*Bounty: 50*

*Bounty: 50*

Suppose that a set of covariates, $X_i$ follows a distribution that is conditional on another variable, $A_i$, for $i in {1, ldots N}$ individuals. For example, $X_i$ can be income, and $A_i$ can be age, defined as young, middle-age, and elderly. Then, suppose that we have:

$$

X_i mid A_i overset{iid}{sim} F

$$

Then, we say that $X_i$ is only i.i.d. within subsets defined by the age variable. Now let $Y(1),Y(0)$ denote the potential outcomes and $T$ the treatment indicator. Suppose that the joint distribution of the potential outcomes, treatment, and covariates are only i.i.d on **subsets** defined by $A_i$, such that

$$

(Y_i(1),Y_i(0),T_i, X_i)mid A_i overset{iid}{sim} F

$$

This scenario appears to hint at a heterogeneous treatment effect, since the distributions of the variables are potentially different given which age is it conditioned on. In example,

$$

tau_i = E[Y_i(1)-Y_i(0)]

$$

where $tau_i$ may be different than a single global $tau$.

In such an example, I am wondering how the unconfoundedness assumption needs to be modified given an observational study. For example, if we assume that the following holds,

$$

(Y_i(1),Y_i(0)) perp T_i mid X_i

$$

will it be enough to identify $tau_i$? In other words, if we have the joint distribution specification above, how will identification and estimation of the average treatment effect be impacted?