0
$\begingroup$

In Angrist & Pischke's book mostly harmless econometrics, they explain that if the treatment in an RCT $D_i$ is randomly assigned, then $D_i$ is independent of potential outcomes and the following holds:

$$ \begin{align} E[Y_i|D_i = 1] - E[Y_i|D_i = 0] &= E[Y_{1i}|D_i = 1] \color{red}{- E[Y_{0i}|D_i = 0]} \\\\ &= E[Y_{1i}|D_i = 1] \color{red}{- E[Y_{0i}|D_i = 1]} \end{align} $$.

The main thing here, as I understand it, is that we can switch the terms in red $E[Y_{0i}|D_i = 0]$ $E[Y_{0i}|D_i = 1]$. But How do I interpret this? I get why we want to randomly assign treatment (because we want the treatment group and control group to be identical), but I don't understand what the above equation says. For me it looks like the expected outcome for the control group is the same even if they get treatment, i.e, $D_i = 1$. I think I am misinterpreting something here.

For completeness. After this, they go on to say that we can simplify further. Starting from the last line above:

$$ \begin{align} E[Y_{1i}|D_i = 1] - E[Y_{0i}|D_i = 1] &= E[Y_{1i} - Y_{0i} | D_i = 1] \\\\ &= E[Y_{1i} - Y_{0i}] \end{align} $$, noting that "The main thing, however, is that random assignment of $D_i$ eliminates selection bias".

Let me know if I need to add some context!

$\endgroup$

1 Answer 1

1
$\begingroup$

an intutive explanation is that given independence of treatment from potential outcomes the average of $Y_0$ in the control group (roughly E[Y_{i0}|D=0]) is as good an estimate of E[$Y_0$] as the average of $Y_0$ in the experimental group. Thus, given the aforementioned independence you can 'exchange' participants in the experimental and the control group; they are just two random samples from the larger population.

Hope that helps. Best Stefan

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.