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

A-Tier
Journal: Journal of Business & Economic Statistics
Year: 2011
Volume: 29
Issue: 1
Pages: 24-39

Authors (2)

Xavier Gabaix (Harvard University) Rustam Ibragimov (not in RePEc)

Score contribution per author:

2.011 = (α=2.01 / 2 authors) × 2.0x A-tier

α: calibrated so average coauthorship-adjusted count equals average raw count

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) = <italic>a</italic> - <italic>b</italic> log(Size), and take <italic>b</italic> 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) = <italic>a</italic> - <italic>b</italic> log(Size). The shift of 1 / 2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent <italic>&#x3b6;</italic> is not the OLS standard error, but is asymptotically (2 / <italic>n</italic>)-super-1 / 2 <italic>&#x3b6;</italic>. 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.

Technical Details

RePEc Handle
repec:taf:jnlbes:v:29:y:2011:i:1:p:24-39
Journal Field
Econometrics
Author Count
2
Added to Database
2026-01-25