Two Stage Least Squares Method (2 SLS)

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Two Stage Least Squares Method

( 2 SLS Method)

The two stage least square (2SLS) is the extension of indirect least squares (ILS) and applicable, when the structural model is over identified. The two stage least squares method systematically creates an instrumental variable to replace the endogenous variable that is appearing as an exogenous variable in an over identified equation.

Let

Y1 =α0 + α1 X1 + α2 Y2 +u1

Y2 =β0 + β1 Y1 + β2 X2 +u2

The above model is transformed into reduced form and find the estimate of Y2 represented by  Y${\stackrel{}{^}}_{\mathrm{2\; by }}$OLS and substitute as instrumental variable as:

$$

Y^2 =β^0 + β^1 Y1 + β^2 X2 

Substitute  Y

Y1 =α0 + α1 X1 + α2 Y^2 +u1

The mechanism of 2 SLS is explained below:

Stage – 1:

·         Apply identification procedure and identify the equations of the model.

·         Transform the structural equations into reduced form equations and estimate the reduced form parameters by OLS method of the endogenous variable appearing as exogenous variable (in the RHS of the equation).

Stage – 2:

·         Replace the endogenous variable appearing as exogenous variable in the equation by their estimates. (like Y1 = f(Y^2, X1, u1) as instrumental variable. Then apply OLS method on the transform equation and estimates the parameters of the equation.

·          Find the estimates of structural parameters of the model from derived reduced form estimates in stage – 3.

 The 2 SLS technique is explained with the help of example given below:

Consider the SEM

Y1 = α0 + α1 Y1 + u1

Y2 = β0 + β1 Y1 + β2 X2 + β3 X3+ u2

Stage – 1: Identification by order condition:

K - k = 2    and    m - 1 = 1

K - k > m - 1

The Y1 = α0 + α1 Y1 + u1 is over identified.

                                                             

                                                           K - k = 0     and    m - 1 = 1

                                                                     K - k < m - 1

The Y2 = β0 + β1 Y1 + β2 X2 + β3 X3+ u2 is under identified.

Identification by rank condition:

 

Identification for eq. 1:

Equation 1 is identified.

Eq. 2 is not identified.

From order and rank condition it is clear that equation 1 is over identified and 2 SLS is applicable, while equation 2 is under identified and no method is available to apply.

Now transformed into reduced form to estimate Y2 as:

Substitute Y1 (from eq. 1) into Y2 (in eq. 2)

Y1 = α0 + α1 X1 + α2 (β0 + β1 Y1 + β2 X2 + u2) + u1

Y1 = α0 + α1 X1 +  (α2 β0 + α2 β1 Y1 + α2 β2 X2 + α2 u2) +u1

Y1 - α2 β1 Y1  = α0 +   α2 β0  + α1 X1+  α2 β2 X2 + α2 u2 +u1

(1- α2 β1 )Y1  = α0 +   α2 β0  + α1 X1+  α2 β2 X2 + α2 u2 +u1

$$

Y2 = π10 + π11 X2 + π12 X2 + u1

Using OLS method and estimated model as:

Y^2 = π^10 + π^11 X2 + π^12 X2 

Stage – 2:

Replace Y2 by Y^2 in equation 1 given below:

Y1 = α0 + α1 Y2 + u1

Y1 = α0 + α1 Y^2 + u1

Estimated this model by again using OLS method as:

The estimated model is given by

                                                            Y^^1 = α^0 + α^1 Y^2 

Assumptions of 2 SLS Model Estimation

i.          The disturbance term of the original structural equations must satisfy the classical assumptions.

a.        E ( ui ) = 0

b.     Var(ui) = σ2

c.         Cov (ui , uj ) = 0    for i # j

d.       E ( X  ui ) = 0

e.    ui ~ N (0,  σ2 )

ii.          The regressors are not perfectly multicollinear.

iii.       The model has been specified correctly.

iv.     The sample size should be large such that the number of observations will be greater than the number of regressors.

 

Practice Question

Consider the SEM

$$

qd = α0 +α1 Pt + α2 Yt +α3 Wt +u1t

qs = β0 + β1 Pt +u2t

The data given below:

qd	Pt	Wt	Yt
4	2	2	6
6	4	3	7
9	3	1	8
3	5	1	4
5	8	3	5
6	9	2	7

Estimate the model by using appropriate method.

Solution:

Stage – 1: Identification by order condition:

                                                K - k = 0        and         m - 1 = 1

As K - k < m - 1, the demand equation is under identified.

                                               K - k = 2        and         m - 1 = 0

As K - k > m - 1, the supply equation is over identified.

Now Identification by rank condition:

For demand equation:

The demand equation is identified.

Now considering the supply equation

The supply model is identified.

From the order and rank conditions, it is concluded that the supply model is over identified.

We need to find the estimate of Pt and use it as an instrumental variable (because Pt is an endogenous variable appearing as exogenous variable in supply model).

Consider the equilibrium status:

qd = qs = q

 β0 + β1 Pt +u2t = α0 +α1 Pt + α2 Yt +α3 Wt + u1t

 β1 Pt - α1 Pt = α0 -  β0 + α2 Yt +α3 Wt + u1t - u2t 

( β1 - α1) Pt = α0 -  β0 + α2 Yt +α3 Wt + u1t - u2t 

$$

Pt = π10 + π11 Yt + π12 Wt + vt

Now we using the given data to estimate π10 , π11 and π12   by OLS.

Stage – 2:

Substitute the values of Yt and Wt from the given table values and estimate P^t, given below:

q t	P^t	qt P^t	P^t2
4	1.73
6	1.06
9	-          2.61
3	4.07
5	4.4
6	0.06
33	8.71	24.36	46.86

Now use P^t as an instrumental variable in supply equation.

qs = β0 +β1 P^t + u2t

Again, using OLS method to estimate  β0 and   β1 

The estimated supply model by 2 SLS is given by;

Practice Question

The market mechanism of a commodity is represented by the following system.

qd = α0 + α1P + α2 Y + u1

qs = β0 +β1P + β2 W+ u2

The relevant observations are as follow:

q	P	Y	W
98	2	154	5
90	3	146	4
73	10	130	1
80	8	140	2
84	7	143	3

Solution: Consider equilibrium and check the identification.

i.                    By Order condition:

                                                       K - k = 1       and       m - 1 = 1

The demand equation is exactly identified.

K - k = 1     and      m - 1 = 1

The supply equation is exactly identified.

ii.                    By Rank condition

Identification for demand model by rank condition:

The demand model is identified.

Now supply equation

The supply model is identified.

From the order and rank conditions, it is concluded that the supply model is over identified.

We need to find the estimate of Pt and use it as an instrumental variable (because Ptis an endogenous variable appearing as exogenous variable in supply model).

Consider the equilibrium status:

qs = qd = q

 β0 +β1P + β2 W+ u2 = α0 + α1P + α2 Y + u1

β1P  - α1P  = α0 -  β0  + α2 Y - β2 W  + u1 - u2 

( β1 - α1) P  = α0 -  β0  + α2 Y - β2 W  + u1 - u2 

Pt =π10 + π11 Yt + π12 Wt + v1

Now we using the given data to estimate π10, π11, π12  by OLS.

The estimated model of  Pt =π10 + π11 Yt + π12 Wt + v1 is given by:

P^t =π^10 + π^11 Yt + π^12 Wt 

P^t = 9.75 - 1.67 Yt +  Wt 

Stage – 2:

Substitute the values of Yt and Wt from the given table values and estimate P^t, given below:

Now use P^t as an instrumental variable in supply equation.

qs = β0 +β1P^ + u2 

Again, using OLS method to estimate β0 and β1.

The estimated supply model by 2 SLS is given by;

                                                                           q^^s = β^0 +β^1P^

q^^s = 6.50 - 0.69 P^

Practice Question

Using 2SLS if possible, to estimate the parameters of the following model.

Y1= α1 +α2 Y2 + α3 X1 +α4 X2 + u1

Y2 =β1+ β2 Y1+ u2

Using the data given below:

Y1	Y2	X1	X2
4	3	1	4
7	6	2	4
8	2	3	8
9	5	4	10
10	7	5	12
11	9	6	20

SolutiSolution:

Identification by order condition:

K - k = 0      and     m - 1 = 1

The first equation is under identified.

K - k = 2      and      m - 1 = 1

The second equation is over identified.

Identification by ranked condition

II

TheTTable of coefficients:

con

Considering equation 1

A is a null matrix, so det (A) = 0

Equation 1 is not identified.

Now considering equation 2:

Equation 2 is identified.

From the order and rank conditions, it is concluded that equation 2 is over identified. So, it is possible to use 2 SLS.

Now we substitute Y2 (from eq. 2) in Y1 (in eq. 1) to estimate Y2 and use it as instrumental variable.

Y1= α1 +α2 Y2 + α3 X1 +α4 X2 + u1

Y1= α1 +α2 ( β1+ β2 Y1+ u2 ) + α3 X1 +α4 X2 + u1

Y1= α1 +α2  β1+α2  β2 Y1+α2  u2  + α3 X1 +α4 X2 + u1

Y1 - α2  β2 Y1= α1 +α2  β1 + α3 X1 +α4 X2 + u1 +α2  u2 

(1 - α2  β2) Y1= α1 +α2  β1 + α3 X1 +α4 X2 + u1 +α2  u2 

Y1 = π11 + π12 X1 +π13 X2 + v1

Now using the given data to estimate the parameters of the above model.

Main Features of 2 SLS

i.          2 SLS can be applied to an individual equation in the system without directly taking with account any other equation in the system.

ii.       2 SLS provide one estimate per parameter in case of overidentified.

iii.     2 SLS is specially designed to handle over identified equation and also applied to exactly identified equation.

iv.      If the value of coefficient of determination in the reduced form equation is high. The OLS estimates and 2 SLS will be very close.

Theorem: Show that ILS and 2 SLS are equivalent when the model is exactly identified.

Proof:                       Consider the simultaneous equation model

 Y1 = α1 +α2 Y2 + u1 → 1

Y2 = β1 + β1 Y1 + β2 X1 + u2 →2

 Equation - 1  is exactly identified.

For the sack of simplicity, we transformed the above model in to deviated forms.

y1 = α2 y2 + u1 ......3

 y2 = β1 y1 + β2 x1 + u2 .....4

Substi     Substitute y2 (from eq. 4) into eq. 3.

y1 = α2 (β1 y1 + β2 x1 + u2) + u1 

y1 =  α2 β1 y1 + α2 β2 x1 + α2 u2 + u1 

y1 - α2 β1 y1 = α2 β2 x1  + u1 + α2 u2

(1 - α2 β1) y1 = α2 β2 x1  + u1 + α2 u2