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.

在本文中，我们介绍了具有异质相似性矩阵的协方差均值回归模型。它不仅将响应的协方差与辅助信息诱导的异质相似矩阵联系起来，而且还建立了响应均值与协变量之间的关系。然而，在这种新的模型设置下，遇到了两个统计推断挑战。第一个挑战是基于标准轮廓似然方法的协方差估计的一致性被打破。因此，我们建议进行调整并制定$Z$-估计和无约束/有约束的普通最小二乘估计方法。我们证明了得到的估计量是一致的且渐近正态的。第二个挑战是测试包含多元均值回归和异质协方差矩阵的协方差均值回归模型的充分性。相应地，我们引入了两个诊断测试统计量，然后获得了它们的理论性质。通过广泛的模拟和美国股票市场股票收益联动的实证研究来说明所提出的估计量和测试。