IV - Is this correct?

Question

I have the following question.

Currently I have an econometric model that uses an instrumental variable, particularly a Bartik instrument (which some people call shift-share). I'm proceding as Mayda et al. (2022) using two instruments for my low and high skilled immigrants, estimating my model with 2SLS. My model would look something like this: $$Y_{i t}=\delta_{i}+\delta_{t}+\beta_{L} \frac{L_{i t}}{P o p_{i t}}+\beta_{H} \frac{H_{i t}}{P o p_{i t}}+\varepsilon_{i t}$$ Being $i$ the municipality and $t$ the time (I also have two-way fixed effects). Currently, I have two instruments, one for each variable, but when I use them the F-statistic of the first-stage is low. However, I have seen that, if I include also the quadratic form of my two instruments, then the F-statistic is high enough. I wanted to ask if this is correct or if I shouldn't use this type of interaction in my model. Any reference is going to be useful.

Thank you for your time.

Answer 1

Currently, I have two instruments, one for each variable, but when I use them the F-statistic of the first-stage is low. However, I have seen that, if I include also the quadratic form of my two instruments, then the F-statistic is high enough.

This depends on what you think the true model is. If the relationship between the endogenous regressor and instrument in the first stage is non-linear and quadratic, then this was correct solution (eg in regression of wages on experience or age you would theoretically expect non-linear effects that’s why age or experience usually enters Mincer equation with also quadratic terms).

However, if there is no reason why the relationship should be quadratic you should not insert the variables in the quadratic form as you would have misspecified mode. Getting the right functional specification is more important then variable not being significant. If first stage is weak consider finding better instruments.