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Concept
Frisch–Waugh–Lovell theorem
Summary
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:
where and are matrices, and are vectors (and is the error term), then the estimate of will be the same as the estimate of it from a modified regression of the form:
where projects onto the orthogonal complement of the of the projection matrix . Equivalently, MX1 projects onto the orthogonal complement of the column space of X1. Specifically,
and this particular orthogonal projection matrix is known as the residual maker matrix or annihilator matrix.
The vector is the vector of residuals from regression of on the columns of .
The most relevant consequence of the theorem is that the parameters in do not apply to but to , that is: the part of uncorrelated with . 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 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.
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.
By 1933, Yule's findings were generally recognized, thanks in part to the detailed discussion of partial correlation and the introduction of his innovative notation in 1907.
Official source
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