  We consider estimation and inference on average treatment effects under
unconfoundedness conditional on the realizations of the treatment variable and
covariates. Given nonparametric smoothness and/or shape restrictions on the
conditional mean of the outcome variable, we derive estimators and confidence
intervals (CIs) that are optimal in finite samples when the regression errors
are normal with known variance. In contrast to conventional CIs, our CIs use a
larger critical value that explicitly takes into account the potential bias of
the estimator. When the error distribution is unknown, feasible versions of our
CIs are valid asymptotically, even when $\sqrt{n}$-inference is not possible
due to lack of overlap, or low smoothness of the conditional mean. We also
derive the minimum smoothness conditions on the conditional mean that are
necessary for $\sqrt{n}$-inference. When the conditional mean is restricted to
be Lipschitz with a large enough bound on the Lipschitz constant, the optimal
estimator reduces to a matching estimator with the number of matches set to
one. We illustrate our methods in an application to the National Supported Work
Demonstration.
