Forecast combination based forecast reconciliation: insights and extensions
In this paper, we build upon and extend along some new directions a recently proposed forecast combination-based approach to the reconciliation of a simple hierarchy (Hollyman et al., 2021)1. In particular, we shed light on the nature and the mathematical derivation of the Level-l Conditional Coherent (LlCC) point forecast reconciliation procedure for an elementary two-level hierarchy. We show that the LlCC procedure is the result of a linearly constrained minimization of a quadratic loss function, with exogenous constraint given by the base forecasts of the top level series of the hierarchy, which is not revised. Endogenous constraints may be considered in the same framework as well, resulting in level conditional reconciled forecasts where both the top and the bottom level forecasts are coherently revised. In addition, we show that the LlCC procedure (i.e., with exogenous constraints, but the result holds in the endogenous case as well) does not guarantee the non-negativity of the reconciled forecasts. This can be an issue in cases where non-negativity is a natural attribute of the variables to be forecast (e.g., sales, tourism flows, etc.). We finally consider two forecasting experiments to evaluate in a fair setting the performance of various cross-sectional forecast combination based point forecast reconciliation procedures vis-à-vis the state-of-the-art procedures. 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, showing that accuracy improvement may be gained by adopting a simple forecast averaging strategy.
1 Hollyman R., Petropoulos F., Tipping M.E., Understanding forecast reconciliation, European Journal of Operational Research, 2021, 294, 149–160