
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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Transcribed Image Text:**b) How would you estimate equation 2? Please describe in detail.**
(Note: In the given text, there are no graphs or diagrams needing explanation. The content is primarily textual, focusing on the estimation of an equation.)
![### System of Equations: Identifiability Analysis
**1. Consider the following system of equations. \( Y_1 \), \( Y_2 \) and \( Y_3 \) are endogenous variables and \( X_1 \), \( X_2 \) and \( X_3 \) are exogenous variables.**
\[
\begin{align*}
Y_1 &= \alpha_1 + \alpha_2 Y_2 + \alpha_3 Y_3 + \alpha_4 X_1 + \alpha_5 X_2 + u_1 \quad &\text{(Eq. 1)} \\
Y_2 &= \beta_1 + \beta_2 Y_1 + \beta_3 Y_3 + \beta_4 X_2 + u_2 \quad &\text{(Eq. 2)} \\
Y_3 &= \gamma_1 + \gamma_2 X_1 + \gamma_3 X_2 + \gamma_4 X_3 + u_3 \quad &\text{(Eq. 3)}
\end{align*}
\]
**(a) Determine if each equation is just identified, over-identified or unidentified. Consider both rank and order conditions.**
**Explanation:**
Analyzing each equation involves applying the *rank* and *order conditions* to determine whether the equations are **just identified**, **over-identified**, or **under-identified**.
### Order Condition:
- **Just Identified:** The number of excluded exogenous variables (from the specific equation) should be equal to the number of endogenous variables minus one.
- **Over-identified:** The number of excluded exogenous variables is greater than the number of endogenous variables minus one.
- **Under-identified (Unidentified):** The number of excluded exogenous variables is less than the number of endogenous variables minus one.
### Rank Condition:
To fully confirm identifiability, the rank condition must also be met, which involves matrix manipulations to ensure that the necessary parameters can be uniquely determined.
### Equations Breakdown:
1. **(Eq. 1):**
- Endogenous Variables: \( Y_2 \), \( Y_3 \)
- Exogenous Variables: \( X_1 \), \( X_2 \)
2. **(Eq. 2):**
- Endogenous Variables: \( Y](https://content.bartleby.com/qna-images/question/7c47eb58-11c8-4f3d-a446-35857f47e26f/87f6ecc0-4876-4874-ae99-d6c81cf73cfc/jsqo7d_thumbnail.png)
Transcribed Image Text:### System of Equations: Identifiability Analysis
**1. Consider the following system of equations. \( Y_1 \), \( Y_2 \) and \( Y_3 \) are endogenous variables and \( X_1 \), \( X_2 \) and \( X_3 \) are exogenous variables.**
\[
\begin{align*}
Y_1 &= \alpha_1 + \alpha_2 Y_2 + \alpha_3 Y_3 + \alpha_4 X_1 + \alpha_5 X_2 + u_1 \quad &\text{(Eq. 1)} \\
Y_2 &= \beta_1 + \beta_2 Y_1 + \beta_3 Y_3 + \beta_4 X_2 + u_2 \quad &\text{(Eq. 2)} \\
Y_3 &= \gamma_1 + \gamma_2 X_1 + \gamma_3 X_2 + \gamma_4 X_3 + u_3 \quad &\text{(Eq. 3)}
\end{align*}
\]
**(a) Determine if each equation is just identified, over-identified or unidentified. Consider both rank and order conditions.**
**Explanation:**
Analyzing each equation involves applying the *rank* and *order conditions* to determine whether the equations are **just identified**, **over-identified**, or **under-identified**.
### Order Condition:
- **Just Identified:** The number of excluded exogenous variables (from the specific equation) should be equal to the number of endogenous variables minus one.
- **Over-identified:** The number of excluded exogenous variables is greater than the number of endogenous variables minus one.
- **Under-identified (Unidentified):** The number of excluded exogenous variables is less than the number of endogenous variables minus one.
### Rank Condition:
To fully confirm identifiability, the rank condition must also be met, which involves matrix manipulations to ensure that the necessary parameters can be uniquely determined.
### Equations Breakdown:
1. **(Eq. 1):**
- Endogenous Variables: \( Y_2 \), \( Y_3 \)
- Exogenous Variables: \( X_1 \), \( X_2 \)
2. **(Eq. 2):**
- Endogenous Variables: \( Y
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