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The Exact Variance of the Average Treatment Effect Estimator in Cluster Randomized Controlled Trials

Yue Fang, Geert Ridder

arXiv 8 Nov 2025 · Econometrics

arXiv:2511.05801 · PDF · DOI · OpenAlex · Extracted main text

Abstract

In cluster randomized controlled trials (CRCT) with a finite populations, the exact design-based variance of the Horvitz-Thompson (HT) estimator for the average treatment effect (ATE) depends on the joint distribution of unobserved cluster-aggregated potential outcomes and is therefore not point-identifiable. We study a common two-stage sampling design-random sampling of clusters followed by sampling units within sampled clusters-with treatment assigned at the cluster level. First, we derive the exact (infeasible) design-based variance of the HT ATE estimator that accounts jointly for cluster- and unit-level sampling as well as random assignment. Second, extending Aronow et al (2014), we provide a sharp, attanable upper bound on that variance and propose a consistent estimator of the bound using only observed outcomes and known sampling/assignment probabilities. In simulations and an empirical application, confidence intervals based on our bound are valid and typically narrower than those based on cluster standard errors.

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Most heavily cited references

The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.

ReferenceIntensityMentionsSectionsMain text
1Abadie, Alberto and Athey, Susan and Imbens, Guido W and Wooldridge,… (2023) When should you adjust standard errors for clustering?1.000103100%
2Su, Fangzhou and Ding, Peng (2021) Model-assisted analyses of cluster-randomized experiments0.9507486%
3Aronow, Peter M and Green, Donald P and Lee, Donald KK (2014) SHARP BOUNDS ON THE VARIANCE IN RANDOMIZED EXPERIMENTS0.93511682%
4Chauvet, Guillaume and Vallée, Audrey-Anne (2020) Inference for two-stage sampling designs0.9285480%
5Dhar, Diva and Jain, Tarun and Jayachandran, Seema (2022) Reshaping adolescents' gender attitudes: Evidence from a school-based experiment in India0.92843100%
6Ohlsson, Esbjörn (1989) Asymptotic normality for two-stage sampling from a finite population0.8434375%
7Abadie, Alberto and Athey, Susan and Imbens, Guido W and Wooldridge,… (2020) Sampling-based versus design-based uncertainty in regression analysis0.64441100%
8Li, Xinran and Ding, Peng (2017) General forms of finite population central limit theorems with applications to causal inference0.6443267%
9Middleton, Joel A and Aronow, Peter M (2015) Unbiased estimation of the average treatment effect in cluster-randomized experiments0.64422100%
10Attanasio, Orazio and Cattan, Sarah and Fitzsimons, Emla and Meghir,… (2020) Estimating the production function for human capital: results from a randomized controlled trial in Colombia0.40511100%

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