# Fuzzy RDD when individuals can affect the probability of treatment

I want to evaluate the effect of getting into a program on earnings. In my case individuals that get a score above $$c$$ can decide weather to get into the program or not. On the other hand those that get below $$c$$ cannot join the program. How can I evaluate the effect of getting into a program on earnings?

I understand that the probability of getting into the program jumps discountinuosly at score $$c$$ and what is tipically done in this framework is to use a small sample of observations around the cutoff and run Fuzzy RDD, or basically 2SLS regression where in the first stage, you regress the binary outcome of partecipating or not to the program on the test score, and than plug the estimated probabilities in the second stage regression where the dependent variable is earnings. How can I pretend that the first stage is valid? I mean, even if I consider only observations around then cutoff, wouldn't be the case that more motivated guys take slightly higher grades, being more likely to get to the program, and earn more because of that?

• The technique you describe assumes data available for at least three classes. Earnings for those below threshold c and who do not qualify for the program. Earnings for those above threshold c and who do complete the program. And earnings for those above threshold c who do not complete the program. Unless there is independent statistical assumptions and relevant data to do regression on "motivation" it is of no concern. Oct 11 at 21:02
• Sorry, I forgot to say that I do not observe motivation, and I do not have any control variable for it. Oct 12 at 7:45