ESSEC Working Papers,
ESSEC Research Center, ESSEC Business School

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.

Keywords: Asymptotic distribution; Sample variance; Functional central limit theorem; Augmented GARCH; Correlation; Sample quantile; Measure of dispersion; Sample mean absolute deviation

JEL-codes: C13; C14; C30

15 pages, June 29, 2019

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