Non-Negative Forecast Reconciliation: Optimal Methods and Operational Solutions

Daniele Girolimetto ()
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Daniele Girolimetto: Department of Statistical Sciences, University of Padova, 35121 Padova, Italy

Forecasting, 2025, vol. 7, issue 4, 1-29

Abstract: In many different applications such as retail, energy, and tourism, forecasts for a set of related time series must satisfy both linear and non-negativity constraints, as negative values are meaningless in practice. Standard regression-based reconciliation approaches achieve coherence with linear constraints, but may generate negative forecasts, reducing interpretability and usability. This paper develops and evaluates three alternative strategies for non-negative forecast reconciliation. First, reconciliation is formulated as a non-negative least squares problem and solved with the operator splitting quadratic program, allowing flexible inclusion of additional constraints. Second, we propose an iterative non-negative reconciliation with immutable forecasts, offering a practical optimization-based alternative. Third, we investigate a family of set-negative-to-zero heuristics that achieve efficiency and interpretability at minimal computational cost. Using the Australian Tourism Demand dataset, we compare these approaches in terms of forecast accuracy and computation time. The results show that non-negativity constraints consistently improve accuracy compared to base forecasts. Overall, set-negative-to-zero achieve near-optimal performance with negligible computation time, the block principal pivoting algorithm provides a good accuracy–efficiency compromise, and the operator splitting quadratic program offers flexibility for incorporating additional constraints in large-scale applications.

Keywords: forecast reconciliation; non-negative forecasts; hierarchical time series; non-negative least squares; set-negative-to-zero (search for similar items in EconPapers)
JEL-codes: A1 B4 C0 C1 C2 C3 C4 C5 C8 M0 Q2 Q3 Q4 (search for similar items in EconPapers)
Date: 2025
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