I have some problems understanding the difference between ATE and ATET and the Selection Bias. To explain what my understanding is I have done the following representation so you can correct me:
We want to see the effect of taking LSD, to do so we have a Group to which we assign the "treatment" so $D=1$ and a group which has no treatment assigned so $D=0$. On the top left corner we can see $Y_{1i}$ and on the right bottom corner $Y_{0i}$. The problem comes now, as far as I can understand because there is no randomization (?) in the sample we have that some guys in the control group that take LSD because they are drug-abusers (red dots) ($E[Y_{0i}|D=1]$). In addition some other guys in the treatment group do not do LSD because they are afraid ($E[Y_{1i}|D=0]$). So:
$ATE = E[Y_{1i}|D=1] – E[Y_{0i}|D=1] + E[Y_{0i}|D=1] – E[Y_{0i}|D=0]$
Where the first two terms are ATE and the last two are Selection Bias.
Why $E[Y_{0i}|D=1]$ is not observable?
Is this explanation correct? Why if we apply randomization in the treatment we manage to get $E[Y_{0i}|D=1] – E[Y_{0i}|D=0] = 0$ ?
Best Answer
$E[Y_{0i}|D=1]$ is not observable because we cannot know the counterfactual, i.e. what outcome the individual would have if they did not receive the treatment as they always get the treatment (D=1). What randomisation does is guarantee that the treatment assignment is statistically independent of potential outcomes, therefore: $E[Y^*_0]=E[Y^*_0|D=1]=E[Y^*_0|D=0]$ and hence $E[Y^*_0]=E[Y^*_0|D=1]-E[Y^*_0|D=0]=0$, i.e. the expected outcome for a non-treated individual is the same if they are in the treated group or if they are in the control group.