# 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.[1][2][3]

The Frisch–Waugh–Lovell theorem states that if the regression we are concerned with is:

${\displaystyle Y=X_{1}\beta _{1}+X_{2}\beta _{2}+u}$

where ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are ${\displaystyle n\times k_{1}}$ and ${\displaystyle n\times k_{2}}$ matrices respectively and where ${\displaystyle \beta _{1}}$ and ${\displaystyle \beta _{2}}$ are conformable, then the estimate of ${\displaystyle \beta _{2}}$ will be the same as the estimate of it from a modified regression of the form:

${\displaystyle M_{X_{1}}Y=M_{X_{1}}X_{2}\beta _{2}+M_{X_{1}}u\!,}$

where ${\displaystyle M_{X_{1}}}$ projects onto the orthogonal complement of the image of the projection matrix ${\displaystyle X_{1}(X_{1}^{\mathsf {T}}X_{1})^{-1}X_{1}^{\mathsf {T}}}$. Equivalently, MX1 projects onto the orthogonal complement of the column space of X1. Specifically,

${\displaystyle M_{X_{1}}=I-X_{1}(X_{1}^{\mathsf {T}}X_{1})^{-1}X_{1}^{\mathsf {T}}.}$

known as the annihilator matrix,[4] or orthogonal projection matrix.[5] 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

1. ^ Frisch, Ragnar; Waugh, Frederick V. (1933). "Partial Time Regressions as Compared with Individual Trends". Econometrica. 1 (4): 387–401. JSTOR 1907330.
2. ^ 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.
3. ^ 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.
4. ^ Hayashi, Fumio (2000). Econometrics. Princeton: Princeton University Press. pp. 18–19. ISBN 0-691-01018-8.
5. ^ Davidson, James (2000). Econometric Theory. Malden: Blackwell. p. 7. ISBN 0-631-21584-0.