Is this possible that Cor(X,Y)=0.99Cor(X, Y)=0.99, Cor(Y,Z)=0.99Cor(Y, Z)=0.99 but Cor(X,Z)=0Cor(X, Z)=0?

Let X, Y, and Z are three random variables. Intuitively, I think that it is impossible to have Cor(X,Y)=0.99, Cor(Y,Z)=0.99 but Cor(X,Z)=0. My intuitive thought is that X and Z are nearly linearly correlated to Y. Hence, they are more or less linearly correlated, which makes the last equality impossible.

I pose this question because of the question and the comments (include my comments) here.

In general, as some others point out, I agree that it is possible that for some ρ>0 we may have Cor(X,Y)=ρ,Cor(Y,Z)=ρ and Cor(X,Z)=0(1).

My questions are:

  1. Do you think that (1) is wrong when ρ is close to 1, e.g., 0.99?
  2. If (1) is wrong when ρ is close to 1, what is the maximum value of ρ so that (1) can be correct?

Answer

The correlation matrix needs to be positive semi-definite with non-negative eigenvalues. The eigenvalues of the correlation matrix are the solutions of
|1λρρρ1λ0ρ01λ|=(1λ)((1λ)22ρ2))=0
so the eigenvalues are 1 and 1±2ρ. These are all non-negative for
12ρ12.

Attribution
Source : Link , Question Author : TrungDung , Answer Author : Jarle Tufto

Leave a Comment