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UNIVERSITEIT GENT GHENT UNIVERSITY

FACULTEIT ECONOMIE EN BEDRIJFSKUNDE FACULTY OF ECONOMICS AND BUSINESS

ADMINISTRATION

ACADEMIC YEAR 2016 – 2017

A Comparison of Dynamic Panel Data Estimators Using Monte Carlo

Simulations and the Firm Growth Model

Masterproef voorgedragen tot het bekomen van de graad van

Master’s Dissertation submitted to obtain the degree of

Master of Science in Business Economics

Bob Mertens

Under the guidance of Peter Beyne

Prof. Philippe Van Cauwenberge

UNIVERSITEIT GENT GHENT UNIVERSITY

FACULTEIT ECONOMIE EN BEDRIJFSKUNDE FACULTY OF ECONOMICS AND BUSINESS

ADMINISTRATION

ACADEMIC YEAR 2016 – 2017

A Comparison of Dynamic Panel Data Estimators Using Monte Carlo

Simulations and the Firm Growth Model

Masterproef voorgedragen tot het bekomen van de graad van

Master’s Dissertation submitted to obtain the degree of


Bob Mertens

Under the guidance of Peter Beyne

Prof. Philippe Van Cauwenberge

Admission to Loan

The author gives permission to make this master’s dissertation available for consultation and

to copy parts of this master’s dissertation for personal use. In the case of any other use, the

limitations of the copyright have to be respected, in particular with regard to the obligation to

state expressly the source when quoting results from this master dissertation.

Bob Mertens, June 2017

A Comparison of Dynamic Panel DataEstimators Using Monte Carlo Simulations

and the Firm Growth Modelby

Bob Mertens

Master’s Dissertation submitted to obtain the academic degree of


Academic 2016–2017 Promoter: Prof. Philippe VAN CAUWENBERGE

Supervisor: Peter Beyne

Faculty of Economics and Business Administration

Ghent University

Department of Accounting, Corporate Finance and Taxation

Samenvatting [Dutch]

In deze masterproef worden 4 schatters vergeleken, gebruikt bij het schatten van dynamis-

che panel data modellen: de Fixed Effects schatter, de Anderson-Hsiao IV schatter, de First-

Differenced GMM schatter en de System GMM schatter. De vergelijking wordt uitgevoerd

op basis van verschillende Monte Carlo simulaties, geımplementeerd in MATLAB. De indeling

van deze thesis is als volgt: In Hoofdstuk 1 worden het doel en de motivatie voor deze thesis

toegelicht. Vervolgens wordt in Hoofdstuk 2 het ’Firm Growth Model’ besproken. Dit dynamisch

panel data model wordt immers gebruikt tijdens de Monte Carlo simulaties en de 4 schatters

zullen dus vergeleken worden bij het schatten van de parameters in dit specifiek model. In

Hoofdstuk 3 wordt de werking van de Fixed Effects schatter besproken, alsook zijn voordelen en

tekortkomingen. Na het behandelen van deze eerder eenvoudige schatter, worden in Hoofdstuk

4 de 3 meer gecompliceerde schatters gepresenteerd die gebruikt worden bij het schatten van

dynamische panel data modellen: de Anderson-Hsiao IV schatter, de First-Differenced GMM

schatter en de System GMM schatter. Ook voor deze schatters worden de werking, voordelen en

tekortkomingen kort besproken. In Hoofdstuk 5 komen we uiteindelijk aan bij de kern van deze

thesis, namelijk de vergelijking van de 4 bestudeerde schatters m.b.v. Monte Carlo simulaties.

De simulatiemethode wordt besproken, tezamen met de resultaten. Deze resultaten tonen dat

de System GMM schatter duidelijk het best presteert bij het schatten van de parameters in het

Firm Growth Model. De System GMM schatter is zowel consistenter als efficienter dan de 3

andere schatters. Ten slotte sluit Hoofdstuk 6 deze thesis af met een conclusie en een laatste

overzicht van de behaalde resultaten.

Keywords

Dynamic Panel Data Models; Firm Growth Model; Fixed Effects estimator; Anderson-Hsiao IV

estimator; First- Differenced GMM estimator; System GMM estimator; Monte Carlo simulation

Preface

This master’s dissertation concludes my Master in Business Economics at Ghent University.

This economical education broadened my horizons and taught me the basics in economics and

business administration. This master, however, is not my first, as I already obtained a degree in

electrical engineering in 2015. Since I like mathematics, this master’s dissertation can be quite

mathematical at some points.

I would like to thank my promoter Prof. Philippe Van Cauwenberge for granting me the oppor-

tunity to write this thesis under his supervision. He introduced me to Peter Beyne, researcher

at the Department of Accounting, Corporate Finance and Taxation, who’s guidance led me to

this particular subject. Many thanks to you as well, Peter.

Lastly, my parents and girlfriend deserve special thanks. My parents gave me the chance to

pursue an extra master in economics after my engineering education and my girlfriend always

supported me during this education.

Bob Mertens, June 2017

CONTENTS i

Contents

List of Abbreviations iii

List of Figures iv

List of Tables v

1 Introduction 1

1.1 The Need for Reliable Dynamic Panel Data Estimators . . . . . . . . . . . . . . . 1

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The Firm Growth Model 3

3 The Fixed Effects Estimator and Bias 5

3.1 Presentation of the Fixed Effects Estimator . . . . . . . . . . . . . . . . . . . . . 5

3.2 The Nickell Bias of the Fixed Effects Estimator . . . . . . . . . . . . . . . . . . . 7

4 Estimation Techniques for Dynamic Panel Data 9

4.1 The Idea of Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 The Anderson-Hsiao Instrument Variable Estimator . . . . . . . . . . . . . . . . 11

4.3 Introduction to the Generalized Method of Moments . . . . . . . . . . . . . . . . 13

4.4 The First-Differenced Generalized Method of Moments Estimator . . . . . . . . . 15

4.5 The System Generalized Method of Moments Estimator . . . . . . . . . . . . . . 18

5 Monte Carlo Simulations 23

5.1 General Set-Up of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Configuration of the Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.3 MATLAB Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

CONTENTS ii

5.4.1 Simulation results for β1 + 1 = 0.1 . . . . . . . . . . . . . . . . . . . . . . 29



5.4.4 Conclusion simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Conclusion 35

A Matlab Implementation of Estimators and Monte Carlo Simulations 37

A.1 FE final.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A.2 AHIV final.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A.3 FDGMM final.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A.4 SGMM final.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A.5 MC simulations final.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.6 MC results processing final.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Bibliography 47

CONTENTS iii

List of Abbreviations

AHIV Anderson-Hsiao Instrument

Variable

FDGMM First-Differenced GMM

FE Fixed Effects

GMM Generalized Method of Moments

IV Instrument Variable

OLS Ordinary Least Squares

SGMM System GMM

LIST OF FIGURES iv

List of Figures

5.1 Schematic diagram representing MATLAB implementation . . . . . . . . . . . . 28

5.2 MC simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.3 MC simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4 MC simulation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.5 MC simulation 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.6 MC simulation 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.7 MC simulation 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.8 MC simulation 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.9 MC simulation 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.10 MC simulation 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.11 MC simulation 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.12 MC simulation 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.13 MC simulation 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

LIST OF TABLES v

List of Tables

5.1 Descriptive statistics of variables ln(S) and ln(A) . . . . . . . . . . . . . . . . . . 26

5.2 Estimates for the population parameters. . . . . . . . . . . . . . . . . . . . . . . 26

5.3 Configuration of simulations to be performed . . . . . . . . . . . . . . . . . . . . 27

INTRODUCTION 1

Chapter 1

Introduction

1.1 The Need for Reliable Dynamic Panel Data Estimators

The use of dynamic panel data models in present-day empirical economic research has become

more and more popular. As the name of these models indicates, dynamic panel data models

make use of panel data and are dynamic. These two components immediately explain the pop-

ularity of these type of models. The first component is the usage of panel data in these models.

Panel data consists of multiple cross-sectional units, e.g. individuals, households, firms, or coun-

tries, observed at different points in time. This type of data delivers an obvious advantage over

cross-sectional data: we cannot estimate dynamic models from observations at a single point in

time, and it is rare for single cross-section surveys to provide sufficient information about earlier

time periods for dynamic relationships to be investigated. It is clear that the combination of

cross-sectional data and time series data allows for richer econometric models and more accurate

conclusions. The second component of these models is the dynamic nature of these models. In

a dynamic models, past observations of the variable of interest can influence the current value

of that variable. In that way, dynamic adjustment processes can be analyzed for a broad base

of cross-sectional units.

The increasing use of dynamic panel data models requires the availability of reliable dynamic

panel data estimators. In this master’s dissertation, 4 dynamic panel data estimators, suggested

in the literature, are compared: the Fixed Effects estimator, the Anderson-Hsiao Instrument

Variable (IV) estimator, the First-Differenced Generalized Method of Moments (GMM) estima-

tor, and the System GMM estimator. The comparison of the estimators is done using Monte

INTRODUCTION 2

Carlo simulations, which are implemented in MATLAB. During this type of simulation, the co-

efficients of a certain dynamic panel data model are estimated repeatedly over a large number of

iterations. In every iteration, a new panel data set is created using a well defined statistical dis-

tribution. The underlying econometric model, used during the simulations, is the ’Firm Growth

Model’. Each of the 4 estimators is then compared in terms of consistency and efficiency.

1.2 Outline

Chapter 2 of this master’s dissertation discusses the ’Firm Growth Model’. This dynamic panel

data model is used during the Monte Carlo simulations. The historical development of this

model is shortly described, and the final form of the model is presented. In Chapter 3, the Fixed

Effects model is explained, and its shortcomings when used in dynamic panel data analysis

are highlighted. After covering the rather simple Fixed Effects estimator, Chapter 4 presents

3 more sophisticated estimators used in dynamic panel data estimation: the Anderson-Hsiao

IV estimator, the First-Differenced GMM estimator, and the System GMM estimator. The 3

estimators are explained and their advantages and disadvantages are highlighted. In Chapter 5

we finally arrive at the core of this master’s dissertation: the Monte Carlo simulations of the 4

presented estimators. The simulation method, implemented in MATLAB, is explained in detail

and the results of the simulations are presented. Chapter 6 concludes this work and gives a last

overview of the accomplished results.

THE FIRM GROWTH MODEL 3

Chapter 2

The Firm Growth Model

During the Monte Carlo simulations carried out in this thesis, the 4 investigated dynamic panel

data estimators will repeatedly estimate the coefficients of a dynamic panel data model. There-

fore, such a model needs to be chosen in order to compare the 4 estimators in terms of consistency

and efficiency. In this study, the ’Firm Growth Model’ will be used for this purpose. The ’Firm

Growth Model’ is frequently used when analyzing dynamic panel data in order to study the in-

fluence of certain economic variables on firm growth. Furthermore, the model was already used

at UGent for analyzing the influence of municipal fiscal policy on firm growth. This model has

been developed by consecutive research by different researchers and is built upon Gibrat’s law,

which summarizes the conventional wisdom on the relationship between firm size and growth.

It asserts that the probability of a proportionate increase in firm size over an interval in time is

the same for all firms, regardless of their size at the beginning of the interval. i.e., Gibrat’s law

does not assumes a relationship between firm growth and firm size. Much early work supported

Gibrat’s law (Hart and Prais, 1956; Simon and Bonini, 1958; Hymer and Pashigian, 1962; Lucas,

1967), but later empirical research found that a relationship between firm growth and size does

exist (Manseld, 1962; Kumar, 1985; Hall, 1987; FitzRoy and Kraft, 1991; Mata, 1994). This

discrepancy between theory and evidence has led to new theories emphasizing managerial effi-

ciency and learning-by-doing as key determinants of firm dynamics. Jovanovic (1982) proposed

a learning model in which firm growth and survival are linked to firm size and age through

an initial level of production efficiency. Firms learn their efficiency level through production

experience. When output is a decreasing convex function of managerial inefficiency, the model

implies that younger firms tend to grow faster than older firms.

THE FIRM GROWTH MODEL 4

Starting from these theories, a number of papers have taken firm size and age as the determinants

of firm growth. Following Evans (1987a,b), the growth variable of interest at time t+1 is modeled

as a function of size and age at time t. This relationship in logarithmic form is expressed as:

Gi,t+1 = ln(F (Ai,tSi,t)) + ui,t (2.1)

where Gi,t+1 = ln(Si,t+1) − ln(Si,t) is the growth variable of interest of firm i (i = 1, 2, ..., N)

in period t + 1 (t = 1, 2, ..., T ), Si,t and Ai,t are the size and age of firm i at time t, and ui,t is

the error term. The error term can be broken down into a firm-specific (µi), a time-specific (λt)

and a random error term (vi,t) as:

ui,t = µi,t + λi,t + vi,t (2.2)

Using a flexible translog functional form for F (.) the relation is written as:

Gi,t+1 = β0 + β1ln(Si,t) + β2ln(Ai,t) + β3(ln(Si,t))2 + β4(ln(Ai,t))

2 + β5ln(Si,t)ln(Ai,t) + ui,t

(2.3)

Eq. 2.3 will be used as econometric model when comparing the different estimators in the Monte

Carlo simulations. During these simulations, the coefficients of the model will be defined in ad-

vance and a dynamic panel data set will be created based upon this model. This is done using a

random input of variables and a random error following a certain statistical distribution. More

on the technicalities of these simulations in Chapter 5.

Before discussing the Monte Carlo simulations, the 4 estimators which are studied in this master’s

dissertation will be presented in the next 2 chapters. In Chapter 3 we take a look at the Fixed

Effects estimator, which is sometimes used in the estimation of dynamic panel data models.

This estimator, however, is not very well suited for usage with this type of models. Therefore,

Chapter 4 presents 3 more suitable estimators to be used when dealing with dynamic panel data

models.

THE FIXED EFFECTS ESTIMATOR AND BIAS 5

Chapter 3

The Fixed Effects Estimator and

Bias

3.1 Presentation of the Fixed Effects Estimator

In this chapter, we discuss the first of 4 estimators studied in this master’s dissertation: the Fixed

Effects (FE) estimator. This estimator serves as a starting point as it is a natural choice when

allowing for individual effects in dynamic models. Both the Ordinary Least Squares (OLS) and

Random Effects (RE) estimator are not suited for estimating dynamic models since the lagged

dependent variable in the right-hand side of the model equation will be positively correlated with

the error-term due to the presence of individual effects. This makes the OLS and RE estimators

inconsistent when used in dynamic modeling. Applying the FE estimator to dynamic panel data

models, however, is not without problems either. When the number of time observation points

in the panel data set is not infinite, a so-called Nickell bias will occur (Nickell, 1981). We will

here shortly introduce the FE estimator in a mathematical manner, and show that a bias occurs

when estimating dynamic models.

We start off by introducing a simplified linear dynamic panel data model, containing explanatory

variables xt as well as the lagged endogenous variable yt−1.

yi,t = ρyi,t−1 + x′i,tβ + αi + εi,t (3.1)

In this model, i = 1..N represents the index for individuals, and t = 1..N the index for years. x′i,t

is a row vector, containing the K explanatory variables for individual i at time point t. ρ is the


unknown parameter of the lagged endogenous variable, and β is the unknown parameter vector

for the K explanatory variables. αi represents the individual specific effects (time invariant).

Further, following assumptions are made: the error term is normally distributed: εi,t ∼ N(0, σ2ε ),

the error term is orthogonal to the exogenous variables: E(x′i,tεi,t) = 0 and the error term is un-

correlated with the lagged endogenous variable: E(yi,t−1εi,t) = 0. We, however, do not assume

the exogenous variables to be uncorrelated with the individual effects: E(x′i,tαi) 6= 0.

This model can easily be expressed in matrix notation as follows:

y = y−1ρ+Xβ +Dα+ ε (3.2)

with

y =

y1

y2

...

yN

yi =

yi,1

yi,2

...

yi,T

X =

X1

X2

...

XN

Xi =

x′i,1

x′i,2

...

x′i,T

D = IN × e e =

1

1

...

1

with dimension T α =

α1

α2

...

αN

From the definition of these matrices it can be seen that y is a (N · T × 1) column vector and

X is a (N · T × K) matrix. The matrix formulation of this model is important, as the same

formulation will be used in order to implement the FE estimator in MATLAB.

In this matrix notation, the dummy matrix D is used to insert the individual specific effects

into the model. This dummy matrix is key, since it will help us to estimate the parameters

in this model using the classic OLS estimator. In order to be able to use the OLS estimator,

the values of the variables in the model have to be ’demeaned’, meaning that their deviations

from the individual means need to be calculated. Once the variables are ’demeaned’, the OLS

estimator can be used, since the individual specific effects are filtered out of the variables. This

’demeaning’ is performed by premultiplying the left-hand side and right-hand side of the model


equation with matrix M :

My = My−1ρ+MXβ +MDα+Mε (3.3)

with

M = INT −D(D′D)−1D′ (3.4)

It can be proven that the term MDα in equation 3.3 is equal to zero, which results in the

disappearance of the individual effects from the model. From this point it is possible to use the

classic OLS estimator to estimate the parameters in the model as follows:

β = (X ′MX)−1X ′My (3.5)

The derivation of equation 3.5 was omitted. This method will be used in the implementation of

the FE estimator in MATLAB.

3.2 The Nickell Bias of the Fixed Effects Estimator

After having presented the Fixed Effects estimator, it is now time to study its theoretical perfor-

mance when estimating dynamic models. It will be shown that the FE estimator exhibits a bias

when dealing with lagged endogenous variables. To come to this, let’s consider the demeaned

lagged endogenous variables y∗i,t−1 and the demeaned error term ε∗i,t:

y∗i,t−1 = yi,t−1 −1

T

t=T∑t=1

yi,t−1 (3.6)

ε∗i,t = εi,t −1

T

t=T∑t=1

εi,t (3.7)

The above 2 expressions result from the scalar representation of equation 3.3. From these 2

expressions it can be conducted that the demeaned lagged endogenous variable correlates with

the demeaned error term, or E(y∗i,t−1ε∗i,t) 6= 0. This due to the fact that the error term ε∗i,t−1 is

contained in y∗i,t−1 with weight 1− 1T , and in ε∗i,t with weight − 1

T . This correlation renders the

FE estimates ρ and β biased. From this, it is also obvious that the correlation decreases with

increasing T , i.e. with increasing number of time observation points. Typical panel data sets,

however, contain a large number of individuals (large N), observed over a rather limited amount

of time (relatively small T). Therefore it is of importance to study the Nickell bias of the FE


estimator for the asymptotic case where N tends to infinity (N →∞), while T stays finite. The

exact calculation of this bias is omitted here. The properties of this bias, however, in case of

infinite N and finite T are as follows:

For N → ∞ and T finite, the asymptotic bias of the estimated parameters (biasρ =

ρ− ρ and biasβ = β − β) has following properties:

1. Increasing for increasing ρ

2. Increasing for increasing N (number of individuals in data set)

3. Increasing for increasing σ2ε (variance of error term)

4. Decreasing for increasing T (number of time observation points in data set)

The Nickell bias completely disappears when the number of time observation points in the panel

data set goes to infinity (T → ∞). This is, however, not achievable in real-world data sets. It

is therefore needed to take the Nickell bias into account. Its presence will be demonstrated in

Chapter 5 when testing the FE estimator using Monte Carlo simulations.

ESTIMATION TECHNIQUES FOR DYNAMIC PANEL DATA 9

Chapter 4

Estimation Techniques for Dynamic

Panel Data

In Chapter 3, the first of 4 estimators studied in this master’s dissertation was discussed, namely

the Fixed Effects estimator. Although simple, the FE estimator exhibits a non-negligible bias

when estimating real-world dynamic panel data models. This bias has led to the development

of more suitable estimators to perform dynamic panel data estimation. This chapter treats

3 of those estimators, which have frequently been proposed in the literature: the Anderson-

Hsiao instrument variable (IV) estimator, the First-Differenced generalized method of moments

(GMM) estimator, and the System GMM estimator. Since each of these estimators makes use

of the principle ’instrumentation’, we will first shortly introduce this technique at the start of

the chapter. Subsequently the 3 mentioned estimators will be presented.

4.1 The Idea of Instrumentation

In Chapter 3, it was shown that a bias occurs in the FE estimates due to the correlation of the

error term with one of the regressors (in this case a lagged dependent variable). The goal of

instrumentation is to prevent this bias resulting from the correlation between the regressor x and

the error term ε. The whole idea of instrumentation in estimators can be summarized as follows:

’Find a variable Z (the instrument), that is highly correlated with the regressor X,

but does not correlate with the error term ε. Then use as new regressor that part

of X which correlates with the instrument Z and is orthogonal to the error term ε.’


The use of instrumentation can be made transparent by expressing it as a two stage process.

In the first stage, the explanatory variable X is regressed on the instrument Z. This leads to a

derived explanatory variable X, which contains the linear dependent part of X on Z. This new

variable X will be uncorrelated with the error term ε, since it only contains the linear dependent

part of X on Z, and Z was chose such that it is uncorrelated with ε. X is then used in the

second stage as new explanatory variable for the main regression to be performed. In matrix

form, this method is executed as follows:

The goal is to estimate the parameter in the regression equation y = Xβ+ε using an instrument

for the explanatory variable X.

First stage: regression of explanatory variable X on instrument variable Z

X = Zγ + ν (4.1)

The regression values X, obtained through OLS estimation of γ are then:

X = Zγ = Z(Z ′Z)−1Z ′X (4.2)

Second stage: usage of variables X in main regression with following regression equation:

y = Xβ + ε (4.3)

β = (XX ′)−1X ′y (4.4)

Again the estimate for the parameter β was obtained using the OLS estimation expression. If

we then insert the result of equation 4.2 into the obtained expression for β (equation 4.4), we

obtain following end result for the estimation of the parameter β:

β = (X ′PX)−1X ′Py (4.5)

with

P = Z(Z ′Z)−1Z ′ (4.6)

This is an important result, since it is the general expression for the instrument variable estimator

and will be used in the next section of this chapter.


4.2 The Anderson-Hsiao Instrument Variable Estimator

The Anderson-Hsiao estimator (Anderson/Hsiao, 1982) is a first estimator which tries to estimate

dynamic panel data models without introducing a bias in the estimates. This estimator uses the

instrumentation principle, explained in the previous section, combined with a differenced form

of the original regression equation. Let’s start again from the linear dynamic model introduced

in Chapter 3 when discussing the Fixed Effects estimator:

yi,t = ρyi,t−1 + x′i,tβ + α+ εi,t (4.7)

For this model, the same conventions are used as in Chapter 3, as well as the same assumptions.

Instead of working with this standard form of the equation, the Anderson-Hsiao estimator is

based on the differenced form of the regression equation:

yi,t − yi,t−1 = ρ(yi,t−1 − yi,t−2) + (x′i,t − x′i,t−1)β + εi,t − εi,t−1 (4.8)

which cancels out the individual effects, which have been assumed to possibly correlate with

the exogenous variables. Next, we can express both the original equation and the differenced

equation in matrix form:

y = y−1ρ+Xβ +Dα+ ε original (4.9)

Fy = Fy−1ρ+ FXβ + FDα+ Fε differenced (4.10)

where the vectors and matrices have the exact same form as in Chapter 3, and with matrix F

defined as:

F = IN × FT and FT =

−1 1 0 ... 0 0

0 −1 1 ... 0 0...

0 0 0 ... −1 1

with dimension (T − 1)× T

As can be deducted from its definition, premultiplication of the data matrices with matrix F

results in the differences of these variables. It is therefore true that FD = 0, which cancels out

the individual fixed effects. This is valuable, since it prevents the correlation between the fixed

effects and the explanatory variable to bias the estimate. We are, however, not out of troubles

yet. If we take at look at the difference of the lagged dependent variable

yi,t−1 − yi,t−2 = ρ(yi,t−2 − yi,t−3) + (x′i,t−1 − x′i,t−2)β + εi,t−1 − εi,t−2 (4.11)


we can see that this differenced lagged dependent variable is correlated with the differenced error

term εi,t−εi,t−1 through the presence of εi,t−1 in expression 4.2: E((yi,t−1−yi,t−2)(εi,t−εi,t−1)) 6=

0. This correlation will render the estimate of the parameters in the model biased.

The presence of the above mentioned correlation brings us to the second technique used in this

estimator, namely instrumentation. Anderson and Hsiao suggested 2 possible instruments to be

used for the differenced lagged dependent variable yi,t−1 − yi,t−2: the level instrument yi,t−2 or

the lagged difference yi,t−2−yi,t−3. These instruments can be expected to have a high correlation

with the differenced lagged dependent variable, and are also expected to be uncorrelated with

the differenced error term. Arellano (1989) analyzed the properties of these 2 instruments and

found the estimator using the level instruments to be superior because of its smaller variance

and no points of singularity. The use of the level instruments also has the added advantage

of losing one time observation point less than is the case for the lagged difference instruments.

This can be of relevance in practical use, since data sets with a large number of individuals but

a limited number of time observation points are commonly encountered.

Combining the first-differencing approach with the instrumentation using level instruments, the

Anderson-Hsiao estimator can be formulated as a standard instrument variable estimator. The

parameters of the model can therefore be estimated using the expressions for the IV estimator,

presented in the previous section:

γ = (X ′PX)−1X ′P y (4.12)

with

P = Z(Z ′Z)−1Z ′ (4.13)

In these expressions, all right-hand side variables (lagged dependent variable and explanatory

variables) are brought together in one matrix X. Their corresponding parameters in the model

are grouped into the row vector γ. The instrumented variables are grouped into one large matrix

as well, namely Z. In this matrix, the differenced lagged dependent variables yi,t−1 − yi,t−2 are

replaced by their level instruments, namely yi,t−2. The complete buildup of these matrices is as


follows:

X =

X1

X2

...

XN

Z =

Z1

Z2

...

ZN

y =

y1

y2

...

yN

with

Xi =

yi,2 − yi,1 x′i,3 − x′i,2yi,3 − yi,2 x′i,4 − x′i,3

......

yi,T−1 − yi,T−2 x′i,T − x′i,T−1

Zi =

yi,1 x′i,3 − x′i,2yi,2 x′i,4 − x′i,3

......

yi,T−2 x′i,T − x′i,T−1

yi =

yi,3 − yi,2

yi,4 − yi,3

...

yi,T − yi,T−1

As you can see, the data in these matrices is grouped per individual, and the blocks of individual

data (containing all time observation points for that individual) are then stacked on top of each

other. This structure is important to highlight, as the implementation of the Anderson-Hsiao

estimator in MATLAB will make use of this structure. The performance of this estimator will

be discussed in Chapter 5, using the results of the Monte Carlo simulations.

4.3 Introduction to the Generalized Method of Moments

In the previous section, the Anderson-Hsiao estimator was presented, which is an instrument

variable estimator. The 2 remaining estimators to be discussed, however, are so-called Gener-

alized Method of Moments (GMM) estimators. This section therefore shortly introduces the

concept of GMM. GMM estimation was formalized by Hansen (1982) and since has become

one of the most widely used methods of estimation in economics and finance. This technique


does not require the complete knowledge of the distribution of the data. Only specified moment

conditions, derived from the underlying model, are needed for GMM estimation. These moment

conditions are functions of the model parameters and the data, such that their expectation

is zero at the true values of the parameters. They are also called ’orthogonality conditions’.

Solving these moment conditions for the model parameters then leads to an estimate for these

parameters.

In order to demonstrate this method, we here represent the simple OLS estimation method as

an application of the method of moments. Using this method, we would like to estimate the

parameter vector β in following model: y = Xβ + ε. The moment condition to start from is in

this case the uncorrelatedness of the explanatory variable and the error term:

E(X ′ε) = 0 (4.14)

This moment condition then needs to be applied to the sample data. We obtain:

1

NX ′(y −Xβ) = 0 (4.15)

When this expression is solved for the parameter vector β, we obtain following well known result

for the OLS estimator:

β = (X ′X)−1X ′y (4.16)

In the same way, the instrument variable estimator can be seen as en application of the method of

moments. This time, the moment condition to start from is the assumption that the instrument

Z is orthogonal to the error term ε:

E(Z ′ε) = 0 (4.17)

Again applying this condition to the sample data leads to following expression:

1

NZ ′(y −Xβ) = 0 (4.18)

Solving this expression for the parameter vector β results in

β = (X ′PX)−1X ′Py (4.19)

with

P = Z(Z ′Z)−1Z ′ (4.20)

which is the same as obtained in the first section of this chapter.


4.4 The First-Differenced Generalized Method of Moments Es-

timator

The First-Differenced GMM estimator was developed by Arrelano and Bond (1991), and has

since become increasingly popular in empirical work using firm level or household data. This

estimator is similar to the Anderson-Hsiao estimator, presented in a previous section, but exploits

additional moment conditions, which enlarges the set of instruments to be used. By introducing

extra moment conditions, this estimators becomes a GMM estimator. To understand this, we

once again start from the same dynamic model used in the presentation of the FE estimator:



Just as was the case for the Anderson-Hsiao estimator, the first-differenced GMM estimator

makes use of the differenced model equation:

yi,t − yi,t−1 = ρ(yi,t−1 − yi,t−2) + (x′i,t − x′i,t−1)β + εi,t − εi,t−1 (4.22)

This again allows us to eliminate the individual specific effects from the model.

To understand where the extra set of moment conditions originates from, we take a look at the

level instruments available for instrumenting the differenced lagged dependent variable yi,t−1 −

yi,t−2 at different time observation points. For t = 3, the equation to be estimated is:

yi,3 − yi,2 = ρ(yi,2 − yi,1) + (x′i,3 − x′i,2)β + εi,3 − εi,2 (4.23)

The available level instrument for yi,2 − yi,1 is just yi,1.

At time observation point t = 4, the equation has following form:

yi,4 − yi,3 = ρ(yi,3 − yi,2) + (x′i,4 − x′i,3)β + εi,4 − εi,3 (4.24)

Now, the available level instruments for yi,3 − yi,2 are yi,1 and yi,2. As can be seen, the time

periods available for instrumentation enlarge up to the time observation point t = T where the

instruments yi,1, yi,2, ..., yi,T−2 are available. The First-Differenced GMM estimator makes use

of all these available instruments, and has therefore an enlarged set of moment conditions to

impose on the available data. This leads in theory to an increased consistency and efficiency of


the estimator.

In matrix formulation, these extra instruments are added to the instrument matrix Z:

Z =

Z1

Z2

...

ZN

with Zi =

yi,1 0 0 ... 0 ... 0 x′i,3 − x′i,20 yi,1 yi,2 ... 0 ... 0 x′i,4 − x′i,3...

...... ...

.... . .

......

0 0 0 ... yi,1 ... yi,T−2 x′i,T − x′i,T−1

This matrix is called a ’GMM-style’ instrument matrix. When other regressors in the model are

endogenous (besides the differenced lagged variable yi,t − yi,t−1), this matrix can be extended

with another set of instrument variables in the same way as for the variable yi,t − yi,t−1. The

estimates of the parameters in the model are eventually obtained by imposing following set of

moment conditions on the data:

E(Zi∆εi) = 0 for i = 1...N (4.25)

with

∆εi =

∆εi,3

∆εi,4...

∆εi,T

This results in a set of (T−1)(T−2)

2 +K meaningful moment conditions per individual, and thus

a total of N( (T−1)(T−2)2 +K) moment conditions. This number typically exceeds the number of

unknown parameters in the model. Therefore, the asymptotically efficient GMM estimator is

obtained by minimizing following expression QN :

QN = (1

N

N∑i=1

Z ′i∆εi)′ WN (

1

N

N∑i=1

Z ′i∆εi) (4.26)

Differentiating this expression with respect to the model parameters γ, and next solving for γ

yields:

γ = (X ′ZWNZ′X)−1X ′ZWNZ

′Xy (4.27)


in which X has following form:

X =

X1

X2

...

XN

with Xi =


......

yi,T−1 − yi,T−2 x′i,T − x′i,T−1

and where WN is the weighting matrix, incorporated to deal with heteroscedasticity of the error

term.

In order to determine the optimal weighting matrix WOPTN , a two-step procedure is used. In

the first step, the First-Differenced GMM estimator is limited to the case of no autocorrelation

in the error term εi,t combined with the homoscedasticity assumption (constant variance of the

error term over different individuals and time observation points). By making these assumptions,

the first-step weighting matrix W s1N can easily be computed as follows:

W s1N = (Z ′GZ)−1 (4.28)

with

G = IN ×GT and GT = F ′TFT =

2 −1 0 ...

−1 2. . . 0

0. . .

. . . −1... 0 −1 2

Once this preliminary weighting matrix is determined, a first estimate of the parameters can be

made using expression 4.27. This leads to an interim estimate for γ:

γs1 = (X ′Z W s1N Z

′X)−1X ′Z W s1N Z

′Xy (4.29)

Using this interim estimate from the first step, the residuals for this first-step regression can be

calculated as follows:

∆ε = Y −Xγs1 (4.30)

These residuals are then used in the second step for calculating the optimal weighting matrix

WOPTN :

WOPTN = (Z ′∆ε∆ε′Z)−1 (4.31)


This optimal weighting matrix is then used for calculating the final estimate of γ:

γs2 = (X ′Z WOPTN Z ′X)−1X ′Z WOPT

N Z ′Xy (4.32)

Again this exact matrix formulation will be used in the implementation of the First-Differenced

GMM estimator in Matlab. This implementation will be discussed in Chapter 5, together with

the performance of the estimator in the Monte Carlo simulations.

Although this estimator makes use of an extended set of instruments, it performs poorly under

certain circumstances. The reason behind this poor performance is the problem of weak instru-

ments, meaning that the instruments are only weakly correlated with the endogenous variables

they are trying to instrument. Blundell and Blond (1998) have demonstrated that the instru-

ments used in the First-Differenced GMM estimator become less informative (and therefore

weaker) when:

� yi,t is close to a random walk (no clear trend observable in the dependent variable).

� The variance of the individual effects σ2α is relatively large in comparison to the variance

of the error term σ2ε .

This problem of weak instruments is redressed by introducing another set of instruments, re-

sulting in the System GMM estimator, presented in the last section of this chapter.

4.5 The System Generalized Method of Moments Estimator

To improve the properties of the First-Differenced GMM estimator, Arrelano and Bover (1995)

and Blundell and Bond (1998) suggested to introduce another set of moment conditions, but

this time for the level model equation, instead of for the differenced model equation. This is

realized by using instruments that are orthogonal to the individual effects. The key here is that,

instead of differencing the regressors to eliminate the individual effects (as was the case for the

Anderson-Hsiao and the First-Differenced GMM estimator), now the instruments are differenced

to make them orthogonal to the individual effects.


In order to illustrate this method, we again start from the same dynamic model used in the

presentation of the FE estimator:



It is this level equation that will be used to come up with an additional set of moment condi-

tions to be used in the System GMM estimator with respect to the First-Differenced estimator.

The level endogenous variables in this equation will be instrumented by differenced instruments,

meaning that yi,t−1 will be instrumented by yi,t−1 − yi,t−2 = ∆yi,t−1. The System GMM esti-

mator exploits this new set of instruments, while retaining the original set for the differenced

equation.

In order to capture all these instruments in one matrix formulation, the System GMM estima-

tor involves building a stacked data set with twice the observations, i.e. in the data of each

individual, the differenced observations go on top and the levels below. This results in following

definition for y+i and X+i :

y+i =

yi,3 − yi,2

yi,4 − yi,3

...

yi,T − yi,T−1

yi,3

yi,4

...

yi,T

and X+i =


......

yi,T−1 − yi,T−2 x′i,T − x′i,T−1yi,2 x′i,3

yi,3 x′i,4...

...

yi,T−1 x′i,T

The instrument matrix per individual Z+i for this system can then be written as:


Z+i =

yi,1 0 0 ... 0 ... 0 0 0 0 0 x′i,3 − x′i,20 yi,1 yi,2 ... 0 ... 0′ 0 0 0 0 x′i,4 − xi,3...

...... ...

.... . .

... 0 0 0 0...

0 0 0 ... yi,1 ... yi,T−2 0 0 0 0 x′i,T − x′i,T−10 0 0 0 0 0 0 ∆yi,2 0 0 0 x′i,3

0 0 0 0 0 0 0 0 ∆yi,3 0 0 x′i,4

0 0 0 0 0 0 0...

.... . .

......′

0 0 0 0 0 0 0 0 0 0 ∆yi,T−1 x′i,T

Using theses matrices, the estimates of the parameters can be obtained by using the same

method as for the First-Differenced GMM estimator. Following set of moment conditions are

now imposed on the data:

E(Z+i ∆ε+i ) = 0 for i = 1...N (4.34)

with

∆ε+i =

∆εi,3

∆εi,4...

∆εi,T

εi,3

εi,4...

εi,T

This results in a large set of ( (T−1)(T−2)2 + (T − 2) + K) meaningful moment conditions per

individual, and thus a total of (N( (T−1)(T−2)2 + (T − 2) +K)) moment conditions. This number

typically largely exceeds the number of unknown parameters in the model. Therefore, the

asymptotically efficient GMM estimator is again obtained by minimizing following expression

QN :

QN = (1

N

N∑i=1

Z+′

i ∆ε+i )′ W+N (

1

N

N∑i=1

Z+′

i ∆ε+i ) (4.35)


Differentiating this expression with respect to the model parameters γ, and next solving for γ

yields:

γ = (X+′Z+W+

NZ+′X+)−1X+′

Z+W+NZ

+′X+y+ (4.36)

and where W+N is the weighting matrix for the Sytem GMM estimator, incorporated to deal

with heteroscedasticity of the error term.

In order to determine the optimal weighting matrix WOPTN , again a two-step procedure is used,

just as was the case for the First-Differenced GMM estimator. In the first step, the System

GMM estimator is limited to the case of no autocorrelation in the error term εi,t combined with

the homoscedasticity assumption (constant variance of the error term over different individuals

and time observation points). Furthermore, the covariance matrix of the individual effects is

assumed to be zero: V [αi] = 0. By making these assumptions, the first-step weighting matrix

W s1N can easily be computed as follows:

W s1N = (Z+′

G+Z+)−1 (4.37)

with

G+ = IN ×G+′

T and G+T =

GD 0

0 GL

where

GD =

2 −1 0 ...

−1 2. . . 0

0. . .

. . . −1... 0 −1 2

and GL =

1 0 ... 0

0 1. . . 0

.... . .

. . . 0

0 0 0 1

Once this preliminary weighting matrix is determined, a first estimate of the parameters can be

made using expression 4.36. This leads to an interim estimate for γ:

γs1 = (X+′Z+ W s1

N Z+′X+)−1X+′

Z+ W s1N Z

+′X+y+ (4.38)

Using this interim estimate from the first step, the residuals for this first-step regression can be

calculated as follows:

∆ε+ = Y + −Xγs1 (4.39)


These residuals are then used in the second step for calculating the optimal weighting matrix

WOPTN :

WOPTN = (Z ′∆ε∆ε′Z)−1 (4.40)

This optimal weighting matrix is then used for calculating the final estimate of γ:

γs2 = (X+′Z+ W s1

N Z+′X+)−1X+′

Z+ W s1N Z

+′X+y+ (4.41)

This exact matrix formulation will be implemented in Matlab, as explained in the next chapter.

MONTE CARLO SIMULATIONS 23

Chapter 5

Monte Carlo Simulations

In this chapter, the core of this master’s dissertation is discussed, namely the implementation

and results of the Monte Carlo simulations. The goal of the Monte Carlo simulations is to

compare the performance of the 4 discussed estimators in estimating the Firm Growth Model

under different circumstances. In order to be able to freely vary the parameters in the simulation,

I have chosen to execute the simulations using the mathematical software package ’MATLAB’.

This environment provides a large freedom when implementing the simulations and enabled me

to set up the simulations for the specific case of the Firm Growth Model. This extra freedom,

however, came at the cost of extra programming time, since everything needed to implemented

from scratch. In the first section of this chapter, the general set-up of the simulation will be

presented. Next, the parameters varied over the different simulations are discussed, as well as

the reasons for this variation. In the third section the MATLAB implementation of the Monte

Carlo simulations is explained. The results of the simulations are presented and discussed in the

last section.

5.1 General Set-Up of Simulations

Monte Carlo (MC) simulations are used to solve problems which are probabilistic in their nature.

Drawing a data sample from a population and estimating the population characteristics from

that sample is an example of a probabilistic problem. A broad definition of Monte Carlo simu-

lations is: ’the repeated sampling from a probabilistic distribution to determine the properties

of some phenomenon’. The standard framework for MC simulations is as follows:


1 Define a domain of possible inputs

2 Generate inputs randomly from a probability distribution over that domain

3 Perform the deterministic calculations necessary to determine searched-for parameters

4 Repeat steps 2-3 over a large amount of iterations

5 Aggregate the results.

This framework can be applied to the study we are performing here, namely the analysis of the

performance of 4 estimators when estimating the parameters in the Firm Growth Model. This

Firm Growth Model was discussed in Chapter 2, and has following form:

Gi,t+1 = β0 + β1ln(Si,t) + β2ln(Ai,t) + β3(ln(Si,t))2 + β4(ln(Ai,t))


where Gi,t+1 = ln(Si,t+1)− ln(Si,t) is the growth variable of interest of firm i (i = 1, 2, ..., N) in

period t+ 1 (t = 1, 2, ..., T ), Si,t and Ai,t are the size and age of firm i at time t, and ui,t is the

error term. Rewriting this equation leads to:

Si,t+1 = β0 + (β1 + 1)ln(Si,t) + β2ln(Ai,t) + β3(ln(Si,t))2 + β4(ln(Ai,t))


It is this form of the equation which will be used during the Monte Carlo simulations.

Following the framework presented above, the procedure for the Monte Carlo simulations per-

formed here is as follows:

1 Specify the possible inputs, i.e. specify the population parameters β1, β2, β3, β4 and β5.

2 Generate a data set by drawing a sample out of this population, i.e. draw random initial

values for ln(Si,1) and ln(Ai,1) from their respective distributions, which will be used to

determine all later values ln(Si,t) and ln(Ai,t) with t = 2...T . Also draw random values

for all error terms ui,t. Then calculate all remaining values in the data set.

3 From the generated data set, calculate estimates for the parameters β1, β2, β3, β4 and β5

using the 4 estimators under study. Save these estimates for later processing.

4 Repeat step 2-3 a large number of times (e.g. D = 1000).

5 Compare the estimated coefficients from all iterations to the population values to determine

the performance of the different estimators.


This simulation procedure will be performed a number of times by varying parameters in the

model. Which parameters will be varied during the different MC simulations is discussed in the

next section.

In the final step of the Monte Carlo simulation, the performance of the different estimators needs

to be determined. In order to do this, performance measures are necessary. In this case, we

evaluated the estimators based upon 2 properties:

� Consistency: meaning that the estimates converge in probability to the population value,

i.e. the distribution of the estimates becomes more and more concentrated near the true

value of the parameter being estimated

� Efficiency: meaning the variance of the estimates is small.

The performance statistics used to measure these properties of the estimators are the following:

� The average absolute bias = 15

∑i=5i=1

1D

∑d=Dd=1 |βi,d − βi|, as a performance statistic for

the consistency of the estimator.

� The average variance = 15

∑i=5i=1

1D

∑d=Dd=1 (βi,d − βi)2, as a performance statistic for the

efficiency of the estimator.

In the definitions of these performance statistics, βi,d represents the estimated parameter βi from

the dth iteration. The first averaging is over all D iteration performed during 1 MC simulation.

The second averaging is over the 5 model parameters (β1 to β5). The word ’average’ in the

definition of the performance statistics therefore implies both averaging actions.

5.2 Configuration of the Simulations

The data sets used in each iteration of the MC simulations are created such that they ap-

proximate a real-world data set as close as possible. In order to achieve this, the population

parameters β1, β2, β3, β4 and β5, and the initial values for ln(Si,1) and ln(Ai,1) need to be

chosen based upon a realistic population. The population sample on which the MC simulations

are based here is the growth data of firms, defined in terms of assets, located in Flemish munic-

ipalities from 2004 to 2013. This data was retrieved from the BELFIRST database published

by Bureau Van Dijk. The sample consists of more then 32 0000 firm-year observations for 69

000 firms and it was gathered and analyzed by P. Van Cauwenberge, P. Beyne and H. Vander


Bauwhede (2016). The descriptive statistics gathered during their analysis will be used here to

set up the MC simulations.

In the creation of the data set during each iteration of the MC simulation, the initial values

for ln(Si,1) and ln(Ai,1) are drawn randomly from a distribution as close as possible to their

real-world distributions. These distributions have following descriptive statistics (as determined

by P. Van Cauwenberge, P. Beyne and H. Vander Bauwhede):

Mean SD Min Median Max

ln(S) 6.3686 1.4655 3.2242 6.2478 10.6868

ln(A) 2.5501 0.8491 0 2.7081 4.0943

Table 5.1: Descriptive statistics of variables ln(S) and ln(A)

Both the distribution of ln(S) and ln(A) are assumed to be normal, since the mean and median

are close to each other and all the data is contained in the interval [Mean−3∗SD;Mean+3∗SD].

Therefore, in each iteration of the MC simulation the initial values ln(Si,1) and ln(Ai,1) are drawn

from the distributions N(6.3686, 1.46552) and N(2.5501, 0.84912) respectively.

For the specification of the population parameters β1, β2, β3, β4 and β5, the research of P. Van

Cauwenberge, P. Beyne and H. Vander Bauwhede is used as well. In their regression analysis

of the real-world data set of Flemish firm growth data (defined in terms of assets), they have

estimated these parameters in the Firm Growth Model. The obtained estimates and their

respective significance levels are:

Parameter Estimate Significance

β1 0.0439 0.0069

β2 -0.0804 0.0084

β3 -0.0048 0.0010

β4 0.0045 0.0011

β5 0.0070 0.0017

Table 5.2: Estimates for the population parameters.

These estimates for β2, β3, β4 and β5 will be used to specify the population in each of the MC


simulations. Parameter β1, however, will be varied in order to study the influence of the param-

eter corresponding to the lagged dependent variable on the performance of the 4 estimators. As

discussed in Chapter 3, the Nickell bias increases for increasing values of the dependent lagged

variable parameter ρ. For the Firm Growth Model used here, this parameter ρ is equal to 1+β1.

In order to visualize this Nickell bias clearly, 3 separate MC simulation will be performed: one

for 1 + β1 = 0.1, one for 1 + β1 = 0.5 and one for 1 + β1 = 0.9.

From Chapter 3 we know as well that not only the lagged dependent variable parameter influ-

ences the magnitude of the Nickell bias. The number of individuals in the data set (N) and

the amount of time observation points (T) influence the Nickell bias as well. Therefore, these 2

parameters are varied as well across different MC simulations. N will take on 2 values, namely

N = 100 and N = 1000. T will be varied across two values as well: T = 5 and T = 10. These

values were chosen such that they are separated sufficiently to cause a performance difference,

while still producing feasible simulation sizes using a desktop computer. In total, we will have

following 12 separate MC simulations:

Simulation number β1 + 1 T N

1 0.1 5 100

2 0.1 5 1000

3 0.1 10 100

4 0.1 10 1000

5 0.5 5 100

6 0.5 5 1000

7 0.5 10 100

8 0.5 10 1000

9 0.9 5 100

10 0.9 5 1000

11 0.9 10 100

12 0.9 10 1000

Table 5.3: Configuration of simulations to be performed

The implementation in MATLAB of these simulation is presented in the next section.


5.3 MATLAB Implementation

MATLAB (short for MATrix LABoratory) is a numerical computing environment, well suited

for performing calculations with matrices. Since all 4 estimators studied in this master’s disser-

tation are presented in matrix formulation, MATLAB was an obvious choice for implementing

these estimators, performing the Monte Carlo simulations and processing the results. The total

implementation of the simulations in MATLAB consists of 4 functions and 2 scripts:

� A function implementing the Fixed Effects estimator (’FE final.m’ ).

� A function implementing the Anderson-Hsiao IV estimator (’AHIV final.m’ ).

� A function implementing the First-Differenced GMM estimator (’FDGMM final.m’ ).

� A function implementing the System GMM estimator (’SGMM final.m’ )

� A script implementing the Monte Carlo Simulations(’MC simulations final.m’ ).

� A script implementing the processing and visualization of the results

(’MC results processing final.m’ ).

The main script in the hierarchy is ’MC simulations final.m’, implementing the Monte Carlo

simulation. It is here that the D=1000 iterations of the simulation are performed. The interplay

of the different functions and scripts is visualized in the figure below.

Figure 5.1: Schematic diagram representing MATLAB implementation


This procedure will be executed for the 12 simulations presented in table 5.3. The exact imple-

mentation of the different MATLAB scripts can be found in the appendix of this thesis.

5.4 Simulation Results

In this section, the results of the Monte Carlo simulations are presented and discussed. The

simulation results are grouped into 3 groups, accoring to the value of the lagged dependent

variable parameter β1 + 1. In each group, the values for N (number of individuals in data set)

and for T (number of time observation points in data set) are varied between 2 values: N = 100

and N = 1000, and T = 5 and T = 10. This creates a total of 4 different MC simulations per

value of β1 + 1.

5.4.1 Simulation results for β1 + 1 = 0.1

FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.1, T=5, N=100

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var

Figure 5.2: MC simulation 1

FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.1, T=5, N=1000

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.1, T=10, N=100

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.1, T=10, N=1000

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var



When studying the results of the 4 MC simulations in which β1 +1 = 0.1, it is clear the the Sys-

tem GMM estimator outperforms the other 3 estimators both in terms of consistency (minimal

average absolute bias) and efficiency (minimal average variance). This estimator in particular

outperforms the First-Differenced GMM estimator in each of the 4 simulations, regardless of

the size of N or T . The use of an extended set of instruments in the System GMM estimator

clearly affects its performance in comparison to the smaller set of instruments used in the First-

Differenced GMM estimator. This is particularly true when the size of the data set is small

(small T and small N).

When the value of T is low (T = 5), the Nickell bias of the Fixed Effects estimator becomes

clearly visible. In the simulations 1 and 2, where T = 5, the FE estimator performs worst,

having the largest bias and variance. Furthermore, this bias becomes larger, relatively to the

bias of the other estimators, when N increases. This is in accordance to the properties of the

Nickell bias presented in Chapter 3. When T = 10, however, the FE estimator is no longer the

worst performer of the 4 estimators. For T = 10, the Anderson-Hsiao estimator exhibits the

largest bias ans variance of all 4 estimator for both N = 100 and N = 1000. The use of a limited

set of level instruments for the differenced lagged dependent variables seems to perform worse

relatively to the other estimators when used in estimating the parameters of the Firm Growth

Model for larger values of T .

Lastly, it can be seen that in general all estimators perform best when the data set is largest.

This is no surprise, since a larger data set enables more accurate estimation because of the larger

amount of information available for estimation.



FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.5, T=5, N=100

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.5, T=5, N=1000

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.5, T=10, N=100

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.5, T=10, N=1000

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


The simulation results for β1 +1 = 0.5 show that also for average values of the lagged dependent

variable parameter, the System GMM estimator performs best. The difference with the other 3

estimators is again largest for small N and small T . The performance of the First-Differenced

estimator is again far below that of the System GMM estimator in all 4 simulations.

The FE estimator only performs worst when T = 5 and N = 1000. Here, the Nickell bias is

most visible, as explained in Chapter 3. In general, however, the FE estimator performs better

for β1 + 1 = 0.5 than for β1 + 1 = 0.1. In the 3 other simulations, the Anderson-Hsiao estimator

is performing the worst. We can therefore conclude that the limited set of instruments, used in


the AHIV estimator becomes weaker when the value of the lagged dependent variable parameter

increases in the Firm Growth Model.


FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.9, T=5, N=100

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.9, T=5, N=1000

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.9, T=10, N=100

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


FE AHIV FDGMM SGMM0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

(1+B1) =0.9, T=10, N=1000

Ave

rage

abs

olut

e bi

as

FE AHIV FDGMM SGMM0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ave

rage

var

ianc

e

Bias Var


In the last set of 4 MC simulations, this time for β1 + 1 = 0.9, there is again a clear winner.

The System GMM estimator outperforms all other estimators and the difference in performance

is larger than for low values of β1 + 1. This difference, however, is not caused by the fact that

the System GMM estimator performs better in case of higher values for β1 + 1. It performs

approximately the same as in the previous two sets of simulations. It is the bad performance of

the other estimators that causes this difference.


The FE estimator performs worse for high values of β1 + 1 because of the larger Nickell bias

for larger values of β1 + 1. In Chapter 3, it was highlighted that the Nickell bias increases for

increasing values of the lagged dependent variable parameter. This phenomenon is now clearly

visible when we compare this set of 4 simulations to the 2 previous sets in which β1 + 1 was

lower. It was also highlighted in Chapter 3 that the Nickell bias increases for increasing values

of N , and decreases for increasing values of T . This can also clearly be seen in this last set of

simulations. Simulation 10 shows the largest Nickell bias, since T is small and N is large. In

simulation 11, the Nickell bias is smallest relatively to the biases of the other estimators, since

here T is larger and N smaller.

The reason for the bad performance of the Anderson-Hsiao and the First-Differenced GMM

estimator has the same origin. At the end of section 4.4, it was stated that the reason be-

hind possible poor performance of the FDGMM estimator is the problem of weak instruments,

meaning that the instruments are only weakly correlated with the endogenous variables they

are trying to instrument. This correlation becomes weaker when the dependent variable is close

to a random walk, which is the case when β1 + 1 gets close to 1. It are therefore these weak

level instruments, used to instrument the differenced lagged dependent variables, that cause the

poor performance of the AHIV and the FDGMM estimator when β1 + 1 is close to 1. In all 4

simulations of this set, the Anderson-Hsiao estimator performs worse than the FDGMM estima-

tor. The FDMM estimator uses a larger set of instruments, and uses therefore more information

when estimating the parameters in the Firm Growth Model. This leads to better estimates than

is the case for the limited set of instruments used by the AHIV estimator.

5.4.4 Conclusion simulation results

When comparing the 4 estimators in this specific case of the Firm Growth Model, we can con-

clude that in general the System GMM estimator performs best, regardless of the sample size and

value of the lagged dependent variable parameter. This estimator exhibits the lowest average

absolute bias and lowest average variance in all 12 Monte Carlo simulations. It can therefore be

concluded that it is most consistent and most efficient in comparison to the other 3 estimators

when estimating the Firm Growth Model.


The Fixed Effects estimator clearly exhibits a large Nickell bias when T is small and N is large.

The Nickell bias is highest when β1+1 gets close to 1. This estimator, however, outperformed the

Anderson-Hsiao estimator in all cases where T = 10, despite being more simple. It is therefore

justified to say that for estimating the Firm Growth Model, the FE estimator can be chosen over

the AHIV estimator when confronted with data sets in which the number of time observation

points is large.

Comparing the AHIV estimator with the First-Differenced GMM estimator, we can see that

the FDGMM estimator is a better performer in all simulations. This is no surprise since the

FDGMM estimator is in a way an extension of the AHIV estimator, having an extended set

of instruments. Because of the same reason, both estimators perform bad when the value of

β1 + 1 gets close to 1. The instruments of both estimators become weak, which leads to biased

estimates.

CONCLUSION 35

Chapter 6

Conclusion

In this master’s dissertation, 4 estimators used for estimating dynamic panel data models were

compared. Monte Carlo simulations were used to perform this comparison, and the Firm Growth

Model was chosen as the dynamic panel data model during the Monte Carlo simulations. This

model was presented in Chapter 2 of this master’s dissertation.

The 4 estimators under study were: the Fixed Effects estimator, the Anderson-Hsiao Instrument

Variable estimator, the First-Differenced GMM estimator and the System GMM estimator. All

4 estimators were presented shortly in Chapters 3 and 4, using matrix formulation. Such a for-

mulation was chosen, since the implementation of the estimators in MATLAB was done using

this format. The theoretical advantages and shortcomings of the 4 estimators were presented in

these chapters as well. For the Fixed Effects estimator, the Nickell bias is the most important

shortcoming, making estimates for data sets with a low number of time observation points and

a high number of individuals unreliable. The Anderson-Hsiao and the First-Differenced GMM

estimator share the problem of weak instruments when the values of the dependent variable

in the model gets close to a random walk. This occurs when the parameter corresponding to

the lagged dpendent variable in the model gets close to 1. These weak instruments causes the

estimates to be unreliable. The System GMM estimator is the most complicated estimator of

the 4 estimators studied here, but has in theory the best performance.

The theoretical properties of the 4 estimators were putted to the test using Monte Carlo sim-

ulations, implemented in the mathematical software package MATLAB. Each estimator was

implemented in a separate MATLAB function, and the simulation script called upon these func-

CONCLUSION 36

tions during execution. In the simulation script, a data set was generated in each iteration of the

Monte Carlo simulation, based upon a real-world firm growth data of firms in Flemish munici-

palities. Processing of the results of each Monte Carlo simulation was done in another separate

script. In this script, the average absolute bias and average variance of the estimators were

calculated, being a performance measure for consistency and efficiency, respectively. In total,

12 separate simulations were performed. Between these simulations, the value of 3 parameters

was changed: the value of the lagged dependent variable parameter (β1 +1), the number of time

observation points in the generated data set (T ) and the number of individuals in the generated

data set (N).

In all 12 Monte Carlo simulations, the System GMM estimator was more consistent and efficient

than the other 3 estimators. We can therefore conclude that the System GMM estimator yields

the best estimates when estimating the parameters in the Firm Growth Model. The Fixed Ef-

fects estimator clearly exhibited a Nickell bias when T was small and N large. The bias was

highest when the value of the lagged dependent variable parameter β1+1 got close to 1. When T

was large and N was small, however, the more simple Fixed Effects estimator outperformed the

more complicated Anderson-Hsiao and First-Differenced GMM estimator. These 2 estimators

performed badly when β1 + 1 got close to 1, because of the weak instruments used in these esti-

mators. The FDGMM estimator always outperformed the AHIV estimator, since the FDGMM

estimator is in a way an extension of the AHIV estimator.

Based upon these results it is safe to say that, when estimating the parameters in the Firm

Growth Model, the System GMM estimator should be the preferred option in order to achieve

the best results.

MATLAB IMPLEMENTATION OF ESTIMATORS AND MONTE CARLO SIMULATIONS 37

Appendix A

Matlab Implementation of

Estimators and Monte Carlo

Simulations

A.1 FE final.m

1 f unc t i on [ Alpha , Beta ] = FE f ina l ( Y,X, n un i t s )

2 % This func t i on implements the Fixed F f f e c t s e s t imator .

3 % Bob Mertens 2017

4

5 % Model in matrix notat ion : Y = D*Alpha + X*Beta + e .

6 % Beta = c o e f f i c i e n t s f o r r e g r e s s o r s ( no i n t e r c e p t c o e f f i c i e n t ! ) .

7 % Alpha = Fixed E f f e c t c o e f f i c i e n t s per c ros s−s e c t i o n a l un i t .

8 % Y = dependent v a r i a b l e s matrix .

9 % X = re g r e s s o r values , same s t ru c tu r e as Y matrix .

10 % Structure : rows grouped per c ros s−s e c t i o n a l unit , and advancing time un i t .

11

12 %% Construct ing aux i l i a r y matr i ce s to be used during c a l c u l a t i o n s

13 D=kron ( eye ( n un i t s ) , ones ( l ength (Y) / n un i t s , 1 ) ) ;

14 P D=D* inv (D’*D) *D’ ;

15 M D=eye ( l ength (Y) )−P D ;

16

17 %% Calcu la t ing Beta and f i x ed e f f e c t s

18 Beta = inv (X’*M D*X) *X’*M D*Y;

19 Alpha = inv (D’*D) *D’ * (P D*Y−P D*X*Beta ) ;

20

21 end


A.2 AHIV final.m

1 f unc t i on [ Beta ] = AHIV final ( Y,X, n un i t s , endo )

2 % This func t i on implements the F i r s t D i f f e r enc ed IV es t imator (Anderson−Hsioa ) .


4

5 % Beta = Calcu lated c o e f f i c i e n t s o f r e g r e s s o r s ( no i n t e r c e p t c o e f f i c i e n t ! ) .

6 % Y = Dependent v a r i a b l e s matrix .

7 % X = Regressor va lue s ( i n c l ud ing au t o r e g r e s s i v e r e g r e s s o r s ) , same s t r u c tu r e as Y matrix .

8 % ( Struc ture : rows grouped per c ros s−s e c t i o n a l unit , and advancing time un i t )

9 % endo : vec to r with s i z e equal to number o f r e g r e s s o r e . 1 f o r endogenous , 0 f o r exogenous

.

10


12 T=length (Y) / n un i t s +1;

13 M d=[ ze ro s (T−2 ,1) diag ( ones (T−2 ,1) ) ]− [ d iag ( ones (T−2 ,1) ) z e r o s (T−2 ,1) ] ;

14 F=kron ( eye ( n un i t s ) ,M d) ;

15

16 %% Create Y d i f f : f i r s t −d i f f e r e n c e d dependent v a r i a b l e s

17 Y d i f f=F*Y;

18

19 %% Create X d i f f : f i r s t −d i f f e r e n c e d r e g r e s s o r v a r i a b l e s

20 X d i f f=F*X;

21

22 %% Creat ing Z : instrument matrix conta in ing a l l instrument v a r i a b l e s

23 Z l=kron ( eye ( n un i t s ) , [ eye (T−2) z e ro s (T−2 ,1) ] ) ;

24 Z= [ ] ;

25 f o r j =1: s i z e (X, 2 )

26 i f endo ( j )==1

27 Z j=Z l *X( : , j ) ;

28 e l s e

29 Z j=X d i f f ( : , j ) ;

30 end

31 Z=[ Z j Z ] ;

32 end

33

34 %% Calcu la t ing Beta .

35 P=Z*(Z ’*Z) ˆ(−1)*Z ’ ;

36 Beta=(X d i f f ’*P*X d i f f ) ˆ(−1)*X di f f ’*P*Y d i f f ;

37 end


A.3 FDGMM final.m

1 f unc t i on [ Beta ] = FDGMM final ( Y,X, n un i t s , endo )

2 % This func t i on implements the F i r s t D i f f e r enc ed GMM est imator ,

3 % using a two−s tep procedure .


5

6 % Beta = c o e f f i c i e n t s f o r r e g r e s s o r s ( no i n t e r c e p t c o e f f i c i e n t ,

7 % aut o r e g r e s s i v e c o e f f i c i e n t s inc luded ) .

8 % Y = dependent v a r i a b l e s vec to r .

9 % X = re g r e s s o r va lue s ( i n c l ud ing au t o r e g r e s s i v e r e g r e s s o r s ) , same s t r u c tu r e as Y matrix .


11 % endo : vec to r with s i z e equal to number o f r e g r e s s o r s , 1 f o r endogenous , 0 f o r exogenous

.

12




16 F=kron ( eye ( n un i t s ) ,M d) ;

17


19 Y d i f f=F*Y;

20


22 X d i f f=F*X;

23


25 Z= [ ] ;

26 f o r j =1: s i z e (X, 2 )

27 i f endo ( j )==1

28 Z j=ze ro s ( (T−2)* n uni t s , (T−2)*(T−1)/2) ;

29 f o r k=1: n un i t s

30 l =0;

31 f o r i =1:T−2

32 i f i==1

33 l =1;

34 e l s e

35 l=l+( i −1) ;

36 end

37 Z j ( ( k−1)*(T−2)+i , l : l+i −1)=X( ( k−1)*(T−1)+1:(k−1)*(T−1)+i , j ) ;

38 end

39 end

40 e l s e

41 Z j=X d i f f ( : , j ) ;

42 end

43 Z=[Z Z j ] ;


44 end

45

46 %% Calcu la te f i r s t −s tep Beta .

47 G=kron ( eye ( n un i t s ) , (M d*M d ’ ) ’ ) ;

48 W s1=inv (Z ’*G*Z) ;

49 Beta s1=inv ( X d i f f ’*Z*W s1*Z ’* X d i f f ) *X di f f ’*Z*W s1*Z ’* Y d i f f ;

50

51 %% Calcu la te second−s tep Beta us ing optimal we ight ing matrix .

52 e d i f f=Y di f f−X d i f f *Beta s1 ;

53 W opt=inv (Z ’* e d i f f * e d i f f ’*Z) ;

54 Beta=inv ( X d i f f ’*Z*W opt*Z ’* X d i f f ) *X di f f ’*Z*W opt*Z ’* Y d i f f ;

55 end


A.4 SGMM final.m

1 f unc t i on [ Beta ] = SGMM final ( Y,X, n un i t s , endo )

2 % This func t i on implements the System GMM est imator , us ing a two−s tep procedure .


4

5 % Beta = c o e f f i c i e n t s f o r r e g r e s s o r s ( no i n t e r c e p t c o e f f i c i e n t ,

6 % dynamic c o e f f i c i e n t s inc luded ) .

7 % Y = dependent v a r i a b l e s vec to r .

8 % X = re g r e s s o r va lue s ( i n c l ud ing au t o r e g r e s s i v e r e g r e s s o r s ) , same s t r u c tu e r as Y matrix .


10 % endo : vec to r with s i z e equal to number o f r e g r e s s o r s , 1 f o r endogenous , 0 f o r exogenous

.

11




15 M l=[ z e r o s ( (T−2) ,1 ) eye (T−2) ] ;

16 M tot=[M d ; M l ] ;

17 F=kron ( eye ( n un i t s ) ,M tot ) ;

18


20 Y tot=F*Y;

21


23 X tot=F*X;

24

25


27 Z= [ ] ;

28 f o r j =1: s i z e (X, 2 )

29 i f endo ( j )==1

30 Z j=ze ro s ( (T−2)* n uni t s , (T−2)*(T−1)/2) ;

31 f o r k=1: n un i t s

32 l =0;

33 f o r i =1:T−2

34 i f i==1

35 l =1;

36 e l s e

37 l=l+( i −1) ;

38 end

39 Z j ( ( k−1)*(T−2)+i , l : l+i −1)=X( ( k−1)*(T−1)+1:(k−1)*(T−1)+i , j ) ;

40 Z j ( ( k−1)*(T−2)*2+(T−2)+i , (T−2)*(T−1)/2+ i )=X tot ( ( k−1)*(T−2)*2+i , j ) ;

41 end

42 end

43 e l s e


44 Z j=X tot ( : , j ) ;

45 end

46 Z=[Z Z j ] ;

47 end

48

49 %% Calcu la te f i r s t −s tep Beta

50 G D=M d*M d ’ ;

51 G L=eye (T−2) ;

52 G T=blkd iag (G D, G L) ;

53 G=kron ( eye ( n un i t s ) ,G T ’ ) ;

54 W s1=inv (Z ’*G*Z) ;

55 Beta s1=inv ( X tot ’*Z*W s1*Z ’* X tot ) *X tot ’*Z*W s1*Z ’* Y tot ;

56

57 %% Calcu la te second−s tep Beta us ing optimal we ight ing matrix .

58 e t o t=Y tot−X tot *Beta s1 ;

59 W opt=inv (Z ’* e t o t * e to t ’*Z) ;

60 Beta=inv ( X tot ’*Z*W opt*Z ’* X tot ) *X tot ’*Z*W opt*Z ’* Y tot ;

61 end


A.5 MC simulations final.m

1 %% Scr i p t implementing the Monte Carlo s imu la t i on s o f 4 e s t imato r s

2 % Model : dynamic Firm Growth Model

3 % Data gene ra t i on : based upon rea l−world populat ion ( Flemish f i rms )


5

6 c l o s e a l l

7 c l e a r a l l

8

9 %% Input parameters f o r MC s imu la t i on

10 T=5; %sample s i z e , time dimension

11 n c=100; %sample s i z e , c ros s−s e c t i o n a l dimension

12 B1=−0.9; %value o f beta 1

13 N=1000; %number o f i t e r a t i o n s in s imu la t i on

14

15 %% Model :

16 % ln ( S i t )= B0 i + (B1+1)* ln ( S i ( t−1) ) + B2*( ln ( S i ( t−1) ) ) ˆ2

17 % + B3* ln ( A i ( t−1) ) + B4*( ln ( A i ( t−1) ) ) ˆ2 + B5* ln ( S i ( t−1) ) * ln ( A i ( t−1) )

18

19 B=[B1+1 −0.0804 −0.0048 0 .0045 0 . 0 0 7 ] ’ ; %Co e f f i c i e n t s f o r r e g r e s s o r s

20

21 Beta es t to t FE=ze ro s ( l ength (B) ,N) ;

22 Beta est tot AHIV=ze ro s ( l ength (B) ,N) ;

23 Beta est tot FDGMM=zero s ( l ength (B) ,N) ;

24 Beta est tot SGMM=zero s ( l ength (B) ,N) ;

25

26 %% N i t e r a t i o n s o f MC s imu la t i on

27 f o r i =1:N

28 i

29 Y=ze ro s ( (T−1)*n c , 1 ) ;

30 X=ze ro s ( (T−1)*n c , 5 ) ;

31 f o r k=1: n c

32 lnS0=normrnd (6 . 3686 , 1 . 4 655 ) ;

33 lnA0=normrnd (2 . 5501 , 0 . 8 491 ) ;

34 lnS1=[ lnS0 lnS0 ˆ2 lnA0 lnA0ˆ2 lnS0 * lnA0 ]*B+normrnd (0 , 1 ) ;

35 lnA1=log ( exp ( lnA0 )+1) ;

36 f o r j =1:(T−1)

37 i f ( j==1)

38 X(( k−1)*(T−1)+j , : ) =[ lnS1 lnS1 ˆ2 lnA1 lnA1ˆ2 lnS1 * lnA1 ] ;

39 Y(( k−1)*(T−1)+j )=X( ( k−1)*(T−1)+j , : ) *B+normrnd (0 , 1 ) ;

40 e l s e

41 lnS=Y( ( k−1)*(T−1)+j−1) ;

42 lnA=log ( exp (X( ( k−1)*(T−1)+j −1 ,3) )+1) ;

43 X(( k−1)*(T−1)+j , : ) =[ lnS lnS ˆ2 lnA lnAˆ2 lnS * lnA ] ;

44 Y(( k−1)*(T−1)+j )=X( ( k−1)*(T−1)+j , : ) *B+normrnd (0 , 1 ) ;


45 end

46 end

47 end

48

49

50 % Calcu la t i on o f model c o e f f i c i e n t s

51 endo = [1 1 0 0 1 ] ;

52 [ Alpha , Beta e s t to t FE ( : , i ) ]= FE f ina l (Y,X, n c ) ;

53 Beta est tot AHIV ( : , i )=AHIV final (Y,X, n c , endo ) ;

54 Beta est tot FDGMM ( : , i )=FDGMM final (Y,X, n c , endo ) ;

55 Beta est tot SGMM ( : , i )=SGMM final (Y,X, n c , endo ) ;

56

57 end

58

59 %% Saving the r e s u l t s

60 date=date ;

61 time=f i x ( c l o ck ) ;

62 name=s p r i n t f ( ’%i %i %i %i %i r e s u l t s B 1 1 %.1 f N %i T %i .mat ’ , time (1 ) , time (2 ) , time (3 ) , time

(4 ) , time (5 ) ,B1+1,n c ,T) ;

63 save (name) ;


A.6 MC results processing final.m

1 %% Scr i p t inmplementing the p ro c e s s i ng and v i s u a l i z a t i o n o f the MC s imu la t i on r e s u l t s .


3

4 c l o s e a l l

5

6 %% Calcu la t ing abso lu t e d i f f e r e n c e between e s t imate s and populat ion parameters .

7 Be t a e s t t o t FE d i f f=abs ( Beta est tot FE−repmat (B, 1 ,N) ) ;

8 Beta e s t t o t AHIV d i f f=abs ( Beta est tot AHIV−repmat (B, 1 ,N) ) ;

9 Beta est tot FDGMM diff=abs (Beta est tot FDGMM−repmat (B, 1 ,N) ) ;

10 Beta est tot SGMM dif f=abs ( Beta est tot SGMM−repmat (B, 1 ,N) ) ;

11

12 %% Calcu la t ing average abso lu t e b i a s o f each parameter

13 Beta e s t t o t FE b i a s=mean( Be t a e s t t o t FE d i f f ’ ) ;

14 Beta es t tot AHIV bias=mean( Beta e s t to t AHIV d i f f ’ ) ;

15 Beta est tot FDGMM bias=mean( Beta est tot FDGMM diff ’ ) ;

16 Beta est tot SGMM bias=mean( Beta est tot SGMM dif f ’ ) ;

17

18 %% Calcu la t ing average abso lu t e b i a s over a l l parameters

19 Beta e s t t o t FE b ia s avg=mean( Be ta e s t t o t FE b i a s ) ;

20 Beta es t to t AHIV bias avg=mean( Beta es t tot AHIV bias ) ;

21 Beta est tot FDGMM bias avg=mean( Beta est tot FDGMM bias ) ;

22 Beta est tot SGMM bias avg=mean( Beta est tot SGMM bias ) ;

23

24 %% Calcu la t ing square o f abso lu t e b i a s e s

25 Be t a e s t t o t FE d i f f 2=Be t a e s t t o t FE d i f f . ˆ 2 ;

26 Beta e s t t o t AHIV d i f f 2=Beta e s t t o t AHIV d i f f . ˆ 2 ;

27 Beta est tot FDGMM diff2=Beta est tot FDGMM diff . ˆ 2 ;

28 Beta est tot SGMM dif f2=Beta est tot SGMM dif f . ˆ 2 ;

29

30 %% Calcu la t ing vara ince per parameter

31 Beta e s t to t FE var=mean( Be t a e s t t o t FE d i f f 2 ’ ) ;

32 Beta est tot AHIV var=mean( Beta e s t to t AHIV d i f f 2 ’ ) ;

33 Beta est tot FDGMM var=mean( Beta est tot FDGMM diff2 ’ ) ;

34 Beta est tot SGMM var=mean( Beta est tot SGMM dif f2 ’ ) ;

35

36 %% Calcu la t ing average vara ince over a l l parameters

37 Beta e s t to t FE var avg=mean( Beta e s t to t FE var ) ;

38 Beta est tot AHIV var avg=mean( Beta est tot AHIV var ) ;

39 Beta est tot FDGMM var avg=mean( Beta est tot FDGMM var ) ;

40 Beta est tot SGMM var avg=mean( Beta est tot SGMM var ) ;

41

42 %% Figure p l o t t i g

43 c = { ’FE ’ , ’AHIV ’ , ’FDGMM’ , ’SGMM’ } ;

44


45 f i g u r e

46

47 a=[ [ Be ta e s t t o t FE b ia s avg Beta es t to t AHIV bias avg Beta est tot FDGMM bias avg

Beta est tot SGMM bias avg ] ’ z e r o s (4 , 1 ) ] ;

48 b=[ z e r o s (4 , 1 ) [ Be ta e s t to t FE var avg Beta est tot AHIV var avg

Beta est tot FDGMM var avg Beta est tot SGMM var avg ] ’ ] ;

49

50 [AX,H1 ,H2 ] =plotyy ( [ 1 : 4 ] , a , [ 1 : 4 ] , b , ’ bar ’ , ’ bar ’ ) ;

51 s e t (H1 , ’ FaceColor ’ , ’ r ’ ) % a

52 s e t (H2 , ’ FaceColor ’ , ’ b ’ ) % b

53

54 s t r=s p r i n t f ( ’ (1+B 1 ) =%.1f , T=%i , N=%i ’ ,B(1) , T, n c ) ;

55 t i t l e ( s t r , ’ FontSize ’ ,16 , ’ FontWeight ’ , ’ bold ’ )

56 lh1=legend (H1 , ’ Bias ’ , ’ Locat ion ’ , ’ northwest ’ ) ;

57 s e t ( lh1 , ’ FontSize ’ ,12) ;

58 lh2=legend (H2 , ’Var ’ , ’ Locat ion ’ , ’ nor theas t ’ ) ;

59 s e t ( lh2 , ’ FontSize ’ ,12) ;

60 s e t (AX, ’ x t i c k l a b e l ’ , c , ’ FontSize ’ ,14) ;

61 ylim (AX(1) , [ 0 1 ] )

62 ylim (AX(2) , [ 0 2 ] )

63 s e t (AX(1) , ’ y t i c k ’ , [ 0 : 0 . 1 2 5 : 1 ] )

64 s e t (AX(2) , ’ y t i c k ’ , [ 0 : 0 . 2 5 : 2 ] )

65 y l ab e l (AX(1) , ’ Average abso lu te b i a s ’ , ’ FontSize ’ ,14)

66 y l ab e l (AX(2) , ’ Average var iance ’ , ’ FontSize ’ ,14)

67 g r id on

BIBLIOGRAPHY 47

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