The Yule-Frisch-Waugh-Lovell Theorem

TL;DR

This paper reviews the development and extensions of the Yule-Frisch-Waugh-Lovell theorem in econometrics, emphasizing its historical origins, recent advancements, and computational inference methods for regression coefficients.

Contribution

It proposes renaming the theorem to honor G. Udny Yule, discusses recent extensions by P. Ding, and introduces computational methods for statistical inference in regressions.

Findings

Highlights Yule's contribution to the theorem's origin

Discusses Ding's extension to covariance matrix comparisons

Introduces computational inference methods for regression coefficients

Abstract

This paper traces the historical and analytical development of what is known in the econometrics literature as the Frisch-Waugh-Lovell theorem. This theorem demonstrates that the coefficients on any subset of covariates in a multiple regression is equal to the coefficients in a regression of the residualized outcome variable on the residualized subset of covariates, where residualization uses the complement of the subset of covariates of interest. In this paper, I suggest that the theorem should be renamed as the Yule-Frisch-Waugh-Lovell (YFWL) theorem to recognize the pioneering contribution of the statistician G. Udny Yule in its development. Second, I highlight recent work by the statistician, P. Ding, which has extended the YFWL theorem to a comparison of estimated covariance matrices of coefficients from multiple and partial, i.e. residualized regressions. Third, I show that, in cases where Ding's results do not apply, one can still resort to a computational method to conduct statistical inference about coefficients in multiple regressions using information from partial regressions.