  We propose a new method for multivariate response regressions where the
elements of the response vector can be 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 propose an 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
using simulations and through three data examples.
