# Imbens and Angrist (1994): the $D_i(z)$ variables

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?

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 $$D_i(z)-D_i(w)$$. $$D_i(z)=D_i(w)=1$$ $$D_i(z)=D_i(w)=0$$ $$D_i(z)>D_i(w)$$ $$D_i(z). Consider the LHS of (1). $$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.