Bias Transmission and Variance Reduction in Two-Stage Quantile Regressions

Abstract

In this paper, we integrate the structural approach and the fitted-value approach of the literature on quantile regression with endogeneity problems, including for the non-constant effect case. First, we derive the asymptotic distribution and the variance-covariance matrix of two-stage quantile estimators under very general conditions on both error terms and exogenous variables. Second, we exhibit a `bias transmission property' from the asymptotic representation of our estimator. Third, we show how structural and fitted-values approaches can be integrated, providing some natural independence hypotheses for instruments. Fourth, we elicit the possibility of non-constant effects models with the fitted-value approach, a situation sometimes believed to be ruled out with this approach. Fifth, when the above independence hypotheses are not met, the fitted-value approach allows the estimation of models that cannot be identified with the structural approach, at the cost of interpretation difficulties. Sixth, we introduce a new technique to improve the efficiency of two-stage quantile regressions. Using a reformulation of the dependent variable, we built a trade-off between an asymptotic bias confined to the intercept (or to coefficients of secondary interest) and a reduction of the variance of the other coefficients estimator. Monte Carlo simulation results show the excellent performance of our variance reduction approach