This document discusses the order and rank conditions for identification of equations in a simultaneous equation model. The order condition states that for an equation to be identified, the number of excluded variables must be greater than or equal to the number of endogenous variables minus one. The rank condition requires that it is possible to construct a non-zero determinant of order G-1 (where G is the number of endogenous variables) from the coefficients of excluded variables. An example simultaneous equation model is provided to demonstrate checking if the order and rank conditions are satisfied for each equation. The first two equations satisfy both conditions and are identified, while the third equation fails the rank condition and is unidentified.

1. By: Garima Gupta
RULES FOR IDENTIFICATION:
ORDER AND RANK CONDITIONS

2. ORDER CONDITION
• A necessary (but not sufficient) condition of
identification.
• For an equation to be identified, the total number of
variables excluded from it must be greater than or
equal to the number of endogenous variables in the
model less 1.

3. K - M ≥ G - 1
K – No. of total variables in the model
M – No. of variables included in a particular
equation
G – Total no. of endogenous variables in the
model

4. • If K-M < G-1 → order condition is not satisfied
(under identified equation)
• If K-M = G-1 → order condition is satisfied
(exactly identified equation)
• If K-M > G-1 → order condition is satisfied
(over identified equation)
If Order Condition is satisfied only then go for Rank
Condition.

5. RANK CONDITION
• In a system of G no. of equations, any
particular equation is identified if it is
possible to construct one non-zero
determinant of order G-1 from the
coefficients excluded from the particular
equation but present in the system.

6. EXAMPLE:
Y1 = 3Y2 - 2X1 + X2 + u1 - (1)
Y2 = Y1 + X3 + u2 - (2)
Y3 = Y1 - Y2 - 2X3 + u3 - (3)
Endogenous variables - Y1, Y2, Y3
Exogenous variables – X1, X2, X3
K = 6
G = 3
G – 1 = 2

7. 1. ORDER CONDITION: K – M ≥ G – 1
For eq 1: M = 4
K – M = 6 – 4 = 2 = G – 1 = 2 (Exactly identified)
For eq 2: M = 3
K – M = 6 – 3 = 3 > G – 1 = 2 ( Over identified)
For eq 3: M = 4
K – M = 6 – 4 = 2 = G – 1 = 2 ( Exactly identified)

8. 2. RANK CONDITION
Step 1:
Y1 - 3Y2 + 2X1 - X2 - u1 = 0 - i
Y2 - Y1 - X3 - u2 = 0 - ii
Y3 - Y1 + Y2 + 2X3 - u3 = 0 - iii

9. Step 2:
Y1 Y2 Y3 X1 X2 X3
1 -3 0 2 -1 0
-1 1 0 0 0 -1
-1 1 1 0 0 2

10. Step 3: Remove the included variables and include those variables that
are absent in the equation but present in the model.
For eq i-
Y3 X3 0 -1
0 0 1 2 2x2 = 1 ( Non-zero determinant)
0 -1
1 2 3X2
Rank condition is fulfilled.

11. For eq ii –
Y3 X1 X2
0 2 -1 0 2
0 0 0 0 0 2X2 = 0
1 0 0 3X3
0 2
1 0 2X2 = - 2 ( Non-zero determinant)
Rank condition is fulfilled.

12. For eq iii –
X1 X2
2 -1 2 -1
0 0 0 0 2X2 = 0
0 0 3X2
0 0
0 0 2X2 = 0
Rank condition is not satisfied.