Basic question about correlation

#1
I'm trying to build an econometric model and have a basic question:

Suppose corr(X,Y)=0.5 and corr(X,Z)=0.5, where X, Y, and Z are RVs. Is the correlation between Y and Z equal to 1? If not, is it possible for Y and Z to be uncorrelated?

My intuition about this is mixed, and I haven't been able to derive any conclusions using the definitions (i.e. cov(X,Y)=E(XY)-e(X).)

Thanks in advance for your help!
 

BioStatMatt

TS Contributor
#2
It is certainly possible that the correlation between Y and Z is equal to one. This occurs when Y = Z. However, if they are different, then it is also possible that these variables are uncorrelated. This occurs when Y and Z are independent. It is likewise possible that the correlation between Y and Z can take on any value between -1 and 1. To see how this works, you might use some software that can generate multivariate (3 variate) normal data. You will have the situation where corr(X1, X2) = 0.5, corr(X1, X3) = 0.5, and corr(X2, X3) = 0 when the covariance matrix is:

4 2 2
2 4 0
2 0 4

You could then plot these data in a 3D plot and see whats going on. So I guess your intuition is correct :)

~Matt