# 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,

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

and this particular orthogonal projection matrix is known as the annihilator matrix.[4][5]

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 ${\displaystyle M_{X_{1}}}$ is unnecessary when the predictor variables are uncorrelated (this never happens in practice): 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.

It is not clear who did prove this theorem first. However, in the context of linear regression, it was known well before Frisch and Waugh paper. In fact, it can be found as section 9, pag.184, in the detailed analysis of partial regressions by George Udny Yule published in 1907.[6]

It is of some interest to notice that, in their paper, Frisch and Waugh use, for the partial regression coefficients, the notation introduced by Yule in his 1907 paper. This was quite well known and used by 1933 as Yule presents a detailed discussion of partial correlation, including, among much else, his 1907 notation and the "Frisch, Waugh and Lovell" theorem, as chapter 10 of his, quite successful, Statistics textbook first issued in 1911 which, by 1932, had reached its tenth edition.[7]

## 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.
6. ^ Yule, George Udny (1907). "On the Theory of Correlation for any Number of Variables, Treated by a New System of Notation". Proceedings of the Royal Society A. 79: 182–193.
7. ^ Yule, George Udny (1932). An Introduction to the Theory of Statistics 10th edition. London: Charles Griffin &Co.