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In a completely randomized experiment, the variances of treatment effectestimators in the finite population are usually not identifiable and hence notestimable. Although some estimable bounds of the variances have beenestablished in the literature, few of them are derived in the presence ofcovariates.  In this paper, the difference-in-means estimator and the Wald estimator areconsidered in the completely randomized experiment with perfect compliance andnoncompliance, respectively. Sharp bounds for the variances of these twoestimators are established when covariates are available.  Furthermore, consistent estimators for such bounds are obtained, which can beused to shorten the confidence intervals and improve the power of tests.Confidence intervals are constructed based on the consistent estimators of theupper bounds, whose coverage rates are uniformly asymptotically guaranteed.  Simulations were conducted to evaluate the proposed methods. The proposedmethods are also illustrated with two real data analyses.

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