Linear Combinations of Time Series Models with Minimal Forecast Variance

In this paper construction of optimal combination of time series forecasts (by quality of prediction or forecast variance evaluation) is considered. In addition, averaging of multiple models' forecasts is in scope of this research as a part of weighted model combination. These approaches are widely used in time series modeling and forecasting. In the theoretical part, functions evaluating forecast variance of ARIMA(p, d, q) models over 1, 2, and 3 steps ahead are considered using psi weights. Property of downward convexity is treated for averaged or weighted combination of several ARIMA(p, d, q), p < 4 model forecasts. Also, forecast com-bination for two models of an arbitrary type is considered. Forecasts take part in weighted combination and weights are counted in the way to minimize evaluation of forecast variance. In the experimental part, weighted combinations (optimal by forecast variance) of ARIMA(p, d, q) models and ADL(p, a) models are built. The quality of combined model forecasts is not worse than the accuracy of treated model forecasts. When studying the combination of forecasts of three models, the forecast variance can both decrease when combined and exceed the forecast variances of individual models, so it is not possible to draw general conclusions when combining more than a pair of models.