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

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  • Xavier Gabaix
  • Rustam Ibragimov

Abstract

A popular way to estimate a Pareto exponent is to run an OLS regression: log (Rank) = c - blog (Size), and take b as an estimate of the Pareto exponent. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and argue that, if one wants to use an OLS regression, one should use the Rank -1/2, and run log (Rank- 1/2) = c-b log (Size). The shift of 1/2 is optimal, and cancels the bias to a leading order. The standard error on the Pareto exponent is not the OLS standard error, but is asymptotically (2/n)^{1/2}b. To obtain this result, we provide asymptotic expansions for the OLS estimate in such log-log rank-size regression with arbitrary shifts in the ranks. The arguments for the asymptotic expansions rely on strong approximations to martingales with the optimal rate and demonstrate that martingale convergence methods provide a natural and conceptually simple framework for deriving the asymptotics of the tail index estimates using the log-log rank-size regressions.

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Bibliographic Info

Paper provided by Harvard - Institute of Economic Research in its series Harvard Institute of Economic Research Working Papers with number 2106.

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Date of creation: 2006
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Handle: RePEc:fth:harver:2106

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Cited by:
  1. Yannis M. Ioannides & Henry G. Overman & Esteban Rossi-Hansberg & Kurt Schmidheiny, 2007. "The Effect of Information and Communication Technologies on Urban Structure," CESifo Working Paper Series 2049, CESifo Group Munich.
  2. Daron Acemoglu & Vasco Carvalho & Asuman Ozdaglar & Alireza Tahbaz-Salehi, 2011. "The Network Origins of Aggregate Fluctuations," Working Papers 587, Barcelona Graduate School of Economics.
  3. Makoto Nirei & Theodoros Stamatiou & Vladyslav Sushko, 2012. "Stochastic Herding in Financial Markets Evidence from Institutional Investor Equity Portfolios," BIS Working Papers 371, Bank for International Settlements.
  4. Maarten Bosker & Steven Brakman & Harry Garretsen & Herman de Jong & Marc Schramm, 2007. "The Development of Cities in Italy 1300 – 1861," CESifo Working Paper Series 1893, CESifo Group Munich.
  5. Valente J. Matlaba & Mark Holmes & Philip McCann & Jacques Poot, 2011. "A Century of the Evolution of the Urban System in Brazil," Working Papers in Economics 11/12, University of Waikato, Department of Economics.
  6. Segarra, Agustí & Teruel, Mercedes, 2012. "An appraisal of firm size distribution: Does sample size matter?," Journal of Economic Behavior & Organization, Elsevier, vol. 82(1), pages 314-328.
  7. Michel DIMOU & Alexandra SCHAFFAR & Zhihong CHEN & Shihe FU, 2008. "La Croissance Urbaine Chinoise Reconsidérée," Region et Developpement, Region et Developpement, LEAD, Universite du Sud - Toulon Var, vol. 27, pages 109-131.
  8. Jeroen Hinloopen & Charles van Marrewijk, 2006. "Comparative Advantage, the Rank-size Rule, and Zipf's Law," Tinbergen Institute Discussion Papers 06-100/1, Tinbergen Institute.
  9. Wen-Tai Hsu & Tomoya Mori & Tony E. Smith, 2014. "Spatial Patterns and Size Distributions of Cities," KIER Working Papers 882, Kyoto University, Institute of Economic Research.
  10. Alexandra SCHAFFAR, 2008. "Regional Income Inequality And Urbanisation Trends In China: 1978-2005," Region et Developpement, Region et Developpement, LEAD, Universite du Sud - Toulon Var, vol. 28, pages 87-110.

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