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?

What is in this Chapter?

• 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

What is in this Chapter?

• 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 x1, x2, 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

9.2 Endogenous and Exogenous 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

9.2 Endogenous and Exogenous Variables

• Consider now the demand and supply mode

q = a1 + b1p + c1 y + u1 demand function

q = a2 + b2p + c2R + u2 supply function (9.2)

• q is the quantity, p the price, y the income, R the rainfall, and u1 and u2 are the error terms

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

9.2 Endogenous and Exogenous Variables

• Since the exogenous variables are independent of the error terms u1 and u2 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)

9.2 Endogenous and Exogenous 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

9.3 The Identification Problem: Identification Through Reduced Form

• 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

9.3 The Identification Problem: Identification Through Reduced Form

• 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.

9.3 The Identification Problem: Identification Through Reduced Form

• We can write equations (9.3) as

where v1 and v2 are error terms and

9.3 The Identification Problem: Identification Through Reduced Form

• 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

9.3 The Identification Problem: Identification Through Reduced Form

• Since are all single-valued function of the ,they are consistent estimates of the corresponding structural parameters.

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

212121 ˆ,ˆ,ˆ,ˆ,ˆ,ˆ ccbbaa

9.3 The Identification Problem: Identification Through Reduced Form

• 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

9.3 The Identification Problem: Identification Through Reduced Form

• Then the reduced from is

9.3 The Identification Problem: Identification Through Reduced Form

or

• In this case and .

• But these is no way of getting estimates of a1, b1, and c1.

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

422 ˆ/ˆˆ b 3212 ˆˆ/ˆˆ ba

9.3 The Identification Problem: Identification Through Reduced Form

• 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.

9.3 The Identification Problem: Identification Through Reduced Form

• Finally, suppose that we have the system

9.3 The Identification Problem: Identification Through Reduced Form

or

• Now we get two estimates of b2.

• One is and the other is , and these need not be equal.

• For each of these we get an estimate of a2, which is .

522 ˆ/ˆˆ b 632 ˆ/ˆˆ b

412 ˆˆ/ˆˆ ba

9.3 The Identification Problem: Identification Through Reduced Form

• On the other hand, we get no estimate for the parameters a1 , b1, c1, and d1 of the demand function.

• 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.

9.3 The Identification Problem: Identification Through Reduced Form

• 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.

9.3 The Identification Problem: Identification Through Reduced Form

• 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

9.3 The Identification Problem: Identification Through Reduced Form

• 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.

9.3 The Identification Problem: Identification Through Reduced Form

• 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; henc

e it is not identified.– In the second equation there are two variables missin

g; thus k>g-1 and the equation is overidentified.

9.3 The Identification Problem: Identification Through Reduced Form

• Illustrative Example– In Table 9.1 data are presented for

demand and supply of pork in the United States for 1922-1941

9.3 The Identification Problem: Identification Through Reduced Form

• Pt, retail price of pork (cents per pound)

• Qt, consumption of pork (pounds per capita)

• Yt, disposable personal income (dollars per capital)

• Zt, “predetermined elements in pork production.”

9.3 The Identification Problem: Identification Through Reduced Form

• 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 b2=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

9.3 The Identification Problem: Identification Through Reduced Form

• The least squares estimates of the demand function are:– Normalized with respect to Q

– Normalized with respect to P

9.3 The Identification Problem: Identification Through Reduced Form

• 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.

9.3 The Identification Problem: Identification Through Reduced Form

• 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 b2=0 or quantity supplied is not responsive to price.

9.3 The Identification Problem: Identification Through Reduced Form

• This is also confirmed by the structural estimate of b2, which show a wrong sign for b2 as well but a coefficient close to zero.

• 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

9.5 Methods of Estimation: 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

9.5 Methods of Estimation: The Instrumental Variable Method

• where x is correlated with u• Then we cannot estimate this equation by ordina

ry 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 wit

h u, we can get a consistent estimator for ß• We replace the condition cov (z, u) = 0 by its sa

mple counterpart

9.5 Methods of Estimation: The Instrumental Variable Method

• This gives

• But can be written as zxzu / zxnzun )/1/()/1(

0)(1

xyzn

9.5 Methods of Estimation: The Instrumental Variable Method

• The probability limit of this expression is

since cov (z, x) ≠0.• Hence plim ,thus proving that is a consi

stent estimator for β.• Note that we require z to be correlated with x so

that cov (z, x) ≠0.

ˆ

9.5 Methods of Estimation: The Instrumental Variable Method

• Now consider the simultaneous equations model

where y1, y2 are endogenous variables, z1, z2, z3 a

re exogenous variables, and u1, u2 are error term.• Since z1 and z2 are independent of u1,

– cov (z1, u1) =0 , cov (z2, u1) =0

• However, y2 is not independent of u1

– cov (y2, u1) ≠0.

9.5 Methods of Estimation: The Instrumental Variable Method

• Since we have three coefficients to estimate, we have to find a variable that is independent of u1.

• Fortunately, in this case we have z3 and cov(z3,u

1)=0.

• z3 is the instrumental variable for y2.

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

9.5 Methods of Estimation: The Instrumental Variable Method

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

9.5 Methods of Estimation: The Instrumental Variable Method

• 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 (b

y the order condition)• Note that the order condition (counting rule) is re

lated 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

9.5 Methods of Estimation: The Instrumental Variable Method

• 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

9.5 Methods of Estimation: The Instrumental Variable Method

• 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 y1 and y2 on z1, z2, z3.

• We obtain the predicted values and use these as instrumental variables.

21 ˆandˆ yy

9.5 Methods of Estimation: The Instrumental Variable Method

• For the estimation of the first equation we use , and for the estimation of the second equation we use .

• We can write and as linear function of z1, z2, z

3.

• Let us write

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

2y

1y

2y1y

9.5 Methods of Estimation: The Instrumental Variable Method

• In the estimation of the first equation in (9.8) we use , z1, z2, and z3 as instruments.

• This is the same as using z1, z2, z3 as instruments because

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

• Thus . Hence using as an instrumental variable is the same as using z3 as an instrumental variable.

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

00ˆ 1312 uzuy 2y

9.5 Methods of Estimation: The Instrumental Variable Method

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

• Earlier, we said that we had a choice between z1 and z2 as instruments for y1.

• The use of gives the optimum weighting.• The normal equations now are

since .

• Thus the optimal weights for z1 and z2 are a11 and a12.

1y

023 uz

9.5 Methods of Estimation: The Instrumental Variable Method

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

9.5 Methods of Estimation: The Instrumental Variable Method

9.5 Methods of Estimation: The Instrumental Variable Method

where Qt= real capital consumption of wine

= price of wine relative to CPI

= price of beer relative to CPI

Yt= real per capital disposable income

At= real per capital advertising expenditure

St= index of storage costs

wtpbtp

• are the endogenous variables • The other variable are exogenous.

wtt PQ and

9.5 Methods of Estimation: The Instrumental Variable Method

• For the estimation of the demand function we have only one instrumental variable St.

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

• 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 Pw not only has the wrong sign but is also significant.

.and,, ttbt AYP

9.5 Methods of Estimation: The Instrumental Variable Method

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

• The coefficient of Pw still has a wrong sign but it is at least not significant.

• 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).

9.6 Methods of Estimation: The Two-Stage Least Squares Method

• The 2SLS method differs the IV method described in Section 9.5 in that the ‘s are used as regressors rather than as instruments, but the two methods give identical estimates.

• Consider the equation to be estimated:

• The other exogenous variables in the system are z2, z3, and z4.

)9.9(111211 uzcyby

y

9.6 Methods of Estimation: The Two-Stage Least Squares Method

• Let be the predicted value of y2 from, a regression on y2 on z1, z2, z3, and z4 (the reduces-form equation).

• Then where v2, the residual, is uncorrelated with each of the regressors, z1, z2, z3, and z4 and hence with as well. (This is the property of least squares regression that we discussed in Chapter 4.)

222 ˆ vyy

2y

2y

9.6 Methods of Estimation: The Two-Stage Least Squares Method

• The normal equations for the efficient IV method are

• Substituting we get222 ˆ vyy

9.6 Methods of Estimation: The Two-Stage Least Squares Method

• But these are the normal equations if we replace y2 by in (9.9) and estimate the

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.

2y

9.6 Methods of Estimation: The Two-Stage Least Squares Method

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

Stage 1. Estimate the reduced-form equations by

OLS and obtain the predicted ‘s.

Stage 2.Replace the right-hand side endogenous

variables by ‘s and estimate the

equation by OLS.

y

y

9.6 Methods of Estimation: The Two-Stage Least Squares Method

• Note that the estimates do not change even if we replace y1 by in equation (9.9).

• Take the normal equations (9.12).• • Now substitute in equations (9.12).• We get

111 ˆ vyy

1y

111 ˆ vyy

9.6 Methods of Estimation: The Two-Stage Least Squares Method

• 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

• 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.

wzcyby 11211 ˆˆ

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.

9.10 Granger Causality

• 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

9.10 Granger Causality

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

• Consider two time series, {yt} and {xt}.

• The series xt fails to Granger cause yt if in a regression of yt on lagged y’s and lagged x’s, the coefficients of the latter are zero.

• That is, consider

• Then if βi=0 (i=1,2,....,k), xt fails to cause yt.

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

9.10 Granger Causality

• 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