Correlation coefficient of X Y,Z

[pepsico007]

the correlation coefficient of X Y is 0.6, and the correlation coefficient of Y Z is 0.7,
what is the range of the correlation coefficient of X and Z?

 

[trinker]

Hi! :welcome: We are glad that you posted here! This looks like a homework question though. Our homework help policy can be found here. We mainly just want to see what you have tried so far and that you have put some effort into the problem. I would also suggest checking out this thread for some guidelines on smart posting behavior that can help you get answers that are better much more quickly.

 

[pepsico007]

I have no idea, I try to derivative it from its formula, like COV(X,Z)=COV(X,Z-Y+Y), but get nothing help. I

 

[BGM]

Let \( \rho = Corr[X, Z] \). Then we have the following correlation matrix for \( X, Y, Z \):

\( \begin{bmatrix} 1 & 0.6 & \rho \\ 0.6 & 1 & 0.7 \\ \rho & 0.7 & 1 \end{bmatrix}\)

I think the question want you to check whether this matrix is positive-definite or not.

One easy way to check is the Sylvester's Criterion:

http://en.wikipedia.org/wiki/Sylvester_criterion

which essentially require you to have a positive determinant. When you compute it, you should obtain a quadratic polynomial in \( \rho \) and thus you should obtain a bound for it.

 

[pepsico007]

Thanks! I am sorry to reply you so late. I think this is the right answer, it is a question of a hedge fund company, but I didn't pass the interview, so I was eager to know the answer. I shouldn't forget that the correlation matrix should be positive-definite.

Let \( \rho = Corr[X, Z] \). Then we have the following correlation matrix for \( X, Y, Z \):

\( \begin{bmatrix} 1 & 0.6 & \rho \\ 0.6 & 1 & 0.7 \\ \rho & 0.7 & 1 \end{bmatrix}\)

I think the question want you to check whether this matrix is positive-definite or not.

One easy way to check is the Sylvester's Criterion:

http://en.wikipedia.org/wiki/Sylvester_criterion

which essentially require you to have a positive determinant. When you compute it, you should obtain a quadratic polynomial in \( \rho \) and thus you should obtain a bound for it.