Instruments for Control Function Approach involving quadratic and interaction terms

[Nitin Jain]

My panel model is of the following form where x1, x2 are the endogenous regressors and z1 and z2 are the moderators.

 
 y = f(x1, x1^2, x1z1, x1z2, x2, x2^2, x2z1, x2z2)

The instrument for x1 and x2 are M1 and M2 (based on averages of similar observations except the one under consideration). As per the paper by Haans et al.(2015) https://onlinelibrary.wiley.com/doi/....1002/smj.2399, I am instrumenting both the linear and quadratic terms of x1, x1^2 with M1 and M1^2 and x2, x2^2 with M2, M2^2.

I am confused if I should consider the instruments for interaction terms - M1z1, M1z2, M2z2, M2z1 - also as the instruments for the linear and quadratic terms. Please advise. Thanks.

[Fei Wang]

I am confused if I should consider the instruments for interaction terms - M1z1, M1z2, M2z2, M2z1 - also as the instruments for the linear and quadratic terms.

Yes you should if the function f(.) is linear and you're using the 2SLS strategy. BTW, the whole procedure seems irrelevant to the Control Function Approach which handles the issue differently.

[Nitin Jain]

Yes you should if the function f(.) is linear and you're using the 2SLS strategy. BTW, the whole procedure seems irrelevant to the Control Function Approach which handles the issue differently.

Thanks, Fei. Control function approach is more appropriate for non-linear functions. The error terms from the first stage would be used in the second stage in it. However, I am not sure if the instruments should be
M1, M1^2, M1z1, M1z2 for each term involving x1 and M2, M2^2, M2z1, M2z2 for each term involving x2.

[Fei Wang]

Nitin, essentially you have two endogenous variables x1 and x2. The two first-stage regressions would be regressing x1 (or x2) on the same set of all exogenous variables: M1, M1^2, M1*z1, M1*z2, M2, M2^2, M2*z1, and M2*z2. Then include the two residuals v1 and v2 in the main model to complete the CF procedure.

I should add that the procedure above applies to continuous x1 and x2. Please refer to Wooldridge's JHR paper (2015) for different cases.

[Nitin Jain]

Nitin, essentially you have two endogenous variables x1 and x2. The two first-stage regressions would be regressing x1 (or x2) on the same set of all exogenous variables: M1, M1^2, M1*z1, M1*z2, M2, M2^2, M2*z1, and M2*z2. Then include the two residuals v1 and v2 in the main model to complete the CF procedure.

I should add that the procedure above applies to continuous x1 and x2. Please refer to Wooldridge's JHR paper (2015) for different cases.

Thank you, Fei. I will surely read Prof. Wooldridge's paper. I think I should consider 8 endogenous variables - 4 for each of x1 and x2. Please check this one - https://onlinelibrary.wiley.com/doi/....1002/smj.2399 - where they have mentioned the instruments for linear and quadratic terms.

[Atik Mohammed]

I am using Callaway and Sant'Anna (2021) to estimate cohort-specific AATs. However, I also like to find out how another variable impacts the group-time ATTs through the treatment variable. Using TWFE, I would probably interact (Post_tXD_i) with the third variable. However, this is not allowed when I use the csdid command. Is there a way I can interact the cohort-specific dummy with another covariate that is possibly time varying to estimate how it impacts on the group-time ATT?