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Application of SimultaneousEquation in Social Sciences
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CONTENT
INTRODUCTION
PROBLEMS OF SIMULTANEOUS EQUATIONS MODELS
STRUCTURAL, REDUCED-FORM AND RECURSIVE MODELS
SIMULTANEOUS EQUATION METHODS
CONCLUSION
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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.
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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.
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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
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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:
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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 ''
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Fig. 1 Interdependence of price and quantity
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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
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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
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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.
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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
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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
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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
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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
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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.
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SIMULTANEOUS EQUATION METHODS
Following are the frequently used single equation
methods:
1) Indirect Least Squares (ILS)
2) Two Stage Least Squares (2 SLS)
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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.
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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.
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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.
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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
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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
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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.
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(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
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(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
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(c) Calculation of 2
22
12
21
1121
2
211
11
2
22
12
21
1121
1
22
12
1
21
11
',
;
inssubstituteweobtaintoTherefore
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(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
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(c) Calculation of 11 and
.identifiedexactlyismodelequationussimultaneo
theonly whens'ands'ofvaluesuniqueto
risegivewillsandtscoefficienstructuralebetween threlationthatnotedbeshouldIt
21
111
22
121
and
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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.
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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.
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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
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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
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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
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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.
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(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.
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Table 2 2SLS estimates for simultaneous equation model
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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)
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***, **, * 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
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Table 2. Continue.
Variables PUBINV PVTINV PROD
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)
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***, **, * Significant at 1, 5, and 10 per cent level respectively.
Figures in parentheses are t values
ICAR (New Delhi) Roy and Pal (2002)
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Table 2. Continue..
Variables PUBINV PVTINV PROD
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
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***, **, * Significant at 1, 5, and 10 per cent level respectively.
Figures in parentheses are t values
ICAR (New Delhi) Roy and Pal (2002)
CONCLUSION
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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
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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.
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