Abstract
Markov chains have become a mainstay in the literature on regional income distribution dynamics and convergence. Despite its growing popularity, the Markov framework has some restrictive characteristics associated with the underlying discretization income distributions. This paper introduces several new approaches designed to mitigate some of the issues arising from discretization. Based on the examination of rank distributions, two new Markov-based chains are developed. The first explores the movement of individual economies through the income rank distribution over time. The second provides insight on the movements of ranks over geographical space and time. These also serve as the foundation for two new tests of spatial dynamics or the extent to which movements in the rank distribution are spatially clustered. An illustration of these new methods is included using income data for the lower 48 US states for the years 1929–2009.
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Notes
The regularity conditions for the existence of the steady state distribution require the chain is irreducible (i.e., all states communicate), although the interpretation of the limiting probabilities is slightly different depending on whether the chain is aperiodic or periodic (Ibe 2009, p. 63).
For an overview of Markov processes and measures see Ibe (2009).
All computations relied on the library PySAL (Rey and Anselin 2010).
To avoid confusion with the states of the Markov chain, the term region is used to represent a US-state in what follows.
Alabama, contiguous to Mississippi, falls in the fifth quintile in Fig. (4a) because of the large interval defining that quintile—a span of 142 years. The FMPT from Mississippi to Alabama is 114.6 years, which just exceeds the lower bound of the fifth quintile.
In the current implementation, the permutation of the set of locations is held constant over all time periods. Additionally, the local version of the spatial rank tests is subject to difficulties associated with multiple comparisons and lack of independence which plague all local statistics (Ord and Getis 2001). Future work is planned to examine the impact of alternative permutation schemes and related distributional issues.
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Acknowledgments
This research was funded in part by NSF Award OCI-1047916, SI2-SSI: Cyber GIS Software INtegration for Sustained Geospatial Innovation. I thank the anonymous referees and the editor for their constructive comments.
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Rey, S. Rank-based Markov chains for regional income distribution dynamics. J Geogr Syst 16, 115–137 (2014). https://doi.org/10.1007/s10109-013-0189-0
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DOI: https://doi.org/10.1007/s10109-013-0189-0