More accurate prediction intervals for exponential smoothing with covariates with applications in electrical load forecasting and sales forecasting

The most widely used prediction intervals in empirical time series analysis are of plug-in type; that is, the empirical estimates of model parameters are inserted into formulae for prediction intervals that are obtained from a theoretical analysis of the time series model. Several authors have pointed out that such model-based prediction intervals are too narrow, that is, that the actual confidence level is smaller than the nominal confidence level. The reason is that the uncertainty contained in the parameter estimates is not taken into account in the prediction interval. We investigate this problem for exponential smoothing under covariates with additive trend and additive season. We determine alternative prediction intervals by analysing a linearisation of the underlying model with linear model methods. Two simulation studies based on electricity load data and on sales data confirm the reservations about the plug-in prediction intervals, whereas the intervals based on the linearisation approach are significantly better, and perform very well, with actual confidence levels close to the nominal.