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A Non-local Perspective on the Power Properties of the CUSUM and CUSUM of Squares Tests for Structural Change

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  • Ai Deng

    () (Department of Economics, Boston University)

  • Pierre Perron

    () (Department of Economics, Boston University)

Abstract

We consider the power properties of the CUSUM and CUSUM of squares tests in the presence of a one-time change in the parameters of a linear regression model. A result due to Ploberger and Krämer (1990) is that the CUSUM of squares test has only trivial asymptotic local power in this case, while the CUSUM test has non-trivial local asymptotic power unless the change is orthogonal to the mean regressor. We argue that such conclusions obtained from a local asymptotic framework are not reliable guides to what happens in finite samples. The approach we take is to derive expansions of the test statistics to order Op(T 1/2) that retain terms related to the magnitude of the change under the alternative hypothesis. This enables us to analyze what happens for non-local to zero breaks. Our theoretical results are able to explain how the power function of the tests can be drastically different depending on whether one deals with a static regression with uncorrelated errors, a static regression with correlated errors, a dynamic regression with lagged dependent variables, or whether a correction for non-Normality is applied in the case of the CUSUM of squares. We discuss in which cases the tests are subject to a non-monotonic power function that goes to zero as the magnitude of the change increases, and uncover some curious properties. All theoretical results are verified to yield good guides to the finite sample power through simulation experiments. We finally highlight the practical importance of our results.

Suggested Citation

  • Ai Deng & Pierre Perron, 2005. "A Non-local Perspective on the Power Properties of the CUSUM and CUSUM of Squares Tests for Structural Change," Boston University - Department of Economics - Working Papers Series WP2005-047, Boston University - Department of Economics.
  • Handle: RePEc:bos:wpaper:wp2005-047
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    References listed on IDEAS

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    1. Ploberger, Werner & Krämer;, Walter, 1990. "The Local Power of the CUSUM and CUSUM of Squares Tests," Econometric Theory, Cambridge University Press, vol. 6(03), pages 335-347, September.
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    4. Ai Deng & Pierre Perron, 2005. "The Limit Distribution of the CUSUM of Square Test Under Genreal MIxing Conditions," Boston University - Department of Economics - Working Papers Series WP2005-046, Boston University - Department of Economics.
    5. Kramer, Walter & Ploberger, Werner & Alt, Raimund, 1988. "Testing for Structural Change in Dynamic Models," Econometrica, Econometric Society, vol. 56(6), pages 1355-1369, November.
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    Cited by:

    1. Juhl, Ted & Xiao, Zhijie, 2009. "Tests for changing mean with monotonic power," Journal of Econometrics, Elsevier, vol. 148(1), pages 14-24, January.
    2. Antonio Rubia & Paulo M.M. Rodrigues, 2010. "The Effects of Additive Outliers and Measurement Errors when Testing for Structural Breaks in Variance," Working Papers w201011, Banco de Portugal, Economics and Research Department.
    3. Pierre Perron & Yohei Yamamoto, 2016. "On the Usefulness or Lack Thereof of Optimality Criteria for Structural Change Tests," Econometric Reviews, Taylor & Francis Journals, vol. 35(5), pages 782-844, May.
    4. Lin Fan & Peter W. Glynn & Markus Pelger, 2018. "Change-Point Testing and Estimation for Risk Measures in Time Series," Papers 1809.02303, arXiv.org.
    5. Xu, Ke-Li, 2013. "Power monotonicity in detecting volatility levels change," Economics Letters, Elsevier, vol. 121(1), pages 64-69.
    6. Yamamoto, Yohei & Tanaka, Shinya, 2015. "Testing for factor loading structural change under common breaks," Journal of Econometrics, Elsevier, vol. 189(1), pages 187-206.
    7. Oka, Tatsushi & Perron, Pierre, 2018. "Testing for common breaks in a multiple equations system," Journal of Econometrics, Elsevier, vol. 204(1), pages 66-85.
    8. Seong Yeon Chang & Pierre Perron, 2018. "A comparison of alternative methods to construct confidence intervals for the estimate of a break date in linear regression models," Econometric Reviews, Taylor & Francis Journals, vol. 37(6), pages 577-601, July.
    9. Seongyeon Chang & Pierre Perron, 2013. "A Comparison of Alternative Methods to Construct to Confidence Intervals for the Estimate of a Break Date in Linear Regression Models," Boston University - Department of Economics - Working Papers Series 2013-023, Boston University - Department of Economics.
    10. Alessandro Casini & Pierre Perron, 2018. "Structural Breaks in Time Series," Papers 1805.03807, arXiv.org.
    11. Kejriwal, Mohitosh, 2009. "Tests for a mean shift with good size and monotonic power," Economics Letters, Elsevier, vol. 102(2), pages 78-82, February.
    12. YAMAZAKI, Daisuke & KUROZUMI, Eiji, 2014. "Improving the Finite Sample Performance of Tests for a Shift in Mean," Discussion Papers 2014-16, Graduate School of Economics, Hitotsubashi University.
    13. Xu, Ke-Li, 2013. "Powerful tests for structural changes in volatility," Journal of Econometrics, Elsevier, vol. 173(1), pages 126-142.
    14. Alessandro Casini, 2018. "Tests for Forecast Instability and Forecast Failure under a Continuous Record Asymptotic Framework," Papers 1803.10883, arXiv.org.
    15. Jean-Yves Pitarakis, 2017. "A Simple Approach for Diagnosing Instabilities in Predictive Regressions," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 79(5), pages 851-874, October.
    16. Nielsen, Bent & Sohkanen, Jouni S., 2011. "Asymptotic Behavior Of The Cusum Of Squares Test Under Stochastic And Deterministic Time Trends," Econometric Theory, Cambridge University Press, vol. 27(04), pages 913-927, August.
    17. Kim, Dukpa & Perron, Pierre, 2009. "Assessing the relative power of structural break tests using a framework based on the approximate Bahadur slope," Journal of Econometrics, Elsevier, vol. 149(1), pages 26-51, April.
    18. Nicholas Mangee, 2016. "Can structural change explain the Meese-Rogoff puzzle?," Journal of Economics and Finance, Springer;Academy of Economics and Finance, vol. 40(2), pages 211-234, April.
    19. Ai Deng & Pierre Perron, 2005. "The Limit Distribution of the CUSUM of Square Test Under Genreal MIxing Conditions," Boston University - Department of Economics - Working Papers Series WP2005-046, Boston University - Department of Economics.
    20. Vanessa Berenguer-Rico & Bent Nielsen, 2015. "Cumulated sum of squares statistics for non-linear and non-stationary regressions," Economics Papers 2015-W09, Economics Group, Nuffield College, University of Oxford.
    21. Cho, Jin Seo & White, Halbert, 2011. "Generalized runs tests for the IID hypothesis," Journal of Econometrics, Elsevier, vol. 162(2), pages 326-344, June.
    22. PERRON, Pierre & YAMAMOTO, Yohei, 2018. "Testing for Changes in Forecasting Performance," Discussion Papers 2018-03, Graduate School of Economics, Hitotsubashi University.
    23. HORIE, Tetsushi & YAMAMOTO, Yohei, 2016. "Testing for Speculative Bubbles in Large-Dimensional Financial Panel Data Sets," Discussion Papers 2016-04, Graduate School of Economics, Hitotsubashi University.
    24. JIANG, Peiyun & KUROZUMI, Eiji, 2017. "Power Properties of the Modified CUSUM Tests," Discussion Papers 2017-05, Graduate School of Economics, Hitotsubashi University.

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    Keywords

    Change-point; Mean shift; Local asymptotic power; Recursive residuals; Dynamic models;

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