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Higher-order Refinements of Small Bandwidth Asymptotics for Density-Weighted Average Derivative Estimators

Matias D. Cattaneo, Max H. Farrell, Michael Jansson, Ricardo Masini

Abstract

The density weighted average derivative (DWAD) of a regression function is a canonical parameter of interest in economics. Classical first-order large sample distribution theory for kernel-based DWAD estimators relies on tuning parameter restrictions and model assumptions that imply an asymptotic linear representation of the point estimator. These conditions can be restrictive, and the resulting distributional approximation may not be representative of the actual sampling distribution of the statistic of interest. In particular, the approximation is not robust to bandwidth choice. Small bandwidth asymptotics offers an alternative, more general distributional approximation for kernel-based DWAD estimators that allows for, but does not require, asymptotic linearity. The resulting inference procedures based on small bandwidth asymptotics were found to exhibit superior finite sample performance in simulations, but no formal theory justifying that empirical success is available in the literature. Employing Edgeworth expansions, this paper shows that small bandwidth asymptotic approximations lead to inference procedures with higher-order distributional properties that are demonstrably superior to those of procedures based on asymptotic linear approximations.

Higher-order Refinements of Small Bandwidth Asymptotics for Density-Weighted Average Derivative Estimators

Abstract

The density weighted average derivative (DWAD) of a regression function is a canonical parameter of interest in economics. Classical first-order large sample distribution theory for kernel-based DWAD estimators relies on tuning parameter restrictions and model assumptions that imply an asymptotic linear representation of the point estimator. These conditions can be restrictive, and the resulting distributional approximation may not be representative of the actual sampling distribution of the statistic of interest. In particular, the approximation is not robust to bandwidth choice. Small bandwidth asymptotics offers an alternative, more general distributional approximation for kernel-based DWAD estimators that allows for, but does not require, asymptotic linearity. The resulting inference procedures based on small bandwidth asymptotics were found to exhibit superior finite sample performance in simulations, but no formal theory justifying that empirical success is available in the literature. Employing Edgeworth expansions, this paper shows that small bandwidth asymptotic approximations lead to inference procedures with higher-order distributional properties that are demonstrably superior to those of procedures based on asymptotic linear approximations.
Paper Structure (15 sections, 5 theorems, 128 equations)

This paper contains 15 sections, 5 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Setup and Assumptions
  3. First-Order Distribution Theory
  4. Distributional Approximation
  5. Variance Estimation
  6. Higher-Order Distribution Theory
  7. Standardized Statistics
  8. Studentized Statistics
  9. Discussion and Bootstrap-Based Inference
  10. Conclusion
  11. Second-Order U-Statistics
  12. Proof of Theorem \ref{['thm:EE-ustat']}
  13. Proof of Theorem \ref{['thm:EE-standard']}
  14. Proof of Theorem \ref{['thm:EE-student']}
  15. Auxiliary Lemma

Key Result

Theorem 1

Suppose Assumptions A:DGP and A:kernel hold with $S \geq P$, and that $nh^{2P}\to 0$ and $nh^{d+2}/\log^9 n \to\infty$. If $\vartheta_\mathsf{v}$ is positive and non-random with $\omega_\mathsf{v}^2/\vartheta_\mathsf{v}^2 \to 1$, then with

Theorems & Definitions (10)

  • Theorem 1: Standardization
  • Theorem 2: Studentization
  • Theorem A.1
  • Corollary A.1
  • Remark A.1
  • Remark A.2
  • Remark A.3
  • Remark A.4
  • Lemma E.1
  • proof