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#### Gary K. Grunwald1, Kais Hamza1 and Rob J. Hyndman2

1. Department of Statistics, The University of Melbourne, VIC 3052, Australia.
2. Department of Mathematics, Monash University, VIC 3800, Australia.

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

Keywords: Bayesian forecasting, exponential family time series, exponentially weighted moving averages, non-negative time series, Poisson time series, power steady model.

Online article