# Optimal Forecast Reconciliation with Uncertainty Quantification

arXiv: https://arxiv.org/abs/2402.06480 · HTML: https://mlsanity.com/paper/2402.06480/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2024.

## Built on by (most cited first, 1 of 1)

- [Hierarchical Time Series Forecasting with Robust Reconciliation](https://mlsanity.com/paper/2510.20383/) (2025)

## Abstract (arXiv)

We propose to estimate the weight matrix used for forecast reconciliation as parameters in a general linear model in order to quantify its uncertainty. This implies that forecast reconciliation can be formulated as an orthogonal projection from the space of base-forecast errors into a coherent linear subspace. We use variance decomposition together with the Wishart distribution to derive the central estimator for the forecast-error covariance matrix. In addition, we prove that distance-reducing properties apply to the reconciled forecasts at all levels of the hierarchy as well as to the forecast-error covariance. A covariance matrix for the reconciliation weight matrix is derived, which leads to improved estimates of the forecast-error covariance matrix. We show how shrinkage can be introduced in the formulated model by imposing specific priors on the weight matrix and the forecast-error covariance matrix. The method is illustrated in a simulation study that shows consistent improvements in the log-score. Finally, standard errors for the weight matrix and the variance-separation formula are illustrated using a case study of forecasting electricity load in Sweden.