Dear all,

I am estimating a reduced form CRE tobit as the first stage model in a control function approach (CFA). My second stage contains all the variables from the first stage as well as the generalized residuals, minus the IV. Now, I have found some papers (see below) where the authors added
- squared variables
- interaction variables
- interaction variables where one of the interactors is the dependent variable from the first stage
- interaction variables where one of the interactors is the generalized residual from the first stage

I am aware that squared coefficients and interaction variables are not useful when estimating tobit. However, I was wondering if it was ok to add the aforementioned types of variables in the second stage even if they do not appear in the first stage. To the best of my knowledge, one includes all exogenous variables in the first-stage of a 2SLS IV regression, because the endogenous variable might be correlated with other independent regressors in the equation and could thusly introduce bias. As the CFA is generally seen as an IV approach, does the same hold true? Why or why not?

Many thanks for your kind attention and help,
Anne

Sources:
Liverpool-Tasie, L. S. O., et al. (2014). Understanding fertilizer use and profitability for rice production across Nigeria’s diverse agro ecological conditions. Selected Paper prepared for presentation at the Annual Bank Conference on Africa, June 23-24th. Washington D.C. , World Bank, https://www.worldbank.org/content/da...pool-tasie.pdf.
Theriault, V., et al. (2018). "Economic incentives to use fertilizer on maize under differing agro-ecological conditions in Burkina Faso." Food Security 10(5): 1263-1277. https://link.springer.com/article/10...571-018-0842-z.
Note: This question was also posted on StackExchange Cross Validated https://stats.stackexchange.com/ques...ction-approach