TITLE: $E(X|Y)=0$ and $E(X|Z)=0$ imply $E(X|Y,Z)=0$?
QUESTION [5 upvotes]: Is it true that
$$
E(X|Y)=0\quad\text{and}\quad E(X|Z)=0
$$
imply $E(X|Y,Z)=0$? From iterated expectations, I understand that $E(X|Y,Z)=0$ implies both $E(X|Y)=0$ and $E(X|Z)=0$ but I don't know how to think about it in the reverse direction.

REPLY [6 votes]: Here is a counterexample: assume that $Y$ and $Z$ are independent, non Dirac, centered, and consider $$X=YZ.$$ Then $E(X\mid Y)=YE(Z)=0$ and $E(X\mid Z)=ZE(Y)=0$ while $E(X\mid Y,Z)=X$.