Graph-Laplacian Variance Estimators for Finely Stratified Experiments

2026-08-10

AI Summary

This study addresses unbiased and robust variance estimation for average treatment effects in finely stratified experiments where only one unit per stratum receives treatment. The authors propose a graph Laplacian–based variance estimator that treats strata as vertices in a graph and aggregates cross-stratum information through edge weights, thereby unifying classical approaches such as paired and complete-graph estimators. Theoretical analysis reveals that bias arises from differences in treatment effects between adjacent strata, and establishes an exact bias identity for degree-calibrated estimators. Building on this insight, they design a regularized graph estimator that balances locality with worst-case robustness. The complete-graph estimator is shown to achieve minimax normalized bias under weak heterogeneity, while the regularized variant effectively trades off bias control and local adaptivity in simulations.

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.

Problem

• finely stratified experiments
• average treatment effect
• variance estimation
• treatment-effect heterogeneity
• design-based inference

Innovation

• graph-Laplacian variance estimator
• finely stratified experiments
• degree-calibrated graph
• minimax optimality
• regularized graph estimator