Frisch–Waugh–Lovell theorem

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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:

where and are and matrices respectively and where and are conformable, 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 annihilator matrix.[4][5]

The vector is the vector of residuals from regression of on the columns of .

The theorem implies that the secondary regression used for obtaining 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]


  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.

Further reading[edit]