This research project 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 will be
analyzed in a simulation study, where we will provide evidence on the efficiency gain of the proposed
method compared to estimation obtained by separate univariate quantile regressions. Several applied fields will be considered in the project, in particular financial, environmental and economics.
The research project that we propose can be embedded in a wide recent literature related with building regression methods applied in different field like economics, finance, envinromental. Very little literature growed with respect to the possibility to extend the univariate approach to a multivariate one due to the difficulty of defining a multivariate quantile. The idea of this project is completly new and no one has already used the MAL distribution representation to extend the Asymmetric Laplace approach of quantile regression in a multivariate setting. In order to accomplish this new approach we need to develop several algorithms to estimate the parameters of the multivariate model by using the EM approach and the well known Mixture Gaussian representation of the MAl distribution. In this way we will build new computational procedures able to tackle the multidimensional framework. By so doing we would be able to account for correlation among variables wich may be relevant when several variables are concerned. Since quantile regression is a very useful tool to asses risk contribution, accounting for multidimensionality allows to consider the interdependence among variables . Moreover the possibility to assess risk feature by using quantile regression haids to apply the methodology in several interesting frameworks. In particular in financial and economics contest the possibility to estimate the risk would be the main core of the present project.