Is Correlation Transitive? Suppose that X, Y, and Z are random variables. X and Y are positively correlated and Y and Z are likewise positively correlated. Does it follow that X and Z must be positively correlated?
 A: Here is a great post by Terence Tao on the topic. Words from the man himself: 

I came across the (important) point that correlation is not necessarily transitive: if $X$ correlates with $Y$, and $Y$ correlates with $Z$, then this does not imply that $X$ correlates with $Z$.

A: We may prove that if the correlations are sufficiently close to 1, then $X$ and $Z$ must be positively correlated.
Let’s assume $C(x,y)$ is the correlation coefficient between $x$ and $y$. Like wise we have $C(x,z)$ and $C(y,z)$. Here is an equation which comes from solving correlation equation mathematically :
$$C(x,y) = C(y,z) C(z,x) - \sqrt{ (1 - C(y,z)^2 ) (1 - C(z,x)^2 ) }$$
Now if we want C(x,y) to be more than zero , we basically want the RHS of above equation to be positive. Hence, you need to solve for :
$$C(y,z) C(z,x) > \sqrt{ (1 - C(y,z)^2 ) (1 - C(z,x)^2 ) }$$
We can actually solve the above equation for both C(y,z) > 0 and C(y,z) < 0 together by squaring both sides. This will finally give the result as C(x,y) is a non zero number if following equation holds true:
$$C(y,z) ^ 2 + C(z,x) ^ 2 > 1$$
Wow, this is an equation for a circle. Hence the following plot will explain everything :

If the two known correlation are in the A zone, the third correlation will be positive. If they lie in the B zone, the third correlation will be negative. Inside the circle, we cannot say anything about the relationship. A very interesting insight here is that even if $C(y,z)$ and $C(z,x)$ are 0.5, $C(x,y)$ can actually also be negative.
