Extrapolation Methods

  • Stanley K. Smith
  • Jeff Tayman
  • David A. Swanson
Part of the The Springer Series on Demographic Methods and Population Analysis book series (PSDE, volume 37)


Trend extrapolation methods are those in which future values of a variable are determined solely by its historical values. These methods have often been used for constructing population projections and can be organized into three categories. Simple extrapolation methods are those that have simple mathematical structures and require data for only two points in time. Complex extrapolation methods require data from a number of points in time, have more complicated mathematical structures, and require statistical estimation of the model’s parameters. Ratio extrapolation methods express the population of a smaller unit as a proportion of the population of a larger unit. In this chapter, we describe and illustrate a number of trend extrapolation methods and evaluate their strengths and weaknesses when used for state and local population projections.


Base Period Extrapolation Method Population Projection ARIMA Model Franklin County 
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  1. Alho, J. M. (1990). Stochastic methods in population forecasting. International Journal of Forecasting, 6, 521–530.CrossRefGoogle Scholar
  2. Alho, J. M., & Spencer, B. D. (1997). The practical specification of the expected error of population forecasts. Journal of Official Statistics, 13, 203–225.Google Scholar
  3. Alho, J. M., & Spencer, B. D. (2005). Statistical demography and forecasting. New York: Springer.Google Scholar
  4. Alinghaus, S. L. (1994). Practical handbook of curve fitting. New York: CRC Press.Google Scholar
  5. Armstrong, J. S. (2001). Extrapolation of time series and cross-sectional data. In J. S. Armstrong (Ed.), Principles of forecasting: A handbook for researchers and practitioners (pp. 217–244). Norwell: Kluwer Academic.CrossRefGoogle Scholar
  6. Bongaarts, J., & Bulatao, R. A. (Eds.). (2000). Beyond six billion: Forecasting the world’s population (pp. 188–217). Washington, DC: National Research Council.Google Scholar
  7. Box, G. E., & Jenkins, G. M. (1976). Time series analysis: Forecasting and control. San Francisco: Holden-Day.Google Scholar
  8. Brass, W. (1974). Perspectives in population prediction: Illustrated by the statistics in England and Wales. Journal of the Royal Statistical Society, A, 137, 532–570.CrossRefGoogle Scholar
  9. Brockwell, P. J., & Davis, R. A. (2002). Introduction to time series and forecasting (2nd ed.). New York: Springer.CrossRefGoogle Scholar
  10. Campbell, P. R. (1996). Population projections for states by age, sex, race, and Hispanic origin: 1995 to 2050. PPL 47. Washington, DC: U.S. Census Bureau.Google Scholar
  11. Carter, L. R., & Lee, R. D. (1986). Joint forecasts of U.S. marital fertility, nuptiality, births, and marriages using time series models. Journal of the American Statistical Association, 81, 902–911.CrossRefGoogle Scholar
  12. Chatfield, C. (2000). Time series forecasting. Boca Raton: Chapman & Hall/CRC.CrossRefGoogle Scholar
  13. Cohen, J. E. (1986). Population forecasts and the confidence intervals for Sweden: A comparison of model-based and empirical approaches. Demography, 23, 105–126.CrossRefGoogle Scholar
  14. Davis, C. H. (1995). Demographic projection techniques for regions and smaller areas. Vancouver: UBC Press.Google Scholar
  15. de Beer, J. (1993). Forecast intervals of net migration: The case of the Netherlands. Journal of Forecasting, 12, 585–599.CrossRefGoogle Scholar
  16. Dickey, D. A., Bell, W. R., & Miller, R. B. (1986). Unit roots in time series models: Tests and implications. American Statistician, 74, 427–431.Google Scholar
  17. Dorn, H. F. (1950). Pitfalls in population forecasts and projections. Journal of the American Statistical Association, 43, 311–334.CrossRefGoogle Scholar
  18. Draper, N. R., & Smith, H. (1981). Applied regression analysis (2nd ed.). New York: Wiley.Google Scholar
  19. Duan, N. (1983). Smearing estimate: A nonparametric retransformation method. Journal of the American Statistical Association, 78, 605–610.CrossRefGoogle Scholar
  20. Elliot, G., Rothenberg, T. J., & Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64, 813–836.CrossRefGoogle Scholar
  21. Gabbour, I. (1993). SPOP: Small area population projection. In R. E. Klosterman, R. K. Brail, & G. B. Earl (Eds.), Spreadsheet models for urban and regional analysis (pp. 69–84). New Brunswick: Rutgers University, Center for Urban Policy Research.Google Scholar
  22. Granger, C. W. (1989). Forecasting in business and economics (2nd ed.). San Diego: Academic.Google Scholar
  23. Granger, C. W., & Newbold, P. (1986). Forecasting economic time series (2nd ed.). San Diego: Academic.Google Scholar
  24. Irwin, R. (1977). Guide for local area population projections. Technical Paper # 39. Washington, DC: U.S. Census Bureau.Google Scholar
  25. Isserman, A. M. (1977). The accuracy of population projections for subcounty areas. Journal of the American Institute of Planners, 43, 247–259.CrossRefGoogle Scholar
  26. Jenkins, G. M. (1979). Practical experiences with modeling and forecasting time series. Jersey: Gwilym Jenkins & Partners (Overseas) Ltd.Google Scholar
  27. Keilman, N., Pham, D. Q., & Hetland, A. (2002). Why population forecasts should be probabilistic-illustrated by the case of Norway. Demographic Research, 6, 409–454.CrossRefGoogle Scholar
  28. Keyfitz, N. (1968). An introduction to the mathematics of population. Reading: Addison Wesley.Google Scholar
  29. Land, K. C. (1986). Methods for national population forecasts: A review. Journal of the American Statistical Association, 81, 888–901.CrossRefGoogle Scholar
  30. Leach, D. (1981). Re-evaluation of the logistic curve for human populations. Journal of the Royal Statistical Society A, 144, 94–103.CrossRefGoogle Scholar
  31. Lee, R. D. (1974). Forecasting births in post-transition populations: Stochastic renewal with serially correlated fertility. Journal of the American Statistical Association, 69, 607–617.CrossRefGoogle Scholar
  32. Ljung, G. M., & Box, G. E. (1978). On a measure of a lack of fit in time series models. Biometrika, 65, 297–303.CrossRefGoogle Scholar
  33. Mahmoud, E. (1984). Accuracy in forecasting: A survey. Journal of Forecasting, 3, 139–159.CrossRefGoogle Scholar
  34. Makridakis, S. G., Wheelwright, S. C., & Hyndman, R. J. (1989). Forecasting: Methods and applications (3rd ed.). New York: Wiley.Google Scholar
  35. Manning, W. G. (1998). The logged dependent variable, heteroscedasticity, and the retransformation problem. Journal of Health Economics, 17, 283–295.CrossRefGoogle Scholar
  36. Marchetti, C., Meyer, P. S., & Ausubel, J. H. (1996). Human population dynamics revisited with the logistic model: How much can be modeled and predicted? Technological Forecasting and Social Change, 52, 1–30.CrossRefGoogle Scholar
  37. McCleary, R., & Hay, R. A. (1980). Applied time series analysis for the social sciences. Beverly Hills: Sage.Google Scholar
  38. McDonald, J. (1979). A time series approach to forecasting Australian total live-births. Demography, 16, 575–601.CrossRefGoogle Scholar
  39. McNown, R., & Rogers, A. (1989). Forecasting mortality: A parameterized time series approach. Demography, 26, 645–660.CrossRefGoogle Scholar
  40. Meyler, A., Kenny, G., & Quinn, T. (1998). Forecasting Irish inflation using ARIMA models. Technical Paper Series 3/RT/98. Dublin: Central Bank and Financial Services Authority of Ireland.Google Scholar
  41. Montgomery, D. C., Jennings, C. J., & Kulahci, M. (2008). Introduction to time series analysis and forecasting. Hoboken: Wiley.Google Scholar
  42. Nelson, C. R. (1973). Applied time series analysis for managerial forecasting. San Francisco: Holden-Day.Google Scholar
  43. Pearl, R., & Reed, L. J. (1920). On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proceedings of the National Academy of Science, 6, 275–287.CrossRefGoogle Scholar
  44. Pflaumer, P. (1992). Forecasting U.S. population totals with the Box-Jenkins approach. International Journal of Forecasting, 8, 329–338.CrossRefGoogle Scholar
  45. Phillips, P. C., & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75, 335–346.CrossRefGoogle Scholar
  46. Pielou, E. C. (1969). An introduction to mathematical ecology. New York: Wiley.Google Scholar
  47. Pittenger, D. B. (1976). Projecting state and local populations. Cambridge, MA: Ballinger Publishing Company.Google Scholar
  48. Pritchett, H. S. (1891). A formula for predicting the population of the United States. Publications of the American Statistical Association, 14, 278–296.CrossRefGoogle Scholar
  49. Rayer, S. (2007). Population forecast error: Does the choice of summary measure of error matter? Population Research and Policy Review, 26, 163–184.CrossRefGoogle Scholar
  50. Romaniuc, A. (1990). Population projection as prediction, simulation and prospective analysis. Population Bulletin of the United Nations, 29, 16–31.Google Scholar
  51. Saboia, J. L. (1974). Modeling and forecasting populations by time series: The Swedish case. Demography, 11, 483–492.CrossRefGoogle Scholar
  52. San Diego Association of Governments. (2011). 2050 regional growth forecast, from .
  53. Sanderson, W. C. (1995). Probability, complexity, and catastrophe in a collapsible model of population development and environmental interactions. Mathematical Population Studies, 5, 259–279.CrossRefGoogle Scholar
  54. Schnaars, S. P. (1986). A comparison of extrapolation models on yearly sales forecasts. International Journal of Forecasting, 2, 71–85.CrossRefGoogle Scholar
  55. Shryock, H. J., & Siegel, J. S. (1973). The methods and materials of demography. Washington, DC: U.S. Government Printing Office.Google Scholar
  56. Sieber, G. A., & Wild, C. J. (1989). Nonlinear regression. New York: Wiley.CrossRefGoogle Scholar
  57. Smith, S. K. & Rayer, S. (2012). Projections of Florida population by county, 2011–2040. Florida Population Studies, Bulletin 162. Gainesville, FL: Bureau of Economic and Business Research, University of Florida.Google Scholar
  58. State of Washington. (2011). Forecast of the state population: November 2011, from .
  59. Stock, J. H., & Watson, M. W. (2003). Introduction to econometrics. Boston: Addison Wesley.Google Scholar
  60. Tayman, J. (2011). Assessing uncertainty in small area forecasts: State of the practice and implementation strategy. Population Research and Policy Review, 30, 781–800.CrossRefGoogle Scholar
  61. Tayman, J., Smith, S. K., & Lin, J. (2007). Precision, bias, and uncertainty for state population forecasts: An exploratory analysis of time series models. Population Research and Policy Review, 26, 347–369.CrossRefGoogle Scholar
  62. Treyz, G. I. (1995). Regional economic modeling: A systematic approach to economic forecasting and policy analysis. Boston: Kluwer Academic.Google Scholar
  63. Voss, P. R., & Kale, B. D. (1985). Refinements to small area projection models: Results of a test based on 128 Wisconsin communities. Paper presented at the Population Association of America, Boston.Google Scholar
  64. Voss, P. R., Palit, C. D., Kale, B. D., & Krebs, H. C. (1981). Forecasting state populations using ARIMA time series techniques. Madison: Applied Population Laboratory, University of Wisconsin.Google Scholar
  65. White, H. R. (1954). Empirical study of the accuracy of selected methods of projecting state populations. Journal of the American Statistical Association, 49, 480–498.Google Scholar
  66. Yaffee, R. A., & McGee, M. (2000). An introduction to time series analysis and forecasting: With applications of SAS and SPSS. San Diego: Academic.Google Scholar
  67. Yule, G. U. (1925). The growth of population and the factors which control it. Journal of the Royal Statistical Society, 38, 1–58.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Stanley K. Smith
    • 1
  • Jeff Tayman
    • 2
  • David A. Swanson
    • 3
  1. 1.Bureau of Economic and Business ResearchUniversity of FloridaGainesvilleUSA
  2. 2.Economics DepartmentUniversity of California-San DiegoSan DiegoUSA
  3. 3.Department of SociologyUniversity of California RiversideRiversideUSA

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