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On Zipf’s law and the bias of Zipf regressions

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  • Christian Schluter

    (AMSE - Aix-Marseille Sciences Economiques - EHESS - École des hautes études en sciences sociales - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique)

Abstract

City size distributions are not strictly Pareto, but upper tails are rather Pareto like (i.e. tails are regularly varying). We examine the properties of the tail exponent estimator obtained from ordinary least squares (OLS) rank size regressions (Zipf regressions for short), the most popular empirical strategy among urban economists. The estimator is then biased towards Zipf's law in the leading class of distributions. The Pareto quantile–quantile plot is shown to offer a simple diagnostic device to detect such distortions and should be used in conjunction with the regression residuals to select the anchor point of the OLS regression in a data-dependent manner. Applying these updated methods to some well-known data sets for the largest cities, Zipf's law is now rejected in several cases.

Suggested Citation

  • Christian Schluter, 2021. "On Zipf’s law and the bias of Zipf regressions," Post-Print hal-02880544, HAL.
    • Handle: RePEc:hal:journl:hal-02880544
    DOI: 10.1007/s00181-020-01879-3
    Note: View the original document on HAL open archive server: https://amu.hal.science/hal-02880544
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    References listed on IDEAS

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    1. Christian Schluter & Mark Trede, 2019. "Size distributions reconsidered," Econometric Reviews, Taylor & Francis Journals, vol. 38(6), pages 695-710, July.
    2. Jan Eeckhout, 2004. "Gibrat's Law for (All) Cities," American Economic Review, American Economic Association, vol. 94(5), pages 1429-1451, December.
    3. Luckstead, Jeff & Devadoss, Stephen, 2014. "Do the world’s largest cities follow Zipf’s and Gibrat’s laws?," Economics Letters, Elsevier, vol. 125(2), pages 182-186.
    4. Xavier Gabaix & Rustam Ibragimov, 2011. "Rank - 1 / 2: A Simple Way to Improve the OLS Estimation of Tail Exponents," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(1), pages 24-39, January.
    5. Ioannides, Yannis & Skouras, Spyros, 2013. "US city size distribution: Robustly Pareto, but only in the tail," Journal of Urban Economics, Elsevier, vol. 73(1), pages 18-29.
    6. Einmahl, J. H.J. & Dekkers, A. L.M. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Other publications TiSEM 81970cb3-5b7a-4cad-9bf6-2, Tilburg University, School of Economics and Management.
    7. Yoshihiko Nishiyama & Susumu Osada & Yasuhiro Sato, 2008. "OLS ESTIMATION AND THE t TEST REVISITED IN RANK‐SIZE RULE REGRESSION," Journal of Regional Science, Wiley Blackwell, vol. 48(4), pages 691-716, October.
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    Cited by:

    1. Rocco Rante & Federico Trionfetti & Priyam Verma, 2024. "The Size Distribution of Cities: Evidence from the Lab," Working Papers hal-04556678, HAL.

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    More about this item

    Keywords

    regular variation; city size distributions; Zipf's law; rank size regression; extreme value index; heavy tails;
    All these keywords.

    JEL classification:

    • R12 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General Regional Economics - - - Size and Spatial Distributions of Regional Economic Activity; Interregional Trade (economic geography)
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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