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Journal of Financial Econometrics Advance Access published online on July 11, 2007
Journal of Financial Econometrics, doi:10.1093/jjfinec/nbm010
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Copyright © The Author 2007. Published by Oxford University Press.
The online version of this article has been published under an open access model. Users are entitled to use, reproduce, disseminate, or display the open access version of this article for non-commercial purposes provided that: the original authorship is properly and fully attributed; the Journal and Oxford University Press are attributed as the original place of publication with the correct citation details given; if an article is subsequently reproduced or disseminated not in its entirety but only in part or as a derivative work this must be clearly indicated. For commercial re-use, please contact journals.permissions@oxfordjournals.org

Positivity Conditions for a Bivariate Autoregressive Volatility Specification

C. Gourieroux
     CREST, and University of Toronto

Address correspondence to C. Gourieroux, CREST, and University of Toronto, Canada, UK, or e-mail: christian.gourieroux{at}ensae.fr


  Abstract

We derive necessary and sufficient conditions for the positive definiteness of the predicted volatility matrix in a bivariate autoregressive volatility specification. These nonlinear inequality restrictions have strong implications in terms of causality between volatilities and covolatilities.

KEYWORDS: GARCH model, nonlinear causality, stochastic volatility, Wishart process

We thanks F. Trojani for helpful discussion. The author gratefully acknowledges financial support of NSERC Canada and of the Chair AXA: "Large Risk in Insurance".

Received July 26, 2006; revised April 17, 2007; accepted April 18, 2007


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