Frisch–Waugh–Lovell Theorem

In econometrics, the Frisch–Waugh–Lovell (FWL) theorem is named after the econometricians Ragnar Frisch, Frederick V. Waugh, and Michael C. Lovell.

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:

:$Y = X_1 \beta_1 + X_2 \beta_2 + u$

where $X_1$ and $X_2$ are matrices, $\beta_1$ and $\beta_2$ are vectors (and $u$ is the error term), then the estimate of $\beta_2$ will be the same as the estimate of it from a modified regression of the form:

:$M_ Y = M_ X_2 \beta_2 + M_ u,$

where $M_$ projects onto the orthogonal complement of the image of the projection matrix $X_1(X_1^X_1)^X_1^$. Equivalently, ''M''_''X''1 projects onto the orthogonal complement of the column space of ''X''₁. Specifically,

:$M_ = I - X_1(X_1^X_1)^X_1^,$

and this particular orthogonal projection matrix is known as the residual maker matrix or annihilator matrix.

The vector $M_ Y$ is the vector of residuals from regression of $Y$ on the columns of $X_1$.

The most relevant consequence of the theorem is that the parameters in $\beta_2$ do not apply to $X_2$ but to $M_ X_2$, that is: the part of $X_2$ uncorrelated with $X_1$. This is the basis for understanding the contribution of each single variable to a multivariate regression (see, for instance, Ch. 13 in ).

The theorem also implies that the secondary regression used for obtaining $M_$ 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.

Moreover, the standard errors from the partial regression equal those from the full regression.

History

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. Yule emphasized the theorem's importance for understanding multiple and partial regression and correlation coefficients, as mentioned in section 10 of the same paper.

Yule 1907 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. The book reached its tenth edition by 1932.

In a 1931 paper co-authored with Mudgett, Frisch explicitly quoted Yule's results. Yule's formulas for partial regressions were quoted and explicitly attributed to him in order to rectify a misquotation by another author. 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 1962, Richard Stone generalized the theorem to apply to an arbitrary number of variables which may be chosen for special analysis in the same way that time was distinguished in Frisch's and Waugh's original formulation. Translation published as

In 1963, Lovell published a proof considered more straightforward and intuitive. In recognition, people generally add his name to the theorem name.

References

Further reading

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{{DEFAULTSORT:Frisch-Waugh-Lovell theorem
Economics theorems
Regression analysis
Theorems in statistics