<p>If <span class="math-container">$u$</span> is normally distributed conditional on <span class="math-container">$x$</span> and <span class="math-container">$Q$</span>, then you could estimate through probit the second equation. Because <span class="math-container">$Q$</span> depends on <span class="math-container">$e$</span>, and <span class="math-container">$e$</span> is correlated with <span class="math-container">$u$</span>, this assumption likely fails (that's your point?).  Also, note that if <span class="math-container">$u \sim N(0, \sigma^2)$</span> then probit really estimates <span class="math-container">$\frac{\gamma}{\sigma}$</span>. (i.e., the ratio of the coefficient to the standard deviation of the error).</p>
<p>If the equation with <span class="math-container">$Q$</span> as an outcome had a right-side variable that is not in the equation with <span class="math-container">$D$</span> 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.</p>

CC BY-SA 4.0
2022-07-14T03:05:39.990Z
Thank you so much, Michael! This is very helpful.
ExcitedSnail
CC BY-SA 4.0
2022-07-14T03:21:32.937Z
Could you give me some reference on the ivprobit that you mentioned?
ExcitedSnail
CC BY-SA 4.0
2022-07-14T06:43:38.403Z
The documentation of the Stata command describes the method: https://www.stata.com/manuals/rivprobit.pdf
Michael Gmeiner
CC BY-SA 4.0
2022-07-13T07:05:11.313Z
2022-07-13T07:05:11.313Z
Michael Gmeiner
<p>Suppose I have the following model:</p>
<p><span class="math-container">$Q=1(x'\beta+e&gt;0)$</span>,</p>
<p><span class="math-container">$D=1(x'\alpha+\gamma Q+u&gt;0)$</span></p>
<p>and I want to estimate <span class="math-container">$\gamma$</span>, the error terms <span class="math-container">$e,u$</span> are jointly normal.</p>
<p>If <span class="math-container">$e$</span> and <span class="math-container">$u$</span> are correlated, can I still estimate <span class="math-container">$\gamma$</span> consistently by running a probit using the second equation only?</p>
<p>Why or why not?</p>
<p>If I cannot, how could I estimate  <span class="math-container">$\gamma$</span> in this case?</p>

<econometrics><labor-economics><identification><simultaneous-equation-model>
How to estimate $\gamma$ in the following model?
ExcitedSnail
# How to estimate $\gamma$ in the following model?

(Asked by: username_0 on 2022-07-13)

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?

**Tags:** econometrics, labor-economics, identification, simultaneous-equation-model

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## 1 answers

### Answer by username_1 on 2022-07-13. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you so much, Michael! This is very helpful.  
> **username_0**: Could you give me some reference on the ivprobit that you mentioned?  
> **username_1**: The documentation of the Stata command describes the method: https://www.stata.com/manuals/rivprobit.pdf  

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