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

TL;DR

This paper develops higher-order theoretical results for small bandwidth asymptotics in density-weighted average derivative estimation, improving inference accuracy beyond classical methods.

Contribution

It provides the first formal Edgeworth expansion-based theory for small bandwidth asymptotics in DWAD estimation, demonstrating superior higher-order properties.

Findings

Small bandwidth asymptotics yield better finite sample inference.

Edgeworth expansions justify higher-order improvements.

Theoretical results outperform classical asymptotic linear methods.

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.