On Bayesian quantile regression using a pseudo-joint asymmetric laplace likelihood

Abstract

We consider a pseudo-likelihood for Bayesian estimation of multiple quantiles as a function of covariates. This arises as a simple product of multiple asymmetric Laplace densities (ALD), each corresponding to a particular quantile. The ALD has already been used in the Bayesian estimation of a single quantile. However, the pseudo-joint ALD likelihood is a way to incorporate constraints across quantiles, which cannot be done if each of the quantiles is modeled separately. Interestingly, we find that the normalized version of the likelihood turns out to be misleading. Hence, the pseudo-likelihood emerges as an alternative. In this note, we show that posterior consistency holds for the multiple quantile estimation based on such a likelihood for a nonlinear quantile regression framework and in particular for a linear quantile regression model. We demonstrate the benefits and explore potential challenges with the method through simulations.