Using an enrollment probability as an instrument

Question

Can someone walk me through whether this makes sense?

1. Individuals can apply to be enrolled in a treatment (a particular program), or not.
2. The probability of actual enrollment for applicants varies over time (e.g., program gradually ramps up to full capacity)
3. Individuals can apply for the program only at a particular assigned point in time (batched cohorts). The time they can apply is selected randomly.

For each individual, we can thus calculate the probability of enrollment, z, at their particular application time (based on a trailing enrollment rate). This is similar to a propensity score.

In this case, is it possible to instrument the treatment with z? The necessary assumption for the exclusion restriction would seem to be that z is not correlated with individual characteristics which in turn are correlated with the outcome. This should be satisfied by the lottery on application timing.

However, I haven't seen many examples of using an enrollment probability or a propensity score as an instrument, and am wondering whether I am missing something.

Answer 1

Yes, in fact what you want to do is known as Fuzzy Regression Discontinuity.

Fuzzy RD is in essence IV where probability of treatment is used as an instrument. Or at least that is one way how to implement Fuzzy RD. You might want to have a look at the Mostly Harmless Econometrics by Angrist & Pischke for details.