Prediction Intervals for Exponential Smoothing State Space Models

The main objective of this paper is to provide analytical expressions for forecast variances that can be used in prediction intervals for the exponential smoothing methods. These expressions are based on state space models with a single source of error that underlie the exponential smoothing methods. Three general classes of the state space models are presented. The first class is the standard linear state space 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. Exact matrix formulas for the forecast variances are found for each of these three classes of models. These formulas are specialized to non-matrix formulas for fifteen state space models that underlie nine exponential smoothing methods, including all the widely used methods. In cases where an ARIMA model also underlies an exponential smoothing method, there is an equivalent state space model with the same variance expression. We also discuss relationships between these new ideas and previous suggestions for finding forecast variances and prediction intervals for the exponential smoothing methods.