Implications of conditional mean independence with respect to two random variables

Question

Suppose $$ E(U|X_1,X_2)=0 $$ Is it true this implies that $E(UX_1)=E(UX_2)=0$ and if yes could you help me to show it?

Answer 1

First, by definition

$E[U|X_1,X_2]=E[U|\sigma(X_1,X_2)]$. Since $X_1$ is measurable in $\sigma(X_1,X_2)$, due to the basic properties of the conditional expectation, we have that

$E[X_1 U|X_1,X_2]=X_1 E[ U|X_1,X_2]=0$.

Likewise, we obtain $E[X_2 U|X_1,X_2]=0$.

Now, for $i=1,2$, $E[X_i U]=E[E[X_i U|\sigma(X_1,X_2)]]=E[0]=0$.

And we obtain the result.