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Bivariate FCLT for the Sample Quantile and Measures of Dispersion for Augmented GARCH(p, q) processes

Author

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  • Marcel Bräutigam

    (LabEx MME-DII - UCP - Université de Cergy Pontoise - Université Paris-Seine, ESSEC Business School, LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique)

  • Marie Kratz

    (ESSEC Business School, LabEx MME-DII - UCP - Université de Cergy Pontoise - Université Paris-Seine)

Abstract

In this note, we build upon the asymptotic theory for GARCH processes, considering the general class of augmented GARCH(p, q) processes. Our contribution is to complement the well-known univariate asymptotics by providing a bivariate functional central limit theorem between the sample quantile and the r-th absolute centred sample moment. This extends existing results in the case of identically and independently distributed random variables. We show that the conditions for the convergence of the estimators in the univariate case suffice even for the joint bivariate asymptotics. We illustrate the general results with various specific examples from the class of augmented GARCH(p, q) processes and show explicitly under which conditions on the moments and parameters of the process the joint asymptotics hold.

Suggested Citation

  • Marcel Bräutigam & Marie Kratz, 2019. "Bivariate FCLT for the Sample Quantile and Measures of Dispersion for Augmented GARCH(p, q) processes," Working Papers hal-02176276, HAL.
    • Handle: RePEc:hal:wpaper:hal-02176276
    Note: View the original document on HAL open archive server: https://hal.science/hal-02176276v1
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    References listed on IDEAS

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    Cited by:

    1. Marcel Brautigam & Marie Kratz, 2020. "The Impact of the Choice of Risk and Dispersion Measure on Procyclicality," Papers 2001.00529, arXiv.org.

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