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ABSTRACT
Modelling population density over time: how spatial distance matters. Regional Studies. This study provides an empirical application of the Bayesian approach for modelling the evolution of population density distribution across time. It focuses on the case of Massachusetts by tracking changes in the importance of spatial distance from Boston concerning citizens’ choices of residence according to data for 1880–90 and 1930–2010. By adopting a Bayesian strategy, results show that Boston reinforced its attractiveness until the 1960s, when the city's accessibility no longer represented the unique determinant of population density distribution. Referring to selected historical evidence, a few possible interpretations are presented to endorse these results.
摘要
模式化随着时间推移的人口密度：空间距离如何具有影响. 区域研究。本研究提供以贝叶斯方法模式化随着时间推移的人口密度分佈之经验应用。本研究聚焦马萨诸塞州的案例，根据 1880 年至 1990 年、以及 1930 年至 2010 年间的数据，追踪市民选择的居住地到波市顿的空间距离的重要性。本研究透过採用贝叶斯策略，结果显示波士顿直至 1960 年代之前，强化了自身吸引力，而该城市的可及性，在当时便不再呈现作为人口密度分佈的特殊决定因素。本文透过指涉选定的历史证据，呈现若干可能的诠释来支持这些研究结果。
RÉSUMÉ
La modélisation de la densité de la population au fil du temps: l'importance de la distance spatiale. Regional Studies. Cette étude fournit une application empirique de l'approche bayésienne d'aborder la modélisation de l’évolution de la distribution de la densité de la population au fil du temps. Puisant dans des données pour les deux périodes allant de 1880 à 1890 et de 1930 à 2010, on met au point l’étude de cas de Massachusetts, suivant les changements de l'importance de la distance spatiale de Boston pour ce qui est du choix de lieu de résidence des habitants. En adoptant une stratégie du type bayésienne, les résultats montrent que Boston a consolidé son attractivité jusqu'aux années 1960, le moment où l'accessibilité de la ville n'a plus représenté le seul déterminant de la distribution de la densité de la population. Se référant à des preuves historiques sélectionnées, on présente quelques interprétations éventuelles afin de confirmer ces résultats.
ZUSAMMENFASSUNG
Modellierung der Bevölkerungsdichte im Laufe der Zeit: die Wichtigkeit der räumlichen Entfernung. Regional Studies. Diese Studie enthält eine empirische Anwendung des bayesschen Ansatzes zur Modellierung der Entwicklung der Verteilung der Bevölkerungsdichte im Laufe der Zeit. Im Mittelpunkt steht der Fall von Massachusetts, wobei die Veränderungen hinsichtlich der Wichtigkeit der räumlichen Entfernung von Boston für die Wohnortwahl der Bürger anhand von Daten für die Zeiträume von 1880 bis 1890 sowie von 1930 bis 2010 nachverfolgt werden. Bei Wahl einer bayesschen Strategie verdeutlichen die Ergebnisse, dass Boston seine Anziehungskraft bis in die sechziger Jahre des 20. Jahrhunderts verstärkte; anschließend war die Erreichbarkeit der Stadt nicht länger der einzige Determinant für die Verteilung der Bevölkerungsdichte. Unter Bezug auf ausgewählte historische Belege werden einige mögliche Interpretationen zur Bestätigung dieser Ergebnisse vorgestellt.
RESUMEN
Modelar la densidad de población con el paso del tiempo: la importancia de la distancia espacial. Regional Studies. Este estudio contiene una aplicación empírica del enfoque bayesiano para modelar la evolución de la distribución de la densidad de población con el paso del tiempo. Nos centramos en el caso de Massachusetts enfocandonos en las variaciones de la importancia de la distancia espacial de Boston en relación a las decisiones de localización de residencia de los ciutadanos en base a los datos de 1880 y 1890 así como de 1930 a 2010. Al adoptar una estrategia bayesiana, los resultados indican que Boston ha reforzado su capacidad de atracción hasta la década de los sesenta, cuando el nivel de acceso a la ciudad ya no representaba el determinante único de la distribución de la densidad de población. Con referencia a la evidencia histórica seleccionada, presentamos algunas posibles interpretaciones para confirmar estos resultados.
ACKNOWLEDGEMENTS
The authors are grateful to three anonymous referees, to D. Cuberes, J. Deveraux, M. Felix and A. Saiz, to the participants at the North American Regional Science Council (NARSC) conference (Atlanta, Georgia, 2013), and to those at the seminar at Universidad Autonoma de Madrid for useful comments and suggestions. They also thank E. Cittadini and J. Gaviglio for technical assistance with the data treatment. All remaining errors are the authors' own responsibility.
DISCLOSURE STATEMENT
No potential conflict of interest was reported by the authors.
SUPPLEMENTAL DATA
Supplemental data for this article can be accessed at http://10.1080/00343404.2015.1110237
ORCiDs
Ilenia Epifani http://orcid.org/0000-0001-9005-1378
Rosella Nicolini http://orcid.org/0000-0002-3331-8926
Notes
1. Boston could be also a pole of attraction for other neighbouring areas close to Massachusetts's borders. However, this study's analysis privileges the historical dimension and avoids any change at the institutional level (e.g., governance issues or differences in civil legislation across US states) that might distort the identification strategy. Focusing only on Massachusetts, it is controlled for, while the Bayesian approach also deliver insights that can be of interest for Massachusetts's neighbouring areas. This issue is discussed in the fourth section.
2. As pointed out by a referee, more flexibility may be obtained in a non-parametric way by, for instance, a mixture of distributions. Nevertheless, this kind of specification is analytically much more demanding.
3. This study is privileged to focus on this measure of distance rather than on other measures such as travel time, for instance. The choice is driven by two orders of reasons. First, the authors do not dispose of available date data about the travel time for so long a period of time. Secondly, year-by-year estimations are being performed. In order to exploit the variability of the travel time (and, hence, the information delivered by this variable), it is necessary to deal with panel-style data.
4. Glaeser and Ward (Citation2009) argue that water is an important amenity for creating recreational spaces and important laws have been passed to protect waterways and wetlands. For more information about water areas in Massachusetts, see Simcox (Citation1992).
5. The gamma-probability density Γ(a, b), with shape a and rate b, has kernel xa−1e−bx on the half-line (0, ∞), and it is equal to zero otherwise. The expected value and variance of a random variable X Γ(a, b) are given by E(X) = a/b and Var(X) = a/b2 (i.e., the ratio between the standard deviation and the mean is Cv(X) = 1/a).
6. For instance, see http://www.bayesian-inference.com/samplesize/. Certainly, a Bayesian hierarchical model overcomes the difficulty of dealing with the problem of over-parameterization, but it requires caution in the definition of the priors in order to make them truly representative and also to preserve the sensitivity of the model to capture changes due to the variations of the parameters over time. Furthermore, the authors are conscious that a small sample size makes difficult a singling out of a possible discrepancy between the prior and sample information because the empirical information is diluted on too large a parametric space.
7. In Bayesian statistics, a γ 100% credible interval for an unknown quantity η is given by:where q(1–γ)/2 and q(1+γ)/2 are posterior quantiles of η.
8. The historical strategy includes only marginal information derived from the posterior distribution given the data of the previous decade. Including information from the past about the dependence structure in the prior distribution when non-normally distributed parameters are involved is a technical issue that is difficult to resolve. The authors leave it for further investigation.
9. The DIC is a generalization of Akaike's information criterion (AIC). It is given by the deviance (i.e., minus twice the likelihood) calculated in the posterior means of all parameters plus twice the effective numbers of parameters pD. Models with a small DIC are preferred over those with a large DIC.
10. Town j in county i has been classified as an outlier at time t if its real population density does not fall within the 95% posterior credible interval of the population density
given Data = Y; Dist; Mix; Z (see Appendix D in the supplemental data online for more details on the Bayesian outliers and the 95% posterior credible intervals of the population densities).
11. Estimates for historical priors can be found in Table B3 in Appendix B in the supplemental data online. Appendix B also has additional statistics.
12. From a statistical viewpoint, this circumstance stems from the low variance of all historical priors, given their dependence on the efficient and concentrated estimates of the previous period.
13. Given the computational technique, in the case of historical priors, the estimates for α(t) should not be considered fully reliable. The algorithm performs poorly in updating the posterior values of α(t), probably since these posteriors becomes increasingly more concentrated around a few values from 1940 onwards and then the algorithm may get struck in the initial values.
14. For a complete discussion about the credible intervals for the estimations of the population density, see Appendix D in the supplemental data online.
15. More detailed data can be found at http://www.forecast-chart.com/.
