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Chapter 9

- Simultaneous Equations Models
- (???????)

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, 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

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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 squaresit

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 ps 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.

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.

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 .

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., k1). - Both equations are identified exactly.

9.3 The Identification Problem Identification

Through Reduced Form

- In equations (9.5), again g2.
- There is no variable missing from the first

equation (i.e., k0) hence it is

underidentified. - There is one variable missing in the second

equation (i.e., k1) 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 kgtg-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

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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 b20, 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 b20 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 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

9.5 Methods of Estimation The Instrumental

Variable Method

- This gives
- But can be written as

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

consistent estimator for ß. - Note that we require z to be correlated with x so

that cov (z, x) ?0.

9.5 Methods of Estimation The 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.

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,u1)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

(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

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.

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, z3. - Let us write
- where the as are obtained from the estimation

of the reduced-form equations by OLS.

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.

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.

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

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

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.

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.

9.6 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.)

9.6 Methods of Estimation The Two-Stage Least

Squares Method

- The normal equations for the efficient IV method

are - Substituting we get

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.

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.

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

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.

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 ys and lagged xs,

the coefficients of the latter are zero. - That is, consider
- Then if ßi0 (i1,2,....,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