application of simultaneous equation in social sciences

8/13/2019 Application of Simultaneous Equation in Social Sciences

1/52

Application of SimultaneousEquation in Social Sciences


2/52

CONTENT

INTRODUCTION

PROBLEMS OF SIMULTANEOUS EQUATIONS MODELS

STRUCTURAL, REDUCED-FORM AND RECURSIVE MODELS

SIMULTANEOUS EQUATION METHODS

CONCLUSION

2


3/52


4/52

But there are situations where such one way

causation in the function is not meaningful. This

occurs if Y (dependent) variable is not only

function of X (explanatory) variable but also X isfunction of Y. There is, therefore, a two-way flow

of influence between Y and (some of) the X which

in turn makes the distinction between dependent

and independent variables a little doubtful.

4


5/52

Under such circumstances, we need to consider

more than one regression equations; one for each

interdependent variable to understand the multi-flow

of influences among the variables.

Thus in short,

A system describing the joint dependence of

variables is called a system of simultaneous equations

or simultaneous equations model.

5


6/52

For example:

Consumption Function:

Income Identity:

Where,

Y = Income

C = consumption expenditure

I = Investment

t = time

U = Stochastic disturbance termAbove equations are simultaneous equations.

ttt uYC 10

ttt ICY

6


7/52

Examples of Simultaneous Equation Models in economics

1. DemandSupply Model

It is well known, the price P of commodity and the

quantity Q sold are determined by the intersection of the

Demand-Supply curve.

Thus, assuming that the Demand-Supply curves are

linear and adding the stochastic disturbance terms u1 and

u2, the empirical demand and supply function may be

written as:

7


8/52

Demand function:

Supply function:

Equilibrium condition:

Where,

= quantity demanded

quantity supplied

and

tt

d

t uPQ 10 01

ttst uPQ 210 01

s

t

d

t QQ

d

tQ

stQ

parametersaresands ''

8


9/52

Fig. 1 Interdependence of price and quantity

9


10/52

2. Keynesian Model of Income determination

Consumption function:

Income identity:

Where,

Y = Income

C = consumption expenditure

I = Investment

t = time

U = Stochastic disturbance term

is marginal propensity to consume (MPC) and is expected to

remain between 0 and 1

ttt uYC 10 10 1

ttt ICY

parametersareand 10

1

10


11/52

3. WagePrice Models

Where,

W= rate of change of money wages

UN = unemployment rate in %P = rate of change in prices

R= rate of change of cost of capital

M = rate of change of price of imported raw material

t = time

u = stochastic disturbance term

ttttt

tttt

uMRWP

uPUNW

23210

1210

11


12/52

PROBLEMS OF SIMULTANEOUS EQUATIONS MODELS

Simultaneous equations models create three distinct

problems.

1) Mathematical completeness of model

Any model is said to be mathematically complete onlywhen it possesses as many independent equations as

endogenous variables. In other words if we happen to

know values of disturbance terms, exogenous variablesand structural parameters, then all the endogenous

variables are uniquely determined.

12


13/52

2) Identification of each equation of the model

Many times it so happens that a given set of values of

disturbance terms and exogenous variables yield the same

values of different endogenous variables included in the

model. It is because the equations are observationally

indistinguishable. What is needed is that parameters of

each equation in the system should be uniquely

determines. Hence certain tests are required to examine

the identification of each equation before its estimation.

13


14/52

3) Statistical estimation of each equation of the model

Since application of Ordinary Least Square (OLS) yields

biased and inconsistent estimates, different statistical

techniques are to be developed to estimate the structuralparameters.

14


15/52

STRUCTURAL, REDUCED-FORM AND RECURSIVE MODELS

Some of the important definitions and notations frequently

used in the estimation of simultaneous equations model.

A) Endogenous and Exogenous variables:

Endogenousvariables are regarded as stochastic and their

values are determined within the model.

Exogenous or Predeterminedvariables are treated as non

stochastic and values are given. Generally the notation Y

symbolises for endogenous and X symbolises the

predetermined/exogenous variables.

15


16/52

B) Structural Models:

A structural model describes the complete structure of

the relationships among the variables.

Structural equations of the model may be expressed in

terms of endogenous variables, exogenous variables and

disturbances.

Structural parameters express the direct effect of each

explanatory variable on the dependent variable. Variables

not appearing in any function explicitly may have an

indirect effect on the dependent variable of the function.

Such effect is known as indirect effect and is taken into

account by simultaneous solution of the system.

16


17/52

For instance, a change in consumption affects the

investment indirectly and is not considered in theconsumption function. The effect of consumption on

investment cannot be measured directly by any structural

parameters, but is measured indirectly by considering the

system as a whole.

In the conventional notation endogenous and

exogenous variables are denoted by Ys and Xs

respectively; while structural parameters or coefficients are

depicted by sand s.

17


18/52


19/52

First method is simply to express the endogenous

variables directly as a function of exogenous variables.

Second method is to solve the structural system ofendogenous variables in terms of the exogenous variables,

structural parameters and the disturbances.

The reduced form, by this procedure would be:

222221

111211

eYP

eYQ

01

11

01

12

01

1001

VUYQ

19


20/52

Comparing the above two methods, the following

relationship between s and the structural parameters

must hold good, i.e.,

0101

2

01

00

VUYP

01

1001

11

01

12

12

01

11

11

VUe

20


21/52

As it may be observed sbear a definite relationship

with the structural parameters. The reduced form

parameters measure the total effect (direct and indirect) of

a change in the exogenous variables on the endogenous

variable. For instance in the above model, 12measures the

total effect of unit change in the disposable income on the

quantity.

01

00

21

01

2

22

01

22

VUe

21


22/52

4) Recursive Models

Because of the interdependence between the

disturbance term and the endogenous variables, the OLS

technique is not appropriate for the estimation of an

equation in the simultaneous equations model. However, in

a special type of simultaneous equations model called

Recursive, Triangular or Casual model, the use of OLS

procedure of estimation is appropriate.

22


23/52

Consider following three equation system.

Xs and Ys are the exogenous and endogenous

variables respectively. The disturbance term follow the

following assumption:

3232131232131303

2222121121202

1212111101

UXXYYY

UXXYY

UXXY

0),(),(),( 323121 UUEUUEUUE

23


24/52

The above assumption is most crucial assumption that

defines recursive model. If this does not hold good, the abovesystem is no longer recursive and recursive OLS is also no longer

valid.

The first equation given above contains only the exogenous

variable right hand side. Since by assumption, the exogenous

variables are independent of U1, first equation satisfy the

crucial assumption of OLS and hence OLS can be applied to this

equation.

1212111101 UXXY

24


25/52

Second equation contains endogenous variable Y1 as

one of explanatory variables along with the non-stochastic

Xs. OLS can be applied to this equation only if it can be

shown that Y1 and U2are independent of each other.

This is true because U1 which affects Y1 is by

assumption uncorrelated with U2, i.e., E(U1, U2) = 0. Y1 infact act as a predetermined variable insofar Y2 is

concerned. Hence OLS can be applied to this equation also.

2222121121202 UXXYY

25


26/52

Similar argument can be stretched to the third

equation because Y1 and Y2are independent of U3.

In this way, in recursive system OLS can be applied to

each equation separately.

3232131232131303 UXXYYY

26


27/52

Fig. 3. Unidirectional flow in the recursive system

All the predetermined variables and U1determines Y1.

Y1and all the predetermined variables and U2determines Y2.Y1, Y2 and all the predetermined variables and U3determines

Y2.

27


28/52

SIMULTANEOUS EQUATION METHODS

Following are the frequently used single equation

methods:

1) Indirect Least Squares (ILS)

2) Two Stage Least Squares (2 SLS)

28


29/52

1. Indirect Least Squares (ILS)

This method is designed to estimate one

equation at a time. The method is named indirect

least squares since it estimates the parameters

indirectly by estimating the reduced form

equations, in which endogenous variables are

expressed only as a function of exogenous variables

and of the error term.

29


30/52

Steps of Indirect Least Squares

Step 1:

Obtain the reduced form equations. These reducedform equations are obtained from structural equations in

such a manner that the dependent variable in each

equation is the only endogenous variable and is a function

solely of the predetermined (exogenous) variable and the

stochastic error term(s).

Step 2:

We apply OLS to the reduced form equations

individually. This is permissible since the explanatoryvariables in this equations predetermined and hence

uncorrelated with the stochastic disturbances. The

estimates thus obtained are consistent.

30


31/52

Step 3:

We obtain estimates of the original structural

coefficients from the estimated reduced form coefficients

obtained in step 2. If an equation is exactly identified,

there is a one - to - one correspondence between thestructural and reduced form coefficients; that is, one can

derive unique estimates of the former from the latter.

Let us take an exactly identified simple demand-

supply model.

31


32/52

Let us take an exactly identified simple demand-supplymodel.

Where,D = Quantity demanded

S = Quantity supplied

P = Price of the commodity

Y = Income

W = Weather index

tt

ttt

ttt

SD

UWPS

UYPD

2210

1210

32


33/52

The above model is mathematically complete , i.e.,

there are three endogenous variables (D, S, P) and three

equations.

First step of ILS is to obtain reduced form of model.

11

21

11

2

11

2

11

00

UUWYP ttt

11

1211

11

21

11

12

11

0110

UUWYD

ttt

2222120

1121110

VWYP

VWYD

ttt

ttt

33


34/52

Where sand Vsdepict the coefficients and disturbances

of the reduced form model. Since the reduced form equations

do not contain endogenous variables, the application of the OLS

method to each reduced form equation lead to unbiased

estimates of s. Hence using sample data on D, P, Y and W, we

may obtain estimates of s.

We are interested in the original structural parameters of

the model. Since the estimates if s are function of these

parameters, we estimate them indirectly from the s through

the following manipulation.

34


35/52

(a) Calculation of 0

0

00

1010

11

00

22

12

20

10

20

1

22

12

00

0110

20

10

;

Hence

and

insestimated

relevantthesubstitutethereforeweobtainto

'

,0

22

12

20

10

20

35


36/52

(b) Calculation of 0

21

11

20

10

20

0

0

00

1010

11

00

21

11

20

10

20

1

21

11

00

0110

20

10

',

;

inssubstituteweobtaintoTherefore

Hence

and

36


37/52

(c) Calculation of 2

22

12

21

1121

2

211

11

2

22

12

21

1121

1

22

12

1

21

11

',

;

inssubstituteweobtaintoTherefore

37


38/52

(d) Calculation of2

.

'Re

,

2

21

11

22

12

22

211

12

2

21

11

22

1222

1

21

11

1

22

12

obtainto

indsubstitutebetoareslevant

38


39/52

(c) Calculation of 11 and

.identifiedexactlyismodelequationussimultaneo

theonly whens'ands'ofvaluesuniqueto

risegivewillsandtscoefficienstructuralebetween threlationthatnotedbeshouldIt

21

111

22

121

and

39


40/52

The following are the assumptions of ILS procedure:

(1) The structural equation must be exactly identified.(2) The disturbance term of the reduced form equation must

satisfy all the assumption of OLS. This is essential because

this method is merely application of OLS to the reduced

form equations.If these assumptions are not fulfilled, the bias in s

will be transmitted to the estimates of the structural

parameters.

It may, therefore, be said that ILS method is based onall the assumptions of OLS along with additional

assumption that the model be exactly identified.

40


41/52

2. Two Stage Least Squares

This method also being a single equation method seek

to remove the defect of existence of the correlation

between the disturbance term and the independent

variable(s) so that when we apply OLS technique to eachstructural equation separately, the simultaneity bias gets

eliminated. Therefore, 2SLS may be considered as an

extension of ILS method.

41


42/52

In ILS interdependence between explanatory variable

and disturbance term is bypassed by applying OLS to

reduced form equation; here we purge the explanatory

variable(s) which is correlated with the error term with its

own estimated value. This is done in two stages.

Let us take demand-supply model.

tt

ttt

ttt

SD

UWPS

UYPD

2210

1210

42


43/52

Reduced form equations are:

Where s and Vs are estimated by applying OLS to

these reduced form equations. This is the first stage of

estimation.

Having estimated s, now replace for different

values of Yt and Wt. Now replace Pt in the structural model

by obtained in the first stage as follows:

2222120

1121110

VWYP

VWYD

ttt

ttt

tP

tP

43


44/52

This is now a transformed model.

Since is based on the estimates from the reduced

form equations, it act as an instrumental variable for the

original data on .

All the structural parameters are estimated by applying

OLS to these transformed equations. This is second stage of

estimation.

2210

1210

UWPS

UYPD

ttt

ttt

P

tP

44


45/52

Features of 2SLS

Following are the features of 2SLS

(1) Unlike ILS, which provides multiple estimates of the

parameters when applied to over-identified equations,

2SLS provides only one estimate per parameter.

(2) Although 2SLS has been specially designed to handle over-

identified equations, this method can also be applied to

exactly identified equations. ILS and 2SLS give identical

results in such situation.

45


46/52

(3) In the application ILS, there is no simple method of

estimation of standard errors of the structural coefficients

from the standard errors of the reduced form coefficients.

But this can be done easily in case of 2SLS estimates

because the structural coefficients are directly estimated

from the second stage (OLS) regressions. However, the

estimates standard errors in the second stage regression

need to be modified.

46

Table 2 2SLS estimates for simultaneous equation model


47/52

Table 2. 2SLS estimates for simultaneous equation model

Variables PUBINV PVTINV PROD

Intercept 4144.952***

(11.147)

-272.164***

(3.567)

-1304.541

(1.276)

PUBINV 0.089***(3.368)

0.837**

(2.182)

PVTINV 0.997***(2.763)

PROD 0.033***(3.831) 0.007**(1.977)

POVR -1.950***(3.367)

POPGR 42.561***(3.393)

2.836

(0.377)

166.236***

(3.096)

LITR 1.979***(7.409)

0.308*

(1.673)

7.633***

(5.843)

47

***, **, * Significant at 1, 5, and 10 per cent level respectively.

Figures in parentheses are t values

ICAR (New Delhi) Roy and Pal (2002)

Table 2 Continue


48/52

Table 2. Continue.


GOVREV 0.007***

(3.688)GRANTS 0.526***

(3.673)

SUBSG -0.169***

(4.0647)

SUBTOT 0.509**(2.413)

SUBINP 0.181***

(7.167)

ToT 4.206***

(6.022)

20.770***

(4.059)

CREDIT 0.072

(1.127)

1.975***

(4.986)

48





49/52

Table 2. Continue..


Cropping Intensity 8.861*(1.949)

STORE 14.245***

(5.595)

RAIND -30.745***

(10.403)

Adjusted R 2 0.832 0.898 0.941

FValue 110.10*** 174.85*** 292.09***

D-W statistics 1.857 1.815 2.081

49




CONCLUSION


50/52

CONCLUSION

The single equation methods are more popular to solve

simultaneous equation models. A unique feature of this

method is that one can estimate a single equation in a multi

equation model without worrying too much about other

equations in the model.

Although OLS is, in general, inappropriate in the context of

simultaneous equation models, it can be applied to the

recursive models where there is a definite but unidirectional

cause-and-effect relationship among the endogenous variables

50


51/52

Unique feature of both ILS and 2SLS is that the estimates

obtained thereof are consistent, that is, as the sample

increase indefinitely the estimates tend tot heir true

population values.

51


52/52

application of simultaneous equation in social sciences

Documents