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Two-way Clustering Robust Variance Estimator in Quantile Regression Models

Ulrich Hounyo, Jiahao Lin

Abstract

We study inference for linear quantile regression with two-way clustered data. Using a separately exchangeable array framework and a projection decomposition of the quantile score, we characterize regime-dependent convergence rates and establish a self-normalized Gaussian approximation. We propose a two-way cluster-robust sandwich variance estimator with a kernel-based density ``bread'' and a projection-matched ``meat'', and prove consistency and validity of inference in Gaussian regimes. We also show an impossibility result for uniform inference in a non-Gaussian interaction regime.

Two-way Clustering Robust Variance Estimator in Quantile Regression Models

Abstract

We study inference for linear quantile regression with two-way clustered data. Using a separately exchangeable array framework and a projection decomposition of the quantile score, we characterize regime-dependent convergence rates and establish a self-normalized Gaussian approximation. We propose a two-way cluster-robust sandwich variance estimator with a kernel-based density ``bread'' and a projection-matched ``meat'', and prove consistency and validity of inference in Gaussian regimes. We also show an impossibility result for uniform inference in a non-Gaussian interaction regime.
Paper Structure (28 sections, 7 theorems, 172 equations, 1 figure, 1 table)

This paper contains 28 sections, 7 theorems, 172 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Two-Way Clustering in Quantile Regression
  3. Model Setting
  4. Asymptotic Distribution
  5. Cluster-Robust Variance Estimator (CRVE)
  6. Estimating $D(\tau)$.
  7. Consistency of Quantile Regression CRVE
  8. Monte Carlo simulation
  9. Empirical Studies
  10. Conclusion
  11. Proof of Theorem \ref{['thm:1']}
  12. Proof of Theorem \ref{['thm:Jacobian']}
  13. Term I, stochastic term at $\widehat{\beta}$.
  14. Term II, Consistency and CLT at $\beta_{0}$.
  15. Term III, plug-in error in expectation.
  16. ...and 13 more sections

Key Result

Theorem 2.1

Let $\mathcal{B}_{0}$ denote the collection of DGPs $\Gamma$ that satisfy Assumptions ass:ahk--as:order of variance. Then uniformly over $\Gamma\in\mathcal{B}_{0}$.

Figures (1)

  • Figure 1: Rejection frequency under different structures The default setting is $d=10$, $G=H=50$, and $\omega_{\bullet}^{X}=\omega_{\bullet}^{e}=1$. Results are based on 10,000 Monte Carlo replicates. The predetermined significance level is 5%.

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 3.1
  • Remark 1
  • Theorem 3.2
  • Proposition 3.1: Impossibility of uniform consistency
  • proof
  • proof
  • proof
  • Lemma D.1: Local stochastic equicontinuity of $\nu_{GH}(\beta)$
  • proof
  • ...and 4 more