Improved residuals for linear regression models under heteroskedasticity of unknown form

TL;DR

This paper introduces PCA residuals for linear regression models to effectively detect heteroskedasticity of unknown form, ensuring residual independence and normality for better model diagnostics.

Contribution

The paper presents a novel spectral analysis-based residual transformation that achieves independence and normality under heteroskedasticity of unknown form.

Findings

Residuals are independent and normally distributed.

The method improves model assumption checking.

Application to real data demonstrates effectiveness.

Abstract

In this work we introduce a new residual for normal linear models that are suitable for situations in which we are dealing with heteroskedasticity of unknown form, they are referred to by principal component analysis (PCA) residuals. These residuals are obtained through a linear transformation of the ordinary residuals, by means of a spectral analysis on a heteroskedasticity-consistent estimator of the covariance matrix. The resulting residuals are independent and normally distributed. These residuals provide a simple way to check several assumptions that underlie the normal linear regression model, as well as model adequacy. Since they are independent and normally distributed, one may apply several results on independent random variables directly to these residuals. Finally, we provide an application to real data to illustrate the usefulness of our residuals.