<p>Let us consider Situation 1.</p>
<p>Let us assume that <span class="math-container">$\rho$</span> is observed. If it does not work when <span class="math-container">$\rho$</span> is observed, there is no reason why it (using a proxy of <span class="math-container">$\rho$</span> as instrument) should work when <span class="math-container">$\rho$</span> is not observed.</p>
<p><span class="math-container">$\rho$</span> is endogenous so we can't just include it as a regressor in an equation. Thus, let us consider IV estimation.</p>
<p>Obvious instruments for the first equation are <span class="math-container">$\mathbf{x}_1$</span>, <span class="math-container">$\mathbf{x}_2$</span>, and <span class="math-container">$\mathbf{x}_3$</span>. Can we use <span class="math-container">$\rho$</span> as an extra instrument? It depends on whether <span class="math-container">$\rho$</span> is relevant and whether <span class="math-container">$\rho$</span> is exogenous in the first equation (i.e., the <span class="math-container">$y_1$</span> equation). First, <span class="math-container">$\rho$</span> is relevant as it is correlated with <span class="math-container">$y_2$</span> (unless the last equation is degenerate). Next, is it exogenous?</p>
<p>To check it, let's go maths and consider the following simple model (without intercepts and exogenous regressors, for simplicity):
<span class="math-container">$$y_1=\alpha y_2 + e_1,\;\; \rho = \beta y_1+e_2,\;\; y_2=\gamma \rho + e_3.$$</span>
Write them in matrix form:
<span class="math-container">$$
\begin{pmatrix}
1 &amp; 0 &amp; -\alpha\\
-\beta &amp; 1 &amp; 0\\
0 &amp; -\gamma &amp; 1
\end{pmatrix}
\begin{pmatrix}
y_1\\ \rho\\ y_2
\end{pmatrix}
=
\begin{pmatrix}
e_1\\ e_2\\ e_3
\end{pmatrix}
.
$$</span>
Using <a href="https://en.wikipedia.org/wiki/Cramer%27s_rule" rel="nofollow noreferrer">Cramer's rule</a>, we get
<span class="math-container">$$\rho = (\beta e_1 + e_2 + \alpha \beta e_3 ) / (1-\alpha\beta\gamma).$$</span>
<span class="math-container">$\rho$</span> is correlated with <span class="math-container">$e_1$</span> unless <span class="math-container">$\beta=0$</span>, so <span class="math-container">$\rho$</span> can't be used as an instrument.</p>
<p><strong>Edit</strong></p>
<p>What happens if <span class="math-container">$y_1$</span> is regressed on <span class="math-container">$y_2$</span> and <span class="math-container">$\rho$</span>? Then will <span class="math-container">$\alpha$</span> be consistently estimated? For simplicity, suppose that <span class="math-container">$e_1, e_2, e_3$</span> are <span class="math-container">$iid$</span> standard normal. Then the OLS estimator vector converges in probability to
<span class="math-container">$$\begin{bmatrix}
 \alpha\\ 0
\end{bmatrix}
+
\begin{bmatrix}
 E(y_2^2) &amp; E(y_2 \rho)\\
 E(y_2\rho) &amp; E(\rho^2)
\end{bmatrix}
^{-1}
\begin{bmatrix}
 E(y_2 e_1)\\ E(\rho e_1)
\end{bmatrix}
.$$</span>
Letting <span class="math-container">$e=(e_1,e_2,e_3)'$</span>, write <span class="math-container">$\rho = e'g$</span> and <span class="math-container">$y_2 = e'h$</span> for some <span class="math-container">$3\times 1$</span> nonrandom vectors <span class="math-container">$g$</span> and <span class="math-container">$h$</span>. Then the second term is
<span class="math-container">$$
\begin{pmatrix}
 h'h &amp; h'g\\ g'h &amp; g'g
\end{pmatrix}
^{-1}
\begin{pmatrix}
h_1\\ g_1
\end{pmatrix}
.
$$</span>
The first element of the above is the determinant inverse times <span class="math-container">$g'gh_1 - g'h g_1 = g'(gh_1 - hg_1)$</span>. Let us calculate it. We have already obtained <span class="math-container">$g$</span>. We now need <span class="math-container">$h$</span>:
<span class="math-container">$$y_2 = (-\beta\gamma e_1 - \gamma e_2 + e_3) / (1-\alpha\beta\gamma).$$</span></p>
<p>Let us ignore the determinant part (common). We can work out with <span class="math-container">$g=(\beta, 1, \alpha\beta)'$</span> and <span class="math-container">$h=(-\beta\gamma, -\gamma, 1)'$</span>, ignoring the determinant. Then
<span class="math-container">$$ gh_1 - hg_1 = [0, 0, -\beta (1+\alpha \beta\gamma) ]' \text{ times a constant}.$$</span>
Thus,
<span class="math-container">$$ g'(gh_1 - hg_1) = -\alpha \beta^2 (1+\alpha\beta\gamma) \text{ times another constant}.$$</span>
The OLS regression of <span class="math-container">$y_1$</span> on <span class="math-container">$y_2$</span> and <span class="math-container">$\rho$</span> gives an inconsistent estimator in general, but gives a consistent estimator of <span class="math-container">$\alpha$</span> if <span class="math-container">$\alpha=0$</span> or <span class="math-container">$\beta=0$</span>. Especially, if <span class="math-container">$\alpha=0$</span>, then the OLS estimator is consistent. That is, we can regress <span class="math-container">$y_1$</span> on <span class="math-container">$y_2$</span> and <span class="math-container">$\rho$</span> if we want to test <span class="math-container">$H_0: \alpha=0$</span>. This is unexpected. Not sure if my algebra is correct.</p>

CC BY-SA 4.0
2021-03-05T02:28:10.707Z
Thanks for the response but I am a bit confused about the ordering. You say "$\rho$ is exogenous so we cannot include it as a regressor in the first stage." But then you process to investigate whether it is exogenous for the purposes of IV. But you just said it is endogenous. So in the derivation of rho in terms of e1, are you just proving the statement that it is endogenous?
Ryan da Silva
CC BY-SA 4.0
2021-03-05T03:22:02.220Z
By "first equation" I meant $y_1 = f(y_2,\mathbf{x}_1) + e_1$, not the first-stage regression equation. Sorry for the confusion. I added "(i.e., the $y_1$ equation)".
chan1142
CC BY-SA 4.0
2021-03-05T03:35:50.117Z
Thanks for the clarification. But what I am asking is, doesn't the reduced form equation for $\rho$ show _both_ why $\rho$ cannot be used as a regressor in $y_1 = f() + e_1$ and why it cannot be used as an instrument?
Ryan da Silva
CC BY-SA 4.0
2021-03-05T08:08:54.240Z
Hmm, that looks like a different issue. If we keep the linear model, the question is whether $E(y_1|y_2, \rho) = \alpha y_2 + \delta \rho$ for some $\delta$. Can this be true? Let me work a bit. I'll write a comment if I find an answer.
chan1142
CC BY-SA 4.0
2021-03-05T02:02:15.313Z
2021-03-06T00:31:38.443Z
2021-03-06T00:31:38.443Z
chan1142
<p>I would like to know if the logic in these two situations is correct.</p>
<p><strong>Situation 1:</strong> Let's say we have a continuous dependent variable, <span class="math-container">$y_1$</span>, that then has a causal impact on an unobserved variable, <span class="math-container">$\rho$</span>. This unobserved variable then has a causal impact on a variable, <span class="math-container">$y_2$</span>, which has a causal impact on <span class="math-container">$y_1$</span>. We want to estimate <span class="math-container">$ \frac{\partial y_1}{\partial y_2}$</span>. So, we have a set of structural equations as follows:</p>
<p><span class="math-container">$$y_1 = f(y_2, \mathbf{x_1})+ e_1$$</span>
<span class="math-container">$$\rho = f(y_1, \mathbf{x_2})+ e_2$$</span>
<span class="math-container">$$y_2 = f(\rho, \mathbf{x_3})+ e_3$$</span></p>
<p>where the <span class="math-container">$\mathbf{x_i}$</span> terms are exogenous and the <span class="math-container">$e_i$</span> terms are errors. By substituting the second equation into the third, we can see that we would have simultaneity and our estimates of the impact of <span class="math-container">$y_2$</span> on <span class="math-container">$y_1$</span> would be biased <em>if we could not control for <span class="math-container">$\rho$</span> or some proxy for it in our estimate of the structural equation for <span class="math-container">$y_1$</span></em>.</p>
<p>I am particularly unsure about this last part in italics. Could we use the proxy for <span class="math-container">$\rho$</span> in the estimate of the structural equation for <span class="math-container">$y_1$</span>, or would we have to do 2SLS, with the proxy for <span class="math-container">$\rho$</span> being included in the first stage but excluded in the second?</p>
<p><strong>Situation 2:</strong> Let's say we have a percentage dependent variable, <span class="math-container">$s_1$</span>. Let's say the complement of <span class="math-container">$s_1$</span> is made up of two other percentages, <span class="math-container">$s_2$</span> and <span class="math-container">$s_3$</span>. Furthermore, let's say that <span class="math-container">$s_3$</span> has a causal impact on an unobserved factor, <span class="math-container">$\rho$</span>, and that <span class="math-container">$\rho$</span> has a causal impact on <span class="math-container">$y_2$</span>, which has a causal impact on <span class="math-container">$s_1$</span>. We want to estimate <span class="math-container">$\frac{\partial s_1}{\partial y_2}$</span>. Thus, we have the following structural equations:</p>
<p><span class="math-container">$$s_1 = f(y_2, \mathbf{x_1})+ e_1$$</span>
<span class="math-container">$$\rho = f(s_3, \mathbf{x_2})+ e_2$$</span>
<span class="math-container">$$y_2 = f(\rho, \mathbf{x_3})+ e_3$$</span></p>
<p>Let's now say that <span class="math-container">$s_2$</span> is more or less constant across observations. Thus, there is generally an inverse relation between <span class="math-container">$s_1$</span> and <span class="math-container">$s_3$</span>. This implies that we can rewrite <span class="math-container">$s_3$</span> in the second structural equations in terms of <span class="math-container">$s_1$</span>:</p>
<p><span class="math-container">$$\rho = f(1 - (s_1 + \bar{s}_2), \mathbf{x_2})+ e_2$$</span></p>
<p>Then, just as in situation 1, by substituting this equation into the structural equation for <span class="math-container">$y_2$</span>, we can see there would be simultaneity and our estimates of the impact of <span class="math-container">$y_2$</span> on <span class="math-container">$y_1$</span> would be biased (once again, I am unsure about whether controlling for <span class="math-container">$\rho$</span> or a proxy for it in our estimate of the structural equation for <span class="math-container">$y_1$</span> would solve this).</p>

<econometrics><regression><endogeneity><simultaneous-equation-model>
Is this an an endogeneity/simultaneity problem?
Ryan da Silva
# Is this an an endogeneity/simultaneity problem?

(Asked by: username_0 on 2021-03-04)

I would like to know if the logic in these two situations is correct.

**Situation 1:** Let's say we have a continuous dependent variable, $y\_1$, that then has a causal impact on an unobserved variable, $\rho$. This unobserved variable then has a causal impact on a variable, $y\_2$, which has a causal impact on $y\_1$. We want to estimate $ \frac{\partial y\_1}{\partial y\_2}$. So, we have a set of structural equations as follows:

$$y\_1 = f(y\_2, \mathbf{x\_1})+ e\_1$$
$$\rho = f(y\_1, \mathbf{x\_2})+ e\_2$$
$$y\_2 = f(\rho, \mathbf{x\_3})+ e\_3$$

where the $\mathbf{x\_i}$ terms are exogenous and the $e\_i$ terms are errors. By substituting the second equation into the third, we can see that we would have simultaneity and our estimates of the impact of $y\_2$ on $y\_1$ would be biased *if we could not control for $\rho$ or some proxy for it in our estimate of the structural equation for $y\_1$*.

I am particularly unsure about this last part in italics. Could we use the proxy for $\rho$ in the estimate of the structural equation for $y\_1$, or would we have to do 2SLS, with the proxy for $\rho$ being included in the first stage but excluded in the second?

**Situation 2:** Let's say we have a percentage dependent variable, $s\_1$. Let's say the complement of $s\_1$ is made up of two other percentages, $s\_2$ and $s\_3$. Furthermore, let's say that $s\_3$ has a causal impact on an unobserved factor, $\rho$, and that $\rho$ has a causal impact on $y\_2$, which has a causal impact on $s\_1$. We want to estimate $\frac{\partial s\_1}{\partial y\_2}$. Thus, we have the following structural equations:

$$s\_1 = f(y\_2, \mathbf{x\_1})+ e\_1$$
$$\rho = f(s\_3, \mathbf{x\_2})+ e\_2$$
$$y\_2 = f(\rho, \mathbf{x\_3})+ e\_3$$

Let's now say that $s\_2$ is more or less constant across observations. Thus, there is generally an inverse relation between $s\_1$ and $s\_3$. This implies that we can rewrite $s\_3$ in the second structural equations in terms of $s\_1$:

$$\rho = f(1 - (s\_1 + \bar{s}\_2), \mathbf{x\_2})+ e\_2$$

Then, just as in situation 1, by substituting this equation into the structural equation for $y\_2$, we can see there would be simultaneity and our estimates of the impact of $y\_2$ on $y\_1$ would be biased (once again, I am unsure about whether controlling for $\rho$ or a proxy for it in our estimate of the structural equation for $y\_1$ would solve this).

**Tags:** econometrics, regression, endogeneity, simultaneous-equation-model

---

## 1 answers

### Answer by username_1 on 2021-03-05. Score: 6 | Accepted answer

Let us consider Situation 1.

Let us assume that $\rho$ is observed. If it does not work when $\rho$ is observed, there is no reason why it (using a proxy of $\rho$ as instrument) should work when $\rho$ is not observed.

$\rho$ is endogenous so we can't just include it as a regressor in an equation. Thus, let us consider IV estimation.

Obvious instruments for the first equation are $\mathbf{x}\_1$, $\mathbf{x}\_2$, and $\mathbf{x}\_3$. Can we use $\rho$ as an extra instrument? It depends on whether $\rho$ is relevant and whether $\rho$ is exogenous in the first equation (i.e., the $y\_1$ equation). First, $\rho$ is relevant as it is correlated with $y\_2$ (unless the last equation is degenerate). Next, is it exogenous?

To check it, let's go maths and consider the following simple model (without intercepts and exogenous regressors, for simplicity):
$$y\_1=\alpha y\_2 + e\_1,\;\; \rho = \beta y\_1+e\_2,\;\; y\_2=\gamma \rho + e\_3.$$
Write them in matrix form:
$$
\begin{pmatrix}
1 & 0 & -\alpha\\
-\beta & 1 & 0\\
0 & -\gamma & 1
\end{pmatrix}
\begin{pmatrix}
y\_1\\ \rho\\ y\_2
\end{pmatrix}
=
\begin{pmatrix}
e\_1\\ e\_2\\ e\_3
\end{pmatrix}
.
$$
Using [Cramer's rule](https://en.wikipedia.org/wiki/Cramer%27s_rule), we get
$$\rho = (\beta e\_1 + e\_2 + \alpha \beta e\_3 ) / (1-\alpha\beta\gamma).$$
$\rho$ is correlated with $e\_1$ unless $\beta=0$, so $\rho$ can't be used as an instrument.

**Edit**

What happens if $y\_1$ is regressed on $y\_2$ and $\rho$? Then will $\alpha$ be consistently estimated? For simplicity, suppose that $e\_1, e\_2, e\_3$ are $iid$ standard normal. Then the OLS estimator vector converges in probability to
$$\begin{bmatrix}
\alpha\\ 0
\end{bmatrix}
+
\begin{bmatrix}
E(y\_2^2) & E(y\_2 \rho)\\
E(y\_2\rho) & E(\rho^2)
\end{bmatrix}
^{-1}
\begin{bmatrix}
E(y\_2 e\_1)\\ E(\rho e\_1)
\end{bmatrix}
.$$
Letting $e=(e\_1,e\_2,e\_3)'$, write $\rho = e'g$ and $y\_2 = e'h$ for some $3\times 1$ nonrandom vectors $g$ and $h$. Then the second term is
$$
\begin{pmatrix}
h'h & h'g\\ g'h & g'g
\end{pmatrix}
^{-1}
\begin{pmatrix}
h\_1\\ g\_1
\end{pmatrix}
.
$$
The first element of the above is the determinant inverse times $g'gh\_1 - g'h g\_1 = g'(gh\_1 - hg\_1)$. Let us calculate it. We have already obtained $g$. We now need $h$:
$$y\_2 = (-\beta\gamma e\_1 - \gamma e\_2 + e\_3) / (1-\alpha\beta\gamma).$$

Let us ignore the determinant part (common). We can work out with $g=(\beta, 1, \alpha\beta)'$ and $h=(-\beta\gamma, -\gamma, 1)'$, ignoring the determinant. Then
$$ gh\_1 - hg\_1 = [0, 0, -\beta (1+\alpha \beta\gamma) ]' \text{ times a constant}.$$
Thus,
$$ g'(gh\_1 - hg\_1) = -\alpha \beta^2 (1+\alpha\beta\gamma) \text{ times another constant}.$$
The OLS regression of $y\_1$ on $y\_2$ and $\rho$ gives an inconsistent estimator in general, but gives a consistent estimator of $\alpha$ if $\alpha=0$ or $\beta=0$. Especially, if $\alpha=0$, then the OLS estimator is consistent. That is, we can regress $y\_1$ on $y\_2$ and $\rho$ if we want to test $H\_0: \alpha=0$. This is unexpected. Not sure if my algebra is correct.

> **Comments:**
> **username_0**: Thanks for the response but I am a bit confused about the ordering. You say "$\rho$ is exogenous so we cannot include it as a regressor in the first stage." But then you process to investigate whether it is exogenous for the purposes of IV. But you just said it is endogenous. So in the derivation of rho in terms of e1, are you just proving the statement that it is endogenous?  
> **username_1**: By "first equation" I meant $y_1 = f(y_2,\mathbf{x}_1) + e_1$, not the first-stage regression equation. Sorry for the confusion. I added "(i.e., the $y_1$ equation)".  
> **username_0**: Thanks for the clarification. But what I am asking is, doesn't the reduced form equation for $\rho$ show _both_ why $\rho$ cannot be used as a regressor in $y_1 = f() + e_1$ and why it cannot be used as an instrument?  
> **username_1**: Hmm, that looks like a different issue. If we keep the linear model, the question is whether $E(y_1|y_2, \rho) = \alpha y_2 + \delta \rho$ for some $\delta$. Can this be true? Let me work a bit. I'll write a comment if I find an answer.  

---

