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This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt = σtZt, where the unobservable volatility σt is a parametric function of (Xt−1, . . . , Xt−p, σt−1, . . . , σt−q) for some p, q ≥ 0, and (Zt) 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 (Xt) 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.

We give the theoretical basis of a possible explanation for two stylized facts observed in long log-return series: the long-range dependence (LRD) in volatility and the integrated GARCH (IGARCH). Both these effects can be explained theoretically if one assumes that the data are nonstationary. © 2004 President and Fellows of Harvard College and the Massachusetts Institute of Technology.

We show that the finite-dimensional distributions of a GARCH process are regularly varying, i.e., the tails of these distributions are Pareto-like and hence heavy-tailed. Regular variation of the joint distributions provides insight into the moment properties of the process as well as the dependence structure between neighboring observations when both are large. Regular variation also plays a vital role in establishing the large sample behavior of a variety of statistics from a GARCH process including the sample mean and the sample autocovariance and autocorrelation functions. In particular, if the 4th moment of the process does not exist, the rate of convergence of the sample autocorrelations becomes extremely slow, and if the 2nd moment does not exist, the sample autocorrelations have non-degenerate limit distributions.