Identification and inference for marginal average treatment effect on the treated with an instrumental variable

In observational studies, treatments are typically not randomized and, therefore, estimated treatment effects may be subject to a confounding bias. The instrumental variable (IV) design plays the role of a quasi-experimental handle because the IV is associated with the treatment and only affects the outcome through the treatment. In this paper, we present a novel framework for identification and inferences, using an IV for the marginal average treatment effect amongst the treated (ETT) in the presence of unmeasured confounding. For inferences, we propose three semiparametric approaches: (i) an inverse probability weighting (IPW); (ii) an outcome regression (OR); and (iii) a doubly robust (DR) estimation, which is consistent if either (i) or (ii) is consistent, but not necessarily both. A closed-form locally semiparametric efficient estimator is obtained in the simple case of a binary IV, and outcome, and the efficiency bound is derived for the more general case.