Can we always write $Z=E(Z|X)+E(Z|Y)+\varepsilon$ where $E(\varepsilon|X)=E(\varepsilon|Y)=0$?

Question

Suppose that we have the random variables $X,Y,Z$. Can we always decompose one of them (say $Z$) as $$ Z=E(Z|X)+E(Z|Y)+\varepsilon $$ where $E(\varepsilon|X)=E(\varepsilon|Y)=0$? (Suppose all expectations exist) This is true when we project on only one random variable (see here). But the proof provided does not seem to work when we project on two random variables. I worry that there is a correlation term missing.

Answer 1

Suppose that you could. Then, taking $E\left(\cdot|X\right)$ of both sides yields

$$E(Z|X) = E(Z|X) + E(E(Z|Y)|X) + E(\varepsilon|X).$$

The first term on the RHS is unchanged since $E(Z|X)$ is already measurable w.r.t. $X$. Then, you have

$$E(\varepsilon|X) = -E(E(Z|Y)|X)$$

but there is no reason why the RHS should be zero. $E(Z|Y)$ is a function of $Y$, but this may still have some arbitrary dependence on $X$.