I have a basic question on the relation between counterfactual outcomes and treatment. In what follows ATE means average treatment effect.

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.

Let $Y=DY_1+(1-D)Y_0$ denote the factual outcome.

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

Under this assumption we can show that ATE is identified (i.e., it is a function of known probability distributions). Indeed,

$$ \text{ATE}\equiv E(Y_1)-E(Y_0)\overbrace{=}^{(Y_0, Y_1) \perp D \Rightarrow Y_1\perp D, Y_0\perp D} E(Y_1|D=1)-E(Y_0|D=0)= \underbrace{E(Y|D=1)-E(Y|D=0)}_{\text{known in an identification exercise}} $$

Question: it seems that to show that ATE is identified 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?

  • $\begingroup$ $Y_{1,i}$ and $Y_{0,i}$ are not observed simultaneously for subject $i$ (fundamental problem of causal inference). We observe, however, $(Y_i,D_i)$ for all $i$. Then the probabilistic assumption should be $(Y_{1,i},Y_{0,i})\perp D_i$. $\endgroup$ – MauOlivares Oct 10 '18 at 21:11

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