Posterior Consistency of Bayesian Quantile Regression Under a Mis-Specified Likelihood Based on Asymmetric Laplace Density

We provide a theoretical justification for the widely used and yet only empirically verified approach of using Asymmetric Laplace Density(ALD) in Bayesian Quantile Regression. We derive sufficient conditions for posterior consistency of the quantile regression parameters even if the true underlying likelihood is not ALD, by considering both the case of random as well as non-random covariates. While existing literature on misspecified models address more general models, our approach of exploiting the specific form of ALD allows for a more direct derivation. We verify that the conditions so derived are satisfied by a wide range of potential true underlying probability distributions. We also show that posterior consistency holds even in the case of improper priors as long as the posterior is well defined. We demonstrate the working of the method using simulations