Control function approach with PPML: should I keep the bilateral structure of the data in both stages?

[Etienne Bacher]

Dear all,

I want to estimate a gravity model using PPML. These are the features of my model:

• the main dependent variable is origin-destination-time varying, Yijt
• the interest variable is destination-time varying, Xjt
• there are some origin-destination controls (distance, etc.) and other destination-time controls (in addition to the interest variable)
• I use origin-time and destination FE

I suspect that my interest variable (destination-time varying) is endogenous, so I found an instrument for it. I apply the control function approach, as advised in this answer: https://www.statalist.org/forums/for...95#post1554595. Basically, in the first step I regress my endogenous variable on the instrument and all the other controls and FE that I have in the main equation. I then extract the predicted residuals and plug them in the second step.

This means that I keep the bilateral structure of my dataset in both stages. However, I was told that it doesn't make sense to include the bilateral controls and bilateral FE in the first stage because the endogenous variable and the instrument are only destination-time varying. This means that I should estimate two completely different specifications in both stages of the control function:

• the first stage would consist only of destination-time variables, time and destination FE, and therefore would consist of j*t observations. I should extract the residuals and merge them back to the full dataset
• the second stage would be identical to what I described before, using i*j*t observations

Which procedure is correct? Should I keep the bilateral structure of the data in both stages or estimate two different specifications?

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Note that I also asked a similar question on CrossValidated (no answer so far): https://stats.stackexchange.com/ques...ntrol-function