RANK CONDITION for IDENTIFICATION Cont. … SEM

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Rank condition of identification

It is possible that an equation is not identified even if the order condition is satisfied, making it a required but not sufficient condition for identification. We require a suitable criterion for identification in order to solve this issue. This issue is resolved by the rank condition, which is considered a sufficient condition.

In a model containing M equations in M endogenous variables, an equation is identified if and only if at least one nonzero determinant of order (M − 1) (M − 1) can be constructed from the coefficients of the variables (both endogenous and predetermined) excluded from that particular equation but included in the other equations of the model.

The rank condition of identification may be proceed as:

i.               Write down the coefficient in tabular form.

ii.            Striking out the coefficient of the row in which the equation under consideration appears.

iii.          Also striking out the columns corresponding to those which are non-zero.

iv.           The entries lift in the table will give only the coefficients of variables included in the system but not included in the equation under consideration.

v.             From all these entries form a possible matrix A (say) of order (M – 1)

vi.           If at least one non-zero determinant can be found, the equation will be identified otherwise not identified.

The following example will clear all steps of the rank condition of identification.

Consider the following model

$$

Y1 = 3 Y2 - 2 X1 + X2 + u1

Y2 = Y3 + X3 + u2 

Y3 = Y1 + Y2 + 2 X3 + u3

These equations can be expressed as:

Y1 -  3 Y2 + 2 X1 -  X2 = u1

Y2 - Y3 - X3 = u2 

Y3 - Y1 - Y2 - 2 X3 = u3

1.  Table of coefficients ignoring u's

2               Let we checking the identification for eq. 2. Striking 2^nd row and the columns in which non-zero coefficients for eq. 2.

we have

i.               From this matrix A, we can obtain (M – 1) non zero possible determinants = M – 1 = 2

Hence there is a determinant of order (M – 1 = 2). The rank condition is satisfied and the second equation is identified.

Proceed in the same manner for other equations.

Practice Question

Apply the order and rank condition to examine the identification of eq. 1 and eq. 2.

qd = α2 Pt + α3 yt + u1

qs = β2 Pt + β3 wt + u2

qd = qs

Solution:

 Identification by order condition

Now by rank condition.

Striking out demand equation: and non zero coefficients of demand equation.

we have 

The order of matrix is 2. M – 1 = 2 - 1 =1

The demand equation is exactly identified.

Now Striking out supply equation: and non zero coefficients of supply equation.

we have

The order of matrix is 2. M – 1 = 2 - 1 =1

The Supply equation is exactly identified.

Practice Question

Consider the following demand-and-supply model for money:

Mdt = β0 + β1 Yt + β2 Rt + β3 Pt + u1t →Demand for money

Mst = α0 + α1 Yt + u2t →Supply for money

where M = money

Y = income

R = rate of interest

P = price

Assume that R and P are predetermined.

a. Is the demand function identified?

b. Is the supply function identified?

c. Which method would you use to estimate the parameters of the identified equation(s)? Why?

d. Suppose we modify the supply function by adding the explanatory

variables Yt−1 and Mt−1. What happens to the identification problem?

Would you still use the method you used in c? Why or why not?

Solution: (a & b) Identification

i.                    Identification by order condition

Consider Mdt and Mst at equilibrium i.e., Mdt = Mst

ii.                    Identification by Rank condition

Demand for money model:

The demand for money is identified. 

Now consider the supply of money equation:

Hence (M – 1) = 3 – 1 = 2 non zero determinants.

The supply of money is identified.

(c): The ILS and 2SLS methods can be applied. (discuss later).

 (d):

Mdt = β0 + β1 Yt + β2 Rt + β3 Pt + u1t

Mst = α0 + α1 Yt + α2 Yt-1 + α3 Mt-1 + u2t 

i.                    Identification by order condition

ii.                    I Table of coefficients 

Identification by Rank condition for supply of money.

The supply for money is identified.

Note: Proceed for demand for money model.

Practice Question

Consider the following simple version of Keynesian model of income determination.

Ct = a0 + a1 Yt + a2 It + u1 →Consumption model 

It = b0 + b1 Yt -1+u2 →Investment model

Tt = c0 + cYt + u3 →Taxation model

Yt = Ct + It + Gt →Definition function

i.               Used order condition and check identification for investment & taxation models.

ii.            Also check identification for investment & taxation models by using rank condition.

Solution:

Using order condition: K (Exogenous variables in the system) = 2 i.e., Yt -1 & Gt

ii.               Using rank condition: Table of structural coefficients ignoring intercepts.

Check identification for investment model:

Thus, we have non – zero determinants. Hence the rank condition is satisfied. The investment model is identified.

Now check identification for taxation model

The taxation model is identified.