We consider estimation and inference on average treatment effects under unconfoundedness conditional on the realizations of the treatment variable and covariates. We derive finite-sample optimal estimators and confidence intervals (CIs) under the assumption of normal errors when the conditional mean of the outcome variable is constrained only by nonparametric smoothness and/or shape restrictions . When the conditional mean is restricted to be Lipschitz with a large enough bound on the Lipschitz constant, we show that the optimal estimator reduces to a matching estimator with the number of matches set to one. In contrast to conventional CIs, our CIs use a larger critical value that explicitly takes into account the potential bias of the estimator. It is needed for correct coverage in finite samples and, in certain cases, asymptotically. We give conditions under which root-n inference is impossible, and we provide versions of our CIs that are feasible and asymptotically valid with unknown error distribution, including in this non-regular case. We apply our results in a numerical illustration and in an application to the National Supported Work Demonstration.
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 the conditional mean . We also derive the minimum smoothness conditions on the conditional mean that are necessary for \sqrt{n . 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.
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We derive finite-sample optimal
Given nonparametric smoothness and/or shape restrictions on the conditional mean of the outcome variable, we derive
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under the assumption of normal errors when
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
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of the outcome variable is constrained only by nonparametric smoothness and/or shape restrictions
. We also derive the minimum smoothness conditions on the conditional mean that are necessary for \sqrt{n
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we show that
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In contrast to conventional CIs, our CIs use a larger critical value that explicitly takes into account the potential bias of the estimator. It is needed for correct coverage in finite samples and, in certain cases, asymptotically. We give conditions under which root-n inference is impossible, and we provide versions of our CIs that are feasible and asymptotically valid with unknown error distribution, including in this non-regular case. We apply our results in a numerical illustration and
We illustrate our methods