  In this paper, we consider estimation of average treatment effect on the
treated (ATT), an interpretable and relevant causal estimand to policy makers
when treatment assignment is endogenous. By considering shadow variables that
are unrelated to the treatment assignment but related to interested outcomes,
we establish identification of the ATT. Then we focus on efficient estimation
of the ATT by characterizing the geometric structure of the likelihood,
deriving the semiparametric efficiency bound for ATT estimation and proposing
an estimator that can achieve this bound. We rigorously establish the
theoretical results of the proposed estimator. The finite sample performance of
the proposed estimator is studied through comprehensive simulation studies as
well as an application to our motivating study.
