A nonparametric binomial likelihood approach for causal inference in instrumental variable models

Instrumental variable methods allow for inference about the treatment effect by controlling for unmeasured confounding in randomized experiments with noncompliance. However, many studies do not consider the observed compliance behavior in the testing procedure, leading to loss of power. In this paper, we propose a novel nonparametric likelihood approach, referred to as the binomial likelihood method, that incorporates information on compliance behavior while overcoming several limitations of previous techniques. Our proposed method produces proper estimates of the counterfactual distribution functions by maximizing the binomial likelihood over the space of distribution functions. Using this we propose two versions of a binomial likelihood ratio test for the null hypothesis of no treatment effect, and study their finite sample and asymptotic properties. We also develop an efficient algorithm for computing our estimates, and apply the method to study the effect of Medicaid coverage on mental health using the Oregon Health Insurance Experiment.