Modeling uncertainty in the covariance matrix for probabilistic forecast reconciliation

Abstract

In minimum trace (MinT) forecast reconciliation, the covariance matrix of the base forecast errors plays a crucial role. Typically, this matrix is estimated and then treated as known. This can lead to an underestimation of the variance of the predictive distribution. To address the problem, we propose a Bayesian reconciliation model that accounts for 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.