arXiv 29 Dec 2022 · Econometrics · 1 citations (OpenAlex)
arXiv:2212.14185 · PDF · DOI · OpenAlex · Extracted main text
The recent thought-provoking paper by Hansen [2022, Econometrica] proved that the Gauss-Markov theorem continues to hold without the requirement that competing estimators are linear in the vector of outcomes. Despite the elegant proof, it was shown by the authors and other researchers that the main result in the earlier version of Hansen's paper does not extend the classic Gauss-Markov theorem because no nonlinear unbiased estimator exists under his conditions. To address the issue, Hansen [2022] added statements in the latest version with new conditions under which nonlinear unbiased estimators exist. Motivated by the lively discussion, we study a fundamental problem: what estimators are unbiased for a given class of linear models? We first review a line of highly relevant work dating back to the 1960s, which, unfortunately, have not drawn enough attention. Then, we introduce notation that allows us to restate and unify results from earlier work and Hansen [2022]. The new framework also allows us to highlight differences among previous conclusions. Lastly, we establish new representation theorems for unbiased estimators under different restrictions on the linear model, allowing the coefficients and covariance matrix to take only a finite number of values, the higher moments of the estimator and the dependent variable to exist, and the error distribution to be discrete, absolutely continuous, or dominated by another probability measure. Our results substantially generalize the claims of parallel commentaries on Hansen [2022] and a remarkable result by Koopmann [1982].
appendix boundary found by appendix_command · 79% of the source is main text. Read the extracted text to check this.
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 | Bruce E Hansen (2022) A modern gauss-markov theorem | 1.000 | 25 | 4 | 100% |
| 2 | Benedikt M Pötscher and David Preinerstorfer (2022) A modern gauss-markov theorem? really? | 1.000 | 13 | 3 | 100% |
| 3 | Reinhardt Koopmann (1982) Parameterschätzung bei a-priori-Information, volume 12 | 1.000 | 12 | 4 | 100% |
| 4 | Stephen Portnoy (2022) Linearity of unbiased linear model estimators | 1.000 | 8 | 3 | 100% |
| 5 | TW Anderson (1962) Least squares and best unbiased estimates | 0.874 | 8 | 2 | 100% |
| 6 | Haskell P Rosenthal (1970) On the subspaces of $L^p$ $(p> 2)$ spanned by sequences of independent random variables | 0.874 | 5 | 2 | 100% |
| 7 | S Gnot, H Knautz, G Trenkler, and R Zmyslony (1992) Nonlinear unbiased estimation in linear models | 0.811 | 4 | 2 | 100% |
| 8 | AC Aitken (1935) On least squares and linear combination of observations | 0.644 | 2 | 2 | 100% |
| 9 | D Fraser (1954) Completeness of order statistics | 0.644 | 2 | 2 | 100% |
| 10 | Wassily Hoeffding (1977) More on incomplete and boundedly complete families of distributions | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 18 scored citations.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Estimation and exclusion restrictions in clustered linear models | 0.405 | 1 | 1 |