Asymptotic properties of the QMLE in a log-linear RealGARCH model with Gaussian errors
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Asymptotic properties of the QMLE in a log-linear RealGARCH model with Gaussian errors Caiya Zhang1 · Kaihong Xu1 · Lianfen Qian2 Received: 12 February 2018 / Revised: 4 September 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract To incorporate the realized volatility in stock return, Hansen et al. (J Appl Econ 27:877–906, 2012) proposed a RealGARCH model and conjectured some theoretical properties about the quasi-maximum likelihood estimation (QMLE) for parameters in a log-linear RealGARCH model without rigorous proof. Under Gaussian errors, this paper derives the detailed proof of the theoretical results including consistency and asymptotic normality of the QMLE, hence it solves the conjectures in Hansen et al. (J Appl Econ 27:877–906, 2012). Keywords RealGARCH model · Quasi-maximum likelihood estimator · Consistency and asymptotic normality
1 Introduction Since the basic GARCH specification was proposed by Bollerslev (1986), GARCHtype models have been widely applied to estimate volatility in finance. But Andersen et al. (2003) pointed out that GARCH-type models are poorly suited for situations where volatility changes rapidly to a new level. The reason is that GARCH-type models are slow in ’catching up’ and takes a while for the conditional variance to reach its new level. With the rapid development of high-frequency financial environment, a number of realized measures of volatility has been introduced, including realized volatility, realized bipower variation, realized range-based volatility, realized kernel volatility, and many other related quantities. For more details, see Andersen and Bollerslev (1998), Barndorff-Nielsen et al. (2004), Martens and Dick (2007), Barndorff-Nielsen et al. (2008) and references therein. As the estimators of conditional volatility, these
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Lianfen Qian [email protected]
1
School of Computer and Computing Science, Zhejiang University City College, Hangzhou, China
2
Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA
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C. Zhang et al.
proposed realized measures have been proved to be consistent and efficient. How to use these realized measures for modeling and forecasting future volatility is an interesting and significant task. Engle (2002) proposed a GARCH-X model that consists of a realized measure in the GARCH equation. Within the GARCH-X framework, the realized measure can not be predicted since it is just an exogenous variable in the model. Hence the volatility can only be one-step ahead predicted, which means this type of models is incomplete. To overcome the above shortcomings of GARCH-X models, Hansen et al. (2012) introduced a new framework that adds a measurement equation which combines the realized measure with the volatility. It was called as Realized GARCH (RealGARCH) model. The theoretical study about the GARCH-type models can be found in a great deal of literature. In the early years, Lee and Hansen (1994) established the asymptotic theory for the quasi-maximum likelihood estimator (QMLE) of
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