Nonparametric kernel estimation of the probability density function of regression errors using estimated residuals

TL;DR

This paper develops a nonparametric kernel method for estimating the density of regression errors using residuals, analyzing its asymptotic properties and optimal bandwidth selection.

Contribution

It introduces a feasible estimator based on residuals, compares it with the unfeasible one, and establishes its asymptotic normality and rate optimality.

Findings

Feasible estimator closely approximates the unfeasible one.

Optimal bandwidth selection improves estimation accuracy.

Asymptotic normality of the estimator is proven.

Abstract

This paper deals with the nonparametric density estimation of the regression error term assuming its independence with the covariate. The difference between the feasible estimator which uses the estimated residuals and the unfeasible one using the true residuals is studied. An optimal choice of the bandwidth used to estimate the residuals is given. We also study the asymptotic normality of the feasible kernel estimator and its rate-optimality.