Including a variable in the outcome equation only with Heckman probit In Stata, we are using heckprobit to address an endogeneity concern we have in our model. The model and the instrument used on the selection equation look great and they are significant.
We found a variable that would be interesting to look at on the outcome equation but it wouldn't make sense to include it on the selection equation.
Question
Is it possible to include a variable on the outcome equation only and exclude it from the selection equation?
This is my first time using this method and I am trying to figure out if I am violating any econometric rule by forcing a variable on the outcome equation only.
I appreciate the help.
 A: As you note, it is the instrument used in the first stage (i.e. selection equation) and not included in the second stage (i.e. treatment equation) that is most important in Heckman models. As Sartori (2003) states, "there should be at least one “extra” explanatory factor that influences selection but not the subsequent outcome of interest."
To answer your question directly, it is fine to not include a variable in the selection equation and only in your treatment equation, if this is what theory predicts. After all, the selection equation predicts likelihood of selecting into the sample. If a variable theoretically only predicts variation in the outcome variable but not likelihood of selecting into the sample, there is no need to include in the selection equation. If you are pressed on this issue further, I would run a robustness test (but not change the main model) with the variable also included in the selection equation and report that the results remain unchanged (assuming that this is truly the case).
