If X is correlated with Y, and Y is correlated with Z, does it follow that X and Z are correlated?
No, not necessarily. That is, the relationship of correlation isn't necessarily transitive.
In a blog post from last year the Fields Medallist, Terrence Tao, discusses the question: "When is Correlation Transitive?", and provides a thorough mathematical answer.
He also provides this simple example of correlation intransitivity:
This is something for students of econometrics to keep in mind!