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Demography

, Volume 50, Issue 1, pp 237–260 | Cite as

Probabilistic Forecasting Using Stochastic Diffusion Models, With Applications to Cohort Processes of Marriage and Fertility

  • Mikko MyrskyläEmail author
  • Joshua R. Goldstein
Article

Abstract

In this article, we show how stochastic diffusion models can be used to forecast demographic cohort processes using the Hernes, Gompertz, and logistic models. Such models have been used deterministically in the past, but both behavioral theory and forecast utility are improved by introducing randomness and uncertainty into the standard differential equations governing population processes. Our approach is to add time-series stochasticity to linearized versions of each process. We derive both Monte Carlo and analytic methods for estimating forecast uncertainty. We apply our methods to several examples of marriage and fertility, extending them to simultaneous forecasting of multiple cohorts and to processes restricted by factors such as declining fecundity.

Keywords

Probabilistic forecasting Diffusion models Cohort models Fertility Marriage 

Notes

Acknowledgements

We wish to credit Ron Lee, who first suggested to JRG that the Hernes diffusion models be fit using a time-series, stochastic diffusion approach.

References

  1. Abbot, A. (1988). Transcending general linear reality. Sociological Theory, 6, 169–186.CrossRefGoogle Scholar
  2. Alho, J. M. (1990). Stochastic methods in population forecasting. International Journal of Forecasting, 6, 521–530.CrossRefGoogle Scholar
  3. Alho, J., Alders, M., Cruijsen, H., Keilman, N., Nikander, T., & Pham, D. Q. (2006). New forecast: Population decline postponed in Europe. Statistical Journal of the United Nations Economic Commission for Europe, 23, 1–10.Google Scholar
  4. Anselin, L. (1988). Spatial econometrics: Methods and models. Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  5. Bernardi, L. (2003). Channels of social influence on reproduction. Population Research and Policy Review, 22, 527–555.CrossRefGoogle Scholar
  6. Billari, F. C., & Prskawetz, A. (2003). Agent-based computational demography. Heidelberg, Germany: Physica Verlag.CrossRefGoogle Scholar
  7. Billari, F. C., & Toulemon, L. (2006). Cohort childlessness forecasts and analysis using the Hernes model. Liverpool, UK: Paper presented at the European Population Conference.Google Scholar
  8. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.CrossRefGoogle Scholar
  9. Christakis, N. A., & Fowler, J. H. (2008). The collective dynamics of smoking in a large social network. The New England Journal of Medicine, 358, 2249–2258.CrossRefGoogle Scholar
  10. Coale, A. J., & McNeil, D. R. (1972). The distribution by age of the frequency of first marriage in a female cohort. Journal of the American Statistical Association, 67, 743–749.CrossRefGoogle Scholar
  11. Cowan, R., & Jonard, N. (2004). Network structure and the diffusion of knowledge. Journal of Economic Dynamics and Control, 28, 1557–1575.CrossRefGoogle Scholar
  12. Diekmann, A. (1989). Diffusion and survival models for the process of entry into marriage. Journal of Mathematical Sociology, 14, 31–44.CrossRefGoogle Scholar
  13. Doreian, P. (1980). Linear models with spatially distributed data: Spatial disturbances or spatial effects? Sociological Methods & Research, 9, 29–60.CrossRefGoogle Scholar
  14. Frances, P. H. (1994). A method to select between Gompertz and logistic trend curves. Technological Forecasting and Social Change, 46, 45–49.CrossRefGoogle Scholar
  15. Goldstein, J. R. (2010). A behavioral Gompertz model for cohort fertility schedules in low and moderate fertility populations (MPIDR Working Paper WP 2010–021). Rostock, Germany: Max Planck Institute for Demographic Research.Google Scholar
  16. Goldstein, J. R., & Kenney, C. T. (2001). Marriage delayed or marriage forgone? New cohort forecasts of first marriage for U.S. women. American Sociological Review, 66, 506–519.CrossRefGoogle Scholar
  17. Griffiths, D. V., & Smith, I. M. (1991). Numerical methods for engineers: A programming approach. Boca Raton, FL: CRC Press.Google Scholar
  18. Gruber, H., & Verboven, F. (2001). The diffusion of mobile telecommunications services in the European Union. European Economic Review, 45, 577–588.CrossRefGoogle Scholar
  19. Hammel, E. A., Mason, C., & Wachter, K. W. (1990). SOCSIM II, A sociodemographic microsimulation program, Rev. 1.0, operating manual (Graduate Group in Demography Working Paper No. 29). Berkeley: University of California, Institute of International Studies, Program in Population Research.Google Scholar
  20. Harvery, D. I., & Leybourne, S. J. (2007). Testing for time series linearity. The Econometrics Journal, 10, 149–165.CrossRefGoogle Scholar
  21. Harvey, A. C. (1984). Time series forecasting based on the logistic curve. Journal of the Operational Research Society, 35, 641–646.Google Scholar
  22. Hernes, G. (1972). The process of entry into first marriage. American Sociological Review, 37, 173–182.CrossRefGoogle Scholar
  23. Hinich, M. J. (1982). Testing for Gaussianity and linearity of a stationary time series. Journal of Time Series Analysis, 3, 169–176.CrossRefGoogle Scholar
  24. Hoem, J. M., Madsen, D., Lovgreen Nielsen, J., Ohlsen, E.-M., Oluf Hansen, H., & Rennermalm, B. (1981). Experiments in modelling recent Danish fertility curves. Demography, 18, 231–244.CrossRefGoogle Scholar
  25. Human Fertility Database. (2011). Max Planck Institute for Demographic Research (Germany) and Vienna Institute of Demography (Austria). Retreived from http://www.humanfertility.org
  26. Ike, S. (2002). A non-stationary stochastic process model of completed marital fertility in Japan. Journal of Mathematical Sociology, 26, 35–55.CrossRefGoogle Scholar
  27. Keilman, N., & Pham, D. Q. (2004). Time series based errors and empirical errors in fertility forecasts in the Nordic countries. International Statistical Review, 72, 5–18.CrossRefGoogle Scholar
  28. Kohler, H.-P. (2001). Fertility and social interaction: An economic perspective. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
  29. Land, K. C., Deane, G., & Blau, J. R. (1991). Religious pluralism and church membership: A spatial diffusion model. American Sociological Review, 56, 237–249.CrossRefGoogle Scholar
  30. Lee, R. D. (1993). Modeling and forecasting the time series of U.S. fertility: Age patterns, range, and ultimate level. International Journal of Forecasting, 9, 187–202.CrossRefGoogle Scholar
  31. Lee, R. D. (1998). Probabilistic approaches to population forecasting. Population and Development Review, 24, 156–190.CrossRefGoogle Scholar
  32. Lee, R. D., & Tuljapurkar, S. (1994). Stochastic population forecasts for the United States: Beyond high, medium, and low. Journal of the American Statistical Association, 89, 1175–1189.CrossRefGoogle Scholar
  33. Li, N., & Wu, Z. (2008, April). Modeling and forecasting first marriage: A latent function approach. Paper presented at the annual meeting of the Population Association of America, New Orleans, LA.Google Scholar
  34. Lutz, W., & Goldstein, J. R. (2004). Introduction: How to deal with uncertainty in population forecasting? International Statistical Review, 72, 1–4.CrossRefGoogle Scholar
  35. Lutz, W., Sanderson, W., & Scherbov, S. (2001). The end of world population growth. Nature, 412, 543–545.CrossRefGoogle Scholar
  36. Mansfield, E. (1961). Technical change and the rate of imitation. Econometrica, 29, 741–766.CrossRefGoogle Scholar
  37. Mar-Molinero, C. (1980). Tractors in Spain: A logistic analysis. Journal of the Operational Research Society, 31, 141–152.Google Scholar
  38. Marsden, P. V., & Friedkin, N. E. (1993). Network studies of social influence. Sociological Methods & Research, 22, 127–151.CrossRefGoogle Scholar
  39. Martin, S. P. (2004). Reassessing delayed and forgone marriage in the United States (Working paper). New York: Russell Sage.Google Scholar
  40. Meade, N., & Islam, T. (2006). Modelling and forecasting the diffusion of innovation—A 25-year review. International Journal of Forecasting, 22, 519–545.CrossRefGoogle Scholar
  41. Øksendael, B. (2003). Stochastic differential equations (6th ed.). Berlin, Germany: Springer.CrossRefGoogle Scholar
  42. Pearl, R., & Reed, L. J. (1920). On the rate of growth of the population of the United States and its mathematical representation. Proceedings of the National Academy of Sciences, 6, 275–288.CrossRefGoogle Scholar
  43. Pollard, J., & Valkovics, E. (1992). The Gompertz distribution and its applications. Genus, 48(3–4), 15–29.Google Scholar
  44. Preston, S. H., Heuveline, P., & Guillot, M. (2001). Demography: Measuring and modeling population processes. Oxford, UK: Blackwell Publishers.Google Scholar
  45. Raffalovich, L. E. (1994). Detrending time series: A cautionary note. Sociological Methods and Research, 22, 492–519.CrossRefGoogle Scholar
  46. Tolnay, S. E., Deane, G., & Beck, E. M. (1996). Vicarious violence: Spatial effects on southern lynchings, 1890–1919. The American Journal of Sociology, 102, 788–815.CrossRefGoogle Scholar
  47. Valente, T. W. (1995). Network models of the diffusion of innovations. Cresskill, NJ: Hampton Press.Google Scholar
  48. Vaupel, J. W. (2010). Biodemography of human ageing. Nature, 464, 536–542.CrossRefGoogle Scholar
  49. Wachter, K. W. (1987). Microsimulation of household cycles. In J. Bongaarts, T. Burch, & K. W. Wachter (Eds.), Family demography: Methods and their application (pp. 215–227). Oxford, UK: Clarendon.Google Scholar
  50. Winsor, C. P. (1932). The Gompertz curve as a growth curve. Proceedings of the National Academy of Sciences of the United States of America, 18(1), 1–8.CrossRefGoogle Scholar
  51. Wu, L. L. (1990). Simple graphical goodness-of-fit tests for hazard rate models. Madison: University of Wisconsin Press.Google Scholar

Copyright information

© Population Association of America 2012

Authors and Affiliations

  1. 1.Research Group Lifecourse Dynamics and Demographic ChangeMax Planck Institute for Demographic ResearchRostockGermany
  2. 2.Max Planck Institute for Demographic ResearchRostockGermany

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