The Annals of Statistics

Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach

Daniel Straumann and Thomas Mikosch

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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, …, Xtp, σt−1, …, σtq) 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.

Article information

Ann. Statist. Volume 34, Number 5 (2006), 2449-2495.

First available in Project Euclid: 23 January 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 62F10: Point estimation 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 91B84: Economic time series analysis [See also 62M10]

Stochastic recurrence equation conditionally heteroscedastic time series GARCH asymmetric GARCH exponential GARCH EGARCH stationarity invertibility quasi-maximum-likelihood estimation consistency asymptotic normality


Straumann, Daniel; Mikosch, Thomas. Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Ann. Statist. 34 (2006), no. 5, 2449--2495. doi:10.1214/009053606000000803.

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