Testing for patterns and structures in covariance and correlation matrices

Covariance matrices of random vectors contain information that is crucial for modelling. Specific structures and patterns of the covariances (or correlations) may be used to justify parametric models, e.g., autoregressive models. Until now, there have been only a few approaches for testing such covariance structures and most of them can only be used for one particular structure. In the present paper, we propose a systematic and unified testing procedure working among others for the large class of linear covariance structures. Our approach requires only weak distributional assumptions. It covers common structures such as diagonal matrices, Toeplitz matrices and compound symmetry, as well as the more involved autoregressive matrices. We exemplify the approach for all these structures. We prove the correctness of these tests for large sample sizes and use bootstrap techniques for a better small-sample approximation. Moreover, the proposed tests invite adaptations to other covariance patterns by choosing the hypothesis matrix appropriately. With the help of a simulation study, we also assess the small sample properties of the tests. Finally, we illustrate the procedure in an application to a real data set.