Bayesian variable selection and estimation in quantile regression using a quantile-specific prior

Asymmetric Laplace (AL) specification has become one of the ideal statistical models for Bayesian quantile regression. In addition to fast convergence of Markov Chain Monte Carlo, AL specification guarantees posterior consistency under model misspecification. However, variable selection under such a specification is a daunting task because, realistically, prior specification of regression parameters should take the quantile levels into consideration. Quantile-specific g-prior has recently been developed for Bayesian variable selection in quantile regression, whereas it comes at a high price of the computational burden due to the intractability of the posterior distributions. In this paper, we develop a novel three-stage computational scheme for the foregoing quantile-specific g-prior, which starts with an expectation-maximization algorithm, followed by Gibbs sampler and ends with an importance re-weighting step that improves the accuracy of approximation. The performance of the proposed procedure is illustrated with simulations and a real-data application. Numerical results suggest that our procedure compares favorably with the Metropolis-Hastings algorithm.