{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,4]],"date-time":"2026-10-04T04:34:32Z","timestamp":1791088472512,"version":"4.1.0"},"reference-count":32,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2012,11,12]],"date-time":"2012-11-12T00:00:00Z","timestamp":1352678400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2013,6]]},"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.1017\/s0266466612000473","type":"journal-article","created":{"date-parts":[[2012,11,12]],"date-time":"2012-11-12T14:26:57Z","timestamp":1352730417000},"page":"545-566","source":"Crossref","is-referenced-by-count":16,"title":["ON MOMENT CONDITIONS FOR QUASI-MAXIMUM LIKELIHOOD ESTIMATION OF MULTIVARIATE ARCH MODELS"],"prefix":"10.1017","volume":"29","author":[{"given":"Marco","family":"Avarucci","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eric","family":"Beutner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paolo","family":"Zaffaroni","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2012,11,12]]},"reference":[{"key":"S0266466612000473_ref12","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/S0047-259X(02)00009-X","article-title":"Asymptotic theory for multivariate GARCH processes","volume":"84","author":"Comte","year":"2003","journal-title":"Journal of Multivariate Analysis"},{"key":"S0266466612000473_ref18","doi-asserted-by":"crossref","first-page":"2044","DOI":"10.1016\/j.jmva.2009.03.011","article-title":"On asymptotic theory for multivariate GARCH models","volume":"100","author":"Hafner","year":"2009","journal-title":"Journal of Multivariate Analysis"},{"key":"S0266466612000473_ref26","volume-title":"Matrix Differential Calculus with Application in Statistics and Econometrics","author":"Magnus","year":"1999"},{"key":"S0266466612000473_ref8","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1086\/261527","article-title":"A capital asset pricing model with time varying covariances","volume":"96","author":"Bollerslev","year":"1988","journal-title":"Journal of Political Economy"},{"key":"S0266466612000473_ref9","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/0304-4076(92)90067-2","article-title":"Stationarity of GARCH processes and of some nonnegative time series","volume":"52","author":"Bougerol","year":"1992","journal-title":"Journal of Econometrics"},{"key":"S0266466612000473_ref2","doi-asserted-by":"crossref","first-page":"2730","DOI":"10.1214\/08-AOS674","article-title":"Asymptotic normality of quasi-maximum likelihood estimator for multidimensional causal processes","volume":"37","author":"Bardet","year":"2009","journal-title":"Annals of Statistics"},{"key":"S0266466612000473_ref1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511810800","volume-title":"Matrix Algebra","author":"Abadir","year":"2005"},{"key":"S0266466612000473_ref16","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1017\/S0266466611000156","article-title":"QML estimation of a class of multivariate asymmetric GARCH models","volume":"28","author":"Francq","year":"2012","journal-title":"Econometric Theory"},{"key":"S0266466612000473_ref7","doi-asserted-by":"crossref","first-page":"498","DOI":"10.2307\/2109358","article-title":"Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH","volume":"72","author":"Bollerslev","year":"1990","journal-title":"Review of Economics and Statistics"},{"key":"S0266466612000473_ref31","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1017\/S0266466608080456","article-title":"On the relation between the VEC and the BEKK multivariate GARCH models","volume":"24","author":"Stelzer","year":"2008","journal-title":"Econometric Theory"},{"key":"S0266466612000473_ref5","doi-asserted-by":"crossref","first-page":"201","DOI":"10.3150\/bj\/1068128975","article-title":"Garch processes: Structure and estimation","volume":"9","author":"Berkes","year":"2003","journal-title":"Bernoulli"},{"key":"S0266466612000473_ref24","doi-asserted-by":"crossref","first-page":"575","DOI":"10.2307\/2171862","article-title":"Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1, 1) and covariance stationary GARCH(1, 1) models","volume":"64","author":"Lumsdaine","year":"1996","journal-title":"Econometrica"},{"key":"S0266466612000473_ref32","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511802256","volume-title":"Asymptotic Statistics","author":"van der Vaart","year":"1998"},{"key":"S0266466612000473_ref14","doi-asserted-by":"crossref","first-page":"605","DOI":"10.3150\/bj\/1093265632","article-title":"Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes","volume":"10","author":"Francq","year":"2004","journal-title":"Bernoulli"},{"key":"S0266466612000473_ref15","doi-asserted-by":"crossref","DOI":"10.1002\/9780470670057","volume-title":"GARCH Models. 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