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

Question

The following statement is true?

Suppose $E(Z\mid X)=0$. Then $E(Z\mid X,Y)=0$.

Any comments would be appreciated.

Answer 1

Not nescessarily.

Suppose $X,Y\overset{iid}\sim\mathcal U\{-1,1\}$ and $Z=XY$

Then $\mathsf E(Z\mid X) = X~\mathsf E(Y) = 0$ but $\mathsf E(Z\mid X,Y)=XY\neq 0$ because $XY\in\{-1,1\}$ .