Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

Functional ARCH and GARCH Models: A Yule-Walker Approach

Version 1 : Received: 11 December 2019 / Approved: 12 December 2019 / Online: 12 December 2019 (05:16:14 CET)
Version 2 : Received: 23 January 2020 / Approved: 31 January 2020 / Online: 31 January 2020 (10:09:07 CET)
Version 3 : Received: 19 May 2020 / Approved: 19 May 2020 / Online: 19 May 2020 (04:33:32 CEST)
Version 4 : Received: 22 September 2020 / Approved: 23 September 2020 / Online: 23 September 2020 (04:32:09 CEST)

How to cite: Kühnert, S. Functional ARCH and GARCH Models: A Yule-Walker Approach. Preprints 2019, 2019120163. Kühnert, S. Functional ARCH and GARCH Models: A Yule-Walker Approach. Preprints 2019, 2019120163.


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).


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


Computer Science and Mathematics, Probability and Statistics

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