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A Larch Vector Valued Process

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  • Paul Doukhan

    (Crest)

  • Gilles Teyssière

    (Crest)

  • Pablo Winant

    (Crest)

Abstract

We introduce a vector version of the LARCH(1) equation yielding asimple approach to various models like bilinear or GARCH models. To this aim weprovide an explicit chaotic expansion of a solution for this ARCH(1) equation, andshow the uniqueness of this solution under reasonable conditions. Independent orN-Markov approximations of this process allow to simulate a trajectory or to derivebounds for their weak dependence coefficients as defined by Doukhan and Louhichi(1999). We finally consider a long range dependent version of this model; in thiscase we provide an existence and uniqueness result.

Suggested Citation

  • Paul Doukhan & Gilles Teyssière & Pablo Winant, 2005. "A Larch Vector Valued Process," Working Papers 2005-49, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2005-49
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    References listed on IDEAS

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    1. Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(1), pages 122-150, February.
    2. Giraitis, Liudas & Surgailis, Donatas, 0. "ARCH-type bilinear models with double long memory," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 275-300, July.
    3. Giraitis, Liudas & Kokoszka, Piotr & Leipus, Remigijus, 2000. "Stationary Arch Models: Dependence Structure And Central Limit Theorem," Econometric Theory, Cambridge University Press, vol. 16(1), pages 3-22, February.
    4. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
    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. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    7. Dedecker, Jérôme & Doukhan, Paul, 2003. "A new covariance inequality and applications," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 63-80, July.
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    Cited by:

    1. Paul Doukhan & Jean-David Fermanian & Gabriel Lang, 2009. "An empirical central limit theorem with applications to copulas under weak dependence," Statistical Inference for Stochastic Processes, Springer, vol. 12(1), pages 65-87, February.
    2. Paul Doukhan & Silika Prohl & Christian Robert, 2011. "Subsampling weakly dependent time series and application to extremes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 447-479, November.
    3. Rohit Deo & Meng-Chen Hsieh & Clifford M. Hurvich & Philippe Soulier, 2007. "Long Memory in Nonlinear Processes," Papers 0706.1836, arXiv.org.
    4. Doukhan, Paul & Wintenberger, Olivier, 2008. "Weakly dependent chains with infinite memory," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 1997-2013, November.

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