  We propose a new method for multivariate response regression and covariance
estimation when elements of the response vector are of mixed types, for example
some continuous and some discrete. Our method is based on a model which assumes
the observable mixed-type response vector is connected to a latent multivariate
normal response linear regression through a link function. We explore the
properties of this model and show its parameters are identifiable under
reasonable conditions. We impose no parametric restrictions on the covariance
of the latent normal other than positive definiteness, thereby avoiding
assumptions about unobservable variables which can be difficult to verify in
practice. To accommodate this generality, we propose a novel algorithm for
approximate maximum likelihood estimation that works "off-the-shelf" with many
different combinations of response types, and which scales well in the
dimension of the response vector. Our method typically gives better predictions
and parameter estimates than fitting separate models for the different response
types and allows for approximate likelihood ratio testing of relevant
hypotheses such as independence of responses. The usefulness of the proposed
method is illustrated in simulations; and one biomedical and one genomic data
example.
