Simultaneous Model2

Simultaneous Equations Models

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 x1,

x2, x3... 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 = 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


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)


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 v1 and v2 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



• 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.

212121ˆ,ˆ,ˆ,ˆ,ˆ,ˆ ccbbaa



• 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

• 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.

422ˆ/ˆˆ b 3212

ˆˆ/ˆˆ ba



• 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



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 .

522ˆ/ˆˆ b 632

ˆ/ˆˆ b

412ˆˆ/ˆˆ ba



• 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.



• 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



• 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.”



• 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



• 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 b2=0 or quantity supplied is not

responsive to price.



• 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



• 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



• This gives

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

0)(1

xyzn



• The probability limit of this expression is

since cov (z, x) ≠0.

• Hence plim ,thus proving that is a

consistent estimator for β.

• Note that we require z to be correlated with x so

that cov (z, x) ≠0.

ˆ


Instrumental Variable Method • Now consider the simultaneous equations model

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



• 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,u1)=0.

• z3 is the instrumental variable for y2.

• 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.


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 (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 y1 and y2 on z1, z2, z3.

• We obtain the predicted values and

use these as instrumental variables. 21ˆandˆ yy



• 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, z3.

• Let us write

where the a’s are obtained from the estimation of

the reduced-form equations by OLS.

2y

1y

2y1y



• 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.

00ˆ1312 uzuy 2y



• 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



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





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

w

tpb

tp

• are the endogenous variables

• The other variable are exogenous.

w

tt PQ and



• 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,, tt

b

t AYP



• 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).


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



• 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.)

222ˆ vyy

2y

2y



• The normal equations for the efficient IV method

are

• Substituting we get 222

ˆ vyy



• 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



• 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



• 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

111ˆ vyy

1y

111ˆ vyy



• 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 11211ˆˆ

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, {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.


• 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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Simultaneous Model2