Nonparametric Inference for Distributional Treatment Effects in Instrumental Variable Models

In observational studies, the causal effect of a treatment on the distribution of outcomes is of interest beyond the average treatment effect. Instrumental variable methods allow for causal inference by controlling for unmeasured confounding. The existing nonparametric method for estimating the effect of the treatment on the distribution of outcomes for compliers has several drawbacks, such as producing estimates that violate the non-decreasing and non-negative properties of cumulative distribution functions. In this paper, we propose a novel nonparametric composite likelihood approach, referred to as the binomial likelihood (BL) method, which overcomes the limitations of the previous techniques and utilizes the advantage of likelihood methods. We show the consistency of the maximum binomial likelihood (MBL) estimators and derive their asymptotic distributions. Next, we develop a computationally efficient algorithm for computing the MBL estimates by combining the expectation-maximization (EM) and the pool-adjacent-violators algorithms (PAVA). Moreover, the BL method can be used to construct a binomial likelihood-ratio test (BLRT) for the null hypothesis of no distributional treatment effect. Asymptotic expansion of the BLRT test is derived and the performance of the BL method is demonstrated in simulation studies. Finally, we apply our method to a study of the effect of Vietnam veteran status on the distribution of civilian annual earnings.