# Point and probabilistic forecast reconciliation for general linearly constrained multiple time series

arXiv: https://arxiv.org/abs/2305.05330 · HTML: https://mlsanity.com/paper/2305.05330/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 May 2023.

## Built on by (most cited first, 5 of 5)

- [Hierarchical Forecasting at Scale](https://mlsanity.com/paper/2310.12809/) (2023)
- [Cross-Temporal Forecast Reconciliation at Digital Platforms with Machine Learning](https://mlsanity.com/paper/2402.09033/) (2024)
- [FoReco and FoRecoML: A Unified Toolbox for Forecast Reconciliation in R](https://mlsanity.com/paper/2604.27696/) (2026)
- [Long-Horizon Forecasting of Complete Financial Statements with Forma](https://mlsanity.com/paper/2608.11327/) (2026)
- [SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization](https://mlsanity.com/paper/2609.33515/) (2026)

## Abstract (arXiv)

Forecast reconciliation is the post-forecasting process aimed to revise a set of incoherent base forecasts into coherent forecasts in line with given data structures. Most of the point and probabilistic regression-based forecast reconciliation results ground on the so called "structural representation" and on the related unconstrained generalized least squares reconciliation formula. However, the structural representation naturally applies to genuine hierarchical/grouped time series, where the top- and bottom-level variables are uniquely identified. When a general linearly constrained multiple time series is considered, the forecast reconciliation is naturally expressed according to a projection approach. While it is well known that the classic structural reconciliation formula is equivalent to its projection approach counterpart, so far it is not completely understood if and how a structural-like reconciliation formula may be derived for a general linearly constrained multiple time series. Such an expression would permit to extend reconciliation definitions, theorems and results in a straightforward manner. In this paper, we show that for general linearly constrained multiple time series it is possible to express the reconciliation formula according to a "structural-like" approach that keeps distinct free and constrained, instead of bottom and upper (aggregated), variables, establish the probabilistic forecast reconciliation framework, and apply these findings to obtain fully reconciled point and probabilistic forecasts for the aggregates of the Australian GDP from income and expenditure sides, and for the European Area GDP disaggregated by income, expenditure and output sides and by 19 countries.