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Title: Simultaneous Equations Models

1
Chapter 9
• Simultaneous Equations Models
• (???????)

2
What is in this Chapter?
• How do we detect this problem?
• What are the consequences?
• What are the solutions?

3
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

4
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

5
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

6
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
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

10
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

11
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

12
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

13
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

14
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)

15
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

16
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

17
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

18
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.

19
9.3 The Identification Problem Identification
Through Reduced Form
• We can write equations (9.3) as
• where v1 and v2 are error terms and

20
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

21
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.

22
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

23
9.3 The Identification Problem Identification
Through Reduced Form
• Then the reduced from is

24
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.

25
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.

26
9.3 The Identification Problem Identification
Through Reduced Form
• Finally, suppose that we have the system

27
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 .

28
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.

29
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.

30
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

31
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.

32
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.

33
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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36
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.

37
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

38
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

39
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.

40
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.

41
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

42
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

43
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

44
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

45
9.5 Methods of Estimation The Instrumental
Variable Method
• This gives
• But can be written as

46
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.

47
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.

48
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

49
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.

50
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

51
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

52
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.

53
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.

54
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.

55
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.

56
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

57
9.5 Methods of Estimation The Instrumental
Variable Method
58
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.

59
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.

60
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).

61
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.

62
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.)

63
9.6 Methods of Estimation The Two-Stage Least
Squares Method
• The normal equations for the efficient IV method
are
• Substituting we get

64
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.

65
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.

66
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

67
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.

68
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.

69
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

70
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.

71
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