Estimating Counterfactual Distribution Functions via Optimal Distribution Balancing with Applications

In this paper, we propose a new method to estimate counterfactual distribution functions via the optimal distribution balancing weights, to avoid estimating the inverse propensity weights, which is sensitive to model specification and easily causes unstable estimates as well as often fails to adequately balance covariates. First, we demonstrate that the estimated weights exactly balance the estimated conditional distributions among the treated, untreated, and combined groups via a well-defined convex optimization problem. Secondly, we show that the resulting estimator of counterfactual distribution function converges weakly to a mean-zero Gaussian process at the parametric rate of the squared root n. Additionally, we show that a properly designed Bootstrap method can be used to obtain confidence intervals for conducting statistical inferences, together with its theoretical justification. Furthermore, with the estimates of counterfactual distribution functions, we provide methods to estimate the quantile treatment effects and test the stochastic dominance relationship between the potential outcome distributions. Moreover, Monte Carlo simulations are conducted to illustrate that the finite sample performance for the proposed estimator is better than the inverse propensity score weighted estimators in many scenarios. Finally, our empirical study revisits the effect of maternal smoking on infant birth weight.