Frisch–Waugh–Lovell theorem
In econometrics, the Frisch–Waugh–Lovell 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:
where and are and matrices respectively and where and are conformable, 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 image 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 annihilator matrix.
The vector is the vector of residuals from regression of on the columns of.
The theorem implies that the secondary regression used for obtaining is unnecessary: 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.
