Insights into regression-based cross-temporal forecast reconciliation

TL;DR

This paper investigates methods for ensuring consistent forecasts across different time levels and cross-sectional units, analyzing theoretical properties and demonstrating improvements in accuracy and computational efficiency through empirical experiments.

Contribution

It provides a comprehensive framework for cross-temporal forecast reconciliation, exploring the relationships between different approaches and identifying conditions for optimal convergence.

Findings

Sequential reconciliation can be equivalent to fully coherent methods under certain conditions.

Iterative reconciliation converges to the optimal solution for specific error covariance patterns.

Significant improvements in memory and computation time are achieved in high-dimensional settings.

Abstract

Cross-temporal forecast reconciliation aims to ensure consistency across forecasts made at different temporal and cross-sectional levels. We explore the relationships between sequential, iterative, and optimal combination approaches, and discuss the conditions under which a sequential reconciliation approach (either first-cross-sectional-then-temporal, or first-temporal-then-cross-sectional) is equivalent to a fully (i.e., cross-temporally) coherent iterative heuristic. Furthermore, we show that for specific patterns of the error covariance matrix in the regression model on which the optimal combination approach grounds, iterative reconciliation naturally converges to the optimal combination solution, regardless the order of application of the uni-dimensional cross-sectional and temporal reconciliation approaches. Theoretical and empirical properties of the proposed approaches are investigated through a forecasting experiment using a dataset of hourly photovoltaic power generation. The study presents a comprehensive framework for understanding and enhancing cross-temporal forecast reconciliation, considering both forecast accuracy and the often overlooked computational aspects, showing that significant improvement can be achieved in terms of memory space and computation time, two particularly important aspects in the high-dimensional contexts that usually arise in cross-temporal forecast reconciliation.