Chapter 6: Endogeneity and Instrumental Variables
(IV) estimator
Advanced Econometrics - HEC Lausanne
Christophe Hurlin
University of Orléans
December 15, 2013
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Section 1
Introduction
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1. Introduction
The outline of this chapter is the following:
Section 2. Endogeneity
Section 3. Instrumental Variables (IV) estimator
Section 4. Two-Stage Least Squares (2SLS)
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1. Introduction
References
Amemiya T. (1985), Advanced Econometrics. Harvard University Press.
Greene W. (2007), Econometric Analysis, sixth edition, Pearson - Prentice
Hil (recommended)
Pelgrin, F. (2010), Lecture notes Advanced Econometrics, HEC Lausanne (a
special thank)
Ruud P., (2000) An introduction to Classical Econometric Theory, Oxford
University Press.
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1. Introduction
Notations: In this chapter, I will (try to...) follow some conventions of
notation.
fY ( y )
probability density or mass function
FY ( y )
cumulative distribution function
Pr ()
probability
y
vector
Y
matrix
Be careful: in this chapter, I don’t distinguish between a random vector
(matrix) and a vector (matrix) of deterministic elements (except in section
2). For more appropriate notations, see:
Abadir and Magnus (2002), Notation in econometrics: a proposal for a
standard, Econometrics Journal.
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Section 2
Endogeneity
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2. Endogeneity
Objectives
The objective of this section are the following:
1
To de…ne the endogeneity issue
2
To study the sources of endogeneity
3
To show the inconsistency of the OLS estimator (endogeneity bias)
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2. Endogeneity
Objectives in this chapter, we assume that the assumption A3
(exogeneity) is violated:
E ( εj X) 6= 0N
1
but the disturbances are spherical:
V ( ε j X ) = σ 2 IN
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2. Endogeneity
The reasons for suspecting E ( εj X) 6= 0 are varied:
1
Errors-in-variables
2
Jointly endogenous variables: the usual example is running
quantities on prices to estimate a demand equation (supply also
a¤ects the determination of equilibrium).
3
Omitted variables: one or more columns in X cannot be included in
the regression because no data on those variables are
available— estimation will be altered to the extent that the missing
variables and the included ones are correlated
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2. Endogeneity
1. Error-in-variables
1
Consider the regression model:
yi = xi > β + εi
2
where E ( εi j xi ) = 0.
One does not observe (y , x ) but (y , x)
yi = yi + vi
xi = xi + wi
with
E (vi ) = E (vi εi ) = E (vi yi ) = E wi> xi
=0
E (wi ) = E (vi wi ) = E (wi εi ) = E (wi yi ) = E (vi xi ) = 0
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2. Endogeneity
1. Error-in-variables (cont’d)
1
The mismeasured regression equation is given by:
yi = xi > β + εi
() yi = xi> β + εi
vi + wi> β
() yi = xi> β + η i
with η i = εi
2
vi + wi> β.
The composite error term η i is not orthogonal to the mismeasured
independent variable xi .
E (η i xi ) 6= 0
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2. Endogeneity
1. Error-in-variables (cont’d)
Indeed, we have:
η i = εi
vi + wi> β.
As a consequence:
E (η i xi ) = E (εi xi )
E (vi xi ) + E wi> β xi
= E wi> β xi
E (η i xi ) 6= 0
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2. Endogeneity
2. Simultaneous equation bias
Consider the demand equation
qd = α1 p + α2 y + ud
where qd , p and y denote respectively the quantity, the price and income.
Unfortunately, the price p is not exogenous or the orthogonality condition
E (ud p ) = 0 is not satis…ed!
E (ud p ) 6= 0
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2. Endogeneity
2. Simultaneous equation bias (cont’d)
Indeed, the supply/demand system can be written as:
qd = α1 p + α2 y + ud
qs = β1 p + us
qd = qp
where E (ud ) = E (us ) = E (us ud ) = E (us y ) = E (ud y ) = 0.
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2. Endogeneity
2. Simultaneous equation bias (cont’d)
Solving qd = qp , the reduced-form equations, which express the
endogenous variables in terms of the exogenous variables, write:
p=
q=
u
α2 y
+ d
β1 α1
β1
us
= π 1 y + w1
α1
β1 α2 y
β ud
+ 1
β1 α1
β1
Therefore
E (ud p ) =
α1 us
= π 2 y + w2
α1
σ2ud
β1
α1
6= 0
This result leads to an overestimated (upward biased) price coe¢ cient.
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2. Endogeneity
3. Omited variables
Consider the true model:
yi = β1 + β2 x1i + β2 x2i + εi
with E (εi ) = E (εi x1i ) = E (εi x2i ) = 0.
If we regress y on a constant and x1 (omitted variable x2 ):
yi = β1 + β2 x1i + µi
µi = β2 x2i + εi
If Cov (x1i , x2i ) 6= 0, then
E (µi x1i ) 6= 0
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2. Endogeneity
Question
What is the consequence of the endogeneity assumption on the OLS
estimator?
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2. Endogeneity
Consider the (population) multiple linear regression model:
y = Xβ + ε
where (cf. chapter 3):
y is a N
1 vector of observations yi for i = 1, .., N
X is a N K matrix of K explicative variables xik for k = 1, ., K and
i = 1, .., N
ε is a N
1 vector of error terms εi .
β = ( β1 ..βK )> is a K
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1 vector of parameters
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2. Endogeneity
The OLS estimator is de…ned as to be:
If we assume that
>
b
β
OLS = X X
1
X> y
E ( ε j X) 6 = 0
Then, we have:
>
b
E β
OLS X = β0 + X X
b
E β
OLS
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1
X> E ( ε j X ) 6 = 0
b
= EX E β
OLS X
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6 = β0
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2. Endogeneity
Theorem (Bias of the OLS estimator)
If the regressors are endogenous, i.e. E ( εj X) 6= 0, the OLS estimator of
β is biased
b
E β
OLS 6 = β0
where β0 denotes the true value of the parameters. This bias is called the
endogeneity bias.
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2. Endogeneity
Remark
1
We saw in Chapter 1 that an estimator may be biased (…nite sample
properties) but asymptotically consistent (ex: uncorrected sample
variance).
2
But in presence of endogeneity, the OLS estimator is also
inconsistent.
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2. Endogeneity
Objectives We assume that:
plim
1 >
X ε = γ 6= 0K
N
1
where
γ = E (xi εi ) 6= 0K
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1
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2. Endogeneity
Given the de…nition of the OLS estimator:
>
b
β
OLS = β0 + X X
We have:
b
plim β
OLS = β0 + plim
1
X> ε
1
1 >
X X
N
plim
1 >
X ε
N
Or equivalently:
b
plim β
OLS = β0 + Q
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1
γ 6 = β0
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2. Endogeneity
Theorem (Inconsistency of the OLS estimator)
If the regressors are endogenous with plim N
estimator of β is inconsistent
where Q = plim N
b
plim β
OLS = β0 + Q
1 X> ε
1
= γ, the OLS
γ
1 X> X.
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2. Endogeneity
Remark
The bias and the inconsistency property is not con…ned to the coe¢ cients
on the endogenous variables.
Consider a case where all but the last variable in X are uncorrelated with ε:
1
0
0
B 0 C
1
C
plim X> ε = γ = B
@ .. A
N
γ
Then we have:
b
plim β
OLS = β0 + Q
1
γ
There is no reason to expect that any of the elements of the last column
of Q 1 will equal to zero.
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2. Endogeneity
Remark (cont’d)
1
2
b
plim β
OLS = β0 + Q
1
γ
The implication is that even though only one of the variables in X is
b
correlated with ε, all of the elements of β
OLS are inconsistent,
not just the estimator of the coe¢ cient on the endogenous variable.
This e¤ects is called smearing; the inconsistency due to the
endogeneity of the one variable is smeared across all of the least
squares estimators.
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2. Endogeneity
Example (Endogeneity, OLS estimator and smearing)
Consider the multiple linear regression model
yi = 0.4 + 0.5xi 1
0.8xi 2 + εi
where εi is i.i.d. with E (εi ) . We assume that the vector of variables
de…ned by wi = (xi 1 : xi 2 : εi ) has a multivariate normal distribution with
wi
with
N (03
1 , ∆)
0
1
1 0.3 0
∆ = @ 0.3 1 0.5 A
0 0.5 1
It means that Cov (εi , xi 1 ) = 0 (x1 is exogenous) but Cov (εi , xi 2 ) = 0.5
(x2 is endogenous) and Cov (xi 1, xi 2 ) = 0.3 (x1 is correlated to x2 ).
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2. Endogeneity
Example (Endogeneity, OLS estimator and smearing (cont’d))
Write a Matlab code to (1) generate S = 1, 000 samples fyi , xi 1 , xi 2 gN
i =1
of size N = 10, 000. (2) For each simulated sample, determine the OLS
estimators of the model
yi = β1 + β2 xi 1 + β3 xi 2 + εi
b = b
β1s b
β2s b
β3s
Denote β
s
>
the OLS estimates obtained from the
simulation s 2 f1, ..S g . (3) compare the true value of the parameters in
the population (DGP) to the average OLS estimates obtained for the S
simulations
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2. Endogeneity
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2. Endogeneity
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2. Endogeneity
Question: What is the solution to the endogeneity issue?
The use of instruments..
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2. Endogeneity
Key Concepts
1
Endogeneity issue
2
Main sources of endogeneity: omitted variables, errors-in-variables,
and jointly endogenous regressors.
3
Endogeneity bias of the OLS estimator
4
Inconsistency of the OLS estimator
5
Smearing e¤ect
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Section 3
Instrumental Variables (IV) estimator
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3. Instrumental Variables (IV) estimator
Objectives
The objective of this section are the following:
1
To de…ne the notion of instrument or instrumental variable
2
To introduce the Instrumental Variables (IV) estimator
3
To study the asymptotic properties of the IV estimator
4
To de…ne the notion of weak instrument
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3. Instrumental Variables (IV) estimator
De…nition (Instruments)
Consider a set of H variables zh 2 RN for h = 1, ..N. Denote Z the N
matrix (z1 : .. : zH ) . These variables are called instruments or
instrumental variables if they satisfy two properties:
H
(1) Exogeneity: They are uncorrelated with the disturbance.
E ( εj Z) = 0N
1
(2) Relevance: They are correlated with the independent variables, X.
E (xik zih ) 6= 0
for h 2 f1, .., H g and k 2 f1, .., K g.
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3. Instrumental Variables (IV) estimator
Assumptions: The instrumental variables satisfy the following properties.
Well behaved data:
plim
1 >
Z Z = QZZ a …nite H
N
H positive de…nite matrix
1 >
Z X = QZX a …nite H
N
K positive de…nite matrix
Relevance:
plim
Exogeneity:
plim
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1 >
Z ε = 0K
N
1
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3. Instrumental Variables (IV) estimator
De…nition (Instrument properties)
We assume that the H instruments are linearly independent:
E Z> Z
is non singular
or equivalently
rank E Z> Z
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=H
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3. Instrumental Variables (IV) estimator
Remark
The exogeneity condition
E ( εi j zi ) = 0 =) E (εi zi ) = 0
with zi = (zi 1 ..ziH )> can expressed as an orthogonality condition or
moment condition
E zi yi xi> β
=0
The sample analog is
1
N
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N
∑
zi yi
xi> β
=0
i =1
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3. Instrumental Variables (IV) estimator
De…nition (Identi…cation)
The system is identi…ed if there exists a unique β = β0 such that:
E zi yi
xi> β
=0
where zi = (zi 1 ..ziH )> . For that, we have the following conditions:
(1) If H < K the model is not identi…ed.
(2) If H = K the model is just-identi…ed.
(3) If H > K the model is over-identi…ed.
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3. Instrumental Variables (IV) estimator
Remark
1
Under-identi…cation: less equations (H) than unknowns (K )....
2
Just-identi…cation: number of equations equals the number of
unknowns (unique solution)...=> IV estimator
3
Over-identi…cation: more equations than unknowns. Two equivalent
solutions:
1
2
Select K linear combinations of the instruments to have a unique
solution )...=> Two-Stage Least Squares
Set the sample analog of the moment conditions as close as possible to
zero, i.e. minimize the distance between the sample analog and zero
given a metric (optimal metric or optimal weighting matrix?) =>
Generalized Method of Moments (GMM).
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3. Instrumental Variables (IV) estimator
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3. Instrumental Variables (IV) estimator
Assumption: Consider a just-identi…ed model
H=K
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3. Instrumental Variables (IV) estimator
Motivation of the IV estimator
By de…nition of the instruments:
plim
1
1 >
Z ε = plim Z> (y
N
N
Xβ) = 0K
1
So, we have:
plim
1 >
Z y=
N
plim
1 >
Z X
N
β
or equivalently
β=
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plim
1 >
Z X
N
1
plim
1 >
Z y
N
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3. Instrumental Variables (IV) estimator
De…nition (Instrumental Variable (IV) estimator)
b of parameters
If H = K , the Instrumental Variable (IV) estimator β
IV
β is de…ned as to be:
1
b = Z> X
β
Z> y
IV
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3. Instrumental Variables (IV) estimator
De…nition (Consistency)
b is
Under the assumption that plim N 1 Z> ε, the IV estimator β
IV
consistent:
p
b !
β
β0
IV
where β0 denotes the true value of the parameters.
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3. Instrumental Variables (IV) estimator
Proof
By de…nition:
So, we have:
b =β +
β
IV
0
1 >
Z X
N
1
b = β + plim 1 Z> X
plim β
IV
0
N
1 >
Z ε
N
1
plim
1 >
Z ε
N
Under the assumption of exogeneity of the instruments
plim
1 >
Z ε = 0K
N
1
So, we have
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b =β
plim β
IV
0
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3. Instrumental Variables (IV) estimator
De…nition (Asymptotic distribution)
b is asymptotically
Under some regularity conditions, the IV estimator β
IV
normally distributed:
p
where
b
N β
IV
d
β0 ! N 0K
QZZ = plim
K K
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1 >
Z Z
N
1, σ
2
QZX1 QZZ QZX1
QZX = plim
K K
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1 >
Z X
N
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3. Instrumental Variables (IV) estimator
De…nition (Asymptotic variance covariance matrix)
b is
The asymptotic variance covariance matrix of the IV estimator β
IV
de…ned as to be:
b
Vasy β
IV
=
σ2
Q 1 QZZ QZX1
N ZX
A consistent estimator is given by
b
b asy β
V
IV
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b 2 Z> X
=σ
1
Z> Z
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X> Z
1
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3. Instrumental Variables (IV) estimator
Remarks
1
If the system is just identi…ed H = K ,
Z> X
1
= X> Z
1
QZX = QXZ
the estimator can also written as
2
b
b asy β
V
IV
b 2 Z> X
=σ
1
Z> Z
1
Z> X
As usual, the estimator of the variance of the error terms is:
b2 =
σ
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b
ε>b
ε
1
=
N K
N K
N
∑
yi
i =1
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b
xi> β
IV
2
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3. Instrumental Variables (IV) estimator
Relevant instruments
1
2
Our analysis thus far has focused on the “identi…cation” condition
for IV estimation, that is, the “exogeneity assumption,” which
produces
1
plim Z> ε = 0K 1
N
A growing literature has argued that greater attention needs to be
given to the relevance condition
plim
1 >
Z X = QZX a …nite H
N
K positive de…nite matrix
with H = K in the case of a just-identi…ed model.
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3. Instrumental Variables (IV) estimator
Relevant instruments (cont’d)
plim
1 >
Z X = QZX a …nite H
N
K positive de…nite matrix
1
While strictly speaking, this condition is su¢ cient to determine the
asymptotic properties of the IV estimator
2
However, the common case of “weak instruments,” is only barely
true has attracted considerable scrutiny.
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3. Instrumental Variables (IV) estimator
De…nition (Weak instrument)
A weak instrument is an instrumental variable which is only slightly
correlated with the right-hand-side variables X. In presence of weak
instruments, the quantity QZX is close to zero and we have
1 >
Z X ' 0H
N
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K
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3. Instrumental Variables (IV) estimator
Fact (IV estimator and weak instruments)
b has a poor
In presence of weak instruments, the IV estimators β
IV
precision (great variance). For QZX ' 0H K , the asymptotic variance
tends to be very large, since:
b
Vasy β
IV
=
σ2
Q 1 QZZ QZX1
N ZX
As soon as N 1 Z> X ' 0H K , the estimated asymptotic variance
covariance is also very large since
b
b asy β
V
IV
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b 2 Z> X
=σ
1
Z> Z
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X> Z
1
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3. Instrumental Variables (IV) estimator
Key Concepts
1
Instrument or instrumental variable
2
Orthogonal or moment condition
3
Identi…cation: just-identi…ed or over-identi…ed model
4
Instrumental Variables (IV) estimator
5
Statistical properties of the IV estimator
6
Weak instrument
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Section 4
Two-Stage Least Squares (2SLS) estimator
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4. Two-Stage Least Squares (2SLS) estimator
Assumption: Consider an over-identi…ed model
H>K
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4. Two-Stage Least Squares (2SLS) estimator
Introduction
If Z contains more variables than X, then much of the preceding derivation
is unusable, because Z> X will be H K with
rank Z> X = K < H
So, the matrix Z> X has no inverse and we cannot compute the IV
estimator as:
1
b = Z> X
β
Z> y
IV
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4. Two-Stage Least Squares (2SLS) estimator
Introduction (cont’d)
The crucial assumption in the previous section was the exogeneity
assumption
1
plim Z> ε = 0K 1
N
1
That is, every column of Z is asymptotically uncorrelated with ε.
2
That also means that every linear combination of the columns of Z
is also uncorrelated with ε, which suggests that one approach would
be to choose K linear combinations of the columns of Z.
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4. Two-Stage Least Squares (2SLS) estimator
Introduction (cont’d)
Which linear combination to choose?
A choice consists in using is the projection of the columns of X in the
column space of Z:
1
b = Z Z> Z
X
Z> X
b for Z, we have
With this choice of instrumental variables, X
b
β
2SLS
=
=
b >X
X
>
1
b >y
X
>
X Z Z Z
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1
1
>
Z X
X> Z Z> Z
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1
Z> y
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4. Two-Stage Least Squares (2SLS) estimator
De…nition (Two-stage Least Squares (2SLS) estimator)
The Two-stage Least Squares (2SLS) estimator of the parameters β is
de…ned as to be:
1 >
b
b>
b y
β
X
2SLS = X X
1
b = Z Z> Z
where X
Z> X corresponds to the projection of the columns
of X in the column space of Z, or equivalently by
b
β
2SLS =
>
>
X Z Z Z
Christophe Hurlin (University of Orléans)
1
1
>
Z X
X> Z Z> Z
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1
Z> y
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4. Two-Stage Least Squares (2SLS) estimator
Remark
By de…nition
1
b
b>
β
2SLS = X X
Since
b = Z Z> Z
X
1
b >y
X
Z> X = PZ X
where PZ denotes the projection matrix on the columns of Z. Reminder:
PZ is symmetric and PZ PZ> = PZ . So, we have
b
β
2SLS
Christophe Hurlin (University of Orléans)
>
1
=
X> PZ X
=
X> PZ PZ X
=
b >X
b
X
>
1
b >y
X
1
b >y
X
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b >y
X
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4. Two-Stage Least Squares (2SLS) estimator
De…nition (Two-stage Least Squares (2SLS) estimator)
The Two-stage Least Squares (2SLS) estimator of the parameters β
can also be de…ned as:
b
b> b
β
2SLS = X X
1
b >y
X
b
It corresponds to the OLS estimator obtained in the regression of y on X.
b
Then, the 2SLS can be computed in two steps, …rst by computing X, then
by the least squares regression. That is why it is called the two-stage LS
estimator.
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4. Two-Stage Least Squares (2SLS) estimator
A procedure to get the 2SLS estimator is the following
Step 1: Regress each explicative variable xk (for k = 1, ..K ) on the H
instruments.
xki = α1 z1i + α2 z2i + .. + αH zHi + vi
Step 2: Compute the OLS estimators b
αh and the …tted values b
xki
b
xki = b
α1 z1i + b
α2 z2i + .. + b
αH zHii
Step 3: Regress the dependent variable y on the …tted values b
xki :
yi = β1 b
x1i + β2 b
x2i + .. + βK b
xKi + εi
b
The 2SLS estimator β
2SLS then corresponds to the OLS estimator
obtained in this model.
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4. Two-Stage Least Squares (2SLS) estimator
Theorem
If any column of X also appears in Z, i.e. if one or more explanatory
(exogenous) variable is used as an instrument, then that column of X is
b
reproduced exactly in X.
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4. Two-Stage Least Squares (2SLS) estimator
Example (Explicative variables used as instrument)
Suppose that the regression contains K variables, only one of which, say,
the K th , is correlated with the disturbances, i.e. E (xKi εi ) 6= 0. We can
use a set of instrumental variables z1 ,..., zJ plus the other K 1 variables
that certainly qualify as instrumental variables in their own right. So,
Z = (z1 : .. : zJ : x1 : .. : xK
1)
Then
b = (x1 : .. : xK
X
1
:b
xK )
where b
xK denotes the projection of xK on the columns of Z.
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4. Two-Stage Least Squares (2SLS) estimator
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4. Two-Stage Least Squares (2SLS) estimator
Key Concepts
1
Over-identi…ed model
2
Two-Stage Least Squares (2SLS) estimator
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End of Chapter 6
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Christophe Hurlin (University of Orléans)
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