Q: Conditional independence preservation Suppose $z_1$ and $(u_2,u_3)$ are independent conditional on $u_1$. Does this imply that $u_2$ and $z_1$ are independent conditional on the joint $(u_1, u_3)$? If not, can I still say something about the conditional correlation of $u_2, z_1$ conditional on $u_1, u_3$?
(I am trying to understand a paper that uses this fact but for the life of me cannot workout the details.)
