For the two-equation simultaneous model (a) Determine the identifiability of each equation with the...

Question

For the two-equation simultaneous model

(a) Determine the identifiability of each equation with the aid of the order and rank conditions for identification.

(b) Obtain the OLS normal equations for both equations. Solve for the OLS estimates.

(c) Obtain the 2SLS normal equations for both equations. Solve for the 2SLS estimates.

(d) Can you estimate these equations using Indirect Least Squares? Explain.

Robert · Accepted Answer

Given:  
?1? = ?12?2? + ?11?1? + ?1?                             (1) 
?2? = ?21?1? + ?22?2? + ?23?3? + ?2?            (2) 
With  
?′? = [
20 0 0
0 20 0
0 0 10
]                  ?′? = [
5 10
40 20
20 30
]                      ?′? = [
3 4
4 8
] 
a) For simplicity we remove t from equation (1) and (2) then we get 
?1 = ?12?2 + ?11?1 + ?1               (3) 
?2 = ?21?1 + ?22?2 + ?23?3 + ?2    (4) 
In order to determine identifiability of each equations, we use the order and rank conditions for 
identification 
G = total number of equations (total number of endogenous variables). 
 K= total number of variables in the model (endogenous and pre-determined). 
M = number of variables, endogenous and pre-determined, in a particular equation. 
Order condition (necessary condition) 
  ? − ? ≥ ? − 1 
Rank Condition (Sufficient Condition) 
A sufficient condition for the identification of a relationship is that the rank of the matrix of 
parameters of all the excluded variables (endogenous and pre-determined) from that equation 
be equal to (G-1). 
For equation (3)  
Order condition : 
K=5 
M=3 
G=2 
K-M=5-3=2 
G-1=1 
Over identified 
Rank condition 
?1         ?2         ?1         ?2          ?3 
 
−1
?21
        
?12
−1
         
?11
0
     
0
?22
       
0
?23

[?22 ?23] 
  This is rank 1 matrix. Thus equation (3) is over identified.  
 For equation (4) 
 Order condition:  
 K=5 
 M=4 
 G=2 
 K-M=1 
 G-M=1 
 Exactly identified 
 Rank Condition: 
 Matrix can be given as [?11]. It is rank 1 matrix. Thus equation (3) is exactly identified. 
b) For the first equation,

For the two-equation simultaneous model (a) Determine the identifiability of each equation with the aid of the order and rank conditions for identification. (b) Obtain the OLS normal equations for...

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