Limit theory for a general class of GARCH models with just barely infinite variance

Abstract

In this article, limit theory is established for a general class of generalized autoregressive conditional heteroskedasticity models given by ɛt = σtηt and σt = f (σt−1, σt−2,…, σt−p, ɛt−1, ɛt−2,…, ɛt−q), when {ɛt} is a process with just barely infinite variance, that is, {ɛt} is a process with infinite variance but in the domain of normal attraction. In particular, we show that under some regular conditions, converges weakly to a Gaussian process. Applications of the asymptotic results to statistical inference, such as unit root test and sample autocorrelation, are also investigated. The obtained result fills in a gap between the classical infinite variance and finite variance in the literature. Further, when applying our limiting result to Dickey–Fuller (DF) test in a unit root model with integrated generalized autoregressive conditional heteroskedasticity (IGARCH) errors, it just confirms the simulation result of Kourogenis and Pittis (2008) that the DF statistics with IGARCH errors converges in law to the standard DF distribution.