GMM Quantile Regression

Abstract

This paper develops generalized method of moments (GMM) estimation and inference procedures for quantile regression models when allowing for general parametric restrictions on the parameters of interest over a set of quantiles. First, we suggest a GMM estimator for simultaneous estimation across multiple quantiles. This estimator exploits a partition of the quantile space, which induces a weighting matrix that is independent of the parameters of interest and the number of partitions. The GMM estimator is designed to estimate a fixed number of quantiles simultaneously, is flexible since it allows for imposing restrictions on the parameters of interest over a set of moments indexed by the quantiles, and accounts for information across quantiles to improve efficiency. Second, we study the properties of the GMM estimator when the number of partitions diverge to infinity, and derive its efficiency bound. Third, we suggest an alternative smooth GMM estimation procedure to be used with many moments. We establish the asymptotic properties of both GMM estimators. These methods have the advantage of being simple to implement in practice. Monte Carlo simulations show numerical evidence of the finite sample properties of the methods. Finally, we apply the proposed methods to estimate the effects of various covariates on birthweight of live infants at the extreme bottom of the conditional distribution.