Testing Weak Cross-Sectional Dependence in Large Panels

Abstract

This article considers testing the hypothesis that errors in a panel data model are weakly cross-sectionally dependent, using the exponent of cross-sectional dependence α, introduced recently in Bailey, Kapetanios, and Pesaran (2012). It is shown that the implicit null of the cross-sectional dependence (CD) test depends on the relative expansion rates of N and T. When T = O(N ^ε), for some 0 < ε ≤1, then the implicit null of the CD test is given by 0 ≤ α < (2 − ε)/4, which gives 0 ≤ α <1/4, when N and T tend to infinity at the same rate such that T/N → κ, with κ being a finite positive constant. It is argued that in the case of large N panels, the null of weak dependence is more appropriate than the null of independence which could be quite restrictive for large panels. Using Monte Carlo experiments, it is shown that the CD test has the correct size for values of α in the range [0, 1/4], for all combinations of N and T, and irrespective of whether the panel contains lagged values of the dependent variables, so long as there are no major asymmetries in the error distribution.

Keywords:

• Diagnostic tests
• Dynamic heterogenous panels
• Exponent of cross-sectional dependence
• Panel data models

JEL Classification:

• C12
• C13
• C33

ACKNOWLEDGMENTS

This article complements an earlier unpublished article entitled “General Diagnostic Tests for Cross Section Dependence in Panels,” which was distributed in 2004 as the Working Paper No. 0435 in Cambridge Working Papers in Economics, Faculty of Economics, University of Cambridge. I am grateful to Natalia Bailey and Majid Al-Sadoon for providing 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 Kapetanios, Ron Smith, Takashi Yamagata, and Aman Ullah for helpful comments.

Notes

¹For empirical applications where economic distance such as trade patterns are used in modelling of spatial correlations see Conley and Topa (2002) and Pesaran et al. (2004).

²The assumption that u _it 's are serially uncorrelated is not restrictive and can be accommodated by including a sufficient number of lagged values of y _it amongst the regressors.

³The main results in the article remain valid even if . But for expositional simplicity, we maintain the assumption that .

⁴Similar results can also be obtained for fixed or random effects models. It suffices if the OLS residuals used in the computation of are replaced with associated residuals from fixed or random effects specifications. But the CD test based on the individual-specific OLS residuals are robust to slope and error-variance heterogeneity, whilst the fixed or random effects residuals are not.

⁵For a proof see Appendix A.2 in Pesaran et al. (2008, pp. 123-124).

⁶This corrects the statement made in error in Pesaran (2004).

⁷These assumptions allow for the inclusion of lagged dependent variables amongst the regressors and can be relaxed further to take account of nonstationary I(1) regressors.

⁸Note that since ψ _iT  > 0 then the order of and will be the same.

⁹In the more general case, where the panel data model contains lagged dependent variables as well as exogenous regressors, the symmetry of error distribution does not seem to be sufficient for the symmetry of the residuals, and the problem requires further investigation.

α is maximal cross-sectional exponent of the errors u _it in the panel data model y _it  = μ _i  + β _i x _it  + u _it , u _it  = γ _i1 f _1t  + γ _i2 f _2t  + σ _iϵϵ _it , i = 1,…, N, t = 1,…, T. α = max(α _j ), where α _j corresponds to the rate at which rises with N (O(N ^{α _j} )), for j = 1, 2 factors.

α is maximal cross-sectional exponent of the errors u _it in the panel data model y _it  = (1 − λ _i1 − λ _i2)μ _i  + λ _i1 y _{i, t−1} + λ _i2 y _{i, t−2} + u _it , u _it  = γ _i1 f _1t  + γ _i2 f _2t  + σ _iϵϵ _it , i = 1,…, N, t = 1,…, T. α = max(α _j ), where α _j corresponds to the rate at which rises with N (O(N ^{α _j} )), for j = 1, 2 factors.