Simultaneous Equations II

7/24/2019 Simultaneous Equations II

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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.

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Simultaneous Equations II