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Refined Cluster Robust Inference

Bulat Gafarov, Takuya Ura

Abstract

It has become standard for empirical studies to conduct inference robust to cluster dependence and heterogeneity. With a small number of clusters, the normal approximation for the $t$-statistics of regression coefficients may be poor. This paper tackles this problem using a critical value based on the conditional Cramér-Edgeworth expansion for the $t$-statistics. Our approach guarantees third-order refinement, regardless of whether a regressor is discrete or not, and, unlike the cluster pairs bootstrap, avoids resampling data. Simulations show that our proposal can make a difference in size control with as few as 10 clusters.

Refined Cluster Robust Inference

Abstract

It has become standard for empirical studies to conduct inference robust to cluster dependence and heterogeneity. With a small number of clusters, the normal approximation for the -statistics of regression coefficients may be poor. This paper tackles this problem using a critical value based on the conditional Cramér-Edgeworth expansion for the -statistics. Our approach guarantees third-order refinement, regardless of whether a regressor is discrete or not, and, unlike the cluster pairs bootstrap, avoids resampling data. Simulations show that our proposal can make a difference in size control with as few as 10 clusters.

Paper Structure

This paper contains 12 sections, 12 theorems, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Inverting a Cramér-Edgeworth Expansion with Clustered Errors
  3. Cramér-Edgeworth Expansion in Assumption \ref{['assn:assumption_edgeworth_expansion']}
  4. Monte Carlo simulations
  5. Design with Binary Regressors
  6. Design with a Skewed Error Distribution
  7. Conclusion
  8. Proofs
  9. Proof of Lemma \ref{['lemma:variace_quadratic']}
  10. Proof of Theorem \ref{['thm:theorem1']}
  11. Proof of Theorem \ref{['thm:sufficient_Cramer']}
  12. Proof of Theorem \ref{['thm:theorem3']}

Key Result

Lemma 2.1

where and

Figures (2)

  • Figure 1: Rejection probabilities for two-sided tests in the bertrand2004much design. $N$ is the number of observations per cluster, $\alpha$ is nominal test size.
  • Figure 2: Rejection probabilities for two-sided tests with a skewed error distribution. $N$ is the number of observations per cluster, $\alpha$ is nominal test size.

Theorems & Definitions (27)

  • Lemma 2.1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Theorem 3
  • proof : Proof of Lemma \ref{['lemma:variace_quadratic']}
  • Lemma A.1
  • proof
  • ...and 17 more