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Cusum tests for changes in the Hurst exponent and volatility of fractional Brownian motion

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  • Bibinger, Markus

Abstract

In this letter, we construct cusum change-point tests for the Hurst exponent and the volatility of a discretely observed fractional Brownian motion. As a statistical application of the functional Breuer–Major theorems by Bégyn (2007) and Nourdin and Nualart (2019), we show under infill asymptotics consistency of the tests and weak convergence to the Kolmogorov–Smirnov law under the no-change hypothesis. The test is feasible and pivotal in the sense that it is based on a statistic and critical values which do not require knowledge of any parameter values. Consistent estimation of the break date under the alternative hypothesis is established. We demonstrate the finite-sample properties in simulations and a data example.

Suggested Citation

  • Bibinger, Markus, 2020. "Cusum tests for changes in the Hurst exponent and volatility of fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 161(C).
    • Handle: RePEc:eee:stapro:v:161:y:2020:i:c:s0167715220300286
    DOI: 10.1016/j.spl.2020.108725
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    1. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
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    3. Jean-François Coeurjolly, 2001. "Estimating the Parameters of a Fractional Brownian Motion by Discrete Variations of its Sample Paths," Statistical Inference for Stochastic Processes, Springer, vol. 4(2), pages 199-227, May.
    4. Bibinger, Markus & Madensoy, Mehmet, 2019. "Change-point inference on volatility in noisy Itô semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 4878-4925.
    5. Coeurjolly, Jean-Francois, 2000. "Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 5(i07).
    6. Lavancier, Frédéric & Leipus, Remigijus & Philippe, Anne & Surgailis, Donatas, 2013. "Detection Of Nonconstant Long Memory Parameter," Econometric Theory, Cambridge University Press, vol. 29(5), pages 1009-1056, October.
    7. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
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    Cited by:

    1. Wang, Xiaohu & Xiao, Weilin & Yu, Jun, 2023. "Modeling and forecasting realized volatility with the fractional Ornstein–Uhlenbeck process," Journal of Econometrics, Elsevier, vol. 232(2), pages 389-415.

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