Have the log-population processes stationary and independent increments? Empirical evidence for Italy, Spain and the USA along more than a century.

Ramos, Arturo (2019): Have the log-population processes stationary and independent increments? Empirical evidence for Italy, Spain and the USA along more than a century.

Abstract

We review the classical Gibrat’s process for the population of city sizes. In particular, we are interested in whether the log-population process has stationary and independent (Gibrat’s Law for cities) increments. We have tested these characteristics for the case of the municipalities of Italy and Spain and the places of USA for a time span of more than one century. The results are clear: stationarity and independence are empirically rejected by standard tests. These results open theoretically the way for the observance of other city size distributions other than the lognormal and the double Pareto lognormal, something that in fact has already happened in the literature.

• Item Type: MPRA Paper
• Language: English
• Keywords: Gibrat’s process; log-population process; stationary increments; independent increments; Italian cities; Spanish cities; USA cities
• Subjects: C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C46 - Specific Distributions ; Specific Statistics; R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R1 - General Regional Economics > R11 - Regional Economic Activity: Growth, Development, Environmental Issues, and Changes; R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R1 - General Regional Economics > R12 - Size and Spatial Distributions of Regional Economic Activity

References

Amaral, L. A. N., Buldyrev, S., Havlin, S., Leschhorn, H., Maas, P., Salinger,M., Stanley, E., and Stanley, M. (1997). Scaling behavior in Economics: I. Empirical results for company growth. Journal de Physique, 7:621–633.

Delli Gatti, D., Di Guilmi, C., Gaffeo, E., Giulioni, G., Gallegati, M., and Palestrini, A. (2005). A new approach to business fluctuations: heterogeneous interacting agents, scaling laws and financial fragility. Journal of Economic Behavior & Organization, 56:489–512.

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

Eeckhout, J. (2009). Gibrat’s law for (all) cities: Reply. American Economic Review, 99:1676–1683.

Feller, W. (1968). An introduction to probability theory and its applications (third edition). Volume I. John Wiley & Sons.

Gabaix, X. (1999). Zipf’s law for cities: An explanation. Quarterly Journal of Economics, 114:739–767.

Gabaix, X. (2009). Power laws in Economics and finance. Annu. Rev. Econ., 2009:255–293.

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

Giesen, K. and Suedekum, J. (2014). City age and city size. European Economic Review, 71:193–208.

Giesen, K., Zimmermann, A., and Suedekum, J. (2010). The size distribution across all cities-double Pareto lognormal strikes. Journal of Urban Economics, 68(2):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., Lanaspa, L., and Sanz-Gracia, F. (2013). Gibrat’s law for cities, growth regressions and sample size. Economics Letters, 118:367–369.

González-Val, R., Lanaspa, L., and Sanz-Gracia, F. (2014). New evidence on Gibrat’s Law for cities. Urban Studies, 51(1):93–115.

González-Val, R., Ramos, A., Sanz-Gracia, F., and Vera-Cabello, M. (2015). Size distribution for all cities: Which one is best? Papers in Regional Science, 94(1):177–197.

Hoeffding,W. (1948). A non-parametric test of independence. The Annals of Mathematical Statistics, 19(4):546–557.

Ioannides, Y. M. and Skouras, S. (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.

Kwong, H. S. and Nadarajah, S. (2019). A note on “Pareto tails and lognormal body of US cities size distribution”. Physica A: Statistical Mechanics and its Applications, 513:55–62.

Kyprianou, A. E. (2006). Introductory lectures on fluctuations of Lévy processes with applications. Springer-Verlag.

Levy,M. (2009). Gibrat’s law for (all) cities: Comment. American Economic Review, 99:1672–1675.

Luckstead, J. and Devadoss, S. (2017). Pareto tails and lognormal body of US cities size distribution. Physica A: Statistical Mechanics and its Applications, 465:573–578.

Luckstead, J., Devadoss, S., and Danforth, D. (2017). The size distribution of all Indian cities. Physica A: Statistical Mechanics and its Applications, 474:237–249.

Lukacs, E. (1970). Characteristic functions. Griffin.

Puente-Ajovín,M. and Ramos, A. (2015). On the parametric description of the French, German, Italian and Spanish city size distributions. The Annals of Regional Science, 54(2):489–509.

Ramos, A. (2017). Are the log-growth rates of city sizes distributed normally? Empirical evidence for the USA. Empirical Economics, 53:1109–1123.

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

Reed, W. J. (2003). The Pareto law of incomes–an explanation and an extension. Physica A, 319:469–486.

Reed, W. J. and Jorgensen,M. (2004). The double Pareto-lognormal distribution–a new parametric model for size distributions. Communications in Statistics-Theory and Methods, 33(8):1733–1753.

Rozenfeld, H. D., Rybski, D., Andrade, J. S., Batty, M., Stanley, H. E., and Makse, H. A. (2008). Laws of population growth. Proceedings of the National Academy of Sciences, 105(48):18702–18707.

Rozenfeld, H. D., Rybski, D., Gabaix, X., and Makse, H. A. (2011). The area and population of cities: new insights from a different perspective on cities. American Economic Review, 101:2205–2225.

Sato, K. I. (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press.

Singh, S. K. and Maddala, G. S. (1976). A function for size distribution of incomes. Econometrica, 44(5):963–970.

Stanley, M. H. R., Amaral, L. A. N., Buldyrev, S. V., Havlin, S., Leschhorn, H., Maass, P., Salinger,M. A., and Stanley, H. E. (1996). Scaling behaviour in the growth of companies. Nature, 379:804–806.

Sutton, J. (1997). Gibrat’s legacy. Journal of Economic Literature, 35:40–59.

Toda, A. A. (2012). The double power law in income distribution: Explanations and evidence. Journal of Economic Behavior & Organization, 84:364–381.