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:

Y = X_1 \beta_1 + X_2 \beta_2 + u \!

where $X_1$ and $X_2$ are $n \times k_1$ and $n \times k_2$ matrices respectively and where $\beta_1$ and $\beta_2$ are conformable, then the estimate of $\beta_2$ will be the same as the estimate of it from a modified regression of the form:

M_{X_1} Y = M_{X_1} X_2 \beta_2 + M_{X_1} u \!,

where $M_{X_1}$ projects onto the orthogonal complement of the image of the projection matrix $X_1(X_1^{\top}X_1)^{-1}X_1^{\top}$ . Equivalently, M_X₁ projects onto the orthogonal complement of the column space of X₁. Specifically,

M_{X_1} = I - X_1(X_1^{\top}X_1)^{-1}X_1^{\top}. \!

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

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.

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