Question

Suppose an investigator has specified the following simultaneous equation model:
\[ y_{1,t} = \alpha_{1,0} + \alpha_{1,2} y_{2,t} + \alpha_{2} M_{t} + \alpha_{3} K_{t} + u_{t} \tag{1} \] \[ y_{2,t} = \alpha_{2,0} + \alpha_{2,1} y_{1,t} + \alpha_{3} I_{t} + v_{t} \tag{2} \] where \( y_{1,t} \) and \( y_{2,t} \) are endogenous variables. \( M_t, K_t, I_t \) are predetermined variables. \( u_t \) and \( v_t \) are residual terms. Based on the order condition of identification, which one of the following statements is correct?

• A. Equations (1) and (2) are exactly identified
• B. Equation (1) is exactly identified and equation (2) is over-identified
• C. Equation (1) is over-identified and equation (2) is exactly identified
• D. Both the equations are under-identified

Hint:

In simultaneous equation models, ensure that the number of predetermined variables excluded from each equation matches the number of endogenous variables for identification.

Answer 1

The Correct Option is B

Step 1: Understanding the identification condition.
The order condition of identification involves checking the number of predetermined variables in the system. A system is considered identified if the number of excluded predetermined variables from each equation equals the number of endogenous variables that need to be solved for. Step 2: Analyzing equation (1).
Equation (1) has \( y_{2,t} \) as an endogenous variable, and it has three predetermined variables \( M_t, K_t \), and \( I_t \) (where \( I_t \) is included in equation (2) and excluded from equation (1)). Thus, equation (1) is exactly identified because it has as many excluded predetermined variables as the number of endogenous variables. Step 3: Analyzing equation (2).
Equation (2) has \( y_{1,t} \) as an endogenous variable and three predetermined variables \( M_t, K_t, I_t \), with two of these variables already appearing in equation (1). This results in over-identification for equation (2) since it has more predetermined variables than needed for identification. Step 4: Conclusion.
The correct answer is (B) because equation (1) is exactly identified, and equation (2) is over-identified.