Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed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 ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound for the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.


ARCH and GARCH processes; ARMA processes; functional data; invertible linear processes; parameter estimation; stationary solutions; Yule-Walker equations


Computer Science and Mathematics, Probability and Statistics

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