Functional ARCH and GARCH models: A Yule-Walker approach

Abstract

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).