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Yes, multivariate GARCH is what you should consider. Imho, you can look at a book, analysis of financial time series, by Ruey S. Tsay, in chapter 10, they discussed multivariate volatility models. You can google this book, download the pdf of second edition, hope it helps.


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You can use the either, as both necessarily are symmetric positive definite; covariance is a personal preference. It's really just a matter of scaling, as $\mathcal{N}(0,\Sigma)$ is distributionally $\sqrt{\Sigma} \mathcal{N}(0,1) $. Correlation would require additional scaling (i.e. multiplication of every $\mathcal{N}(0,\rho)$ element by its respective ...



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