# 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 $n\times k_{2}$ matrices 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 $X_{1}(X_{1}^{\mathsf {T}}X_{1})^{-1}X_{1}^{\mathsf {T}}$ . Equivalently, MX1 projects onto the orthogonal complement of the column space of X1. Specifically,

$M_{X_{1}}=I-X_{1}(X_{1}^{\mathsf {T}}X_{1})^{-1}X_{1}^{\mathsf {T}},$ and this particular orthogonal projection matrix is known as the annihilator matrix.

The vector ${\textstyle M_{X_{1}}Y}$ is the vector of residuals from regression of ${\textstyle Y}$ on the columns of ${\textstyle X_{1}}$ .

The theorem implies that the secondary regression used for obtaining $M_{X_{1}}$ 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.