Are these regression equations consistently estimated, and which ones are over/under/exactly identified?

Question

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?

Answer 1

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.