Modeling the uncertainty on the covariance matrix for probabilistic forecast reconciliation

TL;DR

This paper introduces a Bayesian approach to forecast reconciliation that models covariance matrix uncertainty, resulting in more accurate predictive distributions with improved prediction intervals.

Contribution

It proposes a Bayesian MinT reconciliation method using an Inverse-Wishart prior, leading to multivariate t-distributions and better uncertainty quantification.

Findings

Improved prediction intervals over traditional MinT reconciliation.

Closed-form multivariate t-distribution for reconciled forecasts.

Consistent performance across three tourism datasets.

Abstract

In minimum trace (MinT) forecast reconciliation, the covariance matrix of the base forecasts errors plays a crucial role. Typically, this matrix is estimated and then treated as known. This can lead to underestimation of the variance of the predictive distribution. To address the problem, we propose a Bayesian reconciliation model that accounts for the uncertainty in the estimation of the covariance matrix. By adopting an Inverse-Wishart prior and assuming Gaussian residuals, the reconciled predictive distribution follows a multivariate t-distribution, obtained in closed-form, rather than a multivariate Gaussian distribution. We evaluate our method on three tourism-related datasets, including a new publicly available dataset. Empirical results show that our approach consistently improves prediction intervals compared to MinT reconciliation.