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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 X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are n × k 1 {\displaystyle n\times k_{1}} and n × k 2 {\displaystyle n\times k_{2}} matrices respectively and where β 1 {\displaystyle \beta _{1}} and β 2 {\displaystyle \beta _{2}} are conformable, then the estimate of β 2 {\displaystyle \beta _{2}} will be the same as the estimate of it from a modified regression of the form:
where M X 1 {\displaystyle M_{X_{1}}} projects onto the orthogonal complement of the image of the projection matrix X 1 − 1 X 1 T {\displaystyle X_{1}^{-1}X_{1}^{\mathsf {T}}}. Equivalently, MX1 projects onto the orthogonal complement of the column space of X1. Specifically,