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

Author

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  • 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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    Cited by:

    1. Hayat, Aziz & Bhatti, M. Ishaq, 2013. "Masking of volatility by seasonal adjustment methods," Economic Modelling, Elsevier, vol. 33(C), pages 676-688.
    2. Sarah Gelper & Roland Fried & Christophe Croux, 2010. "Robust forecasting with exponential and Holt-Winters smoothing," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 29(3), pages 285-300.
    3. 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.
    4. Mick Silver, 2006. "Core Inflation Measures and Statistical Issues in Choosing Among Them," IMF Working Papers 06/97, International Monetary Fund.
    5. Jan G. De Gooijer & Rob J. Hyndman, 2005. "25 Years of IIF Time Series Forecasting: A Selective Review," Monash Econometrics and Business Statistics Working Papers 12/05, Monash University, Department of Econometrics and Business Statistics.
    6. 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.
    7. Gould, Phillip G. & Koehler, Anne B. & Ord, J. Keith & Snyder, Ralph D. & Hyndman, Rob J. & Vahid-Araghi, Farshid, 2008. "Forecasting time series with multiple seasonal patterns," European Journal of Operational Research, Elsevier, vol. 191(1), pages 207-222, November.
    8. 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.
    9. Hayat, Aziz & Narayan, Paresh Kumar, 2010. "The oil stock fluctuations in the United States," Applied Energy, Elsevier, vol. 87(1), pages 178-184, January.
    10. Taylor, James W., 2007. "Forecasting daily supermarket sales using exponentially weighted quantile regression," European Journal of Operational Research, Elsevier, vol. 178(1), pages 154-167, April.
    11. Theodosiou, Marina, 2011. "Forecasting monthly and quarterly time series using STL decomposition," International Journal of Forecasting, Elsevier, vol. 27(4), pages 1178-1195, October.
    12. 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).
    13. De Gooijer, Jan G. & Hyndman, Rob J., 2006. "25 years of time series forecasting," International Journal of Forecasting, Elsevier, vol. 22(3), pages 443-473.
    14. 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.
    15. George P. Papaioannou & Christos Dikaiakos & Anargyros Dramountanis & Panagiotis G. Papaioannou, 2016. "Analysis and Modeling for Short- to Medium-Term Load Forecasting Using a Hybrid Manifold Learning Principal Component Model and Comparison with Classical Statistical Models (SARIMAX, Exponential Smoot," Energies, MDPI, Open Access Journal, vol. 9(8), pages 1-40, August.
    16. repec:pal:jorsoc:v:61:y:2010:i:1:d:10.1057_jors.2008.152 is not listed on IDEAS
    17. Mauro Bernardi & Lea Petrella, 2015. "Multiple seasonal cycles forecasting model: the Italian electricity demand," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(4), pages 671-695, November.
    18. Taylor, James W., 2008. "An evaluation of methods for very short-term load forecasting using minute-by-minute British data," International Journal of Forecasting, Elsevier, vol. 24(4), pages 645-658.
    19. Goodwin, Paul & Önkal, Dilek & Thomson, Mary, 2010. "Do forecasts expressed as prediction intervals improve production planning decisions?," European Journal of Operational Research, Elsevier, vol. 205(1), pages 195-201, August.
    20. 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.
    21. Gardner, Everette Jr., 2006. "Exponential smoothing: The state of the art--Part II," International Journal of Forecasting, Elsevier, vol. 22(4), pages 637-666.
    22. 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.

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