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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?

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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.

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  • $\begingroup$ Thank you so much, Michael! This is very helpful. $\endgroup$ Jul 14, 2022 at 3:05
  • $\begingroup$ Could you give me some reference on the ivprobit that you mentioned? $\endgroup$ Jul 14, 2022 at 3:21
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    $\begingroup$ The documentation of the Stata command describes the method: stata.com/manuals/rivprobit.pdf $\endgroup$ Jul 14, 2022 at 6:43
  • $\begingroup$ Thanks, Michael! $\endgroup$ Aug 10, 2022 at 2:12

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