Chapter 2. Dynamic panel data models - univ-orleans.fr · 2. The dynamic panel bias Objectives 1 Introduce the AR(1) panel data model. 2 Derive the semi-asymptotic bias of the LSDV

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.


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.


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.


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


Section 2

The Dynamic Panel Bias


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.



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.



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.



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.



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.



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



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



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 )

!



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 )



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



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



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



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



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



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



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



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.



Let us examine this plim

plimn!∞

1n

n

∑i=1y i ,�1εi

We know that


= 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



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



yi ,t�1 = εi ,t�1 + γεi ,t�2 + γ2εi ,t�3 + ...+ γt�2εi1 +1� γt�1


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



yi ,t�1 = εi ,t�1 + γεi ,t�2 + γ2εi ,t�3 + ...+ γt�2εi1 +1� γt�1


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



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



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

!



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 )

�



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



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



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

!



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

�



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 )

�



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.



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 )

�



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



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



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 ))




(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




(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



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.



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



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


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


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




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



Monte Carlo experiments

How to check these semi-asymptotic formula with Monte Carlosimulations?



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]



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:


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



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



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.











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



Click me!



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



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.



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



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



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.



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)



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)



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


Section 3

The Instrumental Variable (IV) approach


Subsection 3.1

Reminder on IV and 2SLS


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.



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



Endogeneity we assume that the assumption A3 (exogeneity) is violated:

E (εjX) 6= 0N�1

withplim

1NX0ε = E (xj εj ) = γ 6= 0K�1



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.



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.



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



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.



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).



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







Question: What is the solution to the endogeneity issue?

The use of instruments..



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.



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



De�nition (Instrument properties)We assume that the H instruments are linearly independent:

E�Z0Z

�is non singular

or equivalentlyrank

�E�Z0Z

��= H



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



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.



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).





Assumption: Consider a just-identi�ed model

H = K



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



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



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.



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 �



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



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



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



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


with H = K in the case of a just-identi�ed model.



Relevant instruments (cont�d)


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.



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



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



Assumption: Consider an over-identi�ed model

H > K



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



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.



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



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



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


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.



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.



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.



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.



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.


Subsection 3.2

Anderson and Hsiao (1982) IV approach


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.



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.



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)



The K1 +K2 + 3 parameters to estimate are


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.



Remark

If the vector ωi includes a constant term, the associated parameter can beinterpreted as the mean of the (random) individual e¤ects


0ωi + αi + εit

α�i = µ+ αi E (αi ) = 0

ωi(K2,1)

=

[email protected]

1CCA ρ(K2,1)

=

0BB@µρ2...

ρK2

1CCA



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



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.



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










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).










Second step: choice of the instruments and IV estimation


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



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



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.



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



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)

!



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)

!


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










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.



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 .










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


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



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.


Section 4

Generalized Method of Moment (GMM)


4. The GMM approach

Let us consider the same dynamic panel data model as in section 3:


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.


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 )


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 :


for t = 2, ..,T .


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).


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.


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 .


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.


4. The GMM approach





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)


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


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)


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)


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

�


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


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


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.


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


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


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 !!


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


4. The GMM approach





4. The GMM approach





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


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


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.


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


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


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.


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


Subsection 4.1

GMM: a general presentation


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.



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


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



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.



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




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



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




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



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.



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.



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

�



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)



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


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.



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).



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!



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�



Problem 1 How to estimate S?

S =∞

∑j=�∞

E


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...



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.



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�



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)�



Problem 2 bθ = argminθ2Ra

bm (y ; θ)0 S�1 bm (y ; θ)S =

∞

∑j=�∞

E


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



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).



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).



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)



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)


Subsection 4.2

Application to dynamic panel data models


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










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



Let us denote

m (yi , xi ; θ) = Wi � (∆yi � ∆yi ,�1γ� ∆Xi β)

withE (m (yi , xi ; θ)) = 0r



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.



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 .










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.



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










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.










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.



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.



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).


End of Chapter 2

Christophe Hurlin (University of Orléans)


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