The future of urban hierearchy and Zipf law: ARIMA and BATS forecasting

Zipf’s Law is recognized as a power law which is used to identify the extent and the evolution of the urban hierarchies. The existing studies have mostly adopted a retrospective view by analysing the past patterns. However, we would like to shed a light onto the future trajectories. Therefore, the aim of this study is to investigate the future of Urban Hierarchies and Zipf’s Law for the U.S. Metropolitan Statistical Areas (MSA) and the period 1969–2070. Having applied, two forecasting methods; i.“ARIMA (Autoregressive Integrated Moving Average)”, ii. “BATS (Exponential smoothing state space model Box–Cox transformation, ARMA errors, Trend and Seasonal components)” and the estimation of rank-size rule, we obtained crucial conclusions (Box and Jenkins in: Time series analysis: forecasting and control, Holden-Day, San Francisco, 1970; Box et al. in: Time series analysis: forecasting and control, Wiley, New Jersey, 2016; Kinney in Acc Rev 53:48–60, 1978; Hyndman et al. in R package version 8.24.0, https://cran.r-project.org/web/packages/forecast/forecast.pdf , 2025; De Livera in: Automatic forecasting with a modified exponential smoothing state space framework, Department of Econometrics & Business Statistics, Monash University (Working Papers 10/10). https://www.monash.edu/business/econometrics-and-business%20statistics/research/publications/ebs/wp10-10.pdf , 2010; De Livera et al. in: Forecasting time series with complex seasonal patterns using exponential smoothing. (Working paper 15/09), Department of Econometrics & Business Statistics, Monash University. https://robjhyndman.com/papers/ComplexSeasonality.pdf , 2010; De Livera et al. in J Am Stat Assoc 106:1513–1527, 2011). We provide evidence that the Zipf’s Law is observed not to hold over the last century and, if existing conditions hold, it is not expected to be valid in the next 50 years. Pareto exponent is found significantly below the Pareto level, historically, currently and prospectively.