Simultaneous Equations Models
Prepared by Vera Tabakova, East Carolina University

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
Principles of Econometrics, 3rd Edition
Slide 11-2
Figure 11.1 Supply and demand equilibrium
Principles of Econometrics, 3rd Edition
Slide11-3
D em and: Q   1 P   2 X  e d
S upply:
Principles of Econometrics, 3rd Edition
Q  1 P  e s
(11.1)
(11.2)
Slide11-4
E ( e d )  0,
var( e d )   d
E ( e s )  0,
var( e s )   s
2
2
(11.3)
cov( e d , e s )  0
Principles of Econometrics, 3rd Edition
Slide11-5
Figure 11.2 Influence diagrams for two regression models
Principles of Econometrics, 3rd Edition
Slide11-6
Figure 11.3 Influence diagram for a simultaneous equations model
Principles of Econometrics, 3rd Edition
Slide11-7
1 P  es   1 P   2 X  ed
P 
2
 1   1 
X 
ed  es
 1   1 
(11.4)
  1 X  v1
Principles of Econometrics, 3rd Edition
Slide11-8
Q  1 P  es

2
ed  es 
 1 
X 
  es
 1   1  
  1   1 

 1 2
 1   1 
X 
 1e d   1e s
(11.5)
 1   1 
  2 X  v2
Principles of Econometrics, 3rd Edition
Slide11-9
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.
Principles of Econometrics, 3rd Edition
Slide 11-10
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.
Principles of Econometrics, 3rd Edition
Slide 11-11
Figure 11.4 The effect of changing income
Principles of Econometrics, 3rd Edition
Slide11-12
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.
Principles of Econometrics, 3rd Edition
Slide 11-13
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.
Principles of Econometrics, 3rd Edition
Slide 11-14
P  E ( P )  v1   1 X  v1
(11.6)
Q   1  E  P   v1   e s
  1 E  P     1 v1  e s 
(11.7)
  1 E  P   e*
Principles of Econometrics, 3rd Edition
Slide 11-15
Pˆ  ˆ 1 X
Q   1 Pˆ  eˆ*
Principles of Econometrics, 3rd Edition
(11.8)
Slide 11-16
Estimating the (11.8) by least squares generates the so-called twostage 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 Pˆ .

Least squares estimation of the structural equation in which the right-
hand side endogenous variable P is replaced by its predicted value Pˆ .
Principles of Econometrics, 3rd Edition
Slide 11-17
y1   2 y 2   3 y 3   1 x1   2 x 2  e1
1.
(11.9)
Estimate the parameters of the reduced form equations
y 2   12 x1   22 x 2 
  K 2 xK  v2
y 3   13 x1   23 x 2 
  K 3 x K  v3
by least squares and obtain the predicted values.
Principles of Econometrics, 3rd Edition
Slide 11-18
yˆ 2  ˆ 1 2 x1  ˆ 2 2 x 2 
 ˆ K 2 x K
(11.10)
yˆ 3  ˆ 1 3 x1  ˆ 2 3 x 2 
Principles of Econometrics, 3rd Edition
 ˆ K 3 x K
Slide 11-19
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)
y1   2 yˆ 2   3 yˆ 3   1 x1   2 x 2  e1
*
Estimate the parameters of this equation by least squares.
Principles of Econometrics, 3rd Edition
Slide 11-20

The 2SLS estimator is a biased estimator, but it is consistent.

In large samples the 2SLS estimator is approximately normally
distributed.
Principles of Econometrics, 3rd Edition
Slide 11-21

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 tvalues, just like an ordinary least squares regression program.
Principles of Econometrics, 3rd Edition
Slide 11-22

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.
Principles of Econometrics, 3rd Edition
Slide 11-23
D em an d : Q i   1   2 Pi   3 P S i   4 D I i  e i
(11.11)
S u p p ly: Q i   1   2 Pi   3 P Fi  e i
(11.12)
d
s
Principles of Econometrics, 3rd Edition
Slide11-24
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.
Principles of Econometrics, 3rd Edition
Slide11-25
Q i   11   21 P S i   31 D I i   41 P Fi  v i 1
Pi   12   22 P S i   32 D I i   42 P Fi  v i 2
Principles of Econometrics, 3rd Edition
Slide11-26
Principles of Econometrics, 3rd Edition
Slide11-27
Principles of Econometrics, 3rd Edition
Slide11-28
Principles of Econometrics, 3rd Edition
Slide11-29
Pˆi  ˆ 12  ˆ 22 P S i  ˆ 32 D I i  ˆ 42 P Fi
  32.512  1.708 P S i  7.602 D I i  1.354 P Fi
Principles of Econometrics, 3rd Edition
Slide11-30
Principles of Econometrics, 3rd Edition
Slide11-31
Principles of Econometrics, 3rd Edition
Slide11-32
ln  QUAN t    1   2 ln  PRICE t    3 M ON t   4TUE t   5W ED t   6THU t  e t
(11.13)
ln  QUAN t   1   2 ln  PRICE t    3 STORM Yt  et
(11.14)
d
s
Principles of Econometrics, 3rd Edition
Slide11-33
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.
Principles of Econometrics, 3rd Edition
Slide11-34
ln  QUAN t    11   21 M ON t   31TUE t   41W ED t   51THU t   61 STORM Yt  v t 1
(11.15)
ln  PRICE t    12   22 M ON t   32TUE t   42W ED t   52THU t   62 STORM Yt  v t 2
(11.16)
Principles of Econometrics, 3rd Edition
Slide11-35
Principles of Econometrics, 3rd Edition
Slide11-36
Principles of Econometrics, 3rd Edition
Slide11-37

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
ln  PRIC E t   ˆ 12  ˆ 22 M O N t  ˆ 32TU E t  ˆ 42W ED t  ˆ 52TH U t  ˆ 62 STO RM Yt
ln  PRICE t   ˆ 12  ˆ 62 STO RM Yt
Principles of Econometrics, 3rd Edition
Slide11-39
Principles of Econometrics, 3rd Edition
Slide11-40










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
Principles of Econometrics, 3rd Edition
Slide 11-41
Principles of Econometrics, 3rd Edition
Slide 11-42
co v  P , e s   E  P  E  P    e s  E  e s  
 E  P es 
[sin ce E  e s   0 ]
 E   1 X  v1  e s
[su b stitu te fo r P ]
 ed  es 
 E
 es



1 
 1
 E  es
2


1   1
 s
[sin ce  1 X is ex o g en o u s]
(11A.1)
[sin ce e d , e s assu m ed u n co rre lated ]
2

1   1
0
Principles of Econometrics, 3rd Edition
Slide 11-43
b1 
b1 
 Pi   1 Pi  e si 
2
P
 i
 Pi Q i
2
P
 i

 1   

E  b1    1 
Principles of Econometrics, 3rd Edition
(11A.2)

e  1 
2  si
Pi 
Pi

 hi e si
(11A.3)
 E  hi e si   1
Slide 11-44
P Q  1 P  P es
2
E  P Q   1 E  P
1 
b1 
 Q i Pi / N
2
 Pi / N

E  PQ 
E P
E  PQ 
E P
Principles of Econometrics, 3rd Edition
2

2


 1 
2
  E  Pe 
s
E  P es 
E P
2

E  P es 
E P
2

s
2
 1 
 1   1 
E P
2

 1
Slide 11-45