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Instrumental variable methods have been widely used to identify causaleffects in the presence of unmeasured confounding. A key identificationcondition known as the exclusion restriction states that the instrument cannothave a direct effect on the outcome which is not mediated by the exposure inview. In the health and social sciences, such an assumption is often notcredible. To address this concern, we consider identification conditions of thepopulation average treatment effect with an invalid instrumental variable whichdoes not satisfy the exclusion restriction, and derive the efficient influencefunction targeting the identifying functional under a nonparametric observeddata model. We propose a novel multiply robust locally efficient estimator ofthe average treatment effect that is consistent in the union of multipleparametric nuisance models, as well as a multiply debiased machine learningestimator for which the nuisance parameters are estimated using generic machinelearning methods, that effectively exploit various forms of linear or nonlinearstructured sparsity in the nuisance parameter space. When one cannot beconfident that any of these machine learners is consistent at sufficiently fastrates to ensure $\surd{n}$-consistency for the average treatment effect, weintroduce a new criteria for selective machine learning which leverages themultiple robustness property in order to ensure small bias. The proposedmethods are illustrated through extensive simulations and a data analysisevaluating the causal effect of 401(k) participation on savings.

Selective Machine Learning of the Average Treatment Effect with an Invalid Instrumental Variable