arXiv 12 Jul 2023 · Econometrics · 2 citations (OpenAlex)
arXiv:2307.12731 · PDF · DOI · OpenAlex · Extracted main text
In this paper, I discuss three aspects of the Frisch-Waugh-Lovell theorem. First, I show that the theorem holds for linear instrumental variables estimation of a multiple regression model that is either exactly or overidentified. I show that with linear instrumental variables estimation: (a) coefficients on endogenous variables are identical in full and partial (or residualized) regressions; (b) residual vectors are identical for full and partial regressions; and (c) estimated covariance matrices of the coefficient vectors from full and partial regressions are equal (up to a degree of freedom correction) if the estimator of the error vector is a function only of the residual vectors and does not use any information about the covariate matrix other than its dimensions. While estimation of the full model uses the full set of instrumental variables, estimation of the partial model uses the residualized version of the same set of instrumental variables, with residualization carried out with respect to the set of exogenous variables. Second, I show that: (a) the theorem applies in large samples to the K-class of estimators, including the limited information maximum likelihood (LIML) estimator, and (b) the theorem does not apply in general to linear GMM estimators, but it does apply to the two step optimal linear GMM estimator. Third, I trace the historical and analytical development of the theorem and suggest that it be renamed as the Yule-Frisch-Waugh-Lovell (YFWL) theorem to recognize the pioneering contribution of the statistician G. Udny Yule in its development.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Yule, G. U (1907) On the Theory of Correlation for any number of Variables, treated by a New System of Notation | 1.000 | 23 | 4 | 100% |
| 2 | Lovell, M. C (1963) Seasonal Adjustment of Economic Time Series and Multiple Regression Analysis | 1.000 | 13 | 3 | 100% |
| 3 | Giles, D. E. A (1984) Instrumental variables regressions involving seasonal data | 1.000 | 12 | 3 | 100% |
| 4 | Frisch, R. and Waugh, F. V (1933) Partial Time Regressions as Compared with Individual Trends | 0.973 | 25 | 5 | 92% |
| 5 | Greene, W. H (2012) Econometric Analysis | 0.956 | 8 | 4 | 88% |
| 6 | Ding, P (2021) The Frisch-Waugh-Lovell theorem for standard errors | 0.909 | 16 | 5 | 75% |
| 7 | Davidson, R. and MacKinnon, J. G (1993) Estimation and Inference in Econometrics | 0.811 | 4 | 2 | 100% |
| 8 | Strang, G (2006) Linear Algebra and its Applications | 0.794 | 6 | 3 | 50% |
| 9 | Davidson, R. and MacKinnon, J. G (2004) Econometric Theory and Methods | 0.737 | 3 | 2 | 100% |
| 10 | Yule, G. U (1911) An Introduction to the Theory of Statistics | 0.644 | 4 | 1 | 100% |
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