Size Distributions for All Cities: Which One is Best?

González-Val, Rafael and Ramos, Arturo and Sanz, Fernando and Vera-Cabello, María (2013): Size Distributions for All Cities: Which One is Best?

Abstract

This paper analyses in detail the features offered by three distributions used in urban economics to describe city size distributions: lognormal, q-exponential and double Pareto lognormal, and another one of use in other areas of economics: the log-logistic. We use a large database which covers all cities with no size restriction in the US, Spain and Italy from 1900 until 2010, and, in addition, the last available year for the rest of the countries of the OECD. We estimate the previous four density functions by maximum likelihood. To check the goodness of the fit in all periods and for the thirty-four countries we use the Kolmogorov-Smirnov and Cramér-von Mises tests, and compute the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). The results show that the distribution which best fits the data in most of the cases (86.76%) is the double Pareto lognormal.

• Item Type: MPRA Paper
• Language: English
• Keywords: city size distribution; double Pareto lognormal; log-logistic; q-exponential; lognormal
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R0 - General > R00 - General

References

Anderson, G. and Y. Ge (2005). “The size distribution of Chinese cities,” Regional Science and Urban Economics 35, 756–776.

Berry, B. J. L. and A. Okulicz-Kozaryn (2012). “The city size distribution debate: Resolution for US urban regions and megalopolitan areas,” Cities 29, S17–S23.

Black, D. and J. V. Henderson (2003). “Urban evolution in the USA,” Journal of Economic Geography 3, 343–372.

Bosker, M., S. Brakman, H. Garretsen and M. Schramm (2008). “A century of shocks: the evolution of the German city size distribution 1925–1999,” Regional Science and Urban Economics 38, 330–347.

Burnham, K. P. and D. R. Anderson (2002). “Model selection and multimodel inference: A practical information-theoretic approach.” Springer–Verlag.

Burnham, K. P. and D. R. Anderson (2004). “Multimodel inference: Understanding AIC and BIC in model selection,” Sociological Methods and Research 33, 261-304.

Cameron, T. A. (1990). “One-stage structural models to explain city size,” Journal of Urban Economics 27(3), 294–207.

Cheshire, P. (1999). “Trends in sizes and structure of urban areas,” in Handbook of Regional and Urban Economics, Vol. 3, P. Cheshire and E. S. Mills (eds.) Amsterdam: Elsevier Science, Chapter 35, 1339–1373.

Cheshire, P. C. and S. Magrini (2006). “Population growth in European cities: weather matters-but only nationally,” Regional Studies 40(1), 23–37.

Choulakian, V. and M. A. Stephens (2001). “Goodness-of-fit tests for the generalized Pareto distribution,” Technometrics 43, 478–484.

Eeckhout, J. (2004). “Gibrat’s Law for (all) cities,” American Economic Review 94(5), 1429–1451.

Fisk, P. R. (1961). “The graduation of income distributions,” Econometrica 29(2), 171–185.

Gabaix, X. and Y. M. Ioannides (2004). “The evolution of city size distributions,” in Handbook of Urban and Regional Economics, Vol. 4, J. V. Henderson and J. F. Thisse (eds.) Amsterdam: Elsevier Science, North-Holland, 2341–2378.

Gibrat, R. (1931). “Les inégalités économiques,” Paris: Librairie du recueil Sirey.

Giesen, K. and J. Suedekum (2011). “Zipf´s law for cities in the regions and the country,” Journal of Economic Geography 11, 667–686.

Giesen, K. and J. Suedekum (2012a). “The size distribution across all ‘cities’: A unifying approach,” IEB Working Paper 2012/2.

Giesen, K. and J. Suedekum (2012b). “The French overall city size distribution,” Région et Développement 36, 107–126.

Giesen, K., A. Zimmermann and J. Suedekum (2010). “The size distribution across all cities – double Pareto lognormal strikes,” Journal of Urban Economics 68, 129–137.

González-Val, R. (2010). “The evolution of US city size distribution from a long term perspective (1900–2000),” Journal of Regional Science 50, 952–972.

González-Val, R., L. Lanaspa and F. Sanz (2012). “New evidence on Gibrat’s law for cities,” IEB Working Paper 2012/18.

Grimshaw, S. D. (1993). “Computing maximum likelihood estimates for the generalized Pareto distribution,” Technometrics 35, 185–191.

Hosking, J. R. M. and J. R. Wallis (1987). “Parameter and quantile estimation for the generalized Pareto distribution,” Technometrics 29, 339–349.

Hsing, Y. (1990). “A note on functional forms and the urban size distribution,” Journal of Urban Economics 27(1), 73–79.

Hsu, W.-T. (2012). “Central place theory and city size distribution,” The Economic Journal 122, 903–932.

Ioannides, Y. M. and H. G. Overman (2003). “Zipf’s law for cities: an empirical examination,” Regional Science and Urban Economics 33, 127–137.

Ioannides, Y. M. and S. Skouras (2013). “US city size distribution: Robustly Pareto, but only in the tail,” Journal of Urban Economics 73, 18–29.

Kalecki, M. (1945). “On the Gibrat distribution,” Econometrica 13(2), 161–170.

Kamecke, U. (1990). “Testing the rank size rule hypothesis with an efficient estimator,” Journal of Urban Economics 27(2), 222–231.

Levy, M. (2009). “Gibrat´s Law for (All) Cities: Comment,” American Economic Review 99, 1672–1675.

Malacarne, L. C., R. S. Mendes and E. K. Lenzi (2001). “q-exponential distribution in urban agglomeration,” Physical Review E 65(017106), 1–3.

Michaels, G., F. Rauch and S. J. Redding (2012). “Urbanization and structural transformation,” The Quarterly Journal of Economics 127(2), 535–586.

Pareto, V. (1896). “Cours d’économie politique,” Geneva: Droz.

Parr, J. B. and K. Suzuki (1973). “Settlement populations and the lognormal distribution,” Urban Studies 10, 335–352.

Razali, N.M. and Y.B. Wah (2011). “Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests,” Journal of Statistical Modeling and Analytics 2, 21–33.

Reed, W. (2001). “The Pareto, Zipf and other power laws,” Economics Letters 74, 15–19.

Reed, W. (2002). “On the rank-size distribution for human settlements,” Journal of Regional Science 42, 1–17.

Reed, W. and M. Jorgensen (2004). “The double Pareto-lognormal distribution – A new parametric model for size distribution,” Com. Stats. –Theory & Methods 33, 1733–1753.

Richardson, H. W. (1973). “Theory of the distribution of city sizes: Review and prospects,” Regional Studies 7, 239–251.

Rosen, K. and M. Resnick (1980). “The size distribution of cities: an examination of the Pareto law and primacy,” Journal of Urban Economics 8, 165–186.

Rozenfeld, H. D., D. Rybski, X. Gabaix and H. A. Makse (2011). “The Area and Population of Cities: New Insights from a Different Perspective on Cities,” American Economic Review 101(5), 2205–25.

Shalizi, C. (2007). “Maximum likelihood estimation for q-exponential (Tsallis) Distributions,” arXiv: math/0701854v2.

Singh, S. K. and G. S. Maddala (2008). “A function for size distribution of incomes,” in Modeling Income Distributions and Lorenz Curves. Economic Studies in Inequality, Social Exclusion and Well-Being, Volume 5, I, 27–35.

Soo, K. T. (2005). “Zipf’s Law for cities: a cross-country investigation,” Regional Science and Urban Economics 35, 239–263.

Soo, K. T. (2007). “Zipf's Law and urban growth in Malaysia,” Urban Studies 44(1), 1–14.

Tsallis, C. (1988). “Possible generalization of Boltzmann-Gibbs statistics,” Journal of Statistical Physics 52, 479–487.

Zipf, G. K. (1949). “Human behaviour and the principle of least effort,” Cambridge, MA: Addison-Wesley.