Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-17T20:27:35.415Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  01 April 2009

Juan Carlos Escanciano*
Indiana University
*Address correspondence to Juan Carlos Escanciano, Indiana University, Department of Economics, 100 S. Woodlawn, Wylie Hall, Bloomington, IN 47405-7104, U.S.A.; e-mail:


This note proves the consistency and asymptotic normality of the quasi–maximum likelihood estimator (QMLE) of the parameters of a generalized autoregressive conditional heteroskedastic (GARCH) model with martingale difference centered squared innovations. The results are obtained under mild conditions and generalize and improve those in Lee and Hansen (1994, Econometric Theory 10, 29–52) for the local QMLE in semistrong GARCH(1,1) models. In particular, no restrictions on the conditional mean are imposed. Our proofs closely follow those in Francq and Zakoïan (2004, Bernoulli 10, 605–637) for independent and identically distributed innovations.

Notes and Problems
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Berkes, I. & Horváth, L. (2003) The rate of consistency of the quasi-maximum likelihood estimator. Statistics and Probability Letters 61, 133143.CrossRefGoogle Scholar
Berkes, I. & Horváth, L. (2004) The efficiency of the estimators of the parameters in GARCH processes. Annals of Statistics 32, 633655.CrossRefGoogle Scholar
Berkes, I., Horváth, L. & Kokoszka, P. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201227.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307327.CrossRefGoogle Scholar
Brooks, C., Burke, S.P., Heravi, S. & Persand, G. (2005) Autoregressive conditional kurtosis. Journal of Financial Econometrics 3, 399421.CrossRefGoogle Scholar
Dahl, C.M. & Iglesias, E.M. (2007) Asymptotic Normality of the QMLE for Nonstationary GARCH with Serially Dependent Innovations. Manuscript, Michigan State University.Google Scholar
Engle, R. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 9871008.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH. Bernoulli 10, 605637.CrossRefGoogle Scholar
Gallant, A.R., Hsieh, D.A. & Tauchen, G. (1991) On fitting a recalcitrant series: The pound/dollar exchange rate, 1974-1983. In Barnett, W.A., Powell, J. & Tauchen, G. (eds.). Nonparametric and Semiparametric Methods in Econometrics and Statistics, pp. 199240. Cambridge University Press.Google Scholar
Hall, P. & Yao, Q.W. (2003) Inference in ARCH and GARCH models. Econometrica 71, 285317.CrossRefGoogle Scholar
Hansen, B. (1994) Autoregressive conditional density estimation. International Economic Review 35, 705730.CrossRefGoogle Scholar
Harvey, C. & Siddique, A. (1999) Autoregressive conditional skewness. Journal of Financial and Quantitative Analysis 34, 465487.CrossRefGoogle Scholar
Harvey, C. & Siddique, A. (2000) Conditional skewness in asset pricing tests. Journal of Finance 55, 12631296.CrossRefGoogle Scholar
Jensen, S.T. & Rahbek, A. (2004a) Asymptotic normality of the QML estimator of ARCH in the nonstationary case. Econometrica 72, 641646.CrossRefGoogle Scholar
Jensen, S.T. & Rahbek, A. (2004b) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 12031226.CrossRefGoogle Scholar
Jondeau, E. & Rockinger, M. (2003) Conditional volatility, skewness, and kurtosis: Existence, persistence, and comovements. Journal of Economic Dynamics and Control 27, 16991737.CrossRefGoogle Scholar
Lee, S. & Hansen, B. (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 2952.CrossRefGoogle Scholar
Leon, A., Rubio, G. & Serna, G. (2005) Autoregressive conditional volatility, skewness and kurtosis: Quarterly Review of Economics and Finance 45, 599618.CrossRefGoogle Scholar
Li, W.K., Ling, S. & McAleer, M. (2002) A survey of recent theoretical results for time series models with GARCH errors. Journal of Economic Survey 16, 245269.CrossRefGoogle Scholar
Ling, S. (2007) Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/ IGARCH models. Journal of Econometrics 140, 849873.CrossRefGoogle Scholar
Linton, O., Pan, J. & Wang, H. (2009) Estimation for a non-stationary semi-strong GARCH(1,1) model with heavy-tailed errors. Econometric Theory 25, forthcoming.Google Scholar
Lumsdaine, R.L. (1996) Consistency and asymptotic normality of the quasi–maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 6, 575596.CrossRefGoogle Scholar
Mikosch, T. & Straumann, D. (2002) Whittle estimation in a heavy-tailed GARCH(1,1) model. Stochastic Processes and Their Applications 100, 187222.CrossRefGoogle Scholar
Premaratne, G. & Bera, A.K. (2001) Modeling Asymmetry and Excess Kurtosis in Stock Return Data. Working paper 00–123, Department of Economics, University of Illinois, Champaign.CrossRefGoogle Scholar
Robinson, P.M. & Zaffaroni, P. (2006) Pseudo-maximum likelihood estimation of ARCH(∞) models. Annals of Statistics 34, 10491074.CrossRefGoogle Scholar
Straumann, D. (2005) Estimation in Conditionally Heteroscedastic Time Series Models. Lecture Notes in Statistics 181. Springer-Verlag.Google Scholar
Weiss, A.A. (1986) Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2, 107131.CrossRefGoogle Scholar