Prediction intervals for exponential smoothing using two new classes of state space models

Three general classes of state space models are presented, using the single source of error formulation. The first class is the standard linear model with homoscedastic errors, the second retains the linear structure but incorporates a dynamic form of 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 of 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 is examined. The paper concludes with a numerical example using a non-linear model.