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Quantiles as optimal point forecasts

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  • Gneiting, Tilmann

Abstract

Loss functions play a central role in the theory and practice of forecasting. If the loss function is quadratic, the mean of the predictive distribution is the unique optimal point predictor. If the loss is symmetric piecewise linear, any median is an optimal point forecast. Quantiles arise as optimal point forecasts under a general class of economically relevant loss functions, which nests the asymmetric piecewise linear loss, and which we refer to as generalized piecewise linear (GPL). The level of the quantile depends on a generic asymmetry parameter which reflects the possibly distinct costs of underprediction and overprediction. Conversely, a loss function for which quantiles are optimal point forecasts is necessarily GPL. We review characterizations of this type in the work of Thomson, Saerens and Komunjer, and relate to proper scoring rules, incentive-compatible compensation schemes and quantile regression. In the empirical part of the paper, the relevance of decision theoretic guidance in the transition from a predictive distribution to a point forecast is illustrated using the Bank of England’s density forecasts of United Kingdom inflation rates, and probabilistic predictions of wind energy resources in the Pacific Northwest.

Suggested Citation

  • Gneiting, Tilmann, 2011. "Quantiles as optimal point forecasts," International Journal of Forecasting, Elsevier, vol. 27(2), pages 197-207.
  • Handle: RePEc:eee:intfor:v:27:y:2011:i:2:p:197-207
    DOI: 10.1016/j.ijforecast.2009.12.015
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    References listed on IDEAS

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    Cited by:

    1. Zhang, Yao & Wang, Jianxue & Wang, Xifan, 2014. "Review on probabilistic forecasting of wind power generation," Renewable and Sustainable Energy Reviews, Elsevier, vol. 32(C), pages 255-270.
    2. repec:eee:rensus:v:81:y:2018:i:p1:p:1548-1568 is not listed on IDEAS
    3. repec:eee:intfor:v:33:y:2017:i:3:p:662-678 is not listed on IDEAS
    4. Bessa, Ricardo J. & Miranda, V. & Botterud, A. & Zhou, Z. & Wang, J., 2012. "Time-adaptive quantile-copula for wind power probabilistic forecasting," Renewable Energy, Elsevier, vol. 40(1), pages 29-39.
    5. Brenda López Cabrera & Franziska Schulz, 2016. "Time-Adaptive Probabilistic Forecasts of Electricity Spot Prices with Application to Risk Management," SFB 649 Discussion Papers SFB649DP2016-035, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    6. Bartosz Uniejewski & Grzegorz Marcjasz & Rafal Weron, 2017. "On the importance of the long-term seasonal component in day-ahead electricity price forecasting. Part II – Probabilistic forecasting," HSC Research Reports HSC/17/02, Hugo Steinhaus Center, Wroclaw University of Technology.
    7. Gneiting, Tilmann, 2011. "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 746-762.
    8. Xinxin Zhu & Marc Genton & Yingzhong Gu & Le Xie, 2014. "Space-time wind speed forecasting for improved power system dispatch," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 1-25, March.
    9. Andrea Bastianin & Marzio Galeotti & Matteo Manera, 2011. "Forecast Evaluation in Call Centers: Combined Forecasts, Flexible Loss Functions and Economic Criteria," Working Papers 20110301, Università degli Studi di Milano-Bicocca, Dipartimento di Statistica.
    10. Nowotarski, Jakub & Weron, Rafał, 2018. "Recent advances in electricity price forecasting: A review of probabilistic forecasting," Renewable and Sustainable Energy Reviews, Elsevier, vol. 81(P1), pages 1548-1568.
    11. Song, Zhe & Jiang, Yu & Zhang, Zijun, 2014. "Short-term wind speed forecasting with Markov-switching model," Applied Energy, Elsevier, vol. 130(C), pages 103-112.
    12. Brenda López Cabrera & Franziska Schulz, 2017. "Forecasting Generalized Quantiles of Electricity Demand: A Functional Data Approach," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 127-136, January.
    13. Georgios Anastasiades & Patrick McSharry, 2013. "Quantile Forecasting of Wind Power Using Variability Indices," Energies, MDPI, Open Access Journal, vol. 6(2), pages 1-34, February.
    14. Arora, Siddharth & Taylor, James W., 2016. "Forecasting electricity smart meter data using conditional kernel density estimation," Omega, Elsevier, vol. 59(PA), pages 47-59.
    15. Reason Lesego Machete, 2011. "Early Warning with Calibrated and Sharper Probabilistic Forecasts," Papers 1112.6390, arXiv.org, revised Jan 2012.
    16. Sgouropoulos, Nikolaos & Yao, Qiwei & Yastremiz, Claudia, 2015. "Matching a distribution by matching quantiles estimation," LSE Research Online Documents on Economics 57221, London School of Economics and Political Science, LSE Library.
    17. repec:eee:appene:v:210:y:2018:i:c:p:1207-1218 is not listed on IDEAS
    18. Lillestøl, Jostein & Sinding-Larsen, Richard, 2015. "Best estimate reporting with asymmetric loss," Discussion Papers 2015/7, Norwegian School of Economics, Department of Business and Management Science.
    19. Joanna Bruzda, 2016. "Quantile forecasting in operational planning and inventory management – an initial empirical verification," Dynamic Econometric Models, Uniwersytet Mikolaja Kopernika, vol. 16, pages 5-20.
    20. Veiga, Helena & Ruiz, Esther & Gonçalves Mazzeu, Joao Henrique, 2015. "Model uncertainty and the forecast accuracy of ARMA models: A survey," DES - Working Papers. Statistics and Econometrics. WS ws1508, Universidad Carlos III de Madrid. Departamento de Estadística.
    21. Komunjer, Ivana, 2013. "Quantile Prediction," Handbook of Economic Forecasting, Elsevier.
    22. Souhaib Ben Taieb & Raphael Huser & Rob J. Hyndman & Marc G. Genton, 2015. "Probabilistic time series forecasting with boosted additive models: an application to smart meter data," Monash Econometrics and Business Statistics Working Papers 12/15, Monash University, Department of Econometrics and Business Statistics.

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