Chapter 9

Chapter 9

Simultaneous Equations Models

( 聯立方程式模型 )

What is in this Chapter?

• How do we detect this problem?

• What are the consequences?

• What are the solutions?


• In Chapter 4 we mentioned that one of the assumptions in the basic regression model is that the explanatory variables are uncorrelated with the error term

• In this chapter we relax that assumption and consider the case where several variables are jointly determined

– Predetermined vs. jointly determined

– Exogenous vs. Endogenous


• This chapter first discusses the conditions under which equations are estimable in the case of jointly determined variables

(the "identification problem") and methods of estimation

• One major method is that of " instrumental variables "

• Finally, this chapter also discusses causality

9.1 Introduction

• In the usual regression model y is the dependent or determined variable and x

1

, x

2

, x

3

... Are the independent or determining variables

• The crucial assumption we make is that the x's are independent of the error term u

• Sometimes, this assumption is violated: for example, in demand and supply models

9.1 Introduction

• Suppose that we write the demand function as:

• where q is the quantity demanded, p the price, and u the disturbance term which denotes random shifts in the demand function

• In Figure 9.1 we see that a shift in the demand function produces a change in both price and quantity if the supply curve has an upward slope

9.1 Introduction

• If the supply curve is horizontal (i.e., completely price inelastic), a shift in the demand curve produces a change in price only

• If the supply curve is vertical (infinite price elasticity), a shift in the demand curve produces a change in quantity only

9.1 Introduction

• Thus in equation (9.1) the error term u is correlated with p when the supply curve is upward sloping or perfectly horizontal

• Hence an estimation of the equation by ordinary least squares produces inconsistent estimates of the parameters

9.2 Endogenous and Exogenous

Variables

• In simultaneous equations models variables are classified as endogenous and exogenous

• The traditional definition of these terms is that endogenous variables are variables that are determined by the economic model and exogenous variables are those determined from outside


Variables

• Endogenous variables are also called jointly determined and exogenous variables are called predetermined. (It is customary to include past values of endogenous variables in the predetermined group.)

• Since the exogenous variables are predetermined, they are independent of the error terms in the model

• They thus satisfy the assumptions that the x's satisfy in the usual regression model of y on x's


Variables

• Consider now the demand and supply mode q = a

1

+ b

1 p + c

1 y + u

1 demand function q = a

2

+ b

2 p + c

2

R + u

2 supply function (9.2)

• q is the quantity, p the price, y the income, R the rainfall, and u

1 and u

2 are the error terms

• Here p and q are the endogenous variables and y and R are the exogenous variables


Variables

• Since the exogenous variables are independent of the error terms u

1 and u

2 and satisfy the usual requirements for ordinary least squares estimation, we can estimate regressions of p and q on y and R by ordinary least squares, although we cannot estimate equations (9.2)by ordinary least squares

• We will show presently that from these regressions of p and q on y and R we can recover the parameters in the original demand and supply equations (9.2)


Variables

• This method is called indirect least squares —it is indirect because we do not apply least squares to equations (9.2)

• The indirect least squares method does not always work, so we will first discuss the conditions under which it works and how the method can be simplified. To discuss this issue, we first have to clarify the concept of identification

9.3 The Identification Problem:

Identification Through Reduced Form

• We have argued that the error terms u1 and u2 are correlated with p in equations (9.2),and hence if we estimate the equation by ordinary least squares, the parameter estimates are inconsistent

• Roughly speaking, the concept of identification is related to consistent estimation of the parameters

• Thus if we can somehow obtain consistent estimates of the parameters in the demand function, we say that the demand function is identified



• Similarly, if we can somehow get consistent estimates of the parameters in the supply function, we say that the supply function is identified

• Getting consistent estimates is just a necessary condition for identification, not a sufficient condition, as we show in the next section



• If we solve the two equations in(9.2) for q and p in terms of y and R, we get

• These equations are called the reduced-form equations.

• Equation (9.2) are called the structural equations because they describe the structure of the economic system.



• We can write equations (9.3) as where v

1 and v

2 are error terms and



• The π’s are called reduced-form parameters.

• The estimation of the equations (9.4) by ordinary least squares gives us consistent estimates of the reduced form parameters.

• From these we have to obtain consistent estimates of the parameters in


Identification Through Reduced Form of the corresponding structural parameters.

• As mentioned earlier, this method is known as the indirect least squares method.



• It may not be always possible to get estimates of the structural coefficients from the estimates of the reduced-form coefficients, and sometimes we get multiple estimates and we have the problem of choosing between them.

• For example, suppose that the demand and supply model is written as



• Then the reduced from is

or


Identification Through Reduced Form b

ˆ

2



2

 ˆ

4 a

ˆ

2

  ˆ

1

/ b

ˆ

2

 ˆ

3

• But these is no way of getting estimates of a

1

, b

1

, and c

1

.

• Thus the supply function is identified but the demand function is not.



• On the other hand, suppose that we have the model

• Now we can check that the demand function is identified but the supply function is not.



• Finally, suppose that we have the system


Identification Through Reduced Form or

• Now we get two estimates of b

2

.

b

ˆ   ˆ

/

 ˆ these need not be equal.

b

ˆ   ˆ

/

 ˆ

• For each of these we get an estimate of a a

ˆ

2

  ˆ

1

/ b

ˆ  ˆ

4

2

, which



• On the other hand, we get no estimate for the parameters a function.

1

, b

1

, c

1

, and d

1 of the demand

• Here we say that the supply function is overidentified and the demand function is underidentified.

• When we get unique estimates for the structural parameters of an equation fro, the reduced-form parameters, we say that the equation is exactly identified.



• When we get multiple estimates, we say that the equation is overidentified, and when we get no estimates, we say that the equation is underidentified (or not identified).

• There is a simple counting rule available in the linear systems that we have been considering.

• This counting rule is also known as the order condition for identification.



• This rule is as follows: Let g be the number of endogenous variables in the system and k the total number of variables (endogenous and exogenous) missing from the equation under consideration.

• Then



• This condition is only necessary but not sufficient.

• Let us apply this rule to the equation systems we are considering.

• In equations (9.2), g, the number of endogenous variable, is 2 and there is only one variable missing from each equation (i.e., k=1).

• Both equations are identified exactly.



• In equations (9.5), again g=2.

– There is no variable missing from the first equation

(i.e., k=0); hence it is underidentified.

– There is one variable missing in the second equation

(i.e., k=1); hence it is exactly identifies.

• In equation (9.6)

– there is no variable missing in the first equation; hence it is not identified.

– In the second equation there are two variables missing; thus k>g-1 and the equation is overidentified.



• Illustrative Example

– In Table 9.1 data are presented for demand and supply of pork in the United

States for 1922-1941



• P t

, retail price of pork (cents per pound)

• Q t

, consumption of pork (pounds per capita)

• Y t

, disposable personal income (dollars per capital)

• Z t

, “predetermined elements in pork production.”



• The coefficient of Y in the second equation is very close to zero and the variable Y can be dropped from this equation.

• This would imply that b

2

=0, or supply is not responsive to price.

• In any case, solving from the reduced from to the structural from, we get the estimates of the structural equation as



• The least squares estimates of the demand function are:

– Normalized with respect to Q

– Normalized with respect to P



• The structural demand function can also be written in the two forms:

– Normalized with respect to Q

– Normalized with respect to P

• The estimates of the parameters in the demand function are almost the same with the direct least squares method as with the indirect least squares method when the demand function is normalized with respect to P.



• Which is the correct normalization?

• We argued in Section 9.1 that if quantity supplied is not responsive to price, the demand function should be normalized with respect to P.

• We saw that fact the coefficient of Y in the reduced-form equation for Q was close to zero implied that b

2

=0 or quantity supplied is not responsive to price.



• This is also confirmed by the structural estimate of b

2

, which show a wrong sign for b

2 a coefficient close to zero.

as well but

• Dropping P from the supply function and using

OLS, we get the supply function as

9.5 Methods of Estimation: The

Instrumental Variable Method

• In previous sections we discussed the indirect least squares method

– However, this method is very cumbersome if there are many equations and hence it is not often used

– Identification problem

• Here we discuss some methods that are more generally applicable

– The Instrumental Variable Method



• Broadly speaking, an instrumental variable is a variable that is uncorrelated with the error term but correlated with the explanatory variables in the equation

• For instance, suppose that we have the equation y = ßx + u



• where x is correlated with u

• Then we cannot estimate this equation by ordinary least squares

• The estimate of ß is inconsistent because of the correlation between x and u

• If we can find a variable z that is uncorrelated with u, we can get a consistent estimator for ß

• We replace the condition cov (z, u) = 0 by its sample counterpart



1 n

 z ( y

  x )



0

• This gives

 zx

( 1 / n )

 zu /( 1 / n )

 zx



• The probability limit of this expression is since cov (z, x) ≠0.

   consistent estimator for β.

 ˆ

• Note that we require z to be correlated with x so that cov (z, x) ≠0.



• Now consider the simultaneous equations model where y

1

, y

2 are endogenous variables, z

1 are exogenous variables, and u

1

, u

2

, z

2

, z

3 are error term.

• Since z

– cov (z

1 and z

1

, u

1

) =0

2 are independent of u

, cov (z

2

, u

1

) =0

1,

• However, y

– cov (y

2

, u

1

2 is not independent of u

) ≠0.

1



• Since we have three coefficients to estimate, we have to find a variable that is independent of u

1

.

• Fortunately, in this case we have z

3 cov(z

3

,u

1

)=0.

• z

3 is the instrumental variable for y

2

.

and

• Thus, writing the sample counterparts of these three covariances, we have three equations



• The difference between the normal equation for the ordinary least squares method and the instrumental variable method is only in the last equation.



• Consider the second equation of our model

• Now we have to find an instrumental variable for y1 but we have a choice of z1 and z2

• This is because this equation is overidentified

(by the order condition)

• Note that the order condition (counting rule) is related to the question of whether or not we have enough exogenous variables elsewhere in the system to use as instruments for the endogenous variables in the equation with unknown coefficients



• If the equation is underidentified we do not have enough instrumental variables

• If it is exactly identified, we have just enough instrumental variables

• If it is overidentified, we have more than enough instrumental variables

– In this case we have to use weighted averages of the instrumental variables available

– We compute these weighted averages so that we get the most efficient (minimum asymptotic variance) estimator



• It has been shown (proving this is beyond the scope of this book) that the efficient instrumental variables are constructed by regressing the endogenous variables on all the exogenous variables in the system (i.e., estimating the reduced-form equations).

• In the case of the model given by equations (9.8), we first estimate the reduced-form equations by regressing y

1 and y

2 on z

1

, z

2

, z

3

.

y

1 and

2 use these as instrumental variables.



• For the estimation of the first equation we y

ˆ y

ˆ

1 y

ˆ y

ˆ z

2

, z

3

.

• Let us write

1

, where the a’s are obtained from the estimation of the reduced-form equations by OLS.



• In the estimation of the first equation in (9.8) we use , z

1

, z

2

, and z

3 as instruments.

• This is the same as using z

1

, z

2

, z

3 because as instruments

• But the first two terms are zero by virtue of the first two equations in (9.8’).



2 1



3 1

 instrumental variable is the same as using z

3 instrumental variable.

as an

• This is the case with exactly indentified equations where there is no choice in the instruments.



• The case with the second equation in (9.8) is different.

• Earlier, we said that we had a choice between z

1 as instruments for y

1

ˆ

.

and z

2

• The normal equations now are

 z

3 u

2



0

• Thus the optimal weights for z

1 and z

2 are a

11 and a

12

.



Illustrative Example

• Table 9.2 provides data on some characteristics of the wine industry in Australia for 1955-1956 to

1974-1975.

• The demand-supply model for the wine industry




Instrumental Variable Method where Q t

= real capital consumption of wine w p t

= price of wine relative to CPI p t b = price of beer relative to CPI

Y t

= real per capital disposable income

A t

= real per capital advertising expenditure

S t

= index of storage costs

• Q and t

P t w are the endogenous variables

• The other variable are exogenous.



• For the estimation of the demand function we have only one instrumental variable S t

.

• But for the estimation of the supply function we have available three instrumental variables: P t b

, Y t

, and A t

• The OLS estimation of the demand function gave the

.

following results (all variables are in logs and figures in parentheses are t-ratios):

• All the coefficients except that of Y have the wrong signs.

• The coefficient of P w also significant.

not only has the wrong sign but is



• Treating P w as endogenous and using S as an instrument, we get following results:

• The coefficient of P w least not significant.

still has a wrong sign but it is at

• In any case the conclusion we arrive at is that the quantity demanded is not responsive to prices and advertising expenditures but is responsive to income.

• The income elasticity of demand for wine is about

4.0 (significantly greater than unity).


Two-Stage Least Squares Method

• The 2SLS method differs the IV method y

ˆ as regressors rather than as instruments, but the two methods give identical estimates.

• Consider the equation to be estimated: y

1

 b

1 y

2

 c

1 z

1

 u

1

( 9 .

9 )

• The other exogenous variables in the system are z

2

, z

3

, and z

4

.


Two-Stage Least Squares Method y

ˆ

2 from, a regression on y

2 on z

1

, z

2

, z

3

, and z

4

(the reduces-form equation).

y

 ˆ  v

2

, the residual, is uncorrelated with each of the regressors, z

1

, z

2

, z

3

, and z

4 property of least squares regression that we discussed in Chapter 4.)



• The normal equations for the efficient IV method are y

2

 ˆ

2

 v

2



• But these are the normal equations if we replace y

2 y

ˆ

2 equation by OLS.

• This method of replacing the endogenous variables on the right-hand side by their predicted values from the reduced form and estimating the equation by OLS is called the two-stage least squares (2SLS) method.



• The name arises from the fact that OLS is used in two stages:

Stage 1. Estimate the reduced-form equations by y

ˆ

Stage 2.Replace the right-hand side endogenous y

ˆ equation by OLS.



•

• Note that the estimates do not change even if we replace y

1 y

ˆ

1

• Take the normal equations (9.12).

y

1

 y

1

 v

1 y

1

 y

ˆ

1

 v

1

• We get



• The last terms of these two equations are zero and the equations that remain are the normal equations from the OLS estimation of the equation

ˆ

1

 b

1 y

ˆ

2

 c

1 z

1

 w

• Thus in stage 2 of the 2SLS method we can replace all the endogenous variables in the equation by their predicted values from the reduced forms and then estimate the equation by OLS.

9.10 Granger Causality

• Granger starts from the premise that the future cannot cause the present or the past.

• If event A occurs after event B, we know that A cannot cause B.

• At the same time, if A occurs before B, it does not necessarily imply that A causes B.

• For instance, the weatherman's prediction occurs before the rain. This does not mean that the weatherman causes the rain.


• In practice, we observe A and B as time series and we would like to know whether A precedes B, or B precedes A, or they are contemporaneous

• For instance, do movements in prices precede movements in interest rates, or is it the opposite, or are the movements contemporaneous?

• This is the purpose of Granger causality

• It is not causality as it is usually understood


• Granger devised some tests for causality (in the limited sense discussed above) which proceed as follows.

• Consider two time series, {y t

} and {x t

}.

• The series x t of y t fails to Granger cause y t if in a regression on lagged y’s and lagged x’s, the coefficients of the latter are zero.

• That is, consider

• Then if β i

=0 (i=1,2,....,k), x t fails to cause y t

.

• The lag length k is, to some extent, arbitrary.


• Learner suggests using the simple word

"precedence" instead of the complicated words

Granger causality since all we are testing is whether a certain variable precedes another and we are not testing causality as it is usually understood

• However, it is too late to complain about the term since it has already been well established in the econometrics literature. Hence it is important to understand what it means

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