# 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, *M*_{X1} projects onto the orthogonal complement of the column space of *X*_{1}. 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.

## References

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

- Frisch, Ragnar; Waugh, Frederick V. (1933). "Partial Time Regressions as Compared with Individual Trends".
*Econometrica*.**1**(4): 387–401. JSTOR 1907330. - Lovell, M. (1963). "Seasonal Adjustment of Economic Time Series and Multiple Regression Analysis".
*Journal of the American Statistical Association*.**58**(304): 993–1010. doi:10.1080/01621459.1963.10480682. - Mitchell, Douglas W. (1991). "Invariance of results under a common orthogonalization".
*Journal of Economics and Business*.**43**(2): 193–196. doi:10.1016/0148-6195(91)90018-R. - Lovell, M. (2008). "A Simple Proof of the FWL Theorem".
*Journal of Economic Education*.**39**(1): 88–91. doi:10.3200/JECE.39.1.88-91. - Davidson, Russell; MacKinnon, James G. (1993).
*Estimation and Inference in Econometrics*. New York: Oxford University Press. pp. 19–24. ISBN 0-19-506011-3.