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A Stochastic Recurrence Equation Approach to Stationarity and phi-Mixing of a Class of Nonlinear ARCH Models

Author

Listed:
  • Francisco (F.) Blasques

    () (VU Amsterdam, The Netherlands; Tinbergen Institute, The Netherlands)

  • Marc Nientker

    () (VU Amsterdam, The Netherlands)

Abstract

This article generalises the results of Saidi and Zakoian (2006) to a considerably larger class of nonlinear ARCH models with discontinuities, leverage effects and robust news impact curves. We propose a new method of proof for the existence of a strictly stationary and phi-mixing solution. Moreover, we show that any path converges to this solution. The proof relies on stochastic recurrence equation theory and builds on the work of Bougerol (1993) and Straumann (2005). The assumptions that we need for this approach are less restrictive than those imposed in Saidi and Zakoian (2006) and typically found in Markov chain theory, as they require very little from the distribution of the underlying process. Furthermore, they can be stated in a general setting for random functions on a separable Banach space as is done in Straumann and Mikosch (2006). Finally, we state sufficient conditions for the existence of moments.

Suggested Citation

  • Francisco (F.) Blasques & Marc Nientker, 2017. "A Stochastic Recurrence Equation Approach to Stationarity and phi-Mixing of a Class of Nonlinear ARCH Models," Tinbergen Institute Discussion Papers 17-072/III, Tinbergen Institute.
  • Handle: RePEc:tin:wpaper:20170072
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    File URL: https://papers.tinbergen.nl/17072.pdf
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    References listed on IDEAS

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    1. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    2. Creal, Drew & Koopman, Siem Jan & Lucas, André, 2011. "A Dynamic Multivariate Heavy-Tailed Model for Time-Varying Volatilities and Correlations," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(4), pages 552-563.
    3. Saidi, Youssef & Zakoian, Jean-Michel, 2006. "Stationarity and geometric ergodicity of a class of nonlinear ARCH models," MPRA Paper 61988, University Library of Munich, Germany, revised 2006.
    4. Zakoian, Jean-Michel, 1994. "Threshold heteroskedastic models," Journal of Economic Dynamics and Control, Elsevier, vol. 18(5), pages 931-955, September.
    5. Engle, Robert F & Ng, Victor K, 1993. " Measuring and Testing the Impact of News on Volatility," Journal of Finance, American Finance Association, vol. 48(5), pages 1749-1778, December.
    6. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    7. Bougerol, Philippe & Picard, Nico, 1992. "Stationarity of Garch processes and of some nonnegative time series," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 115-127.
    8. Drew Creal & Siem Jan Koopman & André Lucas, 2013. "Generalized Autoregressive Score Models With Applications," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(5), pages 777-795, August.
    9. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    Cited by:

    1. Francisco (F.) Blasques & Siem Jan (S.J.) Koopman & Marc Nientker, 2018. "A Time-Varying Parameter Model for Local Explosions," Tinbergen Institute Discussion Papers 18-088/III, Tinbergen Institute.

    More about this item

    Keywords

    Ergodicity; GARCH-type models; mixing; nonlinear time series; stationarity; stochastic recurrence equations; threshold models;

    JEL classification:

    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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