Graph-Laplacian Variance Estimators for Finely Stratified Experiments

Abstract

This paper considers design-based inference on the average treatment effect in finely stratified experiments, where uncertainty arises only from the randomized treatment assignment. We focus on settings in which units are first stratified into groups of fixed size according to baseline covariates and, then within each group, exactly one unit is assigned to treatment. In this setting, we introduce a class of graph-Laplacian variance estimators in which strata form the vertices of a weighted graph and edge weights determine how between-stratum comparisons are aggregated. The canonical estimator of Imai (2008) corresponds to a complete graph with edge weights normalized so that each stratum has weighted degree one, while a paired-stratum estimator arises from a perfect matching graph. For the subclass of degree-calibrated graphs, in which each vertex has weighted degree one, we derive an exact bias identity showing that the corresponding estimators are upward-biased, with bias governed by squared differences in the true stratum-level treatment effects across adjacent strata. As a result, any such estimator may be used for valid inference. The identity further suggests that paired-stratum estimators constructed from a covariate-based perfect matching can induce small biases when treatment effects vary smoothly with the covariates. Without such smoothness, however, we show that paired-stratum estimators can exhibit large worst-case bias, and that, within the class of degree-calibrated estimators, the complete-graph estimator is minimax optimal for normalized bias under a weak bound on treatment-effect heterogeneity. Motivated by this contrast, we propose a regularized graph estimator that controls worst-case normalized bias while preserving much of the locality of the paired-stratum estimator. Simulations illustrate the resulting tradeoff between locality and worst-case protection.