Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood

Abstract

The paper discusses the asymptotic validity of posterior inference of pseudo-Bayesian quantile regression methods with complete or censored data when an asymmetric Laplace likelihood is used. The asymmetric Laplace likelihood has a special place in the Bayesian quantile regression framework because the usual quantile regression estimator can be derived as the maximum likelihood estimator under such a model, and this working likelihood enables highly efficient Markov chain Monte Carlo algorithms for posterior sampling. However, it seems to be underrecognised that the stationary distribution for the resulting posterior does not provide valid posterior inference directly. We demonstrate that a simple adjustment to the covariance matrix of the posterior chain leads to asymptotically valid posterior inference. Our simulation results confirm that the posterior inference, when appropriately adjusted, is an attractive alternative to other asymptotic approximations in quantile regression, especially in the presence of censored data.

Additive Quantile Regression for Clustered Data with an Application to Children's Physical Activity

“…whereΨ is the scaled variance-covariance matrix of w τ = u τ , v τ , and h 0 andḦ are the terms of order, respectively, 0 and 2 of the above-mentioned Taylor expansion around the modeŵ τ (see Appendix A for more details). When using the asymmetric Laplace as pseudo-likelihood, inference should be confined to point estimation (see for example Yang, Wang and He, 2016). Standard errors of non-random parameters estimates can be calculated using block bootstrap (Efron and Tibshirani, 1998), although this increases the computational cost.…”

bayesQR: A Bayesian Approach to Quantile Regression

“…Recently, both Yang, Wang, and He (2015) and Sriram (2015) proposed a correction to the MCMC iterations to construct asymptotically valid intervals. Their method builds on the work of Chernozhukov and Hong (2003) and Yang and He (2012).…”

“…In these papers, it is shown that the MCMC iterations provide a valid piece of the usual sandwich formula (see e.g., Koenker 2005) that can be used to construct a valid estimate of the limiting covariance matrix of the Gaussian approximation. The correction can be accomplished as follows (Yang et al 2015):…”

“…That is, the bayesQR option normal.approx is set to TRUE by default. Following the suggestion of Yang et al (2015), σ 2 is then set to a fixed value (i.e., σ 2 = 1) which simplifies the Gibbs sampler.…”

The Simulation Study to Test the Performance of Quantile Regression Method With Heteroscedastic Error Variance

“…Therefore, to see the goodness of estimating the parameters based on the bias and variance values simultaneously, represented in the value of Mean Square Error (MSE) [8,9,10], formulated as follows :…”