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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
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
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
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
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
Section 2
The Dynamic Panel Bias
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 6 / 209
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 )
�
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 30 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 31 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 32 / 209
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
!
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 33 / 209
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
�
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 34 / 209
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 )
�
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 35 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 36 / 209
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 )
�
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 37 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 38 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 39 / 209
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 ))
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 40 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 41 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 42 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 43 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 44 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 45 / 209
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
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tive
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(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
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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)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 63 / 209
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
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.
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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.
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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
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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
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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
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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
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)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 113 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 114 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 115 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 116 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 117 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 118 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 119 / 209
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).
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 120 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 121 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 122 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 123 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 124 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 125 / 209
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)
!
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 126 / 209
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)
!
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 127 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 128 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 129 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 130 / 209
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 .
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 131 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 132 / 209
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
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
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 135 / 209
Section 4
Generalized Method of Moment (GMM)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 136 / 209
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
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 )
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 138 / 209
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
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
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 141 / 209
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 .
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 142 / 209
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
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
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)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 145 / 209
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
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
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
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
�
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 149 / 209
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
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
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 152 / 209
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
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 154 / 209
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
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
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
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
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
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
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 161 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 162 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 163 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 164 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 165 / 209
Subsection 4.1
GMM: a general presentation
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 166 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 167 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 168 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 169 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 170 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 171 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 172 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 173 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 174 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 175 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 176 / 209
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
�
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 177 / 209
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)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 178 / 209
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
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 180 / 209
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).
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 181 / 209
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!
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 182 / 209
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�
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 183 / 209
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...
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 184 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 185 / 209
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�
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 186 / 209
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)�
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 187 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 188 / 209
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).
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 189 / 209
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).
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 190 / 209
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)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 191 / 209
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)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 192 / 209
Subsection 4.2
Application to dynamic panel data models
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 193 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 194 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 195 / 209
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
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 197 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 198 / 209
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 .
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 199 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 200 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 201 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 202 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 203 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 204 / 209
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
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 205 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 206 / 209
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.
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 207 / 209
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).
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 208 / 209
End of Chapter 2
Christophe Hurlin (University of Orléans)
C. Hurlin (University of Orléans) Advanced Econometrics II April 2018 209 / 209