Estimation of the Conditional Variance in Paired Experiments

In paired randomized experiments units are grouped in pairs, often based on covariate information, with random assignment within the pairs. Average treatment effects are then estimated by averaging the within-pair differences in outcomes. Typically the variance of the average treatment effect estimator is estimated using the sample variance of the within-pair differences. However, conditional on the covariates the variance of the average treatment effect estimator may be substantially smaller. Here we propose a simple way of estimating the conditional variance of the average treatment effect estimator by forming pairs-of-pairs with similar covariate values and estimating the variances within these pairs-of-pairs. Even though these within-pairs-of-pairs variance estimators are not consistent, their average is consistent for the conditional variance of the average treatment effect estimator and leads to asymptotically valid confidence intervals.