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 )

Littauer Center
Cambridge, MA 02138
United States

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Centre for Economic Policy Research (CEPR)

London
United Kingdom

European Corporate Governance Institute (ECGI)

c/o ECARES ULB CP 114
B-1050 Brussels
Belgium

Rustam Ibragimov (Contact Author)

Harvard University - Department of Economics ( email )

Littauer Center
1805 Cambridge St.
Cambridge, MA 02138
United States
617-496-4795 (Phone)
617-495-7730 (Fax)

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

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