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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.
