In econometrics, the Frisch–Waugh–Lovell (FWL) theorem is named after the econometricians Ragnar Frisch, Frederick V. Waugh, and Michael C.
[1][2][3] The Frisch–Waugh–Lovell theorem states that if the regression we are concerned with is expressed in terms of two separate sets of predictor variables: where
Equivalently, MX1 projects onto the orthogonal complement of the column space of X1.
The most relevant consequence of the theorem is that the parameters in
This is the basis for understanding the contribution of each single variable to a multivariate regression (see, for instance, Ch.
The theorem also implies that the secondary regression used for obtaining
is unnecessary when the predictor variables are uncorrelated: using projection matrices to make the explanatory variables orthogonal to each other will lead to the same results as running the regression with all non-orthogonal explanators included.
[7] The origin of the theorem is uncertain, but it was well-established in the realm of linear regression before the Frisch and Waugh paper.
George Udny Yule's comprehensive analysis of partial regressions, published in 1907, included the theorem in section 9 on page 184.
[8] Yule emphasized the theorem's importance for understanding multiple and partial regression and correlation coefficients, as mentioned in section 10 of the same paper.
[8] Yule 1907 [8] also introduced the partial regression notation which is still in use today.
The theorem, later associated with Frisch, Waugh, and Lovell, and Yule's partial regression notation, were included in chapter 10 of Yule's successful statistics textbook, first published in 1911.
[9] In a 1931 paper co-authored with Mudgett, Frisch explicitly quoted Yule's results.
[10] Yule's formulas for partial regressions were quoted and explicitly attributed to him in order to rectify a misquotation by another author.
[10] Although Yule was not explicitly mentioned in the 1933 paper by Frisch and Waugh, they utilized the notation for partial regression coefficients initially introduced by Yule in 1907, which by 1933 was well known due to the success of Yule's textbook.
In 1963, Lovell published a proof[11] considered more straightforward and intuitive.