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Prediction Intervals for Exponential Smoothing State Space Models

Author

Listed:
  • Hyndman, R.J.
  • Koehler, A.B.
  • Ord, J.K.
  • Snyder, R.D.

Abstract

The main objective of this paper is to provide analytical expression for forecast variances that can be used in prediction intervals for the exponential smoothing methods. These expressions are based on state space models with a single source of error that underlie the exponential smoothing methods. In cases where an ARIMA model also underlies an exponential smoothing method, there is an equivalent state space model with the same variance expression. We also discuss relationships between these new ideas and previous suggestions for finding forecast variances and prediction intervals for the exponential smoothing methods.

Suggested Citation

  • Hyndman, R.J. & Koehler, A.B. & Ord, J.K. & Snyder, R.D., 2001. "Prediction Intervals for Exponential Smoothing State Space Models," Monash Econometrics and Business Statistics Working Papers 11/01, Monash University, Department of Econometrics and Business Statistics.
    • Handle: RePEc:msh:ebswps:2001-11
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    File URL: http://www.buseco.monash.edu.au/ebs/pubs/wpapers/2001/wp11-01.pdf
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    References listed on IDEAS

    as
    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. Hyndman, Rob J. & Koehler, Anne B. & Snyder, Ralph D. & Grose, Simone, 2002. "A state space framework for automatic forecasting using exponential smoothing methods," International Journal of Forecasting, Elsevier, vol. 18(3), pages 439-454.
    3. S. A. Roberts, 1982. "A General Class of Holt-Winters Type Forecasting Models," Management Science, INFORMS, vol. 28(7), pages 808-820, July.
    4. Ord, J.K. & Koehler, A. & Snyder, R.D., 1995. "Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models," Monash Econometrics and Business Statistics Working Papers 4/95, Monash University, Department of Econometrics and Business Statistics.
    5. Chatfield, Chris & Yar, Mohammed, 1991. "Prediction intervals for multiplicative Holt-Winters," International Journal of Forecasting, Elsevier, vol. 7(1), pages 31-37, May.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Taylor, James W., 2003. "Exponential smoothing with a damped multiplicative trend," International Journal of Forecasting, Elsevier, vol. 19(4), pages 715-725.
    2. Athanasopoulos, George & Hyndman, Rob J. & Song, Haiyan & Wu, Doris C., 2011. "The tourism forecasting competition," International Journal of Forecasting, Elsevier, vol. 27(3), pages 822-844, July.
    3. Rob J Hyndman & Muhammad Akram, 2006. "Some Nonlinear Exponential Smoothing Models are Unstable," Monash Econometrics and Business Statistics Working Papers 3/06, Monash University, Department of Econometrics and Business Statistics.
    4. Rob J Hyndman & Maxwell L. King & Ivet Pitrun & Baki Billah, 2002. "Local Linear Forecasts Using Cubic Smoothing Splines," Monash Econometrics and Business Statistics Working Papers 10/02, Monash University, Department of Econometrics and Business Statistics.
    5. J Keith Ord & Ralph D Snyder & Anne B Koehler & Rob J Hyndman & Mark Leeds, 2005. "Time Series Forecasting: The Case for the Single Source of Error State Space," Monash Econometrics and Business Statistics Working Papers 7/05, Monash University, Department of Econometrics and Business Statistics.
    6. Ralph D. Snyder & Anne B. Koehler & Rob J. Hyndman & J. Keith Ord, 2002. "Exponential Smoothing for Inventory Control: Means and Variances of Lead-Time Demand," Monash Econometrics and Business Statistics Working Papers 3/02, Monash University, Department of Econometrics and Business Statistics.
    7. Hyndman, Rob J. & Shahid Ullah, Md., 2007. "Robust forecasting of mortality and fertility rates: A functional data approach," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4942-4956, June.
    8. Hyndman, Rob J. & Khandakar, Yeasmin, 2008. "Automatic Time Series Forecasting: The forecast Package for R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 27(i03).
    9. Hayat, Aziz & Bhatti, M. Ishaq, 2013. "Masking of volatility by seasonal adjustment methods," Economic Modelling, Elsevier, vol. 33(C), pages 676-688.
    10. Mick Silver, 2006. "Core Inflation Measures and Statistical Issues in Choosing Among Them," IMF Working Papers 2006/097, International Monetary Fund.
    11. Snyder, Ralph D. & Koehler, Anne B. & Hyndman, Rob J. & Ord, J. Keith, 2004. "Exponential smoothing models: Means and variances for lead-time demand," European Journal of Operational Research, Elsevier, vol. 158(2), pages 444-455, October.
    12. James W. Taylor, 2004. "Smooth transition exponential smoothing," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 23(6), pages 385-404.
    13. Alysha M De Livera, 2010. "Automatic forecasting with a modified exponential smoothing state space framework," Monash Econometrics and Business Statistics Working Papers 10/10, Monash University, Department of Econometrics and Business Statistics.
    14. Muhammad Akram & Rob J. Hyndman & J. Keith Ord, 2007. "Non-linear exponential smoothing and positive data," Monash Econometrics and Business Statistics Working Papers 14/07, Monash University, Department of Econometrics and Business Statistics.
    15. E. Vercher & A. Corberán-Vallet & J. Segura & J. Bermúdez, 2012. "Initial conditions estimation for improving forecast accuracy in exponential smoothing," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 517-533, July.
    16. Rob Hyndman & Muhammad Akram & Blyth Archibald, 2008. "The admissible parameter space for exponential smoothing models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 407-426, June.
    17. Song, Haiyan & Gao, Bastian Z. & Lin, Vera S., 2013. "Combining statistical and judgmental forecasts via a web-based tourism demand forecasting system," International Journal of Forecasting, Elsevier, vol. 29(2), pages 295-310.
    18. Robert R. Andrawis & Amir F. Atiya, 2009. "A new Bayesian formulation for Holt's exponential smoothing," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 28(3), pages 218-234.
    19. Pim Ouwehand & Rob J. Hyndman & Ton G. de Kok & Karel H. van Donselaar, 2007. "A state space model for exponential smoothing with group seasonality," Monash Econometrics and Business Statistics Working Papers 7/07, Monash University, Department of Econometrics and Business Statistics.

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    More about this item

    Keywords

    Forecast distribution; Holt-Winters method; Structural models;
    All these keywords.

    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods

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