Testing for covariance matrices in time-varying coefficient panel data models with fixed effects

In this paper, we study the tests for sphericity and identity of covariance matrices in time-varying coefficient high-dimensional panel data models with fixed effects. In order to construct the effective test statistics and avoid the influence of the unknown fixed effects, we apply the difference method to eliminate the dependence of the residual sample, and further construct test statistics using the trace estimators of the covariance matrices. For the estimators of the coefficient functions, we use the local linear dummy variable method. Under some regularity conditions, we study the asymptotic property of the estimators and establish the asymptotic distributions of our proposed test statistics without specifying an explicit relationship between the cross-sectional and the time series dimensions. We further show that the test statistics are asymptotic distribution-free. Subsequently simulation studies are carried out to evaluate our proposed methods. In order to assess the performance of our proposed test method, we compare with the existing test methods in panel data linear models with fixed effects.