Hi Everyone,
I have a question. I'm writing a moment evaluator program to try and achieve consistent parameter estimation for treatments with a binary endogenous regressor given an exponential mean function in a severely unbalanced panel format. The form of my model, following STATA's gmm documentation, is an additive error.
Array

Next, it says that I can use this moment condition a few pages below to deal with endogeneity (idiosyncratic error endogeneity for one dummy (0/1) covariate):
Array


This, above, Jeff Wooldridge moment condition, and it is quite easy to program. However, there seems to be a slight contradiction to what STATA says and some of the literature. For example:

1. Cameron and Trivedi says that only the multiplicative error specification ((as in \mu_it = a_i*exp(x_{it}*\beta)*\epsilon_it, not ,\mu_it = a_i*exp(x_{it}*\beta) + \epsilon_it as STATA says...the additive error form) could be used with a different moment condition to deal with endogeneity here: Array



2. Windmeijer seems to suggest the same thing. Multiplicative error specification (as in \mu_{it} = a_i*exp(x_{it}*\beta)*\epsilon_it, not ,\mu_{it} = a_i*exp(x_{it}*\beta) + \epsilon_{it} as STATA says...the additive error form) however, the moment condition seems different than what STATA says above. Array

The Cameron/Trivedi full pdfs is attached.

Questions:
1. Does anyone have any insight into the additive error (\mu_{it} = a_i*exp(x_{it}*\beta) + \epsilon_{it}, vs. the multiplicative error (\mu_{it} = a_i*exp(x_{it}*\beta)*\epsilon_{it}) specification for this approach for dealing with endogeneity of the regressors? Is STATA's additive error form correct with the exponential function and that given moment condition?

2. If so, why does it seem to contradict the fact that only a multiplicative error specification can be used with a different moment condition (Cameron and Trivedi)

Please, if anyone understands or could provide insight, I'd appreciate it.

Thank you.