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Some Properties and Generalizations of Non‐negative Bayesian Time Series Models

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  • Gary K. Grunwald
  • Kais Hamza
  • Rob J. Hyndman

Abstract

We study the most basic Bayesian forecasting model for exponential family time series, the power steady model (PSM) of Smith, in terms of observable properties of one‐step forecast distributions and sample paths. The PSM implies a constraint between location and spread of the forecast distribution. Including a scale parameter in the models does not always give an exact solution free of this problem, but it does suggest how to define related models free of the constraint. We define such a class of models which contains the PSM. We concentrate on the case where observations are non‐negative. Probability theory and simulation show that under very mild conditions almost all sample paths of these models converge to some constant, making them unsuitable for modelling in many situations. The results apply more generally to non‐negative models defined in terms of exponentially weighted moving averages. We use these and related results to motivate, define and apply very simple models based on directly specifying the forecast distributions.

Suggested Citation

  • Gary K. Grunwald & Kais Hamza & Rob J. Hyndman, 1997. "Some Properties and Generalizations of Non‐negative Bayesian Time Series Models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(3), pages 615-626.
    • Handle: RePEc:bla:jorssb:v:59:y:1997:i:3:p:615-626
    DOI: 10.1111/1467-9868.00086
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    Cited by:

    1. Rob J. Hyndman & Lydia Shenstone, 2005. "Stochastic models underlying Croston's method for intermittent demand forecasting," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 24(6), pages 389-402.
    2. Ralph Snyder & Adrian Beaumont & J. Keith Ord, 2012. "Intermittent demand forecasting for inventory control: A multi-series approach," Monash Econometrics and Business Statistics Working Papers 15/12, Monash University, Department of Econometrics and Business Statistics.
    3. Higuchi, Tomoyuki, 1999. "Applications of quasi-periodic oscillation models to seasonal small count time series," Computational Statistics & Data Analysis, Elsevier, vol. 30(3), pages 281-301, May.
    4. Feigin, Paul D. & Gould, Phillip & Martin, Gael M. & Snyder, Ralph D., 2008. "Feasible parameter regions for alternative discrete state space models," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 2963-2970, December.
    5. Ord, J. Keith & Koehler, Anne B. & Snyder, Ralph D. & Hyndman, Rob J., 2009. "Monitoring processes with changing variances," International Journal of Forecasting, Elsevier, vol. 25(3), pages 518-525, July.
    6. Ali Caner Türkmen & Tim Januschowski & Yuyang Wang & Ali Taylan Cemgil, 2021. "Forecasting intermittent and sparse time series: A unified probabilistic framework via deep renewal processes," PLOS ONE, Public Library of Science, vol. 16(11), pages 1-26, November.
    7. 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.
    8. Ralph D. Snyder & Adrian Beaumont, 2007. "A Comparison of Methods for Forecasting Demand for Slow Moving Car Parts," Monash Econometrics and Business Statistics Working Papers 15/07, Monash University, Department of Econometrics and Business Statistics.
    9. Zhen, X. & Basawa, I.V., 2009. "Observation-driven generalized state space models for categorical time series," Statistics & Probability Letters, Elsevier, vol. 79(24), pages 2462-2468, December.
    10. Ralph D. Snyder & Gael M. Martin & Phillip Gould & Paul D. Feigin, 2007. "An Assessment of Alternative State Space Models for Count Time Series," Monash Econometrics and Business Statistics Working Papers 4/07, Monash University, Department of Econometrics and Business Statistics.
    11. Fukasawa, T. & Basawa, I. V., 2002. "Estimation for a class of generalized state-space time series models," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 459-473, December.

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