  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. Constructing a linear combination of IVs solves weak IV problems,
however, there are risks estimating biased causal effects by including some
invalid IVs. In this paper, we use Negative Control Outcomes as auxiliary
variables to select valid IVs. By using NCOs, there are no necessity to specify
not only the set of valid IVs but also invalid one in advance: this point is
different from previous methods. We prove that the estimated causal effects has
the same asymptotic variance as the estimator using Generalized Method of
Moments that has the semiparametric efficiency. Also, we confirm properties of
our method and previous methods through simulations.
