# Is this formula for the Law of Iterated Expectations correct?

I saw two versions of the law of iterated expectations, this one: \begin{align} E(E(Y\vert X)) = E(Y) \end{align} and this one: \begin{align} E(E(Y\vert X_1, X_2)\vert X_1) = E(Y \vert X_1) \end{align}

I was wondering if this one is also correct or not: \begin{align} E(E(Y\vert Z) \vert X) = E(Y\vert X) \end{align}

Here $$X,X_1,X_2,Y,Z$$ are random variables.

The conjecture is false. If $$X=Y$$, $$Y$$ and $$Z$$ are independent, $$E(Y)=0$$ then $$E(Y|X)=Y$$, $$E(Y|Z)=0$$, and $$E(E(Y|Z)|X)=0$$.

The reason the second version you state works but the third one does not is that "the smaller information set dominates", see this elaborate answer: A generalization of the Law of Iterated Expectations

Conditioning on $$X_1,X_2$$ means conditioning on more information than conditioning on $$X_1$$ alone, which implies that first conditioning on $$X_1,X_2$$ and then conditioning on $$X_1$$ is as good as the latter only.

In the third statement, we have no such relationship between $$X$$ and $$Z$$.