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Regular variation of order 1 nonlinear AR-ARCH models

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  • Cline, Daren B.H.

Abstract

We prove both geometric ergodicity and regular variation of the stationary distribution for a class of nonlinear stochastic recursions that includes nonlinear AR-ARCH models of order 1. The Lyapounov exponent for the model, the index of regular variation and the spectral measure for the regular variation all are characterized by a simple two-state Markov chain.

Suggested Citation

  • Cline, Daren B.H., 2007. "Regular variation of order 1 nonlinear AR-ARCH models," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 840-861, July.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:7:p:840-861
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    References listed on IDEAS

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    1. Cline, Daren B. H. & Pu, Huay-min H., 2002. "A note on a simple Markov bilinear stochastic process," Statistics & Probability Letters, Elsevier, vol. 56(3), pages 283-288, February.
    2. Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 2002. "Regular variation of GARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 95-115, May.
    3. de Saporta, BenoI^te, 2005. "Tail of the stationary solution of the stochastic equation Yn+1=anYn+bn with Markovian coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 1954-1978, December.
    4. Borkovec, Milan, 2000. "Extremal behavior of the autoregressive process with ARCH(1) errors," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 189-207, February.
    5. 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.
    6. Hult, Henrik & Lindskog, Filip, 2005. "Extremal behavior of regularly varying stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 115(2), pages 249-274, February.
    7. Cline, Daren B. H. & Pu, Huay-min H., 1998. "Verifying irreducibility and continuity of a nonlinear time series," Statistics & Probability Letters, Elsevier, vol. 40(2), pages 139-148, September.
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    Cited by:

    1. Gawon Yoon, 2010. "Nonlinear mean-reversion to purchasing power parity: exponential smooth transition autoregressive models and stochastic unit root processes," Applied Economics, Taylor & Francis Journals, vol. 42(4), pages 489-496.
    2. Arash Nademi & Rahman Farnoosh, 2014. "Mixtures of autoregressive-autoregressive conditionally heteroscedastic models: semi-parametric approach," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(2), pages 275-293, February.

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