  This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general
conditionally heteroscedastic time series model of multiplicative form
$X_t=\sigma_tZ_t$, where the unobservable volatility $\sigma_t$ is a parametric
function of $(X_{t-1},...,X_{t-p},\sigma_{t-1},... ,\sigma_{t-q})$ for some
$p,q\ge0$, and $(Z_t)$ is standardized i.i.d. noise. We assume that these
models are solutions to stochastic recurrence equations which satisfy a
contraction (random Lipschitz coefficient) property. These assumptions are
satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH
processes. Exploiting the contraction property, we give conditions for the
existence and uniqueness of a strictly stationary solution $(X_t)$ to the
stochastic recurrence equation and establish consistency and asymptotic
normality of the QMLE. We also discuss the problem of invertibility of such
time series models.
