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Exponential smoothing and non-negative data


  • Muhammad Akram

    () (Department of Econometrics and Business Statistics)

  • Rob J Hyndman

    () (Department of Econometrics and Business Statistics,Monash University)

  • J. Keith Ord

    () (McDonough School of Business,Georgetown University)


The most common forecasting methods in business are based on exponential smoothing and the most common time series in business are inherently non-negative. Therefore it is of interest to consider the properties of the potential stochastic models underlying exponential smoothing when applied to non-negative data. We explore exponential smoothing state space models for non-negative data under various assumptions about the innovations, or error, process. We first demonstrate that prediction distributions from some commonly used state space models may have an infinite variance beyond a certain forecasting horizon. For multiplicative error models which do not have this flaw, we show that sample paths will converge almost surely to zero even when the error distribution is non-Gaussian. We propose a new model with similar properties to exponential smoothing, but which does not have these problems, and we develop some distributional properties for our new model. We then explore the implications of our results for inference, and compare the short-term forecasting performance of the various models using data on the weekly sales of over three hundred items of costume jewelry. The main findings of the research are that the Gaussian approximation is adequate for estimation and one-step-ahead forecasting. However, as the forecasting horizon increases, the approximate prediction intervals become increasingly problematic. When the model is to be used for simulation purposes, a suitably specified scheme must be employed.

Suggested Citation

  • Muhammad Akram & Rob J Hyndman & J. Keith Ord, 2008. "Exponential smoothing and non-negative data," Working Papers 2008-003, The George Washington University, Department of Economics, Research Program on Forecasting.
  • Handle: RePEc:gwc:wpaper:2008-003

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    References listed on IDEAS

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    Cited by:

    1. Snyder, Ralph D. & Ord, J. Keith & Beaumont, Adrian, 2012. "Forecasting the intermittent demand for slow-moving inventories: A modelling approach," International Journal of Forecasting, Elsevier, vol. 28(2), pages 485-496.
    2. Ralph D. Snyder & J. Keith Ord & Adrian Beaumont, 2010. "Forecasting the Intermittent Demand for Slow-Moving Items," Working Papers 2010-003, The George Washington University, Department of Economics, Research Program on Forecasting, revised Mar 2011.
    3. de Silva, Ashton J, 2010. "Forecasting Australian Macroeconomic variables, evaluating innovations state space approaches," MPRA Paper 27411, University Library of Munich, Germany.
    4. 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.

    More about this item


    forecasting; time series; exponential smoothing; positive-valued processes; seasonality; state space models.;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling

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