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CONTENTS
ACKNOWLEDGEMENT 2
ABSTRACT 3
INTRODUCTION 4
ECONOMIC THEORY 5
MODEL JUSTIFICATION 7
DATA USED 8
ANALYSIS 9
CONCLUSION 16
REFERENCE 17
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ACKNOWLEDGEMENT
We are grateful to the faculty of Department of Economics (Jadavpur University) for
their unwavering support and cooperation. Working on this project has given us the
opportunity to gather immense knowledge regarding econometric tools and
economic analysis that will surely benefit us significantly in our careers in the future.
We thank our professor Dr. Arpita Dhar immensely for setting us this task of
preparing and presenting this project. We are extremely grateful and thankful to her
for her tireless guidance without which it would not have been possible for us to
make progress in our endeavour. We also take this opportunity to thank our
department for providing us with a functioning computer laboratory and library
facilities which helped us to fulfil all our needs regarding our project. Moreover, we
are also grateful to our friends and families for their constant support and help.
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ABSTRACT
A casual look at the published empirical work in business and economics will
reveal that many economic relationships are of the single-equation type.In such models,
one variable (the dependent variable Y) is expressed as a linear function of one or more
other variables (the explanatory variables, the X’s). In such models an implicit
assumption is that the cause-and-effect relationship, if any, between Y and the X’s is
unidirectional: The explanatory variables are the cause and the dependent variable is
the effect.
However, there are situations where there is a two-way flow of influence among
economic variables; that is, one economic variable affects another economic variable(s)
and is, in turn, affected by it (them). Thus, we need to consider twoequations. And this
leads usto consider simultaneous-equation models, models in which there is morethan
one regression equation, one for each interdependent variable.
Keywords: Nonseparablemodels,simultaneousequations, multicollinearity.
JEL classification. : C3.
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INTRODUCTION
Single equation models, i.e., models in which there was a single dependent
variable Y and one or more explanatory variables, the X’s, the emphasis was on
estimating and/or predicting the average value of Y conditional upon the fixed values of
the X variables. The cause-and-effect relationship, if any, in such models therefore ran
from the X’s to the Y.
But in many situations, such a one-way or unidirectional cause-and-effect
relationship is not meaningful. This occurs if Y is determined by the X’s, and some of the
X’s are, in turn, determined by Y. In short, there is a twoway, or simultaneous,
relationship between Y and (some of) the X’s, which makes the distinction between
dependent and explanatory variables of dubious value. It is better to lump together a set
of variables that can be determined simultaneously by the remaining set of variables—
precisely what is done in simultaneous-equation models. In such models there is
more than one equation—one for each of the mutually, or jointly, dependent or
endogenous variables. And unlike the single-equation models, in the simultaneous-
equation models one may not estimate the parameters of a single equation without
taking into account information provided by other equations in the system.
In quite a similar view as of our single equation models , our conventional
Classical Linear Model Assumption , is that all the explanatory variables of an
equation are strictly unrelated. But when we start dealing with Simultaneity Bias it
very likely gives rise to the problem of related explanatory variables, precisely the
problem of Multicollinearity. In this particular problem the individual regression
parameters are not estimable with sufficient precision because of high standard errors
which often occurs due to highly intercorrelated regressors. In econometric literature
Multicollinearity is one of the misunderstoood conceptions since high
intercorrelations among the explanatory variables are neither necessary nor
sufficient to cause the multicollinearity problem rather the best indicators of the
problem are the t-ratios of the individual coefficients.
Thus our endeavour is to resolve the problem of simultaneity, if present , as well
as to emphasise on the fact that intercorrelation between explanatory variables is
neither necessary nor sufficient for the existence of Multicollinearity rather we should
concentrate more on the model’s significance.
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ECONOMIC THEORY OF SIMULTANEOUS EQUATION
SYSTEM
An obvious reason for the endogeneity of explanatory variables in a regression
model is simultaneity: that is, one or more of the explanatory variables are jointly
determined with the dependent variable. Models of this sort are known as
simultaneous equations models (SEMs), and they are widely utilized in both applied
microeconomics and macroeconomics. Each equation in a SEM should be a behavioral
equation which describes how one or more economic agents will react to shocks or
shifts in the exogenous explanatory variables, ceteris paribus. The simultaneously
determined variables often have an equilibriuminterpretation, and we consider that
these variables are only observed when the underlying model is in equilibrium.
For instance, a demand curve relating the quantity demanded to the price of a
good, as well as income, the prices of substitute commodities, etc. conceptually would
express that quantity for a range of prices. But the only price-quantity pair that we
observe is that resulting from market clearing, where the quantities supplied and
demanded were matched, and an equilibrium price was struck. In the context of labor
supply, we might relate aggregate hours to the average wage and additional explanatory
factors:
where the unit of observation might be the county. This is a structural equation, or
behavioral equation, relating labor supply to its causal factors: that is, it reacts the
structure of the supply side of the labor market. This equation resembles many that we
have considered earlier, and we might wonder why there would be any difficulty in
estimating it. But if the data relate to an aggregate such as the hours worked at the
county level, in response to the average wage in the county this equation poses
problems that would not arise if, for instance, the unit of observation was the individual,
derived from a survey. Although we can assume that the individual is a price- (or wage-)
taker, we cannot assume that the average level of wages is exogenous to the labor
market in Suffolk County. Rather, we must consider that it is determined within the
market, affected by broader economic conditions. We might consider that the z variable
expresses wage levels in other areas, which would cet.par. have an effect on the supply
of labor in Suffolk County; higher wages in Middlesex County would lead to a reduction
in labor supply in the Suffolk County labor market, cet. par.
To complete the model, we must add a specification of labor demand:
where we model the quantity demanded of labor as a function of the average wage and
additional factors that might shift the demand curve. Since the demand for labor is a
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derived demand, dependent on the cost of other factors of production, we might include
some measure of factor cost (e.g. the cost of capital) as this equation's z variable.
In this case, we would expect that a higher cost of capital would trigger
substitution of labor for capital at every level of the wage, so that
Note that the supply equation represents the behavior of workers in the
aggregate, while the demand equation represents the behavior of employers in the
aggregate. In equilibrium, we would equate these two equations, and expect that at
some level of equilibrium labor utilization and average wage that the labor market
is equilibrated. These two equations then constitute a simultaneous equations model
(SEM) of the labor market. Neither of these equations may be consistently estimated via
OLS, since the wage variable in each equation is correlated with the respective error
term. How do we know this? Because these two equations can be solved and rewritten
as two reduced form equations in the endogenous variables hi and wi. Each of those
variables will depend on the exogenous variables in the entire system-z1 and z2-as well
as the structural errors ui and vi:
In general, any shock to either labor demand or supply will affect both the
equilibrium quantity and price (wage). Even if we rewrote one of these equations to
place the wage variable on the left hand side, this problem would persist: both
endogenous variables in the system are jointly determined by the exogenous variables
and structural shocks. Another implication of this structure is that we must have
separate explanatory factors in the two equations. If z1 = z2; for instance, we would not
be able to solve this system and uniquely identify its structural parameters. There must
be factors that are uniqueto each structural equation that, for instance, shift the supply
curve without shifting the demand curve. The implication here is that even if we only
care about one of these structural equations-for instance, we are tasked with modelling
labor supply, and have no interest in working with the demand side of the market-we
must be able to specify the other structural equations of the model. We need not
estimatethem, but we must be able to determine what measures they would contain.
For instance, consider estimating the relationship between murder rate, number
of police, and wealth for a number of cities. We might expect that both of those factors
would reduce the murder rate, cet.par.: more police are available to apprehend
murderers, and perhaps prevent murders, while we might expect that lower-income
cities might have greater unrest and crime. But can we reasonably assume that the
number of police (per capita) is exogenous to the murder rate? Probably not, in the
sense that cities striving to reduce crime will spend more on police. Thus we might
consider a second structural equation that expressed the number of police per capita as
a function of a number of factors. We may have no interest in estimating this equation
(which is behavioral, re effecting the behavior of city officials), but if we areto
consistently estimate the former equation-the behavioral equation reeffecting the
behaviorof murderers{we will have to specify the second equation as well, and collect
data for its explanatory factors.
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MODEL AND ITS ECONOMIC JUSTIFICATION
We have to deal with either seemingly unrelated regression equation or
Simultaneous Equation System where we have to search for the problem of related
explanatory variables and the factors which might give rise to the problem of
multicollinearity.
So we have taken into consideration a Simultaneous Equation System :-
Money demand :- Md=β0 + β1Yt + β2Rt + β3 Pt+ u1t
Money supply :-Ms = α0 + α1Yt + u2t
Where M= Money
Y= Income / GDP (gdp)
R=Treasury Bill Rate (tbrate)
P=Consumer price index(cpi)
This model has a justifiable economic implication. The first equation shows that
Money Demand is the dependent variable and it depends on GDP , Interest rate or
treasury bill rate and consumer price index. The economy’s gross domestic product
obviously defines the money demand since people will be demanding money based on
the income generated in the economy.
Money demand depends on interest rate since if the interest rate is lower people
will be demanding more money and if the interest rate is higher people will be
demanding less money. Theconsumer price index states the consumption basket of the
consumers and hence it depends on price and income of the consumers and on how
much money is available with them thus in return making money demand dependent on
consumer price index.
The second equation shows that Money Supply depends on the economy’s
income only because depending on the income generated the financial authority decides
whether to restrict money supply or not.
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ANALYSIS
Since we have simultaneous equation model so it is highly likely that we will
have a problem of identification and the model can be just identified,over-identified or
under-identified. For dealing with problems of identification we use methods of Indirect
least squares for just identified models and method of two stage and three stage least
square for over identified and under identified models.
In our simultaneous equation model we have M2(Money demand) and GDP(gdp) as the
endogenous variables and treasury bill rate (tbrate) and consumer price index (cpi) as
the exogenous variables. Since GDP is itself a dependent variable thus it gives rise to
identification problem of simultaneity and from the model it is evident that here we
have a problem of overidentification. Overidentification can be resolved by computing
two stage least square regression of the entire simultaneous equation model for which it is
required to obtain the reduced form of GDP in terms of the two exogenous variables tbrate
and cpi.
. ge ey = 2662.307 + 34.62386*cpi - 59.71936 t r e
_cons 2662.307 237.2587 11.22 0.000 2175.493 3149.122
cpi 34.62386 1.375115 25.18 0.000 31.80236 37.44537
tbrtae -59.71936 22.66009 -2.64 0.014 -106.214 -13.2247
gdp Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 67039652.9 29 2311712.17 Root MSE = 299.52
Adj R-squared = 0.9612
Residual 2422168.66 27 89709.9504 R-squared = 0.9639
Model 64617484.2 2 32308742.1 Prob > F = 0.0000
F( 2, 27) = 360.15
Source SS df MS Number of obs = 30
. reg gdp tbrtae cpi
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Now we are generating gdp10 which will be dependent on only the two exogenous
variables and hence if we replace gdp by gdp10 then the simultaneity problem will be
resolved since the gdp present in the equation of money demand will no longer be
correlated with the error term and neither the other explanatory variables
gen gdp10 = 2662.307 + 34.62386*cpi - 59.71936*tbrate
gdp10 will be dealt as an instrument of gdp and thus for computing 2SLS regression we
regress m2 on tbrate cpi and the replaced value of gdp which is gdp10. As for example
if Yt is an endogenous variable it can be expressed by two variables – one is the
estimated Yt or Yt^ which is free of the error term and the estimated error Vt^.
Yt = Yt^+ Vt^
The instrumental variable regression / 2SLS regression
Instruments: tbrtae cpi gdp10
Instrumented: gdp
_cons -1839.792 359.1825 -5.12 0.000 -2576.774 -1102.81
cpi 7.665082 5.819632 1.32 0.199 -4.275816 19.60598
tbrtae 0 (omitted)
gdp .5948185 .1629644 3.65 0.001 .2604433 .9291938
m2 Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 42797787 29 1475785.76 Root MSE = 128.64
Adj R-squared = 0.9888
Residual 446782.742 27 16547.509 R-squared = 0.9896
Model 42351004.2 2 21175502.1 Prob > F = 0.0000
F( 2, 27) = 1276.91
Source SS df MS Number of obs = 30
Instrumental variables (2SLS) regression
. ivreg m2 (gdp=gdp10) tbrtae cpi
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While computing the instrumental variable regression or the two stage least
square regression we take gdp10 as the instrumented gdp. Since we have data available
only for money demand so we run the two stage least square regression on the equation
of money demand replacing gdp by the instrument gdp10. Here we see that we get the
desired signs of all the coefficients and also the variables are quite significant.. So the
2SLS regression gives us desired results and is successful in abolishing the problem of
over identification.
Now we come to the Three Stage Least Square Regression
After conducting the three stage least square regression we see that all the
coefficients have desired signs and all the variables are significant at 1% level of
significance and only tbrate is significant at 5% level of significance. The adjusted R2
value is also high showing that the goodness of fit is highly significant.
Exogenous variables: gdp tbrate cpi
Endogenous variables: m2
_cons -1132.728 180.7951 -6.27 0.000 -1487.08 -778.3761
cpi 16.8606 2.17858 7.74 0.000 12.59066 21.13054
tbrate -15.86045 8.135697 -1.95 0.051 -31.80612 .0852237
gdp .3292353 .0616228 5.34 0.000 .2084567 .4500138
m2
m2 Coef. Std. Err. z P>|z| [95% Conf. Interval]
m2 30 3 95.90556 0.9936 4623.01 0.0000
Equation Obs Parms RMSE "R-sq" chi2 P
Three-stage least-squares regression
. reg3 ( m2 gdp tbrate cpi)
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MULTIC0LLINEARITY AND ITS INDICATOR
The term multicollinearity is due to Ragnar Frisch. Originally it meant the
existence of a “perfect,” or exact, linear relationship among some or all explanatory
variables of a regression model. For the k-variable regression involving explanatory
variable X1, X2, . . . , Xk(where X1 = 1 for all observationsto allow for the intercept term),
an exact linear relationship is said toexist if the following condition is satisfied:
λ1X1 + λ2X2 +· · ·+λkXk= 0
whereλ1, λ2, . . . , λkare constants such that not all of them are zero
simultaneously.
Today, however, the term multicollinearity is used in a broader sense to include
the case of perfect multicollinearity as well as the case where the X variables are
intercorrelated but not perfectly so, as follows :
λ1X1 + λ2X2 +· · ·+λ2Xk + vi= 0
wherevi is a stochastic error term.
If multicollinearity is perfect , the regression coefficients of the X variables are
indeterminate and their standard errors are infinite. If multicollinearity is less than
perfect, the regression coefficients, although determinate, possess large standard errors
(in relation to the coefficients themselves), which means the coefficients cannot be
estimated with great precision or accuracy.
There are several sources of multicollinearity :-
1. The data collection method employed, for example, sampling over a limited
range of the values taken by the regressors in the population.
2. Constraints on the model or in the population being sampled. For example, in
the regression of electricity consumption on income (X2) and house size (X3) there is a
physical constraint in the population in that families with higher incomes generally
have larger homes than families with lower incomes.
3. Model specification, for example, adding polynomial terms to a regression
model, especially when the range of the X variable is small.
4. An overdetermined model. This happens when the model has more
explanatory variables than the number of observations. This could happen
in medical research where there may be a small number of patients about whom
information is collected on a large number of variables.
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For our simultaneous equation model its quite likely to have a multicollinearity
problem since explanatory variable of one of the equations is the dependent variable of
the other and that is why it’s quite probable that the explanatory variable has a relation
with the other explanatory variables and error term. In our example gdp is the
explanatory cum dependent variable which gives rise to both the problem of
simultaneity and multicollinearity. As is evident from the regression table below we see
that gdp has correlation with tbrate and cpi and all the coefficients are noticeably
significant. So our equation of money demand will obviously have problem of
multicollinearity.
Here we are providing individual regression of Money Demand with each of its
explanatory variable and we see that for all the 3 cases the coefficients give desired
values at significant confidence intervals. We have conducted this test in order to see
whether when we run a combined regression the values get hampered or not because of
multicollinearity.
_cons -2195.468 126.646 -17.34 0.000 -2454.89 -1936.045
gdp .7911096 .0211633 37.38 0.000 .7477586 .8344606
m2 Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 42797787 29 1475785.76 Root MSE = 173.28
Adj R-squared = 0.9797
Residual 840727.688 28 30025.9889 R-squared = 0.9804
Model 41957059.3 1 41957059.3 Prob > F = 0.0000
F( 1, 28) = 1397.36
Source SS df MS Number of obs = 30
. reg m2 gdp
. ge ey = 2662.307 + 34.62386*cpi - 59.71936 t r e
_cons 2662.307 237.2587 11.22 0.000 2175.493 3149.122
cpi 34.62386 1.375115 25.18 0.000 31.80236 37.44537
tbrtae -59.71936 22.66009 -2.64 0.014 -106.214 -13.2247
gdp Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 67039652.9 29 2311712.17 Root MSE = 299.52
Adj R-squared = 0.9612
Residual 2422168.66 27 89709.9504 R-squared = 0.9639
Model 64617484.2 2 32308742.1 Prob > F = 0.0000
F( 2, 27) = 360.15
Source SS df MS Number of obs = 30
. reg gdp tbrtae cpi
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_cons 3411.099 611.5584 5.58 0.000 2158.379 4663.82
tbrtae -153.0497 85.76094 -1.78 0.085 -328.7231 22.62359
m2 Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 42797787 29 1475785.76 Root MSE = 1171.5
Adj R-squared = 0.0701
Residual 38426955.7 28 1372391.27 R-squared = 0.1021
Model 4370831.3 1 4370831.3 Prob > F = 0.0852
F( 1, 28) = 3.18
Source SS df MS Number of obs = 30
. reg m2 tbrtae
_cons -548.9967 80.42128 -6.83 0.000 -713.7322 -384.2612
cpi 28.80403 .7313971 39.38 0.000 27.30583 30.30223
m2 Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 42797787 29 1475785.76 Root MSE = 164.64
Adj R-squared = 0.9816
Residual 758942.463 28 27105.088 R-squared = 0.9823
Model 42038844.5 1 42038844.5 Prob > F = 0.0000
F( 1, 28) = 1550.96
Source SS df MS Number of obs = 30
. reg m2 cpi
_cons -1132.728 194.2051 -5.83 0.000 -1531.922 -733.5337
cpi 16.8606 2.340171 7.20 0.000 12.05031 21.67089
tbrtae -15.86045 8.73914 -1.81 0.081 -33.82401 2.103111
gdp .3292353 .0661935 4.97 0.000 .1931725 .465298
m2 Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 42797787 29 1475785.76 Root MSE = 103.02
Adj R-squared = 0.9928
Residual 275936.319 26 10612.9353 R-squared = 0.9936
Model 42521850.6 3 14173950.2 Prob > F = 0.0000
F( 3, 26) = 1335.54
Source SS df MS Number of obs = 30
. reg m2 gdp tbrtae cpi
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But we see even when we compute a combined regression the factors remain
significantly different from zero with coefficients having desired values and signs.
Though this is an elementary way of checking whether multicollinearity poses a big
problem or not but we can somehow observe that multicllinearity is not a big adversity
in this case.
But a more conventional and methodical approach is to find out the extent and
intensity of the multicollinearity problem for which we have to conduct the Variance
Inflation Factor(VIF) test where
VIF = 1/1-R2i
Where R2i gives the goodness of fit or the degree of association of the ith variable
with all the other explanatory variables and thus higher the value of R2i higher is the
degree of association between the explanatory variables which states multicollinearity.
So if VIF value is high that means the extent of the problem is high and if the
value is low then the extent of the problem is low.
Here we see that none of the VIF values are high which says that the degree of
multicollinearity is not much and also we see that while regressing we get the desired
signs of the coefficients and also the variables are significant with mostly 1% level of
significance and for tbrate we have 10% level of significance. So we need not consider
the problem of multicollinearity even if it is present in the model but has not much
adverse effect.
Herein once again our model proves that though VIF indicates high inter-
correlation among the explanatory variables still our t-ratios are significant showing
that related regressosr is not an efficient clause determining multi-collinearity. The
model becomes acceptable even after the presence of the problem of multi-collinearity.
Mean VIF 18.39
tbrtae 1.34 0.744731
cpi 26.15 0.038247
gdp 27.68 0.036130
Variable VIF 1/VIF
. estat vif
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CONCLUSION
Thus in our simultaneous equation model we see that as per our pre conceived
idea there arises a problem of identification. We have a problem of over identification
and we resolve it through instrumental variable regression where we take the help of
two stage and three stage least square regression to find out the desired estimates. As
per our model we find that our regression estimates are significant in each case and the
R2s are high showing that the regression result fits the model properly. In the same
model we have to check for the presence of multicollinearity.
Since our model comprises of money demand , money supply , GDP , interest rate
and consumer price index so its very well evident that if we deal with the equation of
money demand where GDP , interest rate and consumer price index are the explanatory
variables , we are likely to get a relation between the explanatory variables and
specially when GDP is the explanatory variable for the equation of money supply too.
GDP(variable name gdp) is the variable which gives rise to both the problem of
simultaneity and multicolllinearity.
Thus elementarily we conduct the individual regression of the dependent
variable money demand with each of its independent variable and find that all the
regressors are significant and when we conduct the combined regression then also we
find that the regressors remain significant and the level of significance also remains the
same. So from simple regression we can conclude that multicollinearity is not posing a
big problem though we are sure that it is present. Thus we carry on the more
conventional VIF test where we see that VIF estimates are quite high rather more than
5 except for tbrate. From our usual knowledge of econometrics we would say that there
is high collinearity between the estimates but in reality we see that multi collinearity is
not creating a problem for the regression which is running smoothly with proper and
significant estimates.
.
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