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On joint testing of changes in conditional mean and variance functions of stationary and ergodic time series

Author

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  • K. Ghoudi

    (United Arab Emirates University)

  • N. Laïb

    (CY Cergy Paris Université)

Abstract

Detecting changes in the behavior of time series is an important issue in statistical analysis. This paper introduces a framework for simultaneously testing changes in the conditional mean and variance functions of stationary and ergodic time series. These tests are based on functionals of weighted cumulative sum (CUSUM) processes. In a semi-parametric framework, we establish the weak convergence of this family of processes under the null hypothesis of no change-point, assuming stationarity and allowing the errors to exhibit dependence while forming a martingale difference sequence. Test statistics, derived as functionals of these processes, are shown to have distribution-free limits. This approach unifies many existing change-point tests. Monte Carlo simulations are conducted to illustrate the validity of our approach and to compare the proposed tests with existing methods from the change-point literature. Additionally, the testing procedures are applied to two real data examples.

Suggested Citation

  • K. Ghoudi & N. Laïb, 2025. "On joint testing of changes in conditional mean and variance functions of stationary and ergodic time series," Statistical Papers, Springer, vol. 66(5), pages 1-36, August.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:5:d:10.1007_s00362-025-01728-4
    DOI: 10.1007/s00362-025-01728-4
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    References listed on IDEAS

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    1. Kilani Ghoudi & Naâmane Laïb & Mohamed Chaouch, 2023. "Joint parametric specification checking of conditional mean and volatility in time series models with martingale difference innovations," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 35(1), pages 88-121, January.
    2. Sangyeol Lee & Siyun Park, 2001. "The Cusum of Squares Test for Scale Changes in Infinite Order Moving Average Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(4), pages 625-644, December.
    3. Deng, Ai & Perron, Pierre, 2008. "The Limit Distribution Of The Cusum Of Squares Test Under General Mixing Conditions," Econometric Theory, Cambridge University Press, vol. 24(3), pages 809-822, June.
    4. Sangyeol Lee & Jeongcheol Ha & Okyoung Na & Seongryong Na, 2003. "The Cusum Test for Parameter Change in Time Series Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(4), pages 781-796, December.
    5. Escanciano, J. Carlos, 2010. "Asymptotic Distribution-Free Diagnostic Tests For Heteroskedastic Time Series Models," Econometric Theory, Cambridge University Press, vol. 26(3), pages 744-773, June.
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    9. Haejune Oh & Sangyeol Lee, 2018. "On score vector- and residual-based CUSUM tests in ARMA–GARCH models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 27(3), pages 385-406, August.
    10. Stefan Richter & Weining Wang & Wei Biao Wu, 2023. "Testing for parameter change epochs in GARCH time series," The Econometrics Journal, Royal Economic Society, vol. 26(3), pages 467-491.
    11. Michael Messer, 2022. "Bivariate change point detection: Joint detection of changes in expectation and variance," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 886-916, June.
    12. Ghoudi, Kilani & Kulperger, Reg J. & Rémillard, Bruno, 2001. "A Nonparametric Test of Serial Independence for Time Series and Residuals," Journal of Multivariate Analysis, Elsevier, vol. 79(2), pages 191-218, November.
    13. Kilani Ghoudi & Bruno Rémillard, 2018. "Serial independence tests for innovations of conditional mean and variance models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(1), pages 3-26, March.
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