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Chapter 6: Endogeneity and Instrumental Variables(IV) estimator
Advanced Econometrics - HEC Lausanne
Christophe Hurlin
University of Orléans
December 15, 2013
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 1 / 68
Section 1
Introduction
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 2 / 68
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)
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 3 / 68
1. Introduction
References
Amemiya T. (1985), Advanced Econometrics. Harvard University Press.
Ruud P., (2000) An introduction to Classical Econometric Theory, OxfordUniversity Press.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 4 / 68
1. Introduction
Notations: In this chapter, I will (try to...) follow some conventions ofnotation.
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 section2). For more appropriate notations, see:
Abadir and Magnus (2002), Notation in econometrics: a proposal for astandard, Econometrics Journal.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 5 / 68
Section 2
Endogeneity
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 6 / 68
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)
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 7 / 68
2. Endogeneity
Objectives in this chapter, we assume that the assumption A3(exogeneity) is violated:
E (εjX) 6= 0N�1
but the disturbances are spherical:
V (εjX) = σ2IN
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 8 / 68
2. Endogeneity
The reasons for suspecting E (εjX) 6= 0 are varied:
1 Errors-in-variables
2 Jointly endogenous variables: the usual example is runningquantities on prices to estimate a demand equation (supply alsoa¤ects the determination of equilibrium).
3 Omitted variables: one or more columns in X cannot be included inthe regression because no data on those variables areavailable� estimation will be altered to the extent that the missingvariables and the included ones are correlated
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 9 / 68
2. Endogeneity
1. Error-in-variables
1 Consider the regression model:
y �i = x�>i β+ εi
where E ( εi j x�i ) = 0.2 One does not observe (y �, x�) but (y , x)
yi = y �i + vi xi = x�i +wi
withE (vi ) = E (vi εi ) = E (viy �i ) = E
�w>i x
�i
�= 0
E (wi ) = E (viwi ) = E (wi εi ) = E (wiy �i ) = E (vix�i ) = 0
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 10 / 68
2. Endogeneity
1. Error-in-variables (cont�d)
1 The mismeasured regression equation is given by:
y �i = x�>i β+ εi
() yi = x>i β+ εi � vi +w>i β
() yi = x>i β+ ηi
with ηi = εi � vi +w>i β.2 The composite error term ηi is not orthogonal to the mismeasuredindependent variable xi .
E (ηixi ) 6= 0
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 11 / 68
2. Endogeneity
1. Error-in-variables (cont�d)
Indeed, we have:ηi = εi � vi +w>i β.
As a consequence:
E (ηixi ) = E (εixi )�E (vixi ) +E�w>i β xi
�= E
�w>i β xi
�E (ηixi ) 6= 0
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 12 / 68
2. Endogeneity
2. Simultaneous equation bias
Consider the demand equation
qd = α1p + α2y + ud
where qd , p and y denote respectively the quantity, the price and income.
Unfortunately, the price p is not exogenous or the orthogonality conditionE (udp) = 0 is not satis�ed!
E (udp) 6= 0
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 13 / 68
2. Endogeneity
2. Simultaneous equation bias (cont�d)
Indeed, the supply/demand system can be written as:
qd = α1p + α2y + ud
qs = β1p + us
qd = qp
where E (ud ) = E (us ) = E (usud ) = E (usy) = E (udy) = 0.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 14 / 68
2. Endogeneity
2. Simultaneous equation bias (cont�d)
Solving qd = qp , the reduced-form equations, which express theendogenous variables in terms of the exogenous variables, write:
p =α2y
β1 � α1+ud � usβ1 � α1
= π1y + w1
q =β1α2y
β1 � α1+
β1ud � α1usβ1 � α1
= π2y + w2
Therefore
E (udp) =σ2ud
β1 � α16= 0
This result leads to an overestimated (upward biased) price coe¢ cient.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 15 / 68
2. Endogeneity
3. Omited variables
Consider the true model:
yi = β1 + β2x1i + β2x2i + εi
with E (εi ) = E (εix1i ) = E (εix2i ) = 0.
If we regress y on a constant and x1 (omitted variable x2):
yi = β1 + β2x1i + µi
µi = β2x2i + εi
If Cov (x1i , x2i ) 6= 0, then
E (µix1i ) 6= 0
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 16 / 68
2. Endogeneity
Question
What is the consequence of the endogeneity assumption on the OLSestimator?
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 17 / 68
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 andi = 1, ..,N
ε is a N � 1 vector of error terms εi .
β = (β1..βK )> is a K � 1 vector of parameters
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 18 / 68
2. Endogeneity
The OLS estimator is de�ned as to be:
bβOLS = �X>X��1 X>yIf we assume that
E (εjX) 6= 0Then, we have:
E�bβOLS ���X� = β0 +
�X>X
��1 �X>E (εjX)
�6= 0
E�bβOLS� = EX
�E�bβOLS ���X�� 6= β0
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 19 / 68
2. Endogeneity
Theorem (Bias of the OLS estimator)
If the regressors are endogenous, i.e. E (εjX) 6= 0, the OLS estimator ofβ is biased
E�bβOLS� 6= β0
where β0 denotes the true value of the parameters. This bias is called theendogeneity bias.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 20 / 68
2. Endogeneity
Remark
1 We saw in Chapter 1 that an estimator may be biased (�nite sampleproperties) but asymptotically consistent (ex: uncorrected samplevariance).
2 But in presence of endogeneity, the OLS estimator is alsoinconsistent.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 21 / 68
2. Endogeneity
Objectives We assume that:
plim1NX>ε = γ 6= 0K�1
whereγ = E (xi εi ) 6= 0K�1
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 22 / 68
2. Endogeneity
Given the de�nition of the OLS estimator:
bβOLS = β0 +�X>X
��1 �X>ε
�We have:
plim bβOLS = β0 + plim�1NX>X
��1� plim
�1NX>ε
�
Or equivalently:plim bβOLS = β0 +Q
�1γ 6= β0
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 23 / 68
2. Endogeneity
Theorem (Inconsistency of the OLS estimator)
If the regressors are endogenous with plim N�1X>ε = γ, the OLSestimator of β is inconsistent
plim bβOLS = β0 +Q�1γ
where Q = plim N�1X>X.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 24 / 68
2. Endogeneity
Remark
The bias and the inconsistency property is not con�ned to the coe¢ cientson the endogenous variables.
Consider a case where all but the last variable in X are uncorrelated with ε:
plim1NX>ε = γ =
0BB@00..γ
1CCAThen we have:
plim bβOLS = β0 +Q�1γ
There is no reason to expect that any of the elements of the last columnof Q�1 will equal to zero.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 25 / 68
2. Endogeneity
Remark (cont�d)
plim bβOLS = β0 +Q�1γ
1 The implication is that even though only one of the variables in X iscorrelated with ε, all of the elements of bβOLS are inconsistent,not just the estimator of the coe¢ cient on the endogenous variable.
2 This e¤ects is called smearing; the inconsistency due to theendogeneity of the one variable is smeared across all of the leastsquares estimators.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 26 / 68
2. Endogeneity
Example (Endogeneity, OLS estimator and smearing)Consider the multiple linear regression model
yi = 0.4+ 0.5xi1 � 0.8xi2 + εi
where εi is i .i .d . with E (εi ) . We assume that the vector of variablesde�ned by wi = (xi1 : xi2 : εi ) has a multivariate normal distribution with
wi � N (03�1,∆)
with
∆ =
0@ 1 0.3 00.3 1 0.50 0.5 1
1AIt means that Cov (εi , xi1) = 0 (x1 is exogenous) but Cov (εi , xi2) = 0.5(x2 is endogenous) and Cov (xi1,xi2) = 0.3 (x1 is correlated to x2).
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 27 / 68
2. Endogeneity
Example (Endogeneity, OLS estimator and smearing (cont�d))
Write a Matlab code to (1) generate S = 1, 000 samples fyi , xi1, xi2gNi=1of size N = 10, 000. (2) For each simulated sample, determine the OLSestimators of the model
yi = β1 + β2xi1 + β3xi2 + εi
Denote bβs = �bβ1s bβ2s bβ3s�> the OLS estimates obtained from the
simulation s 2 f1, ..Sg . (3) compare the true value of the parameters inthe population (DGP) to the average OLS estimates obtained for the Ssimulations
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 28 / 68
2. Endogeneity
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 29 / 68
2. Endogeneity
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 30 / 68
2. Endogeneity
Question: What is the solution to the endogeneity issue?
The use of instruments..
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 31 / 68
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
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 32 / 68
Section 3
Instrumental Variables (IV) estimator
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 33 / 68
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
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 34 / 68
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 �Hmatrix (z1 : .. : zH ) . These variables are called instruments orinstrumental variables if they satisfy two properties:
(1) Exogeneity: They are uncorrelated with the disturbance.
E (εjZ) = 0N�1
(2) Relevance: They are correlated with the independent variables, X.
E (xikzih) 6= 0
for h 2 f1, ..,Hg and k 2 f1, ..,Kg.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 35 / 68
3. Instrumental Variables (IV) estimator
Assumptions: The instrumental variables satisfy the following properties.
Well behaved data:
plim1NZ>Z = QZZ a �nite H �H positive de�nite matrix
Relevance:
plim1NZ>X = QZX a �nite H �K positive de�nite matrix
Exogeneity:
plim1NZ>ε = 0K�1
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 36 / 68
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 equivalentlyrank
�E�Z>Z
��= H
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 37 / 68
3. Instrumental Variables (IV) estimator
Remark
The exogeneity condition
E ( εi j zi ) = 0 =) E (εizi ) = 0
with zi = (zi1..ziH )> can expressed as an orthogonality condition or
moment conditionE�zi�yi � x>i β
��= 0
The sample analog is
1N
N
∑i=1
�zi�yi � x>i β
��= 0
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 38 / 68
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 � x>i β
��= 0
where zi = (zi1..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.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 39 / 68
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 ofunknowns (unique solution)...=> IV estimator
3 Over-identi�cation: more equations than unknowns. Two equivalentsolutions:
1 Select K linear combinations of the instruments to have a uniquesolution )...=> Two-Stage Least Squares
2 Set the sample analog of the moment conditions as close as possible tozero, i.e. minimize the distance between the sample analog and zerogiven a metric (optimal metric or optimal weighting matrix?) =>Generalized Method of Moments (GMM).
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 40 / 68
3. Instrumental Variables (IV) estimator
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 41 / 68
3. Instrumental Variables (IV) estimator
Assumption: Consider a just-identi�ed model
H = K
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 42 / 68
3. Instrumental Variables (IV) estimator
Motivation of the IV estimator
By de�nition of the instruments:
plim1NZ>ε = plim
1NZ> (y�Xβ) = 0K�1
So, we have:
plim1NZ>y =
�plim
1NZ>X
�β
or equivalently
β =
�plim
1NZ>X
��1plim
1NZ>y
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 43 / 68
3. Instrumental Variables (IV) estimator
De�nition (Instrumental Variable (IV) estimator)
If H = K , the Instrumental Variable (IV) estimator bβIV of parametersβ is de�ned as to be: bβIV = �Z>X��1 Z>y
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 44 / 68
3. Instrumental Variables (IV) estimator
De�nition (Consistency)
Under the assumption that plim N�1Z>ε, the IV estimator bβIV isconsistent: bβIV p! β0
where β0 denotes the true value of the parameters.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 45 / 68
3. Instrumental Variables (IV) estimator
Proof
By de�nition: bβIV = β0 +
�1NZ>X
��1 � 1NZ>ε
�So, we have:
plimbβIV = β0 +
�plim
1NZ>X
��1 �plim
1NZ>ε
�Under the assumption of exogeneity of the instruments
plim1NZ>ε = 0K�1
So, we haveplim bβIV = β0 �
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 46 / 68
3. Instrumental Variables (IV) estimator
De�nition (Asymptotic distribution)
Under some regularity conditions, the IV estimator bβIV is asymptoticallynormally distributed:
pN�bβIV � β0
�d! N
�0K�1, σ2Q�1ZXQZZQ
�1ZX
�where
QZZK�K
= plim1NZ>Z QZX
K�K= plim
1NZ>X
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 47 / 68
3. Instrumental Variables (IV) estimator
De�nition (Asymptotic variance covariance matrix)
The asymptotic variance covariance matrix of the IV estimator bβIV isde�ned as to be:
Vasy
�bβIV � = σ2
NQ�1ZXQZZQ
�1ZX
A consistent estimator is given by
bVasy
�bβIV � = bσ2 �Z>X��1 �Z>Z� �X>Z��1
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 48 / 68
3. Instrumental Variables (IV) estimatorRemarks
1 If the system is just identi�ed H = K ,�Z>X
��1=�X>Z
��1QZX = QXZ
the estimator can also written as
bVasy
�bβIV � = bσ2 �Z>X��1 �Z>Z� �Z>X��12 As usual, the estimator of the variance of the error terms is:
bσ2 = bε>bεN �K =
1N �K
N
∑i=1
�yi � x>i bβIV �2
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 49 / 68
3. Instrumental Variables (IV) estimator
Relevant instruments
1 Our analysis thus far has focused on the �identi�cation�conditionfor IV estimation, that is, the �exogeneity assumption,�whichproduces
plim1NZ>ε = 0K�1
2 A growing literature has argued that greater attention needs to begiven to the relevance condition
plim1NZ>X = QZX a �nite H �K positive de�nite matrix
with H = K in the case of a just-identi�ed model.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 50 / 68
3. Instrumental Variables (IV) estimator
Relevant instruments (cont�d)
plim1NZ>X = QZX a �nite H �K positive de�nite matrix
1 While strictly speaking, this condition is su¢ cient to determine theasymptotic properties of the IV estimator
2 However, the common case of �weak instruments,� is only barelytrue has attracted considerable scrutiny.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 51 / 68
3. Instrumental Variables (IV) estimator
De�nition (Weak instrument)A weak instrument is an instrumental variable which is only slightlycorrelated with the right-hand-side variables X. In presence of weakinstruments, the quantity QZX is close to zero and we have
1NZ>X ' 0H�K
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 52 / 68
3. Instrumental Variables (IV) estimator
Fact (IV estimator and weak instruments)
In presence of weak instruments, the IV estimators bβIV has a poorprecision (great variance). For QZX ' 0H�K , the asymptotic variancetends to be very large, since:
Vasy
�bβIV � = σ2
NQ�1ZXQZZQ
�1ZX
As soon as N�1Z>X ' 0H�K , the estimated asymptotic variancecovariance is also very large since
bVasy
�bβIV � = bσ2 �Z>X��1 �Z>Z� �X>Z��1
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 53 / 68
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
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 54 / 68
Section 4
Two-Stage Least Squares (2SLS) estimator
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 55 / 68
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 derivationis 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 IVestimator as: bβIV = �Z>X��1 Z>y
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 57 / 68
4. Two-Stage Least Squares (2SLS) estimator
Introduction (cont�d)
The crucial assumption in the previous section was the exogeneityassumption
plim1NZ>ε = 0K�1
1 That is, every column of Z is asymptotically uncorrelated with ε.
2 That also means that every linear combination of the columns of Zis also uncorrelated with ε, which suggests that one approach wouldbe to choose K linear combinations of the columns of Z.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 58 / 68
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 thecolumn space of Z: bX = Z �Z>Z��1 Z>XWith this choice of instrumental variables, bX for Z, we have
bβ2SLS =�bX>X��1 bX>y
=
�X>Z
�Z>Z
��1Z>X
��1X>Z
�Z>Z
��1Z>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 isde�ned as to be: bβ2SLS = �bX>X��1 bX>ywhere bX = Z �Z>Z��1 Z>X corresponds to the projection of the columnsof X in the column space of Z, or equivalently by
bβ2SLS = �X>Z �Z>Z��1 Z>X��1 X>Z �Z>Z��1 Z>y
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 60 / 68
4. Two-Stage Least Squares (2SLS) estimatorRemark
By de�nition bβ2SLS = �bX>X��1 bX>ySince bX = Z �Z>Z��1 Z>X = PZXwhere PZ denotes the projection matrix on the columns of Z. Reminder:PZ is symmetric and PZP>Z = PZ . So, we have
bβ2SLS =�X>P
>ZX��1 bX>y
=�X>P
>ZPZX
��1 bX>y=
�bX>bX��1 bX>yChristophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 61 / 68
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 βcan also be de�ned as:
bβ2SLS = �bX>bX��1 bX>yIt corresponds to the OLS estimator obtained in the regression of y on bX.Then, the 2SLS can be computed in two steps, �rst by computing bX, thenby the least squares regression. That is why it is called the two-stage LSestimator.
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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 Hinstruments.
xki = α1z1i + α2z2i + ..+ αH zHi + vi
Step 2: Compute the OLS estimators bαh and the �tted values bxkibxki = bα1z1i + bα2z2i + ..+ bαH zHii
Step 3: Regress the dependent variable y on the �tted values bxki :
yi = β1bx1i + β2bx2i + ..+ βKbxKi + εi
The 2SLS estimator bβ2SLS then corresponds to the OLS estimatorobtained in this model.
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4. Two-Stage Least Squares (2SLS) estimator
TheoremIf 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 isreproduced exactly in bX.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 64 / 68
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 canuse a set of instrumental variables z1,..., zJ plus the other K � 1 variablesthat certainly qualify as instrumental variables in their own right. So,
Z = (z1 : .. : zJ : x1 : .. : xK�1)
Then bX = (x1 : .. : xK�1 : bxK )where bxK denotes the projection of xK on the columns of Z.
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 65 / 68
4. Two-Stage Least Squares (2SLS) estimator
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 66 / 68
4. Two-Stage Least Squares (2SLS) estimator
Key Concepts
1 Over-identi�ed model
2 Two-Stage Least Squares (2SLS) estimator
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 67 / 68
End of Chapter 6
Christophe Hurlin (University of Orléans)
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 68 / 68