Cluster-Robust Inference for Quadratic Forms

Michal Koles\'ar, Pengjin Min, Wenjie Wang and Yichong Zhang

Papers from arXiv.org

Abstract: This paper studies inference for quadratic forms of linear regression coefficients with clustered data and many covariates. Our framework covers three important special cases: instrumental variables regression with many instruments and controls, inference on variance components, and testing multiple restrictions in a linear regression. Na\"{\i}ve plug-in estimators are known to be biased. We study a leave-one-cluster-out estimator that is unbiased, and provide sufficient conditions for its asymptotic normality. For inference, we establish the consistency of a leave-three-cluster-out variance estimator under primitive conditions. In addition, we develop a novel leave-two-cluster-out variance estimator that is computationally simpler and guaranteed to be conservative under weaker conditions. Our analysis allows cluster sizes to diverge with the sample size, accommodates strong within-cluster dependence, and permits the dimension of the covariates to diverge with the sample size, potentially at the same rate.

Date: 2026-02, Revised 2026-02
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