Estimation Techniques

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Estimation Techniques

The identification condition is basic requirement, to check whether the structural parameters can be retrieved from the reduced form parameters.

In cases where an equation or system is exactly identified, the indirect least square (ILS) method is the most suitable method, and methods such as the two-stage least squares (2SLS) method, the three-stage least squares (3SLS) method, the instrumental variable technique, and the maximum likelihood estimation method are used to estimate the over identified equation or system. However, there is no estimating technique for the under identified equation or system in question.

These methods of estimation are not appropriate for estimating the parameters of reduced forms. The OLS method can be used to estimate the reduced form model, and the reduced form parameters can be used to determine the structural parameters.

Indirect Least Squares Estimation Method

(ILS Method)

When a system or an equation of the model is exactly identified, the indirect least squares estimation method is appropriate. This method is usually employed when one of the system's endogenous variables is a function of other endogenous and exogenous variables.

Consider the demand – supply model:

Qd = α0 + α1 Pt + α2 Yt + u1t 

Qs = β0 + β1 Pt + β2 Wt + u2t

The steps involved in ILS method:

Step – 1: Identification status of an equation

Step – 2: Transformed the structural form model into reduced form model.

that is endogenous variables is a function of exogenous variables.

Step – 3: Apply OLS method on the reduced form model on each equation separately. Estimate the reduced form parameters (π⌢ij ) denoted by .πij

Step – 4: Obtain the estimates of structural parameters from the relationships of structural and reduced form estimates.

Step – 5: Re transformed the reduced model in to structural model.

Assumptions of ILS Estimation

i.               The structural equations must be exactly identified.

ii.            The disturbance term of the reduced form model must satisfy all the assumptions OLS method.

iii.          The exogenous variables of the model must not be perfectly col linear.

 Practice Question

Consider the demand – supply model: 

Qd = α0 + α1 Pt + α2 Yt + u1t

Qs = β0 + β1 Pt + β2 Wt + u2t

Qd = Qs

identify the model and suggest / apply the  estimation method.

Solution:

Step – 1: Identification of the equations.

Using order condition of Identification 

Using rank condition

The table of coefficients ignoring intercepts

Identification for demand equation

As M - 1 = 1 of non zero determinants. The demand equation is identified.

Now using rank condition for supply eq.

The supply eq. is also identified.

Step – 2: Transform structural parameters into reduced form parameters.

Consider the equilibrium condition.

Qd = Qs

                                α0 + α1 Pt + α2 Yt + u1t = β0 + β1 Pt + β2 Wt + u2t

   α1 Pt - β1 Pt+ = β0 -  α0 + β2 Wt - α2 Yt  + u2t -  u1t

(α1 - β1) Pt+ = β0 -  α0 + β2 Wt - α2 Yt  + u2t -  u1t

We know that Qd = Qs substitute Pt in demand equation,

                                              Qd = α0 + α1 Pt + α2 Yt + u1t

In equation A and B endogenous variable is a function of exogenous variables. The OLS method can be applied to estimates the parameters.

That’s

 Now related structural parameters to reduced form parameters, using eq. C & eq. D.

Hence the structural parameters are obtained from reduced form parameters.

The estimated models are:

Practice Question

Consider the model.

Y1t = β10 + β12 Y2t + γ11 X1t + u1t

Y2t = β20 + β21 Y1t + u2t

Produces the following structural equations.

$$

Y1t = 4 + 8 X1t

Y2t = 2 + 12 X1t 

Which structural coefficients can be obtained from reduced form coefficients? Demonstrate your contention.

Solution: First we use identification procedure:

Identification by order condition:

for equation - 1:

K - k = o  and m - 1 = 1

The second equation is over identified.

for equation - 1:

K - k = 1  and m - 1 = 1

The first equation is exactly identified..

Identification by rank condition:

The table of structural parameters:

Identification of eq. 1:

The null matrix is obtained, so Det (A) = 0

Equation – 1 is not identified.

Identification of eq. 2:

he equation – 2 is identified.

According to order and rank condition, equation – 2 is exactly identified, so we use ILS method to estimate equation – 2.

Substitute  in equation 1, given below:

Y1t = β10 + β12 Y2t + γ11 X1t + u1t

Y1t = β10 + β12 ( β20 + β21 Y1t + u2t ) + γ11 X1t + u1t

Y1t = β10 +( β20  β12 +  β12  β21 Y1t +  β12  u2t ) + γ11 X1t + u1t

 

Y1t = β10 + β20  β12 +  β12  β21 Y1t + γ11 X1t + u1t  +  β12  u2t 

                              Y1t  -  β12  β21 Y1t = β10 + β20  β12+ γ11 X1t + u1t  +  β12  u2t 

                            ( 1  -  β12  β21 ) Y1t = β10 + β20  β12+ γ11 X1t + u1t  +  β12  u2t 

In equation 1 & 2, endogenous variable is a function of exogenous variable. So, OLS method can be used to estimate the reduced form parameters.

Now obtaining the estimates of structural parameters from reduced form in eq. 3 & 4.

Practice Question

Data on three variables (quantity, price, weather conditions) are given below:

Quantity (Qt)	Price (Pt)	Weather condition (Wt )
4	2	2
6	4	3
9	3	1
3	5	1
5	8	3

Determines reduced form parameters and structural parameters using the following demand supply models.

Qd = α1 + α2 Pt + u1t 

Qs =β1 +β2 Pt +β3 Wt + u2t

Qd  = Qs

Solution: First we check identification

Identification by order condition: 

K - k = 1

m - 1 = 1

As K - k = m - 1, the demand equation is exactly identified.

K - k = 0

m - 1 = 0

As K - k = m - 1, the supply equation is exactly identified.

Now using rank condition:

Checking identification for demand model:

The demand model is identified.

Checking identification for supply model:

The supply model is also identified.

From order and rank conditions it is observed that both models are exactly identified. Now transform structural model into reduced form (called single equation model).

Qd = Qs 

α1 + α2 Pt + u1t = β1 +β2 Pt +β3 Wt + u2t

 α2 Pt - β2 Pt  = β1 - α1 + β3 Wt + u2t -  u1t

( α2  - β2  )Pt  = β1 - α1 + β3 Wt + u2t -  u1t

Equation – 1 & 2 are single equation model, so we can use OLS method to estimate the parameters.

The estimated demand model is given by

Practice Question

From the model

Y1t = β10 +β12 Y2t +γ11 X1t + u1t

Y2t = β20 +β21 Y1t +γ22 X2t +u2t

The following reduced-form equations are obtained:

$Y_{1t} = 10 +11 Y_{2t} +12 X_{1t} + w_t$

$$\begin{gathered} \text{ \\ <br>} \end{gathered}$$

$$\begin{gathered} \text{ \\ Y2t = 20 +21 Y1t +22 X2t +vt} \end{gathered}$$/div

a. Are the structural equations identified?

b. What happens to identification if it is known a priori that γ11 = 0?

Solution:

a.       First, we check the identification status:

Identification by order condition:

K - k = 1 and m - 1 = 1

The first equation is exactly identified.

K - k = 1 and m - 1 = 1

The second equation is exactly identified.

Identification by rank condition:

Checking identification status for eq. 1:

Equation. 1 is identified.

Checking identification status for eq. 2:

Equation 2 is also identified.

From order and rank condition it is clear that both the equations are exactly identified. So we can use OLS method to estimate reduced form parameters.

Now transformed in to reduced form by substituting  (from eq.2) in eq. 1:

Y1t = β10 +β12 ( β20 +β21 Y1t +γ22 X2t +u2t ) +γ11 X1t + u1t

Y1t = β10 + β12 β20 + β12 β21 Y1t +β12 γ22 X2t +β12  u2t  +γ11 X1t + u1t

Y1t -  β12 β21 Y1t  = β10 + β12 β20 ++β12 γ22 X2t +β12  u2t  +γ11 X1t + u1t

                     ( 1 -  β12 β21 ) Y1t  = β10 + β12 β20 ++β12 γ22 X2t +γ11 X1t + u1t  + β12  u2t 

Identification by order condition:

equation - 1:

K - k = 1 and   m - 1 = 1

Equation - 1 is exactly identified.

equation - 2:

K - k = 0 and   m - 1 = 1

Equation - 2 is under identified.

Identification by rank condition:

Rank condition for equation – 1:

Rank condition for equation – 2:

Practice Question

Consider the following model:

Rt = β0 + β1 Mt + β2 Yt +u1t

 

Yt = α0 +α1 Rt + u2t

where Mt (money supply) is exogenous, Rt is the interest rate, and  is GDP.

1. How would you justify the model? b. Are the equations identified?

Solution: Identification

Order condition:

K - k = 0   and   m - 1 = 1

The equation - 1 is under identified.

K - k = 1    and    m - 1 = 1

The equation - 2 is exactly identified.

identification by Rank condition:

Table of coefficients:

Identification for equation – 1 by rank condition:

A is null matrix. So, eq. 1 is not identified.

Identification for equation – 2 by rank condition:

Practice Question

Consider the SEM

:Rt = β0 + β1 Mt + β2 Yt + β3 Yt-1+u1t

Yt = α0 +α1 Rt + u2t

Find out if the system is identified.

Solution:

Identification by order condition:

K - k = 0   and   m - 1 = 1

Equation - 1 is under identified.

K - k = 2   and   m - 1 = 1

Equation - 2 is over identified.

Identification Rank condition:

Identification of eq.1 by rank condition:

A is null matrix. So, eq. 1 is not identified.

Identification of eq.2 by rank condition:

From order and rank conditions it is clear that eq. 1 is under identified and eq. 2 is over identified. Now equation 2 can be solved by 2 SLS.