Frisch–Waugh–Lovell theorem

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:

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,

known as the annihilator matrix,[1] or orthogonal projection matrix.[2] 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.


  1. Hayashi, Fumio (2000). Econometrics. Princeton: Princeton University Press. pp. 18–19. ISBN 0-691-01018-8.
  2. Davidson, James (2000). Econometric Theory. Malden: Blackwell. p. 7. ISBN 0-631-21584-0.
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