Unraveling Residualization: enhancing its application and exposing its relationship with the FWL theorem

TL;DR

This paper clarifies the residualization procedure, explores its relationship with the FWL theorem, and promotes its correct application for better interpretation in statistical modeling.

Contribution

It provides a detailed analysis of residualization, its potential uses beyond multicollinearity mitigation, and clarifies its distinction from the FWL theorem for improved application.

Findings

Residualization can be used to analyze isolated effects of variables.

Residualization and FWL theorem yield the same estimates but differ in interpretation.

The paper includes a real data example illustrating these concepts.

Abstract

The residualization procedure has been applied in many different fields to estimate models with multicollinearity. However, there exists a lack of understanding of this methodology and some authors discourage its use. This paper aims to contribute to a better understanding of the residualization procedure to promote an adequate application and interpretation of it among statistics and data sciences. We highlight its interesting potential application, not only to mitigate multicollinearity but also when the study is oriented to the analysis of the isolated effect of independent variables. The relation between the residualization methodology and the Frisch-Waugh-Lovell (FWL) theorem is also analyzed, concluding that, although both provide the same estimations, the interpretation of the estimated coefficients is different. These different interpretations justify the application of the residualization methodology regardless of the FWL theorem. A real data example is presented for a better illustration of the contribution of this paper.