2019/06/21 by Marcel Bräutigam, Bräutigam, Marcel, Marie Kratz +1
Economics, Econometrics and Finance · Mathematics · #Applied mathematics #Asymptotic analysis #Autoregressive conditional heteroskedasticity #Bivariate analysis #Central limit theorem #Computer science #Copula (linguistics) #Dispersion (optics) #Econometrics #Economics #Estimator #FOS: Mathematics #Financial Risk and Volatility Modeling #Independent and identically distributed random variables #Joint probability distribution #Mathematics #Multivariate statistics #Physics #Quantile #Quantile regression #Random variable #Sample (material) #Statistical Distribution Estimation and Applications #Statistical Methods and Inference #Statistics #Statistics Theory (math.ST) #Univariate #Volatility (finance) #Weak convergence #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1906.09332
16 pages, 1 figure, 3 tables; Changes to previous version: Precised conditions in Lemma 7 and Proposition 8. Corrected (and simplified) Step 3 in the proof of Theorem 3
openalex publication_date 2019/06/21 · arxiv created 2019/12/22 · arxiv updated 2019/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, 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 joint (bivariate) functional central limit theorem of 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.