I have a portfolio of 4 assets. I also have returns for 3 indices. I want to get the multivariate betas for these 4 assets-based on these assets. I only have the 7 x 7 covariance matrix estimated by a Exponential Weighted Moving Average Model (EWMA). How would I get these multivariate betas? I know from a univariate percpective I would use the covarience(Asset1Return,Index1Return)/Var(Index1) but this only gives the uni-variate beta.
2 Answers
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With this solution you have to split your covariance matrix somewhat, but it should give you a vector with betas based on you conditional covariances.
Example with two indexes, $x1$ and $x2$, and one asset $y$.
$$[\sigma_{y,x1}, \sigma_{y, x2}]\begin{bmatrix} \sigma_{x1}^2 & \sigma_{x1,x2} \\ \sigma_{x1,x2} & \sigma_{x2}^2 \end{bmatrix}^{-1}$$
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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 in $X$ if you want to also have a constant, but this does not matter. The OLS solution would be $$\beta = (X'X)^{-1}(X'Y)$$
You can see the analogy to the univariate cov/var formula that you describe. The $X'Y$ part corresponds to the $(3\times 4)$ asset-factor covariances, while the $X'X$ parts corresponds to the $(3\times 3)$ factor-factor covariances.
Therefore I would say that your large $(7\times 7)$ EWMA covariance matrix can be partitioned as $$\left( \begin{array}{cc} X'X & X'Y \\ Y'X & Y'Y \end{array} \right) $$ From which you can pick the right elements to calculate the betas.
