Simultaneity problem: use of Instrumental Variables and test The following model jointly determines monthly child support payments and monthly visitation rights for divorced couples with children:

$$support = d_1 + g_1 visits + d_1 finc + d_1 fremarr + d_1 dist + u_1$$
$$visits = d_2 + g_2 support + d_2 mremarr + d_2 dist + u_2$$

For expository purposes, assume that children live with their mothers, so that fathers pay child support. Thus, the first equation is the father’s ‘‘reaction function’’: it describes the amount of child support paid for any given level of visitation rights and the other exogenous variables finc (father’s income), fremarr (binary indicator if father remarried), and dist (miles currently between the mother and father). Similarly, the second equation is the mother’s reaction function: it describes visitation rights for a given amount of child support; mremarr is a binary indicator for whether the mother is remarried.

a. Discuss identification of each equation.

b. How would you estimate each equation using a single-equation method?

c. How would you test for endogeneity of visits in the father’s reaction function?



I'd like to get some help with the whole question and to have my reasoning on it checked, thank you:

a)identification not possible due to simultaneity (which causes endogeneity)

b)via instrumental variable (possible choice (fremarr and mremarr) or ( dist)

c) my only idea was to estimate visits via OLS then use Hausman test between OLS and IV

Is the approach correct? How can I answer C in an alternative way?
To clarify notation, and treating summarily the other variables we have a system

$$y = a_0 +a_1x + \mathbf w_1'\gamma_1 + u_1 \\
x = b_0 + b_1y + \mathbf w_2'\gamma_2 + u_2$$

If $b_1 \neq 0$, then solving for $x$ we obtain

$$ x = b_0 + b_1(a_0 +a_1x + \mathbf w_1'\gamma_1+ u_1) + \mathbf w_2'\gamma_2 + u_2 \\
\implies x = c + \mathbf w_1'\frac {\gamma_1}{1-a_1b_1}+\frac {b_1}{1-a_1b_1}u_1 + \mathbf w_2'\frac {\gamma_2}{1-a_1b_1}+\frac {1}{1-a_1b_1}u_2$$

and  $x$ is correlated with $u_1$ in the first equation, and so it is endogenous (we would obtain an analogous result if we solved for $y$ -it would be endogenous in the second equation). Historically, this is the "original" endogeneity problem, the one realized back in the '20s when looking at market equilibrium, where the assumption of market-clearing created the endogeneity. Namely it is not the usual "omitted variables" endogeneity problem, but the form of endogeneity due to "simultaneity".  

So endogeneity of $x$ hinges on $b_1\neq 0$. But running OLS on the second equation and testing whether $b_1=0$ is not valid, because, as said, $y$ is endogenous in the second equation, so the OLS estimation is unreliable, being inconsistent.
This is a bit subtle. We want to test $b_1 =0$, so under the null hypothesis $y$ does not belong in the second equation. But inference will use $y$ itself. This means that if the null is rejected, the rejection is not reliable, because if the null is not true, then $b_1 \neq 0$ and automatically the estimator, t-tests etc become inconsistent.

The question asks to test for endogeneity in the first equation, so the evident thing to do is to search for an instrument, or instruments, for $x$ and then, for example, apply the Hausman test in the first equation.

Assume that we applied the Hausman test in the second equation, as the OP suggested. This would mean searching for instruments for $y$. Here if the null is not rejected, it would mean that $y$ is not correlated with $u_2$. But from this it does not necessarily follow that $b_1\neq 0$, and so it does not guarantee that $x$ is uncorrelated with $u_1$. If the null is rejected, it means that $y$ is correlated with $u_2$. But this could happen for any other reasons, not necessarily due to the simultaneity.
So the endogeneity tests should be conducted on the first equation. 

Note that in reality, the Hausman test  in IV estimation has as its null hypothesis that both the OLS and the IVE estimator are consistent. If the null is rejected, it only tells us that at least one of the two is inconsistent. It cannot tell us which one, and it cannot guarantee that not both are inconsistent: The Hausman test is more informative and specific in its conclusions when the null is not rejected, rather than when the null is rejected. (This is a point usually not clearly stressed in the literature, although side remarks and footnotes do exist to that effect).