Posterior Consistency of Bayesian Quantile Regression Based on the Misspecified Asymmetric Laplace Density

We explore an asymptotic justification for the widely used and empirically verified approach of assuming an asymmetric Laplace distribution (ALD) for the response in Bayesian Quantile Regression. Based on empirical findings, Yu and Moyeed (2001) argued that the use of ALD is satisfactory even if it is not the true underlying distribution. We provide a justification to this claim by establishing posterior consistency and deriving the rate of convergence under the ALD misspecification. Related literature on misspecified models focuses mostly on i.i.d. models which in the regression context amounts to considering i.i.d. random covariates with i.i.d. errors. We study the behavior of the posterior for the misspecified ALD model with independent but non identically distributed response in the presence of non-random covariates. Exploiting the specific form of ALD helps us derive conditions that are more intuitive and easily seen to be satisfied by a wide range of potential true underlying probability distributions for the response. Through simulations, we demonstrate our result and also find that the robustness of the posterior that holds for ALD fails for a Gaussian formulation, thus providing further support for the use of ALD models in quantile regression.