Title: Envelope Quantile Regression
Quantile regression offers a valuable complement of classical mean regression for robust and comprehensive data analysis in a variety of applications. We propose a novel envelope quantile regression method (EQR) that adapts a nascent technique called enveloping (Cook, Li, and Chiaromonte, 2010) to improve the efficiency of standard quantile regression. The new method aims to identify material and immaterial information in a quantile regression model and use only the material information for estimation. By excluding the immaterial part, the EQR method has the potential to substantially reduce the estimation variability. Unlike many envelope approaches based on the likelihood framework, our proposed estimator is defined through a set of nonsmooth estimating equations. We facilitate the estimation via the generalized method of moments (GMM) and derive the asymptotic normality of the proposed estimator by applying empirical process techniques. We further establish that EQR is asymptotically more efficient than (or at least as asymptotically efficient as) the standard quantile regression estimators without imposing stringent conditions. The effectiveness of the proposed method is demonstrated via Monte-Carlo simulations and real data examples.