Extrapolation for Time-Series and Cross-Sectional Data

  • J. Scott Armstrong

Abstract

Extrapolation methods are reliable, objective, inexpensive, quick, and easily automated. As a result, they are widely used, especially for inventory and production forecasts, for operational planning for up to two years ahead, and for long-term forecasts in some situations, such as population forecasting. This paper provides principles for selecting and preparing data, making seasonal adjustments, extrapolating, assessing uncertainty, and identifying when to use extrapolation. The principles are based on received wisdom (i.e., experts’ commonly held opinions) and on empirical studies. Some of the more important principles are:
  • In selecting and preparing data, use all relevant data and adjust the data for important events that occurred in the past.

  • Make seasonal adjustments only when seasonal effects are expected and only if there is good evidence by which to measure them.

  • When extrapolating, use simple functional forms. Weight the most recent data heavily if there are small measurement errors, stable series, and short forecast horizons. Domain knowledge and forecasting expertise can help to select effective extrapolation procedures. When there is uncertainty, be conservative in forecasting trends. Update extrapolation models as new data are received.

  • To assess uncertainty, make empirical estimates to establish prediction intervals.

  • Use pure extrapolation when many forecasts are required, little is known about the situation, the situation is stable, and expert forecasts might be biased.

Keywords

Acceleration adaptive parameters analogous data asymmetric errors base rate Box-Jenkins Combining Conservatism contrary series cycles damping decomposition discontinuities domain knowledge experimentation exponential smoothing functional form judgmental adjustments M-Competition measurement error moving averages nowcasting prediction intervals projections random walk seasonality simplicity tracking signals trends uncertainty updating 

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References

  1. Allen, P. G. R. Fildes (2001), “Econometric forecasting,” in J. S. Armstrong (ed.), Principles of Forecasting. Norwell, MA: Kluwer Academic Publishers.Google Scholar
  2. Alonso, W. (1968), “Predicting with imperfect data,” Journal of the American Institute of Planners, 34, 248–255.CrossRefGoogle Scholar
  3. Armstrong, J. S. (1970), “An application of econometric models to international marketing,” Journal of Marketing Research, 7, 190–198. Full-text at hops.wharton.upenn. edu/forecast.Google Scholar
  4. Armstrong, J. S. (1978, 1985, 2nd ed.), Long-Range Forecasting: From Crystal Ball to Computer. New York: John Wiley. 1985 edition available in full-text at http://hops.wharton.upenn.edu/forecastGoogle Scholar
  5. Armstrong, J. S. F. Collopy (2000). “Speculations about seasonal factors,” in full text at hops.wharton.upenn.edu/forecast/seasonalfactors.pdf.Google Scholar
  6. Armstrong, J. S., M. Adya F. Collopy (2001), “Rule-based forecasting: Using judgment in time-series extrapolation,” in J. S. Armstrong (ed.), Principles of Forecasting. Norwell, MA: Kluwer Academic Publishers.Google Scholar
  7. Armstrong, J. S. F. Collopy (2001), “Identification of asymmetric prediction intervals through causal forces,” Journal of Forecasting (forthcoming).Google Scholar
  8. Armstrong, J. S. F. Collopy (1998), “Integration of statistical methods and judgment for time-series forecasting: Principles from empirical research,” in G. Wright and P.Google Scholar
  9. Goodwin, Forecasting with Judgment. Chichester: John Wiley, pp. 269–293. Full-text at hops.wharton.upenn.edu/forecast.Google Scholar
  10. Armstrong, J. S. F. Collopy (1992), “Error measures for generalizing about forecasting methods: Empirical comparisons” (with discussion), International Journal of Forecasting, 8, 69–111. Full-text at hops.wharton.upenn.edu/forecast.Google Scholar
  11. Armstrong, J. S. E. J. Lusk (1983), “The accuracy of alternative extrapolation models: Analysis of a forecasting competition through open peer review,” Journal of Forecasting, 2, 259–311 (Commentaries by seven authors with replies by the original authors of Makridakis et al. 1982). Full-text at hops.wharton.upenn.edu/forecast.Google Scholar
  12. Box, G. E. G. Jenkins ( 1970; 3`d edition published in 1994), Time-series Analysis, Forecasting and Control. San Francisco: Holden Day.Google Scholar
  13. Brown, R. G. (1959), “Less-risk in inventory estimates,” Harvard Business Review, 37, July—August, 104–116.Google Scholar
  14. Brown, R. G. (1962), Smoothing, Forecasting and Prediction. Englewood Cliffs, N.J.: Prentice Hall.Google Scholar
  15. Burns, A. F. W. C. Mitchell (1946), Measuring Business Cycles. New York: National Bureau of Economic Research.Google Scholar
  16. Cairncross, A. (1969), “Economic forecasting,” Economic Journal, 79, 797–812.CrossRefGoogle Scholar
  17. Chatfield, C. (2001), “Prediction intervals for time-series,” in J. S. Armstrong (ed.), Principles of Forecasting. Norwell, MA: Kluwer Academic Publishers.Google Scholar
  18. Collopy, F. J. S. Armstrong (1992a), “Rule-based forecasting: Development and validation of an expert systems approach to combining time series extrapolations,” Management Science, 38, 1374–1414.CrossRefGoogle Scholar
  19. Collopy, F. J. S. Armstrong (1992b), “Expert opinions about extrapolation and the mystery of the overlooked discontinuities,” International Journal of Forecasting, 8, 575–582. Full-text at hops.wharton.upenn.edu/forecast.Google Scholar
  20. Croston, J. D. (1972), “Forecasting and stock control for intermittent demand,” Operational Research Quarterly, 23 (3), 289–303.CrossRefGoogle Scholar
  21. Dalrymple, D. J. B. E. King (1981), “Selecting parameters for short-term forecasting techniques,” Decision Sciences, 12, 661–669.CrossRefGoogle Scholar
  22. Dewey, E. R. E. F. Dakin (1947), Cycles: The Science of Prediction. New York: Holt.Google Scholar
  23. Dorn, H. F. (1950), “Pitfalls in population forecasts and projections,” Journal of the American Statistical Association, 45, 311–334.CrossRefGoogle Scholar
  24. Fildes, R., M. Hibon, S. Makridakis N. Meade (1998), “Generalizing about univariate forecasting methods: Further empirical evidence,” International Journal of Forecasting, 14, 339–358. (Commentaries follow on pages 359–366.)Google Scholar
  25. Fildes, R. S. Makridakis (1995), “The impact of empirical accuracy studies on timeseries analysis and forecasting,” International Statistical Review, 63, 289–308.CrossRefGoogle Scholar
  26. Findley, D. F., B. C. Monsell W. R. Bell (1998), “New capabilities and methods of the X-12 ARIMA seasonal adjustment program,” Journal of Business and Economic Statistics, 16, 127–152.Google Scholar
  27. Gardner, E. S. Jr. (1985), “Exponential smoothing: The state of the art, ” Journal of Forecasting, 4, 1–28. (Commentaries follow on pages 29–38.)Google Scholar
  28. Gardner, E. S. Jr. (1988), “A simple method of computing prediction intervals for time-series forecasts,” Management Science, 34, 541–546.CrossRefGoogle Scholar
  29. Gardner, E. S. Jr. D. G. Dannenbring (1980), “Forecasting with exponential smoothing: Some guidelines for model selection,” Decision Sciences, 11, 370–383.CrossRefGoogle Scholar
  30. Gardner, E. S. Jr. E. McKenzie (1985), “Forecasting trends in time-series, ” Management Science, 31, 1237–1246.CrossRefGoogle Scholar
  31. Groff, G. K. (1973), “Empirical comparison of models for short range forecasting,” Management Science, 20, 22–31.CrossRefGoogle Scholar
  32. Hajnal, J. (1955), “The prospects for population forecasts, ” Journal of the American Statistical Association, 50, 309–327.CrossRefGoogle Scholar
  33. Hill, T.P. (1998), “The first digit phenomenon,” American Scientist, 86, 358–363.Google Scholar
  34. Hogan, W. P. (1958), “Technical progress and production functions,” Review of Economics and Statistics, 40, 407–411.CrossRefGoogle Scholar
  35. Ittig, P. (1997), “A seasonal index for business,” Decision Sciences, 28, 335–355.CrossRefGoogle Scholar
  36. Kirby, R.M. (1966), “A comparison of short and medium range statistical forecasting methods,” Management Science, 13, B202 - B210.CrossRefGoogle Scholar
  37. MacGregor, D. (2001), “Decomposition for judgmental forecasting and estimation,” in J.S. Armstrong (ed.), Principles of Forecasting. Norwell, MA: Kluwer Academic Publishers.Google Scholar
  38. Makridakis, S., A. Andersen, R. Carbone, R. Fildes, M. Hibon, R. Lewandowski, J. Newton, E. Parzen R. Winkler (1982), “The accuracy of extrapolation (time-series) methods: Results of a forecasting competition,” Journal of Forecasting, 1, 111–153.CrossRefGoogle Scholar
  39. Makridakis, S., A. Andersen, R. Carbone, R. Fildes, M. Hibon, R. Lewandowski, J. Newton, E. Parzen R. Winkler (1984), The Forecasting Accuracy of Major Time-series Methods. Chichester: John Wiley.Google Scholar
  40. Makridakis, S., C. Chatfield, M. Hibon, M. Lawrence, T. Mills, K. Ord L. F. Simmons (1993), “The M2-Competition: A real-time judgmentally based forecasting study,” International Journal of Forecasting, 9, 5–22. (Commentaries by the authors follow on pages 23–29)Google Scholar
  41. Makridakis, S. M. Hibon (1979), “Accuracy of forecasting: An empirical investigation,” (with discussion), Journal of the Royal Statistical Society: Series A, 142, 97–145.Google Scholar
  42. Makridakis, S. M. Hibon (2000), “The M3-Competition: Results, conclusions and implications,” International Journal of Forecasting, 16, 451–476.CrossRefGoogle Scholar
  43. Makridakis, S., M. Hibon, E. Lusk M. Belhadjali (1987), “Confidence intervals,” International Journal of Forecasting, 3, 489–508.CrossRefGoogle Scholar
  44. Makridakis, S. R. L. Winkler (1989), “Sampling distributions of post-sample forecasting errors,” Applied Statistics, 38, 331–342.CrossRefGoogle Scholar
  45. Meade, N. T. Islam (2001), “Forecasting the diffusion of innovations: Implications for time-series extrapolation,” in J. S. Armstrong (ed.), Principles of Forecasting. Norwell, MA: Kluwer Academic Publishers.Google Scholar
  46. Meese, R. J. Geweke (1984), “A comparison of autoregressive univariate forecasting procedures for macroeconomic time-series, ”Journal of Business and Economic Statistics, 2, 191–200.Google Scholar
  47. Nelson, C.R. (1972), “The prediction performance of the FRB-MIT-Penn model of the U.S. economy,” American Economic Review, 5, 902–917.Google Scholar
  48. Nevin, J. R. (1974), “Laboratory experiments for estimating consumer demand: A validation study,” Journal of Marketing Research, 11, 261–268.Google Scholar
  49. Newbold, P. C. W. J. Granger (1974), “Experience with forecasting univariate time-series and the combination of forecasts,” Journal of the Royal Statistical Society: Series A, 137, 131–165.CrossRefGoogle Scholar
  50. Nigrini, M. (1999), “I’ve got your number,” Journal of Accountancy, May, 79–83.Google Scholar
  51. Rao, A. V. (1973), “A comment on `Forecasting and stock control for intermittent demands’,” Operational Research Quarterly, 24, 639–640.CrossRefGoogle Scholar
  52. Reilly, R. R. G. T. Chao (1982), “Validity and fairness of some alternative employee selection procedures,” Personnel Psychology, 35, 1–62.CrossRefGoogle Scholar
  53. Robertson, I. T. R. S. Kandola (1982), “Work sample tests: Validity, adverse impact and applicant reaction,” Journal of Occupational Psychology, 55, 171–183CrossRefGoogle Scholar
  54. Sanders, N. L. Ritzman (2001), “Judgmental adjustments of statistical forecasts,” in J. S. Armstrong (ed.), Principles of Forecasting. Norwell, MA: Kluwer Academic Publishers.Google Scholar
  55. Schnaars, S. P. (1984), “Situational factors affecting forecast accuracy,” Journal ofMarketing Research, 21, 290–297.CrossRefGoogle Scholar
  56. Scott, S. (1997), “Software reviews: Adjusting from X-11 to X-12, ” International Journal of Forecasting, 13, 567–573.CrossRefGoogle Scholar
  57. Shiskin, J. (1965), The X-11 variant of the census method II seasonal adjustment program. Washington, D.C.: U.S Bureau of the Census.Google Scholar
  58. Simon, J. (1981), The Ultimate Resource. Princeton: Princeton University Press.Google Scholar
  59. Simon, J. (1985), “Forecasting the long-term trend of raw material availability, ” International Journal of Forecasting, 1, 85–109 (includes commentaries and reply).Google Scholar
  60. Smith, M. C. (1976), “A comparison of the value of trainability assessments and other tests for predicting the practical performance of dental students,” International Review of Applied Psychology, 25, 125–130.CrossRefGoogle Scholar
  61. Smith, S. K. (1997), “Further thoughts on simplicity and complexity in population projection models,” International Journal of Forecasting, 13, 557–565.CrossRefGoogle Scholar
  62. Smith, S. K. T. Sincich (1988), “Stability over time in the distribution of population forecast errors,” Demography, 25, 461–474.CrossRefGoogle Scholar
  63. Smith, S. K. T. Sincich (1990), “The relationship between the length of the base period and population forecast errors,” Journal of the American Statistical Association, 85, 367–375.CrossRefGoogle Scholar
  64. Solow, R. M. (1957), “Technical change and the aggregate production function,” Review of Economics and Statistics, 39, 312–320.CrossRefGoogle Scholar
  65. Sutton, J. (1997), “Gibrat’s legacy,” Journal of Economic Literature, 35, 40–59.Google Scholar
  66. Tashman, L. J. J. Hoover (2001), “An evaluation of forecasting software,” in J. S. Armstrong (ed.), Principles of Forecasting. Norwell, MA: Kluwer Academic Publishers.Google Scholar
  67. Tashman, L. J. J. M. Kruk (1996), “The use of protocols to select exponential smoothing procedures: A reconsideration of forecasting competitions,” International Journal of Forecasting, 12, 235–253.CrossRefGoogle Scholar
  68. Tessier, T. H. J. S. Armstrong (1977), “Improving current sales estimates with econometric methods,” in full text at hops.wharton.upenn.edu/forecast.Google Scholar
  69. Webby, R., M. O’Connor M. Lawrence (2001), “Judgmental time-series forecasting using domain knowledge,” in J. S. Armstrong (ed.), Principles of Forecasting. Norwell, MA: Kluwer Academic Publishers.Google Scholar
  70. Willemain, T. R., C. N. Smart, J. H. Shocker P. A. DeSautels (1994), “Forecasting intermittent demand in manufacturing: A comparative evaluation of Croston’s method,” International Journal of Forecasting, 10, 529–538.CrossRefGoogle Scholar
  71. Williams, D. W. D. Miller (1999), “Level-adjusted exponential smoothing for modeling planned discontinuities,” International Journal of Forecasting, 15, 273–289.CrossRefGoogle Scholar
  72. Williams, W. H L. Goodman (1971), “A simple method for the construction of empirical confidence limits to economic forecasts,” Journal of the American Statistical Association, 66, 752–754.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • J. Scott Armstrong
    • 1
  1. 1.The Wharton SchoolUniversity of PennsylvaniaUSA

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