Abstract
The authors propose a new method for estimating the power-law exponents of firm size variables. Their focus is on how to empirically identify a range in which a firm size variable follows a power-law distribution. On the one hand, as is well known a firm size variable follows a power-law distribution only beyond some threshold. On the other hand, in almost all empirical exercises, the right end part of a distribution deviates from a power-law due to finite size effects. The authors modify the method proposed by Malevergne et al. (2011). In this way they can identify both the lower and the upper thresholds and then estimate the power-law exponent using observations only in the range defined by the two thresholds. They apply this new method to various firm size variables, including annual sales, the number of workers, and tangible fixed assets for firms in more than thirty countries.
References
Axtell, R. (2001). Zipf Distribution of U.S. Firm Sizes. Science, 293: 1818–1820. urlhttp://linkage.rockefeller.edu/wli/zipf/axtell01.pdf.Search in Google Scholar
Clauset, A., Shalizi, C., and Newman, M. (2009). Power-Law Distributions in Empirical Data. SIAM Review, 51: 661–703. urlhttp://arxiv.org/abs/arxiv:0706.1062.Search in Google Scholar
Cobb, C., and Douglas, P. (1928). A Theory of Production. American Economics Review, 18: 139–165. urlhttp://www.aeaweb.org/aer/top20/18.1.139-165.pdf.Search in Google Scholar
Coronel-Brizio, H., and Hernandez-Montoya, A. (2005). On Fitting the Pareto– Levy Distribution to Stock Market Index Data: Selecting a Suitable Cutoff Value. Physica A, 354: 437–449. urlhttp://arxiv.org/abs/cond-mat/0411161.Search in Google Scholar
Coronel-Brizio, H., and Hernandez-Montoya, A. (2010). The Anderson-Darling Test of Fit for the Power Law Distribution from Left Censored Samples. Physica A: Statistical Mechanics and its Applications, 389(3): 3508–3515. urlhttp://arxiv.org/abs/1004.0417.Search in Google Scholar
del Castillo, J., and Puig, P. (1999). The Best Test of Exponentiality against Singly Truncated Normal Alternatives. Journal of the American Statistical Association, 94(446): 529–532.Search in Google Scholar
Fujiwara, Y., Guilmi, C. D., Aoyama, H., Gallegati, M., and Souma, W. (2004). Do Pareto–Zipf and Gibrat Laws Hold True? An Analysis with European Firms. Physica A, 335: 197–216. urlhttp://chemlabs.nju.edu.cn/Literature/Zipf/Do%20Pareto-Zipf%20and%20Gibrat%20laws%20hold%20true-An%20analysis%20with%20European%20firms%20.pdf.Search in Google Scholar
Gaffeo, E., Gallegati, M., and Palestrinib, A. (2003). On the Size Distribution of Firms: Additional Evidence from the G7 Countries. Physica A, 324: 117–123. urlhttp://www.mendeley.com/research/on-the-size-distribution-offirms-additional-evidence-from-the-g7-countries/.Search in Google Scholar
Geyer, C. (1994). On the Asymptotics of Constrained M-Estimation. The Annals of Statistics, 22: 1993–2010. urlhttp://www.jstor.org/stable/2242495.Search in Google Scholar
Hisano, R., and Mizuno, T. (2011). Sales Distribution of Consumer Electronics. Physica A, 390: 309–318. urlhttp://arxiv.org/abs/1004.0637.Search in Google Scholar
Jessen, A. H., and Mikosch, T. (2006). Regularly Varying Functions. Publications de l’Institut Mathematique, Nouvelle serie, 80(94): 171–192.Search in Google Scholar
Malevergne, Y., Pisarenko, V., and Sornette, D. (2011). Testing the Pareto against the Lognormal Distributions with the Uniformly Most Powerful Unbiased Test Applied to the Distribution of Cities. Physical Review E, 83: 036111. urlhttp://www.mendeley.com/research/testing-pareto-against-lognormal-distributionsuniformly-most-powerful-unbiased-test-applied-distribution-cities/.Search in Google Scholar
Mizuno, T., Katori, M., Takayasu, H., and Takayasu, M. (2006). Statistical and Stochastic Laws in the Income of Japanese Companies. In H. Takayasu (Ed.), Empirical Science of Financial Fluctuations: The Advent of Econophysics. urlhttp://arxiv.org/ftp/cond-mat/papers/0308/0308365.pdf.Search in Google Scholar
Newman, M. (2005). Power Laws, Pareto Distributions and Zipf’s law. Contemporary Physics, 46: 323–351. urlhttp://arxiv.org/PS_cache/cond-mat/pdf/0412/0412004v3.pdf.Search in Google Scholar
Okuyama, K., Takayasu, M., and Takayasu, H. (1999). Zipf’s Law in Income Distribution of Companies. Physica A, 269: 125–131. urlhttp://www.mendeley.com/research/zipfs-law-income-distribution-companies/.Search in Google Scholar
Pareto, V. F. D. (1897). Cours d’Economique Politique. London: Macmillan.Search in Google Scholar
Ramsden, J., and Kiss-Haypál, G. (2000). Company Size Distribution in Different Countries. Physica A, 277: 220–227.Search in Google Scholar
Self, S., and Liang, K.-Y. (1987). Asymptotic Properties of Maximum Likelhood Estimatiors and Likelihood Ratio Test Under Nonstandard Conditions. Journal of the American Statistical Association, 82(398): 605–610. urlhttp://www.jstor.org/stable/2289471.Search in Google Scholar
Stanley, M. H. R., Buldyrev, S. V., Havlin, S., Mantegna, R. N., Salinger, M. A., and Stanley, H. E. (1995). Zipf plots and the size distribution of firms. Economics Letters, 49: 453–457. urlhttp://ideas.repec.org/a/eee/ecolet/v49y1995i4p453-457.html.Search in Google Scholar
Zhang, J., Chen, Q., and Wang, Y. (2009). Zipf distribution in top Chinese Firms and an Economic Explanation. Physica A, 388: 2020–2024. urlhttp://www.sciencedirect.com/science/article/pii/S0378437109000806.Search in Google Scholar
© 2011 Shouji Fujimoto et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.