Rank-1/2: A simple way to improve the OLS estimation of tail exponents
File(s)IbragimovGabaix.pdf (381.86 KB)
Accepted version
Author(s)
Gabaix, Xavier
Ibragimov, Rustam
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.
Date Issued
2011-01-01
Date Acceptance
2011-01-01
Citation
Journal of Business and Economic Statistics, 2011, 29 (1), pp.24-39
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
National Science Foundation
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000286407200003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Grant Number
SES-0820124
Subjects
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
Publication Status
Published
Date Publish Online
2012-01-01