Forecasts reconciliation for hierarchical time series forecasting problem

The hierarchical time series forecasting problem is researched. Time series forecasts must satisfy the physical constraints and the hierarchical structure. In this paper, a new algorithm for hierarchical time series forecasts reconciliation is proposed. The algorithm is called GTOp (Game-theoretically optimal reconciliation). It guarantees that the quality of reconciled forecasts is not worse than the quality of self-dependent forecasts. This approach is based on Nash equilibrium search for the antagonistic game and turns the forecasts reconciliation problem into the optimization problem with equality and inequality constraints. It is proved that the Nash equilibrium in pure strategies exists in the game if some assumptions about the hierarchical structure, the physical constraints, and the loss function are satisfied. The algorithm performance is demonstrated for different types of hierarchical structures of time series.