3
$\begingroup$

I'm referring to the Econometrica (1994) paper by Imbens and Angrist. In particular, I don't really understand the random variables $D_i(z)$ (defined on page 468). Could you please provide me with intuition about them and explain why, specifically, for $z\neq w$, $$ D_i(z)-D_i(w) $$ can only be either $1$ and $-1$? I understand that, in order to get $1$ or $-1$, the authors are looking at 2 scenarios $0-1$ and $1-0$. But why are the remaining scenarios $1-1$ and $0-0$ excluded?

$\endgroup$
2
$\begingroup$

I hope this meets your idea for intuition, but equation 1 comes from using the Law of Total Expectation with the independence condition (Condition 1). There are four possible values of $D_i(z)-D_i(w)$. $$D_i(z)=D_i(w)=1$$ $$D_i(z)=D_i(w)=0$$ $$D_i(z)>D_i(w)$$ $$D_i(z)<D_i(w)$$. Consider the LHS of (1). $$E[(D_i(z)-D_i(w))*(Y_i(1)-Y_i(0))]$$ From the four possibilities above it is clear that this expectation only takes on a value other than 0 when the last two are true, where the third option gives a value of 1 and the fourth a value of -1. So then using the Law of Total Expectation with the independence condition yields the equality from the paper.

| improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks for the answer. Let me study it a bit and I will get back to you. $\endgroup$ – yurnero Feb 5 '19 at 14:51
  • $\begingroup$ Sounds good. If you have any questions let me know. $\endgroup$ – soccer_stats Feb 5 '19 at 15:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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