Bias Transmission and Variance Reduction in Two-Stage Quantile Regression

In this paper, we propose a variance reduction method for quantile regressions with endogeneity problems. First, we derive the asymptotic distribution of two-stage quantile estimators based on the fitted-value approach under very general conditions on both error terms and exogenous variables. Second, we exhibit a bias transmission property derived from the asymptotic representation of our estimator. Third, using a reformulation of the dependent variable, we improve the efficiency of the two-stage quantile estimators by exploiting a trade-off between an asymptotic bias confined to the intercept estimator and a reduction of the variance of the slope estimator. Monte Carlo simulation results show the excellent performance of our approach. In particular, by combining quantile regressions with first-stage trimmed least-squares estimators, we obtain more accurate slope estimates than 2SLS, 2SLAD and other estimators for a broad range of distributions.