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Cluster-Robust Inference for Quadratic Forms

Michal Kolesár, Pengjin Min, Wenjie Wang, Yichong Zhang

TL;DR

The paper develops a debiased inference framework for quadratic forms $\theta = \pi' A_0 \gamma$ under clustered, high-dimensional data. It establishes the asymptotic normality of a leave-one-cluster-out estimator and introduces two cluster-robust variance estimators: a consistent leave-three-clusters-out (L3CO) and a conservative leave-two-clusters-out (L2CO), with primitive rate conditions that allow diverging cluster sizes and flexible within-cluster dependence. The approach unifies IV with many instruments/controls, variance components, and testing many linear restrictions, while remaining computationally feasible through leave-out algebra and robust residual calculations. Simulation results show that L3CO/L2CO provide reliable size control where standard methods fail, while preserving meaningful power in challenging high-dimensional, clustered settings.

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. Naïve 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.

Cluster-Robust Inference for Quadratic Forms

TL;DR

The paper develops a debiased inference framework for quadratic forms under clustered, high-dimensional data. It establishes the asymptotic normality of a leave-one-cluster-out estimator and introduces two cluster-robust variance estimators: a consistent leave-three-clusters-out (L3CO) and a conservative leave-two-clusters-out (L2CO), with primitive rate conditions that allow diverging cluster sizes and flexible within-cluster dependence. The approach unifies IV with many instruments/controls, variance components, and testing many linear restrictions, while remaining computationally feasible through leave-out algebra and robust residual calculations. Simulation results show that L3CO/L2CO provide reliable size control where standard methods fail, while preserving meaningful power in challenging high-dimensional, clustered settings.

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. Naïve 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.
Paper Structure (43 sections, 12 theorems, 442 equations, 3 figures, 3 tables)

This paper contains 43 sections, 12 theorems, 442 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Instrumental Variables Regression with Many Instruments and Controls
  4. Variance and Covariance Components in Linear Regressions
  5. Testing Many Linear Restrictions
  6. Asymptotic Normality
  7. Variance Estimator
  8. Leave-three-clusters-out Variance Estimator
  9. Leave-two-clusters-out Variance Estimator
  10. Practical Guidance
  11. Simulation
  12. Design I: Homogeneous Treatment Effect
  13. Design II: Heterogeneous Treatment Effect, Saturated
  14. Design III: Heterogeneous Treatment Effect, Approximated
  15. Remarks
  16. ...and 28 more sections

Key Result

Lemma 2.1

The bias-correction matrix $C$ that minimizes $\text{tr}(C'C)$ subject to (i) $\mathop{\mathrm{Bdiag}}\nolimits(C_{g,g})=\mathop{\mathrm{Bdiag}}\nolimits(A_{g,g})$ (ii) $CW=0$, and (iii) $W'C=0$ is given by $C_{\rm KR}=M\mathop{\mathrm{Bdiag}}\nolimits(\Lambda)M$, where $\mathop{\mathrm{bvec}}\nolim

Figures (3)

  • Figure 1: Power curves for the CNT23 simulation. Top row: 5% significance level. Bottom row: 10% significance level.
  • Figure 2: Power curves for the Yap24 simulation.
  • Figure 3: Power curves for the judge design simulation.

Theorems & Definitions (33)

  • Lemma 2.1
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Theorem 3.1
  • ...and 23 more