I have a basic question on the relation between counterfactual outcomes and treatment.

Notation: Let $D$ denote the treatment, taking value in $\{0,1\}$ where $D=0$ means untreated and $D=1$ means treated.

Let $Y_0, Y_1$ denote the counterfactual outcomes.

Perfect randomisation assumption: $(Y_0, Y_1) \perp D$

Under this assumption we can show that ATE=ATT (average treatment effect = average treatment effect among treated). Indeed,

$$ \text{ATT}\equiv E(Y_1-Y_0|D=1)=E(Y_1|D=1)-E(Y_0|D=1)=E(Y_1)-E(Y_0)\equiv \text{ATE} $$ where the last step comes from the fact that $(Y_0, Y_1) \perp D$ $\Rightarrow $ $Y_1\perp D, Y_0\perp D$

Question: it seems that to show ATE=ATT it is sufficient to have marginal independence (i.e., $Y_1\perp D, Y_0\perp D$) rather than joint independence (i.e., $(Y_0, Y_1) \perp D$). Is this correct?


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