An IID test for functional time series with applications to high-frequency VIX index data

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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.
Original languageEnglish
Article number25
Pages (from-to)1-25
Number of pages25
JournalRisks
Volume13
Issue number2
DOIs
Publication statusPublished - Feb 2025

Bibliographical note

© 2025 by the authors. Licensee MDPI, Basel, Switzerland. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

Keywords

  • BDS test
  • functional GARCH model
  • functional autoregressive model
  • independence test
  • VIX index

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