Cross-Temporal Probabilistic Forecast Reconciliation
Forecast reconciliation is a post-forecasting process that maps a set of incoherent forecasts into coherent forecasts which satisfy a given set of linear constraints for a multivariate time series. In this paper we extend the state-of-the-art cross-sectional probabilistic forecast reconciliation to the cross-temporal framework, where temporal constraints are also considered. We develop a non parametric bootstrap and a parametric Gaussian approach to draw samples from an incoherent cross-temporal distribution. The multi-step residuals are used for a better estimation of the covariance matrix, specifically in the time dimension where the usual one-step residuals fail. To address the high-dimensionality issues, we propose four alternatives for the covariance matrix by exploiting the two-fold nature (cross-sectional and temporal) of the cross-temporal structure and we introduce the idea of overlapping residuals. A simulation study is performed to investigate the theoretical and empirical proprieties of the different approaches. Finally, we consider two empirical forecasting experiments using the Australian GDP and the Australian Tourism Demand datasets to evaluate the feasibility and the performance of the proposed procedures. For these applications, the optimal cross-temporal reconciliation approaches significantly outperform the base forecasts according to the Continuous Ranked Probability Score and to the Energy Score. These findings demonstrate the effectiveness of the proposed techniques in improving the accuracy of probabilistic forecasting models. Overall, the paper expands and unifies the notation for cross-sectional, temporal and cross-temporal reconciliation and investigate the probabilistic cross-temporal framework in more detail.