  Quantile regression permits describing how quantiles of a scalar response
variable depend on a set of predictors. Because a unique definition of
multivariate quantiles is lacking, extending quantile regression to
multivariate responses is somewhat complicated. In this paper, we describe a
simple approach based on a two-step procedure: in the first step, quantile
regression is applied to each response separately; in the second step, the
joint distribution of the signs of the residuals is modeled through multinomial
regression. The described approach does not require a multidimensional
definition of quantiles, and can be used to capture important features of a
multivariate response and assess the effects of covariates on the correlation
structure. We apply the proposed method to analyze two different datasets.
