This paper considers an empirical likelihood method to estimate the parameters of the quantile regression (QR) models and to construct conﬁdence regions that are accurate in ﬁnite samples. To achieve the higher-order reﬁnements, we smooth the estimating equations for the empirical likelihood. We show that the smoothed empirical likelihood (SEL) estimator is ﬁrst-order asymptotically equivalent to the standard QR estimator and establish that conﬁdence regions based on the smoothed empirical likelihood ratio have coverage errors of order n –1 and may be Bartlett-corrected to produce regions with an error of order n –2 , where n denotes the sample size. We further extend these results to censored quantile regression models. Our results are extensions of the previous results of Chen and Hall (1993) to the regression contexts. Monte Carlo experiments suggest that the smoothed empirical likelihood conﬁdence regions may be more accurate in small samples than the conﬁdence regions that can be constructed from the smoothed bootstrap method recently suggested by Horowitz (1998).
Whang, Yoon-Jae, "Smoothed Empirical Likelihood Methods for Quantile Regression Models" (2004). Cowles Foundation Discussion Papers. 1729.