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

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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.

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  • 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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    1. Barbara Rossi & Atsushi Inoue, 2012. "Out-of-Sample Forecast Tests Robust to the Choice of Window Size," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 30(3), pages 432-453, April.
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    5. West, Kenneth D, 1996. "Asymptotic Inference about Predictive Ability," Econometrica, Econometric Society, vol. 64(5), pages 1067-1084, September.
    6. Peter Reinhard Hansen & Allan Timmermann, 2012. "Choice of Sample Split in Out-of-Sample Forecast Evaluation," CREATES Research Papers 2012-43, Department of Economics and Business Economics, Aarhus University.
    7. Todd Clark & Michael McCracken, 2005. "Evaluating Direct Multistep Forecasts," Econometric Reviews, Taylor & Francis Journals, vol. 24(4), pages 369-404.
    8. Hansen, Bruce E., 1992. "Convergence to Stochastic Integrals for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 8(4), pages 489-500, December.
    9. McCracken, Michael W., 2007. "Asymptotics for out of sample tests of Granger causality," Journal of Econometrics, Elsevier, vol. 140(2), pages 719-752, October.
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    Cited by:

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    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. Zhu, Yinchu & Timmermann, Allan, 2022. "Conditional rotation between forecasting models," Journal of Econometrics, Elsevier, vol. 231(2), pages 329-347.
    10. Timmermann, Allan & Zhu, Yinchu, 2019. "Comparing Forecasting Performance with Panel Data," CEPR Discussion Papers 13746, C.E.P.R. Discussion Papers.
    11. Andrew B. Martinez, 2020. "Extracting Information from Different Expectations," Working Papers 2020-008, The George Washington University, Department of Economics, H. O. Stekler Research Program on Forecasting.
    12. Kothari, Pratik & O’Doherty, Michael S., 2023. "Job postings and aggregate stock returns," Journal of Financial Markets, Elsevier, vol. 64(C).
    13. 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.
    14. Allan Timmermann, 2018. "Forecasting Methods in Finance," Annual Review of Financial Economics, Annual Reviews, vol. 10(1), pages 449-479, November.
    15. Jean-Yves Pitarakis, 2020. "A Novel Approach to Predictive Accuracy Testing in Nested Environments," Papers 2008.08387, arXiv.org, revised Oct 2023.
    16. 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.
    17. 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.
    18. Timmermann, Allan & Zhu, Yinchu, 2021. "Conditional Rotation Between Forecasting Models," CEPR Discussion Papers 15917, C.E.P.R. Discussion Papers.
    19. Qu, Ritong & Timmermann, Allan & Zhu, Yinchu, 2023. "Comparing forecasting performance in cross-sections," Journal of Econometrics, Elsevier, vol. 237(2).
    20. 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.

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    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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