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Equivalence Between Out‐of‐Sample Forecast Comparisons and Wald Statistics

Author

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  • Peter Reinhard Hansen
  • Allan Timmermann

Abstract

We demonstrate the asymptotic equivalence between commonly used test statistics for out‐of‐sample forecasting performance and conventional Wald statistics. This equivalence greatly simplifies the computational burden of calculating recursive out‐of‐sample test statistics and their critical values. For the case with nested models, we show that the limit distribution, which has previously been expressed through stochastic integrals, has a simple representation in terms of χ-super-2‐distributed random variables and we derive its density. We also generalize the limit theory to cover local alternatives and characterize the power properties of the test.

Suggested Citation

  • Peter Reinhard Hansen & Allan Timmermann, 2015. "Equivalence Between Out‐of‐Sample Forecast Comparisons and Wald Statistics," Econometrica, Econometric Society, vol. 83, pages 2485-2505, November.
    • Handle: RePEc:wly:emetrp:v:83:y:2015:i::p:2485-2505
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    Cited by:

    1. Wang, Cindy S.H. & Fan, Rui & Xie, Yiqiang, 2023. "Market systemic risk, predictability and macroeconomics news," Finance Research Letters, Elsevier, vol. 56(C).
    2. Firmin Doko Tchatoka & Qazi Haque, 2023. "On bootstrapping tests of equal forecast accuracy for nested models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 42(7), pages 1844-1864, November.
    3. Philippe Goulet Coulombe & Maxime Leroux & Dalibor Stevanovic & Stéphane Surprenant, 2022. "How is machine learning useful for macroeconomic forecasting?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(5), pages 920-964, August.
    4. Martin, Ian W.R. & Nagel, Stefan, 2022. "Market efficiency in the age of big data," Journal of Financial Economics, Elsevier, vol. 145(1), pages 154-177.
    5. Christis Katsouris, 2023. "Limit Theory under Network Dependence and Nonstationarity," Papers 2308.01418, arXiv.org, revised Aug 2023.
    6. Hunt, Ian, 2022. "In-sample tests of predictability are superior to pseudo-out-of-sample tests, even when data mining," International Journal of Forecasting, Elsevier, vol. 38(3), pages 872-877.
    7. Andrew Martinez, 2017. "Testing for Differences in Path Forecast Accuracy: Forecast-Error Dynamics Matter," Working Papers (Old Series) 1717, Federal Reserve Bank of Cleveland.
    8. Ciner, Cetin, 2019. "Do industry returns predict the stock market? A reprise using the random forest," The Quarterly Review of Economics and Finance, Elsevier, vol. 72(C), pages 152-158.
    9. Arasteh, Abdollah, 2025. "A data-driven prediction method for multi-period portfolio optimization using the real options approach," Finance Research Letters, Elsevier, vol. 80(C).
    10. Zhu, Yinchu & Timmermann, Allan, 2022. "Conditional rotation between forecasting models," Journal of Econometrics, Elsevier, vol. 231(2), pages 329-347.
    11. Schreindorfer, David & Sichert, Tobias, 2025. "Conditional risk and the pricing kernel," Journal of Financial Economics, Elsevier, vol. 171(C).
    12. Timmermann, Allan & Zhu, Yinchu, 2019. "Comparing Forecasting Performance with Panel Data," CEPR Discussion Papers 13746, C.E.P.R. Discussion Papers.
    13. Andrew B. Martinez, 2020. "Extracting Information from Different Expectations," Working Papers 2020-008, The George Washington University, The Center for Economic Research.
    14. Kothari, Pratik & O’Doherty, Michael S., 2023. "Job postings and aggregate stock returns," Journal of Financial Markets, Elsevier, vol. 64(C).
    15. Allan Timmermann, 2018. "Forecasting Methods in Finance," Annual Review of Financial Economics, Annual Reviews, vol. 10(1), pages 449-479, November.
    16. Pitarakis, Jean-Yves, 2025. "A Novel Approach To Predictive Accuracy Testing In Nested Environments," Econometric Theory, Cambridge University Press, vol. 41(1), pages 35-78, February.
    17. Gonçalves, Sílvia & McCracken, Michael W. & Perron, Benoit, 2017. "Tests of equal accuracy for nested models with estimated factors," Journal of Econometrics, Elsevier, vol. 198(2), pages 231-252.
    18. Rui Liu, 2019. "Forecasting Bond Risk Premia with Unspanned Macroeconomic Information," Quarterly Journal of Finance (QJF), World Scientific Publishing Co. Pte. Ltd., vol. 9(01), pages 1-62, March.
    19. Knut Are Aastveit & Claudia Foroni & Francesco Ravazzolo, 2017. "Density Forecasts With Midas Models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(4), pages 783-801, June.
    20. Timmermann, Allan & Zhu, Yinchu, 2021. "Conditional Rotation Between Forecasting Models," CEPR Discussion Papers 15917, C.E.P.R. Discussion Papers.
    21. Qu, Ritong & Timmermann, Allan & Zhu, Yinchu, 2023. "Comparing forecasting performance in cross-sections," Journal of Econometrics, Elsevier, vol. 237(2).
    22. Daniel Borup & Martin Thyrsgaard, 2017. "Statistical tests for equal predictive ability across multiple forecasting methods," CREATES Research Papers 2017-19, Department of Economics and Business Economics, Aarhus University.
    23. Alessandro Morico & Ovidijus Stauskas, 2025. "Robust Tests for Factor-Augmented Regressions with an Application to the novel EA-MD-QD Dataset," Papers 2504.08455, arXiv.org, revised Nov 2025.

    More about this item

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

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