Variance of the sum of correlated variables

Question

If the variance of two correlated variables is: $$Var(r_1+r_2)=\sigma^2_1+\sigma^2_2+2\textrm{cov}(r_1,r_2)=\sigma^2_1+\sigma^2_2+2\rho\sigma_1\sigma_2$$ where $r_1$ and $r_2$ are vectors, then what is the multivariate representation of this.

So, if $R_1$ and $R_2$ both denote a matrix we get $$Var(R_1+R_2)=\Sigma_1+\Sigma_2+...$$ where $\Sigma_i$ denotes the covariance matrix for $R_i$.

Anyone knows how to fill in the dots?

Answer 1

... if $R_1$ and $R_2$ both denote a matrix we get

$R_1$ and $R_2$ should be vectors.

Then $Var(R_1+R_2)=\Sigma_1+\Sigma_2+2\rho_{12}\sqrt{\Sigma_1} \sqrt{ \Sigma_2}$

where $\Sigma_i$ denotes the variance matrix of $R_1$ and $R_2$ respectively.

Answer 2

When working with multivariate variances and covariances, it's good to keep this notational advice in mind.

I'll stick with your notation and use $\operatorname{Var}(R)$ to denote the (co)variance matrix of the random vector $R$, i.e. $\operatorname{Var}(R)=\operatorname{cov}(R,R)$.

Then

\begin{align} \operatorname{Var}(R_1+R_2)&=\operatorname{cov}(R_1+R_2,R_1+R_2) \\ &= \operatorname{cov}(R_1,R_1)+ \operatorname{cov}(R_1,R_2)+ \operatorname{cov}(R_2,R_1)+ \operatorname{cov}(R_2,R_2) \\ &= \operatorname{Var}(R_1)+ \operatorname{Var}(R_2)+ 2\operatorname{cov}(R_1,R_2)\;, \end{align}

so the multivariate case is exactly analogous to the univariate case.