Forecast combination-based forecast reconciliation: Insights and extensions

In this paper, we build upon a recently proposed forecast combination-based approach to the reconciliation of a simple hierarchy (Hollyman R., Petropoulos F., Tipping M.E., Understanding forecast reconciliation, European Journal of Operational Research, 2021, 294, 149–160) and extend it in some new directions. In particular, we provide insights into the nature and mathematical derivation of the level-l conditional coherent (LlCC) point forecast reconciliation procedure for an elementary two-level hierarchy. We show that: (i) the LlCC procedure is the result of a linearly constrained minimization of a quadratic loss function, with an exogenous constraint given by the base forecast of the top level series of the hierarchy, which is not revised; and (ii) endogenous constraints may also be considered in the same framework, thereby resulting in level conditional reconciled forecasts where both the top and the bottom level time series forecasts are coherently revised. In addition, we show that the LlCC procedure (i.e., with exogenous constraints but the result also holds in the endogenous case) does not guarantee the non-negativity of the reconciled forecasts, which can be an issue in cases when non-negativity is a natural attribute of the variables that need to be forecast (e.g., sales and tourism flows). Finally, we consider two forecasting experiments to evaluate the performance of various cross-sectional forecast combination-based point forecast reconciliation procedures (vis-à-vis the state-of-the-art procedures) in a fair setting. In this framework, due to the crucial role played by the (possibly different) models used to compute the base forecasts, we re-interpret the combined conditional coherent reconciliation procedure (CCCH) of Hollyman et al. (2021) as a forecast pooling approach, and show that accuracy improvements may be obtained by adopting a simple forecast averaging strategy.