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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: p.zaffaroni@imperial.ac.uk.
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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.

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