Bivariate FCLT for the Sample Quantile and Measures of Dispersion for Augmented GARCH(p, q) processes
Marcel Bräutigam and
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Marcel Bräutigam: LabEx MME-DII - UCP - Université de Cergy Pontoise - Université Paris-Seine, ESSEC Business School - 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: SID - Information Systems, Decision Sciences and Statistics Department - Essec Business School, LabEx MME-DII - UCP - Université de Cergy Pontoise - Université Paris-Seine
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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.
Keywords: asymptotic distribution; (sample) variance; functional central limit theorem; (augmented) GARCH; correlation; (sample) quantile; measure of dispersion; (sample) mean absolute deviation (search for similar items in EconPapers)
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