Probabilistic forecast reconciliation: Properties, evaluation and score optimisation

Anastasios Panagiotelis, Puwasala Gamakumara, George Athanasopoulos and Rob Hyndman

European Journal of Operational Research, 2023, vol. 306, issue 2, 693-706

Abstract: We develop a framework for forecasting multivariate data that follow known linear constraints. This is particularly common in forecasting where some variables are aggregates of others, commonly referred to as hierarchical time series, but also arises in other prediction settings. For point forecasting, an increasingly popular technique is reconciliation, whereby forecasts are made for all series (so-called base forecasts) and subsequently adjusted to cohere with the constraints. We extend reconciliation from point forecasting to probabilistic forecasting. A novel definition of reconciliation is developed and used to construct densities and draw samples from a reconciled probabilistic forecast. In the elliptical case, we prove that true predictive distributions can be recovered using reconciliation even when the location and scale of base predictions are chosen arbitrarily. Reconciliation weights are estimated to optimise energy or variogram score. The log score is not considered since it is improper when comparing unreconciled to reconciled forecasts, a result also proved in this paper. Due to randomness in the objective function, optimisation uses stochastic gradient descent. This method improves upon base forecasts in simulated and empirical data, particularly when the base forecasting models are severely misspecified. For milder misspecification, extending popular reconciliation methods for point forecasting results in similar performance to score optimisation.

Keywords: Forecasting; Scoring rules; Hierarchical time series; Stochastic gradient descent (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (9)

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Related works:
Working Paper: Probabilistic Forecast Reconciliation: Properties, Evaluation and Score Optimisation (2020)
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Persistent link: https://EconPapers.repec.org/RePEc:eee:ejores:v:306:y:2023:i:2:p:693-706

DOI: 10.1016/j.ejor.2022.07.040

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