  Standard causal inference characterizes treatment effect through averages,
but the counterfactual distributions could be different in not only the central
tendency but also spread and shape. To provide a comprehensive evaluation of
treatment effects, we focus on estimating quantile treatment effects (QTEs).
Existing methods that invert a nonsmooth estimator of the cumulative
distribution functions forbid inference on probability density functions
(PDFs), but PDFs can reveal more nuanced characteristics of the counterfactual
distributions. We adopt a semiparametric conditional distribution regression
model that allows inference on any functionals of counterfactual distributions,
including PDFs and multiple QTEs. To account for the observational nature of
the data and ensure an efficient model, we adjust for a double balancing score
that augments the propensity score with individual covariates. We provide a
Bayesian estimation framework that appropriately propagates modeling
uncertainty. We show via simulations that the use of double balancing score for
confounding adjustment improves performance over adjusting for any single score
alone, and the proposed semiparametric model estimates QTEs more accurately
than other semiparametric methods. We apply the proposed method to the North
Carolina birth weight dataset to analyze the effect of maternal smoking on
infant's birth weight.
