Chapter 11
Simultaneous Equations Models
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 1
Chapter Contents






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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 2
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
Chapter 11: Simultaneous Equations Models
Page 3
11.1
A Supply and Demand Model
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 4
11.1
A Supply and
Demand Model
A very simple supply and demand model might
look like:
Eq. 11.1
Demand: Q  1P  2 X  ed
Eq. 11.2
Supply: Q  1P  es
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 5
11.1
A Supply and
Demand Model
Principles of Econometrics, 4th Edition
FIGURE 11.1 Supply and demand equilibrium
Chapter 11: Simultaneous Equations Models
Page 6
11.1
A Supply and
Demand Model
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 7
11.1
A Supply and
Demand Model
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 8
11.1
A Supply and
Demand Model
For the error terms, we have:
E (ed )  0, var(ed )   d2
Eq. 11.3
E (es )  0, var(es )   2s
cov(ed , es )  0
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 9
11.1
A Supply and
Demand Model
An ‘‘influence diagram’’ is a graphical
representation of relationships between model
components
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 10
11.1
A Supply and
Demand Model
FIGURE 11.2 Influence diagrams for two regression models
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 11
11.1
A Supply and
Demand Model
FIGURE 11.3 Influence diagram for a simultaneous equations model
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 12
11.1
A Supply and
Demand Model
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 13
11.2
The Reduced-Form Equations
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 14
11.2
The Reduced-Form
Equations
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 15
11.2
The Reduced-Form
Equations
To solve for P, set Q in the demand and supply
equations to be equal:
1P  es  1P  2 X  ed
– Solve for P:
Eq. 11.4
2
ed  es
P
X
1  1 
1  1 
 1 X  v1
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
(11.4)
Page 16
11.2
The Reduced-Form
Equations
Solving for Q:
Q  1P  es
 2
ed  es 
 1 
X
  es
1  1  
  1  1 
Eq. 11.5
1 2
1ed  1es

X
1  1 
1  1 
(11.5)
 2 X  v2
– The parameters π1 and π2 in Eqs. 11.4 and 11.5 are
called reduced-form parameters. The error terms
v1 and v2 are called reduced-form errors
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 17
11.3
The Failure of Least Squares
Estimation
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 18
11.3
The Failure of Least
Squares Estimation
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, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 19
11.4
The Identification Problem
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 20
11.4
The Identification
Problem
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 21
11.4
The Identification
Problem
Principles of Econometrics, 4th Edition
FIGURE 11.4 The effect of changing income
Chapter 11: Simultaneous Equations Models
Page 22
11.4
The Identification
Problem
It is the absence of variables in one equation that
are present in another equation that makes
parameter estimation possible
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 23
11.4
The Identification
Problem
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 24
11.4
The Identification
Problem
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 25
11.5
Two-Stage Least Squares Estimation
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 26
11.5
Two-Stage Least
Squares Estimation
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 27
11.5
Two-Stage Least
Squares Estimation
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 righthand side of the equation is correlated with the
error term es.
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 28
11.5
Two-Stage Least
Squares Estimation
The reduced-form model is:
P  E ( P)  v1  1 X  v1
Eq. 11.6
Suppose we know π1.
– Then through substitution:
Q  1  E  P   v1   es
Eq. 11.7
Principles of Econometrics, 4th Edition
 1 E  P    1v1  es 
Chapter 11: Simultaneous Equations Models
Page 29
11.5
Two-Stage Least
Squares Estimation
We can estimate π1 using ˆ1 from the reducedform equation for P
– A consistent estimator for E(P) is:
Pˆ  ˆ 1 X
– Then:
Eq. 11.8
Principles of Econometrics, 4th Edition
Q  1Pˆ  e*
Chapter 11: Simultaneous Equations Models
Page 30
11.5
Two-Stage Least
Squares Estimation
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 31
11.5
Two-Stage Least
Squares Estimation
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 Pˆ
2. 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, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 32
11.5
Two-Stage Least
Squares Estimation
11.5.1
The General TwoStage Least Squares
Estimation
Procedure
Suppose the first structural equation in a system of
M simultaneous equations is:
Eq. 11.9
Principles of Econometrics, 4th Edition
y1  2 y2  3 y3  1 x1  2 x2  e1
Chapter 11: Simultaneous Equations Models
Page 33
11.5
Two-Stage Least
Squares Estimation
11.5.1
The General TwoStage Least Squares
Estimation
Procedure
If this equation is identified, then its parameters
can be estimated in the two steps:
1. Estimate the parameters of the reduced-form
equations
y2  12 x1  22 x2 
  K 2 xK  v2
y3  13 x1  23 x2 
  K 3 xK  v3
by least squares and obtain the predicted
values
Eq. 11.10
Principles of Econometrics, 4th Edition
yˆ 2  ˆ 12 x1  ˆ 22 x2 
yˆ3  ˆ 13 x1  ˆ 23 x2 
Chapter 11: Simultaneous Equations Models
 ˆ K 2 xK
 ˆ K 3 xK
Page 34
11.5
Two-Stage Least
Squares Estimation
11.5.1
The General TwoStage Least Squares
Estimation
Procedure
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:
y1  2 yˆ2  3 yˆ3  1 x1  2 x2  e1*
• Estimate the parameters of this equation by
least squares
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 35
11.5
Two-Stage Least
Squares Estimation
11.5.2
The Properties of the
Two-Stage Least
Squares Estimators
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 36
11.5
Two-Stage Least
Squares Estimation
11.5.2
The Properties of the
Two-Stage Least
Squares Estimators
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 37
11.6
An Example of Two-Stage Least
Squares Estimation
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 38
11.6
An Example of
Two-Stage Least
Squares Estimation
Consider a supply and demand model for truffles:
Eq. 11.11
Demand: Qi  1  2 Pi  3 PSi  4 DIi  eid
Eq. 11.12
Supply: Qi  1  2 Pi  3 PFi  eis
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 39
11.6
An Example of
Two-Stage Least
Squares Estimation
11.6.1
Identification
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 40
11.6
An Example of
Two-Stage Least
Squares Estimation
11.6.2
The Reduced-Form
Equations
The reduced-form equations are:
Qi  11  21 PSi  31 DI i  41PFi  vi1
Pi  12  22 PSi  32 DI i  42 PFi  vi 2
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 41
11.6
An Example of
Two-Stage Least
Squares Estimation
Table 11.1 Representative Truffle Data
11.6.2
The Reduced-Form
Equations
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 42
11.6
An Example of
Two-Stage Least
Squares Estimation
Table 11.2a Reduced Form for Quantity of Truffles (Q)
11.6.2
The Reduced-Form
Equations
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 43
11.6
An Example of
Two-Stage Least
Squares Estimation
Table 11.2b Reduced Form for Price of Truffles (P)
11.6.2
The Reduced-Form
Equations
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 44
11.6
An Example of
Two-Stage Least
Squares Estimation
11.6.3
The Structural
Equations
From Table 11.2b we have:
Pˆ  ˆ 12  ˆ 22 PS  ˆ 32 DI  ˆ 42 PF
 32.512  1.708PS  7.603DI  1.354PF
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 45
11.6
An Example of
Two-Stage Least
Squares Estimation
Table 11.3a 2SLS Estimates for Truffle Demand
11.6.3
The Structural
Equations
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 46
11.6
An Example of
Two-Stage Least
Squares Estimation
Table 11.3b 2SLS Estimates for Truffle Supply
11.6.3
The Structural
Equations
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 47
11.7
Supply and Demand at the Fulton
Fish Market
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 48
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
Chapter 11: Simultaneous Equations Models
Page 49
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:
Eq. 11.13
Eq. 11.14
ln  QUANt   1  2 ln  PRICEt   3 MONt  4TUEt  5WEDt
 6THUt  etd
• α2 is the price elasticity of demand
– The supply equation is:
ln  QUANt   1 2 ln  PRICEt  3STORMYt  ets
• β2 is the price elasticity of supply
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 50
11.7
Supply and Demand
at the Fulton Fish
Market
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, 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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 51
11.7
Supply and Demand
at the Fulton Fish
Market
11.7.2
The Reduced-Form
Equations
The reduced-form equations specify each
endogenous variable as a function of all
exogenous variables:
Eq. 11.15
Eq. 11.16
ln  QUANt   11  21MONt  31TUEt  41WEDt  51THU t
 61STORMYt  vt1
ln  PRICEt   12  22 MONt  32TUEt  42WEDt  52THU t
Principles of Econometrics, 4th Edition
 62 STORMYt  vt 2
Chapter 11: Simultaneous Equations Models
Page 52
11.7
Supply and Demand
at the Fulton Fish
Market
Table 11.4a Reduced Form for ln(Quantity) Fish
11.7.2
The Reduced-Form
Equations
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 53
11.7
Supply and Demand
at the Fulton Fish
Market
Table 11.4b Reduced Form for ln(Price) Fish
11.7.2
The Reduced-Form
Equations
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 54
11.7
Supply and Demand
at the Fulton Fish
Market
11.7.2
The Reduced-Form
Equations
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
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 55
11.7
Supply and Demand
at the Fulton Fish
Market
11.7.2
The Reduced-Form
Equations
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:
ln  PRICEt   ˆ 12  ˆ 22 MONt  ˆ 32TUEt  ˆ 42WEDt  ˆ 52THU t
 ˆ 62 STORMYt
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 56
11.7
Supply and Demand
at the Fulton Fish
Market
11.7.2
The Reduced-Form
Equations
If the coefficients on the daily indicator variables
are all identically zero, then:
ln  PRICEt   ˆ 12  ˆ 62 STORMYt
– If we replace ln(PRICE) in the supply equation
(Eq. 11.14) with this predicted value, there will
be exact collinearity between ln  PRICE 
and the variable STORMY, which is already in
the supply equation
– Two-stage least squares will fail
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 57
11.7
Supply and Demand
at the Fulton Fish
Market
Table 11.5 2SLS Estimates for Fish Demand
11.7.3
Two-Stage Least
Squares Estimation
of Fish Demand
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 58
Key Words
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 59
Keywords
endogenous
variables
exogenous
variables
identification
Principles of Econometrics, 4th Edition
reduced-form
equation
reduced-form
errors
reduced-form
parameters
Chapter 11: Simultaneous Equations Models
simultaneous
equations
structural
parameters
two-stage least
squares
Page 60
Appendices
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 61
11A
An Algebraic
Explanation of the
Failure of Least
Squares
The covariance between P and es for the supply
and demand model is:
cov  P, es   E  P  E  P   es  E  es  
Eq. 11A.1
 E  Pes 
[since E  es   0]
 E  1 X  v1  es
[substitute for P]
e e 
 E  d s  es
 1  1 

 E  es2 
1  1
[since 1 X is exogenous]
[since ed , es assumed uncorrelated]
 2s

0
1  1
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 62
11A
An Algebraic
Explanation of the
Failure of Least
Squares
The least squares estimator of the supply equation
is:
PQ

i i
b1 
2
P
 i
Eq. 11A.2
– Substituting, we get:
Eq. 11A.3
 Pi 
Pi 1Pi  esi 

b1 
 1   
e  1   hi esi
2
2  si
 Pi
  Pi 
where
Principles of Econometrics, 4th Edition
hi  Pi
2
P
i
Chapter 11: Simultaneous Equations Models
Page 63
11A
An Algebraic
Explanation of the
Failure of Least
Squares
The expected value of the least squares estimator
is:
E  b1   1   E  hiesi   1
– Also:
PQ  1P 2  Pes
E  PQ   1E  P 2   E  Pes 
1 
Principles of Econometrics, 4th Edition
E  PQ 
EP
2


E  Pes 
E  P2 
Chapter 11: Simultaneous Equations Models
Page 64
11A
An Algebraic
Explanation of the
Failure of Least
Squares
In large samples, as N→∞,
2
E
PQ
E
Pe

Q
P
/
N





s
s  1  1 
i i
b1 

 1 
 1 
 1
2
2
2
2
EP 
EP 
EP 
 Pi / N
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 65
11B
2SLS Alternatives
There has always been great interest in alternatives
to the standard IV/2SLS estimator
– The search for better alternatives was energized
by the discovery of the problems weak
instruments pose for the usual IV/2SLS
estimator
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 66
11B
2SLS Alternatives
11B.1
The k-Class of
Estimators
Suppose the first structural equation within a
system is:
y1  α2 y2  β1 x1  β2 x2  e1
Eq. 11B.1
– If this equation is identified, then its parameters
can be estimated
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 67
11B
2SLS Alternatives
11B.1
The k-Class of
Estimators
The parameters of the reduced-form equation are
consistently estimated by least squares, so that:
Eq. 11B.2
Principles of Econometrics, 4th Edition
E  y2   πˆ 12 x1  πˆ 22 x2 
Chapter 11: Simultaneous Equations Models
 πˆ K 2 xK
Page 68
11B
2SLS Alternatives
11B.1
The k-Class of
Estimators
The reduced form residuals are:
– Or:
Eq. 11B.3
vˆ2  y2  E  y2 
E  y2   y2  vˆ2
– The two-stage least squares estimator is an IV
estimator using E  y2  as an instrument
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 69
11B
2SLS Alternatives
11B.1
The k-Class of
Estimators
The k-class of estimators is a unifying framework.
– A k-class estimator is an IV estimator using
instrumental variable y2  kvˆ2
– It is called a class of estimators because it
represents the least squares estimator if k = 0
and the 2SLS estimator if k = 1
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 70
11B
2SLS Alternatives
11B.2
The LIML
Estimator
The LIML estimator is one of the oldest estimators
for an equation within a system of simultaneous
equations, or any equation with an endogenous
variable on the righthand side
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 71
11B
2SLS Alternatives
11B.2
The LIML
Estimator
The equation
y1  α2 y2  β1 x1  β2 x2  e1
is in normalized form, meaning that we have
chosen one variable to appear as the dependent
variable
– In general the first equation can be written in
implicit form as
α1 y1  α2 y2  β1x1  β2 x2  e1  0
– There is no rule that says y1has to be the
dependent variable in the first equation.
– Normalization amounts to setting α1or α2 to the
value -1
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 72
11B
2SLS Alternatives
11B.2
The LIML
Estimator
We can write:
y*  β1x1  β2 x2  e1  θ1x1  θ2 x2 
Eq. 11B.4
– The exogenous variables x3, ... , xK were
omitted
– If included, then:
Eq. 11B.5
Principles of Econometrics, 4th Edition
y*  θ1x1 
 θK xK 
Chapter 11: Simultaneous Equations Models
Page 73
11B
2SLS Alternatives
11B.2
The LIML
Estimator
The least variance ratio estimator chooses α1and
α2 so that the ratio of the sum of squared residuals
from Eq. 11B.4 relative to the sum of squared
residuals from Eq. 11B.5 is as small as possible
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 74
11B
2SLS Alternatives
11B.2
The LIML
Estimator
Define:
SSE from regression of y* on x1 , x2
l
1
*
SSE from regression of y on x1 , , xK
Eq. 11B.6
– The minimum value of l , call it lˆ , results in
the LIML estimator when used as k in the kclass estimator
– That is, use k  lˆ when forming the instrument
y2  kvˆ2 and the resulting IV estimator is the
LIML estimator
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 75
11B
2SLS Alternatives
11B.2.1
Fuller’s Modified
LIML
A uses the k-class value:
k  lˆ 
Eq. 11B.7
a
N K
– The value a is a constant
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 76
11B
2SLS Alternatives
11B.2.1
Fuller’s Modified
LIML
A value a = 4 leads to an estimator that minimizes
the ‘‘mean square error’’ of estimation
– If we are estimating some parameter δ using an
estimator δˆ , then the mean square error of
estimation is
  
 var  δˆ    E  δˆ   δ 


 var  δˆ   bias  δˆ  


MSE δˆ  E δˆ  δ
2
2
2
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 77
11B
2SLS Alternatives
11B.2.2
Advantages of
LIML
Some findings are:
– For the 2SLS estimator the amount of bias is an
increasing function of the degree of over
identification.
• The distributions of the 2SLS and least
squares estimators tend to become similar
when overidentification is large.
• LIML has the advantage over 2SLS when
there are a large number of instruments.
– The LIML estimator converges to normality
faster than the 2SLS estimator and is generally
more symmetric
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 78
11B
2SLS Alternatives
Table 11B.1 Critical Values for the Weak Instrument Test Based
on LIML Test Size (5% level of significance)
11B.2.3
Stock-Yogo Weak
IV Tests for LIML
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 79
11B
2SLS Alternatives
Table 11B.2 Critical Values for the Weak Instrument Test Based
on Fuller-k Relative Bias (5% level of significance)
11B.2.3
Stock-Yogo Weak
IV Tests for LIML
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 80
11B
2SLS Alternatives
11B.2.3a
Testing for Weak
Instruments with
LIML
Eq. 11B.8
We estimate the HOURS supply equation:
HOURS  β1  β2 MTR  β3 EDUC  β4 KIDSL6  β5 NWIFEINC  e
The LIML estimates are given in Table 11B.3
– The models we consider are:
Model 1: endogenous: MTR; IV : EXPER
Model 2: endogenous: MTR; IV : EXPER, EXPER 2 , LARGECITY
Model 3: endogenous: MTR, EDUC; IV : MOTHEREDUC , FATHEREDUC
Model 4: endogenous: MTR, EDUC; IV : MOTHEREDUC , FATHEREDUC , EXPER
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 81
11B
2SLS Alternatives
Table 11B.3 LIML Estimations
11B.2.3a
Testing for Weak
Instruments with
LIML
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 82
11B
2SLS Alternatives
Table 11B.4 Fuller (a = 1) Estimations
11B.2.3b
Testing for Weak
Instruments with
Fuller Modified
LIML
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 83
11B
2SLS Alternatives
11B.3
Monte Carlo
Simulation Results
We previously carried out a Monte Carlo
simulation to explore the properties of the
IV/2SLS estimators
– Now we employ the same experiment, adding
aspects of the new estimators we have
introduced
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 84
11B
2SLS Alternatives
Table 11B.5 Monte Carlo Simulation Results
11B.3
Monte Carlo
Simulation Results
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 85
11B
2SLS Alternatives
11B.3
Monte Carlo
Simulation Results
The mean square error measures how close the
estimates are to the true parameter value
– For the IV/2SLS estimator, the empirical mean
square error is:
 

10000
ˆ
MSE β2  m1 βˆ 2m  β2
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models

2
10,000
Page 86
11B
2SLS Alternatives
11B.3
Monte Carlo
Simulation Results
We can conclude that the Fuller modified LIML
has lower mean square error than the IV/2SLS
estimator in each experiment, and when the
instruments are weak, the improvement is large
Principles of Econometrics, 4th Edition
Chapter 11: Simultaneous Equations Models
Page 87