The role of propensity score structure in asymptotic efficiency of estimated conditional quantile treatment effect

When a strict subset of covariates is given, we propose conditional quantile treatment effect ((Formula presented.)) to offer, compared with the unconditional quantile treatment effect ((Formula presented.)) and conditional average treatment effect ((Formula presented.)), a more complete and informative view of the heterogeneity of treatment effects via the quantile sheet that is a function of the given covariates and quantile levels. Even though either one or both (Formula presented.) and (Formula presented.) are not significant, (Formula presented.) could still show some impact of the treatment on the upper and lower tails of subpopulations' (defined by the covariates subset) distribution. To the best of our knowledge, this is the first to consider such a low-dimensional conditional quantile treatment effect in the literature. We focus on deriving the asymptotic normality of propensity score-based estimators under parametric, nonparametric, and semiparametric structure. We make a systematic study on the estimation efficiency to check the importance of propensity score structure and the essential differences from the unconditional counterparts. The derived unique properties can answer: what is the general ranking of these estimators? how does the affiliation of the given covariates to the set of covariates of the propensity score affect the efficiency? how does the convergence rate of the estimated propensity score affect the efficiency? and why would semiparametric estimation be worth of recommendation in practice? The simulation studies are conducted to examine the performances of these estimators. A real data example is analyzed for illustration and some new findings are acquired.