It's Not So Easy to be LATE

We all know about the four assumptions needed in order to interpret an IV estimate as a local average treatment effect (see below if you've forgotten). But I think we economists tend to focus on the independence assumption (and the first stage),  and we sort of pay little mind to the exclusion assumption. It turns out that the exclusion restriction can be a big deal! See here. I really like the simple examples. Challenge to blog readers: Can you think of a real world example of a potential IV that satisfies the first three assumptions but not monotonicity?