Reduction of time series forecast variance by the case of ArDL(p, q) models

Combination of time series forecasts or time series models is often considered nowadays in various research papers and is usually a good technique in practice. Still it has got weak theoretical explanation and there’s a lot of work that is going to be done in this field. In the research optimal combination of forecasters in terms of minimal variance is investigated. These results don’t depend on nature of combined models but in the practical part ArDL models describing connections between pairs of time series are considered. Weighted sum of models’ predictions is used as a new prediction with variance that should be less than individual models’ predictions variance. Optimal weights for combination of two models are calculated theoretically. They are based on correlation of combined models and their forecasts’ variance. Optimal model combination variance (variance of the combined models with optimal weights) has been investigated and it has been shown that it doesn’t exceed variances of the combined models. Practical experiments and conditions under which combined time series model is expected to improve forecasts of original ones are in scope of this research. Forecasts of two ArDL models are combined in order to construct forecast with lower variety. This approach can be used in case of more models but theoretical expression of optimal weights is going to be a part of further research. Russian macroeconomical time series statistics and economic data of the USA are used as experimental time series. Combined models have got predictions close to the forecasts of the best models and better than the worst models used in combinations. It’s shown that optimal combination should make predictions of high quality if combined predictors describe different aspects of investigated time series and correlation of their forecasts isn’t very high. Otherwise, it turns into mean forecast of two models.