LM b and in particular CD p tests are reliable alternatives to the directed tests in terms of size. Let us now discuss the results under the power scenarios. An examination of able 7 shows that, under scenario S, only CD p keeps up with the CD ρ X and the CDw X tests in terms of power. he covariance-based version CDσ X is less powerful than the cross-correlation based ones, though not by much, while still being more powerful than the LM a or LM b tests. 8 is only visible for small. he advantage over the CD p test But it is not the power per se that is most interesting in our setup. Examining able 8 with the size of the slope parameter test computed with standard errors chosen by the cross-correlation tests, we notice that it can be severely oversized if using LM a or LM b as pretests, with sizes e.g. up to 30% for = 0 and = 200. In contrast, the CD tests including CD p which can be seen as a particular case of the directed tests, see Section 2.2 come all close to holding size of the combined testing procedure, 8 he differences in power depend on the strength of the regressor cross-correlation, but are still visible when the regressor cross-dependence is weak, while the domination of CD ρ X and CDw X is quite visible when the regressor cross-dependence is strong; the exact figures are available from the authors upon request 2

22 LM F RE LM a LM b CD p J CD σ X CDρ X CDw X able 3: Size: heterogenous coefficients; for details see the text LM F RE LM a LM b CD p J CD σ X CDρ X CDw X able 4: Size: common regressors prompting the use of a generalized matrix inverse and adjusted degrees of freedom; for details see the text 22

23 LM F RE LM a LM b CD p J CD σ X CDρ X CDw X able 5: Size: Weak cross-dependence of exponent α = 0.5; for details see the text LM F RE LM a LM b CD p J CD σ X CDρ X CDw X able 6: Size: Weak cross-dependence of exponent α = 0.75; for details see the text 23