Correlation of sum

Question

What is Corr(X+Y, Z) in terms of Corr(X, Z) and Corr(Y, Z)? If there is no direct formula, is there any relationship in terms of > or <? Also, the same thing for rank correlation and Kendall's Tau?

Answer 1

Note $\textrm{Cov}(X+Y,Z)=\textrm{Cov}(X,Z)+\textrm{Cov}(Y,Z)$ using the (bi)linearity of covariance.

This does not translate into correlations without a bit of a mess. You can say:

$$\textrm{Cor}(X+Y,Z)\sqrt{\textrm{Var}(X+Y)}=\textrm{Cor}(X,Z)\sqrt{\textrm{Var}(X)}+\textrm{Cor}(Y,Z)\sqrt{\textrm{Var}(Y)}$$

By manipulating the variances, particularly of $X+Y$, this provides a way of illustrating possible relationships:

• for example with $(X,Y,Z)=(0,0,0)$ or $(50,50,50)$ or $(100,100,100)$ with equal probability, then $$\textrm{Cor}(X+Y,Z)=1<2=\textrm{Cor}(X,Z)+\textrm{Cor}(Y,Z)$$

• while with $(X,Y,Z)=(0,100,0)$ or $(100,0,0)$ or $(51,51,100)$ with equal probability, you get $$\textrm{Cor}(X+Y,Z)=1 > 0.023 \approx \textrm{Cor}(X,Z)+\textrm{Cor}(Y,Z)$$