rho Range

There are 3 random variables $$x,y,z$$ . Suppose $\mathrm{Corr}(x,y)=0.8$ and $\mathrm{Corr}(x,z)=0.8$ . What are the minimum and maximum possible values of $\mathrm{Corr}(y,z)=\rho$ ?

Answer

The correlation matrix must be positive semi-definite:

P=\begin{pmatrix} 1 & 0.8 & 0.8\\ 0.8 & 1 & \rho\\ 0.8 & \rho & 1 \end{pmatrix}\succeq 0.

Using the principal minors condition (all principal minors $\ge 0$ ):

The $2\times 2$ principal minor for $\{y,z\}$ gives
$\det\begin{pmatrix}1&\rho\\ \rho&1\end{pmatrix}=1-\rho^2\ge 0 \quad\Rightarrow\quad -1\le \rho\le 1.$
The $3\times 3$ principal minor gives
$\det(P)=\det\begin{pmatrix} 1 & 0.8 & 0.8\\ 0.8 & 1 & \rho\\ 0.8 & \rho & 1 \end{pmatrix} =\frac{(1-\rho)(25\rho-7)}{25}\ge 0.$
Since $\rho\le 1$ , we must have $25\rho-7\ge 0$ , so $\rho\ge \frac{7}{25}=0.28$ .

Therefore,

0.28 \le \rho \le 1.