## Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equations approach.(English)Zbl 1108.62094

Summary: 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 $$(X_{t-1},\dots,X_{t-p},\sigma_{t -1},\dots,\sigma_{t-q})$$ for some $$p,q\geq 0$$, 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 Lip-schitz coefficient) property. These assumption 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.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F12 Asymptotic properties of parametric estimators 60H30 Applications of stochastic analysis (to PDEs, etc.) 62P05 Applications of statistics to actuarial sciences and financial mathematics 60H25 Random operators and equations (aspects of stochastic analysis) 62F10 Point estimation 62M05 Markov processes: estimation; hidden Markov models 91B84 Economic time series analysis
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