How can I convert from a sample Correlation matrix to a sample Covariance matrix?

Question

I have seen the relation that for correlation matrix $R$, I can compute $V=DRD$ such that D is a diagonal matrix with standard deviations along the diagonal. Why does this hold? does this hold for sample correlation and covariance or only population? Any references on this would help as well.

Answer 1

The short answer is Yes. But it depends on what you mean by "hold" in "does this hold?"

There is the population covariance matrix $V=DRD$ for which there is the following example:

$$D=\left( \begin{array}{ccc} \sigma _1 & 0 & 0 \\ 0 & \sigma _2 & 0 \\ 0 & 0 & \sigma _3 \\ \end{array} \right)$$

$$R=\left( \begin{array}{ccc} 1 & \rho _{12} & \rho _{13} \\ \rho _{12} & 1 & \rho _{23} \\ \rho _{13} & \rho _{23} & 1 \\ \end{array} \right)$$

$$V=DRD=\left( \begin{array}{ccc} \sigma _1^2 & \rho _{12} \sigma _1 \sigma _2 & \rho _{13} \sigma _1 \sigma _3 \\ \rho _{12} \sigma _1 \sigma _2 & \sigma _2^2 & \rho _{23} \sigma _2 \sigma _3 \\ \rho _{13} \sigma _1 \sigma _3 & \rho _{23} \sigma _2 \sigma _3 & \sigma _3^2 \\ \end{array} \right)$$

If one has estimates of the parameters based on a sample (say $\hat{D}$ and $\hat{R}$), then $\hat{D} \hat{R} \hat{D}$ certainly gives you an estimate of $V$ (rather than "the" estimate of $V$).