IDEAS home Printed from https://ideas.repec.org/p/msh/ebswps/2006-3.html
   My bibliography  Save this paper

Some Nonlinear Exponential Smoothing Models are Unstable

Author

Listed:
  • Rob J Hyndman
  • Muhammad Akram

Abstract

This paper discusses the instability of eleven nonlinear state space models that underly exponential smoothing. Hyndman et al. (2002) proposed a framework of 24 state space models for exponential smoothing, including the well-known simple exponential smoothing, Holt's linear and Holt-Winters' additive and multiplicative methods. This was extended to 30 models with Taylor's (2003) damped multiplicative methods. We show that eleven of these 30 models are unstable, having infinite forecast variances. The eleven models are those with additive errors and either multiplicative trend or multiplicative seasonality, as well as the models with multiplicative errors, multiplicative trend and additive seasonality. The multiplicative Holt-Winters' model with additive errors is among the eleven unstable models. We conclude that: (1) a model with a multiplicative trend or a multiplicative seasonal component should also have a multiplicative error; and (2) a multiplicative trend should not be mixed with additive seasonality.

Suggested Citation

  • 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.
  • Handle: RePEc:msh:ebswps:2006-3
    as

    Download full text from publisher

    File URL: http://www.buseco.monash.edu.au/ebs/pubs/wpapers/2006/wp3-06.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    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. Taylor, James W., 2003. "Exponential smoothing with a damped multiplicative trend," International Journal of Forecasting, Elsevier, vol. 19(4), pages 715-725.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kourentzes, Nikolaos & Athanasopoulos, George, 2019. "Cross-temporal coherent forecasts for Australian tourism," Annals of Tourism Research, Elsevier, vol. 75(C), pages 393-409.
    2. Kourentzes, Nikolaos & Athanasopoulos, George, 2021. "Elucidate structure in intermittent demand series," European Journal of Operational Research, Elsevier, vol. 288(1), pages 141-152.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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).
    2. 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.
    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. 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.
    5. 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.
    6. Kim, Jae H. & Wong, Kevin & Athanasopoulos, George & Liu, Shen, 2011. "Beyond point forecasting: Evaluation of alternative prediction intervals for tourist arrivals," International Journal of Forecasting, Elsevier, vol. 27(3), pages 887-901, July.
    7. 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.
    8. 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.
    9. Hayat, Aziz & Bhatti, M. Ishaq, 2013. "Masking of volatility by seasonal adjustment methods," Economic Modelling, Elsevier, vol. 33(C), pages 676-688.
    10. 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.
    11. 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.
    12. James W. Taylor, 2004. "Smooth transition exponential smoothing," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 23(6), pages 385-404.
    13. Kourentzes, Nikolaos & Petropoulos, Fotios & Trapero, Juan R., 2014. "Improving forecasting by estimating time series structural components across multiple frequencies," International Journal of Forecasting, Elsevier, vol. 30(2), pages 291-302.
    14. de Silva, Ashton J, 2010. "Forecasting Australian Macroeconomic variables, evaluating innovations state space approaches," MPRA Paper 27411, University Library of Munich, Germany.
    15. Niematallah Elamin & Mototsugu Fukushige, 2016. "Forecasting extreme seasonal tourism demand," Discussion Papers in Economics and Business 16-23, Osaka University, Graduate School of Economics.
    16. Corberán-Vallet, Ana & Bermúdez, José D. & Vercher, Enriqueta, 2011. "Forecasting correlated time series with exponential smoothing models," International Journal of Forecasting, Elsevier, vol. 27(2), pages 252-265, April.
    17. Zeynep Ozsut Bogar & Askiner Gungor, 2023. "Forecasting Waste Mobile Phone (WMP) Quantity and Evaluating the Potential Contribution to the Circular Economy: A Case Study of Turkey," Sustainability, MDPI, vol. 15(4), pages 1-38, February.
    18. R Fildes & K Nikolopoulos & S F Crone & A A Syntetos, 2008. "Forecasting and operational research: a review," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(9), pages 1150-1172, September.
    19. Meira, Erick & Cyrino Oliveira, Fernando Luiz & Jeon, Jooyoung, 2021. "Treating and Pruning: New approaches to forecasting model selection and combination using prediction intervals," International Journal of Forecasting, Elsevier, vol. 37(2), pages 547-568.
    20. Fernández-Amador, Octavio & Francois, Joseph F. & Oberdabernig, Doris A. & Tomberger, Patrick, 2020. "The methane footprint of nations: Stylized facts from a global panel dataset," Ecological Economics, Elsevier, vol. 170(C).

    More about this item

    Keywords

    Exponential smoothing; forecast variance; nonlinear models; prediction intervals; stability; state space models.;
    All these keywords.

    JEL classification:

    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:msh:ebswps:2006-3. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Professor Xibin Zhang (email available below). General contact details of provider: https://edirc.repec.org/data/dxmonau.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.