Abstract: In a completely randomized experiment, the variances of the treatment effect estimators in a finite population are usually not identifiable, and hence not estimable. Although some estimable bounds of such variances have been established in the literature, few are derived in the presence of covariates. We consider the difference-in-means estimator and the Wald estimator in completely randomized experiments with perfect compliance and noncompliance, respectively. We also establish sharp bounds for the variances of these two estimators when covariates are available. Furthermore, we obtain consistent estimators for such bounds that can be used to shorten the confidence intervals and improve the power of tests. The confidence intervals are constructed based on the consistent estimators of the upper bounds, and have coverage rates that are uniformly asymptotically guaranteed. We use analyses based on simulations and real data to evaluate and demonstrate the proposed methods.

Key words and phrases: Causal inference, partial identification, potential outcome, randomized experiment.