The admissible parameter space for exponential smoothing models

  • Rob J. Hyndman
  • Muhammad Akram
  • Blyth C. Archibald
Article

Abstract

We discuss the admissible parameter space for some state space models, including the models that underly exponential smoothing methods. We find that the usual parameter restrictions (requiring all smoothing parameters to lie between 0 and 1) do not always lead to stable models. We also find that all seasonal exponential smoothing methods are unstable as the underlying state space models are neither reachable nor observable. This instability does not affect the forecasts, but does corrupt the state estimates. The problem can be overcome with a simple normalizing procedure. Finally we show that the admissible parameter space of a seasonal exponential smoothing model is much larger than that for a basic structural model, leading to better forecasts from the exponential smoothing model when there is a rapidly changing seasonal pattern.

Keywords

Exponential smoothing Invertibility Observability Parameter space Reachability Stability State space models Structural models 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Archibald B.C. (1984). Invertible region of Holt–Winters’ model, Working paper 31/1984, School of Business Administration. Halifax, Dalhousie UniversityGoogle Scholar
  2. Archibald B.C. (1990). Parameter space of the Holt-Winters’ model. International Journal of Forecasting 6, 199–229CrossRefGoogle Scholar
  3. Archibald B.C. (1991). Invertible region of damped trend, seasonal, exponential smoothing model, Working paper 10/1991, School of Business Administration. Halifax, Dalhousie UniversityGoogle Scholar
  4. Hannan E.J., Deistler M. (1988). The statistical theory of linear systems. New York, WileyMATHGoogle Scholar
  5. Harvey A.C. (1989). Forecasting, structural time series models and the Kalman filter. Cambridge, Cambridge University PressGoogle Scholar
  6. Hyndman R.J., Koehler A.B., Ord J.K., Snyder R.D. (2005). Prediction intervals for exponential smoothing state space models. Journal of Forecasting 24, 17–37CrossRefMathSciNetGoogle Scholar
  7. Hyndman R.J., Koehler A.B., Snyder R.D., Grose S. (2002). A state space framework for automatic forecasting using exponential smoothing methods. International Journal of Forecasting 18(3): 439–454CrossRefGoogle Scholar
  8. Lawton R. (1998). How should additive Holt-Winters’ estimates be corrected?. International Journal of Forecasting 14, 393–403CrossRefGoogle Scholar
  9. Makridakis S., Wheelwright S.C., Hyndman R.J. (1998). Forecasting: methods and applications (3rd ed.) New York, WileyGoogle Scholar
  10. McClain J.O., Thomas L.J. (1973). Response–variance tradeoffs in adaptive forecasting. Operations Research 21, 554–568MATHMathSciNetGoogle Scholar
  11. Ord J.K., Koehler A.B., Snyder R.D. (1997). Estimation and prediction for a class of dynamic nonlinear statistical models. Journal of American Statistical Association 92, 1621–1629MATHCrossRefGoogle Scholar
  12. Ralston A. (1965). A first course in numerical analysis. New York, McGraw-HillMATHGoogle Scholar
  13. Roberts S.A. (1982). A general class of Holt-Winters type forecasting models. Management Science 28(8): 808–820MATHMathSciNetCrossRefGoogle Scholar
  14. Snyder R.D., Forbes C.S. (2003). Reconstructing the Kalman filter for stationary and non-stationary time series. Studies in nonlinear dynamics and econometrics 7(2): 1–18Google Scholar
  15. Snyder R.D., Ord J.K., Koehler A.B. (2001). Prediction intervals for ARIMA models. Journal of Business and Economics Statists 19(2): 217–225CrossRefMathSciNetGoogle Scholar
  16. Sweet A.L. (1985). Computing the variance of the forecast error for the Holt-Winters seasonal models. Journal of Forecasting 4, 235–243CrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2007

Authors and Affiliations

  • Rob J. Hyndman
    • 1
  • Muhammad Akram
    • 1
  • Blyth C. Archibald
    • 2
  1. 1.Department of Econometrics and Business StatisticsMonash UniversityClaytonAustralia
  2. 2.School of Business AdministrationDalhousie UniversityHalifaxCanada

Personalised recommendations