A robust test for serial correlation in panel data models

Abstract

We consider a new nonparametric test for serial correlation of unknown form in the estimated residuals of a panel regression model, where individual and time effects can be fixed or random, and the panel data can be balanced or unbalanced. Our test is robust against potential weak error cross-sectional dependence and error serial dependence in higher-order moments. This is in contrast to existing tests for serial correlation in panel data models, which assume error components to be cross-sectionally and serially independent. Our test has an asymptotic N(0, 1) distribution under the null hypothesis and is consistent against serial correlation of unknown form. No common alternative is assumed and hence our test allows for substantial inhomogeneity in serial correlation across individuals. A simulation study highlights the merits of the proposed test relative to a variety of existing tests in the literature. We apply the new test to the empirical study of Wolfers on the relationship between unilateral divorce laws and divorce rates and find strong evidence against serial uncorrelatedness even controlling for the fixed effect.

Keywords:

• Cross-sectional dependence
• hypothesis testing
• Kernel function
• panel data model
• power spectrum density
• robust
• serial correlation

JEL Classifications::

• C12
• C14
• C23

Acknowledgments

We thank the editor, Esfandiar Maasoumi, the associate editor and four referees for careful and constructive comments. We thank seminar participants at the University of Rochester and the Symposium on Modern Statistics at Xiamen University for their useful comments and discussions. We also thank Chihwa Kao for providing the Gauss codes on computing Hong and Kao (2004) test for serial correlation. Any remaining errors are solely ours.

Notes

1 Lee (2014) considers generalized spectrum tests for the correct specification of linear panel data models. The tests are robust to potential conditional heteroskedasticity.

2 The two-way fixed effect estimator is the OLS estimator of the transformed model $Y_{it}-{\overline{Y}}_i-{\overline{Y}}_t+\overline{Y}={(X_{it}-{\overline{X}}_i-{\overline{X}}_t+\overline{X})}^{\prime }\beta +({\epsilon }_{it}-{\overline{\epsilon }}_i-{\overline{\epsilon }}_t+\overline{\epsilon }),$ where ${\overline{A}}_i=T_i^{-1}{\sum }_{t=1}^{T_i}{A}_{it},{\overline{A}}_t=N^{-1}{\sum }_{i=1}^N{A}_{it},$ $\overline{A}=N^{-1}{\sum }_{i=1}^N{\overline{A}}_i$ and $A=Y,X$ or $\epsilon .$

3 Alternatively, $\left\{{\epsilon }_{it}\right\}$ may be assumed to be a mixing random field, which is defined as the set of random variables $\{{\epsilon }_z,z\in Z^2\}$ on $\{\Omega ,Ϝ,P\},$ where $\{\Omega ,Ϝ,P\}$ denotes the standard probability triple and Z² denotes the two-dimensional lattice of integers: $Z^2=\{(i,t)|i=1,2,\dots ,t=1,2,\dots \}$ (see, e.g., Deo, 1975).

4 Assumption A.2 is stronger than the assumption imposed in Bai (2003) and Chudik, Pesaran, and Tosetti (2011). We test the serial correlation by comparing the estimated spectral density function under ${\mathbb{H}}_0\text{and}{\mathbb{H}}_A$ for all i. Our test statistic involves four summations. To bound the higher-order terms, we need to use a kind of mixing conditions in both cross-sectional and time dimensions. However, it is hard to impose the mixing condition in the cross-sectional dimension without natural ordering for cross-sectional indices. Hence we impose Assumption A.2 for theoretical derivation. We check whether our test is robust to potential cross-sectional dependence via simulation and the numerical results show that our test is indeed robust to potential conditional heteroscedasticity and cross-sectional dependence.

5 Note that N can’t be larger than $T^2.$

6 We have tried the Parzen kernel for $k(\cdot )$ and obtained similar results.