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What is the condition for underlying stochastic volatility processes to give a consistent covariance matrix?

I read in Hull that in order to have a consistent covariance matrix, volatility parameters should be estimated using same model. Does that mean, for example, if I am using a Garch(1,1) model with some parameters, I should use the same parameters for all underlying? Or it is just enough to have Garch(1,1) and not necessarily the same parameters.

In either case, what would be a solution if the underlyings obviously fall under different models?

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  • $\begingroup$ A consistent resp. valid covariance matrix has to be positive-definite (in fact: non singular). Should I elaborate on that? $\endgroup$
    – Richi Wa
    Apr 3, 2013 at 8:09
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    $\begingroup$ If you are interested you can look at this question for requirements for a positive-definite covariance matrix. $\endgroup$
    – Richi Wa
    Apr 3, 2013 at 11:22
  • $\begingroup$ I know it should be positive definite. My question is that how should I make sure the stochastic volatility models that I estimate for individual stocks does not disturb positive definite character of the matrix. Assume stock A,B are Garch(1,1) while stock C has constant volatility. As I perceived from Hull book this might make the covariance matrix inconsistent ( non-positive definite). $\endgroup$ Apr 3, 2013 at 23:44
  • $\begingroup$ I thought of using a constant correlation matrix and generating covariance matrix using this and individual volatilities. But I think you should let the correlation changes too when the stochastic volatility changes. For instance, when stock A becomes highly volatile probably its correlation structure breaks down and becomes uncorrelated. In a nutshell, how would you reconcile your estimated volatility models in a covariance matrix. $\endgroup$ Apr 3, 2013 at 23:50
  • $\begingroup$ Very good question! $\endgroup$
    – Richi Wa
    Apr 4, 2013 at 7:40

2 Answers 2

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I am not an expert in this field, but it would be best to consider a full multivariate GARCH model. This paper by Engle and Sheppard should be a good start.

I think the constant correlation matrix approach is covered to a certain extent too. I hope this helps.

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  • $\begingroup$ Multivariate GARCH seems to be a good fit to your problem. Do look at this answer for some further information and warnings. $\endgroup$
    – Bob Jansen
    May 4, 2013 at 9:23
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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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