Determine range of $\rho$ in correlation matrix

Question

I was asked a question and I was wondering if you can help solve it.

Given are $3$ random variables $A$, $B$and $C$ all having the same correlation. So the correlation matrix looks like:

$$\begin{pmatrix}A& B& C\\ 1 & \rho & \rho\\ \rho& 1 & \rho\\ \rho & \rho& 1\end{pmatrix}$$

What is the possible range of $\rho$?

Thank you.

Answer 1

Lets start with a few observations on possible $\rho$s.

• $\rho$ can be $0$ or uncorrelated.
• $\rho$ can be $1$ or $A$, $B$, $C$ are the same Random Variable.
• $\rho$ cannot be $-1$ as you cannot have $A=-B$, $B=-C$ and $C=-A$.

We can prove that the set must be connected and thus an interval. Consider a set of Random Variables $X_A$, $X_B$ and $X_C$ such that they have correlation $\rho$. Then consider a 2nd set of Random Variables $Y_A$, $Y_B$ and $Y_C$ such that they are all independent and independent of the $X$'s. Then you can construct any correlation between $\rho$ and $0$ using $Z_i=\alpha X_i + (1-\alpha) Y_i$ for $\alpha\in[0,1]$. Hence we can construct any positive correlation as we know a $\rho$ of $1$ is possible.

Now we just need to find the maximal negative $\rho$. For this I don't have an answer. Though I know that it must be at least $1\over n-1$ where $n$ is the number of Random Variables in the original problem. We can see this by considering a set of Random Variables $X_{AB}$, $X_{AC}$ and $X_{BC}$ which are all i.i.d. Then let $A=X_{AB}-X_{AC}$, $B=X_{BC}-X_{AB}$ and $C=X_{AC}-X_{BC}$. These each have a $-{1\over 2}$ correlation with each other. We see that this construction generalizes for any number of random variables. My intuition is that this is the bound for the most negative $\rho$ but I do not have a proof of this.

Answer 2

A value $\rho$ is admissible in size $n$ if and only if the corresponding matrix $M_\rho$ is symmetric nonnegative. Note that $M_\rho=(1-\rho)I+\rho J$ where $J$ is the matrix of ones hence $J^2=nJ$ and the eigenvalues of $J$ are $n$ and $0$. The eigenvalues of $M_\rho$ are $1-\rho+0\cdot\rho=1-\rho$ and $1-\rho+n\rho$, hence $M_\rho$ is a covariance matrix if and only if $\rho\leqslant1$ and $(n-1)\rho\geqslant-1$, that is, if and only if $$ -1/(n-1)\leqslant\rho\leqslant1. $$ For three random variables, the case $\rho=1$ is realized when $A=B=C$, the case $\rho=0$ is realized when $(A,B,C)$ is independent, and the case $\rho=-1/2$ is realized when $(A,B,C)$ are, say, independent standard normal random variables $(X,Y,Z)$ conditioned on $X+Y+Z=0$.