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Prediction intervals for exponential smoothing using two new classes of state space models

Author

Listed:
  • Anne B. Koehler

    (Miami University, USA)

  • Rob J. Hyndman

    (Monash University, Australia)

  • Ralph D. Snyder

    (Monash University, Australia)

  • J. Keith Ord

    (Georgetown University, USA)

Abstract

Three general classes of state space models are presented, using the single source of error formulation. The first class is the standard linear model with homoscedastic errors, the second retains the linear structure but incorporates a dynamic form of heteroscedasticity, and the third allows for non-linear structure in the observation equation as well as heteroscedasticity. These three classes provide stochastic models for a wide variety of exponential smoothing methods. We use these classes to provide exact analytic (matrix) expressions for forecast error variances that can be used to construct prediction intervals one or multiple steps ahead. These formulas are reduced to non-matrix expressions for 15 state space models that underlie the most common exponential smoothing methods. We discuss relationships between our expressions and previous suggestions for finding forecast error variances and prediction intervals for exponential smoothing methods. Simpler approximations are developed for the more complex schemes and their validity examined. The paper concludes with a numerical example using a non-linear model. Copyright © 2005 John Wiley & Sons, Ltd.

Suggested Citation

  • Anne B. Koehler & Rob J. Hyndman & Ralph D. Snyder & J. Keith Ord, 2005. "Prediction intervals for exponential smoothing using two new classes of state space models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 24(1), pages 17-37.
  • Handle: RePEc:jof:jforec:v:24:y:2005:i:1:p:17-37
    DOI: 10.1002/for.938
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    File URL: http://hdl.handle.net/10.1002/for.938
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    References listed on IDEAS

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    1. Koehler, Anne B. & Snyder, Ralph D. & Ord, J. Keith, 2001. "Forecasting models and prediction intervals for the multiplicative Holt-Winters method," International Journal of Forecasting, Elsevier, vol. 17(2), pages 269-286.
    2. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    3. Yar, Mohammed & Chatfield, Chris, 1990. "Prediction intervals for the Holt-Winters forecasting procedure," International Journal of Forecasting, Elsevier, vol. 6(1), pages 127-137.
    4. Chatfield, Chris & Yar, Mohammed, 1991. "Prediction intervals for multiplicative Holt-Winters," International Journal of Forecasting, Elsevier, vol. 7(1), pages 31-37, May.
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