Probabilistic forecast reconciliation under the Gaussian framework

TL;DR

This paper investigates probabilistic forecast reconciliation under the Gaussian assumption, proving that MinT optimizes the logarithmic score and outperforms OLS in this context, supported by simulations and real data analysis.

Contribution

It provides theoretical proof that MinT minimizes the logarithmic score for Gaussian predictive distributions, extending point forecast reconciliation methods to probabilistic forecasts.

Findings

MinT minimizes the logarithmic score for Gaussian distributions.

MinT's logarithmic score is lower than OLS for each marginal predictive density.

Simulation and real data confirm theoretical results.

Abstract

Forecast reconciliation of multivariate time series is the process of mapping a set of incoherent forecasts into coherent forecasts to satisfy a given set of linear constraints. Commonly used projection matrix based approaches for point forecast reconciliation are OLS (ordinary least squares), WLS (weighted least squares), and MinT (minimum trace). Even though point forecast reconciliation is a well-established field of research, the literature on generating probabilistic forecasts subject to linear constraints is somewhat limited. Available methods follow a two-step procedure. Firstly, it draws future sample paths from the univariate models fitted to each series in the collection (which are incoherent). Secondly, it uses a projection matrix based approach or empirical copula based reordering approach to account for contemporaneous correlations and linear constraints. The projection matrices are estimated either by optimizing a scoring rule such as energy or variogram score, or simply using a projection matrix derived for point forecast reconciliation. This paper proves that (a) if the incoherent predictive distribution is Gaussian then MinT minimizes the logarithmic scoring rule; and (b) the logarithmic score of MinT for each marginal predictive density is smaller than that of OLS. We show these theoretical results using a set of simulation studies. We also evaluate them using the Australian domestic tourism data set.