A Covariance Regression Model

Classical regression analysis relates the expectation of a response variable to a linear combination of explanatory variables. In this article, we propose a covariance regression model that parameterizes the covariance matrix of a multivariate response vector as a parsimonious quadratic function of explanatory variables. The approach is analogous to the mean regression model, and is similar to a factor analysis model in which the factor loadings depend on the explanatory variables. Parameter estimation for the model is made simple via a random-effects representation and either an EM-algorithm or a Gibbs sampling scheme. The proposed methodology provides a simple but flexible representation of heteroscedasticity across the levels of an explanatory variable, and gives better calibrated prediction regions when compared to a homoscedastic model. Some key words: heteroscedasticity, MCMC, positive definite cone, random effects. 1