Rules for Identification Cont.…SEM

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Rules for Identification

It is possible to identify something either from its structural form or its reduced form. It is possible to examine an equation's identification in a model of simultaneous equations by finding out its structural parameters from reduced form parameters, but this procedure is laborious and time-consuming.

There are the two approaches listed below, both of which make it simpler to apply and verify the identity. These techniques are;

1.      Order Condition

2.      Rank Condition

Order Condition

For identification, the order criterion is a necessary but insufficient condition. Let's introduce the following notations to state this condition:

There are M endogenous variables and K endogenous variables in the system. Let m and k are the endogenous and exogenous variables in an equation being examine for identification.

The identification status is depended on the following conditions:

i.               The equation is exactly identified, when

K - k = m - 1

ii.            The equation is over identified, when

K - k > m - 1

iii.          The equation is under identified, when

K - k < m - 1

Practice Question

State the identification status of the following by using order condition.

$$

Qd = α1 P + α2 I + u1

Qs = β1 P + u2

Solution: The number of exogenous variables in the model is; K = 1 (i.e., I)

Considering the demand equation:

The number of endogenous variables in demand equation is; m = 2 (i.e., Q & P)

The number of exogenous variables in demand equation is; k = 1 (i.e., I)

According to order condition:

K - k = 0

m - 1 = 1

K- k < m - 1

The demand equation is under identified.

Now considering the supply equation;

The number of endogenous variables in demand equation is; m = 2 (i.e., Q & P)

The number of exogenous variables in demand equation is; k = 0

According to order condition:

K - k = 1

m - 1 = 1

K- k = m - 1

The supply equation is exactly identified.

Practice Question

Apply the order condition to examine the identification of the following model of simultaneous equations.

$Y_{\mathrm{1 }}= {\alpha }_{\mathrm{0 }}+ {\alpha }_{\mathrm{1 }}{X}_{\mathrm{1 }}+ {\alpha }_{\mathrm{2 }}{Y}_{\mathrm{2 }}+ {\alpha }_{\mathrm{3 }}{Y}_{\mathrm{3 }}+ {\alpha }_{\mathrm{4 }}{X}_{\mathrm{2 }}+ u_1$

$$\begin{gathered} \text{ \\ Y2 = β0 + β1 X1 + β2 X2 + β3 Y1 + β4 Y3 + u2} \end{gathered}$$

$$\begin{gathered} \text{ \\ Y3 = γ0 + γ1 Y1 + γ2 Y2 + γ3 X1 + γ4 X2 + u3} \end{gathered}$$

$$\begin{gathered} \text{ \\ } \end{gathered}$$

Solution:

The No, of endogenous variables in the system (OR model): M = 3 (i.e. Y1,  Y2, Y1 ).

The No, of exogenous variables in the system (OR model): K = 2 (i.e.X1, X2).

Equation – 1:

The No, of endogenous variables in the equation – 1: m = 3 (i.e. Y1,  Y2, Y1 ).

The No, of exogenous variables in the equation – 1: k = 2 (i.e.X1, X2).

According to identification condition:

K - k = 2 - 2 = 0

m - 1 = 3 - 1 = 2

K - k < m - 1

Equation – 1 is under identified.

Equation – 2:

The No, of endogenous variables in the equation – 1: m = 3 (i.e. Y1,  Y2, Y1 ).

The No, of exogenous variables in the equation – 1: k = 2 (i.e.X1, X2).

According to identification condition:

K - k = 2 - 2 = 0

m - 1 = 3 - 1 = 2

K - k < m - 1

Equation – 2 is under identified.

Similarly proceed for equation – 3.

Practice Question

A researcher develops the demand supply models for a particular period of time of a particular area as:

Qd = α2 Pt + α3 Yt + α4 Wt + u1

Qs = β2 Pt + u2

Examine the identification by using order condition.

Solution:

The No, of endogenous variables in the system (OR model): M = 2 (i.e. Qd, Pt).

The No, of exogenous variables in the system (OR model): K = 2 (i.e. Yt, Wt).

Equation – 1 (Demand model)

The No, of endogenous variables in the equation – 1: m = 2 (i.e. Qd, Pt).

The No, of exogenous variables in the equation – 1: k = 2 (i.e.Yt, Wt).

K - k = 0

m - 1 = 1

K - k < m - 1

The demand model is under identified.

Equation – 2 (supply model)

The No, of endogenous variables in the equation – 1: m = 2 (i.e.Qd, Pt).

The No, of exogenous variables in the equation – 1: k = 0 

K - k = 2

m - 1 = 1

K - k > m - 1

The supply model is over identified.

Practice Question

Apply counting rule examine the identification for the following simultaneous equation system.

Qd = α1 + α2 Pt + α3 Yt + ϵ1t 

Qs = β1 + β2 Pt + β3 Wt + ϵ2t

Qd = Qs

Solution:

The No, of endogenous variables in the system (OR model): M = 2 (i.e. Qt, Pt).

The No, of exogenous variables in the system (OR model): K = 2 (i.e. Yt, Wt).

Equation – 1 (Demand model)

The No, of endogenous variables in the equation – 1: m = 2 (i.e. Qt, Pt).

The No, of exogenous variables in the equation – 1: k = 1 (i.e.Yt).

K - k = 1

m - 1 = 1

K - k = m - 1

The Demand model is exactly identified.

Equation – 2 (supply model)

The No, of endogenous variables in the equation – 1: m = 2 (i.e. Qt, Pt).

The No, of exogenous variables in the equation – 1: k = 1 (i.e. Wt).

K - k = 1

m - 1 = 1

K - k = m -1

The Supply model is exactly identified

Practice Question

Using order condition and test whether the following simultaneous equations are identified, over identified or under identified.

Qd = α0 + α1 Pt + α2 lt + α3 Rt + ϵ1t 

Qs = β0 + β1 Pt + β2 Pt-1 + ϵ2t

Solution: At equilibrium:

Qd = Qs

The No, of exogenous variables in the system (OR model): K = 3 (i.e. ,lt, Rt, Pt-1 ).

 Equation – 1 (Demand model)

The No, of endogenous variables in the equation – 1: m = 2 (i.e.lt, Rt).

The No, of exogenous variables in the equation – 1: k = 2 (i.e.Qd, Pt).

K - k = 1

m - 1 = 1

The demand model is exactly identified.

Equation – 2 (Supply model)

The No, of endogenous variables in the equation – 1: m = 2 (i.e. Qt, Pt).

The No, of exogenous variables in the equation – 1: k = 1 (i.e.Pt-1).

K - k = 2

 m - 1 = 1

K - k > m - 1

The supply model is over identified.