Sharp bounds for variance of treatment effect estimators in the finite population in the presence of covariates

In the completely randomized experiment, the variances of treatment effect estimators in the finite population are usually not identifiable and hence not estimable. Although some estimable bounds of the variances have been established in the literature, few of them are derived in the presence of covariates. In this paper, the difference-in-means estimator and the Wald estimator are considered in the completely randomized experiment with perfect compliance and noncompliance, respectively. Sharp bounds for the variances of these two estimators are established when covariates are available. Furthermore, consistent estimators for such bounds are obtained, which can be used to shorten the confidence intervals and improve power of tests. Simulations were conducted to evaluate the proposed methods. The proposed methods are also illustrated with two real data analyses.