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2020 Functional ARCH and GARCH models: A Yule-Walker approach
Sebastian Kühnert
Electron. J. Statist. 14(2): 4321-4360 (2020). DOI: 10.1214/20-EJS1778


Conditional heteroskedastic financial time series are commonly modelled by (G)ARCH processes. ARCH$(1)$ and GARCH were recently established in $C[0,1]$ and $L^{2}[0,1]$. This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of (G)ARCH processes for any order in $C[0,1]$ and $L^{p}[0,1]$. It deduces explicit asymptotic upper bounds of estimation errors for the shift term, the complete (G)ARCH operators and the projections of ARCH operators on finite-dimensional subspaces. The operator estimaton is based on Yule-Walker equations, and estimating the GARCH operators also involves a result estimating operators in invertible linear processes being valid beyond the scope of (G)ARCH. Moreover, our results regarding (G)ARCH can be transferred to functional AR(MA).


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Sebastian Kühnert. "Functional ARCH and GARCH models: A Yule-Walker approach." Electron. J. Statist. 14 (2) 4321 - 4360, 2020.


Received: 1 January 2020; Published: 2020
First available in Project Euclid: 12 December 2020

MathSciNet: MR4187136
Digital Object Identifier: 10.1214/20-EJS1778

Primary: 47B38 , 60G10 , 62F12

Keywords: ARCH , ARMA , functional data , functional principal components , functional time series , GARCH , invertible linear processes , parameter estimation; stationary solutions , Yule-Walker equation


Vol.14 • No. 2 • 2020
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