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ON MOMENT CONDITIONS FOR QUASI-MAXIMUM LIKELIHOOD ESTIMATION OF MULTIVARIATE ARCH MODELS

Published online by Cambridge University Press:  12 November 2012

Marco Avarucci ,
Eric Beutner  and
Paolo Zaffaroni
Marco Avarucci
Affiliation:
Maastricht University
Eric Beutner
Affiliation:
Maastricht University
Paolo Zaffaroni*
Affiliation:
Imperial College London and University of Rome “La Sapienza”
*
*Address correspondence to Paolo Zaffaroni, Imperial College Business School, Imperial College London, South Kensington campus, London SW7 2AZ; e-mail: [email protected].
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Abstract

This paper questions whether it is possible to derive consistency and asymptotic normality of the Gaussian quasi-maximum likelihood estimator (QMLE) for possibly the simplest multivariate GARCH model, namely, the multivariate ARCH(1) model of the Baba, Engle, Kraft, and Kroner form, under weak moment conditions similar to the univariate case. In contrast to the univariate specification, we show that the expectation of the log-likelihood function is unbounded, away from the true parameter value, if (and only if) the observable has unbounded second moment. Despite this nonstandard feature, consistency of the Gaussian QMLE is still warranted. The same moment condition proves to be necessary and sufficient for the stationarity of the score when evaluated at the true parameter value. This explains why high moment conditions, typically bounded sixth moment and above, have been used hitherto in the literature to establish the asymptotic normality of the QMLE in the multivariate framework.

Type
Articles
Information
Econometric Theory , Volume 29 , Issue 3 , June 2013 , pp. 545 - 566
Copyright
Copyright © Cambridge University Press 2012 

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Footnotes

We thank Anders Rahbek. We also thank two anonymous referees for their very careful reading and useful comments. The usual disclaimers apply. Paolo Zaffaroni acknowledges the ESRC grant RES-000-22-3219 for financial support.

References

Abadir, K.M. & Magnus, J.R. (2005) Matrix Algebra. Econometric Exercises 1. Cambridge University Press.CrossRefGoogle Scholar
Bardet, J.M. & Wintenberger, O. (2009) Asymptotic normality of quasi-maximum likelihood estimator for multidimensional causal processes. Annals of Statistics 37, 2730–2759.CrossRefGoogle Scholar
Bauwens, L., Laurent, S., & Romboust, J. (2006) Multivariate GARCH models: A survey. Journal of Applied Econometrics 21, 79–109.CrossRefGoogle Scholar
Berkes, I. & Horvàth, L. (2004) The efficiency of the estimators of the parameters in GARCH processes. Annals of Statistics 32, 633–655.Google Scholar
Berkes, I., Horvàth, L., & Kokoszka, P. (2003) Garch processes: Structure and estimation. Bernoulli 9, 201–227.CrossRefGoogle Scholar
Billingsley, P. (1995) Probability and Measure. Wiley.Google Scholar
Bollerslev, T. (1990) Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH. Review of Economics and Statistics 72, 498–505.CrossRefGoogle Scholar
Bollerslev, T., Engle, R., & Woolridge, J. (1988) A capital asset pricing model with time varying covariances. Journal of Political Economy 96, 116–131.CrossRefGoogle Scholar
Bougerol, P. & Picard, N. (1992a) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.CrossRefGoogle Scholar
Bougerol, P. & Picard, N. (1992b) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 1714–1729.CrossRefGoogle Scholar
Boussama, F., Fuchs, F., & Stelzer, R. (2011) Stationarity and geometric ergodicity of BEKK multivariate GARCH models. Stochastic Processes and Their Applications 121, 2331–2360.Google Scholar
Comte, F. & Lieberman, O. (2003) Asymptotic theory for multivariate GARCH processes. Journal of Multivariate Analysis 84, 61–84.CrossRefGoogle Scholar
Engle, R.F. & Kroner, K.F. (1995) Multivariate simultaneous generalized ARCH. Econometric Theory 11, 122–150.CrossRefGoogle Scholar
Francq, C. &Zakoïan, J.-M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637.CrossRefGoogle Scholar
Francq, C. &Zakoïan, J.-M. (2010) GARCH Models. Structure, Statistical Inference and Financial Applications. Wiley.CrossRefGoogle Scholar
Francq, C. &Zakoïan, J.-M. (2012) QML estimation of a class of multivariate asymmetric GARCH models. Econometric Theory 28, 179–206.CrossRefGoogle Scholar
Hafner, C.M. & Preminger, A. (2009a) Asymptotic theory for a factor GARCH model. Econometric Theory 25, 336–363.CrossRefGoogle Scholar
Hafner, C.M. & Preminger, A. (2009b) On asymptotic theory for multivariate GARCH models. Journal of Multivariate Analysis 100, 2044–2054.CrossRefGoogle Scholar
Harville, D. (1997) Matrix Algebra from a Statistician’s Perspective. Springer-Verlag.CrossRefGoogle Scholar
Horn, R. & Johnson, C. (1985) Topics in Matrix Analysis. Cambridge University Press.CrossRefGoogle Scholar
Jensen, S. & Rahbek, A. (2004a) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 1203–1226.CrossRefGoogle Scholar
Jensen, S. & Rahbek, A. (2004b) Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641–646.CrossRefGoogle Scholar
Lee, S. & Hansen, B. (1994) Asymptotic theory for the GARCH(1, 1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.CrossRefGoogle Scholar
Lumsdaine, R. (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1, 1) and covariance stationary GARCH(1, 1) models. Econometrica 64, 575–596.CrossRefGoogle Scholar
Lütkepohl, H. (1996) Handbook of Matrices. Wiley.Google Scholar
Magnus, J. & Neudecker, H. (1999) Matrix Differential Calculus with Application in Statistics and Econometrics. Wiley.Google Scholar
Nelson, D. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Roberts, L.A. (1995) On the existence of moments of ratios of quadratic forms. Econometric Theory 11, 750–774.CrossRefGoogle Scholar
Robinson, P.M. & Zaffaroni, P. (2006) Pseudo-maximum likelihood estimation for ARCH(∞) models. Annals of Statistics 34, 1049–1074.CrossRefGoogle Scholar
Silvennoinen, A. & Teräsvirta, T. (2008) Multivariate GARCH models. In Andersen, T., Davis, R., & Stǎricǎ, J.-P. (eds.), Handbook of Financial Time Series. Springer-Verlog.Google Scholar
Stelzer, R. (2008) On the relation between the VEC and the BEKK multivariate GARCH models. Econometric Theory 24, 1131–1136.CrossRefGoogle Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.CrossRefGoogle Scholar