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October 2006 Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach
Daniel Straumann, Thomas Mikosch
Ann. Statist. 34(5): 2449-2495 (October 2006). DOI: 10.1214/009053606000000803


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.


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Daniel Straumann. Thomas Mikosch. "Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach." Ann. Statist. 34 (5) 2449 - 2495, October 2006.


Published: October 2006
First available in Project Euclid: 23 January 2007

zbMATH: 1108.62094
MathSciNet: MR2291507
Digital Object Identifier: 10.1214/009053606000000803

Primary: 60H25
Secondary: 62F10 , 62F12 , 62M05 , 62M10 , 91B84

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

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 5 • October 2006
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