At What Level Should One Cluster Standard Errors in Paired and Small-Strata Experiments?

In clustered paired experiments, randomization units, say villages, are matched into pairs, and one unit of each pair is randomly assigned to treatment. To estimate the treatment effect, researchers often regress their outcome on the treatment and pair fixed effects, clustering standard errors at the unit-of-randomization level. We show that the variance estimator in this regression may be severely downward biased: under constant treatment effect, its expectation equals 1/2 of the true variance. Instead, researchers should cluster at the pair level. Using simulations, we show that those results extend to clustered stratified experiments with few units per strata.