Two Stage Least Square Techniques in Simultaneous Equation Model

A Semester Paper on

Eco 602 : Applied Quantitative Techniques

Submitted to Central Department of Economics (CEDECON) for the

Internal Assessment of Partial Fulfillment of Master of Philosophy

(M. Phil 2010) in Economics

Submitted by

Bigyan Shrestha

Roll No. 21

March, 2010

Table of Contents

1. Introduction 1

2. Objectives 1

3. Two Stage Least Square Method 2

4. Assumption of Two Stage Least Square 3

5. Simultaneous equation model – Keynesian model 4

6. Estimation of parameters of Keynesian model in the context of

Nepal using Eview program 6

7. Conclusion 8

References 9

Page: 1

1. Introduction When variables under study are dependent upon each other or value of the variables are

affected by the another endogenous variables, the variables are called simultaneously

related and the model describing the relationship among the variables are called

simultaneous equations model. The variables may appear as endogenous variable in some

equations and explanatory variables in other equations of the model. For example, if the

model of two variables X and Y are taken as;

Y = f(x); and

X=f(y)

In the above case, Y is determined by X as shown in (i), but X is also dependent on Y, which

makes above model a simultaneous equations model.

If ordinary least square (OLS) estimates is applied to get the parameter of the simultaneous

equations model, estimates of the parameter are both biased and inconsistent. The bias

arising due to the simultaneous dependence of the variables is called simultaneous

equations bias.

There are several methods for the purpose of estimates of unbiased and consistent

parameters of the simultaneous equations model, the most common are:

1. The reduced form method or Indirect least square method

2. The methods of Instrumental variables

3. Two stage least squares

4. Limited information maximum likelihood

5. The mixed estimation method

6. Three stage least squares

7. Full information maximum likelihood

Two stage least square method has been developed by the Theil and independently by

Bassman. It is a single equation method being applied to one equation of the system at a

time. It has provided satisfactory result for the estimate of parameters and has been

Page: 2

accepted as the most important of the single equation techniques for estimation of

overidentified models.

2. Objective

The objective of this paper is to present the two stage least square techniques in estimating

the parameters of the simultaneous equations model and determine the parameters of the

simple Keynesian model in the context of Nepalese economy taking the data from 1988 to

2008 using the two stage least square techniques. And evaluate the parameters obtained.

3. Two Stage Least Square Method Two stage least square estimates is the method applied to estimate the parameters of the

simultaneous equations model. This is a single equation method being applied to one

equation of the system at a time. This method aims at the elimination of the simultaneous

equations bias as far as possible. In this method ordinary least square is applied in two

stages.

In the first stage, least square method is applied to reduced form equations in order to

obtain an estimate of exact and the random component of exogenous variable; and

In the second stage, endogenous variable appearing in the model as explanatory variable is

replaced by the estimated value from the first stage estimates and ordinary least square is

again applied to obtain estimates of the structural parameters.

Methods of determination of parameters through two stage least square method is

presented below:

Let the ith structural equation is of the general form

�� = ��� �� � � ����� � �+ + ��� �� � � �� �� � �+ + + �� ���� � ��

Where Yi’s will denote endogenous variables (I = 1,2,………G)

Xi’s will denote predetermined variables (I = 1,2………….K)

b’s will represent the coefficient of endogenous variables

� . … will represent the coefficient of predetermined variables.

In the first stage, we apply ordinary least squares to the reduced form equations to obtain

estimates of the π’s given below:

Page: 3

�� = ��� �� � ���� ��� �+ + + ��� ��� �� � ��

�� = ��� �� � ������� �+ + + ��� ����� � ��

… …… …... ……. …… ..

… …… …... ……. …… ..

�� = ��� �� � ���� ��� �+ + + ��� ��� �� � ��

Using the reduced from coefficients obtained in first stage, we obtain a set of estimated

(computed) values for the endogenous variables: ��� � ����+ + + + ��� .

In the second stage, we substitute the �� ’s into the structural equation and obtain the

transformed functions,

�� = ��� ��� � ������� ��+ + ��� ��� � � �� �� � �+ + + �� ���� � ��

Applying ordinary least square method to the transformed structural equation we obtain

the 2 stage least square estimate of the structural parameters.

4. Assumptions of Two Stage Least Squares

a) The disturbance term u of the original structural equations must satisfy the usual

stochastic assumptions of zero mean, constant variance and zero covariance.

b) The error term of the reduced form equations v’s must satisfy the usual stochastic

assumptions, that is v has zero mean, constant variance, zero covariance, and must be

independent of the exogenous variable of the whole structural model.

c) The explanatory variables are not perfectly multicolinear and all macro variables are

properly aggregated.

d) Specification of the model is correct so far as the exogenous variables are concerned.

e) The sample is large enough, and in particular that the number of observations is greater

than the number of predetermined variables in the structural system.

5. Simultaneous Equation Model – Keynesian Macro Economic Model

Simple Keynesians macroeconomic model shows the relationship between consumption,

investment and Income. Mathematically, the model is given by:

Page: 4

�� = ��� � ��� �� � ������ � ��� …………………………………….(1)

�� = ��� � ��� �� � ������ � ���………………………………………(2)

�� = ��� � ��� � ���………………………………………………………….(3)

Where Ct = consumption for the period t

I t = Investment during the period t

Y t = Income for the period t and

G t = Government expenditure for the period t

u’s are random variables

a’s and b’s are parameters of the equations.

In this model, consumption, investment and Income are endogenous variable which are

interdependent. In the first equation, income determines consumption. In the second

model Income during the year and prior year income determines the Investment.

Consumption, Investment and Government expenditure during the year determines the

Income, which is shown by third equations. Hence, the above model is in the form of

simultaneous equation models.

Lagged consumption, lagged income and government expenditure are exogenous variables

in the model.

For the estimates of the parameters of the above simultaneous equations model, we apply

the 2 stage least square techniques, which provides the unbiased and consistent estimates

for the simultaneous equation model.

We take the data for the 20 year period from 1988 to 2008 relating to Nepalese economy

and estimate the parameters of the above Keynes model to estimate the parameters of the

model.

Page: 5

Table 1 National Income Data

Rs. In Million

Year Gross National Product (GNP)

Consumption (C) Investment (I) Government Expenditure (G)

1988 233,340.49 180,310.78 21,815.21 43,293.81

1989 244,125.66 200,013.88 20,934.40 39,224.63

1990 260,118.21 207,579.50 29,929.61 41,969.82

1991 271,492.04 216,498.68 33,793.36 39,669.07

1992 281,168.97 214,696.22 41,054.00 42,920.90

1993 305,408.52 231,632.97 43,077.58 44,152.57

1994 316,301.58 235,036.00 47,023.30 49,899.77

1995 330,690.31 250,781.03 50,369.96 53,231.82

1996 348,871.98 264,693.14 50,649.95 54,291.92

1997 360,343.93 271,713.44 50,260.50 59,403.26

1998 380,894.57 285,947.49 44,661.49 58,730.92

1999 405,227.82 297,198.86 48,394.53 62,340.68

2000 443,220.00 348,989.00 66,687.00 59,091.00

2001 441,615.01 347,398.46 65,830.61 59,557.27

2002 458,968.25 346,798.32 69,889.82 60,075.63

2003 480,309.69 335,778.28 70,458.71 62,253.14

2004 500,888.14 332,388.98 66,133.90 62,292.37

2005 522,194.93 327,430.27 70,085.58 59,870.05

2006 542,885.61 313,962.36 65,664.21 61,991.88

2007 573,852.78 305,240.97 64,960.42 63,852.78

2008 608,201.11 287,410.27 57,881.19 70,029.70 Src: Economic Survey 2001,2009

Page: 6

6. Estimation of parameters of Keynesian model in the context of

Nepal using Eview program

The two stage least square technique is applied for estimate of parameters of the model

and run with the above data in E-Views. The output of the E - view program is as follow:

For the first structural equation (Consumption Function)

Interpretation of results,

From the above outcome, it shows that constant and coefficient of GNP are not significant

at 5% level as shown by t test. Also from standard error of coefficients, constant and GNP

coefficient doesn’t explain the variation in Consumption [s.e(coefficient)>½ of coefficient]. R

squared is greater than 0.9, which shows that fitted line explains 90% of the total variation

of consumption around there mean. From value of F statistics, we can say that GNP and

lagged consumption are significant explanatory variable in explaining consumption.

Overall, the fitted line is not good fit. Constant and coefficient of GNP are not significant.

Page: 7

For the equation 2, investment function

Interpretation of result

Coefficient of all three variables are not significant as shown by t statistics and standard

error of coefficient. The observed values of statistics shows the no relationship in explaining

investment function.

Hence, the data of Nepal doesn’t fit the Keynesian general model of income stated in

equation 1,2 and 3.

Page: 8

7. Conclusion Two stage least square estimates is the method applied to estimate the parameters of the

simultaneous equations model. This method has been developed by Theil and

independently by Basmann.

This is a single equation method being applied to one equation of the system at a time. This

method aims at the elimination of the simultaneous equations bias as far as possible. In this

method ordinary least square is applied in two stages.

In the first stage, least square method is applied to reduced form equations in order to

obtain an estimate of exact and the random component of exogenous variable; and

In the second stage, endogenous variable appearing in the model as explanatory variable is

replaced by the estimated value from the first stage estimates and ordinary least square is

again applied to obtain estimates of the structural parameters.

In the context of Nepalese economy, on fitting the Keynesian macroeconomic model using

two stage least square method, it is found that the model is not good fit to the actual data

obtained.

Page: 9

REFERENCES

A. Koutsoyiannis, Theory of Econometrics, Second Edition (2004), Palgrave

Ministry of Finance, Economic Survey, 2001, 2009

D. Gujarati, Basic Econometrics, Fourth Edition (2004), McGraw Hill Companies