  Quantile regression is a powerful data analysis tool that accommodates
heterogeneous covariate-response relationships. We find that by coupling the
asymmetric Laplace working likelihood with appropriate shrinkage priors, we can
deliver posterior inference that automatically adapts to possible sparsity in
quantile regression analysis. After a suitable adjustment on the posterior
variance, the posterior inference provides asymptotically valid inference under
heterogeneity. Furthermore, the proposed approach leads to oracle asymptotic
efficiency for the active (nonzero) quantile regression coefficients and
super-efficiency for the non-active ones. By avoiding the need to pursue
dichotomous variable selection, the Bayesian computational framework
demonstrates desirable inference stability with respect to tuning parameter
selection. Our work helps to uncloak the value of Bayesian computational
methods in frequentist inference for quantile regression.
