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 expressed in terms of two separate sets of predictor variables:
where and are matrices, and are vectors (and is the error term), then the estimate of will be the same as the estimate of it from a modified regression of the form:
where projects onto the orthogonal complement of the image of the projection matrix . Equivalently, MX1 projects onto the orthogonal complement of the column space of X1. Specifically,
and this particular orthogonal projection matrix is known as the
The vector is the vector of residuals from regression of on the columns of .
The most relevant consequence of the theorem is that the parameters in do not apply to but to , that is: the part of uncorrelated with . This is the basis for understanding the contribution of each single variable to a multivariate regression (see, for instance, Ch. 13 in [6]).
The theorem also implies that the secondary regression used for obtaining is unnecessary when the predictor variables are uncorrelated: 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.
Moreover, the standard errors from the partial regression equal those from the full regression.[7]
History
The origin of the theorem is uncertain, but it was well-established in the realm of linear regression before the Frisch and Waugh paper.
By 1933, Yule's findings were generally recognized[
In a 1931 paper co-authored with Mudgett, Frisch cited Yule's results.[10] Yule's formulas for partial regressions were quoted and explicitly attributed to him in order to rectify a misquotation by another author.[10] Although Yule was not explicitly mentioned in the 1933 paper by Frisch and Waugh, they utilized the notation for partial regression coefficients initially introduced by Yule in 1907, which was widely accepted by 1933[original research?].
In 1963, Lovell published a proof[11] considered more straightforward and intuitive. In recognition, people generally add his name to the theorem name.
References
- JSTOR 1907330.
- .
- S2CID 154907484.
- ISBN 0-691-01018-8.
- ISBN 0-631-21584-0.
- ^ Mosteller, F.; Tukey, J. W. (1977). Data Analysis and Regression a Second Course in Statistics. Addison-Wesley.
- ^ Peng, Ding (2021). "The Frisch--Waugh--Lovell theorem for standard errors". Statistics and Probability Letters. 168: 108945.
- ^ .
- ^ Yule, George Udny (1932). An Introduction to the Theory of Statistics 10th edition. London: Charles Griffin &Co.
- ^ .
- .
Further reading
- 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.
- 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. ISBN 9780262337465.