Inferences for Partially Conditional Quantile Treatment Effect Model

In this paper, a new model, termed as the partially conditional quantile treatment effect (PCQTE) model, is proposed to characterize the heterogeneity of treatment effect conditional on some predetermined variable(s). We show that the partially conditional quantile treatment effect is identified under the assumption of selection on observables, which leads to a semiparametric estimation procedure in two steps: First, parametric estimation of the propensity score function and then, nonparametric estimation of conditional quantile treatment effect. Under some regularity conditions, the consistency and asymptotic normality of the proposed semiparametric estimator are derived. In addition, a specification test is seminally proposed in quantile regression literature, to test whether there exits heterogeneity for PCQTE across sub-populations, a consistent test, based on the Cramer-von Mises type criterion. The asymptotic properties of the proposed test statistic are investigated, including consistency and asymptotic normality. Finally, the performance of the proposed methods is illustrated through Monte Carlo experiments and an empirical application on estimating the effect of the first-time mother's smoking during pregnancy on the baby's birth weight conditional on mother's age and testing whether the partially conditional quantile treatment effect varies across different mother's age.