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Robust Inference for Convex Pairwise Difference Estimators

Matias D. Cattaneo, Michael Jansson, Kenichi Nagasawa

TL;DR

This paper develops bandwidth-robust distribution theory and bootstrap-based inference for a broad class of convex pairwise difference estimators, which minimize a kernel-weighted convex objective over pairs of observations with similar covariates. The authors introduce three interlocking pillars: (i) small-bandwidth Gaussian approximations that hold under substantially weaker localization, (ii) a debiasing procedure via generalized jackknifing that permits larger bandwidths while preserving convexity, and (iii) a bandwidth-adjusted bootstrap that corrects for bandwidth-induced variance distortions. The contributions yield asymptotic normality in several regimes and provide a practical, robust inference framework applicable to partially linear models such as the PLR, PLL, and PLT variants, with explicit guidance on bootstrap implementation and variance correction. The work has important implications for empirical practice by enabling reliable inference across a wide range of bandwidth choices and by offering pathways for future extensions to higher-order expansions, generated regressors, and non-smooth objective functions.

Abstract

This paper develops distribution theory and bootstrap-based inference methods for a broad class of convex pairwise difference estimators. These estimators minimize a kernel-weighted convex-in-parameter function over observation pairs that are similar in terms of certain covariates, where the similarity is governed by a localization (bandwidth) parameter. While classical results establish asymptotic normality under restrictive bandwidth conditions, we show that valid Gaussian and bootstrap-based inference remains possible under substantially weaker assumptions. First, we extend the theory of small bandwidth asymptotics to convex pairwise estimation settings, deriving robust Gaussian approximations even when a smaller than standard bandwidth is used. Second, we employ a debiasing procedure based on generalized jackknifing to enable inference with larger bandwidths, while preserving convexity of the objective function. Third, we construct a novel bootstrap method that adjusts for bandwidth-induced variance distortions, yielding valid inference across a wide range of bandwidth choices. Our proposed inference method enjoys demonstrable more robustness, while retaining the practical appeal of convex pairwise difference estimators.

Robust Inference for Convex Pairwise Difference Estimators

TL;DR

This paper develops bandwidth-robust distribution theory and bootstrap-based inference for a broad class of convex pairwise difference estimators, which minimize a kernel-weighted convex objective over pairs of observations with similar covariates. The authors introduce three interlocking pillars: (i) small-bandwidth Gaussian approximations that hold under substantially weaker localization, (ii) a debiasing procedure via generalized jackknifing that permits larger bandwidths while preserving convexity, and (iii) a bandwidth-adjusted bootstrap that corrects for bandwidth-induced variance distortions. The contributions yield asymptotic normality in several regimes and provide a practical, robust inference framework applicable to partially linear models such as the PLR, PLL, and PLT variants, with explicit guidance on bootstrap implementation and variance correction. The work has important implications for empirical practice by enabling reliable inference across a wide range of bandwidth choices and by offering pathways for future extensions to higher-order expansions, generated regressors, and non-smooth objective functions.

Abstract

This paper develops distribution theory and bootstrap-based inference methods for a broad class of convex pairwise difference estimators. These estimators minimize a kernel-weighted convex-in-parameter function over observation pairs that are similar in terms of certain covariates, where the similarity is governed by a localization (bandwidth) parameter. While classical results establish asymptotic normality under restrictive bandwidth conditions, we show that valid Gaussian and bootstrap-based inference remains possible under substantially weaker assumptions. First, we extend the theory of small bandwidth asymptotics to convex pairwise estimation settings, deriving robust Gaussian approximations even when a smaller than standard bandwidth is used. Second, we employ a debiasing procedure based on generalized jackknifing to enable inference with larger bandwidths, while preserving convexity of the objective function. Third, we construct a novel bootstrap method that adjusts for bandwidth-induced variance distortions, yielding valid inference across a wide range of bandwidth choices. Our proposed inference method enjoys demonstrable more robustness, while retaining the practical appeal of convex pairwise difference estimators.

Paper Structure

This paper contains 20 sections, 6 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Motivating Examples
  3. Partially Linear Regression Model
  4. Partially Linear Logit Model
  5. Partially Linear Tobit Model
  6. Distributional Approximation and Bootstrap Inference
  7. Regularity Conditions
  8. Small Bandwidth Asymptotics
  9. Debiasing
  10. Bootstrapping
  11. Proofs and Other Technical Results
  12. A Useful Lemma
  13. Proof of Theorems \ref{['Theorem: Asymptotic Distribution']} and \ref{['Theorem: Generalized Jackknifing']}
  14. Proof of Theorem \ref{['Theorem: Bootstrapping']}
  15. Verifying Assumption \ref{['Assumption: Bias of thetahat']}
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose Assumptions Assumption: Convergence of M and Assumption: Asymptotic Distribution hold. If $n^2h_n^d\to\infty$ and if $h_n\to 0$, then

Theorems & Definitions (7)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Lemma 1
  • proof