A reliable ensemble forecasting modeling approach for complex time series with distributionally robust optimization

Accurate ensemble forecast results provide strong support for management decisions. While the existing ensemble forecasting model ignores the requirement of directional accuracy making its prediction direction is usually error-prone. To address this issue, we develop a convex directional prediction error measure and subsequently construct a reliable ensemble forecasting model that minimizes the horizontal prediction error with the bounded directional prediction error. Mathematically, we provided the proper range of the required directional error level. Furthermore, we find the classical ensemble forecasting model is a special case of our study when the directional error level is larger than the upper bound we gave in this study. To deal with the possible deterioration of the individual model over time, we also considered its worst possible prediction performance by introducing the distributionally robust optimization (DRO) technique into the proposed reliable ensemble forecasting model. Technically, we showed that the DRO-based reliable ensemble forecasting model is convex and can be reformulated into a second-order cone problem which can be readily solved by off-the-shelf solvers. Finally, the effectiveness of the proposed reliable ensemble forecasting model and the DRO-based reliable ensemble forecasting model were validated on three different datasets, e.g., the exchange rate, the oil price, and the country risk index. To sum up, we construct a reliable ensemble forecasting model to simultaneously control the horizontal prediction error and directional prediction error of ensemble forecasting, and thus enhance the reliability of ensemble forecasting.