[Nitin Jain]

I read a bit too quickly. Because you regress the interactions on the nonlinear functions it is not the same as the "forbidden regression." But the better way is to use X1hat*X3 as the IV for X1*X3 and X2hat*X3 as the IV for X2*X3. And you shouldn't have exclusion restrictions in your first stages.

reg X1 Z1 Z2
predict X1hat
reg X2 Z1 Z2
predict X2hat
ivregress 2sls Y X3 (X1 X2 c.X1#c.X3 c.X2#c.X3 = Z1 Z2 c.X1hat#c.X3 c.X2hat#c.X3, vce(r)

Thank you, Prof. Wooldridge. Here Z1 is the instrument for X1 and Z2 for X2. Should we not keep just the relevant instrument instead of both the instruments when predicting X1hat and X2hat?

Also, could you please suggest a reference that we could cite while using this approach for interaction terms with endogenous variables? Thanks.

[Andrew Musau]

In linear regression, if a RHS variable is truly "irrelevant", its coefficient is zero. Whether you include or exclude it from the regression does not affect the coefficients of the other regressors. While the degrees of freedom may differ with inclusion or exclusion, this is negligible when dealing with a few regressors and a sufficiently large sample size. I recall a discussion of this method in Jeff's MIT Press book, so you can find the relevant chapter and cite it.