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

Question

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?

Answer 1

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.