Forecasting models and prediction intervals for the multiplicative Holt–Winters method

A new class of models for data showing trend and multiplicative seasonality is presented. The models allow the forecast error variance to depend on the trend and/or the seasonality. It is shown that each of these models has essentially the same updating equations and forecast functions as the multiplicative Holt–Winters method, whether or not the error variation in the model is constant. Although the different models produce identical updating relationships for the point forecast, the prediction intervals, of course, depend on the structure of the error variance and so it is essential to be able to choose the most appropriate form of model. Two methods for model selection are presented and examined by simulation. For the most common case of series with an upward trend, we recommend using a model with variance dependent on both the trend and seasonal elements.