Prediction Intervals for Exponential Smoothing Using Two New Classes State Space Models

  • Hydman R
  • Koehler A
  • Ord J
 et al. 
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Three general classes state space models are presented, using the single
source error formulation. The first class is the standard linear model with
homoscedastic errors, the second retains the linear structure but incorporates a
dynamic form 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 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.

Author-supplied keywords

  • forecast distribution forecast interval forecast e

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  • Rob J Hydman

  • Anne B Koehler

  • J Keith Ord

  • Ralph D Snyder

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