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There are 2 issues that come to mind What is the correct definition of semi-covariance $$\frac{1}{n}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\min \left( {{r_i},0} \right)} } \min \left( {{r_j},0} \right)$$ $$\frac{1}{n}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\min \left( {{r_i}{r_j},0} \right)} }$$ 2. Can you get a positive ...

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Say that you did the calculations in the classic regression way. If you stick the returns of your 4 asset returns in a $(T\times 4)$ matrix $Y$, and your 3 factor returns in a $(T\times 3)$ matrix $X$, then your betas would solve the multiple regressions, collected in a $(3\times 4)$ matrix $$Y = X\cdot \beta + \epsilon$$ You could also add a column of ones ...

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Not sure your question is about having a process for covariance or to have multivariate GARCH. The standard viewpoint on a stochastic volatility for covariance is to use a Whishart process. See for instance Philipov, A. and M. E. Glickman (2006, July) Multivariate stochastic volatility via wishart processes. Journal of Business & Economic Statistics 24 ...

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I think you're looking for multivariate GARCH models of which this is an overview paper. Multivariate GARCH models have one big drawback: they are pretty hard to estimate due to the number of correlations. This paper by Caporin and McAleer might be of interest in that regard.

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