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Cross-temporal Probabilistic Forecast Reconciliation

Author

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  • Daniele Girolimetto
  • George Athanasopoulos
  • Tommaso Di Fonzo
  • Rob J Hyndman

Abstract

Forecast reconciliation is a post-forecasting process that involves transforming 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 current state-of-the-art cross-sectional probabilistic forecast reconciliation approach to encompass a cross-temporal framework, where temporal constraints are also applied. Our proposed methodology employs both parametric Gaussian and non-parametric bootstrap approaches to draw samples from an incoherent crosstemporal distribution. To improve the estimation of the forecast error covariance matrix, we propose using multi-step residuals, especially in the time dimension where the usual one-step residuals fail. To address high-dimensionality issues, we present four alternatives for the covariance matrix, where we exploit the twofold nature (cross-sectional and temporal) of the cross-temporal structure, and introduce the idea of overlapping residuals. We evaluate the proposed methods through a detailed simulation study that investigates their theoretical and empirical properties. We further assess the effectiveness of the proposed cross-temporal reconciliation approach by applying it to two empirical forecasting experiments, using the Australian GDP and the Australian Tourism Demand datasets. For both applications, we show that the optimal cross-temporal reconciliation approaches significantly outperform the incoherent base forecasts in terms of the Continuous Ranked Probability Score and the Energy Score. Overall, our study expands and unifies the notation for cross-sectional, temporal and cross-temporal reconciliation, thus extending and deepening the probabilistic cross-temporal framework. The results highlight the potential of the proposed cross-temporal forecast reconciliation methods in improving the accuracy of probabilistic forecasting models.

Suggested Citation

  • Daniele Girolimetto & George Athanasopoulos & Tommaso Di Fonzo & Rob J Hyndman, 2023. "Cross-temporal Probabilistic Forecast Reconciliation," Monash Econometrics and Business Statistics Working Papers 6/23, Monash University, Department of Econometrics and Business Statistics.
  • Handle: RePEc:msh:ebswps:2023-6
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    File URL: https://www.monash.edu/business/ebs/research/publications/ebs/2023/wp06-2023.pdf
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    References listed on IDEAS

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    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Hyndman, Rob J. & Khandakar, Yeasmin, 2008. "Automatic Time Series Forecasting: The forecast Package for R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 27(i03).
    3. Kourentzes, Nikolaos & Athanasopoulos, George, 2019. "Cross-temporal coherent forecasts for Australian tourism," Annals of Tourism Research, Elsevier, vol. 75(C), pages 393-409.
    4. Jeon, Jooyoung & Panagiotelis, Anastasios & Petropoulos, Fotios, 2019. "Probabilistic forecast reconciliation with applications to wind power and electric load," European Journal of Operational Research, Elsevier, vol. 279(2), pages 364-379.
    5. Shanika L. Wickramasuriya & George Athanasopoulos & Rob J. Hyndman, 2019. "Optimal Forecast Reconciliation for Hierarchical and Grouped Time Series Through Trace Minimization," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(526), pages 804-819, April.
    6. Schäfer Juliane & Strimmer Korbinian, 2005. "A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-32, November.
    7. Spiliotis, Evangelos & Petropoulos, Fotios & Kourentzes, Nikolaos & Assimakopoulos, Vassilios, 2020. "Cross-temporal aggregation: Improving the forecast accuracy of hierarchical electricity consumption," Applied Energy, Elsevier, vol. 261(C).
    8. Souhaib Ben Taieb & James W. Taylor & Rob J. Hyndman, 2017. "Coherent Probabilistic Forecasts for Hierarchical Time Series," Monash Econometrics and Business Statistics Working Papers 3/17, Monash University, Department of Econometrics and Business Statistics.
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    Cited by:

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    Keywords

    coherent; GDP; linear constraints; multivariate time series; temporal aggregation; tourism flows;
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