Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach
Loading...
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form $X_t=σ_tZ_t$, where the unobservable volatility $σ_t$ is a parametric function of $(X_{t-1},...,X_{t-p},σ_{t-1},... ,σ_{t-q})$ 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.
Published at http://dx.doi.org/10.1214/009053606000000803 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Published at http://dx.doi.org/10.1214/009053606000000803 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)