<p>None of the structural equations could be estimated consistently by OLS because simultaneity results in bias. Showing this is a little arduous, so I will only give one example. Consider the first equation. We will show that <span class="math-container">$Cov(Y_{3t}, \epsilon_{1t}) \ne 0$</span>.</p>
<p><span class="math-container">$$Y_{3t} = \beta_{31} +\beta_{32}Y_{1t}+\epsilon_{3t}$$</span>
Let us plug in equation 1.
<span class="math-container">$$Y_{3t} = \beta_{31} +\beta_{32}(\beta_{11}+\beta_{12}Y_{2t}+\beta_{13}Y_{3t} + \beta_{14}X_{1t}+\beta_{15}X_{2t}+\epsilon_{1t})+\epsilon_{3t}$$</span>
Let us plug in for <span class="math-container">$Y_{2t}$</span> using equation 2.
<span class="math-container">$$Y_{3t} = \beta_{31} +\beta_{32}(\beta_{11}+\beta_{12}(\beta_{21}+\beta_{22}Y_{3t}+ \beta_{23}X_{1t} + \epsilon_{2t})+\beta_{13}Y_{3t} + \beta_{14}X_{1t}+\beta_{15}X_{2t}+\epsilon_{1t})+\epsilon_{3t}$$</span></p>
<p>Rearraging and solving for <span class="math-container">$Y_{3t}$</span> gives us the reduced form equation of <span class="math-container">$Y_{3t}$</span>, <span class="math-container">$$Y_{3t} = \gamma_0 +\gamma_1 X_{1t} +\gamma_2 X_{2t} +v_t $$</span></p>
<p>Where <span class="math-container">$v_t = \frac{\beta_{32}\beta_{12}\epsilon_{2t} + \beta_{32}\beta_{13}\epsilon_{1t} + \epsilon_{3t}}{1-\beta_{32}\beta_{12}\beta_{22}-\beta_{32}\beta_{13}}$</span></p>
<p>The <span class="math-container">$\gamma$</span> coefficients are combinations of <span class="math-container">$\beta$</span>. I'm not going to solve for them.</p>
<p>Let us now consider the covariance of interest.
<span class="math-container">$$Cov(Y_{3t}, \epsilon_{1t}) $$</span>
Plug in using the reduced form for <span class="math-container">$Y_{3t}$</span> that we derived.
<span class="math-container">$$Cov(\gamma_0 +\gamma_1 X_{1t} +\gamma_2 X_{2t} +v_t, \epsilon_{1t}) $$</span>
Because <span class="math-container">$v_t$</span> contains <span class="math-container">$\epsilon_{1t}$</span>, this covariance will in general be nonzero. Thus, OLS estimation of the first structural equation is biased and inconsistent. This will be true with the other structural equations as well, but I'll leave the derivation to you or someone else.</p>
<p>It looks to me like you can estimate the second equation by using <span class="math-container">$X_2$</span> as an instrument for <span class="math-container">$Y_3$</span> and you can also estimate the third equation by using <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> as instruments for <span class="math-container">$Y_1$</span>.</p>
<p>For the second equation, the first stage would be, <span class="math-container">$$Y_{3t} = \delta_{20}+\delta_{21}X_{2t}+u_{2t} $$</span> You would attain predicted values, <span class="math-container">$\hat{Y}_{3t}$</span>, then estimate equation 2 with those predicted values rather than the data of <span class="math-container">$Y_{3t}$</span>.</p>
<p>For the third structural equation, the first stage would be, <span class="math-container">$$Y_{1t} = \delta_{30}+\delta_{31}X_{1t}+\delta_{32}X_{2t}+u_{3t} $$</span> You would attain predicted values, <span class="math-container">$\hat{Y}_{3t}$</span>, then estimate equation 3 with those predicted values rather than the data of <span class="math-container">$Y_{1t}$</span>.</p>
<p>Regrading the first equation, there are no excluded instruments, so I think the parameters are not identified.</p>
<p>Reduced form equations in general are &quot;an outcome of interest is on the left and covariates that are exogeneous are on the right&quot;. Estimating a reduced form by OLS is consistent. To claim it is &quot;efficient&quot; would require making the Gauss-Markov assumptions of no serial correlation and homoskedasticity, then OLS is the most efficient linear unbiased estimator.  If you are considering &quot;efficiency&quot; in the context of all estimators, not just linear estimators, you would need to also assume the errors are normally distributed for OLS to be efficient, as OLS is equivalent to MLE in such a case.</p>

CC BY-SA 4.0
2022-05-31T14:57:52.080Z
2022-05-31T14:57:52.080Z
Michael Gmeiner
<p>Consider the structural system of simultational equations where the where
Y variables are endogenous, and the X variables are exogenous. The errors may be correlated contemporaneously between equations but not over
time either within or between equations. The errors have a mean of 0 but different variances.
<span class="math-container">\begin{align}
Y_{1t}&amp;= \beta_{11} + \beta_{12}Y_{2t} +\beta_{13}Y_{3t}+\beta_{14}X_{1t} +\beta_{15}X_{2t} + \epsilon_{1t},\\
Y_{2t} &amp;= \beta_{21} + \beta_{22}Y_{3t} +\beta_{23}X_{1t}+\epsilon_{2t},\\
Y_{3t} &amp;= \beta_{31} + \beta_{32}Y_{1t}+ \epsilon_{3t},
\end{align}</span>
where you can rewrite the structural equations as:
<span class="math-container">\begin{align}
Y_{1t} - \beta_{12}Y_{2t} -\beta_{13}Y_{3t} &amp;= \beta_{11} +\beta_{14}X_{1t} +\beta_{15}X_{2t} + \epsilon_{1t},\\
Y_{2t}- \beta_{22}Y_{3t} &amp;= \beta_{21} +\beta_{23}X_{1t}+\epsilon_{2t},\\
Y_{3t}-\beta_{32}Y_{1t} &amp;= \beta_{31}+ \epsilon_{3t}.
\end{align}</span>
Obviously the first equation is over under identified, the second one is exactly identified and the third one is over identified.
We can estimate the reduced-form coefficients by OLS because there are no endogenous variables on the right-hand side. Reduced form equations represent each endogenous variable as a function of only exogenous variables. Which means there can  be no endogenous variables in the RHS.</p>
<ol>
<li>Some questions: Are the coefficients in the first equation estimated consistently by OLS?
Why?</li>
<li>Can the reduced form coefficients be estimated EFFICIENTLY by OLS applied each equation
separately?
Why?</li>
<li>How would you estimate each of the identified equations?</li>
</ol>

CC BY-SA 4.0
2022-05-30T16:16:48.550Z
Is there a typo in your third equation that has $Y_{1t}$ on the left and the right?
Michael Gmeiner
CC BY-SA 4.0
2022-05-31T01:34:11.340Z
Hi, I fixed the issue.
DaveJohnson123
CC BY-SA 4.0
2022-05-31T05:49:34.763Z
Welcome. In Q1, what do you mean by "the first equation"? Is it the reduced-form equation for $Y_{1t}$? I'm asking because the first structural equation is under-identified, as you wrote, so its estimation is impossible without some external instruments.
chan1142
CC BY-SA 4.0
2022-05-31T07:20:23.840Z
Without any assumption on the random vector $\epsilon$, you can obtain any results you want.
Bertrand
<econometrics><regression>
Are these regression equations consistently estimated, and which ones are over/under/exactly identified?
DaveJohnson123
# Are these regression equations consistently estimated, and which ones are over/under/exactly identified?

(Asked by: username_0 on 2022-05-30)

Consider the structural system of simultational equations where the where
Y variables are endogenous, and the X variables are exogenous. The errors may be correlated contemporaneously between equations but not over
time either within or between equations. The errors have a mean of 0 but different variances.
\begin{align}
Y\_{1t}&= \beta\_{11} + \beta\_{12}Y\_{2t} +\beta\_{13}Y\_{3t}+\beta\_{14}X\_{1t} +\beta\_{15}X\_{2t} + \epsilon\_{1t},\\
Y\_{2t} &= \beta\_{21} + \beta\_{22}Y\_{3t} +\beta\_{23}X\_{1t}+\epsilon\_{2t},\\
Y\_{3t} &= \beta\_{31} + \beta\_{32}Y\_{1t}+ \epsilon\_{3t},
\end{align}
where you can rewrite the structural equations as:
\begin{align}
Y\_{1t} - \beta\_{12}Y\_{2t} -\beta\_{13}Y\_{3t} &= \beta\_{11} +\beta\_{14}X\_{1t} +\beta\_{15}X\_{2t} + \epsilon\_{1t},\\
Y\_{2t}- \beta\_{22}Y\_{3t} &= \beta\_{21} +\beta\_{23}X\_{1t}+\epsilon\_{2t},\\
Y\_{3t}-\beta\_{32}Y\_{1t} &= \beta\_{31}+ \epsilon\_{3t}.
\end{align}
Obviously the first equation is over under identified, the second one is exactly identified and the third one is over identified.
We can estimate the reduced-form coefficients by OLS because there are no endogenous variables on the right-hand side. Reduced form equations represent each endogenous variable as a function of only exogenous variables. Which means there can be no endogenous variables in the RHS.

1. Some questions: Are the coefficients in the first equation estimated consistently by OLS?
   Why?
2. Can the reduced form coefficients be estimated EFFICIENTLY by OLS applied each equation
   separately?
   Why?
3. How would you estimate each of the identified equations?

**Tags:** econometrics, regression

> **Comments:**
> **username_1**: Is there a typo in your third equation that has $Y_{1t}$ on the left and the right?  
> **username_0**: Hi, I fixed the issue.  
> **username_2**: Welcome. In Q1, what do you mean by "the first equation"? Is it the reduced-form equation for $Y_{1t}$? I'm asking because the first structural equation is under-identified, as you wrote, so its estimation is impossible without some external instruments.  
> **username_3**: Without any assumption on the random vector $\epsilon$, you can obtain any results you want.  

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## 1 answers

### Answer by username_1 on 2022-05-31. Score: 0 | Accepted answer

None of the structural equations could be estimated consistently by OLS because simultaneity results in bias. Showing this is a little arduous, so I will only give one example. Consider the first equation. We will show that $Cov(Y\_{3t}, \epsilon\_{1t}) \ne 0$.

$$Y\_{3t} = \beta\_{31} +\beta\_{32}Y\_{1t}+\epsilon\_{3t}$$
Let us plug in equation 1.
$$Y\_{3t} = \beta\_{31} +\beta\_{32}(\beta\_{11}+\beta\_{12}Y\_{2t}+\beta\_{13}Y\_{3t} + \beta\_{14}X\_{1t}+\beta\_{15}X\_{2t}+\epsilon\_{1t})+\epsilon\_{3t}$$
Let us plug in for $Y\_{2t}$ using equation 2.
$$Y\_{3t} = \beta\_{31} +\beta\_{32}(\beta\_{11}+\beta\_{12}(\beta\_{21}+\beta\_{22}Y\_{3t}+ \beta\_{23}X\_{1t} + \epsilon\_{2t})+\beta\_{13}Y\_{3t} + \beta\_{14}X\_{1t}+\beta\_{15}X\_{2t}+\epsilon\_{1t})+\epsilon\_{3t}$$

Rearraging and solving for $Y\_{3t}$ gives us the reduced form equation of $Y\_{3t}$, $$Y\_{3t} = \gamma\_0 +\gamma\_1 X\_{1t} +\gamma\_2 X\_{2t} +v\_t $$

Where $v\_t = \frac{\beta\_{32}\beta\_{12}\epsilon\_{2t} + \beta\_{32}\beta\_{13}\epsilon\_{1t} + \epsilon\_{3t}}{1-\beta\_{32}\beta\_{12}\beta\_{22}-\beta\_{32}\beta\_{13}}$

The $\gamma$ coefficients are combinations of $\beta$. I'm not going to solve for them.

Let us now consider the covariance of interest.
$$Cov(Y\_{3t}, \epsilon\_{1t}) $$
Plug in using the reduced form for $Y\_{3t}$ that we derived.
$$Cov(\gamma\_0 +\gamma\_1 X\_{1t} +\gamma\_2 X\_{2t} +v\_t, \epsilon\_{1t}) $$
Because $v\_t$ contains $\epsilon\_{1t}$, this covariance will in general be nonzero. Thus, OLS estimation of the first structural equation is biased and inconsistent. This will be true with the other structural equations as well, but I'll leave the derivation to you or someone else.

It looks to me like you can estimate the second equation by using $X\_2$ as an instrument for $Y\_3$ and you can also estimate the third equation by using $X\_1$ and $X\_2$ as instruments for $Y\_1$.

For the second equation, the first stage would be, $$Y\_{3t} = \delta\_{20}+\delta\_{21}X\_{2t}+u\_{2t} $$ You would attain predicted values, $\hat{Y}\_{3t}$, then estimate equation 2 with those predicted values rather than the data of $Y\_{3t}$.

For the third structural equation, the first stage would be, $$Y\_{1t} = \delta\_{30}+\delta\_{31}X\_{1t}+\delta\_{32}X\_{2t}+u\_{3t} $$ You would attain predicted values, $\hat{Y}\_{3t}$, then estimate equation 3 with those predicted values rather than the data of $Y\_{1t}$.

Regrading the first equation, there are no excluded instruments, so I think the parameters are not identified.

Reduced form equations in general are "an outcome of interest is on the left and covariates that are exogeneous are on the right". Estimating a reduced form by OLS is consistent. To claim it is "efficient" would require making the Gauss-Markov assumptions of no serial correlation and homoskedasticity, then OLS is the most efficient linear unbiased estimator. If you are considering "efficiency" in the context of all estimators, not just linear estimators, you would need to also assume the errors are normally distributed for OLS to be efficient, as OLS is equivalent to MLE in such a case.


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