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An IID Test for Functional Time Series with Applications to High-Frequency VIX Index Data

Author

Listed:
  • Xin Huang

    (Commonwealth Bank of Australia, 11 Harbour St, Sydney, NSW 2000, Australia)

  • Han Lin Shang

    (Department of Actuarial Studies and Business Analytics, Macquarie University, Sydney, NSW 2109, Australia)

  • Tak Kuen Siu

    (Department of Actuarial Studies and Business Analytics, Macquarie University, Sydney, NSW 2109, Australia)

Abstract

To address a key issue in functional time series analysis on testing the randomness of an observed series, we propose an IID test for functional time series by generalizing the Brock–Dechert–Scheinkman (BDS) test, which is commonly used for testing nonlinear independence. Similarly to the BDS test, the proposed functional BDS test can be used to evaluate the suitability of prediction models as a model specification test and to detect nonlinear structures as a nonlinearity test. We establish asymptotic results for the test statistic of the proposed test in a generic separate Hilbert space and show that it enjoys the same asymptotic properties as those for the univariate case. To address the practical issue of selecting hyperparameters, we provide the recommended range of the hyperparameters. Using empirical data on the VIX index, empirical studies are conducted that feature the applications of the proposed test to evaluate the adequacy of the fAR ( 1 ) and fGARCH ( 1 , 1 ) models in fitting the daily curves of cumulative intraday returns (CIDR) of the index. The results reveal that the proposed test remedies some shortcomings of the existing independence test. Specifically, the proposed test can detect nonlinear temporal structures, while the existing test can only detect linear structures.

Suggested Citation

  • Xin Huang & Han Lin Shang & Tak Kuen Siu, 2025. "An IID Test for Functional Time Series with Applications to High-Frequency VIX Index Data," Risks, MDPI, vol. 13(2), pages 1-25, January.
  • Handle: RePEc:gam:jrisks:v:13:y:2025:i:2:p:25-:d:1580573
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    References listed on IDEAS

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