Probabilistic forecast reconciliation: cross-temporal framework
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. Classical reconciliation (bottom-up, top-down and middle out) methods address the issue of incoherent forecasts in a hierarchical structure by forecasting only one level and then propagating these forecasts along the other levels of the structure. These approaches ignore useful information available at other levels. Consequently, in the last decade, hierarchical forecasting and forecast reconciliation have significantly evolved to include modern least squares-based reconciliation techniques in the cross-sectional framework, later extended to temporal and cross-temporal framework. Recently, Di Fonzo & Girolimetto (2023) suggested a unified reconciliation step that considers both the cross-sectional and temporal dimensions, instead of dealing with them separately, utilizing the entire cross-temporal hierarchy. However, they focus on point forecasting, and do not consider distributional or probabilistic forecasts (Gneiting & Katzfuss 2014). In the cross-sectional framework, Panagiotelis et al. (2023) made a significant contribution by formalizing cross-sectional probabilistic reconciliation using the geometric framework for point forecast reconciliation and giving useful insights on the computation of the forecasts. In this work we extend the state-of-the-art cross-sectional probabilistic forecast reconciliation to the cross-temporal framework, where temporal constraints are also considered. We expand and unify the notation for cross-sectional, temporal and cross-temporal reconciliation and investigate the probabilistic cross-temporal framework in more detail. A non parametric bootstrap and a parametric Gaussian approach to draw samples from an incoherent cross-temporal distribution are developed. The multi-step residuals are used for a better estimation of the covariance matrix, specifically in the time dimension where the in-sample 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 consider overlapping residuals. A simulation study is performed to investigate the theoretical and empirical proprieties of the different approaches. The methodological contributions are implemented in the FoReco package for R (Girolimetto & Di Fonzo 2023). 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. The results show the effectiveness of the proposed techniques in improving the accuracy of probabilistic forecasts.
References
Di Fonzo, T. & Girolimetto, D. (2023), ‘Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives’, International Journal of Forecasting 39(1), 39–57.
Gneiting, T. & Katzfuss, M. (2014), ‘Probabilistic Forecasting’, Annual Review of Statistics and Its Application 1(1), 125–151.
Girolimetto, D. & Di Fonzo, T. (2023), FoReco: Point Forecast Reconciliation. R package v0.2.6. URL: https://danigiro.github.io/FoReco/
Panagiotelis, A., Gamakumara, P., Athanasopoulos, G. & Hyndman, R. J. (2023), ‘Probabilistic forecast reconciliation: Properties, evaluation and score optimisation’, European Journal of Operational Research 306(2), 693–706.