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

Author

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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.

Suggested Citation

  • Xavier Gabaix & Rustam Ibragimov, 2006. "Log(Rank-1/2): A Simple Way to Improve the OLS Estimation of Tail Exponents," Harvard Institute of Economic Research Working Papers 2106, Harvard - Institute of Economic Research.
  • Handle: RePEc:fth:harver:2106
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    1. Yannis M. Ioannides & Henry G. Overman & Esteban Rossi-Hansberg & Kurt Schmidheiny, 2008. "The effect of information and communication technologies on urban structure," Economic Policy, CEPR;CES;MSH, vol. 23, pages 201-242, April.
    2. 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.
    3. 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.
    4. 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.
    5. Daron Acemoglu & Vasco M. Carvalho & Asuman Ozdaglar & Alireza Tahbaz‐Salehi, 2012. "The Network Origins of Aggregate Fluctuations," Econometrica, Econometric Society, vol. 80(5), pages 1977-2016, September.
    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. João Nicolau & Paulo M.M. Rodrigues, 2015. "A New Regression-Based Tail Index Estimator: An Application to Exchange Rates," Working Papers w201514, Banco de Portugal, Economics and Research Department.
    8. Valente J. Matlaba & Mark J. Holmes & Philip McCann & Jacques Poot, 2013. "A Century Of The Evolution Of The Urban System In Brazil," Review of Urban & Regional Development Studies, Wiley Blackwell, vol. 25(3), pages 129-151, November.
    9. 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.
    10. Zelai Xu & Nong Zhu, 2014. "City size distribution in China: are large cities dominant?," CIRANO Working Papers 2014s-04, CIRANO.
    11. Alexandra SCHAFFAR & Sotiris PAVLEAS, 2014. "The Evolution Of The Greek Urban Centers: 1951-2011," Region et Developpement, Region et Developpement, LEAD, Universite du Sud - Toulon Var, vol. 39, pages 87-104.
    12. 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.

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