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

34 Pages Posted: 7 Feb 2006 Last revised: 19 Jun 2009

See all articles by Xavier Gabaix

Xavier Gabaix

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

Rustam Ibragimov

Harvard University - Department of Economics

Multiple version iconThere are 2 versions of this paper

Date Written: May 2009

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)= c - blog (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) c - blog (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.

Keywords: power law, heavy-tailedness, OLS log-log rank-size regression, bias, standard errors, Zipf's law

JEL Classification: C13, C14, C16

Suggested Citation

Gabaix, Xavier and Ibragimov, Rustam, Rank-1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents (May 2009). Harvard Institute of Economic Research Discussion Paper No. 2106, Available at SSRN: https://ssrn.com/abstract=881759 or http://dx.doi.org/10.2139/ssrn.881759

Xavier Gabaix

Harvard University - Department of Economics ( email )

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National Bureau of Economic Research (NBER)

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Centre for Economic Policy Research (CEPR)

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European Corporate Governance Institute (ECGI)

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Rustam Ibragimov (Contact Author)

Harvard University - Department of Economics ( email )

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HOME PAGE: http://www.economics.harvard.edu/faculty/ibragimov/ibragimov.html

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