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}},

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 $M_{X_{1}}$ 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.

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.

Davidson, Russell; MacKinnon, James G. (1993). Estimation and Inference in Econometrics. New York: Oxford University Press. pp. 19–24. ISBN 0-19-506011-3.
Davidson, Russell; MacKinnon, James G. (2004). Econometric Theory and Methods. New York: Oxford University Press. pp. 62–75. ISBN 0-19-512372-7.
Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome (2017). "Multiple Regression from Simple Univariate Regression" (PDF). The Elements of Statistical Learning : Data Mining, Inference, and Prediction (2nd ed.). New York: Springer. pp. 52–55. ISBN 978-0-387-84857-0.
Ruud, P. A. (2000). An Introduction to Classical Econometric Theory. New York: Oxford University Press. pp. 54–60. ISBN 0-19-511164-8.
Stachurski, John (2016). A Primer in Econometric Theory. MIT Press. pp. 311–314.

[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.

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References

Further reading