Rank-1/2: A simple way to improve the OLS estimation of tail exponents

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Title: Rank-1/2: A simple way to improve the OLS estimation of tail exponents
Authors: Gabaix, X
Ibragimov, R
Item Type: Journal Article
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 ζ is not the OLS standard error, but is asymptotically (2 / n)1 / 2ζ. 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.
Issue Date: 1-Jan-2011
Date of Acceptance: 1-Jan-2011
URI: http://hdl.handle.net/10044/1/67780
DOI: https://dx.doi.org/10.1198/jbes.2009.06157
ISSN: 0735-0015
Publisher: Taylor & Francis
Start Page: 24
End Page: 39
Journal / Book Title: Journal of Business and Economic Statistics
Volume: 29
Issue: 1
Copyright Statement: © 2011 American Statistical Association. This is an Accepted Manuscript of an article published by Taylor & Francis in Journal of Business & Economic Statistics on 1 Jan 2011, available online: https://dx.doi.org/10.1198/jbes.2009.06157
Sponsor/Funder: National Science Foundation
Funder's Grant Number: SES-0820124
Keywords: Social Sciences
Science & Technology
Physical Sciences
Economics
Social Sciences, Mathematical Methods
Statistics & Probability
Business & Economics
Mathematical Methods In Social Sciences
Mathematics
Bias
Heavy-tailedness
OLS log-log rank-size regression
Power law
Standard errors
Zipf's law
PARETO WEALTH DISTRIBUTION
LEAST-SQUARES ESTIMATORS
PARTIAL SUMS
ZIPFS LAW
CITIES
SIZE
APPROXIMATION
REGRESSION
GROWTH
PRICES
01 Mathematical Sciences
14 Economics
15 Commerce, Management, Tourism And Services
Econometrics
Publication Status: Published
Online Publication Date: 2012-01-01
Appears in Collections:Imperial College Business School



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