I don't understanding the estimation of IV thoroughoutly. The following is my puzzle.
In the first stage, we have $$D_i = \alpha + \beta Z_i + u_i$$
then we get predicotrs $\hat{D_i}$.
In my mind, $\hat{D_i}$ is a mixture of "compliers, always-takers, never-takers and Defiers" given the IV variable, $Z$
Given a specific suition:
We want to test the effect of a medicien on health, so we have some people to take medicien and see how they will feel after taking it. The confounder is health condition. To aviod it, we randomly arrange for some people to take medication. But there are always-takers and never-takers and no defiers. Even if some people are arranged to take medication, they may not take it, and vice versa. And I think they are influenced by their own health status, the confounder mentioned before. People who never take medicine may choose not to do so because they are in good health. And always-takers may be the people who are not so healthy.
See $\hat{D}$ again, it is a mixture of compliers, always-takers and never-takers. Also, we know . Can we say that $\hat{D}$ is not indepent on the potential outcomes?
In this way, can we estimate the real effect? $$Y_i = \pi + \rho^{IV} \hat{D}_i + e_i$$
I don't know where my reasoning is wrong. Thank you a lot!