Rank - 1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents. Gabaix, X. & Ibragimov, R. Journal of Business & Economic Statistics, 29(1):24–39, 2011. ![Rank - 1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents [link]](https://bibbase.org/img/filetypes/link.svg "link")
Link doi abstract bibtex Despite the availability of more sophisticated methods, a popular way to estimate a Pareto exponent is still to run an OLS regression: log(Rank) = a - b log(Size), and take b as an estimate of the Pareto exponent. The reason for this popularity is arguably the simplicity and robustness of this method. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and propose that, if one wants to use an OLS regression, one should use the Rank - 1/2, and run log(Rank - 1/2) = a - b log(Size). The shift of 1/2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent $ζ$ is not the OLS standard error, but is asymptotically (2/n) 1/2 $ζ$. Numerical results demonstrate the advantage of the proposed approach over the standard OLS estimation procedures and indicate that it performs well under dependent heavy-tailed processes exhibiting deviations from power laws. The estimation procedures considered are illustrated using an empirical application to Zipf's law for the United States city size distribution.
@article{GabaixIbragimov2011,
title = {Rank - 1/2: A Simple Way to Improve the {{OLS}} Estimation of Tail Exponents},
author = {Gabaix, Xavier and Ibragimov, Rustam},
year = {2011},
journal = {Journal of Business \& Economic Statistics},
volume = {29},
number = {1},
pages = {24--39},
doi = {10.1198/jbes.2009.06157},
url = {https://doi.org/10.1198/jbes.2009.06157},
abstract = {Despite the availability of more sophisticated methods, a popular way to estimate a Pareto exponent is still to run an OLS regression: log(Rank) = a - b log(Size), and take b as an estimate of the Pareto exponent. The reason for this popularity is arguably the simplicity and robustness of this method. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and propose that, if one wants to use an OLS regression, one should use the Rank - 1/2, and run log(Rank - 1/2) = a - b log(Size). The shift of 1/2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent {$\zeta$} is not the OLS standard error, but is asymptotically (2/n) 1/2 {$\zeta$}. Numerical results demonstrate the advantage of the proposed approach over the standard OLS estimation procedures and indicate that it performs well under dependent heavy-tailed processes exhibiting deviations from power laws. The estimation procedures considered are illustrated using an empirical application to Zipf's law for the United States city size distribution.},
keywords = {Methods of Estimation of Wealth Inequality}
}
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