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Chapter 11
Simultaneous Equations Models
Prepared by Vera Tabakova, East Carolina University
Chapter 11: Simultaneous Equations Models
11.1 A Supply and Demand Model
11.2 The Reduced Form Equations
11.3 The Failure of Least Squares
11.4 The Identification Problem
11.5 Two-Stage Least Squares Estimation
11.6 An Example of Two-Stage Least Squares Estimation
11.7 Supply and Demand at the Fulton Fish Market
Slide 11-2Principles of Econometrics, 3rd Edition
11.1 A Supply and Demand Model
Figure 11.1 Supply and demand equilibrium
Slide11-3Principles of Econometrics, 3rd Edition
11.1 A Supply and Demand Model
Slide11-4Principles of Econometrics, 3rd Edition
(11.1)
(11.2)
1 2Demand: dQ P X e
1Supply: sQ P e
11.1 A Supply and Demand Model
Slide11-5Principles of Econometrics, 3rd Edition
(11.3)
2
2
( ) 0, var( )
( ) 0, var( )
cov( , ) 0
d d d
s s s
d s
E e e
E e e
e e
11.1 A Supply and Demand Model
Figure 11.2 Influence diagrams for two regression models
Slide11-6Principles of Econometrics, 3rd Edition
11.1 A Supply and Demand Model
Figure 11.3 Influence diagram for a simultaneous equations model
Slide11-7Principles of Econometrics, 3rd Edition
11.2 The Reduced Form Equations
Slide11-8Principles of Econometrics, 3rd Edition
(11.4)
1 1 2s dP e P X e
2
1 1 1 1
1 1
d se eP X
X v
11.2 The Reduced Form Equations
Slide11-9Principles of Econometrics, 3rd Edition
(11.5)
1
21
1 1 1 1
1 2 1 1
1 1 1 1
2 2
s
d ss
d s
Q P e
e eX e
e eX
X v
11.3 The Failure of Least Squares
Slide 11-10Principles of Econometrics, 3rd Edition
The least squares estimator of parameters in a structural
simultaneous equation is biased and inconsistent because of the
correlation between the random error and the endogenous
variables on the right-hand side of the equation.
11.4 The Identification Problem
In the supply and demand model given by (11.1) and (11.2)
the parameters of the demand equation, 1 and 2, cannot be
consistently estimated by any estimation method, but
the slope of the supply equation, 1, can be consistently estimated.
Slide 11-11Principles of Econometrics, 3rd Edition
11.4 The Identification Problem
Figure 11.4 The effect of changing income
Slide11-12Principles of Econometrics, 3rd Edition
11.4 The Identification Problem
Slide 11-13Principles of Econometrics, 3rd Edition
A Necessary Condition for Identification: In a system of M
simultaneous equations, which jointly determine the values of M
endogenous variables, at least M–1 variables must be absent from
an equation for estimation of its parameters to be possible. When
estimation of an equation’s parameters is possible, then the
equation is said to be identified, and its parameters can be
estimated consistently. If less than M–1 variables are omitted
from an equation, then it is said to be unidentified and its
parameters can not be consistently estimated.
11.4 The Identification Problem
Slide 11-14Principles of Econometrics, 3rd Edition
Remark: The two-stage least squares estimation procedure is
developed in Chapter 10 and shown to be an instrumental
variables estimator. The number of instrumental variables required
for estimation of an equation within a simultaneous equations
model is equal to the number of right-hand-side endogenous
variables. Consequently, identification requires that the number of
excluded exogenous variables in an equation be at least as large as
the number of included right-hand-side endogenous variables. This
ensures an adequate number of instrumental variables.
11.5 Two-Stage Least Squares Estimation
Slide 11-15Principles of Econometrics, 3rd Edition
(11.6)
(11.7)
1 1 1( )P E P v X v
1 1
1 1 1
1 *
s
s
Q E P v e
E P v e
E P e
11.5 Two-Stage Least Squares Estimation
Slide 11-16Principles of Econometrics, 3rd Edition
(11.8)
1ˆ ˆ P X
1 *ˆ ˆQ P e
11.5 Two-Stage Least Squares Estimation
Estimating the (11.8) by least squares generates the so-called two-
stage least squares estimator of β1, which is consistent and
asymptotically normal. To summarize, the two stages of the
estimation procedure are:
Least squares estimation of the reduced form equation for P and the
calculation of its predicted value
Least squares estimation of the structural equation in which the right-
hand side endogenous variable P is replaced by its predicted value
Slide 11-17Principles of Econometrics, 3rd Edition
ˆ.P
ˆ.P
11.5.1 The General Two-Stage Least Squares Estimation Procedure
1. Estimate the parameters of the reduced form equations
by least squares and obtain the predicted values.
Slide 11-18Principles of Econometrics, 3rd Edition
(11.9)1 2 2 3 3 1 1 2 2 1y y y x x e
2 12 1 22 2 2 2
3 13 1 23 2 3 3
K K
K K
y x x x v
y x x x v
11.5.1 The General Two-Stage Least Squares Estimation Procedure
Slide 11-19Principles of Econometrics, 3rd Edition
(11.10)2 12 1 22 2 2
3 13 1 23 2 3
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
K K
K K
y x x x
y x x x
11.5.1 The General Two-Stage Least Squares Estimation Procedure
2. Replace the endogenous variables, y2 and y3, on the right-hand side
of the structural (11.9) by their predicted values from (11.10)
Estimate the parameters of this equation by least squares.
Slide 11-20Principles of Econometrics, 3rd Edition
*1 2 2 3 3 1 1 2 2 1ˆ ˆy y y x x e
11.5.2 The Properties of the Two-Stage Least Squares Estimator
The 2SLS estimator is a biased estimator, but it is consistent.
In large samples the 2SLS estimator is approximately normally
distributed.
Slide 11-21Principles of Econometrics, 3rd Edition
11.5.2 The Properties of the Two-Stage Least Squares Estimator
The variances and covariances of the 2SLS estimator are unknown in
small samples, but for large samples we have expressions for them
which we can use as approximations. These formulas are built into
econometric software packages, which report standard errors, and t-
values, just like an ordinary least squares regression program.
Slide 11-22Principles of Econometrics, 3rd Edition
11.5.2 The Properties of the Two-Stage Least Squares Estimator
If you obtain 2SLS estimates by applying two least squares
regressions using ordinary least squares regression software, the
standard errors and t-values reported in the second regression are not
correct for the 2SLS estimator. Always use specialized 2SLS or
instrumental variables software when obtaining estimates of
structural equations.
Slide 11-23Principles of Econometrics, 3rd Edition
11.6 An Example of Two-Stage Least Squares Estimation
Slide11-24Principles of Econometrics, 3rd Edition
(11.11)
(11.12)
1 2 3 4Demand: di i i i iQ P PS DI e
1 2 3Supply: si i i iQ P PF e
11.6.1 Identification
The rule for identifying an equation is that in a system of M equations
at least M 1 variables must be omitted from each equation in order
for it to be identified. In the demand equation the variable PF is not
included and thus the necessary M 1 = 1 variable is omitted. In the
supply equation both PS and DI are absent; more than enough to
satisfy the identification condition.
Slide11-25Principles of Econometrics, 3rd Edition
11.6.2 The Reduced Form Equations
Slide11-26Principles of Econometrics, 3rd Edition
11 21 31 41 1
12 22 32 42 2
i i i i i
i i i i i
Q PS DI PF v
P PS DI PF v
11.6.2 The Reduced Form Equations
Slide11-27Principles of Econometrics, 3rd Edition
11.6.2 The Reduced Form Equations
Slide11-28Principles of Econometrics, 3rd Edition
11.6.2 The Reduced Form Equations
Slide11-29Principles of Econometrics, 3rd Edition
11.6.3 The Structural Equations
Slide11-30Principles of Econometrics, 3rd Edition
12 22 32 42ˆ ˆ ˆ ˆ ˆ
32.512 1.708 7.602 1.354
i i i i
i i i
P PS DI PF
PS DI PF
11.6.3 The Structural Equations
Slide11-31Principles of Econometrics, 3rd Edition
11.6.3 The Structural Equations
Slide11-32Principles of Econometrics, 3rd Edition
11.7 Supply and Demand at the Fulton Fish Market
Slide11-33Principles of Econometrics, 3rd Edition
(11.13)
(11.14)
1 2 3 4 5 6ln ln dt t t t t t tQUAN PRICE MON TUE WED THU e
t 1 2 3ln ln st t tQUAN PRICE STORMY e
11.7.1 Identification
The necessary condition for an equation to be identified is that in this
system of M = 2 equations, it must be true that at least M – 1 = 1
variable must be omitted from each equation. In the demand equation
the weather variable STORMY is omitted, while it does appear in the
supply equation. In the supply equation, the four daily dummy
variables that are included in the demand equation are omitted.
Slide11-34Principles of Econometrics, 3rd Edition
11.7.2 The Reduced Form Equations
Slide11-35Principles of Econometrics, 3rd Edition
(11.15)
(11.16)
11 21 31 41 51 61 1ln t t t t t t tQUAN MON TUE WED THU STORMY v
12 22 32 42 52 62 2ln t t t t t t tPRICE MON TUE WED THU STORMY v
11.7.2 The Reduced Form Equations
Slide11-36Principles of Econometrics, 3rd Edition
11.7.2 The Reduced Form Equations
Slide11-37Principles of Econometrics, 3rd Edition
11.7.2 The Reduced Form Equations
To identify the supply curve the daily dummy variables must be jointly significant. This implies that at least one of their coefficients is statistically different from zero, meaning that there is at least one significant shift variable in the demand equation, which permits us to reliably estimate the supply equation.
To identify the demand curve the variable STORMY must be statistically significant, meaning that supply has a significant shift variable, so that we can reliably estimate the demand equation.
Principles of Econometrics, 3rd Edition Slide11-38
11.7.2 The Reduced Form Equations
Principles of Econometrics, 3rd Edition Slide11-39
12 22 32 42 52 62ˆ ˆ ˆ ˆ ˆ ˆln t t t t t tPRICE MON TUE WED THU STORMY
12 62ˆ ˆln t tPRICE STORMY
11.7.3 Two-Stage Least Squares Estimation of Fish Demand
Principles of Econometrics, 3rd Edition Slide11-40
Keywords
Slide 11-41Principles of Econometrics, 3rd Edition
endogenous variables exogenous variables Fulton Fish Market identification reduced form equation reduced form errors reduced form parameters simultaneous equations structural parameters two-stage least squares
Chapter 11 Appendix
Slide 11-42Principles of Econometrics, 3rd Edition
Appendix 11A An Algebraic Explanation of the
Failure of Least Squares
Appendix 11A An Algebraic Explanation of the Failure of Least Squares
Principles of Econometrics, 3rd Edition Slide 11-43
(11A.1)
1 1
11 1
2
1 1
cov ,
[since 0]
[substitute for ]
[since is exogenous]
[since , assumed uncorre
s s s
s s
s
d ss
s
d s
P e E P E P e E e
E Pe E e
E X v e P
e eE e X
E ee e
2
1 1
lated]
0s
Appendix 11A An Algebraic Explanation of the Failure of Least Squares
Principles of Econometrics, 3rd Edition Slide 11-44
(11A.2)
(11A.3)
1 2i i
i
PQb
P
11 1 12 2
i i si isi i si
i i
P P e Pb e h e
P P
1 1 1i siE b E h e
Appendix 11A An Algebraic Explanation of the Failure of Least Squares
Principles of Econometrics, 3rd Edition Slide 11-45
21
21
1 2 2
s
s
s
PQ P Pe
E PQ E P E Pe
E PQ E Pe
E P E P
21 1
1 1 1 12 2 2 2
/
/s si i
i
E PQ E PeQ P Nb
P N E P E P E P