1
$\begingroup$

I need to find min/max possible values of $q$ in the following situation.

"Consider random variables $x, y$, and $z$. Suppose it is known that $\operatorname{Corr}[x, y]=\operatorname{Corr}[x, z]=$ $0.5$. Denote $q=\operatorname{Corr}[y, z]$."

I can easily see that max is $1$ if Y = Z. However, can't find how to approach this problem to find minimum. Any hint?

$\endgroup$

1 Answer 1

1
$\begingroup$

I ll shamelessly refer to my answer to a super similar question with only different numbers.

Let $X,Y,Z$ be three random variables such that the correlation coefficients $\rho_{XY}=0.2, \rho_{YZ}=0.2$, what values can $\rho_{XZ}$ take?

You re right about the upper bound which is $1$. The constraint on lower bound is the covatiance matrix of the 3 variables shall be (semi) positive definite.

Then we could apply Sylvester theorem to derive the criterion for the covariance to be positive definite.


More specifically, normalizing variables won't change their correlation or positive definiteness of the covariance, so we consider the normalized variables $x',y'z'$. $$Cov([x',y',z'])= \begin{bmatrix} 1&0.5&0.5\\ 0.5&1&q\\ 0.5&q&1\\ \end{bmatrix} $$ By Generalized Sylvester Theorem we need to ensure all principal minor to be non-negative definite. $$ 1-q^2\geq 0 \\ \det Cov=1-q^2+\frac 12(q-1)\geq 0 $$ You get $-0.5\leq q\leq 1$

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .