Minimization of Forecast Variance Using an Example of ETS Models

Abstract—Construction of a combined model of time series (for two models of the same type that exhibit additivity, for example, ARIMA) or a combined forecast of models (in the absence of additivity, for example, for ETS models) providing minimization of the estimated forecast variance is considered. As distinct from alternative models of time series in which the forecast variance is estimated using the Student test, the ARIMA and ETS models allow construction of a function that is related to the parameters of model. Thus, it is possible to estimate the value of the confidence interval for the forecast and construct combinations of models with a minimum estimate of the width of the interval depending on the parameters of the combination. The theoretical part of the work studies linear combinations of forecasts of two models, in which the estimate of forecast variance is minimized (regardless of the type of model). The Hessian of the function for estimating the forecast variance is obtained for construction of a linear combination of forecasts. It is analyzed under the conditions for extremum (zero first derivatives of the function for estimating the variance of the forecast for the combined models). Then, the Hessian is estimated for several groups of ETS models, and the conditions for the presence of a minimum of the estimated forecast variance at a stationary point are considered versus parameters of models.