Functional central limit theorem for ARCH(∞) models with weakly dependent innovations

Abstract

Since the seminal work of Engle (1982) and Bollerslev (1986), many ARCH-type models have been suggested and examined to explain a variety of stylized facts of financial and economic time series. Various popular ARCH-type models can be expressed as ARCH(∞) models. In this paper, we study the stationarity and functional central limit theorem (FCLT) for ARCH(∞) models, because statistical inferences for ARCH(∞) sequences require the study of asymptotics of various statistics concerned. Most previous results are obtained under independent and identically distributed (i.i.d.) innovation processes. But the i.i.d. assumption on innovations substantially restricts the flexibility of the models. In addition, many authors have shown that the i.i.d. assumption can be weakened to mild conditions. We consider the ARCH(∞) model where the innovation processes are strictly stationary and λ-weakly dependent instead of independent and identically distributed. We provide sufficient conditions for the existence of a unique stationary and λ-weakly dependent Volterra series type solution to the given ARCH(∞) process. The FCLT for the stationary and λ-weakly dependent solution is also obtained by adding weak dependence coefficients condition on innovations and condition on ARCH(∞) parameters. The FCLT for GARCH(p; q) model with λ-weakly dependent innovations is considered as an example.