Abstract
Quantile regression provides a convenient framework for analyzing the impact of covariates on the complete conditional distribution of a response variable instead of only the mean. While frequentist treatments of quantile regression are typically completely nonparametric, a Bayesian formulation relies on assuming the asymmetric Laplace distribution as auxiliary error distribution that yields posterior modes equivalent to frequentist estimates. In this paper, we utilize a location-scale mixture of normals representation of the asymmetric Laplace distribution to transfer different flexible modelling concepts from Gaussian mean regression to Bayesian semiparametric quantile regression. In particular, we will consider high-dimensional geoadditive models comprising LASSO regularization priors and mixed models with potentially non-normal random effects distribution modeled via a Dirichlet process mixture. These extensions are illustrated using two large-scale applications on net rents in Munich and longitudinal measurements on obesity among children. The impact of the likelihood misspecification that underlies the Bayesian formulation of quantile regression is studied in terms of simulations.
Highlights
Quantile regression allows to determine the influence of covariates on the conditional quantiles of the distribution of a dependent variable
For classical linear quantile regression as introduced by Koenker and Bassett (1978), estimation of the quantile-specific regression coefficients βτ relies on minimizing the sum of asymmetrically weighted absolute deviations (AWADs)
While the asymmetric Laplace distribution (1.3) provides a convenient way to express quantile regression in a Bayesian framework based on an auxiliary error distribution, it complicates inference based on Markov chain Monte Carlo (MCMC) simulations due to the inherent nondifferentiability of the check function ρτ
Summary
Quantile regression allows to determine the influence of covariates on the conditional quantiles of the distribution of a dependent variable. Our aim is to make flexible components in semiparametric regression models, such as nonlinear effects, spatial effects, LASSO regularized coefficient blocks or non-normal random effects, applicable in the context of quantile regression. These are typically difficult to combine with linear programming or other direct maximization approaches. The rest of this paper is organized as follows: In Section 2, we first introduce the location-scale mixture representation of the asymmetric Laplace distribution and present a generic MCMC simulation algorithm for Bayesian quantile regression with conditionally Gaussian priors.
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