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

Marcel Bräutigam (1, 2, 3) , Marie Kratz (2, 1)

1 LabEx MME-DII
2 ESSEC Business School
3 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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.

Mots clés: functional central limit theorem; (augmented) GARCH; correlation; (sample) quantile; measure of dispersion; (sample) mean absolute deviation; (sample) variance; asymptotic distribution

Domaines: Statistiques [math.ST]; Économie et finance quantitative [q-fin]

Marcel Bräutigam, Marie Kratz. Bivariate FCLT for the Sample Quantile and Measures of Dispersion for Augmented GARCH(p, q) processes. 2019. ⟨hal-02176276⟩