Possible values of Correlation Coefficient.

Question

There are 3 random variables $X,Y,Z$, suppose the correlation coefficient of each pair is the same, ie. $$\begin{equation} Corr(X,Y) = Corr(Y,Z)=Corr(Z,X)=r, \end{equation}$$ what are the possible values of $r$?

Frankly, I have no idea about how to solve it. Could you please give me some hint? Thanks a lot.

Answer 1

The correlation matrix is

$$\begin{bmatrix} 1 & r & r \\ r & 1 & r \\ r & r & 1\end{bmatrix}$$

Let's check when is it positive semidefinite.

We require $1-r^2 \ge 0$ and

\begin{align} (1-r^2)-r(r-r^2)+r(r^2-r)&=2r(r^2-r)+(1-r^2) \\&= 2r^2(r-1)+(1-r)(1+r) \\ &=(1-r)(-2r^2+1+r) \\ &= (1-r)(-r+1)(2r+1) \\ &=(1-r)^2(2r+1) \ge 0 \end{align}

Hence we just require $2r+1 \ge 0$.