In the notation of the unconfounded assumption, does $$\left(Y(0),Y(1)\right)\perp W \mid X $$
$$ f(Y(0),Y(1), W\mid X)=f(Y(0),Y(1)\mid X)\cdot f(W\mid X)$$ ?
I can prove that the second line if the set of random variables $(Y(0),Y(1))$ is independent of $W$ given $X$: $$f(Y(0),Y(1), W\mid X)=f(Y(0),Y(1)\mid W,X)\cdot f(W|X)$$ by the conditioning rule. Since $f((Y(0),Y(1)|W,X)$ does not depend on $W$ by the assumption stated, the result is obtained. But every signle paper omits this discussion. I am studying this treatment effect literature by myself, so I need to understabd this fundamental assumption based on my econometrics knowledge.