Valid Instrumental Variables Selection Methods using Auxiliary Variable and Constructing Efficient Estimator

In observational studies, we are usually interested in estimating causal effects between treatments and outcomes. When some covariates are not observed, an unbiased estimator usually cannot be obtained. In this paper, we focus on instrumental variable (IV) methods. By using IVs, an unbiased estimator for causal effects can be estimated even if there exists some unmeasured covariates. IV methods are useful, however, they sometimes suffer from weak IV and invalid IV problems. In this paper, we propose the moment type estimator which overcomes the major IV problems at once. To achieve this, we consider the situation where some auxiliary variables such as the Negative Control Outcomes can be used. One of the important points of our proposed method is that there are no necessity to specify not only the set of valid IVs but also the proportion of them in advance: this point is different from previous methods. We prove the proposed estimator has the same asymptotic variance as Generalized Method of Moments; the semiparametric efficiency. Also, we confirm properties of our method and previous methods through simulations.