simultaneous equations ii
TRANSCRIPT
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Simultaneous Equations
Chapter 11 entitled Violating Assumption Four:
Simultaneous Equations from the book by Peter Kennedy A
Guide to Econometrics Wiley-Blackwell
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Lets assume the following econometric model (1):
Q is the equilibrium quantity exchanged on the market, P is the
equilibrium price and Y is income of consumers.
The variables Q and P are endogenous and Y is exogenous.
s and s are parameters, us are random disturbances, and t
represents a specific period of time.
Note that both relations are needed for determining the values of
the two endogenous variables, so that system is one ofsimultaneous equations.
Equations given in model (1) are called the structural form of the
model under study. This form is derived from economic theory.
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The structural equations can be solved for the endogenous variables
to give (2):
The solution in (2) is called the reduced form of the model.
The reduced form equations show how the endogenous variables are
jointly dependent on the predetermined variables and thedisturbances of the system.
We can see that the values of Q and P are fully determined by Yand
us. The value of Yis determined outside of the market in question
and to be in no way influenced by P and Q. The coefficient of Yin the reduced form equation for Q represents a
total effect of Yon Q.
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This effect consists of a direct effect of Y on Q given by the coefficienta3 of the demand equation in (1) and of an indirect effect of Y on Qthrough P defined as
From the point of statistical inference, the single relevantcharacteristic of the simultaneous equation system, and the one thatrequires special consideration, is the appearance of endogenousvariables among the explanatory variables of at least some of thestructural equations.
This leads to problems because the endogenous variables are, ingeneral, correlated with the disturbance of the equation in whichthey appear.
Consider the supply-demand model of (1). In both equations the
endogenous P appears as an explanatory variable. But from (2) wecan see that P is correlated with both disturbances because thefollowing is true:
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The Identification Problem
Knowledge of the reduced form of a system of equations is not
always sufficient to allow us to discern the value of the parameters in
the original set of structural equations.
The problem of whether we can determine the structural equations,
given knowledge of the reduced form, is called the identification
problem.
Note that knowledge of the structural parameters is not absolutely
necessary ifprediction or forecasting is our primary purpose,
because forecasts can be obtained through the reduced-form
equations directly.
A system of simultaneous equations is said to be complete if it
contains at least as many independent equations as endogenous
variables.
For identification of the entire model, it is necessary for the model
to be complete and for each equation in it to be identified.
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We shall say that an equation is unidentified if there is no way of
estimating all the structural parameters from the reduced form.
An equation is identified if it is possible to obtain values of the
parameters from the reduced-form equation system.
An equation is exactly identified if a unique parameter value exists
and overidentified if more than one value is obtained for some
parameters.
Consider first the following supply-demand time series model (3) in
which there are no predetermined variables:
When we try to estimate model (3) using market data we obtainmeaningless results.
There is no way to ascertain the true supply and demand slopes given
only the equilibrium data.
Both demand and supply in (3) are unidentified.
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This situation is depicted in the following figure.
From the above figure it is apparent that any pair of demand and
supply curves which are intersecting at point E could be the truedemand and supply curves.
In other words there is an infinite number of structural models
(demand and supply curves) which are consistent with the same
reduced form (equilibrium value of P and Q). Note that the reduced form equations of the structural model (3)
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Consider the following example of supply-demand system (4):
Note that in this case income (Y) determines demand and becauseincome varies over time, we must account for the fact that the
demand curve shifts over time. This situation is depicted in the
following figure.
From the above figure we can see that the equilibrium values traceout the path of the underlying supply curve.
The supply curve is identified because the supply parameters can be
deduced from the reduced form. Notice that it is the movement of Y
over time for the identification of the supply equation.
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The supply equation is identified because the exogenous variable Y
was excluded from the supply equation.
The demand equation is unidentified because prior information is not
available which allows for the unique determination of the demand
relationship.
Now consider another model in which the supply relationship is
determined by the temperature Tin the region and the demand
curve is not, then the prior information about the excluded
exogenous variable (temperature) in the demand equation wouldallow the demand curve to be identified.
In this case both the demand and supply curve are identified. The
following demand and supply model (5) has just this property:
In this case it is not easy to depict the supply-demand relationship
graphically.
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Note that if Tand Yvary over time both demand and supplyrelationships will shift.
The structural values of the demand and supply parameters can bedetermined uniquely.
The shifting of the supply curve associated with changes in Thelps toplot out the demand curve at the same time that shifts in demand(changes in Y) plot out the supply curve.
To reiterate, certain exogenous variables (which appear in thestructural system) are excluded from individual equations within thesystem that allows the equations to be identified.
As a final model (5), consider the one given bellow, in which thedemand curve is a function not only of income but also of wealth.
Supply:
Demand:
In this case the demand curve shits through time due to changes intwo variables. The supply equation is overidentified because thereare two exogenous variables in the system which are excluded from
the supply equation.
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The conditions for identification of one equation in a simultaneous system
of equations can be generalized in two rules: 1) order condition and 2) rank
condition.
Note that the order condition is a necessary condition but not a sufficient
one and technically speaking the rank condition must also be checked. Notethat the rank condition much more complicated than the order condition.
The order condition states that: For an equation to be identified the total
number of variables excluded from it but included in other equations must
be at least as great as the number of equations of the system less one.
Let G=total number of equations (=total number of endogenous variables)
K= number of variables in the model (endogenous and predetermined)
M= number of variables, endogenous and exogenous, included in a
particular equation.
Then the order condition can be expressed as:
(K-M) (G-1)
(excluded variables) (total number of equations -1)
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TWO-STAGE LEAST SQUARES (2SLS)
Two-stage least squares (2SLS) provides a useful estimation procedure
for obtaining the values of structural parameters in overidentified
equations.
Two-stage least squares (2SLS) estimation uses the information available
from the specification of an equation system to obtain a unique
estimate for each structural parameter.
The first stage of 2SLS involves the creation of an instrument, while the
second stage involves a variant of instrumental-variable estimation.
Lets describe very briefly the workings of 2SLS.
Consider the supply-demand model (5), which was provided previously.
The structural model and the resulting reduced form equations are
provided bellow:
(5.1)
(5.2)
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The supply equation of (5.1) is overidentified so estimation of the
reduced form equations (5.2) will not yield unique parameter
estimates.
The 2SLS procedure works as follows:
1. In the first step the reduced-form equation for is estimated
using ordinary least squares (OLS). This is accomplice by regressingp on
all the predetermined variables in the equation system.
From the first stage regression, the fitted values of the dependent
variable are determined. The fitted values will by construction be
independent of the error terms .
Thus, the first stage process allows us to construct a variable which is
linearly related to the predetermined model variables (through OLS)
and which is purged of any correlation with the error term in thesupply equation.
It seems reasonable to view this newly created variable as an
instrument.
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2. In the second-stage regression, the supply equation of the
structural model is estimated by replacing the variable with the first-
stage fitted variable .
The use of OLS in this second stage will yield a consistent estimator of
the supply parameter If additional predetermined variables appear
in the supply equation, 2SLS would also estimate those parameters
consistently.
The 2SLS procedure is quite easy to use and is frequently used when
overidentified equations are present. We also can use 2SLS when anequation is exactly identified.
When the supply equation is not identified, e.g. when the variables y
and wappear in the supply equation, then 2SLS is impossible. This
happens because the fitted-value variable, , used in the secondstage-regression is a weighted average (or linear combination) of the
predetermined variables in the system. When one attempts to
regress in the second stage, the perfect collinearity
between the three repressors will make estimation impossible.
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Seemingly Unrelated-equations (SUR)
The SUR model consists of a series of equations linked because the
error terms across equations are correlated.
Consider the two-equation model
We assume that the model is a cross-section model and that N
observations are available in the model. Under the assumption that and are correlated for identical
cross-section units, we can improve upon the efficiency of ordinary
least squares by writing the equation system as one combined
equation, estimating that equation using generalized least-squaresestimation.
In order to write the system as one large equation rather than two
smaller equations, it is necessary to distinguish between observations
associated with the first equation, and observations associated with
the second equation of the system.
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To do so, we shall relabel the observations, arbitrarily assigning
observations 1 to N to the first-equation variables and observations
N+1 to 2N to the second-equation variables. We now define four
new variables.
With this new notation the combined equation can be written as
Applying the generalized least-square procedure to the abovetransformed equation allows us to obtain parameter estimates for
and .
Since the algebra involved is substantial we shall simply present the
results as provided bellow:
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The application of generalized least squares necessitates obtainingestimates of the error covariances between equations.
These estimates are obtained by first estimating each single equation usingordinary least squares.
The variances and covariances of the estimated residuals then provideconsistent estimators of the error variances ad covariances.
In our two-equation example, we would estimate
THREE STAGE LEAST SQUARES (3SLS)
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THREE-STAGE LEAST SQUARES (3SLS)
Three-stage least squares (3SLS) is a system method, that is itapplied to all the equations of the model at the same time and givesestimates of all the parameters simultaneously.
It involves the application of the method of least squares in threesuccessive stages.
It utilizes more information than the single-equation techniques, thatis, it takes into account the entire structure of the model with all therestrictions that this structure imposes on the values of theparameters.
The single-equation techniques make use only of the variablesappearing in the particular equation, but they ignore the restrictionsset by the structure on the coefficients of other equations, as well as
the contemporaneous dependence on the random terms of thevarious equations.
In simultaneous equations models it is almost certain that the errorterm of any equation will be correlated with the error term of other
equations. This fact is ignored by single-equation methods.
Th l i i h f d i f 2SLS d
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Three-stage least squares is a straightforward extension of 2SLS and
involves the application of least squares in three stages.
The first two stages are the same as 2SLS except that we deal with
the reduced-form of all the equations of the system.
The third stage involves the application of generalized least squares,
that is, the application of least squares to a set of transformed
equations, in which the transformation required is obtained from the
reduced-form residuals of the previous stage.
Suppose that we are left with a system in G endogenous and K
predetermined variables. There are G equations in the system of the
form:
P l i l i h i b h K d i d i bl
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Pre-multiplying each equation by the Kpredetermined variables we
obtain a system of KG equations, that is, we have Kforms for each
one of the G equations.
The set of the K forms for the first structural equations is
Set of K forms for the second structural equation
Th t f K f f th Gth t t l ti i
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The set of K forms for the Gth structural equation is
We observe that the disturbances of these equations are
heteroscedastic, since the composite random termtend to change together with thex variables.
Hence the appropriate method for the estimation of the parameters
of the system is generalized least squares.
The transformation required involves the variances and thecovariances of the original error terms which however are
unknown.
We may obtain an estimate of these covariances by applying 2SLS to
each one of the structural equations of the original model.
B d th 3SLS ti ti t h i h th f ll i
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Based on the 3SLS estimation techniques we have the following
three stages of estimation:
Stage I. In the first stage we estimate the reduced-form of all the
equations of the system
We thus obtain estimated values of the endogenous variables,
Stage II. We substitute the above calculated values of theendogenous variables in the right-hand side of the structural
equations and apply least squares to the transformed equations.
We thus obtain the 2SLS of the bs and the s which we use for the
estimation of the error term of the various equations.
W fi d t f G t d b th l f l f th
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We find a set of G errors computed by the usual formula of the
covariane
The complete set of the variances and the covariances of the error
terms is as follows
Stage III. We use the above variances and covariances of the error
terms in order to obtain the transformations of the original variablesfor the application of generalized least squares (GLS).
Th t ti f th t ti f th thi d t b
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The presentation of the computations of the third stage becomes
extremely complicated with the use of simple alebra and
summations and will not be presented here.
However it should be restated that the third stage involves the
application of generalized least squares to a set of transformed
equations. In general lines the transformation is achieved as follows:
1. First each single equation of the system is multiplied through
(transformed) by the transpose of the matrix of observations on all
the exogenous variables in the system.
2. Then all the equations to be estimated stacked one on top of the
other, and this stock is rewritten as a single, very large, equation.
3. Finally, GLS estimation is applied to this giant equation taking into
consideration the previous two stages.
To summarize the process
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To summarize the process:
In the first stage of the process, the reduced form of the model
system is estimated.
The fitted values of the endogenous variables are then used to get
2SLS estimates of all the equations of the system.
Once the 2SLS parameters have been calculated, the residuals of
each equation are used to estimate the cross-equation variances
and covariances.
In the third and final stage of the estimation process, generalized
least-squares parameter estimates are obtained.
The 3SLS procedure can be shown to yield more efficient parameter
estimates than the 2SLS because it takes into account cross-equation
correlation.