Forecast combination based forecast reconciliation: insights and extensions

Abstract

In a recent paper, while elucidating the links between forecast combination and cross-sectional forecast reconciliation, Hollyman et al. (2021) have proposed a forecast combination-based approach to the reconciliation of a simple hierarchy. A new Level Conditional Coherent (LCC) point forecast reconciliation procedure was developed, and it was shown that the simple average of a set of LCC, and bottom-up reconciled forecasts (called Combined Conditional Coherent, CCC) results in good performance as compared to those obtained through the state-of-the-art cross-sectional reconciliation procedures. In this paper, we build upon and extend this proposal along some new directions. (1) We shed light on the nature and the mathematical derivation of the LCC reconciliation formula, showing that it is the result of an exogenously linearly constrained minimization of a quadratic loss function in the differences between the target and the base forecasts with a diagonal associated matrix. (2) Endogenous constraints may be considered as well, resulting in level conditional reconciled forecasts of all the involved series, where both the upper and the bottom time series are coherently revised. We show that even in this framework it is still valid the interesting interpretation given by Hollyman et al. ( 2021) of the reconciliation formula as the combination of direct (base) and indirect forecasts, the latter ones depending on the accounting relationships linking upper and bottom series. (3) As the LCC 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, we argue that -when non-negativity is a natural attribute of the variables to be forecast -its interpretation as an 'unbiased top-down reconciliation procedure' leaves room for some doubts. (4) The new procedures are used in a forecasting experiment on the classical Australian Tourism Demand (Visitor Nights) dataset. Due to the crucial role played by the (possibly different) models used to compute the base forecasts, we re-interpret the CCC reconciliation of Hollyman et al. (2021) as a forecast pooling approach, showing that accuracy improvement may be gained by adopting a simple forecast averaging strategy.

Optimal reconciliation with immutable forecasts

“…Third, our approach generalizes to grouped hierarchies. More importantly, from the mathematical point of view, the LCC approach reconciles the base forecasts within a low-level hierarchy -the conditional level and the bottom level -using the variance scaling approach (Hyndman et al, 2016;Wickramasuriya et al, 2019) and keeping the conditional level immutable, which is also pointed out by Di Fonzo and Girolimetto (2021). Consequently, LCC approach would only incorporate information of bottom level.…”

“…The estimator entirely ignores information of middle level and covariance. Furthermore, if y A is set to be immutable, LCC would only consider the small hierarchy y = [y A , y AA , y AB ] (Di Fonzo and Girolimetto, 2021) and ignore information from other nodes, which can be captured by our proposed method. Besides, LCC can not be used when a subset of bottom-level series is immutable.…”

“…Our proposed method also extends Hollyman et al (2021) and Di Fonzo and Girolimetto (2021) in three more important ways. First, it is combined with the non-negative reconciliation approach in Wickramasuriya et al (2020) via the imposition of inequality constraints.…”

“…Subsequent work by Di Fonzo and Girolimetto (2021) further explore the theoretical properties of the method proposed by Hollyman et al (2021).…”

“…We extend this literature by proposing a method for optimal reconciliation that keeps forecasts of a subset of series unchanged or "immutable". In contrast to Hollyman et al (2021) and Di Fonzo and Girolimetto (2021), the immutable series in our proposed method may come from different levels of the hierarchy. This may be particularly attractive when both the top-level forecasts and the forecasts of some key individual product lines are subject to judgmental adjustments by experts and therefore should not be altered by the reconciliation process.…”