If Cov(X,Y|X>0) > 0 and Cov(X,Y|X<0) >0 then Cov(X,Y) > 0?

Question

If $Cov(X,Y|X>0) > 0$ and $Cov(X,Y|X<0) >0$ then $Cov(X,Y) > 0$?

Any idea?

I dont know how to think this question. I would answer yes. Iam imagine that as $X>0$, when X increase Y increase with X thats why $Cov(X,Y|X>0) > 0$. Am I right?

Answer 1

Let $(X,Y)$ be uniform on the four points $(2, -2)$, $(3, -1)$, $(-2, 2)$, and $(-3, 1)$.

Then $\text{Cov}(X,Y)$ is negative, but the two conditional covariances are positive.

[Plot the four points to gain intuition for how to construct other counterexamples.]