Forecast Intervals in ARCH Exponential Smoothing

Exponential smoothing (ES) with ARCH (autoregressive conditionally heteroscedastic) and GARCH (generalized ARCH) errors are introduced. This is done for a large class of ES methods, those for which the forecasts are obtained using a set of additive updating formulas, and also those for which an ARIMA form had been previously established. The class includes HOLT and WINTERS methods but also methods involving a damped trend. The transition from homoscedastic ES to ES-GARCH methods is based on the updating formulas by relaxing the assumption that the errors constitute a white noise process. It is assumed instead that the error process is a martingale difference, with the usual GARCH representation for the conditional variance. The device for deriving the results is the underlying ARIMA-GARCH representation of the ES-GARCH methods. The problems which are discussed are the following: (a) the estimation of the smoothing constants; (b) the computation of the forecast intervals. Examples demonstrating the new approach are given. They are related to finance and marketing.