  This paper proposes a maximum-likelihood approach to jointly estimate
marginal conditional quantiles of multivariate response variables in a linear
regression framework.
  We consider a slight reparameterization of the Multivariate Asymmetric
Laplace distribution proposed by Kotz et al (2001) and exploit its
location-scale mixture representation to implement a new EM algorithm for
estimating model parameters. The idea is to extend the link between the
Asymmetric Laplace distribution and the well-known univariate quantile
regression model to a multivariate context, i.e. when a multivariate dependent
variable is concerned. The approach accounts for association among multiple
responses and study how the relationship between responses and explanatory
variables can vary across different quantiles of the marginal conditional
distribution of the responses. A penalized version of the EM algorithm is also
presented to tackle the problem of variable selection. The validity of our
approach is analyzed in a simulation study, where we also provide evidence on
the efficiency gain of the proposed method compared to estimation obtained by
separate univariate quantile regressions. A real data application is finally
proposed to study the main determinants of financial distress in a sample of
Italian firms.
