  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. Using a random-effects representation, parameter
estimation for the model is straightforward using either an EM-algorithm or an
MCMC approximation via Gibbs sampling. The proposed methodology provides a
simple but flexible representation of heteroscedasticity across the levels of
an explanatory variable, improves estimation of the mean function and gives
better calibrated prediction regions when compared to a homoscedastic model.
