  Correctly identifying treatment effects in observational studies is very
difficult due to the fact that the outcome model or the treatment assignment
model must be correctly specified. Taking advantages of semiparametric models
in this article, we use single-index models to establish the outcome model and
the treatment assignment model, which can allow the link function to be
unbounded and have unbounded support. The link function is regarded as a point
in an infinitely dimensional function space, and we can estimate the link
function and the index parameter simultaneously. The sieve method is used to
approximate the link function and obtain the estimator of the average treatment
effect by the simple linear regression. We establish the asymptotic properties
of the proposed estimator. The finite-sample performance of the proposed
estimator is evaluated through simulation studies and an empirical example.
