Scandinavian Journal of Statistics, Vol. 36: 157–170, 2009 doi: 10.1111/j.1467-9469.2008.00624.x  2008 Board of the Foundation of the Scandinavian Journal of Statistics. Published by Blackwell Publishing Ltd. Normal Mixture Quasi-maximum Likelihood Estimator for GARCH Models TAEWOOK LEE Department of Information Statistics, Hankuk University of Foreign Studies SANGYEOL LEE Department of Statistics, Seoul National University ABSTRACT. The generalized autoregressive conditional heteroscedastic (GARCH) model has been popular in the analysis of financial time series data with high volatility. Conventionally, the parameter estimation in GARCH models has been performed based on the Gaussian quasi-maximum likelihood. However, when the innovation terms have either heavy-tailed or skewed distributions, the quasi-maximum likelihood estimator (QMLE) does not function well. In order to remedy this defect, we propose the normal mixture QMLE (NM-QMLE), which is obtained from the normal mixture quasi-likelihood, and demonstrate that the NM-QMLE is consistent and asymptotically normal. Finally, we present simulation results and a real data analysis in order to illustrate our findings. Key words: asymptotic normality, consistency, GARCH, normal mixture, quasi-maximum likelihood estimator 1. Introduction Since Bollerslev (1986), the generalized autoregressive conditional heteroscedastic (GARCH) model has been widely used in the analysis of financial time series with high volatility. During the past decades,researchers have extensively studiedthe problem of estimating the parameters in GARCH models. Earlier, the innovations in GARCH models were assumed to follow a standard normal distribution and the Gaussian maximum likelihood estimator (MLE) was employed to estimate the GARCH parameters. Although many authors pointed out strong evidence against the normality assumption through empirical studies, the Gaussian MLE method was applied to non-Gaussian innovation models due to its simplicity (cf. Bollers- lev & Wooldridge, 1992; Gouri´ eroux, 1997; Mikosch & St˘ aric˘ a, 2000). With regard to the asymptotic inference for the quasi-MLE (QMLE), see Weiss (1986), Lee & Hansen (1994), Lumsdaine (1996), Berkes et al. (2003), Berkes & Horv´ ath (2003, 2004), Hall & Yao (2003), Francq & Zako¨ ıan (2004) and Jensen & Rahbek (2004). As the QMLE is unstable when the innovations in GARCH models have a skewed and heavy-tailed distribution as reported by Hall & Yao (2003), researchers have developed methods to correct this defect of the QMLE. In the literature, there are two approaches correcting this defect: the adoption of the GARCH models with their innovations following a distribution that reflects their non-normal behaviour and the use of mixture type models, which are not GARCH models by definition. The latter approach was motivated by con- sidering that in contrast to the normal GARCH models, the estimators based on the first approach have a severe defect to lose their analytic tractability, and the normal mixture model is a functional tool for the modelling of heavy-tailed and skewed distributions (cf. McLachlan & Peel, 2000). The first approach was used in Bollerslev (1987), who considered Student’s t-GARCH models; Berkes & Horv´ ath (2004), who considered GARCH models with innovations following two-sided exponential distributions; and Mittnik & Rachev (2000) and Carr & Wu (2003), who considered GARCH models with innovations following stable