Fixed effects Poisson unbalanced panel with Binary endogenous regressor: GMM moment evaluator program

[AJ Williamson]

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.

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):

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:

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.

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.

[Phil Bromiley]

You didn't get a quick answer. You'll increase your chances of a helpful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex. We also do not encourage posting pictures or asking us to open files. If it is not worth your trouble to try to explain the problem yourself, you can't expect us to donate our time to you to help answer it. Folks won't open files due to virus concerns.

Regarding your questions - At the end of the documentation for every Stata command is a section laying out the main equations used in the estimator. Additive versus multiplicative errors certainly have a lot to do with your theory and substantive problem (which we don't know).

You're far more likely to get a useful answer if you post a real problem you have rather than asking us to interpret a paper for you.