A Nonparametric Likelihood Approach for 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, which can lead to a loss of power. In this paper, we propose a novel nonparametric likelihood approach, referred to as the binomial likelihood (BL) method, that incorporates information on compliance behavior while overcoming several limitations of previous techniques and utilizing the advantages of likelihood methods. 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. We show that both versions are more powerful to detect any distributional change than existing methods in finite sample cases, and are asymptotically equivalent to the two-sample Anderson-Darling test. We also develop an efficient algorithm for computing our estimates, and apply the binomial likelihood method to a study of the effect of Medicaid coverage on mental health using the Oregon Health Insurance Experiment.