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Testing for Threshold Diffusion

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  • Fei Su
  • Kung-Sik Chan

Abstract

The threshold diffusion model assumes a piecewise linear drift term and a piecewise smooth diffusion term, which constitutes a rich model for analyzing nonlinear continuous-time processes. We consider the problem of testing for threshold nonlinearity in the drift term. We do this by developing a quasi-likelihood test derived under the working assumption of a constant diffusion term, which circumvents the problem of generally unknown functional form for the diffusion term. The test is first developed for testing for one threshold at which the drift term breaks into two linear functions. We show that under some mild regularity conditions, the asymptotic null distribution of the proposed test statistic is given by the distribution of certain functional of some centered Gaussian process. We develop a computationally efficient method for calibrating the p-value of the test statistic by bootstrapping its asymptotic null distribution. The local power function is also derived, which establishes the consistency of the proposed test. The test is then extended to testing for multiple thresholds. We demonstrate the efficacy of the proposed test by simulations. Using the proposed test, we examine the evidence of nonlinearity in the term structure of a long time series of U.S. interest rates.

Suggested Citation

  • Fei Su & Kung-Sik Chan, 2017. "Testing for Threshold Diffusion," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(2), pages 218-227, April.
  • Handle: RePEc:taf:jnlbes:v:35:y:2017:i:2:p:218-227
    DOI: 10.1080/07350015.2015.1073594
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    References listed on IDEAS

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    1. Dieter Sondermann, 2006. "Introduction to Stochastic Calculus for Finance," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-34837-5, October.
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    Cited by:

    1. Antoine Lejay & Paolo Pigato, 2017. "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data," Working Papers hal-01669082, HAL.
    2. Antoine Lejay & Paolo Pigato, 2019. "A Threshold Model For Local Volatility: Evidence Of Leverage And Mean Reversion Effects On Historical Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    3. Antoine Lejay & Paolo Pigato, 2020. "Maximum likelihood drift estimation for a threshold diffusion," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 609-637, September.
    4. Heiko Rachinger & Edward M. H. Lin & Henghsiu Tsai, 2024. "A bootstrap test for threshold effects in a diffusion process," Computational Statistics, Springer, vol. 39(5), pages 2859-2872, July.
    5. Dingwen Zhang, 2024. "Determining the Number and Values of Thresholds for Multi-regime Threshold Ornstein–Uhlenbeck Processes," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3581-3626, November.
    6. Giannerini, Simone & Goracci, Greta & Rahbek, Anders, 2024. "The validity of bootstrap testing for threshold autoregression," Journal of Econometrics, Elsevier, vol. 239(1).

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