Maximum likelihood estimation of a probit model with sample selection

[Mat Log]

Hello, I'm using the command heckprobit to fit a probit model with sample selection. In Stata documentation it is written:

"For the model to be well identified, the selection equation should have at least one variable that is not in the probit equation. Otherwise, the model is identified only by functional form, and the coefficients have no structural interpretation."

My understanding is that we still need to assume functional forms for the error terms (i.e. normal distributions) to identify the model even if we have a variable in the selection equation that is not in the outcome equation. I need help to understand what is really missing when such a variable is not available.
Thanks

[Phil Bromiley]

You'll increase your chances of a useful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex. You didn't get a quick answer. Many are not willing to invest time in trying to guess exactly what you're doing.

This is covered in any elementary econometrics text. You need a variable in the selection equation that does not appear in the outcome equation to identify the model. It is similar to the need for an exogenous instrumental variable to identify a simultaneous equation. While models can be identified in theory by functional form assumptions, these are somewhat questionable.

[Mat Log]

Thank you very much for your answer and I apologize if I have not followed the FAQ.

I agree that it is covered in several textbooks but what I would like to understand is what is exactly the value added of having an extra variable in the selection equation when we use a parametric model? How does it "boost" identification exactly? Using the command heckprobit, we still assume functional forms for the error terms (i.e. joint normal distributions) to identify the model even if we have a variable in the selection equation that is not in the outcome equation.

Thank you once again.