Does $E(X \mid Y,Z)=0$ imply $E(X \mid Y)=0$?

Question

Does $E(X \mid Y,Z)=0$ imply $E(X \mid Y)=0$?

In other words, if we have $E(X \mid Y,Z)=0$ then can we also say $E(X \mid Y)=0$?

Answer 1

Yes, by the law of iterated expectations (with a slight generalization):

$\mathbb{E}\left(X\mid Y\right)=\mathbb{E}\left[\mathbb{E}\left(X\mid Y,Z\right)\mid Z\right]=\mathbb{E}\left[0\mid Z\right]=0$.