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Principles of Econometrics, 4th Edition Page 1 Chapter 11: Simultaneous Equations Models
Lecture note 4 Simultaneous Equations Models
Principles of Econometrics, 4th Edition Page 2 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 Estimation
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
Chapter Contents
Principles of Econometrics, 4th Edition Page 3 Chapter 11: Simultaneous Equations Models
We will consider econometric models for data that
are jointly determined by two or more economic
relations
– These simultaneous equations models differ
from those previously studied because in each
model there are two or more dependent
variables rather than just one
– Simultaneous equations models also differ from
most of the econometric models we have
considered so far, because they consist of a set
of equations
Principles of Econometrics, 4th Edition Page 4 Chapter 11: Simultaneous Equations Models
11.1
A Supply and Demand Model
Principles of Econometrics, 4th Edition Page 5 Chapter 11: Simultaneous Equations Models
A very simple supply and demand model might
look like:
11.1 A Supply and
Demand Model
1 2Demand: dQ P X e
1Supply: sQ P e
Eq. 11.1
Eq. 11.2
Principles of Econometrics, 4th Edition Page 6 Chapter 11: Simultaneous Equations Models
11.1 A Supply and
Demand Model FIGURE 11.1 Supply and demand equilibrium
Principles of Econometrics, 4th Edition Page 7 Chapter 11: Simultaneous Equations Models
It takes two equations to describe the supply and
demand equilibrium
– The two equilibrium values, for price and
quantity, P* and Q*, respectively, are
determined at the same time
– In this model the variables P and Q are called
endogenous variables because their values are
determined within the system we have created
11.1 A Supply and
Demand Model
Principles of Econometrics, 4th Edition Page 8 Chapter 11: Simultaneous Equations Models
The endogenous variables P and Q are dependent
variables and both are random variables
The income variable X has a value that is
determined outside this system
– Such variables are said to be exogenous, and
these variables are treated like usual ‘‘x’’
explanatory variables
11.1 A Supply and
Demand Model
Principles of Econometrics, 4th Edition Page 9 Chapter 11: Simultaneous Equations Models
For the error terms, we have:
11.1 A Supply and
Demand Model
2
2
( ) 0, var( )
( ) 0, var( )
cov( , ) 0
d d d
s s s
d s
E e e
E e e
e e
Eq. 11.3
Principles of Econometrics, 4th Edition Page 10 Chapter 11: Simultaneous Equations Models
An ‘‘influence diagram’’ is a graphical
representation of relationships between model
components
11.1 A Supply and
Demand Model
Principles of Econometrics, 4th Edition Page 11 Chapter 11: Simultaneous Equations Models
11.1 A Supply and
Demand Model FIGURE 11.2 Influence diagrams for two regression models
Principles of Econometrics, 4th Edition Page 12 Chapter 11: Simultaneous Equations Models
11.1 A Supply and
Demand Model FIGURE 11.3 Influence diagram for a simultaneous equations model
Principles of Econometrics, 4th Edition Page 13 Chapter 11: Simultaneous Equations Models
The fact that P is an endogenous variable on the
right-hand side of the supply and demand
equations means that we have an explanatory
variable that is random
– This is contrary to the usual assumption of
‘‘fixed explanatory variables’’
– The problem is that the endogenous regressor P
is correlated with the random errors, ed and es,
which has a devastating impact on our usual
least squares estimation procedure, making the
least squares estimator biased and inconsistent
11.1 A Supply and
Demand Model
Principles of Econometrics, 4th Edition Page 14 Chapter 11: Simultaneous Equations Models
11.2
The Reduced-Form Equations
Principles of Econometrics, 4th Edition Page 15 Chapter 11: Simultaneous Equations Models
The two structural equations Eqs. 11.1 and 11.2
can be solved to express the endogenous variables
P and Q as functions of the exogenous variable X.
– This reformulation of the model is called the
reduced form of the structural equation system
11.2 The Reduced-Form
Equations
Principles of Econometrics, 4th Edition Page 16 Chapter 11: Simultaneous Equations Models
To solve for P, set Q in the demand and supply
equations to be equal:
– Solve for P:
11.2 The Reduced-Form
Equations
1 1 2s dP e P X e
2
1 1 1 1
1 1
d se eP X
X v
Eq. 11.4
Principles of Econometrics, 4th Edition Page 17 Chapter 11: Simultaneous Equations Models
Solving for Q:
– The parameters p1 and p2 in Eqs. 11.4 and 11.5 are
called reduced-form parameters. The error terms
v1 and v2 are called reduced-form errors
11.2 The Reduced-Form
Equations
Eq. 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
Principles of Econometrics, 4th Edition Page 18 Chapter 11: Simultaneous Equations Models
11.3
The Failure of Least Squares
Estimation
Principles of Econometrics, 4th Edition Page 19 Chapter 11: Simultaneous Equations Models
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
11.3 The Failure of Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 20 Chapter 11: Simultaneous Equations Models
11.4
The Identification Problem
Principles of Econometrics, 4th Edition Page 21 Chapter 11: Simultaneous Equations Models
In the supply and demand model given by Eqs.
11.1 and 11.2:
– The parameters of the demand equation, α1and
α2, cannot be consistently estimated by any
estimation method
– The slope of the supply equation, β1, can be
consistently estimated
11.4 The Identification
Problem
Principles of Econometrics, 4th Edition Page 22 Chapter 11: Simultaneous Equations Models
11.4 The Identification
Problem FIGURE 11.4 The effect of changing income
Principles of Econometrics, 4th Edition Page 23 Chapter 11: Simultaneous Equations Models
It is the absence of variables in one equation that
are present in another equation that makes
parameter estimation possible
11.4 The Identification
Problem
Principles of Econometrics, 4th Edition Page 24 Chapter 11: Simultaneous Equations Models
A general rule, which is called a necessary condition for identification of an equation, is:
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 fewer than M - 1variables are omitted from an equation, then it is said to be unidentified, and its parameters cannot be consistently estimated
11.4 The Identification
Problem
Principles of Econometrics, 4th Edition Page 25 Chapter 11: Simultaneous Equations Models
The identification condition must be checked
before trying to estimate an equation
– If an equation is not identified, then changing
the model must be considered before it is
estimated
11.4 The Identification
Problem
Principles of Econometrics, 4th Edition Page 26 Chapter 11: Simultaneous Equations Models
11.5
Two-Stage Least Squares Estimation
Principles of Econometrics, 4th Edition Page 27 Chapter 11: Simultaneous Equations Models
The most widely used method for estimating the
parameters of an identified structural equation is
called two-stage least squares
– This is often abbreviated as 2SLS
– The name comes from the fact that it can be
calculated using two least squares regressions
11.5 Two-Stage Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 28 Chapter 11: Simultaneous Equations Models
Consider the supply equation discussed previously
– We cannot apply the usual least squares
procedure to estimate β1 in this equation
because the endogenous variable P on the right-
hand side of the equation is correlated with the
error term es.
11.5 Two-Stage Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 29 Chapter 11: Simultaneous Equations Models
The reduced-form model is:
Suppose we know π1.
– Then through substitution:
11.5 Two-Stage Least
Squares Estimation
1 1 1( )P E P v X v Eq. 11.6
1 1
1 1 1
s
s
Q E P v e
E P v e
Eq. 11.7
Principles of Econometrics, 4th Edition Page 30 Chapter 11: Simultaneous Equations Models
We can estimate π1 using from the reduced-
form equation for P
– A consistent estimator for E(P) is:
– Then:
11.5 Two-Stage Least
Squares Estimation
Eq. 11.8
1̂
1ˆ ˆ P X
1 *ˆQ P e
Principles of Econometrics, 4th Edition Page 31 Chapter 11: Simultaneous Equations Models
Estimating Eq. 11.8 by least squares generates the
so-called two-stage least squares estimator of β1,
which is consistent and normally distributed in
large samples
11.5 Two-Stage Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 32 Chapter 11: Simultaneous Equations Models
The two stages of the estimation procedure are:
1. Least squares estimation of the reduced-form
equation for P and the calculation of its
predicted value
2. Least squares estimation of the structural
equation in which the right-hand-side
endogenous variable P is replaced by its
predicted value
11.5 Two-Stage Least
Squares Estimation
P̂
P̂
Principles of Econometrics, 4th Edition Page 33 Chapter 11: Simultaneous Equations Models
Suppose the first structural equation in a system of
M simultaneous equations is:
11.5 Two-Stage Least
Squares Estimation
11.5.1 The General Two-
Stage Least Squares
Estimation
Procedure
1 2 2 3 3 1 1 2 2 1y y y x x e Eq. 11.9
Principles of Econometrics, 4th Edition Page 34 Chapter 11: Simultaneous Equations Models
If this equation is identified, then its parameters
can be estimated in the two steps:
1. Estimate the parameters of the reduced-form
equations
by least squares and obtain the predicted
values
11.5 Two-Stage Least
Squares Estimation
11.5.1 The General Two-
Stage Least Squares
Estimation
Procedure
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
2 12 1 22 2 2
3 13 1 23 2 3
늿 늿
늿 늿K K
K K
y x x x
y x x x
Eq. 11.10
Principles of Econometrics, 4th Edition Page 35 Chapter 11: Simultaneous Equations Models
If this equation is identified, then its parameters
can be estimated in the two steps (Continued):
2. Replace the endogenous variables, y2 and y3,
on the right-hand side of the structural Eqs.
11.9 by their predicted values from Eqs.
11.10:
• Estimate the parameters of this equation by
least squares
11.5 Two-Stage Least
Squares Estimation
11.5.1 The General Two-
Stage Least Squares
Estimation
Procedure
*
1 2 2 3 3 1 1 2 2 1ˆ ˆy y y x x e
Principles of Econometrics, 4th Edition Page 36 Chapter 11: Simultaneous Equations Models
The properties of the two-stage least squares
estimator are as follows:
– The 2SLS estimator is a biased estimator, but it
is consistent
– In large samples the 2SLS estimator is
approximately normally distributed
11.5 Two-Stage Least
Squares Estimation
11.5.2 The Properties of the
Two-Stage Least
Squares Estimators
Principles of Econometrics, 4th Edition Page 37 Chapter 11: Simultaneous Equations Models
The properties of the two-stage least squares
estimator are as follows (Continued):
– The variances and covariances of the 2SLS
estimator are unknown in small samples, but
for large samples we have expressions for them
that we can use as approximations
– 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
11.5 Two-Stage Least
Squares Estimation
11.5.2 The Properties of the
Two-Stage Least
Squares Estimators
Principles of Econometrics, 4th Edition Page 38 Chapter 11: Simultaneous Equations Models
11.6
An Example of Two-Stage Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 39 Chapter 11: Simultaneous Equations Models
Consider a supply and demand model for truffles:
11.6 An Example of
Two-Stage Least
Squares Estimation
1 2 3 4Demand: d
i i i i iQ P PS DI e
1 2 3Supply: s
i i i iQ P PF e
Eq. 11.11
Eq. 11.12
Principles of Econometrics, 4th Edition Page 40 Chapter 11: Simultaneous Equations Models
The rule for identifying an equation is:
– 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; 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
11.6 An Example of
Two-Stage Least
Squares Estimation
11.6.1 Identification
Principles of Econometrics, 4th Edition Page 41 Chapter 11: Simultaneous Equations Models
The reduced-form equations are:
11.6 An Example of
Two-Stage Least
Squares Estimation
11.6.2 The Reduced-Form
Equations
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
Principles of Econometrics, 4th Edition Page 42 Chapter 11: Simultaneous Equations Models
11.6 An Example of
Two-Stage Least
Squares Estimation
11.6.2 The Reduced-Form
Equations
Table 11.1 Representative Truffle Data
Principles of Econometrics, 4th Edition Page 43 Chapter 11: Simultaneous Equations Models
11.6 An Example of
Two-Stage Least
Squares Estimation
11.6.2 The Reduced-Form
Equations
Table 11.2a Reduced Form for Quantity of Truffles (Q)
Principles of Econometrics, 4th Edition Page 44 Chapter 11: Simultaneous Equations Models
11.6 An Example of
Two-Stage Least
Squares Estimation
11.6.2 The Reduced-Form
Equations
Table 11.2b Reduced Form for Price of Truffles (P)
Principles of Econometrics, 4th Edition Page 45 Chapter 11: Simultaneous Equations Models
From Table 11.2b we have:
11.6 An Example of
Two-Stage Least
Squares Estimation
11.6.3 The Structural
Equations
12 22 32 42ˆ ˆ ˆ ˆ ˆ
32.512 1.708 7.603 1.354
P PS DI PF
PS DI PF
Principles of Econometrics, 4th Edition Page 46 Chapter 11: Simultaneous Equations Models
11.6 An Example of
Two-Stage Least
Squares Estimation
11.6.3 The Structural
Equations
Table 11.3a 2SLS Estimates for Truffle Demand
Principles of Econometrics, 4th Edition Page 47 Chapter 11: Simultaneous Equations Models
11.6 An Example of
Two-Stage Least
Squares Estimation
11.6.3 The Structural
Equations
Table 11.3b 2SLS Estimates for Truffle Supply
Principles of Econometrics, 4th Edition Page 48 Chapter 11: Simultaneous Equations Models
11.7
Supply and Demand at the Fulton
Fish Market
Principles of Econometrics, 4th Edition Page 49 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
If supply is fixed, with a vertical supply curve,
then price is demand-determined
– Higher demand leads to higher prices but no
increase in the quantity supplied
– If this is true, then the feedback between prices
and quantities is eliminated
– Such models are said to be recursive and the
demand equation can be estimated by ordinary
least squares rather than the more complicated
two-stage least squares procedure
Principles of Econometrics, 4th Edition Page 50 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
A key point is that ‘‘simultaneity’’ does not require
that events occur at a simultaneous moment in
time
– Specify the demand equation for this market as:
• α2 is the price elasticity of demand
– The supply equation is:
• β2 is the price elasticity of supply
1 2 3 4 5
6
ln ln
t t t t t
d
t t
QUAN PRICE MON TUE WED
THU e
Eq. 11.13
t 1 2 3ln ln s
t t tQUAN PRICE STORMY e Eq. 11.14
Principles of Econometrics, 4th Edition Page 51 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
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, and it does appear in the
supply equation
– In the supply equation, the four daily indicator
variables that are included in the demand
equation are omitted
11.7.1 Identification
Principles of Econometrics, 4th Edition Page 52 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
The reduced-form equations specify each
endogenous variable as a function of all
exogenous variables:
11.7.2 The Reduced-Form
Equations
11 21 31 41 51
61 1
ln
t t t t t
t t
QUAN MON TUE WED THU
STORMY v
12 22 32 42 52
62 2
ln
t t t t t
t t
PRICE MON TUE WED THU
STORMY v
Eq. 11.15
Eq. 11.16
Principles of Econometrics, 4th Edition Page 53 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
11.7.2 The Reduced-Form
Equations
Table 11.4a Reduced Form for ln(Quantity) Fish
Principles of Econometrics, 4th Edition Page 54 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
11.7.2 The Reduced-Form
Equations
Table 11.4b Reduced Form for ln(Price) Fish
Principles of Econometrics, 4th Edition Page 55 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
The key reduced-form equation is Eq. 11.16 for ln(PRICE):
– To identify the supply curve, the daily indicator variables must be jointly significant
– To identify the demand curve, the variable STORMY must be statistically significant
• The supply has a significant shift variable, so that we can reliably estimate the demand equation
• The two-stage least squares estimator performs very poorly if the shift variables are not strongly significant
11.7.2 The Reduced-Form
Equations
Principles of Econometrics, 4th Edition Page 56 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
To implement two-stage least squares, we take the
predicted value from the reduced-form regression
and include it in the structural equations in place
of the right-hand-side endogenous variable:
11.7.2 The Reduced-Form
Equations
12 22 32 42 52
62
ˆ ˆ ˆ ˆ ˆln
ˆ
t t t t t
t
PRICE MON TUE WED THU
STORMY
Principles of Econometrics, 4th Edition Page 57 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
If the coefficients on the daily indicator variables
are all identically zero, then:
– If we replace ln(PRICE) in the supply equation
(Eq. 11.14) with this predicted value, there will
be exact collinearity between
and the variable STORMY, which is already in
the supply equation
– Two-stage least squares will fail
11.7.2 The Reduced-Form
Equations
12 62ˆ ˆln t tPRICE STORMY
ln PRICE
Principles of Econometrics, 4th Edition Page 58 Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand
at the Fulton Fish
Market
11.7.3 Two-Stage Least
Squares Estimation
of Fish Demand
Table 11.5 2SLS Estimates for Fish Demand
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