Rank-1/2: A Simple Way to Improve the Ols Estimation of Tail Exponents

37 Pages Posted: 14 Sep 2007 Last revised: 10 Jun 2023

See all articles by Rustam Ibragimov

Rustam Ibragimov

Harvard University - Department of Economics

Xavier Gabaix

Harvard University - Department of Economics; National Bureau of Economic Research (NBER); Centre for Economic Policy Research (CEPR); European Corporate Governance Institute (ECGI)

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Date Written: September 2007

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 U.S. city size distribution.

Suggested Citation

Ibragimov, Rustam and Gabaix, Xavier, Rank-1/2: A Simple Way to Improve the Ols Estimation of Tail Exponents (September 2007). NBER Working Paper No. t0342, Available at SSRN: https://ssrn.com/abstract=1014025

Rustam Ibragimov

Harvard University - Department of Economics ( email )

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Xavier Gabaix (Contact Author)

Harvard University - Department of Economics ( email )

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