We are heavily discussing how to analyze our experiment.
There is a binary treatment and a binary dependent variable. Therefore I think I should not use OLS but rather probit/logistic regression.
It is kind of a medical experiment where we invited a random subset (assigned to treatment) of our study group (all of whom would theoretically benefit from the treatment and the treatment effect) to come to my university for a guided self-measurement of vital functions. People answer yes and no to this. Those who said no went into the control group, as in addition to those who were already previously in the control group. We kept track of this.
In randomized order, I asked those people who said "yes" whether they would come to the university on a certain day for the treatment (measurement of vital functions). Of course, other days were also possible. Only some actually came to the measurement = only some of the assigned treatment, that agreed, actually received treatments. Others might have come, but at some point, we had no more appointments available. All the people who didn't actually physically came went into the control group.
Some time later, we observed the binary dependent variable (some attrition).
I wanted to estimate an instrumental variable IV model with the intention to treat (ITT) (measured before the first question whether people would come) as an instrument. And the actual treatment (people came to us) is instrumented with it. I know this only gives Local Average Treatment Effects (LATE) but I cannot improve on this, I suppose.
Now I have read that it comes to the forbidden regression problem in Woolsrodge Econometric analysis of cross-section and panel data on page 236 of the 2010 edition. I could attach a photo but not sure about copyright.
Question: The statistical advisor at my university said that I can estimate 2 logistic models and still use two stage least squares. And this leaves me confused! Isn't that a non-linear model. What is the correct way out of here.
Also, is there a Stata (or R) command to do so? Edit: Forgot to mention, I know of the ivprobit command in Stata, but it says its only for continuous instrument and I only have a binary/boolean instrument.
My alternative would be using matching, right? I found this question: Comparing the effect of a treatment that was optional for its receivers