  Quantile regression models are a powerful tool for studying different points
of the conditional distribution of univariate response variables. Their
multivariate counterpart extension though is not straightforward, starting with
the definition of multivariate quantiles. We propose here a flexible Bayesian
quantile regression model when the response variable is multivariate, where we
are able to define a structured additive framework for all predictor variables.
We build on previous ideas considering a directional approach to define the
quantiles of a response variable with multiple-outputs and we define
noncrossing quantiles in every directional quantile model. We define a Markov
Chain Monte Carlo (MCMC) procedure for model estimation, where the noncrossing
property is obtained considering a Gaussian process design to model the
correlation between several quantile regression models. We illustrate the
results of these models using two data sets: one on dimensions of inequality in
the population, such as income and health; the second on scores of students in
the Brazilian High School National Exam, considering three dimensions for the
response variable.
