Suppose we have random variables Y, D and X, where Y is independent of D conditional on X (Y⊥D|X). If there is another variable Z=f(X), where f(.) is a measurable real function, my question is: (1) under what conditions can we have Y⊥D|Z ?; (2) do we need the sigma-algebra σ(Z) belongs to
σ(X), so σ(Z) is sub-σ-algebra of σ(X)?


This is crucial to casual inference in econometrics and statistics, where we want to know if the conditional independent assumption (CIA) condition can be relaxed.



Relaxing conditional independent assumption
A sufficient condition to have $Y⊥D|Z$ is that $f$ is injective. The sharp condition (if $Y$ and $D$ are not specified) is


$(\*)$ $\sigma(X)$ should be contained in the completion of $\sigma(Z)$.


If $(\*)$ holds, then conditioning on $X$ is equivalent to conditioning on $Z$.
If (\*) does not hold, then there is an event $A \in \sigma(X)$ such that $0<P(A|Z)<1$.
Take $Y=D=1\_A$. Then $Y⊥D|X$ but $Y $ and $D$ are dependent given $Z$.


