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Calibration tests for multivariate Gaussian forecasts

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  • Wei, Wei
  • Balabdaoui, Fadoua
  • Held, Leonhard

Abstract

Forecasts by nature should take the form of probabilistic distributions. Calibration, the statistical consistency of forecast distributions and observations, is a central property of good probabilistic forecasts. Calibration of univariate forecasts has been widely discussed, and significance tests are commonly used to investigate whether a prediction model is miscalibrated. However, calibration tests for multivariate forecasts are rare. In this paper, we propose calibration tests for multivariate Gaussian forecasts based on two types of the Dawid–Sebastiani score (DSS): the multivariate DSS (mDSS) and the individual DSS (iDSS). Analytic results and simulation studies show that the tests have sufficient power to detect miscalibrated forecasts with incorrect mean or incorrect variance. But for forecasts with incorrect correlation coefficients, only the tests based on mDSS are sensitive to miscalibration. As an illustration, we apply the methodology to weekly data on Norovirus disease incidence among males and females in Germany, in 2011–2014. The results further show that tests for multivariate forecasts are useful tools and superior to univariate calibration tests for correlated multivariate forecasts.

Suggested Citation

  • Wei, Wei & Balabdaoui, Fadoua & Held, Leonhard, 2017. "Calibration tests for multivariate Gaussian forecasts," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 216-233.
    • Handle: RePEc:eee:jmvana:v:154:y:2017:i:c:p:216-233
    DOI: 10.1016/j.jmva.2016.11.005
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    References listed on IDEAS

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    1. Diebold, Francis X & Gunther, Todd A & Tay, Anthony S, 1998. "Evaluating Density Forecasts with Applications to Financial Risk Management," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 863-883, November.
    2. Wei Wei & Leonhard Held, 2014. "Calibration tests for count data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(4), pages 787-805, December.
    3. Tilmann Gneiting & Larissa Stanberry & Eric Grimit & Leonhard Held & Nicholas Johnson, 2008. "Rejoinder on: Assessing probabilistic forecasts of multivariate quantities, with an application to ensemble predictions of surface winds," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(2), pages 256-264, August.
    4. Tilmann Gneiting & Larissa Stanberry & Eric Grimit & Leonhard Held & Nicholas Johnson, 2008. "Assessing probabilistic forecasts of multivariate quantities, with an application to ensemble predictions of surface winds," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(2), pages 211-235, August.
    5. Tilmann Gneiting & Fadoua Balabdaoui & Adrian E. Raftery, 2007. "Probabilistic forecasts, calibration and sharpness," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(2), pages 243-268, April.
    6. L. Held & K. Rufibach & F. Balabdaoui, 2010. "A Score Regression Approach to Assess Calibration of Continuous Probabilistic Predictions," Biometrics, The International Biometric Society, vol. 66(4), pages 1295-1305, December.
    7. Francis X. Diebold & Jinyong Hahn & Anthony S. Tay, 1999. "Multivariate Density Forecast Evaluation And Calibration In Financial Risk Management: High-Frequency Returns On Foreign Exchange," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 661-673, November.
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    3. Bhaumik, Dulal K. & Nordgren, Rachel K., 2019. "Prediction and calibration for multiple correlated variables," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 313-327.

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