Inference on covariance-mean regression

In this article, we introduce a covariance-mean regression model with heterogeneous similarity matrices. It not only links the covariance of responses to heterogeneous similarity matrices induced by auxiliary information, but also establishes the relationship between the mean of responses and covariates. Under this new model setting, however, two statistical inference challenges are encountered. The first challenge is that the consistency of the covariance estimator based on the standard profile likelihood approach breaks down. Hence, we propose an adjustment and develop the Z-estimation and unconstrained/constrained ordinary least squares estimation methods. We demonstrate that the resulting estimators are consistent and asymptotically normal. The second challenge is testing the adequacy of the covariance-mean regression model comprising both the multivariate mean regression and the heterogeneous covariance matrices. Correspondingly, we introduce two diagnostic test statistics and then obtain their theoretical properties. The proposed estimators and tests are illustrated via extensive simulations and an empirical example study of the stock return comovement in the US stock market.