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Principles of Econometrics, 4th Edition Page 1Chapter 11: Simultaneous Equations Models
Chapter 11Simultaneous Equations Models
Walter R. Paczkowski Rutgers University
Principles of Econometrics, 4th Edition Page 2Chapter 11: Simultaneous Equations Models
11.1 A Supply and Demand Model11.2 The Reduced-Form Equations11.3 The Failure of Least Squares Estimation11.4 The Identification Problem11.5 Two-Stage Least Squares Estimation11.6 An Example of Two-Stage Least Squares
Estimation11.7 Supply and Demand at the Fulton Fish
Market
Chapter Contents
Principles of Econometrics, 4th Edition Page 3Chapter 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 4Chapter 11: Simultaneous Equations Models
11.1 A Supply and Demand Model
Principles of Econometrics, 4th Edition Page 5Chapter 11: Simultaneous Equations Models
A very simple supply and demand model might look like:
11.1A 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 6Chapter 11: Simultaneous Equations Models
11.1A Supply and
Demand ModelFIGURE 11.1 Supply and demand equilibrium
Principles of Econometrics, 4th Edition Page 7Chapter 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
11.1A Supply and
Demand Model
Principles of Econometrics, 4th Edition Page 8Chapter 11: Simultaneous Equations Models
The endogenous variables P and Q are dependent variables and both are random variablesThe 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
11.1A Supply and
Demand Model
Principles of Econometrics, 4th Edition Page 9Chapter 11: Simultaneous Equations Models
For the error terms, we have:
11.1A 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 10Chapter 11: Simultaneous Equations Models
An ‘‘influence diagram’’ is a graphical representation of relationships between model components
11.1A Supply and
Demand Model
Principles of Econometrics, 4th Edition Page 11Chapter 11: Simultaneous Equations Models
11.1A Supply and
Demand Model FIGURE 11.2 Influence diagrams for two regression models
Principles of Econometrics, 4th Edition Page 12Chapter 11: Simultaneous Equations Models
11.1A Supply and
Demand Model FIGURE 11.3 Influence diagram for a simultaneous equations model
Principles of Econometrics, 4th Edition Page 13Chapter 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
11.1A Supply and
Demand Model
Principles of Econometrics, 4th Edition Page 14Chapter 11: Simultaneous Equations Models
11.2 The Reduced-Form Equations
Principles of Econometrics, 4th Edition Page 15Chapter 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
11.2The Reduced-Form
Equations
Principles of Econometrics, 4th Edition Page 16Chapter 11: Simultaneous Equations Models
To solve for P, set Q in the demand and supply equations to be equal:
– Solve for P:
11.2The 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
(11.4)
Principles of Econometrics, 4th Edition Page 17Chapter 11: Simultaneous Equations Models
Solving for Q:
– 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
11.2The 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
(11.5)
Principles of Econometrics, 4th Edition Page 18Chapter 11: Simultaneous Equations Models
11.3 The Failure of Least Squares
Estimation
Principles of Econometrics, 4th Edition Page 19Chapter 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
11.3The Failure of Least Squares Estimation
Principles of Econometrics, 4th Edition Page 20Chapter 11: Simultaneous Equations Models
11.4 The Identification Problem
Principles of Econometrics, 4th Edition Page 21Chapter 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
11.4The Identification
Problem
Principles of Econometrics, 4th Edition Page 22Chapter 11: Simultaneous Equations Models
11.4The Identification
Problem FIGURE 11.4 The effect of changing income
Principles of Econometrics, 4th Edition Page 23Chapter 11: Simultaneous Equations Models
It is the absence of variables in one equation that are present in another equation that makes parameter estimation possible
11.4The Identification
Problem
Principles of Econometrics, 4th Edition Page 24Chapter 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
11.4The Identification
Problem
Principles of Econometrics, 4th Edition Page 25Chapter 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
11.4The Identification
Problem
Principles of Econometrics, 4th Edition Page 26Chapter 11: Simultaneous Equations Models
11.5 Two-Stage Least Squares Estimation
Principles of Econometrics, 4th Edition Page 27Chapter 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
11.5Two-Stage Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 28Chapter 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.
11.5Two-Stage Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 29Chapter 11: Simultaneous Equations Models
The reduced-form model is:
Suppose we know π1.– Then through substitution:
11.5Two-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 30Chapter 11: Simultaneous Equations Models
We can estimate π1 using from the reduced-form equation for P – A consistent estimator for E(P) is:
– Then:
11.5Two-Stage Least
Squares Estimation
Eq. 11.8
1̂
1ˆ ˆ P X
1 *ˆQ P e
Principles of Econometrics, 4th Edition Page 31Chapter 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
11.5Two-Stage Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 32Chapter 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
11.5Two-Stage Least
Squares Estimation
P̂
P̂
Principles of Econometrics, 4th Edition Page 33Chapter 11: Simultaneous Equations Models
Suppose the first structural equation in a system of M simultaneous equations is:
11.5Two-Stage Least
Squares Estimation
11.5.1The 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 34Chapter 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
11.5Two-Stage Least
Squares Estimation
11.5.1The 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 vy x x x v
2 12 1 22 2 2
3 13 1 23 2 3
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ
K K
K K
y x x xy x x x
Eq. 11.10
Principles of Econometrics, 4th Edition Page 35Chapter 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
11.5Two-Stage Least
Squares Estimation
11.5.1The 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 36Chapter 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
11.5Two-Stage Least
Squares Estimation
11.5.2The Properties of the
Two-Stage Least Squares Estimators
Principles of Econometrics, 4th Edition Page 37Chapter 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
11.5Two-Stage Least
Squares Estimation
11.5.2The Properties of the
Two-Stage Least Squares Estimators
Principles of Econometrics, 4th Edition Page 38Chapter 11: Simultaneous Equations Models
11.6 An Example of Two-Stage Least
Squares Estimation
Principles of Econometrics, 4th Edition Page 39Chapter 11: Simultaneous Equations Models
Consider a supply and demand model for truffles:
11.6An Example of Two-Stage Least Squares
Estimation
1 2 3 4Demand: di i i i iQ P PS DI e
1 2 3Supply: si i i iQ P PF e
Eq. 11.11
Eq. 11.12
Principles of Econometrics, 4th Edition Page 40Chapter 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
11.6An Example of Two-Stage Least Squares
Estimation
11.6.1Identification
Principles of Econometrics, 4th Edition Page 41Chapter 11: Simultaneous Equations Models
The reduced-form equations are:
11.6An Example of Two-Stage Least Squares
Estimation
11.6.2The 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 vP PS DI PF v
Principles of Econometrics, 4th Edition Page 42Chapter 11: Simultaneous Equations Models
11.6An Example of Two-Stage Least Squares
Estimation
11.6.2The Reduced-Form
Equations
Table 11.1 Representative Truffle Data
Principles of Econometrics, 4th Edition Page 43Chapter 11: Simultaneous Equations Models
11.6An Example of Two-Stage Least Squares
Estimation
11.6.2The Reduced-Form
Equations
Table 11.2a Reduced Form for Quantity of Truffles (Q)
Principles of Econometrics, 4th Edition Page 44Chapter 11: Simultaneous Equations Models
11.6An Example of Two-Stage Least Squares
Estimation
11.6.2The Reduced-Form
Equations
Table 11.2b Reduced Form for Price of Truffles (P)
Principles of Econometrics, 4th Edition Page 45Chapter 11: Simultaneous Equations Models
From Table 11.2b we have:
11.6An Example of Two-Stage Least Squares
Estimation
11.6.3The Structural
Equations
12 22 32 42ˆ ˆ ˆ ˆ ˆ
32.512 1.708 7.603 1.354P PS DI PF
PS DI PF
Principles of Econometrics, 4th Edition Page 46Chapter 11: Simultaneous Equations Models
11.6An Example of Two-Stage Least Squares
Estimation
11.6.3The Structural
Equations
Table 11.3a 2SLS Estimates for Truffle Demand
Principles of Econometrics, 4th Edition Page 47Chapter 11: Simultaneous Equations Models
11.6An Example of Two-Stage Least Squares
Estimation
11.6.3The Structural
Equations
Table 11.3b 2SLS Estimates for Truffle Supply
Principles of Econometrics, 4th Edition Page 48Chapter 11: Simultaneous Equations Models
11.7 Supply and Demand at the Fulton
Fish Market
Principles of Econometrics, 4th Edition Page 49Chapter 11: Simultaneous Equations Models
11.7Supply 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 50Chapter 11: Simultaneous Equations Models
11.7Supply 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
dt t
QUAN PRICE MON TUE WED
THU e
Eq. 11.13
t 1 2 3ln ln st t tQUAN PRICE STORMY e Eq. 11.14
Principles of Econometrics, 4th Edition Page 51Chapter 11: Simultaneous Equations Models
11.7Supply 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
11.7.1Identification
Principles of Econometrics, 4th Edition Page 52Chapter 11: Simultaneous Equations Models
11.7Supply and Demand
at the Fulton Fish Market
The reduced-form equations specify each endogenous variable as a function of all exogenous variables:
11.7.2The 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 53Chapter 11: Simultaneous Equations Models
11.7Supply and Demand
at the Fulton Fish Market
11.7.2The Reduced-Form
Equations
Table 11.4a Reduced Form for ln(Quantity) Fish
Principles of Econometrics, 4th Edition Page 54Chapter 11: Simultaneous Equations Models
11.7Supply and Demand
at the Fulton Fish Market
11.7.2The Reduced-Form
Equations
Table 11.4b Reduced Form for ln(Price) Fish
Principles of Econometrics, 4th Edition Page 55Chapter 11: Simultaneous Equations Models
11.7Supply 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
11.7.2The Reduced-Form
Equations
Principles of Econometrics, 4th Edition Page 56Chapter 11: Simultaneous Equations Models
11.7Supply 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:
11.7.2The Reduced-Form
Equations
12 22 32 42 52
62
ˆ ˆ ˆ ˆ ˆlnˆ
t t t t t
t
PRICE MON TUE WED THUSTORMY
Principles of Econometrics, 4th Edition Page 57Chapter 11: Simultaneous Equations Models
11.7Supply 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
11.7.2The Reduced-Form
Equations
12 62ˆ ˆln t tPRICE STORMY
ln PRICE
Principles of Econometrics, 4th Edition Page 58Chapter 11: Simultaneous Equations Models
11.7Supply and Demand
at the Fulton Fish Market
11.7.3Two-Stage Least
Squares Estimation of Fish Demand
Table 11.5 2SLS Estimates for Fish Demand
Principles of Econometrics, 4th Edition Page 59Chapter 11: Simultaneous Equations Models
Key Words
Principles of Econometrics, 4th Edition Page 60Chapter 11: Simultaneous Equations Models
endogenous variablesexogenous variablesidentification
Keywords
reduced-form equationreduced-form errorsreduced-form parameters
simultaneous equationsstructural parameterstwo-stage least squares
Principles of Econometrics, 4th Edition Page 61Chapter 11: Simultaneous Equations Models
Appendices
Principles of Econometrics, 4th Edition Page 62Chapter 11: Simultaneous Equations Models
The covariance between P and es for the supply and demand model is:
11AAn Algebraic
Explanation of the Failure of
Least Squares
Eq. 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
sd 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
Principles of Econometrics, 4th Edition Page 63Chapter 11: Simultaneous Equations Models
The least squares estimator of the supply equation is:
– Substituting, we get:
where
11AAn Algebraic
Explanation of the Failure of
Least Squares
Eq. 11A.2 1 2i i
i
PQb
P
11 1 12 2
i i si isi i si
i i
P P e Pb e h eP P
Eq. 11A.3
2i i ih P P
Principles of Econometrics, 4th Edition Page 64Chapter 11: Simultaneous Equations Models
The expected value of the least squares estimator is:
– Also:
11AAn Algebraic
Explanation of the Failure of
Least Squares
1 1 1i siE b E h e
21
21
1 2 2
s
s
s
PQ P Pe
E PQ E P E Pe
E PQ E PeE P E P
Principles of Econometrics, 4th Edition Page 65Chapter 11: Simultaneous Equations Models
In large samples, as N→∞,
11AAn Algebraic
Explanation of the Failure of
Least Squares
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
Principles of Econometrics, 4th Edition Page 66Chapter 11: Simultaneous Equations Models
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
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 67Chapter 11: Simultaneous Equations Models
Suppose the first structural equation within a system is:
– If this equation is identified, then its parameters can be estimated
11B.1The k-Class of
Estimators
1 2 2 1 1 2 2 1α β βy y x x e Eq. 11B.1
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 68Chapter 11: Simultaneous Equations Models
The parameters of the reduced-form equation are consistently estimated by least squares, so that:
11B.1The k-Class of
Estimators
2 12 1 22 2 2ˆ ˆ ˆπ π πK KE y x x x Eq. 11B.2
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 69Chapter 11: Simultaneous Equations Models
The reduced form residuals are:
– Or:
– The two-stage least squares estimator is an IV estimator using as an instrument
11B.1The k-Class of
Estimators
2 2 2v̂ y E y
Eq. 11B.3 2 2 2ˆE y y v
2E y
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 70Chapter 11: Simultaneous Equations Models
The k-class of estimators is a unifying framework.– A k-class estimator is an IV estimator using
instrumental variable – It is called a class of estimators because it
represents the least squares estimator if k = 0 and the 2SLS estimator if k = 1
11B.1The k-Class of
Estimators
2 2ˆy kv
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 71Chapter 11: Simultaneous Equations Models
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
11B.2The LIML Estimator
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 72Chapter 11: Simultaneous Equations Models
The equation
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
– 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
11B.2The LIML Estimator
1 2 2 1 1 2 2 1α β βy y x x e
1 1 2 2 1 1 2 2 1α α β β 0y y x x e
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 73Chapter 11: Simultaneous Equations Models
We can write:
– The exogenous variables x3, ... , xK were omitted
– If included, then:
11B.2The LIML Estimator
*1 1 2 2 1 1 1 2 2β β θ θy x x e x x Eq. 11B.4
*1 1θ θK Ky x x Eq. 11B.5
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 74Chapter 11: Simultaneous Equations Models
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
11B.2The LIML Estimator
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 75Chapter 11: Simultaneous Equations Models
Define:
– The minimum value of , call it , results in the LIML estimator when used as k in the k-class estimator
– That is, use when forming the instrument and the resulting IV estimator is the
LIML estimator
11B.2The LIML Estimator
*1 2
*1
from regression of on , 1 from regression of on , , K
SSE y x xlSSE y x x
Eq. 11B.6
l l̂
ˆk l
2 2ˆy kv
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 76Chapter 11: Simultaneous Equations Models
A uses the k-class value:
– The value a is a constant
11B.2.1Fuller’s Modified
LIML
Eq. 11B.7 ˆ ak lN K
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 77Chapter 11: Simultaneous Equations Models
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 , then the mean square error of estimation is
11B.2.1Fuller’s Modified
LIML
2
2
2
ˆ ˆδ δ δ
ˆ ˆvar δ δ δ
ˆ ˆvar δ δ
MSE E
E
bias
δ̂
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 78Chapter 11: Simultaneous Equations Models
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
11B.2.2Advantages of
LIML
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 79Chapter 11: Simultaneous Equations Models
11B.2.3Stock-Yogo Weak IV Tests for LIML
Table 11B.1 Critical Values for the Weak Instrument Test Based on LIML Test Size (5% level of significance)
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 80Chapter 11: Simultaneous Equations Models
11B.2.3Stock-Yogo Weak IV Tests for LIML
Table 11B.2 Critical Values for the Weak Instrument Test Based on Fuller-k Relative Bias (5% level of significance)
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 81Chapter 11: Simultaneous Equations Models
We estimate the HOURS supply equation:
The LIML estimates are given in Table 11B.3– The models we consider are:
11B.2.3aTesting for Weak Instruments with
LIML
1 2 3 4 5β β β β 6 βHOURS MTR EDUC KIDSL NWIFEINC e Eq. 11B.8
2
Model 1: endogenous: ; :
Model 2: endogenous: ; : , ,Model 3: endogenous: , ; : ,Model 4: endogenous: , ; : , ,
MTR IV EXPER
MTR IV EXPER EXPER LARGECITYMTR EDUC IV MOTHEREDUC FATHEREDUCMTR EDUC IV MOTHEREDUC FATHEREDUC EXPER
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 82Chapter 11: Simultaneous Equations Models
11B.2.3aTesting for Weak Instruments with
LIML
Table 11B.3 LIML Estimations11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 83Chapter 11: Simultaneous Equations Models
11B.2.3bTesting for Weak Instruments with Fuller Modified
LIML
Table 11B.4 Fuller (a = 1) Estimations11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 84Chapter 11: Simultaneous Equations Models
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
11B.3Monte Carlo Simulation
Results
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 85Chapter 11: Simultaneous Equations Models
11B.3Monte Carlo Simulation
Results
Table 11B.5 Monte Carlo Simulation Results11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 86Chapter 11: Simultaneous Equations Models
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:
11B.3Monte Carlo Simulation
Results
2100002 2 21
ˆ ˆβ β β 10,000mmMSE
11B2SLS
Alternatives
Principles of Econometrics, 4th Edition Page 87Chapter 11: Simultaneous Equations Models
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
11B.3Monte Carlo Simulation
Results
11B2SLS
Alternatives
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