Cluster-Robust Inference for Quadratic Forms

TL;DR

This paper develops new cluster-robust inference methods for quadratic forms in linear regression, addressing bias issues and allowing for complex data structures with many covariates and large clusters.

Contribution

It introduces unbiased leave-one-cluster-out estimators and novel variance estimators that are computationally efficient and valid under broad conditions.

Findings

Leave-one-cluster-out estimator is unbiased and asymptotically normal.

New variance estimators are consistent and conservative under weak conditions.

Methods accommodate diverging cluster sizes and high-dimensional covariates.

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.