Dummy Variables in IV designs

[Francesco Migliore]

Hi to all,
I have a question about the validity of an IV design obtained using a dummy variable. I want to analyze the impact of being a populist municipality on abstention rates. I use a model with Year and Province FE

reghdfe abs_I populist_m $controls, a(Year pp) vce(robust)

. I found moreover that being a populist municipality in time t is related with the share of disaffected voters in t-1. I estimate with the same fixed effect model as before

reghdfe populist_m past_sh $control, a(Year pp) vce(robust)

. I used so the code

ivregress 2sls abs_I i.Year i.pp $controls (populist_m=past_sh), vce(robust)

but I found inconsistent estimates.

I think is that the problem is that populist_m is a dummy variable and so I should not use a fixed effect model to estimate the relation with the past share of disaffected voters. In fact I estimated the residuals of

reghdfe populist_m past_sh $control, a(Year pp) vce(robust)

and I notice that are negative. In this occasion I would have used a probit model to calculate how an increase in the share of disaffected voters affected the probability of observing a populist_municipality. However, it makes sense to use the predicted values of this equation, which would be

probit populist_m past_a i.Year i.pp $controls

, in my main specification? And if so, what values should I use after my probit equation?