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:

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 $X_{1}(X_{1}^{\mathsf {T}}X_{1})^{-1}X_{1}^{\mathsf {T}}$ . Equivalently, M_X₁ projects onto the orthogonal complement of the column space of X₁. Specifically,

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

↑ 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.
↑ 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.
↑ 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.

Literature

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.
Davidson, Russell; MacKinnon, James G. (1993). Estimation and Inference in Econometrics. New York: Oxford University Press. pp. 19–24. ISBN 0-19-506011-3.
Stachurski, John (2016). A Primer in Econometric Theory. MIT Press. pp. 311–314.

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