# Frisch–Waugh–Lovell theorem

Frisch–Waugh–Lovell theorem (Redirected from FWL theorem) 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: {displaystyle Y=X_{1}beta _{1}+X_{2}beta _{2}+u} where {displaystyle X_{1}} and {displaystyle X_{2}} are {displaystyle ntimes k_{1}} and {displaystyle ntimes k_{2}} matrices respectively and where {displaystyle beta _{1}} and {displaystyle beta _{2}} are conformable, then the estimate of {displaystyle beta _{2}} will be the same as the estimate of it from a modified regression of the form: {displaystyle M_{X_{1}}Y=M_{X_{1}}X_{2}beta _{2}+M_{X_{1}}u,} where {displaystyle M_{X_{1}}} projects onto the orthogonal complement of the image of the projection matrix {displaystyle X_{1}(X_{1}^{mathsf {T}}X_{1})^{-1}X_{1}^{mathsf {T}}} . Equivalently, MX1 projects onto the orthogonal complement of the column space of X1. Specifically, {displaystyle 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 {displaystyle 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.

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] 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. ^ 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. ^ 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. 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. hide vte Least squares and regression analysis Computational statistics Least squaresLinear least squaresNon-linear least squaresIteratively reweighted least squares Correlation and dependence Pearson product-moment correlationRank correlation (Spearman's rhoKendall's tau)Partial correlationConfounding variable Regression analysis Ordinary least squaresPartial least squaresTotal least squaresRidge regression Regression as a statistical model Linear regression Simple linear regressionOrdinary least squaresGeneralized least squaresWeighted least squaresGeneral linear model Predictor structure Polynomial regressionGrowth curve (statistics)Segmented regressionLocal regression Non-standard Nonlinear regressionNonparametricSemiparametricRobustQuantileIsotonic Non-normal errors Generalized linear modelBinomialPoissonLogistic Decomposition of variance Analysis of varianceAnalysis of covarianceMultivariate AOV Model exploration Stepwise regressionModel selection Mallows's CpAICBICModel specificationRegression validation Background Mean and predicted responseGauss–Markov theoremErrors and residualsGoodness of fitStudentized residualMinimum mean-square errorFrisch–Waugh–Lovell theorem Design of experiments Response surface methodologyOptimal designBayesian design Numerical approximation Numerical analysisApproximation theoryNumerical integrationGaussian quadratureOrthogonal polynomialsChebyshev polynomialsChebyshev nodes Applications Curve fittingCalibration curveNumerical smoothing and differentiationSystem identificationMoving least squares Regression analysis categoryStatistics category Mathematics portalStatistics outlineStatistics topics Categories: Economics theoremsRegression analysisTheorems in statistics

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