Frisch–Waugh–Lovell theorem

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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:

 Y = X_1 \beta_1 + X_2 \beta_2 + u \!

where  X_1 and  X_2 are  n \times k_1 and  n \times k_2 respectively and where  \beta_1 and  \beta_2 are conformable, then the estimate of  \beta_2 will be the same as the estimate of it from a modified regression of the form:

 M_{X_1} Y = M_{X_1} X_2 \beta_2 + M_{X_1} u \!,

where  M_{X_1} projects onto the orthogonal complement of the image of the projection matrix  X_1(X_1'X_1)^{-1}X_1' . Equivalently, MX1 projects onto the orthogonal complement of the column space of X1. Specifically,

 M_{X_1} = I - X_1(X_1'X_1)^{-1}X_1'. \!

This result implies that all these secondary regressions are 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.