Exponent of cross-sectional dependence for residuals

In this paper we focus on estimating the degree of cross-sectional dependence in the error terms of a classical panel data regression model. For this purpose we propose an estimator of the exponent of cross-sectional dependence denoted by alpha, which is based on the number of non-zero pair-wise cross correlations of these errors. We prove that our estimator, tilde(alpha), is consistent and derive the rate at which tilde(alpha) approaches its true value. We evaluate the finite sample properties of the proposed estimator by use of a Monte Carlo simulation study. The numerical results are encouraging and supportive of the theoretical findings. Finally, we undertake an empirical investigation of alpha for the errors of the CAPM model and its Fama-French extensions using 10-year rolling samples from S&P 500 securities over the period Sept 1989 - May 2018.