currently I am studying the model of potential outcomes and am still confused after reading the other answers on the other threads but also books treating this model.
How can the potential outcomes $(Y_{0i},Y_{1i})$ be independent of the treatment variable $T$?
Formally: $(Y_ {0i},Y_{1i}) \perp\!\!\!\perp T$ whereby $i$ is the $i$-th unit.
I know that we make this assumption to be able to eliminate the selection bias: $E[Y_{0i} | T_i=1] - E[Y_{0i} | T_i=0]$.
But how can the potential outcomes be independent of the treatment variable if we want to check for a causal relationship between $Y$ and $T$?
Looking at the switching equation: $Y_i=Y_{0i}+(Y_{1i}-Y_{0i})\cdot T_i$ with $T_i \in \{0,1\}$
$Y_i=Y_{0i}$ for $T_i=0$,
$Y_i=Y_{1i}$ for $T_i=1$
So I can not see why we can assume that the potential outcomes are independent of $T$ if the observed outcome we get is $Y_{0i}$ or $Y_{1i}$ depending on the treatment variable $T$.
The switching equations says that the potential outcome that will be assigned to our observed result $Y_i$ is dependent on the treatment variable $T$.
But assuming $(Y_ {0i},Y_{1i}) \perp\!\!\!\perp T$ while looking at the switching equation seems like a contradiction to me.
It looks like I am missing something here.
Would be happy if someone could explain me where my misconception lies.