chapter 2. dynamic panel data models - univ-orleans.fr · 2. the dynamic panel bias objectives 1...
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Chapter 2. Dynamic panel data modelsSchool of Economics and Management - University of Geneva
Christophe Hurlin, Université of Orléans
University of Orléans
April 2018
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 1 / 209
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1. Introduction
De�nition (Dynamic panel data model)We now consider a dynamic panel data model, in the sense that it contains(at least) one lagged dependent variables. For simplicity, let us consider
yit = γyi ,t�1 + β0xit + α�i + εit
for i = 1, .., n and t = 1, ..,T . α�i and λt are the (unobserved) individualand time-speci�c e¤ects, and εit the error (idiosyncratic) term withE(εit ) = 0, and E(εit εjs ) = σ2ε if j = i and t = s, and E(εit εjs ) = 0otherwise.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 2 / 209
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1. Introduction
Remark
In a dynamic panel model, the choice between a �xed-e¤ects formulationand a random-e¤ects formulation has implications for estimation that areof a di¤erent nature than those associated with the static model.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 3 / 209
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1. Introduction
Dynamic panel issues
1 If lagged dependent variables appear as explanatory variables, strictexogeneity of the regressors no longer holds. The LSDV is no longerconsistent when n tends to in�nity and T is �xed.
2 The initial values of a dynamic process raise another problem. Itturns out that with a random-e¤ects formulation, the interpretationof a model depends on the assumption of initial observation.
3 The consistency property of the MLE and the GLS estimator alsodepends on the way in which T and n tend to in�nity.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 4 / 209
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Introduction
The outline of this chapter is the following:
Section 1: Introduction
Section 2: Dynamic panel bias
Section 3: The IV (Instrumental Variable) approach
Subsection 3.1: Reminder on IV and 2SLS
Subsection 3.2: Anderson and Hsiao (1982) approach
Section 4: The GMM (Generalized Method of Moment) approach
Subsection 4.1: General presentation of GMM
Subsection 4.2: Application to dynamic panel data models
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 5 / 209
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Section 2
The Dynamic Panel Bias
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 6 / 209
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2. The dynamic panel bias
Objectives
1 Introduce the AR(1) panel data model.
2 Derive the semi-asymptotic bias of the LSDV estimator.
3 Understand the sources of the dynamic panel bias or Nickell�s bias.
4 Evaluate the magnitude of this bias in a simple AR(1) model.
5 Asses this bias by Monte Carlo simulations.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 7 / 209
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2. The dynamic panel bias
Dynamic panel bias
1 The LSDV estimator is consistent for the static model whether thee¤ects are �xed or random.
2 On the contrary, the LSDV is inconsistent for a dynamic panel datamodel with individual e¤ects, whether the e¤ects are �xed or random.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 8 / 209
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2. The dynamic panel bias
De�nition (Nickell�s bias)The biais of the LSDV estimator in a dynamic model is generaly known asdynamic panel bias or Nickell�s bias (1981).
Nickell, S. (1981). Biases in Dynamic Models with Fixed E¤ects,Econometrica, 49, 1399�1416.
Anderson, T.W., and C. Hsiao (1982). Formulation and Estimation ofDynamic Models Using Panel Data, Journal of Econometrics, 18, 47�82.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 9 / 209
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2. The dynamic panel bias
De�nition (AR(1) panel data model)
Consider the simple AR(1) model
yit = γyi ,t�1 + α�i + εit
for i = 1, .., n and t = 1, ..,T . For simplicity, let us assume that
α�i = α+ αi
to avoid imposing the restriction that ∑ni=1 αi = 0 or E (αi ) = 0 in the
case of random individual e¤ects.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 10 / 209
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2. The dynamic panel bias
Assumptions
1 The autoregressive parameter γ satis�es
jγj < 1
2 The initial condition yi0 is observable.
3 The error term satis�es with E (εit ) = 0, and E (εit εjs ) = σ2ε if j = iand t = s, and E (εit εjs ) = 0 otherwise.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 11 / 209
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2. The dynamic panel bias
Dynamic panel bias
In this AR(1) panel data model, we will show that
plimn!∞
bγLSDV 6= γ dynamic panel bias
plimn,T!∞
bγLSDV = γ
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 12 / 209
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2. The dynamic panel bias
The LSDV estimator is de�ned by (cf. chapter 1)
bαi = y i � bγLSDV y i ,�1bγLSDV =
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1)2
!�1
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1) (yit � y i )
!
x i =1T
T
∑t=1xit y i =
1T
T
∑t=1yit y i ,�1 =
1T
T
∑t=1yi ,t�1
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 13 / 209
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2. The dynamic panel bias
De�nition (bias)The bias of the LSDV estimator is de�ned by:
bγLSDV � γ =
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1)2
!�1
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1) (εit � εi )
!
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 14 / 209
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2. The dynamic panel bias
The bias of the LSDV estimator can be rewritten as:
bγLSDV � γ =
n∑i=1
T∑t=1(yi ,t�1 � y i ,�1) (εit � εi ) / (nT )
n∑i=1
T∑t=1(yi ,t�1 � y i ,�1)2 / (nT )
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 15 / 209
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2. The dynamic panel bias
Let us consider the numerator. Because εit are (1) uncorrelated with α�iand (2) are independently and identically distributed, we have
plimn!∞
1nT
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1) (εit � εi )
= plimn!∞
1nT
T
∑t=1
n
∑i=1yi ,t�1εit| {z }
N1
� plimn!∞
1nT
T
∑t=1
n
∑i=1yi ,t�1εi| {z }
N2
� plimn!∞
1nT
T
∑t=1
n
∑i=1y i ,�1εit| {z }
N3
+ plimn!∞
1nT
T
∑t=1
n
∑i=1y i ,�1εi| {z }
N4
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 16 / 209
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2. The dynamic panel bias
Theorem (Weak law of large numbers, Khinchine)
If fXig , for i = 1, ..,m is a sequence of i.i.d. random variables withE (Xi ) = µ < ∞, then the sample mean converges in probability to µ:
1m
m
∑i=1Xi
p! E (Xi ) = µ
or
plimm!∞
1m
m
∑i=1Xi = E (Xi ) = µ
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 17 / 209
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2. The dynamic panel bias
By application of the WLLN (Khinchine�s theorem)
N1 = plimn!∞
1nT
n
∑i=1
T
∑t=1yi ,t�1εit = E (yi ,t�1εit )
Since (1) yi ,t�1 only depends on εi ,t�1, εi ,t�2 and (2) the εit areuncorrelated, then we have
E (yi ,t�1εit ) = 0
and �nally
N1 = plimn!∞
1nT
n
∑i=1
T
∑t=1yi ,t�1εit = 0
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 18 / 209
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2. The dynamic panel bias
For the second term N2, we have:
N2 = plimn!∞
1nT
n
∑i=1
T
∑t=1yi ,t�1εi
= plimn!∞
1nT
n
∑i=1
εiT
∑t=1yi ,t�1
= plimn!∞
1nT
n
∑i=1
εiTy i ,�1 as y i ,�1 =1T
T
∑t=1yi ,t�1
= plimn!∞
1n
n
∑i=1
εiy i ,�1
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 19 / 209
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2. The dynamic panel bias
In the same way:
N3 = plimn!∞
1nT
n
∑i=1
T
∑t=1y i ,�1εit = plim
n!∞
1nT
n
∑i=1y i ,�1
T
∑t=1
εit = plimn!∞
1n
n
∑i=1y i ,�1εi
N4 = plimn!∞
1nT
n
∑i=1
T
∑t=1y i ,�1εi = plim
n!∞
1nTT
n
∑i=1y i ,�1εi = plim
n!∞
1n
n
∑i=1y i ,�1εi
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 20 / 209
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2. The dynamic panel bias
The numerator of the bias expression can be rewritten as
plimn!∞
1nT
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1) (εit � εi )
= 0|{z}N1
� plimn!∞
1n
n
∑i=1
εiy i ,�1| {z }N2
� plimn!∞
1n
n
∑i=1y i ,�1εi| {z }
N3
+ plimn!∞
1n
n
∑i=1y i ,�1εi| {z }
N4
= � plimn!∞
1n
n
∑i=1y i ,�1εi
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 21 / 209
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2. The dynamic panel bias
SolutionThe numerator of the expression of the LSDV bias satis�es:
plimn!∞
1nT
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1) (εit � εi ) = � plim
n!∞
1n
n
∑i=1y i ,�1εi
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 22 / 209
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2. The dynamic panel bias
Remark
bγLSDV � γ =
n∑i=1
T∑t=1(yi ,t�1 � y i ,�1) (εit � εi ) / (nT )
n∑i=1
T∑t=1(yi ,t�1 � y i ,�1)2 / (nT )
plimn!∞
1nT
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1) (εit � εi ) = � plim
n!∞
1n
n
∑i=1y i ,�1εi
If this plim is not null, then the LSDV estimator bγLSDV is biased when ntends to in�nity and T is �xed.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 23 / 209
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2. The dynamic panel bias
Let us examine this plim
plimn!∞
1n
n
∑i=1y i ,�1εi
We know that
yit = γyi ,t�1 + α�i + εit
= γ2yi ,t�2 + α�i (1+ γ) + εit + γεi ,t�1
= γ3yi ,t�3 + α�i�1+ γ+ γ2
�+ εit + γεi ,t�1 + γ2εi ,t�2
= ...
= γtyi0 +1� γt
1� γα�i + εit + γεi ,t�1 + γ2εi ,t�2 + ...+ γt�1εi1
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 24 / 209
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2. The dynamic panel bias
For any time t, we have:
yit = εit + γεi ,t�1 + γ2εi ,t�2 + ...+ γt�1εi1
+1� γt
1� γα�i + γtyi0
For yi ,t�1, we have:
yi ,t�1 = εi ,t�1 + γεi ,t�2 + γ2εi ,t�3 + ...+ γt�2εi1
+1� γt�1
1� γα�i + γt�1yi0
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 25 / 209
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2. The dynamic panel bias
yi ,t�1 = εi ,t�1 + γεi ,t�2 + γ2εi ,t�3 + ...+ γt�2εi1 +1� γt�1
1� γα�i + γt�1yi0
Summing yi ,t�1 over t, we get:
T
∑t=1yi ,t�1 = εi ,T�1 +
1� γ2
1� γεi ,T�2 + ...+
1� γT�1
1� γεi1
+(T � 1)� Tγ+ γT
(1� γ)2α�i +
1� γT
1� γyi0
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 26 / 209
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2. The dynamic panel bias
yi ,t�1 = εi ,t�1 + γεi ,t�2 + γ2εi ,t�3 + ...+ γt�2εi1 +1� γt�1
1� γα�i + γt�1yi0
Proof: We have (each lign corresponds to a date)
T
∑t=1yi ,t�1 = yi ,T�1 + yi ,T�2 + ..+ yi ,1 + yi ,0
= εi ,T�1 + γεi ,T�2 + ..+ γT�2εi1 +1� γT�1
1� γα�i + γT�1yi0
+εi ,T�2 + γεi ,T�3 + ...+ γT�3εi1 +1� γT�2
1� γα�i + γT�2yi0
+..
+εi ,1 +1� γ1
1� γα�i + γyi0
+yi0
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 27 / 209
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2. The dynamic panel bias
Proof (ct�d): For the individual e¤ect α�i , we have
α�i1� γ
�1� γ+ 1� γ2 + ...+ 1� γT�1
�=
α�i1� γ
�T � 1� γ� γ2 � ..� γT�1
�=
α�i1� γ
�T � 1� γT
1� γ
�=
α�i�T � Tγ� 1+ γT
�(1� γ)2
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 28 / 209
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2. The dynamic panel bias
So, we have
y i ,�1 =1T
T
∑t=1yi ,t�1
=1T
�εi ,T�1 +
1� γ2
1� γεi ,T�2 + ...+
1� γT�1
1� γεi1
+
�T � Tγ� 1+ γT
�(1� γ)2
α�i +1� γT
1� γyi0
!
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 29 / 209
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2. The dynamic panel bias
Finally, the plim is equal to
plimn!∞
1n
n
∑i=1y i ,�1εi
= plimn!∞
1n
n
∑i=1
�1T
�εi ,t�1 +
1� γ2
1� γεi ,t�2 + ...+
1� γT�1
1� γεi1
+
�T � Tγ� 1+ γT
�(1� γ)2
α�i +1� γT
1� γyi0
!� 1T(εi1 + ...+ εiT )
�
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2. The dynamic panel bias
Because εit are i.i.d, by a law of large numbers, we have:
plimn!∞
1n
n
∑i=1y i ,�1εi
= plimn!∞
1n
n
∑i=1
�1T
�εi ,T�1 +
1� γ2
1� γεi ,T�2 + ...+
1� γT�1
1� γεi1
+
�T � Tγ� 1+ γT
�(1� γ)2
α�i +1� γT
1� γyi0
!� 1T(εi1 + ...+ εiT )
�=
σ2εT 2
�1� γ
1� γ+1� γ2
1� γ+ ...+
1� γT�1
1� γ
�=
σ2εT 2
�T � Tγ� 1+ γT
�(1� γ)2
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2. The dynamic panel bias
Theorem
If the errors terms εit are i.i.d.�0, σ2ε
�, we have:
plimn!∞
1nT
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1) (εit � εi )
= �plimn!∞
1n
n
∑i=1y i ,�1εi
= � σ2εT 2
�T � Tγ� 1+ γT
�(1� γ)2
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2. The dynamic panel bias
By similar manipulations, we can show that the denominator of bγLSDVconverges to:
plimn!∞
1nT
n
∑i=1
T
∑t=1(yi ,t�1 � y i ,�1)2
=σ2ε
1� γ2
1� 1
T� 2γ
(1� γ)2��T � Tγ� 1+ γT
�T 2
!
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2. The dynamic panel bias
So, we have :
plimn!∞
(bγLSDV � γ)
= plimn!∞
�1nT
n∑i=1
T∑t=1(yi ,t�1 � y i ,�1) (εit � εi )
1nT
n∑i=1
T∑t=1(yi ,t�1 � y i ,�1)2
= �σ2εT 2(T�T γ�1+γT )
(1�γ)2
σ2ε1�γ2
�1� 1
T �2γ
(1�γ)2� (T�T γ�1+γT )
T 2
�
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2. The dynamic panel bias
This semi-asymptotic bias can be rewriten as:
plimn!∞
(bγLSDV � γ)
= ��T � Tγ� 1+ γT
��1�γ1+γ
� �T 2 � T � 2γ
(1�γ)2� (T � Tγ� 1+ γT )
�= � (1+ γ)
�T � Tγ� 1+ γT
�(1� γ)
�T 2 � T � 2γ
(1�γ)2� (T � Tγ� 1+ γT )
�
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2. The dynamic panel bias
FactIf T also tends to in�nity, then the numerator converges to zero, anddenominator converges to a nonzero constant σ2ε /
�1� γ2
�, hence the
LSDV estimator of γ and αi are consistent.
FactIf T is �xed, then the denominator is a nonzero constant, and bγLSDV andbαi are inconsistent estimators when n is large.
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2. The dynamic panel bias
Theorem (Dynamic panel bias)
In a dynamic panel AR(1) model with individual e¤ects, thesemi-asymptotic bias (with n) of the LSDV estimator on the autoregressiveparameter is equal to:
plimn!∞
(bγLSDV � γ) = �(1+ γ)
�T � Tγ� 1+ γT
�(1� γ)
�T 2 � T � 2γ
(1�γ)2� (T � Tγ� 1+ γT )
�
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2. The dynamic panel bias
Theorem (Dynamic panel bias)
For an AR(1) model, the dynamic panel bias can be rewriten as :
plimn!∞
(bγLSDV � γ) = � 1+ γ
T � 1
�1� 1
T1� γT
1� γ
���1� 2γ
(1� γ) (T � 1)
�1� 1� γT
T (1� γ)
���1
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2. The dynamic panel bias
FactThe dynamic bias of bγLSDV is caused by having to eliminate the individuale¤ects α�i from each observation, which creates a correlation of order(1/T ) between the explanatory variables and the residuals in thetransformed model
(yit � y i ) = γ
0B@yi ,t�1 � y i ,�1|{z}depends on past value of εit
1CA+
0@εit � εi|{z}depends on past value of εit
1A
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2. The dynamic panel bias
Intuition of the dynamic bias
(yit � y i ) = γ (yi ,t�1 � y i ,�1) + (εit � εi )
with cov (y i ,�1, εi ) 6= 0 since
cov (y i ,�1, εi ) = cov
1T
T
∑t=1yi ,t�1,
1T
T
∑t=1
εit
!
= cov
1T
T
∑t=1yi ,t�1,
1T
T
∑t=1
εit
!=
1T 2cov ((yi1 + ...+ yiT�1) , (εi1 + ...+ εiT ))
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2. The dynamic panel bias
Intuition of the dynamic bias
(yit � y i ) = γ (yi ,t�1 � y i ,�1) + (εit � εi ) with cov (y i ,�1, εi ) 6= 0
If we approximate yit by εit (in fact yit also depend on εit�1, εt�2, ...) thenwe have
cov (y i ,�1, εi ) =1T 2cov ((yi1 + ...+ yiT�1) , (εi1 + ...+ εiT ))
' 1T 2(cov (εi ,1, εi ,1) + ...+ (cov (εi ,T�1, εi ,T�1)))
' (T � 1) σ2εT 2
6= 0
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2. The dynamic panel bias
Intuition of the dynamic bias
(yit � y i ) = γ (yi ,t�1 � y i ,�1) + (εit � εi ) with cov (y i ,�1, εi ) 6= 0
If we approximate yit by εit then we have
cov (y i ,�1, εi ) =(T � 1) σ2ε
T 2
By taking into account all the interaction terms, we have shown that
plimn!∞
1n
n
∑i=1y i ,�1εi = cov (y i ,�1, εi ) =
σ2εT 2
�(T � 1) γ� 1+ γT
�(1� γ)2
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2. The dynamic panel bias
Remarks
plimn!∞
(bγLSDV � γ) = � 1+ γ
T � 1
�1� 1
T1� γT
1� γ
���1� 2γ
(1� γ) (T � 1)
�1� 1� γT
T (1� γ)
���11 When T is large, the right-hand-side variables become asymptoticallyuncorrelated.
2 For small T , this bias is always negative if γ > 0.
3 The bias does not go to zero as γ goes to zero.
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2. The dynamic panel bias
0 0.2 0.4 0.6 0.8 10.3
0.25
0.2
0.15
0.1
0.05
0
Sem
iasy
mpt
otic
bias
Dynam ic panel bias
T=10T=30T=50T=100
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2. The dynamic panel bias
0 0.2 0.4 0.6 0.8 10.2
0
0.2
0.4
0.6
0.8
1
sem
iasy
mpt
otic
bias
T=10
True value ofplim of the LSDV estimator
0 0.2 0.4 0.6 0.8 10.2
0
0.2
0.4
0.6
0.8
1
sem
iasy
mpt
otic
bias
T=30
True value ofplim of the LSDV estimator
0 0.2 0.4 0.6 0.8 10.2
0
0.2
0.4
0.6
0.8
1
sem
iasy
mpt
otic
bias
T=50
True value ofplim of the LSDV estimator
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sem
iasy
mpt
otic
bias
T=100
True value ofplim of the LSDV estimator
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2. The dynamic panel bias
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9120
100
80
60
40
20
0
rela
tive
bias
(in
%)
Dynam ic bias for T=10 (in % of the true value)
T=10T=30T=50T=100
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2. The dynamic panel bias
Monte Carlo experiments
How to check these semi-asymptotic formula with Monte Carlosimulations?
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2. The dynamic panel bias
Step 1: parameters
Let assume that γ = 0.5, σ2ε = 1 and εiti .i .d .� N (0, 1) .
Simulate n individual e¤ects α�i once at all. For instance, we can use auniform distribution
α�i � U[�1,1]
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2. The dynamic panel bias
Step 2: Monte Carlo pseudo samples
Simulate n (typically n = 1, 000) i.i.d. sequences fεitgTt=1 for a givenvalue of T (typically T = 10)
Generate n sequences fyitgTt=1 for i = 1, .., n with the model:
yit = γyi ,t�1 + α�i + εit
Repeat S times the step 2 in order to generate S = 5, 000 sequencesny (s)it
oTt=1
for s = 1, ..,S for each cross-section unit i = 1, ..., n
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2. The dynamic panel bias
Step 3: LSDV estimates on pseudo series
For each pseudo sample s = 1, ...,S , consider the empirical model
y sit = γy si ,t�1 + αi + µit i = 1, .., n t = 1, ...T
and compute the LSDV estimates bγsLSDV .Compute the average bias of the LSDV estimator bγLSDV based onthe S Monte Carlo simulations
av .bias =1S
S
∑s=1bγsLSDV � γ
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2. The dynamic panel bias
Step 4: Semi-asymptotic bias
1 Repeat this experiment for various cross-section dimensions n:when n increases,the average bias should converge to
plimn!∞
(bγLSDV � γ) = � 1+ γ
T � 1
�1� 1
T1� γT
1� γ
���1� 2γ
(1� γ) (T � 1)
�1� 1� γT
T (1� γ)
���12 Repeat this this experiment for various time dimensions T : when Tincreases,the average bias should converge to 0.
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2. The dynamic panel bias
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2. The dynamic panel bias
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2. The dynamic panel bias
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2. The dynamic panel bias
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2. The dynamic panel bias
0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38
hat
0
50
100
150
200
250
300
350
Num
ber o
f sim
ulat
ions
Histogram of the LSDV estimates for=0.5, T=10 and n=1000
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2. The dynamic panel bias
Click me!
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2. The dynamic panel bias
0 200 400 600 800 1000Sample size n
0.18
0.175
0.17
0.165
0.16
0.155
0.15
Theoretical semiasymptotic biasMC average bias
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2. The dynamic panel bias
Question: What is the importance of the dynamic bias in micro-panels?�Macroeconomists should not dismiss the LSDV bias as
insigni�cant. Even with a time dimension T as large as 30, we�nd that the bias may be equal to as much 20% of the true valueof the coe¢ cient of interest.� (Judson et Owen, 1999, page 10)
Judson R.A. and Owen A. (1999), Estimating dynamic panel data models: aguide for macroeconomists. Economics Letters, 1999, vol. 65, issue 1, 9-15.
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2. The dynamic panel bias
Finite Sample results (Monte Carlo simulations)
n T γ Avg. bγLSDV Avg. bias
10 10 0.5 0.3282 �0.171850 10 0.5 0.3317 �0.1683100 10 0.5 0.3338 �0.166210 50 0.5 0.4671 �0.032950 50 0.5 0.4688 �0.0321100 50 0.5 0.4694 �0.0306
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2. The dynamic panel bias
Finite Sample results (Monte Carlo simulations)
n T γ Avg. bγLSDV Avg. bias
10 10 �0.3 �0.3686 �0.068650 10 �0.3 �0.3743 �0.0743100 10 �0.3 �0.3753 �0.075310 50 �0.3 �0.3134 �0.013450 50 �0.3 �0.3133 �0.0133100 50 �0.5 �0.3142 �0.0142
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2. The dynamic panel bias
Fact (smearing e¤ect)The LSDV for dynamic individual-e¤ects model remains biased with theintroduction of exogenous variables if T is small; for details of thederivation, see Nickell (1981); Kiviet (1995).
yit = α+ γyi ,t�1 + β0xit + αi + εit
In this case, both estimators bγLSDV and bβLSDV are biased.
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2. The dynamic panel bias
What are the solutions?
Consistent estimator of γ can be obtained by using:
1 ML or FIML (but additional assumptions on yi0 are necessary)
2 Feasible GLS (but additional assumptions on yi0 are necessary)
3 LSDV bias corrected (Kiviet, 1995)
4 IV approach (Anderson and Hsiao, 1982)
5 GMM approach (Arenallo and Bond, 1985)
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2. The dynamic panel bias
What are the solutions?
Consistent estimator of γ can be obtained by using:
1 ML or FIML (but additional assumptions on yi0 are necessary)
2 Feasible GLS (but additional assumptions on yi0 are necessary)
3 LSDV bias corrected (Kiviet, 1995)
4 IV approach (Anderson and Hsiao, 1982)
5 GMM approach (Arenallo and Bond, 1985)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 64 / 209
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2. The dynamic panel bias
Key Concepts Section 2
1 AR(1) panel data model
2 Semi-asymptotic bias
3 Dynamic panel bias (Nickell�s bias)
4 Monte Carlo experiments
5 Magnitude of the dynamic panel bias
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Section 3
The Instrumental Variable (IV) approach
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Subsection 3.1
Reminder on IV and 2SLS
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3.1 Reminder on IV and 2SLS
Objectives
1 De�ne the endogeneity bias and the smearing e¤ect.
2 De�ne the notion of instrument or instrumental variable.
3 Introduce the exogeneity and relevance properties of an instrument.
4 Introduce the notion of just-identi�ed and over-identi�ed systems.
5 De�ne the IV estimator and its asymptotic variance.
6 De�ne the 2SLS estimator and its asymptotic variance.
7 De�ne the notion of weak instrument.
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3.1 Reminder on IV and 2SLS
Consider the (population) multiple linear regression model:
y = Xβ+ ε
y is a N � 1 vector of observations yj for j = 1, ..,N
X is a N �K matrix of K explicative variables xjk for k = 1, .,K andj = 1, ..,N
β = (β1..βK )0 is a K � 1 vector of parameters
ε is a N � 1 vector of error terms εi with (spherical disturbances)
V (εjX) = σ2IN
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3.1 Reminder on IV and 2SLS
Endogeneity we assume that the assumption A3 (exogeneity) is violated:
E (εjX) 6= 0N�1
withplim
1NX0ε = E (xj εj ) = γ 6= 0K�1
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3.1 Reminder on IV and 2SLS
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= β
where β denotes the true value of the parameters. This bias is called theendogeneity bias.
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3.1 Reminder on IV and 2SLS
Theorem (Inconsistency of the OLS estimator)
If the regressors are endogenous with plim N�1X0ε = γ, the OLSestimator of β is inconsistent
plim bβOLS = β+Q�1γ
where Q = plim N�1X0X.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 72 / 209
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3.1 Reminder on IV and 2SLS
Proof: Given the de�nition of the OLS estimator:
bβOLS =�X0X
��1 X0y=
�X0X
��1 X0 (Xβ+ ε)
= β+�X0X
��1 �X0ε�We have:
plim bβOLS = β+ plim�1NX0X
��1� plim
�1NX0ε�
= β+Q�1γ 6= β
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3.1 Reminder on IV and 2SLS
Remarks
plim bβOLS = β+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 e¤ect: the inconsistency due to theendogeneity of the one variable is smeared across all of the leastsquares estimators.
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3.1 Reminder on IV and 2SLS
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).
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3.1 Reminder on IV and 2SLS
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�0 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
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3.1 Reminder on IV and 2SLS
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3.1 Reminder on IV and 2SLS
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3.1 Reminder on IV and 2SLS
Question: What is the solution to the endogeneity issue?
The use of instruments..
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3.1 Reminder on IV and 2SLS
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 (xjkzjh) 6= 0
for h 2 f1, ..,Hg and k 2 f1, ..,Kg.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 80 / 209
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3.1 Reminder on IV and 2SLS
Assumptions: The instrumental variables satisfy the following properties.
Well behaved data:
plim1NZ0Z = QZZ a �nite H �H positive de�nite matrix
Relevance:
plim1NZ0X = QZX a �nite H �K positive de�nite matrix
Exogeneity:
plim1NZ0ε = 0K�1
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3.1 Reminder on IV and 2SLS
De�nition (Instrument properties)We assume that the H instruments are linearly independent:
E�Z0Z
�is non singular
or equivalentlyrank
�E�Z0Z
��= H
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3.1 Reminder on IV and 2SLS
The exogeneity condition
E ( εj j zj ) = 0 =) E (εjzj ) = 0H
can expressed as an orthogonality condition or moment condition
E
0@ zj(H ,1)
�yj � x0jβ
�(1,1)
1A = 0H(H ,1)
So, we have H equations and K unknown parameters β
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 83 / 209
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3.1 Reminder on IV and 2SLS
De�nition (Identi�cation)
The system is identi�ed if there exists a unique vector β such that:
E�zj�yj � x0jβ
��= 0
where zj = (zj1..zjH )0 . 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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3.1 Reminder on IV and 2SLS
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 (2SLS)
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).
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3.1 Reminder on IV and 2SLS
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 86 / 209
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3.1 Reminder on IV and 2SLS
Assumption: Consider a just-identi�ed model
H = K
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3.1 Reminder on IV and 2SLS
Motivation of the IV estimator
By de�nition of the instruments:
plim1NZ0ε = plim
1NZ0 (y�Xβ) = 0K�1
So, we have:
plim1NZ0y =
�plim
1NZ0X
�β
or equivalently
β =
�plim
1NZ0X
��1plim
1NZ0y
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 88 / 209
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3.1 Reminder on IV and 2SLS
De�nition (Instrumental Variable (IV) estimator)
If H = K , the Instrumental Variable (IV) estimator bβIV of parametersβ is de�ned as to be: bβIV = �Z0X��1 Z0y
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3.1 Reminder on IV and 2SLS
De�nition (Consistency)
Under the assumption that plim N�1Z0ε = 0, the IV estimator bβIV isconsistent: bβIV p! β
where β denotes the true value of the parameters.
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3.1 Reminder on IV and 2SLS
Proof: By de�nition:
bβIV = �Z0X��1 Z0y = β+
�1NZ0X
��1 � 1NZ0ε�
So, we have:
plimbβIV = β+
�plim
1NZ0X
��1 �plim
1NZ0ε�
Under the assumption of exogeneity of the instruments
plim1NZ0ε = 0K�1
So, we haveplim bβIV = β �
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3.1 Reminder on IV and 2SLS
De�nition (Asymptotic distribution)
Under some regularity conditions, the IV estimator bβIV is asymptoticallynormally distributed:
pN�bβIV � β
�d! N
�0K�1, σ2Q�1ZXQZZQ
�1ZX
�where
QZZK�K
= plim1NZ0Z QZX
K�K= plim
1NZ0X
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3.1 Reminder on IV and 2SLS
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 �Z0X��1 �Z0Z� �X0Z��1
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3.1 Reminder on IV and 2SLS
Remarks
1 If the system is just identi�ed H = K ,�Z0X
��1=�X0Z
��1QZX = QXZ
the estimator can also written as
bVasy
�bβIV � = bσ2 �Z0X��1 �Z0Z� �Z0X��12 As usual, the estimator of the variance of the error terms is:
bσ2 = bε0bεN �K =
1N �K
N
∑i=1
�yi � x0i bβIV �2
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3.1 Reminder on IV and 2SLS
Relevant instruments
1 Our analysis thus far has focused on the �identi�cation�conditionfor IV estimation, that is, the �exogeneity assumption,�whichproduces
plim1NZ0ε = 0K�1
2 A growing literature has argued that greater attention needs to begiven to the relevance condition
plim1NZ0X = QZX a �nite H �K positive de�nite matrix
with H = K in the case of a just-identi�ed model.
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3.1 Reminder on IV and 2SLS
Relevant instruments (cont�d)
plim1NZ0X = 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.
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3.1 Reminder on IV and 2SLS
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
1NZ0X ' 0H�K
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3.1 Reminder on IV and 2SLS
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�1Z0X ' 0H�K , the estimated asymptotic variancecovariance is also very large since
bVasy
�bβIV � = bσ2 �Z0X��1 �Z0Z� �X0Z��1
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3.1 Reminder on IV and 2SLS
Assumption: Consider an over-identi�ed model
H > K
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3.1 Reminder on IV and 2SLS
Introduction
If Z contains more variables than X, then much of the preceding derivationis unusable, because Z0X will be H �K with
rank�Z0X
�= K < H
So, the matrix Z0X has no inverse and we cannot compute the IVestimator as: bβIV = �Z0X��1 Z0y
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3.1 Reminder on IV and 2SLS
Introduction (cont�d)
The crucial assumption in the previous section was the exogeneityassumption
plim1NZ0ε = 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.
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3.1 Reminder on IV and 2SLS
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 �Z0Z��1 Z0XWith this choice of instrumental variables, bX for Z, we have
bβ2SLS =�bX0X��1 bX0y
=�X0Z
�Z0Z
��1 Z0X��1 X0Z �Z0Z��1 Z0y
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3.1 Reminder on IV and 2SLS
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 = �bX0X��1 bX0ywhere bX = Z �Z0Z��1 Z0X corresponds to the projection of the columns ofX in the column space of Z, or equivalently by
bβ2SLS = �X0Z �Z0Z��1 Z0X��1 X0Z �Z0Z��1 Z0y
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3.1 Reminder on IV and 2SLS
Remark
By de�nition bβ2SLS = �bX0X��1 bX0ySince bX = Z �Z0Z��1 Z0X = PZXwhere PZ denotes the projection matrix on the columns of Z. Reminder:PZ is symmetric and PZP0Z = PZ . So, we have
bβ2SLS =�X0P0ZX
��1 bX0y=
�X0P0ZPZX
��1 bX0y=
�bX0bX��1 bX0yC. Hurlin (University of Orléans) Advanced Econometrics II April 2018 104 / 209
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3.1 Reminder on IV and 2SLS
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 = �bX0bX��1 bX0yIt 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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3.1 Reminder on IV and 2SLS
A procedure to get the 2SLS estimator is the following
Step 1: Regress each explicative variable xk (for k = 1, ..K ) on the Hinstruments.
xkj = α1z1j + α2z2j + ..+ αH zHj + vj
Step 2: Compute the OLS estimators bαh and the �tted values bxkjbxkj = bα1z1j + bα2z2j + ..+ bαH zHj
Step 3: Regress the dependent variable y on the �tted values bxki :
yj = β1bx1j + β2bx2j + ..+ βKbxKj + εj
The 2SLS estimator bβ2SLS then corresponds to the OLS estimatorobtained in this model.
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3.1 Reminder on IV and 2SLS
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.
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3.1 Reminder on IV and 2SLS
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.
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3.1 Reminder on IV and 2SLS
Key Concepts SubSection 3.1
1 Endogeneity bias and smearing e¤ect.
2 Instrument or instrumental variable.
3 Exogeneity and relavance properties of an instrument.
4 Instrumental Variable (IV) estimator.
5 Two-Stage Least Square (2SLS) estimator.
6 Weak instrument.
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Subsection 3.2
Anderson and Hsiao (1982) IV approach
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3.2 Anderson and Hsiao (1982) IV approach
Objectives
1 Introduce the IV approach of Anderson and Hsiao (1982).
2 Describe their 4 steps estimation procedure.
3 Introduce the �rst di¤erence transformation of the dynamic model.
4 Describe their choice of instruments.
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3.2 Anderson and Hsiao (1982) IV approach
Consider a dynamic panel data model with random individual e¤ects:
yit = γyi ,t�1 + β0xit + ρ
0ωi + αi + εit
αi are the (unobserved) individual e¤ects,
xit is a vector of K1 time-varying explanatory variables,
ωi is a vector of K2 time-invariant variables.
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3.2 Anderson and Hsiao (1982) IV approach
Assumption: we assume that the component error term vit = εit + αi
E (εit ) = 0, E (αi ) = 0
E (εit εjs ) = σ2ε if j = i and t = s, 0 otherwise.
E (αiαj ) = σ2α if j = i , 0 otherwise.
E (αixit ) = 0, E (αiωi ) = 0 (exogeneity assumption for ωi )
E (εitxit ) = 0, E (εitωi ) = 0 (exogeneity assumption for xit)
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3.2 Anderson and Hsiao (1982) IV approach
The K1 +K2 + 3 parameters to estimate are
yit = γyi ,t�1 + β0xit + ρ
0ωi + αi + εit
1 γ the autoregressive parameter,
2 β is the K1 � 1 vector of parameters for the time-varying explanatoryvariables,
3 ρ is the K2 � 1 vector of parameters for the time-invariant variables,
4 σ2ε and σ2α the variances of the error terms.
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3.2 Anderson and Hsiao (1982) IV approach
Remark
If the vector ωi includes a constant term, the associated parameter can beinterpreted as the mean of the (random) individual e¤ects
yit = γyi ,t�1 + β0xit + ρ
0ωi + αi + εit
α�i = µ+ αi E (αi ) = 0
ωi(K2,1)
=
1CCA ρ(K2,1)
=
0BB@µρ2...
ρK2
1CCA
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3.2 Anderson and Hsiao (1982) IV approach
Vectorial form:
yi = yi ,�1γ+ Xi β+ω0iρe + αie + εi
εi , yi and yi ,�1 are T � 1 vectors (T is the adjusted sample size),
Xi a T �K1 matrix of time-varying explanatory variables,
ωi is a K2 � 1 vector of time-invariant variables,
e is the T � 1 unit vector, and
E (αi ) = 0 E�
αix0it
�= 0 E
�αiω
0i
�= 0
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3.2 Anderson and Hsiao (1982) IV approach
In the dynamic panel data models context:
The Instrumental Variable (IV) approach was �rst proposed byAnderson and Hsiao (1982).
They propose an IV procedure with 2 choices of instruments and 4steps to estimate γ, β, ρ and σ2ε .
Anderson, T.W., and C. Hsiao (1982). Formulation and Estimation ofDynamic Models Using Panel Data, Journal of Econometrics, 18, 47�82.
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3.2 Anderson and Hsiao (1982) IV approach
The Anderson and Hsiao (1982) IV approach
1 First step: �rst di¤erence transformation
2 Second step: choice of instruments and IV estimation of γ and β
3 Third step: estimation of ρ
4 Fourth step: estimation of the variances σ2α and σ2ε
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3.2 Anderson and Hsiao (1982) IV approach
The Anderson and Hsiao (1982) IV approach
1 First step: �rst di¤erence transformation
2 Second step: choice of instruments and IV estimation of γ and β
3 Third step: estimation of ρ
4 Fourth step: estimation of the variances σ2α and σ2ε
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3.2 Anderson and Hsiao (1982) IV approach
First step: �rst di¤erence transformation
Taking the �rst di¤erence of the model, we obtain for t = 2, ..,T .
(yit � yi ,t�1) = γ (yi ,t�1 � yi ,t�2) + β0(xit � xi ,t�1) + εit � εi ,t�1
The �rst di¤erence transformation leads to "lost" one observation.
But, it allows to eliminate the individual e¤ects (as the Withintransformation).
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3.2 Anderson and Hsiao (1982) IV approach
The Anderson and Hsiao (1982) IV approach
1 First step: �rst di¤erence transformation
2 Second step: choice of instruments and IV estimation of γ and β
3 Third step: estimation of ρ
4 Fourth step: estimation of the variances σ2α and σ2ε
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3.2 Anderson and Hsiao (1982) IV approach
Second step: choice of the instruments and IV estimation
(yit � yi ,t�1) = γ (yi ,t�1 � yi ,t�2) + β0(xit � xi ,t�1) + εit � εi ,t�1
A valid instrument zit should satisfy
E (zit (εit � εi ,t�1)) = 0 Exogeneity property
E (zit (yi ,t�1 � yi ,t�2)) 6= 0 Relevance property
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3.2 Anderson and Hsiao (1982) IV approach
Anderson and Hsiao (1982) propose two valid instruments:
1 First instrument: zi ,t = yi ,t�2
E (yi ,t�2 (εit � εi ,t�1)) = 0 Exogeneity property
E (yi ,t�2 (yi ,t�1 � yi ,t�2)) 6= 0 Relevance property
2 Second instrument: zi ,t = (yi ,t�2 � yi ,t�3)
E ((yi ,t�2 � yi ,t�3) (εit � εi ,t�1)) = 0 Exogeneity property
E ((yi ,t�2 � yi ,t�3) (yi ,t�1 � yi ,t�2)) 6= 0 Relevance property
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3.2 Anderson and Hsiao (1982) IV approach
Remarks
The initial �rst di¤erences model includes K1 + 1 regressors.
The regressor (yi ,t�1 � yi ,t�2) is endogeneous.
The regressors (xit � xi ,t�1) are assumed to be exogeneous.
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3.2 Anderson and Hsiao (1982) IV approach
De�nition (Instruments)
Anderson and Hsiao (1982) consider two sets of K1 + 1 instruments, inboth cases the system is just identi�ed (IV estimator):
zi(K1+1,1)
=
yi ,t�2(1,1)
: (xit � xi ,t�1)(1,K1)
0!0
zi(K1+1,1)
=
(yi ,t�2 � yi ,t�3)
(1,1): (xit � xi ,t�1)
(1,K1)
0!0
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3.2 Anderson and Hsiao (1982) IV approach
IV estimator with the �rst set of instruments� bγIVbβIV�=�Z0X
��1 Z0y = n
∑i=1
T
∑t=2
(yi ,t�1 � yi ,t�2) yi ,t�2 yi ,t�2 (xit � xi ,t�1)
0
(xit � xi ,t�1) yi ,t�2 (xit � xi ,t�1) (xit � xi ,t�1)0
!!�1
�
n
∑i=1
T
∑t=2
�yi ,t�2
xit � xi ,t�1
�(yi ,t � yi ,t�1)
!
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3.2 Anderson and Hsiao (1982) IV approach
IV estimator with the second set of instruments� bγIVbβIV�=�Z0X
��1 Z0y = n
∑i=1
T
∑t=3
(yi ,t�1 � yi ,t�2) (yi ,t�2 � yi ,t�3) (yi ,t�2 � yi ,t�3) (xit � xi ,t�1)0
(xit � xi ,t�1) (yi ,t�2 � yi ,t�3) (xit � xi ,t�1) (xit � xi ,t�1)0
!�1
�
n
∑i=1
T
∑t=3
�yi ,t�2 � yi ,t�3xit � xi ,t�1
�(yi ,t � yi ,t�1)
!
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3. Instrumental variable (IV) estimators
Remarks
1 The �rst estimator (with zit = yi ,t�2) has an advantage over thesecond one (with zit = yi ,t�2 � yi ,t�3), in that the minimum numberof time periods required is two, whereas the �rst one requires T � 3.
2 In practice, if T � 3, the choice between both depends on thecorrelations between (yi ,t�1 � yi ,t�2) and yi ,t�2 or (yi ,t�2 � yi ,t�3)=> relevance assumption.
Anderson, T.W., and C. Hsiao (1981). Estimation of Dynamic Models withError Components, Journal of the American Statistical Association, 76,598�606
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3.2 Anderson and Hsiao (1982) IV approach
The Anderson and Hsiao (1982) IV approach
1 First step: �rst di¤erence transformation
2 Second step: choice of instruments and IV estimation of γ and β
3 Third step: estimation of ρ
4 Fourth step: estimation of the variances σ2α and σ2ε
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3.2 Anderson and Hsiao (1982) IV approach
Third stepyit = γyi ,t�1 + β
0xit + ρ
0iωi + αi + εit
Given the estimates bγIV and bβIV , we can deduce an estimate of ρ,the vector of parameters for the time-invariant variables ωi .
Let us consider, the following equation
y i � bγIV y i ,�1 � bβ0IV x i = ρ0ωi + vi i = 1, ..., n
with vi = αi + εi .
The parameters vector ρ can simply be estimated by OLS.
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3.2 Anderson and Hsiao (1982) IV approach
De�nition (parameters of time-invariant variables)A consistent estimator of the parameters ρ is given by
bρ(K2,1)
=
n
∑i=1
ωiω0i
!�1 n
∑i=1
ωihi
!
with hi = y i � bγIV y i ,�1 � bβ0IV x i .
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3.2 Anderson and Hsiao (1982) IV approach
The Anderson and Hsiao (1982) IV approach
1 First step: �rst di¤erence transformation
2 Second step: choice of instruments and IV estimation of γ and β
3 Third step: estimation of ρ
4 Fourth step: estimation of the variances σ2α and σ2ε
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3.2 Anderson and Hsiao (1982) IV approach
Fourth step: estimation of the variances
De�nition
Given bγIV , bβIV , and bρ, we can estimate the variances as follows:bσ2ε = 1
n (T � 1)T
∑t=2
n
∑i=1
bε2itbσ2α = 1
n
n
∑i=1
�y i � bγIV y i ,�1 � bβ0IV x i � bρ0zi�2 � 1
Tbσ2ε
with
bεit = (yi ,t � yi ,t�1)� bγIV (yi ,t�1 � yi ,t�2)� bβ0IV (xi ,t � xi ,t�1)C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 133 / 209
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3.2 Anderson and Hsiao (1982) IV approach
TheoremThe instrumental-variable estimators of γ, β, and σ2ε are consistent whenn (correction of the Nickell bias), or T , or both tend to in�nity.
plimn,T!∞
bγIV = γ plimn,T!∞
bβIV = β plimn,T!∞
bσ2ε = σ2ε
The estimators of ρ and σ2α are consistent only when n goes to in�nity.
plimn!∞
bρ = ρ plimn!∞
bσ2α = σ2α
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 134 / 209
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3.2 Anderson and Hsiao (1982) IV approach
Key Concepts SubSection 3.2
1 Anderson and Hsiao (1982) IV approach.
2 The 4 steps of the estimation procedure.
3 First di¤erence transformation of the dynamic panel model.
4 Tow choices of instrument.
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Section 4
Generalized Method of Moment (GMM)
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4. The GMM approach
Let us consider the same dynamic panel data model as in section 3:
yit = γyi ,t�1 + β0xit + ρ
0ωi + αi + εit
αi are the (unobserved) individual e¤ects,
xit is a vector of K1 time-varying explanatory variables,
ωi is a vector of K2 time-invariant variables.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 137 / 209
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4. The GMM approach
Assumptions: we assume that the component error term vit = εit + αi
E (εit ) = 0, E (αi ) = 0
E (εit εjs ) = σ2ε if j = i and t = s, 0 otherwise.
E (αiαj ) = σ2α if j = i , 0 otherwise.
E (αixit ) = 0, E (αiωi ) = 0 (exogeneity assumption for ωi )
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4. The GMM approach
De�nition (First di¤erence model)The GMM estimation method is based on a model in �rst di¤erences, inorder to swip out the individual e¤ects αi and th variables ωi :
(yit � yi ,t�1) = γ (yi ,t�1 � yi ,t�2) + β0(xit � xi ,t�1) + εit � εi ,t�1
for t = 2, ..,T .
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 139 / 209
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4. The GMM approach
Intuition of the moment conditions
Notice that yi ,t�2 and (yi ,t�2 � yi ,t�3) are not the only validinstruments for (yi ,t�1 � yi ,t�2).
All the lagged variables yi ,t�2�j , for j � 0, satisfy
E (yi ,t�2�j (εi ,t � εi ,t�1)) = 0 Exogeneity property
E (yi ,t�2�j (yi ,t�1 � yi ,t�2)) 6= 0 Relevance property
Therefore, they all are legitimate instruments for (yi ,t�1 � yi ,t�2).
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 140 / 209
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4. The GMM approach
Intuition of the moment conditions
The m+ 1 conditions
E (yi ,t�2�j (εi ,t � εi ,t�1)) = 0 for j = 0, 1, ..,m
can be used as moment conditions in order to estimate
θ =�
β,γ, ρ, σ2α, σ2ε
�Arellano, M., and S. Bond (1991). �Some Tests of Speci�cation for PanelData: Monte Carlo Evidence and an Application to Employment Equations,�Review of Economic Studies, 58, 277�297.
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4. The GMM approach
Remark: The moment conditions
E (yi ,t�2�j (εi ,t � εi ,t�1)) = 0 for j = 0, 1, ..,m
mean that there exists a vector of parameters (true value)
θ0 =�
β00,γ0, ρ
00, σ
2α0, σ
2ε0
�0such that
E�yi ,t�2�j �
�∆yit � γ0∆yi ,t�1 � β
00∆xit
��= 0
where ∆ = (1� L) and L denotes the lag operator .
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4. The GMM approach
We consider two alternative assumptions on the explanatory variables xit
1 The explanatory variables xit are strictly exogeneous.
2 The explanatory variables xit are pre-determined.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 143 / 209
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4. The GMM approach
We consider two alternative assumptions on the explanatory variables xit
1 The explanatory variables xit are strictly exogeneous.
2 The explanatory variables xit are pre-determined.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 144 / 209
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4. The GMM approach
Assumption: exogeneity
We assume that the time varying explanatory variables xit are strictlyexogeneous in the sense that:
E�x0it εis
�= 0 8 (t, s)
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4. The GMM approach
De�nition (moment conditions)For each period, we have the following orthogonal conditions
E (qit∆εit ) = 0, t = 2, ..,T
qit(t�1+TK1,1)
=�yi0, yi1, .., yi ,t�2, x
0i
�0where x
0i =
�x0i1, .., x
0iT
�, ∆ = (1� L) and L denotes the lag operator
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 146 / 209
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4. The GMM approach
Example (moment conditions)
The condition E (qit∆εit ) = 0, qit = (yi0, yi1, .., yi ,t�2, x 0i )0at time t = 2
becomes
E
qi2
(1+TK1,1)∆εi2(1,1)
!= E
��yi0x 0i
�(εi2 � εi1)
�= 0(1+TK1,1)
where x0i =
�x0i1, .., x
0iT
�. At time t = 3, we have
E
qi3
(2+TK1,1)∆εi3(1,1)
!= E
0@0@ yi0yi1x 0i
1A (εi3 � εi2)
1A = 0(2+TK1,1)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 147 / 209
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4. The GMM approach
Under the exogeneity assumption, what is the number of momentconditions?
E (qit∆εit ) = 0, t = 2, ..,T
Time Number of moment conditions
t = 2 1+ TK1t = 3 2+ TK1... ...
t = T T � 1+ TK1total T (T � 1) (K1 + 1/2)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 148 / 209
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4. The GMM approach
Proof: the total number of moment conditions is equal to
r = 1+ TK1 + 2+ TK1..+ TK1 + (T � 1)= T (T � 1)K1 + 1+ 2+ ..+ (T � 1)
= T (T � 1)K1 +T (T � 1)
2
= T (T � 1)�K1 +
12
�
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4. The GMM approach
Stacking the T � 1 �rst-di¤erenced equations in matrix form, we have
∆yi(T�1,1)
= ∆yi ,�1(T�1,1)
γ(1,1)
+ ∆Xi(T�1,K1)
β(K1,1)
+ ∆εi(T�1,1)
i = 1, ..,N
where :
∆yi(T�1,1)
=
0BB@yi2 � yi1yi3 � yi2..
yiT � yi ,T�1
1CCA ∆yi ,�1(T�1,1)
=
0BB@yi1 � yi0yi2 � yi1..
yiT�1 � yi ,T�2
1CCA
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 150 / 209
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4. The GMM approach
Stacking the T � 1 �rst-di¤erenced equations in matrix form, we have
∆yi(T�1,1)
= ∆yi ,�1(T�1,1)
γ(1,1)
+ ∆Xi(T�1,K1)
β(K1,1)
+ ∆εi(T�1,1)
i = 1, ..,N
where :
∆Xi(T�1,K1)
=
0BB@xi2 � xi1xi3 � xi2..
xiT � xi ,T�1
1CCA ∆εi(T�1,1)
=
0BB@εi2 � εi1εi3 � εi2..
εiT � εi ,T�1
1CCA
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 151 / 209
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4. The GMM approach
De�nition (moment conditions)
The conditions E (qit∆εit ) = 0 for t = 2, ..,T , can be written as
E
Wi
(r ,T�1)∆εi
(T�1,1)
!= 0(m,1)
Wi =
0BBBBBB@
qi2(1+TK1,1)
0 ... 0
0 qi3(2+TK1,1)
..0 .. qiT
(T�1+TK1,1)
1CCCCCCAwhere r = T (T � 1) (K1 + 1/2) is the number of moment conditions.
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4. The GMM approach
Example (moment conditions, vectorial form)At time t = 2, we have
E (qi2∆εi2) = E
��yi0x 0i
�(εi2 � εi1)
�= 0
In a vectorial form we have the �rst set of 1+ TK1 moment conditions
E (Wi∆εi ) = E
0BB@� qi2(1+TK1,1)
0 ... 0�0BB@
εi2 � εi1εi3 � εi2..
εiT � εi ,T�1
1CCA1CCA = 0
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 153 / 209
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4. The GMM approach
Example (moment conditions, vectorial form)At time t = 3, we have
E (qi3∆εi3) = E
0@0@ yi0yi1x 0i
1A (εi3 � εi2)
1A = 0
In a vectorial form we have the second set of 2+ TK1 moment conditions
E (Wi∆εi ) = E
0BB@� 0 qi3(2+TK1,1)
... 0�0BB@
εi2 � εi1εi3 � εi2..
εiT � εi ,T�1
1CCA1CCA = 0
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4. The GMM approach
ExampleFor T = 10 et K1 = 0 (without explicative variable), we have
r =T (T � 1)
2= 45 orthogonal conditions
ExampleFor T = 50 et K1 = 0 (without explicative variable), we have
r =T (T � 1)
2= 1225 orthogonal conditions !!
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 155 / 209
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4. The GMM approach
0 10 20 30 40 50 60 70 80 90 1000
500
1000
1500
2000
2500
3000
3500
4000
4500
5000Number of orthogonal conditions
T
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 156 / 209
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4. The GMM approach
We consider two alternative assumptions on the explanatory variables xit
1 The explanatory variables xit are strictly exogeneous.
2 The explanatory variables xit are pre-determined.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 157 / 209
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4. The GMM approach
We consider two alternative assumptions on the explanatory variables xit
1 The explanatory variables xit are strictly exogeneous.
2 The explanatory variables xit are pre-determined.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 158 / 209
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4. The GMM approach
Assumption: pre-determination
We assume that the time varying explanatory variables xit arepre-determined in the sense that:
E�x 0it εis
�= 0 if t � s
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 159 / 209
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4. The GMM approach
In this case, we have
E (qit∆εit ) = 0, t = 2, ..,T
qit(t�1+tK1,1)
=
0B@yi0, yi1, .., yi ,t�2, x 0i1, .., x 0i ,t�2| {z }not T
1CA0
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 160 / 209
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4. The GMM approach
De�nitionThe conditions E (qit∆εit ) = 0 for t = 2, ..,T , can be written as
E
Wi
(r ,T�1)∆εi
(T�1,1)
!= 0(m,1)
Wi =
0BBBBBB@
qi2(1+K1,1)
0 ... 0
0 qi3(2+2K1,1)
..0 .. qiT
(T�1+(T�1)K1,1)
1CCCCCCAwhere r = T (T � 1) (K1 + 1) /2 is the number of moment conditions.
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4. The GMM approach
Proof: the total number of moment conditions is equal to
r = 1+K1 + 2+K1..+ (T � 1)K1 + (T � 1)= (1+K1) (1+ 2+ ...+ (T � 1))
= (1+K1)T (T � 1)
2
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4. The GMM approach
0 10 20 30 40 50 60 70 80 90 1000
5000
10000
15000Number of orthogonal conditions (K1=1)
T
X exogeneousX predetermined
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4. The GMM approach
FactWhatever the assumption made on the explanatory variable, the number ofothogonal conditions (moments) r is much larger than the number ofparameters, e.g. K1 + 1. Thus, Arellano and Bond (1991) suggest ageneralized method of moments (GMM) estimator.
Arellano, M., and S. Bond (1991). �Some Tests of Speci�cation for PanelData: Monte Carlo Evidence and an Application to Employment Equations,�Review of Economic Studies, 58, 277�297.
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4. The GMM approach
We will exploit the moment conditions
E (Wi∆εi ) = 0
to estimate by GMM the parameters θ =�γ, β0
�0 in∆yi = ∆yi ,�1γ+ ∆Xi β+ ∆εi i = 1, .., n
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Subsection 4.1
GMM: a general presentation
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4.1 GMM: a general presentation
De�nitionThe standard method of moments estimator consists of solving theunknown parameter vector θ by equating the theoretical moments withtheir empirical counterparts or estimates.
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4.1 GMM: a general presentation
1 Suppose that there exist relations m (yt ; θ) such that
E (m (yt ; θ0)) = 0
where θ0 is the true value of θ and m (yt ; θ0) is a r � 1 vector.2 Let bm (y , θ) be the sample estimates of E (m (yt ; θ)) based on nindependent samples of yt
bm (y , θ) = 1n
n
∑t=1m (yt ; θ)
3 Then the method of moments consit in �nding bθ, such thatbm �y ,bθ� = 0
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4.1 GMM: a general presentationIntuition of the GMM
Consider the moment conditions such that
E (m (yt ; θ0)) = 0
Under some regularity assumptions, 8θ 2 Θ
bm (y , θ) = 1n
n
∑t=1m (yt ; θ)
p! E (m (yt ; θ))
In particular bm (y , θ0) p! E (m (yt ; θ0)) = 0
So, the GMM consists in �nding bθ such thatbm �y ,bθ� = 0 =) bθ p! θ0
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4.1 GMM: a general presentation
Fact (just identi�ed system)
If the number r of equations (moments) is equal to the dimension a of θ, itis in general possible to solve for bθ uniquely. The system is just identi�ed.
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4.1 GMM: a general presentation
Example (classical method of moment)
Consider a random variable yt � t (v). We want to estimate v from ani.i.d. sample fy1, ..yng. We know that:
µ2 = E�y2t�= V (yt ) =
vv � 2
If µ2 is known, then v can be identi�ed as:
v =2E�y2t�
E (y2t )� 1
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4.1 GMM: a general presentation
Example (classical method of moment)
Now let us consider the sample variance bµ2,Tbµ2 = 1
n
n
∑t=1y2t
p�! µ2
So, a consistent estimate (classical method of moment) of v is de�ned by:
bv = 2bµ2bµ2 � 1
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4.1 GMM: a general presentation
Example (classical method of moment)Another way to write the problem consists in de�ning
m (yt ; v) = y2t �v
v � 2
By de�nition, we have:
E (m (yt ; v)) = E
�y2t �
vv � 2
�= 0
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4.1 GMM: a general presentation
Example (classical method of moment)
The moment condition (r = 1) is
E (m (yt ; v)) = E
�y2t �
vv � 2
�= 0
The empirical counterpart is
bm (y ; v) = 1n
n
∑t=1m (yt ; v) =
1n
n
∑i=1
�y2t �
vv � 2
�So, the estimator bv of the classical method of moment is de�ned by:
bm (y ; bv) = 0 , bv = 2bµ2bµ2 � 1 p�! v =2E�y2t�
E (y2t )� 1
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4.1 GMM: a general presentation
De�nition (over-identi�ed system)
If the number of moments r is greater than the dimension of θ, the systemof non linear equation bm (y ; bv) = 0, in general, has no solution. Thesystem is over-identi�ed.
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4.1 GMM: a general presentation
If the system is over-identi�ed, it is then necessary to minimize some norm(or distance measure) of bm (y ; θ) in order to �nd bθ :
q (y , θ) = bm (y ; θ)0 S�1 bm (y ; θ)where S�1 is a positive de�nite (weighting) matrix.
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4.1 GMM: a general presentation
Example (weigthing matrix)
Consider a random variable yt � t (v). We want to estimate v from ani.i.d. sample fy1, ..yng. We know that:
µ2 = E�y2t�=
vv � 2
µ4 = E�y4t�=
3v2
(v � 2) (v � 4)The two moment conditions (r = 2) can be written as
E (m (yt ; v)) = E
y2t � v
v�2y4t � 3v 2
(v�2)(v�4)
!=
�00
�
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4.1 GMM: a general presentation
Example (weigthing matrix)
It is impossible to �nd a unique value bv such thatbm (y ; bv) = 1
n
n
∑t=1m (yt ; bv) = 1
n ∑nt=1 y
2t � bvbv�2
1n ∑n
t=1 y2t � 3bv 2
(bv�2)(bv�4)!=
�00
�So, we have to impose some weights to the two moment conditions
bm (y ; v)0 S�1 bm (y ; v)
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4.1 GMM: a general presentation
Example (weigthing matrix)Let us assume that
S�1 =�1 00 2
�then we have
bm (y ; v)0 S�1 bm (y ; v) =
1n
n
∑t=1y2t �
vv � 2
!2
+2
1n
n
∑t=1y2t �
3v2
(v � 2) (v � 4)
!2
It is now possible to �nd bv such that bm (y ; v)0 S�1 bm (y ; v) = 0C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 179 / 209
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4.1 GMM: a general presentation
De�nition (GMM estimator)
The GMM estimator bθ minimizes the following criteriabθ = argmin
θ2Raq (y , θ)(1,1)
= argminθ2Ra
bm (y ; θ)0(1,r )
S�1(r ,r )
bm (y ; θ)(r ,1)
where S�1 is the optimal weighting matrix.
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4.1 GMM: a general presentation
What is the optimal weigthing matrix?
bθ = argminθ2Ra
q (y , θ)(1,1)
= argminθ2Ra
bm (y ; θ)0(1,r )
S�1(r ,r )
bm (y ; θ)(r ,1)
The optimal choice (if there is no autocorrelation of m (y ; θ0)) of Sturns out to be
S(r ,r )
= E
m (y ; θ0)(r ,1)
m (y ; θ0)(1,r )
0!
The matrix S corresponds to variance-covariance matrix of the vectorm (y ; θ0).
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4.1 GMM: a general presentation
De�nition (Optimal weighting matrix)In the general case, the optimal weighting matrix is the inverse of thelong-run variance covariance matrix of m (yt ; θ0).
S(r ,r )
=∞
∑j=�∞
E
m (yt ; θ0)
(r ,1)m (yt�j ; θ0)
(1,r )
0!
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4.1 GMM: a general presentation
Remark
The optimal weighting matrix is
S =∞
∑j=�∞
E�m (yt ; θ0)m (yt�j ; θ0)
0�We can replace the unknow value θ0 by the GMM estimator θ̂ and theoptimal weighting matrix becomes
S =∞
∑j=�∞
E
�m�yt ;bθ�m �yt�j ;bθ�0�
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4.1 GMM: a general presentation
Problem 1 How to estimate S?
S =∞
∑j=�∞
E
�m�yt ;bθ�m �yt�j ;bθ�0�
A �rst solution (too) simple solution consits in using the empiricalcounterparts of variance and covariances
bS = n�2∑
j=�(n�2)bΓj
bΓj = 1n
n
∑t=j+2
m�yt ;bθ�m �yt�j ;bθ�0
But, this estimator may be no positive de�nite...
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4.1 GMM: a general presentation
Solution (Non-parametric kernel estimators)A positive de�nite kernel estimator for S has been proposed by Newey andWest (1987) and is de�ned as
bS = bΓ0 + q
∑j=1
�1� j
q + 1
��bΓj + bΓ0j�
bΓj = 1n
n
∑t=j+2
m�yt ;bθ�m �yt�j ;bθ�0
where q is a bandwidth parameter and K (j) = 1� j/ (q + 1) a Bartlettkernel function.
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4.1 GMM: a general presentation
Example (Newey and West kernel estimator)
bS = bΓ0 + q
∑j=1
�1� j
q + 1
��bΓj + bΓ0j�If q = 2 then we have
bS = bΓ0 + 23 �bΓ1 + bΓ01�+ 13 �bΓ2 + bΓ02�
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4.1 GMM: a general presentation
Other estimators => other kernel functions
bS = bΓ0 + q
∑j=1K�
jq + 1
��bΓj + bΓ0j�1 Gallant (1987): Parzen kernel
K (u) =
8<:1� 6 juj2 + 6 juj32 (1� juj)30
if 0 � juj � 1/2if 1/2 � juj � 1otherwise
2 Andrews (1991): quadratic spectral kernel
K (u) =3
(6πu/5)2
�sin (6πu/5)(6πu/5)
� cos (6πu/5)�
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4.1 GMM: a general presentation
Problem 2 bθ = argminθ2Ra
bm (y ; θ)0 S�1 bm (y ; θ)S =
∞
∑j=�∞
E
�m�yt ;bθ�m �yt�j ;bθ�0�
1 In order to compute bθ, we have to know S�1.2 In order to compute S , we have to know bθ... a circularity issue
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4.1 GMM: a general presentation
Solutions
1 Two-step GMM: Hansen (1982)
2 Iterative GMM: Ferson and Foerster (1994)
3 Continuous-updating GMM: Hansen, Heaton and Yaron (1996),Stock and Wright (2000), Newey and Smith (2003), Ma (2002).
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4.1 GMM: a general presentation
Solutions
1 Two-step GMM: Hansen (1982)
2 Iterative GMM: Ferson and Foerster (1994)
3 Continuous-updating GMM: Hansen, Heaton and Yaron (1996),Stock and Wright (2000), Newey and Smith (2003), Ma (2002).
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4.1 GMM: a general presentation
Two-step GMM
Step 1: put the same weight to the r moment conditions by using anidentity weighting matrix
S0 = Ir
Compute a �rst consistent (but not e¢ cient) estimator bθ0bθ0 = argminθ2Ra
bm (y ; θ)0 S�10 bm (y ; θ)= argmin
θ2Rabm (y ; θ)0 bm (y ; θ)
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4.1 GMM: a general presentation
Two-step GMM
Step 2: Compute the estimator for the optimal weighting matrix bS1bS1 = bΓ0 + q
∑j=1K�
jq + 1
��bΓj + bΓ0j�
bΓj = 1n
n
∑t=j+2
m�yt ;bθ0�m �yt�j ;bθ0�0
Finally, compute the e¢ cient GMM estimator bθ1 asbθ1 = argminθ2Ra
bm (y ; θ)0 bS�11 bm (y ; θ)
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Subsection 4.2
Application to dynamic panel data models
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4.2 Application to dynamic panel data models
Various GMM estimators (i.e. moment conditions) have been proposed fordynamic panel data models
1 Arellano and Bond (1991): GMM estimator
2 Arellano and Bover (1995): GMM estimator
3 Ahn and Schmidt (1995): GMM estimator
4 Blundell and Bond (2000): a system GMM estimator
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4.2 Application to dynamic panel data models
Various GMM estimators (i.e. moment conditions) have been proposed fordynamic panel data models
1 Arellano and Bond (1991): GMM estimator
2 Arellano and Bover (1995): GMM estimator
3 Ahn and Schmidt (1995): GMM estimator
4 Blundell and Bond (2000): a system GMM estimator
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4.2 Application to dynamic panel data models
Problem
Let us consider the dynamic panel data model in �rst di¤erences
∆yi = ∆yi ,�1γ+ ∆Xi β+ ∆εi i = 1, .., n
We want to estimate the K1 + 1 parameters θ =�γ, β0
�0.For that, we have r = T (T � 1) (K1 + 1/2) moment conditions (ifxit are strictly exogeneous)
E (Wi∆εi ) = E (Wi � (∆yi � ∆yi ,�1γ� ∆Xi β)) = 0r
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 196 / 209
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4.2 Application to dynamic panel data models
Let us denote
m (yi , xi ; θ) = Wi � (∆yi � ∆yi ,�1γ� ∆Xi β)
withE (m (yi , xi ; θ)) = 0r
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4.2 Application to dynamic panel data models
De�nition (Arellano and Bond (1991) GMM estimator)
The Arellano and Bond GMM estimator of θ =�γ, β0
�0 isbθ = argmin
θ2RK1+1
1n
n
∑i=1m (yi , xi ; θ)
!0S�1
1n
n
∑i=1m (yi , xi ; θ)
!
or equivalently
bθ = argminθ2RK1+1
1n
n
∑i=1
∆ε0iW0i
!S�1
1n
n
∑i=1Wi∆εi
!
with S = E (m (y ; θ0) m (y ; θ0))0.
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4.2 Application to dynamic panel data models
Under the assumption of non-autocorrelation, the optimal weightingmatrix can be expressed as
S = E
1n2
n
∑i=1Wi∆εi∆ε0iW
0i
!
In the general case, S is the long-run variance covariance matrix ofn�2 ∑n
i=1Wi∆εi∆ε0iW0i .
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4.2 Application to dynamic panel data models
Various GMM estimators (i.e. moment conditions) have been proposed fordynamic panel data models
1 Arellano and Bond (1991): GMM estimator
2 Arellano and Bover (1995): GMM estimator
3 Ahn and Schmidt (1995): GMM estimator
4 Blundell and Bond (2000): a system GMM estimator
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4.2 Application to dynamic panel data models
In addition to the previous moment conditions, Arellano and Bover (1995)also note that E (v i ) = E (εi + αi ) = 0, where
v i = y i � γy i ,�1 � β0x i � ρ0ωi
Therefore, if instruments eqi exist (for instance, the constant 1 is a validinstrument) such that
E (eqiv i ) = 0then a more e¢ cient GMM estimator can be derived by incorporating thisadditional moment condition.
Arellano, M., and O. Bover (1995). �Another Look at the InstrumentalVariable Estimation of Error-Components Models,� Journal of Econometrics,68, 29�51.
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4.2 Application to dynamic panel data models
De�nitionArellano and Bond (1991) consider the following moment conditions
E (m (yi , xi ; θ)) = E (Wi (∆yi � ∆yi ,�1γ� ∆Xi β)) = 0
De�nitionArellano and Bover (1995) consider additional moment conditions
E (m (yi , xi ; θ)) = E�eqi �y i � γy i ,�1 � β0x i � ρ0ωi
��= 0
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4.2 Application to dynamic panel data models
Various GMM estimators (i.e. moment conditions) have been proposed fordynamic panel data models
1 Arellano and Bond (1991): GMM estimator
2 Arellano and Bover (1995): GMM estimator
3 Ahn and Schmidt (1995): GMM estimator
4 Blundell and Bond (2000): a system GMM estimator
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4.2 Application to dynamic panel data models
Apart from the previous linear moment conditions, Ahn and Schmidt(1995) note that the homoscedasticity condition on E
�ε2it�implies the
following T � 2 linear conditions
E (yit∆εi ,t+1 � yi ,t+1∆εi ,t+1) = 0 t = 1, ..,T � 2
Combining these restrictions to the previous ones leads to a more e¢ cientGMM estimator.
Ahn, S.C., and P. Schmidt (1995). �E¢ cient Estimation of Models forDynamic Panel Data,� Journal of Econometrics, 68, 5�27.
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4.2 Application to dynamic panel data models
Various GMM estimators (i.e. moment conditions) have been proposed fordynamic panel data models
1 Arellano and Bond (1991): GMM estimator
2 Arellano and Bover (1995): GMM estimator
3 Ahn and Schmidt (1995): GMM estimator
4 Blundell and Bond (2000): a system GMM estimator
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4.2 Application to dynamic panel data models
De�nition (system GMM)
The system GMM (Blundell and Bond) was invented to tackle the weakinstrument problem. It consists in considering both the equation in leveland in �rst di¤erences
E (yit ,�s∆εit ) = 0 E (xi ,t�s∆εit ) = 0 Di¤erence equation
Following additional moments are explored:
E (∆yit ,�s (α�i + εit )) = 0 E (∆xi ,t�s (α�i + εit )) = 0 Level equation
Blundell and Bond, S. (2000): GMM Estimation with persistent panel data:an application to production functions. Econometric Reviews,19(3), 321-340.
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4.2 Application to dynamic panel data models
Remarks
1 While theoretically it is possible to add additional moment conditionsto improve the asymptotic e¢ ciency of GMM, it is doubtful howmuch e¢ ciency gain one can achieve by using a huge number ofmoment conditions in a �nite sample.
2 Moreover, if higher-moment conditions are used, the estimator can bevery sensitive to outlying observations.
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4.2 Application to dynamic panel data models
Remarks
1 Through a simulation study, Ziliak (1997) has found that thedownward bias in GMM is quite severe as the number of momentconditions expands, outweighing the gains in e¢ ciency.
2 The strategy of exploiting all the moment conditions for estimation isactually not recommended for panel data applications. For furtherdiscussions, see Judson and Owen (1999), Kiviet (1995), andWansbeek and Bekker (1996).
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End of Chapter 2
Christophe Hurlin (University of Orléans)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 209 / 209