Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives

TL;DR

This paper introduces a new iterative method and a closed-form solution for cross-temporal forecast reconciliation, improving the accuracy of hierarchical time series forecasts by simultaneously addressing both cross-sectional and temporal hierarchies.

Contribution

It presents a novel iterative reconciliation procedure and a closed-form optimal forecast expression that jointly handle cross-sectional and temporal constraints, advancing existing methods.

Findings

The proposed methods outperform single-dimension reconciliation procedures.

The iterative approach overcomes weaknesses of previous two-step methods.

Empirical evaluation on Australian GDP data demonstrates improved forecast accuracy.

Abstract

Forecast reconciliation is a post-forecasting process aimed to improve the quality of the base forecasts for a system of hierarchical/grouped time series (Hyndman et al., 2011). Contemporaneous (cross-sectional) and temporal hierarchies have been considered in the literature, but - except for Kourentzes and Athanasopoulos (2019) - generally these two features have not been fully considered together. Adopting a notation able to simultaneously deal with both forecast reconciliation dimensions, the paper shows two new results: (i) an iterative cross-temporal forecast reconciliation procedure which extends, and overcomes some weaknesses of, the two-step procedure by Kourentzes and Athanasopoulos (2019), and (ii) the closed-form expression of the optimal (in least squares sense) point forecasts which fulfill both contemporaneous and temporal constraints. The feasibility of the proposed procedures, along with first evaluations of their performance as compared to the most performing `single dimension' (either cross-sectional or temporal) forecast reconciliation procedures, is studied through a forecasting experiment on the 95 quarterly time series of the Australian GDP from Income and Expenditure sides considered by Athanasopoulos et al. (2019).