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

TL;DR

This paper derives sharp bounds for the variances of treatment effect estimators in finite populations with covariates, providing methods to improve inference in randomized experiments.

Contribution

It introduces the first sharp variance bounds for estimators considering covariates and develops consistent estimators to enhance confidence intervals and hypothesis testing.

Findings

Sharp variance bounds are established for estimators with covariates.

Consistent estimators for bounds improve confidence interval accuracy.

Simulation and real data demonstrate the effectiveness of the proposed methods.

Abstract

In a 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 the power of tests. Confidence intervals are constructed based on the consistent estimators of the upper bounds, whose coverage rates are uniformly asymptotically guaranteed. Simulations were conducted to evaluate the proposed methods. The proposed methods are also illustrated with two real data analyses.