Working paper

Cluster-Robust Inference for Quadratic Forms

Michal Kolesár, Pengjin Min, Wenjie Wang, Yichong Zhang

Preparing for submission

Abstract

We study inference for quadratic forms of regression coefficients with many covariates. We develop a unifying framework that allows for clustering with diverging cluster sizes, and covers design-based inference on treatment effects, model-based inference on variance components, testing many linear restrictions, and instrumental variables models with many instruments and controls. We show that, in the presence of many controls, existing methods do not yield valid inference in the design-based setting. We develop a novel jackknife variance estimator that yields valid, but conservative inference, and a leave-three-cluster-out estimator that is more computationally involved but is asymptotically exact, up to a nonnegative heterogeneity term that cannot be avoided. For the model-based applications, we propose a different jackknife variance estimator that leads to conservative inference in general, but is asymptotically exact if the number of controls grows more slowly than the sample size. A leave-three-cluster-out variance estimator is asymptotically exact. We illustrate our methods in several applications.