# How to estimate $\gamma$ in the following model?

Suppose I have the following model:

$$Q=1(x'\beta+e>0)$$,

$$D=1(x'\alpha+\gamma Q+u>0)$$

and I want to estimate $$\gamma$$, the error terms $$e,u$$ are jointly normal.

If $$e$$ and $$u$$ are correlated, can I still estimate $$\gamma$$ consistently by running a probit using the second equation only?

Why or why not?

If I cannot, how could I estimate $$\gamma$$ in this case?

If $$u$$ is normally distributed conditional on $$x$$ and $$Q$$, then you could estimate through probit the second equation. Because $$Q$$ depends on $$e$$, and $$e$$ is correlated with $$u$$, this assumption likely fails (that's your point?). Also, note that if $$u \sim N(0, \sigma^2)$$ then probit really estimates $$\frac{\gamma}{\sigma}$$. (i.e., the ratio of the coefficient to the standard deviation of the error).
If the equation with $$Q$$ as an outcome had a right-side variable that is not in the equation with $$D$$ as the outcome, then that variable could serve as an instrument and you could perform IV Probit. At present, there isn't such a variable, and thus IV probit isn't an option.