L1–GARCH MODELS: PARAMETER ESTIMATIONS,

PERFORMANCE MEASURES AND ITS APPLICATIONS

SHAMSUL RIJAL BIN MUHAMMAD SABRI

THESIS SUBMITTED IN FULFILMENT OF THE

REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

FACULTY OF SCIENCE

UNIVERSITY OF MALAYA

KUALA LUMPUR

2008

UNIVERSITI MALAYA

ORIGINAL LITERARY WORK DECLARATION Name of Candidate: Shamsul Rijal bin Muhammad Sabri I.C./Passport No.:- 750602-07-5767 Registration / Matric No.:- SHB030004 Name of Degrees:- Doctor of Philosophy (PhD) Title of Project Paper / Research Report / Dissertation / Thesis (“this Work”): L1 – GARCH Models: Parameter Estimations, Performance Measures and Its Applications Field of Study: Applied Statistics I do solemnly and sincerely declare that: (1) I am the sole author/writer of this Work; (2) This Work is original; (3) Any use of any work in which copyright exists was done by way of fair dealing

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(5) I hereby assign all and every rights in the copyright to this Work to the University of Malaya (“UM”), who henceforth shall be owner of the copyright in this Work and that any reproduction or use in any form or by any means whatsoever is prohibited without the written consent of UM having been first had and obtained;

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Candidate’s Signature

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ABSTRAK

Sejak kegagalan terhadap data berfrekuensi tinggi dalam penyelidikan, pakar ekonomi telah

menyelidiki beberapa kaedah alternatif untuk mengatasi masalah permodelan

heteroskedastik. Salah satu model yang baru diperkenalkan ialah kumpulan L1–GARCH

teguh. Buat masa ini, model ini diperkenalkan berdasarkan teori statistik dan pencapaian

dalam penganggaran parameter model tersebut.

Dalam tesis ini, kajian mendalam terhadap pencapaian model L1–GARCH ini dalam bentuk

penganggaran varians bersyaratnya dilaksanakan. Penganggaran model ini diperkemaskan

dengan mengaplikasikan analisis pengeksploran data terlebih dahulu sebelum diteruskan

dengan model L1–GARCH. Bagi memahami kelakuan anggaran parameter-parameter ini,

kajian simulasi dijalankan. Keputusan yang diperolehi ialah, model klasik GARCH(1,1)

adalah terbaik dalam menganggar varians bersyarat dengan ketiadaan data terpencil. Walau

bagaimanapun, model L1–GARCH(1,1) menunjukkan kemajuan dalam penganggaran

apabila peratusan data terpencil meningkat.

Satu lagi permasalahan berkaitan dengan permodelan heteroskedastik ialah kebagusan

model tersebut. Dalam model GARCH, ujian yang biasa dilaksanakan terhadap model

tersebut ialah ujian taburan kenormalan dengan menggunakan model klasik Jarque-Bera

( ). Ujian ini walaubagaimanapun terlalu mudah dipengaruhi oleh kehadiran data

terpencil seterusnya menyebabkan kegagalannya menerima andaian taburan kenormalan.

Di sini, kami memperkenalkan Jarque-Bera teguh ( ) yang mana ia (i) kurang peka

terhadap kehadiran data terpencil dan (ii) boleh mengesan sisihan daripada andaian taburan

normal biasa, lazimnya taburan berekor tebal. Dalam menilai pencapaian dan ,

JB

RJB

JB RJB

iii

kami membandingkan ujian statistik dengan kehadiran data terpencil dan kesimetrian

berekor tebal; di sini kami menggunakan kajian simulasi untuk menghitung kuasa dalam

menolak andaian ujian hipotesis nul (data bertabur secara normal). Keputusan yang

diperolehi menunjukkan mampu menghasilkan (i) dapatan yang baik bagi menangani

kewujudan data terpencil dan (ii) secekap dengan statistik teguh yang pernah dikemukakan

oleh beberapa penyelidik lain apabila berlaku andaian yang menyisih daripada normal.

RJB

Kajian ini juga turut mengaplikasikan model L1–ARCH yang teguh untuk menganggarkan

kemeruapan (ketidaktentuan) inflasi bagi negara-negara ASEAN-5. Sebagai pelengkap,

penilaian dibuat terhadap hubungan antara kemeruapan inflasi dan pertumbuhan ekonomi

dengan menggunakan model regresi teguh. Dari kajian ini, terdapat bukti yang meluas yang

menyokong hipotesis di mana ketidaktentuan kenaikan inflasi mendorong ke arah kejatuhan

pertumbuhan ekonomi. Hubungan positif di antara kemeruapan inflasi dan pertumbuhan

ekonomi adalah konsisten dengan kajian-kajian lain, yang dilaksanakan pada kebayakan

negara perindustrian lain. Keputusan ini sejajar dengan konjektur Friedman yang mana

kemeruapan mengaitkan pertukaran matawang akan memberi tekanan terhadap aktiviti

ekonomi sebenar.

Di dalam kajian ini, pengaturcaraan SPLUS digunakan bagi menjalankan (a) ujian simulasi

terhadap ujian ; (b) penganggaran parameter model LRJB 1–GARCH; (c) perbandingan

kecekapan varians bersyarat antara GARCH(1,1) biasa dan L1–GARCH(1,1) dan (d)

menganggar parameter-parameter model L1–ARCH bagi kemeruapan inflasi negara-negara

ASEAN-5.

iv

ABSTRACT

Since the failure of the high frequency data appears in many researches, economists have

explored some alternative methods to overcome heteroscedastic modeling. One such model

that has been introduced recently is robust L1–GARCH family. These tools have been

introduced initially due to the statistical properties and their performance in parameter

estimates.

In this thesis, we further explore the performance of the L1–GARCH particularly, in

estimating conditional variances. The estimation is enhanced by first employing an

exploratory data analysis before proceeding with L1–GARCH. To better understand the

behavior of the estimates as well as the performance of these methods, simulation studies

were carried out. The results obtained suggest that, ordinary GARCH(1,1) performs well in

estimating conditional variances in the absence of outliers or contaminants in the data.

However, L1–GARCH(1,1) outperform the GARCH(1,1) in estimating the conditional

variances in the presence of outliers.

Another problem that persists with heteroscedastic modeling is that of the goodness-of-fit

test. In GARCH models, the most common adequacy test used is the classical Jarque-Bera

( ) test. This test however, is known to be extremely sensitive to outliers and hence a

single outlier may lead to failure of normality assumption. To overcome this, we introduce

robust ( ) measures that are (i) less sensitive to the presence of outliers and (ii) able

to detect the departure from the usual normal distribution (symmetric heavy tailed). In

assessing the performance of and , the test statistics are compared, in the presence

JB

JB RJB

JB RJB

v

of outliers and symmetric heavy tailed alternatives; here we conduct simulation studies to

calculate the power of rejecting the null hypothesis of the test (the data is normal

distributed). Our simulations demonstrate that the are able to yield (i) good result to

overcome the presence of outliers of the data and (ii) as efficient as other robust statistics

that were introduced by others previously when the existence of assumption other than

normal distribution.

RJBs

We also apply robust L1–ARCH model to compute the uncertainty of inflation of ASEAN–

5 countries. For completion, we examine the relationship between uncertainty of inflation

and their economic growth using robust regression models. We find overwhelming

statistical evidence supporting the hypothesis that increased inflation uncertainty lead to

slow down in economic growth. This positive association between inflation uncertainty and

growth is consistent with earlier studies, conducted for the major industrialized countries.

This finding is in line with Friedman’s hypothesis that suggests uncertainty concerning

regime changes depresses real economic activity.

Throughout this thesis, the SPLUS programming language is used to run (a) simulation

tests towards tests; (b) estimating parameters of LRJB 1–GARCH model; (c) comparison

of efficiency of conditional variance between ordinary GARCH(1,1) and L1–GARCH(1,1)

and (d) estimating parameters of L1–ARCH model for inflation uncertainty amongst

ASEAN-5 countries.

vi

ACKNOWLEDGEMENT

I would like to express my sincere appreciation to the University Science of Malaysia

(USM) and the Government of Malaysia for continuously sponsoring me towards being a

fellow of the Academic Staff Training Scheme (ASTS) of the USM.

I would like to express my sincere grateful to the following

1. Professor Dr. Nor Aishah binti Hamzah of the University of Malaya (UM), who has

supervised me towards completing my Ph.D.

2. Professor Dr. Ahmad Zubaidi bin Baharumshah of the University Putra Malaysia

(UPM), who advises me in application of statistics that contributes to better

understanding of current economic issues.

3. My beloved wife, Nurul Anisah binti Jaapar and children who understand my

current situation.

4. My mother, Norma binti Ahmad and my siblings.

5. Muhammad Fathilah bin Muhd. Yusuff and family.

6. My father-in-law, Jaapar bin Dollah and mother-in-law, Asiah binti Othman, and

my wife’s siblings.

7. And those who have supported me materially and spiritually.

It is hard to achieve at this stage without all of your assistance.

vii

TABLE OF CONTENTS

Abstrak iii

Abstract v

Acknowledgement vii

Table of Contents viii

List of Figures xi

List of Tables xii

1. Chapter 1 – Introduction 1

1.1 Introduction 1

1.2 When are ARCH and GARCH models applicable? 2

1.3 Problems with ordinary GARCH 8

1.4 The GARCH family 10

1.4.1 Inflation, inflation uncertainty and economic growth 11

1.4.2 Stock Market and Exchange Rates 13

1.5 Software and Computation 14

1.6 Problem Statements and Objectives of the Study 15

2 Chapter 2 – GARCH family 17

2.1 Introduction 17

2.2 Properties of GARCH family 18

2.3 Maximum Likelihood Estimation (MLE) of the GARCH ( Model )qp, 20

2.4 GARCH tests 24

2.4.1 Normality test 24

viii

2.4.1.1 Measures of Skewness and Kurtosis 25

2.4.1.2 Classical and Robust Jarque – Bera

statistics

32

2.4.1.3 The Robust Jarque–Bera ( ) test

statistics

RJB 35

2.4.1.4 Critical Values and Power of the New

tests RJB

40

2.4.2 Other test statistics 50

3. Chapter 3 – Least Absolute Deviation (LAD) for GARCH models 54

3.1 Introduction 54

3.2 L1–ARCH 55

3.2.1 Model 55

3.2.2 Parameter Estimation 55

3.2.3 Asymptotic normal distribution of Parameter

estimates

57

3.3 L1–GARCH 58

3.4 L1–(G)ARCH – Peng and Yao (2003) 60

3.5 Alternative estimations for L1–GARCH parameters 61

3.6 Bootstrap Standard Error 64

3.7 Testing on the autocorrelations of the L1–(G)ARCH models – L1–

residuals autocorrelation and robust portmanteau test

64

4. Chapter 4 – Study on the Performance of Conditional Variances – L1–

GARCH and GARCH – a comparison.

68

4.1 Introduction 68

ix

4.2 Measures of performance 71

4.3 Relationship between and . 21tσ 2

2tσ 75

4.4 Outlier diagnostic procedures 76

4.5 Outlier detection and estimation of conditional variances 77

4.6 Results 80

5 Chapter 5 – Inflation Uncertainty and Economic Growth: Evidence from LAD

ARCH Model

85

5.1 Introduction 85

5.2 The volatility model: Robust L1–ARCH model 88

5.3 Data 90

5.4 Empirical Results 92

6 Chapter 6 – Discussion 101

7 References 105

8 Appendix 113

Appendix 1 – Sample data of volatility 113

Appendix 2 – Asymptotic Normal Distribution for Median Statistic and

Bahadur Representation for L1–GARCH model

118

Appendix 3 – Programming language in S–PLUS for simulation study 125

Appendix 4 – Inflation Uncertainty and Economic Growth: Evidence

from LAD ARCH Model

162

Appendix 5 – Tables of Distributions and their Properties 169

x

LIST OF FIGURES

Figure 1.1 The time series plot for 756 data of return 2

Figure 1.2 The time series plot of residuals of volatility data in Appendix

1

5

Figure 1.3 Density plot of residuals 6

Figure 2.1 Quantile – quantile (Q-Q) plots for Jarque – Bera ( s)

statistics

JB 38

Figure 4.1 Boxplot of Average Absolute Errors of parameter estimates 74

Figure 4.2 Conditional variances of GARCH(1,1) (solid lines) model and

the contaminated conditional variances (dotted lines)

78

Figure 4.3 Boxplots for and for non contaminated data, 2tAEσ 2

tAAEσ

0=γ

82

Figure 4.4 Boxplots for and for 1% contaminated data 2tAEσ 2

tAAEσ 83

Figure 4.5 Boxplots for and for 5% contaminated data 2tAEσ 2

tAAEσ 83

Figure 5.1 Inflation Uncertainty and Economic Growth: LTS and OLS fits 98

Figure A2.1 Density plot for 1000 values of L which has been generated

with 1000 size of data

120

Figure A4.1a L1–ARCH Estimates of Inflation Uncertainties – Indonesia 166

Figure A4.1b L1–ARCH Estimates of Inflation Uncertainties – Malaysia 166

Figure A4.1c L1–ARCH Estimates of Inflation Uncertainties – Philippines 167

Figure A4.1d L1–ARCH Estimates of Inflation Uncertainties – Singapore 167

Figure A4.1e L1–ARCH Estimates of Inflation Uncertainties – Thailand 168

xi

LIST OF TABLES

Table 1.1 Statistical summaries of fitted models for data from Table A1.1

in Appendix 1

4

Table 2.1 Empirical Estimates of average mean, , and using

1,000 simulated with various distributions

J 2SK 2KR 26

Table 2.2 Empirical Estimates of average mean, , and using

1,000 simulated with various distributions

J 2SK 2KR 30

Table 2.3 Estimated variance for skewness and kurtosis using 10,000

simulated standard normal distribution

34

Table 2.4 Critical region at 5% for and robust JB JBs 41

Table 2.5 Power of rejection for tests – tests among different types of

distributions

JB 42

Table 2.6 Power of rejection for tests – tests among various

conditions of outliers

JB 46

Table 4.1a Average estimated parameters of (4.2a), L1–AR(1)–ARCH(2)

model

73

Table 4.1b Average estimated parameters of (4.2b), L1–AR(1)–

GARCH(1,1) model

73

Table 4.1c Average estimated parameters of (4,2c), L1–AR(1)–

GARCH(1,1) model

73

Table 4.2 Statistical summary for 2tiAEσ 84

Table 5.1 Summary statistics of inflation rates data for ASEAN-5 92

xii

Table 5.2 Standardized absolute L1–residual autocorrelations, sr~ 92

Table 5.3 Parameter estimates and tests for model: GARCH(1,1) model 93

Table 5.4 L1–ARCH models for each inflation rates for ASEAN-5

countries

95

Table 5.5 Parameter Estimates of the simplest model, 1980-2003 97

Table 5.6 The comparison of method of estimates between OLS and

LTS, for the equation (5.5)

100

Table A1.1 Sample data of volatility discussed in Chapter 1 113

Table A2.1 The statistics of 1000 replicated data of

( )[ ] ( ) ( )[ ]MFMFMML XX −−−= *2/*

119

Table A3.1 Average and Median of values of when 1l 96.1=c 127

Table A3.2 Average and Median of values of when 1l 50.2=c 127

Table A3.3 Average and Median of values of when 1l 00.3=c 127

Table A3.4 Average and Median of values of when 1l 50.3=c 127

Table A3.5 Average and Median of values of when 1l 00.4=c 128

Table A3.6 Average and Median of values of when 2l 96.1=c 128

Table A3.7 Average and Median of values of when 2l 50.2=c 128

Table A3.8 Average and Median of values of when 2l 00.3=c 128

Table A3.9 Average and Median of values of when 2l 50.3=c 129

Table A3.10 Average and Median of values of when 2l 00.4=c 129

Table A3.11 Estimated mean for skewness and kurtosis using 10,000

simulated standard Normal distribution

133

Table A3.12 Estimated correlation between skewness and kurtosis 134

xiii

Table A4.1 Consumer Price Index (CPI) for Asean–5 countries 162

Table A4.2 Growth of Gross Domestic Product (GDP) (%) for Asean-5

(1980 – 2003)

165

Table A5.1 Distributions and their Properties – Density function, Mean and

Median

169

Table A5.2 Distributions and their Properties – Variance, Skewness and

Kurtosis

171

xiv

1

CHAPTER 1

Introduction

1.1 Introduction

Since Engle (1982) introduced the Autoregressive Conditional Heteroscedastic (ARCH)

process, there has been an increase in the number of theoretical and applied works where

the conditional variances and scales are allowed to change as a function of past errors (see

Fiorentini et al (1996), Koenker and Zhao (1996), Jiang et al (2001) and Park (2002a)). The

ARCH model and its extensions have been widely applied in finance and econometrics

such as stock returns, inflation uncertainty and exchange rates (see Jiang et al (2001)). The

Generalized ARCH (GARCH), which was developed by Bollerslev (1986) and has evolved

for more than two decades, has been widely used in explaining the behavior of financial

time series data.

In measuring risk, forecasting the volatility of a time series is of interest (see Franses and

Ghijsels (1999)). In addition, forecasting financial market volatility has been the subject of

one of the most active bodies of literature in finance (see Park (2002a)). The GARCH

model which computes the conditional variance is an important tool measuring the

volatility, called the risk measurement. From the macroeconomic point of view, several

researchers have focused on several aspects of how individuals respond to inflation

uncertainty. This has brought about the importance of such measurement and has lead to

the development and applications of (G)ARCH processes (see Engle (1986) on the U.K.

inflation estimates, and Apergis (2003), Apergis (2005) for OECD countries).

2

Quantifying uncertainty in financial time series is relatively new in modern finance theory.

Malik (2005), for instance, supports that, the treatment of the exchange rates, using the

fascinating tools of GARCH modeling would make this research a valuable addition to the

existing literature on the volatility of the exchange rates. Further, Wang et al (2001) argued

that, contemporary modeling of exchange rate time series results in the widespread use of

GARCH family.

1.2 When are ARCH and GARCH models applicable?

Gujarati (2003) in his text book, defines the volatility clustering as follows

“periods in which their prices show wide swings for an extended time period followed by

periods in which there is relative calm”.

Consider the data of returns given in Appendix 1. This time series data of size 756=T , is

presented in Figure 1.1. This figure visualizes the definition of volatility cluster of the time

series data quoted from Appendix 1.

Figure 1.1 The time series plot for 756 data of return

3

Consider the general time series model:

(1.1) ttty εμ +=−1tψ|

where the residual, tε , is usually being normally distributed with zero mean and variance,

2σ for all Tt ,,1…= ; the conditional mean, tμ , in equation (1.1) can be of the

autoregressive (AR), moving average (MA) or the combination of autoregressive moving

average (ARMA), and many more; and finally …,, 21 −−− = tt yy1tψ . If we consider the

AR( q ), then we have qttt yy −− +++= ααωμ …1 and for ARMA( pq, ), we have

pttqttt yy −−−− ++++++= βεβεααωμ …… 11 . If we allow the simplest model to fit the

data given in Appendix 1, 0007619.0ˆˆ == ωμ t . If we plot the time series data of the

residuals, that is the difference between the crude data and mean, we obtain a similar plot as

Figure 1.1. In addition, the standard error, 012111.0ˆ =Tσ is bigger than its mean,

implying that the simplest model of tty εω += is unsuitable for the volatility data under

study. We are also considering the model of AR(1) and ARMA(1,1) and are presented in

Table 1.1 below.

4

Table 1.1 Statistical summaries of fitted models for data from Table A1.1 in Appendix 1

Model ( ) ω=−1tψ|tyE ( ) ω=−1tψ|tyE 1−+ tyα

( ) ω=−1tψ|tyE

11 −− ++ tty βεα ω 0.0008

(0.0012) 0.0008

(0.0012) 0.0009

(0.0014) α - -0.0312

(0.0364) -0.1784 (1.0975)

β - - -0.1466 (1.1033)

JB 1978.514 [0.000]

1982.000 [0.000]

1985.309 [0.000]

10Q 16.288 [0.092]

544.955 [0.000]

14.996 [0.132]

210Q 548.131

[0.000] 212.540 [0.000]

212.821 [0.000]

Note: The figures in parentheses indicate the standard error of parameter estimates. The figure in bracket shows the p-values of estimated statistics.

In Table 1.1, ω , α and β are the estimated parameters, JB is the Jarques–Bera test of

normality distribution for estimated residuals whereas 10Q and 210Q are autocorrelation

tests (Ljung – Box test and Portmanteau test) for residuals and squared residuals

respectively. Details of the measurements of autocorrelation and normal tests are discussed

in Chapter 2 of the thesis. The JB statistic is asymptotically 22χ and the Q -statistic at M

lags is 2Mχ . Here, the JB test suggests that the residuals are not normally distributed for all

models. In addition, if we take 5%, significance level, that is 307.182%5,10 =χ , the Ljung–

Box test, 10Q suggests that the residuals are autocorrelated only at AR(1) model, but the

Portmanteau suggests that the squared residuals are autocorrelated for all models. The

autocorrelated of the squared residuals may suggest there must be a relationship between

the present and previous values of the second order moment of the residuals.

5

If we take the squared residuals (by considering the simplest model) of the data, the time

series plot of the residuals can be shown in Figure 1.2.

Figure 1.2 The time series plot of residuals of volatility data in Appendix 1

The problem triggered here is the squared or absolute residuals are autocorrelated. Further,

the density plot of the residuals shown in Figure 1.3 is indicative of non-normal distribution

with excess kurtosis (sometimes called leptokurtosis). The skewness and kurtosis of the

residuals are 0.3092 and 10.9909 respectively, suggesting a heavy-tailed distribution (for

normal distribution, skewness and kurtosis are 0 and 3, respectively). Equation (1.1),

assumes that the variance is constant for the entire time series. However, Figure 1.2

illustrates that volatility clustering is present for this time series data. Therefore, models

that are associated with fitting the conditional mean such as AR, MA, ARMA or even

Autoregressive Integrated Moving Average (ARIMA) are insufficient for the volatility fit.

Hence, the need for some other alternative model is inevitable.

6

Figure 1.3 Density plot of residuals

-0.2 -0.1 0.0 0.1 0.2 0.3

02

46

810

12

skewnesskurtosis

0.30910.991

As an alternative, we may impose additional assumption in equation (1.1). Here, the

residuals may be modified to include,

(1.2) ttt uσε =

where tu is white noise (with zero mean and unit variance) and the residuals are distributed

with zero mean and standard deviation tσ , where tσ is a function consisting of its previous

…,, 22

21 −− tt σσ . Here, the squared of tσ in (1.2), 2

tσ is called conditional variance, may be

expressed in the form that is similar to AR, MA, ARMA and many more. This is the insight

of GARCH modeling, comprising two main parts: conditional mean and conditional

variance.

In financial data, volatility clustering is evidence as reported by many researchers. Horváth

and Liese (2004) supported that in analyzing data from financial markets ARMA processes

7

for instance, and their modifications, cannot be fitted to such data in a satisfactory manner.

The main reason is that ARMA processes model the conditional expectation as a linear

function with variance assumed constant and homogeneous for all series. Thus, standard

ARMA processes fail to fit such data. The basic idea of ARCH and GARCH consist of

developing a model for the conditional variance. This approach seems to be consistent with

atypical data from finance. The basic idea for constructing ARCH processes is to model the

volatility, and subsequently study the behavior of volatility. GARCH was introduced

mainly to allow a more flexible lag structure compared with the ARCH specification (see

Fiorentini et al (1996)). Bollerslev (1986) fitted the US GNP deflator using GARCH(1,1),

where as Engle and Kraft used ARCH(8). The reducing of ( )∞ARCH to ( )pq,GARCH ,

where ∞<+ pq , means that less parameter estimates is required by the GARCH model

than those of ARCH, yet sufficient to explain the time series uncertainty (see Bollerslev

(1986) in discussing the Engle and Kraft (1983) ARCH(8) model).

An important feature of a GARCH model is that it can be fitted to data with excess kurtosis

(see Franses and Ghijsels (1999)). Wang et al (2001) early supported that, GARCH models

have been shown not only to capture volatility clustering, but also to accommodate some of

the leptokurtosis (i.e. thick tails) commonly found in exchange rates and time series data.

Park (2002a) verified that financial economists have built sophisticated models for

forecasting volatility, such as ARCH and GARCH models. The GARCH models are now

the most widely used models to forecast the time-varying volatility observed in many

financial returns, simply because they are able to handle financial data with heavy-tailed

distribution and volatility clustering. A detail discussion of the standard GARCH model can

be provided in Chapter 2.

8

1.3 Problems with ordinary GARCH

Despite the advantages and dynamics of the GARCH model, it is frequently observed that

the estimated residuals from GARCH model continue to face the problem with excess

kurtosis (see Franses and Ghijsels (1999), Park (2002a) and Hall and Yao (2003)).

Furthermore, some evidence recognized that financial returns tend to form thicker tails that

is far away than normal distributions (see Jiang et al (2001)). In addition, Park (2002b)

claimed that, asymmetric impact of shocks on volatility can vary with levels of returns.

Thus, GARCH models with conditional normal errors assumptions, generally fail to

sufficiently capture leptokurtosis which is evident in asset returns (see Wang et al (2001)).

Early successes of (G)ARCH modeling of financial time series were confined to the case of

normal errors, for which an explicit conditional likelihood function is readily available to

facilitate estimation of parameters in the model (see Hall and Yao (2003)). However, in

practice, the actual distribution error term is typically unknown. Therefore, misspecification

of the form of the conditional distribution in likelihood estimation can create serious

problems for parameter estimations and conditional prediction intervals (see Koenker and

Zhao (1996), Jiang et al (2001) and Hall and Yao (2003)). Investigation of non-normal

cases has been partly driven by empirical evidence that financial time series are heavy-

tailed. For instance, the increased attention focused on distributional properties (particularly

tail thickness) when estimating exchange rates models has lead to the widespread adoption

of non-normal conditional error distributions, the most commonly used is the Student-t.

The Student-t model has thicker tails than the normal, with skewness remains at zero (see

Wang, Fawson, Barret and Mcdonald (2001)).

9

Another possible cause to the failure of the ordinary GARCH is the presence of additive

outliers (AOs), which are not captured by the model. Note that, neglecting AOs could lead

to biased parameter estimates in conditional mean equations, and subsequently result in

biasness to sample forecasts (see Franses and Ghijsels (1999)).

With the problems in hand, it is probably worthwhile to investigate some general modeling

techniques without specific distribution assumptions. In earlier study, conditional Gaussian

likelihood has motivated parameter estimators, called maximum quasi-likelihood estimators

(see Hall and Yao (2003), Bollerslev and Wooldridge (1992), Bollerslev et al (1992) and

Fiorentini et al (1996)). Another alternative uses the method of estimation based on least

absolute deviation (LAD) method thus resulting in robust estimation procedures (see Jiang

et al (2001), Park (2002a) and Peng and Yao (2003)). Consequently, these findings lead to

the perception that the GARCH models may have unavoidable limitations in forecasting the

volatility of exchange rate returns (see Park (2002a)). This motivation has led us to

investigate methods that are less sensitive to normality assumption (see Koenker and Zhao

(1996)), enabling us to overcome such problems. This leads to semiparametric (G)ARCH

models in which the error distributions are unknown (see Hall and Yao (2003)).

Since the forecasts from any time series model are based on the historical patterns and if

the parameters of the model are very dependent on a few typical observations that are

isolated or non repeatable events, the quality of the forecasts can be expected to be poor

(see Pena (1990) and Chen and Liu (1993)). Some examples of non-repeatable events

include implementation of a new regulation, major changes in political or economic policy,

or the occurrence of a disaster in time series concept may be referred as additive outliers

(AOs). Earlier works of Pena (1990), Chen and Liu (1993) and Franses and Ghijsels (1999)

10

and well as some other outlier detection tools which are able to recognize, minimize the

impact of suspicious observations and measure the change that produces in relevant

features of the model such as the estimated parameter values or the forecasts (see Pena

(1990)) could be employed in detecting AOs.

1.4 The GARCH family

Over the years, the GARCH family has become more efficient in fitting the volatility data

as they consist of the second order moment that measures the time-variant of the volatility

data. The initial studies by Engle (1982) and Bollerslev (1986) turn out to be the better

models for volatility (financial) data as the residuals of the data form fatter tailed. The

maximum likelihood estimation (MLE), is a natural approach to employ, when the

standardized residual is normal distributed (see Bollerslev and Wooldridge (1992), Franses

and Ghijsels (1999), Jiang et al (2001), Park (2002a), Dijk et al (1999), Koenker and Zhao

(1996), Peng and Yao (2003), Hall and Yao (2003), Duchesne (2004) and Horvath and

Liese (2004) and many more).

The linear model of the conditional variance itself has its limitation and hence the GARCH

itself may fail to fit some financial data especially in high frequency data. This leads to

empirical findings that indicate the weakness of imposing ordinary GARCH model;

subsequent development and modification of GARCH include the following: Nelson

(1990) performed EGARCH that is the conditional variance being exponentially formed,

Engle et al (1987) with their ARCH-M, Engle and Rivera (1991) with semiparametric

ARCH, Engle and Bollerslev (1986) with Integrated GARCH (IGARCH), Engle et al

(1990) with factor-ARCH, Baillie et al (1996) with Fractionally GARCH (FIGARCH) and

Bolllerslev and Ghysels (1996) with Periodic ARCH. Recent development of robust

11

GARCH models includes the L1–ARCH by Jiang et al (2001), L1–GARCH by Park (2002a)

and Peng and Yao (2003) and Lp–ARCH by Horváth and Liese (2004).

It is known that the GARCH family has good fit for many econometric data and this tool is

now widely used to explain some current economic situation. The most popular finance

economic data that have been considered in various studies are inflation uncertainty, stock

returns, and exchange rates.

1.4.1 Inflation, inflation uncertainty and economic growth

The previous traditional studies discussed the relationship between inflation and growth

(see Motley (1998), Fischer (1993), Temple (2000) for some examples). Since GARCH

family was introduced recently to compute uncertainty, the relationship between inflation,

inflation uncertainty and economic growth have been widely discussed (see Fischer (1993),

Tommassi (1994), Barro (1996), Judson and Orphanides (1996), Greir and Perry (2000),

Dotsey and Sarte (2000), Apergis (2005)). Many articles support that inflation is positively

correlated with its uncertainty, but uncertainty on the other hand affects other factors

negatively, such as economic growth and other real activities (see Fischer (1993), Grier and

Perry (1996), Grier and Perry (2000), Judson and Orphanides (1996) Tommassi (1994)).

The estimated conditional variance from the family of GARCH models is used in this thesis

to evaluate inflation uncertainties. For the U.S. inflation studies, Hwang (2001), use various

types of ARFIMA-GARCH models and found that inflation is negatively, but weakly

affected by its uncertainty, and a high rate of inflation does not necessarily imply a high

uncertainty of inflation.

12

Grier and Perry (1996) studied the relationship between inflation, inflation uncertainty and

relative price dispersion (RPD). In their study, briefly, they computed the RPD such that,

( )∑=

−=n

ititt nRPD

1

2/1 ππ

where, tπ represents the aggregate inflation rate, itπ is the price change in individual

commodity (in their study, they listed 10 categories of commodities, such as, furniture,

other durables, food, clothing, housing, household operations, transportation, medical care,

and other services). They estimated the inflation uncertainty using bivariate GARCH(1,1)-

M, and thus regressed the RPD with the inflation and its estimated uncertainty to assess the

relationship above. They found less evident that, trend inflation influences the RPD, but the

combination of inflation and inflation uncertainty influence the relative price dispersion

effectively. In this thesis, we shall adopt the L1–ARCH approach to study the behavior of

inflation uncertainty of ASEAN-5 data.

In studying the effects of real and nominal uncertainty on inflation and output growth, Grier

and Perry (2000), suggested that there is no evidence that higher inflation uncertainty or

higher output growth uncertainty raises the inflation rate. They also found that, there is no

evidence that the higher the uncertainty of growth, the higher the growth rate. Also, the

higher inflation uncertainty lowers the output growth was reported. The GARCH-M

method is used to estimate the uncertainty of inflation and growth.

In another study, Daal et al (2005) examined the relationship between inflation and

inflation uncertainty for developed and emerging countries, such as, Asian, G7, Latin

American and Middle East, by using autoregressive moving average with asymmetric

13

power GARCH (ARMA–PGARCH(1,1)) to estimate the inflation uncertainty for each

country. For this, Granger causality test (which will not be discussed in this thesis) is used

to test whether the inflation causes inflation uncertainty vise versa. They found strong

evidence – for Latin American countries, positive inflationary shocks have stronger impacts

on inflation uncertainty.

Panel data is also used for the above study. Apergis (2004) conducted comprehensive study

of the relationship between inflation and output growth for panel data of G7 countries. The

panel data estimation was carried out by using Error Correction Vector Autoregressive

(ECVAR) (which will not be discussed in this thesis). The causality tests were done to

recognize the relationship between inflation and inflation uncertainty, inflation uncertainty

and growth, and inflation and growth. The earlier study discovered that, the inflation affects

output growth, and inflation causes inflation uncertainty.

Berument and Dincer (2005) used GARCH to compute the inflation uncertainty for G7

countries. By using full information maximum likelihood method with extended lags, they

found that the inflation causes inflation uncertainty except for Canada, France, Japan, the

UK and the US.

1.4.2 Stock Market and Exchange Rates

The use of GARCH in measuring uncertainties also has wide applications in stock market

and exchange rates studies. As the stock returns form volatility, to study the behavior of

this series – forecasting the returns and their volatility and study the crisis that effect the

high spike of the volatility, the GARCH family offer dynamic tools as they do not also

contain the first order moment, but also second order, which leads to study widely, the

14

current global issues on this matter. Interesting applications can be found in Franses and

Ghijsels (1999), Jiang et al (2001) and Mun (2005).

In exchange rates studies, the risk of foreign exchange rates of return has been the subject

in modern financial analysis. Examples of the application of GARCH in exchange rates

studies can be found in Engle and Bollerslev (1986), Park (2002a and b) and Tambakis and

Royen (2002).

1.5 Software and Computation

A substantial part of the research work and amount of research time is devoted to the study

of the properties of various methods used to estimate the parameters of models: ARCH,

GARCH and its robust versions. In performing such tasks, many problems surfaced as far

as computation is concerned.

Many statistical software that are available for computation of parameter estimates and

forecasting for the various models considered, do not reveal the complete computational

programmes. The hidden subroutines made it more difficult for users to edit and modify

the existing programmes when new materials are to be incorporated, thus demanding new

programmes to be written.

Throughout this research, I have used the S-programming language for computation of

parameter estimates, generating data sets as well as conducting simulation studies. The S-

programming language has been around for more than two decades and is widely used in S-

Plus and Open Source R, particularly developed for statistical data analysis. The computer

programmes developed for this research work is included under Appendix 3.

15

1.6 Problem Statements and Objectives of the Study

In section 1.3, we have discussed the disadvantages of classical GARCH which uses least

squares method of estimation. Several models with various assumption of the distributions

and techniques of estimates of parameters have been introduced. In this thesis, we allow the

distribution assumed to be other than the standard normal. Therefore, instead of relying on

least squares (LS), we use least absolute deviation (LAD) as an alternative estimation

procedure.

The properties and method of estimation for ordinary GARCH, is reviewed and discussed

in Chapter 2. We also discuss the statistical tests on normality and autcorrelation of the

estimated innovation. We demonstrate that the classical Jarque-Bera test is sensitive in the

presence of outliers, thus rejecting the normality assumption (see Gel and Gastwirth (2007)

for further explanation). In this chapter, we propose new tests of normality using robust

measure of skewness and kurtosis as well as some robust measures which utilizes a filter

function which is bounded and continuous with the hope that these measures are less

influenced by outliers.

Some authors (Engle (1982), Bollerslev (1986), Fiorentini et al (1996), Jiang et al (2001)

and many more) systematically derived the algorithm of estimating (G)ARCH parameters.

Park (2002a), Hitomi and Kagihara (2001) and Peng and Yao (2003), for example,

introduced the method of estimation of parameters which minimizes the Least Absolute

Deviation (LAD) objective function for nonlinear and GARCH models without extension

of deriving their method of estimations. In this thesis, due to the computational demanding,

we discuss such friendly algorithm, using the SPLUS software, of estimating the

16

parameters for L1–(G)ARCH models in Chapter 3. We also discuss the inference of the L1–

ARCH and GARCH with further details in Appendix 2.

Park (2002a) compares his robust GARCH model with those ordinary GARCH and random

walk model with computing Mean Absolute Error and Root Mean Square Error of one-

step-ahead-forecast errors. Others compare their performances in estimating parameters

(see Jiang et al (2001) and Peng and Yao (2003)). In our thesis, we are interested in

assessing the performance of estimation of conditional variances between L1–GARCH

compared to ordinary GARCH. To the best of our knowledge, no previous studies have

examined the difference in the performance of these two volatility models. There are slight

differences between conditional scale (standard deviation) for L1–(G)ARCH and the

ordinary GARCH, as the assumption in distributional and the models itself. In this thesis,

we derive the relationship between these conditional scales, so as to enable us to assess the

goodness of L1–GARCH model, particularly in the presence of AOs. Details of these topics

as well as the simulation studies to enhance our understanding of behavior of two estimates

are given in Chapter 4 of this thesis.

Finally, in Chapter 5, we apply robust L1–ARCH model to construct the ASEAN–5

inflation uncertainty. We also look at the cross sectional relationship between inflation,

inflation uncertainty and economic growth among panel data of the countries involved:

Indonesia, Malaysia, Philippines, Singapore and Thailand. The main purpose is to show

how the robust L1–ARCH model may be applied to empirical work.

17

CHAPTER 2

GARCH Family

2.1 Introduction

In this chapter, we present and discuss the properties of ordinary linear GARCH models

and its method of parameter estimation. The properties of (G)ARCH models can be found

in Engle (1982) , Bollerslev (1986), Weiss (1986) and Hamilton (1994), while the method

of parameter estimation and its algorithm are discussed by Bollerslev (1986), Greene

(2003), Gujarati (2003), Fiorentini et al (1996). The use of these models in analyzing

volatility in time series data can be referred to Zivot and Wang (2001). We also provide

GARCH tests based on their normality and autocorrelation tests. In addition, we introduce

our robust Jarque-Bera test that is less sensitive to the existing of outliers.

Engle (1982) and Bollerslev (1986) provide a detail account on the method of maximum

likelihood of estimation (MLE) for ordinary ARCH and GARCH parameters respectively.

Bollerslev (1986) and Fiorentini et al (1996) employ the Berndt, Hall, Hall, and Hausman

(BHHH) algorithm introduced by Berndt et al (1974), to speed up the iterative part so that

convergence of the objective function can be achieved in fewer iterations.

18

2.2 Properties of GARCH family

Consider a simple linear GARCH( pq, ) of (1.1) where,

(2.1) tttttt uy σμεμ +=+=

and

2211

22110

2ptptqtqtt −−−− ++++++= σβσβεαεαασ ……

It is easier to explain from equation (2.1),

(2.2) 2211

22110 ptptqtqtttt uy −−−− +++++++= σβσβεαεααμ ……

with tu as random variable with ( ) 0=tuE and ( ) 12 =tuE . To ensure the validity of second

term of equation (2.1), that is 2 0tσ > for 1, ,t T= … , the conditions, 0≥iα , qi ≤≤1 ,

0≥jβ , pj ≤≤1 and 00 >α are imposed. Here, tε is a random variable referred as

residuals, tu as the white noise or standardized residuals1, and 2tσ as the conditional

variance at time t . Briefly, the GARCH model consists of two components: the conditional

expectation (sometimes called conditional mean) ( ) ttyE μ=−1tψ| , and the conditional

variance ( ) 22 | ttE σε =−1tψ of equation (2.1) with tu assumed as the white noise. Equation

(2.1) indicates the conditional variance, ( ) 22 | ttE σε =−1tψ of this equation forms an

Autoregressive Moving Average (ARMA) process which usually is called linear GARCH

model.

1 In some literature such as Jiang et al (2001) and Park (2002, 1), the white noise might be defined as innovation as it may be derived from some alternative distribution (other than standard normal distribution).

19

Bollerslev (1986) formulates the unconditional variance, 2σ of GARCH model as

(2.3) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛+−

==

∑∑==

p

jj

q

ii

tE

11

022

1 βα

αεσ

and exist if 111

<+∑∑==

p

jj

q

ii βα .

If we assume that white noise is Gaussian distributed (i.e. ( )1,0~ Nut ), then the residuals,

tε , is also Gaussian distributed: ( )2,0~ tt N σε . Thus, the covariance between the white

noise and residuals is

(2.4) ( )⎩⎨⎧

≠=

=stst

uE tst 0

σε

and the covariance for residuals,

(2.5) ( )⎩⎨⎧

≠=

=stst

Cov tst 0

,2σ

εε

When the underlying distribution is known (here, being the assumption of normal

distribution), a natural method of parameter estimation to be used is that of maximum

likelihood estimation. Details of the derivation can be found in Engle (1982), Bollerslev

(1986), Greene (2003), Gujarati (2003), Hamilton (1994), and Fiorentini et al (1996).

20

Consider the following AR model for (1.1),

(2.6) ttktktt ybybby εε +=++++= −− bxTt…110

where tε is assumed as ( )20~ σε ,Nt . In practice, however, this is not usually the case as

residual may not be stationary, and may take the form of ( )20~ tt ,N σε . For this,

Bollerslev (1986) introduces the GARCH ( )pq, model, that is,

(2.7) γzTt=++++++= −−−−

2211

22110

2ptptqtqtt σβσβεαεαασ ……

When 0=p , (2.7) becomes an ARCH ( )q model (see Engle, 1982) :

(2.8) 22110

2qtqtt −− +++= εαεαασ …

2.3 Maximum Likelihood Estimation (MLE) of the GARCH ( )qp, Model

Consider model (2.6) and (2.7), with the assumption, ( )20~ tt ,N σε . Let T be the number

of observations and define vectors, ( )( )',, 1110 ×+= kkbbb …b ,

( )( )',...,,,...,,

11110 ×++=

pqpq ββαααγ , ( )( )',,,1 111 ×+−−= kktt yy …tx and

( )( )',,,,,,1

1122

122

1 ×++−−−−=pqpttqtt σσεε ……tz , for Tt ,,1…= , and k number of independent

variables. As equation (2.6) forms AR process, Engle (1982) and Bollerslev (1986) suggest

an ordinary least squares (OLS) to estimate b , initially. The estimated error, tε is

computed as bx'tˆˆ −= tt yε . From equations (2.6) and (2.7) above, we define:

21

(2.9a) tt gγ

zγ

=+= ∑=

−p

j

jtj

t

dd

dd

1

22 σβσ

(2.9b) txb

−=dd tε

(2.9c) tit db

xb

=+−= ∑∑=

−=

−−

p

jjtj

q

iiti

t

dd

dd

1

2

1

2

2 σβεασ

(2.9d) 12

2

−=t

ttv

σε

(2.9e) 4

222

t

tttt

ddv

σεσε tt dx

b+

−=

The log-likelihood function for a sample of T observations is

(2.10) ( ) ( ) ( ) ( )∑∑==

⎥⎦

⎤⎢⎣

⎡+−−=⎥

⎦

⎤⎢⎣

⎡++−=

T

t t

tt

T

t t

tt

TL1

2

22

12

22 ln

212ln

2ln2ln

21ln

σεσπ

σεσπ

MLE aims at estimating γ which maximizes equation (2.10). This may be attained by

differentiating the equation (2.10) respect to γ and substituting equation (2.9a) into

differentiated (2.10) to solve the following equation:

(2.11) vG'ggγ

tt ==⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑∑

==

T

t

t

t

T

t t

t

t

vLdd

12

12

2

2 2211

21ln

σσε

σ

where G is a matrix containing elements ⎟⎟⎠

⎞⎜⎜⎝

⎛22

1

tσtg whereas v is a vector containing

⎟⎠

⎞⎜⎝

⎛2tv . Taking the expectation of equation (2.11), this yields

22

(2.12) ( ) ( ) 1pq1tt 0gg

γ ×++==

==⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∑∑T

tt

t

T

t t

t

t

vEELddE

12

12

2

2 211

21ln

σσε

σ

To solve this, we will resort to the Newton-Raphson method containing Hessian matrix and

first derivatives vector. The Newton-Raphson method forms as follows:

(2.13) ( ) ( )

( )m

m1m

γγ'γγγ ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

−

+ LddL

dddE lnlnˆˆ

12

which requires the second derivative of (2.10) with respect to γ , as the Hessian matrix.

But, Greene (2003) suggests method of scoring as the Hessian, instead. Fiorentini at el

(1996) called this Hessian as information matrix. This may result the equation (2.13) being

(2.14) ( ) ( ) ( ) ( ) ( )mm1m vG'GG'γγ 1ˆˆ −+ +=

where, '

gggG T21⎟⎟⎠

⎞⎜⎜⎝

⎛= 22

221

,,,2

1

Tσσσ… and ( )',,,

21

21 Tvvv …=v .

We also aim, by using MLE, estimating b which maximizes equation (2.10). This may be

attained by differentiating the equation (2.10) respect to b and substituting equations

(2.9b), (2.9c) and (2.9d) into differentiated (2.10) to solve the following equations:

(2.15) ∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

T

t tt

t

t vLdd

122 2

1ln tt dxb σσ

ε

Taking the expectation of equation (2.15), this results

23

(2.16) ( ) 1k1tt 0dxb ×+

=

=⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟

⎠⎞

⎜⎝⎛ ∑

T

t tt

t

t vELddE

122 2

1lnσσ

ε

According to Greene (2003), the second derivatives of equation (2.10) respect to b are

(2.17) ∑∑∑∑====

⎟⎠⎞

⎜⎝⎛+−−−=

T

t t

tT

t t

tT

t t

tT

t t ddvvL

ddd

12

14

14

12

2

21

211ln ttttttt d'

bd'dx'dx'x

b'b σσσε

σ

∑∑==

−−T

t tt

tT

t t

t

142

2

14

121

tttt d'dd'xσσ

εσε

This leads us to derive the expectation of the second derivative as

(2.18) ∑∑==

+=⎟⎟⎠

⎞⎜⎜⎝

⎛−

T

t t

T

t t

Ldd

dE1

41

2

2 1211ln tttt d'dx'x

b'b σσ

To estimate the parameters, b , we again use the Newton-Raphson method which results in

(2.19) ( ) ( )

( )m1

m1m

bb'bbb ⎟

⎠⎞

⎜⎝⎛

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

−

+

dLdL

dddE lnln

2

To speed up the estimation process, Bollerslev (1986), Fiorentini et al (1996) and others

employs the BHHH algorithm define as follows

(2.20) ( ) ( ) ( ) ⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=+

θ'θθ'θθθ m1m

ddLd

ddLdm lnln

-12

λ

where, ( )'γ',b'θ = and ( )mλ is step–size at the m th –iteration, used to speed up the

convergence of parameter estimates.

24

The normal asymptotic for the GARCH parameters are as follows

(2.21) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−→−

-12 ln,ˆθ'θ

0θθdd

LdN

2.4 GARCH tests

The estimated white noise (or sometimes called innovation) (see equation (2.1)), tu are

being computed as:

γz'bx'

t

t

ˆ

ˆ

ˆˆ

ˆ −== t

t

tt

yuσε

To test the adequacy of the GARCH model, the Jarque-Bera test, motivated by Bera and

Jarque (1981) and Bera and Jarque (1982), is usually employed in applied work. Whilst the

test is widely used, we note that the JB -test is well known to be highly sensitive to even a

single outlier.

2.4.1 Normality test

The Jarque–Bera ( JB ) test for normality is one of the most popular goodness-of-fit tests

employed in economics. Accordingly, the JB test statistic utilized the information of the

sample skewness and sample kurtosis; in particular the test is the sum of squared of the

third and squared of the fourth moments. Before proceeding to the topic of robust JB test

statistics, let us reconsider some of the possible measures of skewness and kurtosis

developed in the literature.

25

2.4.1.1 Measures of Skewness and Kurtosis

Let tu , Tt ,,1…= , be a set of independent and identically distributed random samples

with mean μ , median M , and variance 2σ . The classical estimates of skewness, iSK and

kurtosis, iKR for tu are given as follows:

(2.22) ∑=

⎟⎠⎞

⎜⎝⎛ −

=T

t

tuT

SK1

3

1 ˆˆ1

σμ

and 3ˆ

ˆ11

4

1 −⎟⎠⎞

⎜⎝⎛ −

= ∑=

T

t

tuT

KRσμ

where, ∑=

=T

ttu

T 1

1μ and ( )∑=

−=T

ttu

T 1

22 ˆ1ˆ μσ

As μ and σ are subject to even a single outlier or departures from the underlying

distribution, the resulting skewness and kurtosis as in equation (2.22) can be very large. In

order to limit such influence, the following measures were proposed by several authors (for

examples, see Kim and White (2003) and Gel and Gastwirth (2007)).

The measure of skewness and kurtosis which are the third and fourth sample moments

respectively are sensitive to small departures from normality, thus resulting in a non-robust

JB test statistic. To overcome the sensitivity of departures from normal distribution, Kim

and White (2003) reviewed the various measures of skewness and kurtosis developed by

Bowley (1920), Groeneveld and Meeden (1984), Kendall and Stuart (1977), Moors (1988)

and Crow and Siddiqui (1967) that are less sensitive to outliers and apply them to S&P500

index. Gel and Gastwirth (2007) proposed a modified skewness and kurtosis by replacing

σ from equation (2.19) into average absolute deviation from the median (MAAD), thus

deriving the robust JB test statistic, which subsequently provides a powerful class of test

26

for testing several alternatives to normality. The Gel and Gastwirth (2007) skewness and

kurtosis are formed as follows

(2.23) ∑=

⎟⎠⎞

⎜⎝⎛ −

==T

t

t

Ju

TJm

SK1

3

33

2

ˆ1 μ

and

3ˆ13

1

4

44

2 −⎟⎠⎞

⎜⎝⎛ −

=−= ∑=

T

t

t

Ju

TJmKR

μ

where ( )∑=

−=T

tutu

Tm

1

33 ˆ1 μ , ( )∑

=

−=T

tutu

Tm

1

44 ˆ1 μ and ∑

=

−=T

tt Mu

TJ

1

2/π . Here,

( )tM median u= .

Table 2.1 Empirical Estimates of average mean, J , 2SK and 2KR using 1,000 simulated with various distributions. Distribution Size Mean J 2SK 2KR 30 0.0073 0.9731 -0.0125 -0.0321 Normal 50 0.0074 0.9813 0.0151 0.0024 100 -0.0039 0.9927 -0.0050 0.0153 30 0.0073 1.3432 0.0106 13.2447 t3 50 0.0141 1.3620 0.6399 27.6961 100 -0.0035 1.3694 -0.1932 35.1357 30 0.0187 1.7030 -0.0034 1.3581 Logistic 50 0.0016 1.7065 -0.0344 1.6708 100 -0.0065 1.7305 -0.0058 1.7730 30 1.6187 1.3508 6.8584 45.9279 Lognormal 50 1.6325 1.3811 9.7004 87.5126 100 1.6463 1.4020 12.6852 151.9723 30 -0.0062 1.1812 -0.0728 16.5026 0.95N(0,1)+0.05N(0,5) 50 -0.0056 1.1862 -0.0484 24.0949 100 0.0009 1.1943 -0.1259 31.3083 30 0.1084 1.0531 0.9291 5.2420 0.95N(0,1)+0.05exp(1/2) 50 0.1059 1.0585 1.0811 6.7973

100 0.0963 1.0695 1.2397 8.8742 30 0.0836 1.0415 0.8571 6.1021 0.95N(0,1)+0.05lognormal 50 0.0809 1.0387 1.4486 15.9707

100 0.0827 1.0431 1.7808 23.0312

27

Even though J is fairly robust, these skewness and kurtosis are subject to influence by

outliers. Table 2.1 displays the results of 2SK and 2KR which illustrate the sensitiveness in

the presence of outliers and finally fail to allow normality. Since we cannot directly apply

these measures, we explore other possibilities using other measures of robust skewness and

kurtosis.

Kim and White briefly reviewed other possible robust following skewness and kurtosis

measures:

Robust Measure of Skewness reviewed by Kim and White (2003)

(a) Bowley (1920):

(2.24) 13

2133 QQ

Q2QQ−

−+=SK

where, ( )4/Q 1 iFi−= , 3,2,1=i .

Here, ( )aF 1− is inverse cumulative distribution function at a , 10 << a . In other form iQ

is known as quartile function. For standard normal distribution, it is known that

6745.0Q1 −= , 0Q2 = and 6745.0Q 3 = . Thus, for symmetric distribution, 03 =SK .

(b) Groeneveld and Meeden (1984):

(2.25) ( )2

24

QQˆ

−

−=

tuESK

μ

28

The estimates of the absolute deviation in (2.25) is given by

( ) ∑=

−=−T

ttt u

TuE

122 Q1Qˆ

For symmetric distribution, 04 =SK .

(c) Kendall and Stuart (1977):

Here, the robust skewness is obtained by replacing the absolute deviation in (2.25) by the

estimated standard deviation, σ , resulting in:

(2.26) σ

μˆQˆ 2

5−

=SK

Robust Measures of Kurtosis reviewed by Kim and White (2003)

(a) Moors (1988) considers the statistics:

(2.27) ( ) ( )

26

1357

EEEEEE

−−+−

where ( )8/1 iFEi−= , 7,,1…=i .

For Gaussian distribution, equation (2.27) is approximately 1.23. Thus, Moors proposed the

measure of kurtosis as:

(2.28) ( ) ( )

23.126

13573 −

−−+−

=EE

EEEEKR

29

(b) Crow and Siddiqui (1967):

(2.29) ( ) ( )( ) ( )25.075.0

025.0975.011

11

−−

−−

−−

FFFF

Under Gaussian distribution, equation (2.29) is approximately 2.91. Thus Crow and

Siddiqui measure of kurtosis is given by:

(2.30) ( ) ( )( ) ( ) 91.2

25.075.0025.0975.0

11

11

4 −−−

= −−

−−

FFFFKR

It is also interested when observing the standardized white noise by employing filter

function which is bounded and continuous. We propose a modification of the skewness and

kurtosis applied to the standardized residuals. Here, we replace the conventional measures

of skewness and kurtosis of equation (2.22). Using a ψ - filter function, we let the robust

skewness ( RSK ) and kurtosis, ( RKR ) being

(2.31) ∑=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc

uuT

RSK1

3

1 ˆ1

σψ

and 11

4

1 ˆ1 l

uuT

RKRT

t

tc −⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑= σ

ψ

Also, we consider the following function by modifying Gel and Gastwirth (2007) skewness

and kurtosis, that is

(2.32) ∑=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc J

uuT

RSK1

3

21 ψ

30

and 21

4

21 l

Juu

TRKR

T

t

tc −⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

ψ

The simple filter function may take the form of (Winsorization),

( ) ( )⎩

⎨⎧

≥×<

=cttcctt

tsgn

ψ

for some constants, c , 1l and 2l .

We take note that, under normality assumption, 3limlim 21 ==∞→∞→

llcc

. For various values of c ,

we obtain 1l and 2l via simulation, to illustrate the following Table 2.22.

Table 2.2 The values of robust kurtosis c

1.96 2.5 3.0 3.5 4.0 1l 2.02 2.64 2.89 2.97 2.99

2l 2.02 2.64 2.89 2.98 3.00

Based on the equations (2.31), (2.32) and Table 2.2, we may derive the following skewness

and kurtosis

(2.33) ∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc

uuT

SK1

3

96.16 ˆ1

σψ , 96.1=c

∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc

uuT

SK1

3

50.27 ˆ1

σψ , 500.2=c

2 The small simulation in SPLUS can be referred in Appendix 3.1. The estimated values of l is shown in Table A4.1 and Table A4.2.

31

∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc

uuT

SK1

3

00.38 ˆ1

σψ , 00.3=c

∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc

uuT

SK1

3

50.39 ˆ1

σψ , 500.3=c

∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc

uuT

SK1

3

00.410 ˆ1

σψ , 00.4=c

∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc J

uuT

SK1

3

96.1111 ψ , 96.1=c

∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc J

uuT

SK1

3

50.2121 ψ , 500.2=c

∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc J

uuT

SK1

3

00.3131 ψ , 00.3=c

∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc J

uuT

SK1

3

50.3141 ψ , 500.3=c

∑=

= ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=T

t

tc J

uuT

SK1

3

00.4151 ψ , 00.4=c

02.2~1

1

4

96.15 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ , 96.1=c

64.2~1

1

4

50.26 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ 500.2=c

89.2~1

1

4

00.37 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ 00.3=c

97.2~1

1

4

50.38 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ 500.3=c

32

99.2~1

1

4

00.49 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ 00.4=c

02.2~1

1

4

96.110 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ , 96.1=c

64.2~1

1

4

50.211 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ 500.2=c

89.2~1

1

4

00.312 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ 00.3=c

98.2~1

1

4

50.313 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ 500.3=c

and 00.3~1

1

4

00.414 −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= ∑=

=

T

t

tc

MuT

KRσ

ψ 00.4=c

These measures are robust and less sensitive with outliers and the breakdown point of these

estimators are dependent on the type of ψ -function used.

2.4.1.2 Classical and Robust Jarque – Bera statistics

Let Tuu ,,1 … be samples randomly selected from a Gaussian distribution. Under the null

hypothesis of normality, the sample skewness and kurtosis in equation (2.22) are

asymptotically independent and normally distributed such that,

(2.34) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡→⎟⎟

⎠

⎞⎜⎜⎝

⎛24006

,00

1

1 NKRSK

T

Gel and Gastwirth (2007) proposed another asymptotic normal of their robust skewness and

kurtosis in equation (2.23), that is,

33

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡→⎟⎟

⎠

⎞⎜⎜⎝

⎛D

CN

KRSK

T0

0,

00

2

2

Here, the values of asymptotic variances for robust skewness and kurtosis, C and D are

being computed through simulation study. By adopting this procedure, we compute the

asymptotic variances for all skewness and kurtosis statistics. For 15,,1…=i , 14,,1…=j ,

and suppose ( )21 ,~,, σμNuu T… , then,

(2.35) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡→⎟⎟

⎠

⎞⎜⎜⎝

⎛

j

i

j

i

DC

NKRSK

T0

0,

00

where, iC and jD are positive constants.

The variances iC and jD in equation (2.35) vary for small to moderate sample size. To

determine the variances for small to moderate sample size, we run a simulation study where

we generate T data derived from a Gaussian distribution. The measures of skewness and

kurtosis, discussed earlier are being computed. By taking K replication, each of size T , we

obtain K values of iSKT and jKRT , respectively. Using these K values, we calculate

the variances3 of these iSKT and jKRT . Table 2.3 displays the variance estimates of

iSKT and jKRT for small to moderate sample sizes. The mean of each measure can be

found in Appendix 3.2.

3 We provide the SPLUS programming language for this simulation in Appendix 3.2.

34

Table 2.3 Estimated variance for skewness and kurtosis using 10,000 simulated standard normal distribution.

Size, T 50 100 200 500 1000

1SKT 5.27 5.54 5.79 5.90 5.83

2SKT 6.21 6.04 6.05 6.00 5.88

3SKT 1.74 1.80 1.75 1.84 1.82

4SKT 0.85 0.85 0.87 0.91 0.89

5SKT 0.53 0.54 0.55 0.58 0.57

6SKT 1.22 1.18 1.17 1.22 1.20

7SKT 3.04 3.00 3.06 3.09 3.06

8SKT 4.35 4.48 4.62 4.65 4.60

9SKT 4.83 5.17 5.40 5.49 5.44

10SKT 4.94 5.36 5.63 5.78 5.72

11SKT 1.23 1.18 1.17 1.22 1.20

12SKT 3.17 3.06 3.09 3.10 3.06

13SKT 4.80 4.71 4.73 4.69 4.63

14SKT 5.68 5.62 5.62 5.58 5.48

15SKT 6.05 5.95 5.93 5.90 5.78

1KRT 17.80 20.60 22.29 23.08 24.00

2KRT 61.74 60.15 58.65 57.72 59.09

3KRT 3.23 3.24 3.19 3.19 3.12

4KRT 12.99 13.68 14.53 14.47 15.15

5KRT 1.59 1.79 1.81 1.86 1.87

6KRT 4.62 4.73 4.88 4.81 5.01

7KRT 10.51 11.12 11.27 11.08 11.51

8KRT 14.67 16.85 17.41 17.74 18.45

9KRT 16.16 19.37 20.51 21.44 22.19

10KRT 3.86 3.87 3.91 3.83 3.81

11KRT 16.99 17.10 17.15 16.64 17.16

12KRT 35.07 35.74 35.27 34.40 35.51

13KRT 49.38 50.57 49.00 48.47 49.90

14KRT 56.88 57.51 55.63 55.13 56.46

35

From the above results and tables in Appendix A3.3 – A3.4f, we found similar results that,

the variance of 1SK and 1KR are close to their theoretical values (i.e. 61 =C and 241 =D )

while remaining results are as follows; 62 =C , 8.13 =C , 9.04 =C , 6.05 =C , 2.16 =C ,

37 =C , 6.48 =C , 5.59 =C , 7.510 =C , 2.111 =C , 1.312 =C , 7.413 =C , 6.514 =C ,

9.515 =C , 602 =D 1.33 =D , 154 =D , 87.15 =D , 56 =D , 5.117 =D , 5.188 =D ,

229 =D , 8.310 =D , 1711 =D , 3512 =D , 5013 =D and 5614 =D . The estimated

correlation in Table A4.4a – A4.4d in the Appendix 3 provide supports that the skewness

and kurtosis are not correlated.

2.4.1.3 The robust Jarque–Bera ( RJB ) test statistics

Let Tuu ,,1 … be samples randomly selected from a Gaussian distribution. Using the above

notation for skewness and kurtosis, the Jarque-Bera ( JB ) test statistic is:

(2.36) 21

211 246

KRTSKTJB +=

Corollary 1: Under the null hypothesis of normality, the classical Jarque–Bera ( )1JB test

statistics is 22χ -distributed.

To complement these measures, we also use other combinations of robust skewness and

kurtosis measure of (2.23) – (2.33) to measure the impact of RJB , as follows:

(2.37a) 22

2

22

22 KR

DTSK

CTJB +=

(2.37b) 23

3

23

33 KR

DTSK

CTJB +=

36

(2.37c) 23

3

24

44 KR

DTSK

CTJB +=

(2.37d) 23

3

25

55 KR

DTSK

CTJB +=

(2.37e) 24

4

23

36 KR

DTSK

CTJB +=

(2.37f) 24

4

24

47 KR

DTSK

CTJB +=

(2.37g) 24

4

25

58 KR

DTSK

CTJB +=

(2.37h) 25

5

26

69 KR

DTSK

CTJB +=

(2.37i) 26

6

27

710 KR

DTSK

CTJB +=

(2.37j) 27

7

28

811 KR

DTSK

CTJB +=

(2.37k) 28

8

29

912 KR

DTSK

CTJB +=

(2.37l) 29

9

210

1013 KR

DTSK

CTJB +=

(2.37m) 210

10

211

1114 KR

DTSK

CTJB +=

(2.37n) 211

11

212

1215 KR

DTSK

CTJB +=

(2.37o) 212

12

213

1316 KR

DTSK

CTJB +=

(2.37p) 213

13

214

1417 KR

DTSK

CTJB += and

37

(2.37q) 214

14

215

1518 KR

DTSK

CTJB +=

Corollary 2: Under the null hypothesis of normality, the JB test from equation (2.36) and

robust Jarque – Bera tests from equations (2.37a – q) are asymptotically 22χ -distribution.

To illustrate this, we run 10,000 replications simulation by generating 100 sizes of data of

normal distribution, to compute 10,000 various JB s statistics (equation (2.36) and (2.37a –

q)). Figure 2.1 are the 22χ -distribution Q-Q plot of JB s. All plots indicate that, they

approximately follow 22χ -distribution4.

4 J. Jiang et al (2001) constructed the Q-Q plot to demonstrate that the robust portmanteau statistic at lag 6 follows asymptotically 2

6χ -distribution.

38

Figure 2.1 Quantile – quantile (Q-Q) plots for Jarque – Bera ( JB s) statistics.

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

1 S

ampl

e

0 2 4 6 8 10 12

020

6010

0

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

2 S

ampl

e

0 2 4 6 8 10 12

020

6010

0

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

3 S

ampl

e

0 2 4 6 8 10 12

05

1020

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

4 S

ampl

e

0 2 4 6 8 10 12

05

1020

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

5 S

ampl

e

0 2 4 6 8 10 12

05

1020

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

6 S

ampl

e

0 2 4 6 8 10 12

05

1525

39

Figure 2.1 continues…..

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

7 S

ampl

e

0 2 4 6 8 10 12

05

1525

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

8 S

ampl

e

0 2 4 6 8 10 12

05

1525

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

9 S

ampl

e

0 2 4 6 8 10 12

05

1020

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

10 S

ampl

e

0 2 4 6 8 10 12

05

1015

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

11 S

ampl

e

0 2 4 6 8 10 12

05

1015

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

12 S

ampl

e

0 2 4 6 8 10 12

05

1525

40

Figure 2.1 continues…..

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

13 S

ampl

e

0 2 4 6 8 10 12

05

1525

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

14 S

ampl

e

0 2 4 6 8 10 12

05

1015

20

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

15 S

ampl

e

0 2 4 6 8 10 12

05

1020

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

16 S

ampl

e

0 2 4 6 8 10 12

05

1525

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

17 S

ampl

e

0 2 4 6 8 10 12

010

2030

Quantiles of Chi-squared-2

Qua

ntile

s of

JB

18 S

ampl

e

0 2 4 6 8 10 12

010

2030

40

2.4.1.4 Critical Values and Power of the New RJB tests

For small to moderate sample size, the asymptotic 22χ -approximation for JB and RJB are

not sufficiently accurate5. Using 10,000 simulated samples with various sample sizes, T ,

we obtain approximations to the exact critical values as shown in Table 2.46.

5 Gel and Gastwirth (2007) indicate similar findings. 6 Appendix 3.3 presents the SPLUS programming language for setting the critical region of Normal rejection at 5%.

41

Table 2.4 Critical region at 5% for JB and robust JBs T 30 50 100 200

1JB 4.374 4.749 5.436 5.623 2JB 7.078 6.662 6.791 6.185 3JB 5.758 6.021 5.972 6.106 4JB 5.857 5.976 5.992 5.919 5JB 5.638 5.797 5.827 5.794 6JB 5.085 5.304 5.635 5.878 7JB 5.253 5.390 5.519 5.774 8JB 5.047 5.209 5.346 5.638 9JB 5.559 5.678 5.892 5.966 10JB 5.376 5.524 5.846 5.916 11JB 4.908 5.011 5.475 5.703 12JB 4.081 4.867 5.406 5.466 13JB 3.793 4.424 5.415 5.692 14JB 6.085 5.989 6.113 5.959 15JB 6.079 5.748 5.887 5.903 16JB 6.859 6.459 6.050 5.888 17JB 8.078 7.093 6.610 5.911 18JB 7.358 6.921 6.998 6.241

Note: The results obtained are based on 10,000 replications. The critical value of 2,2 αχ at

05.0=α is 5.99.

From Table 2.4, we found all critical regions for small sample size for all JB s are far away

from the critical value, 99.52%5,2 =χ . The increasing of sample size leads to convergence to

this critical value. Here, we found that, 4JB , 9JB , 10JB , 14JB , 15JB and 17JB are close to

5.99. With the calculated critical value, we can now use for assessing the power of classical

JB and other 17 types of robust JBs , ( )RJBs as defined in equations (2.36) and (2.37a -

q). We test the robustness of the JB estimators under several different conditions, namely

when the samples are derived from the following distributions: Gaussian, logistic,

42

exponent, lognormal and mixture distributions7. Table 2.5 presents the results of the

simulation study for different generated distributions.

Table 2.5 Power of rejection for JB tests – tests among different types of distributions

T Method of JB N(0,1) t3 t10 logistic EXP(1/2

) Lognorm

al 1JB 0.0499 0.5176 0.1560 0.1905 1.0000 0.9552 2JB 0.0515 0.5508 0.1623 0.2037 0.9998 0.9239 3JB 0.0491 0.1112 0.0596 0.0699 0.4690 0.3151 4JB 0.0501 0.1370 0.0603 0.0729 0.9902 0.8345 5JB 0.0504 0.1076 0.0566 0.0672 0.9845 0.6419

6JB 0.0491 0.3241 0.0736 0.1033 0.5988 0.5582

7JB 0.0466 0.3334 0.0721 0.1003 0.9919 0.8759

8JB 0.0470 0.3185 0.0685 0.0929 0.9892 0.8182

30 9JB 0.0500 0.3040 0.0992 0.1026 1.0000 0.9903

10JB 0.0507 0.2287 0.0681 0.0823 1.0000 0.9735 11JB 0.0489 0.3775 0.1291 0.1628 1.0000 0.9546 12JB 0.0507 0.5167 0.1568 0.1892 1.0000 0.9659 13JB 0.0501 0.5109 0.1541 0.1863 1.0000 0.9679 14JB 0.0446 0.0670 0.0512 0.0563 1.0000 0.7577 15JB 0.0506 0.2127 0.0828 0.1088 0.9999 0.7554

16JB 0.0489 0.3675 0.1137 0.1558 0.9999 0.8414

17JB 0.0517 0.5521 0.1615 0.2037 0.9994 0.9193

18JB 0.0517 0.5516 0.1626 0.2045 0.9998 0.9220Note: These simulation assessments are done based on 10,000 replications. 7 Appendix 3.4 presents the SPLUS programming language for computing the power of rejection beyond critical values generated from different size and JB statistics, whereas Appendix 5 represents the properties of the tested distributions – their density function, mean, median, variance, skewness and kurtosis.

43

Table 2.5 continues…

T Method of JB N(0,1) t3 t10 logistic EXP(1/2

) Lognorm

al 1JB 0.0524 0.7105 0.2126 0.2644 1.0000 0.9981 2JB 0.0511 0.7424 0.2250 0.2849 0.9997 0.9919 3JB 0.0473 0.1243 0.0661 0.0712 0.4468 0.4465 4JB 0.0515 0.1600 0.0675 0.0736 0.9890 0.9709 5JB 0.0506 0.1260 0.0637 0.0676 0.9809 0.8869

6JB 0.0522 0.4018 0.0889 0.1177 0.5816 0.7097

7JB 0.0487 0.4149 0.0866 0.1170 0.9915 0.9796

8JB 0.0485 0.3943 0.0840 0.1122 0.9871 0.9515

50 9JB 0.0530 0.4536 0.1178 0.1292 1.0000 1.0000

10JB 0.0508 0.2931 0.0892 0.1053 1.0000 0.9999 11JB 0.0478 0.4481 0.1289 0.1766 1.0000 0.9992 12JB 0.0513 0.6950 0.2122 0.2631 1.0000 0.9988 13JB 0.0522 0.7050 0.2098 0.2604 1.0000 0.9990 14JB 0.0510 0.0847 0.0546 0.0583 1.0000 0.9962 15JB 0.0513 0.2677 0.1059 0.1312 1.0000 0.9430

16JB 0.0464 0.5112 0.1478 0.2030 0.9999 0.9619

17JB 0.0500 0.6816 0.2108 0.2696 0.9995 0.9812

18JB 0.0513 0.7435 0.2256 0.2861 0.9997 0.9914

44

Table 2.5 continues…

T Method of JB N(0,1) t3 t10 logistic EXP(1/2

) Lognorm

al 1JB 0.0516 0.9076 0.3082 0.3924 1.0000 1.0000 2JB 0.0495 0.9285 0.3196 0.4223 0.9995 1.0000 3JB 0.0542 0.1758 0.0665 0.0875 0.4531 0.7379 4JB 0.0495 0.2125 0.0672 0.0834 0.9896 0.9999 5JB 0.0512 0.1730 0.0627 0.0786 0.9812 0.9973

6JB 0.0486 0.6327 0.1114 0.1737 0.5458 0.9324

7JB 0.0483 0.6565 0.1140 0.1731 0.9907 1.0000

8JB 0.0498 0.6435 0.1095 0.1703 0.9860 0.9995

100 9JB 0.0485 0.6671 0.1550 0.1787 1.0000 1.0000

10JB 0.0468 0.3813 0.0975 0.1119 1.0000 1.0000 11JB 0.0466 0.5360 0.1590 0.2385 1.0000 1.0000 12JB 0.0482 0.7588 0.2515 0.3474 1.0000 1.0000 13JB 0.0505 0.8849 0.3032 0.3857 1.0000 1.0000 14JB 0.0448 0.1022 0.0520 0.0563 1.0000 1.0000 15JB 0.0455 0.3798 0.1101 0.1688 1.0000 1.0000

16JB 0.0489 0.7307 0.2004 0.3101 1.0000 0.9995

17JB 0.0476 0.8739 0.2649 0.3817 0.9998 0.9990

18JB 0.0507 0.9241 0.3189 0.4246 0.9995 0.9998

45

Table 2.5 continues…

T Method of JB N(0,1) t3 t10 logistic EXP(1/2

) Lognorm

al 1JB 0.0474 0.9928 0.4613 0.5852 1.0000 1.0000 2JB 0.0480 0.9969 0.4875 0.6342 0.9998 1.0000 3JB 0.0510 0.2474 0.0759 0.0917 0.4440 0.9465 4JB 0.0540 0.3079 0.0803 0.0977 0.9905 1.0000 5JB 0.0538 0.2629 0.0754 0.0935 0.9827 0.9999

6JB 0.0457 0.8588 0.1483 0.2594 0.5367 0.9964

7JB 0.0476 0.8694 0.1493 0.2626 0.9908 1.0000

8JB 0.0473 0.8644 0.1447 0.2607 0.9843 1.0000

200 9JB 0.0493 0.8822 0.2128 0.2443 1.0000 1.0000

10JB 0.0460 0.4910 0.1029 0.1377 1.0000 1.0000 11JB 0.0466 0.6045 0.2124 0.3406 1.0000 1.0000 12JB 0.0484 0.8430 0.3635 0.5183 1.0000 1.0000 13JB 0.0470 0.9462 0.4428 0.5757 1.0000 1.0000 14JB 0.0493 0.1339 0.0543 0.0732 1.0000 1.0000 15JB 0.0443 0.5607 0.1488 0.2585 1.0000 1.0000

16JB 0.0457 0.9245 0.2989 0.4704 1.0000 1.0000

17JB 0.0480 0.9850 0.4039 0.5858 0.9999 1.0000

18JB 0.0475 0.9940 0.4736 0.6300 0.9998 1.0000Note: These simulation assessments are done based on 10,000 replications.

When no outliers (or contaminants) occur in the data, the classical JB is the most efficient

test-statistic with the power of rejection close to 0.05. The results showed that the 1JB and

2JB successfully reject the null hypothesis when the data were selected from a non-

Gaussian distribution, particularly those that ranges from moderate to heavy-tailed ( t -

distribution and logistic) and symmetric (lognormal and exponent) distributions. However,

the increasing values of c s, that are 3, 3.5 and 4 (see 11JB , 12JB , 13JB , 16JB , 17JB and

18JB ), successfully distinguish the normality from heavy-tailed distributions. This indicates

that the larger the values of c , the robust JB tests become as sensitive as 1JB and 2JB . All

46

tests reject normality for symmetric distributions. Other robust JBs , ( RJBs ) such as 6JB ,

7JB and 8JB successfully reject −3t distribution with the increasing size of data.

Table 2.6 Power of rejection for JB tests – tests among various conditions of outliers

T Method of JB

0.99N(0,1)+

0.01N(0,5)

0.99N(0,1)+

0.01exp(1/2)

0.99N(0,1)+

0.01lognormal

0.95N(0,1)+

0.05N(0,5)

0.95N(0,1)+

0.05exp(1/2)

0.95N(0,1)+

0.05lognormal

1JB 0.1721 0.0953 0.0774 0.5151 0.2421 0.1785 2JB 0.1771 0.0973 0.0786 0.5159 0.2409 0.1791 3JB 0.0522 0.0493 0.0527 0.0649 0.0567 0.0537 4JB 0.0636 0.0519 0.0555 0.1018 0.0685 0.0634 5JB 0.0555 0.0492 0.0527 0.0633 0.0563 0.0532

6JB 0.0843 0.0616 0.0553 0.2488 0.0990 0.0878

7JB 0.0923 0.0605 0.0568 0.2698 0.1084 0.0903

8JB 0.0854 0.0576 0.0532 0.2479 0.0971 0.0850

30 9JB 0.1522 0.0856 0.0731 0.4257 0.1947 0.1475

10JB 0.1090 0.0645 0.0571 0.2555 0.1225 0.1063 11JB 0.0835 0.0676 0.0598 0.2594 0.1405 0.1054 12JB 0.1726 0.0956 0.0781 0.5149 0.2422 0.1795 13JB 0.1726 0.0953 0.0781 0.5132 0.2422 0.1790 14JB 0.0521 0.0495 0.0474 0.0504 0.0482 0.0529 15JB 0.0497 0.0517 0.0540 0.0807 0.0636 0.0534

16JB 0.0718 0.0610 0.0538 0.2157 0.1075 0.0815

17JB 0.1768 0.0971 0.0783 0.5154 0.2402 0.1789

18JB 0.1774 0.0976 0.0790 0.5160 0.2409 0.1795Note: The number of simulation is 10,000.

47

Table 2.6 continues…

T Method of JB

0.99N(0,1)+

0.01N(0,5)

0.99N(0,1)+

0.01exp(1/2)

0.99N(0,1)+

0.01lognormal

0.95N(0,1)+

0.05N(0,5)

0.95N(0,1)+

0.05exp(1/2)

0.95N(0,1)+

0.05lognormal

1JB 0.2457 0.1303 0.0975 0.7013 0.3575 0.2595 2JB 0.2430 0.1291 0.0938 0.6952 0.3522 0.2548 3JB 0.0495 0.0512 0.0517 0.0591 0.0534 0.0520 4JB 0.0583 0.0557 0.0560 0.1075 0.0717 0.0648 5JB 0.0481 0.0520 0.0523 0.0645 0.0574 0.0511

6JB 0.0647 0.0554 0.0520 0.2472 0.1058 0.0769

7JB 0.0723 0.0561 0.0524 0.2739 0.1191 0.0878

8JB 0.0627 0.0544 0.0489 0.2457 0.1066 0.0757

50 9JB 0.2170 0.1086 0.0886 0.6290 0.2861 0.2047

10JB 0.1736 0.0892 0.0774 0.3976 0.2104 0.1654 11JB 0.0993 0.0716 0.0565 0.3226 0.1644 0.1216 12JB 0.2302 0.1257 0.0947 0.6671 0.3474 0.2489 13JB 0.2455 0.1298 0.0978 0.6990 0.3566 0.2579 14JB 0.0768 0.0493 0.0569 0.1338 0.0663 0.0702 15JB 0.0466 0.0505 0.0494 0.0783 0.0716 0.0566

16JB 0.0695 0.0604 0.0471 0.2928 0.1483 0.0935

17JB 0.1472 0.0944 0.0675 0.5206 0.2587 0.1701

18JB 0.2433 0.1292 0.0938 0.6955 0.3524 0.2548

48

Table 2.6 continues…

T Method of JB

0.99N(0,1)+

0.01N(0,5)

0.99N(0,1)+

0.01exp(1/2)

0.99N(0,1)+

0.01lognormal

0.95N(0,1)+

0.05N(0,5)

0.95N(0,1)+

0.05exp(1/2)

0.95N(0,1)+

0.05lognormal

1JB 0.3872 0.1854 0.1301 0.8968 0.5511 0.4003 2JB 0.3826 0.1814 0.1266 0.8924 0.5401 0.3897 3JB 0.0525 0.0560 0.0533 0.0621 0.0553 0.0587 4JB 0.0622 0.0569 0.0552 0.1146 0.0840 0.0743 5JB 0.0502 0.0518 0.0523 0.0659 0.0628 0.0577

6JB 0.0556 0.0560 0.0553 0.2873 0.1067 0.0798

7JB 0.0724 0.0623 0.0595 0.3227 0.1399 0.0982

8JB 0.0589 0.0576 0.0557 0.2951 0.1230 0.0812

100 9JB 0.3389 0.1482 0.1112 0.8519 0.4395 0.3267

10JB 0.2905 0.1253 0.1055 0.6626 0.3408 0.2653 11JB 0.1791 0.0948 0.0824 0.3718 0.2655 0.1996 12JB 0.1663 0.0990 0.0724 0.6274 0.3501 0.2331 13JB 0.3517 0.1769 0.1224 0.8669 0.5352 0.3751 14JB 0.1041 0.0565 0.0567 0.2788 0.0831 0.0946 15JB 0.0402 0.0515 0.0546 0.0544 0.0668 0.0612

16JB 0.0631 0.0645 0.0543 0.3481 0.1862 0.1061

17JB 0.1479 0.0926 0.0647 0.6890 0.3399 0.2026

18JB 0.3257 0.1643 0.1106 0.8602 0.5082 0.3428

49

Table 2.6 continues…

T Method of JB

0.99N(0,1)+

0.01N(0,5)

0.99N(0,1)+

0.01exp(1/2)

0.99N(0,1)+

0.01lognormal

0.95N(0,1)+

0.05N(0,5)

0.95N(0,1)+

0.05exp(1/2)

0.95N(0,1)+

0.05lognormal

1JB 0.5996 0.2853 0.2098 0.9890 0.7768 0.5966 2JB 0.5927 0.2790 0.2071 0.9875 0.7690 0.5875 3JB 0.0514 0.0554 0.0497 0.0651 0.0553 0.0550 4JB 0.0693 0.0617 0.0610 0.1266 0.1035 0.0796 5JB 0.0554 0.0578 0.0553 0.0760 0.0781 0.0561

6JB 0.0648 0.0565 0.0544 0.3218 0.1193 0.0727

7JB 0.0808 0.0607 0.0605 0.3668 0.1630 0.0959

8JB 0.0677 0.0584 0.0555 0.3314 0.1410 0.0750

200 9JB 0.5169 0.2131 0.1587 0.9764 0.6329 0.4837

10JB 0.4603 0.1834 0.1418 0.9015 0.5103 0.4035 11JB 0.3124 0.1286 0.1117 0.4302 0.4015 0.3013 12JB 0.2273 0.1250 0.1080 0.7040 0.5260 0.3355 13JB 0.3662 0.1883 0.1423 0.9279 0.6804 0.4611 14JB 0.1461 0.0666 0.0676 0.5265 0.1146 0.1390 15JB 0.0538 0.0512 0.0562 0.0453 0.0797 0.0607

16JB 0.0693 0.0679 0.0615 0.4677 0.2795 0.1198

17JB 0.1929 0.1129 0.0862 0.8749 0.5346 0.2859

18JB 0.3941 0.1868 0.1321 0.9658 0.6875 0.4444Note: The number of simulation is 10,000.

Table 2.6 displays the power of rejection in the presence of contamination within the data

set. When a small percentage (1%) of contamination is observed, the robust Jarque–Bera

from combinations of skewness and kurtosis that were reviewed by Kim and White (2003)

are less sensitive to small percentage of outliers (see 3JB , 4JB , 5JB , 6JB , 7JB and 8JB ).

Unfortunately, these tests are also less sensitive to those which are totally not normally

distributed. The newly proposed RJB , namely 139 JBJB − and 1814 JBJB − are also being

observed to search suitable values of c that are outperform the 1JB as well as 2JB of Gel

and Gastwirth (2007) when a small percentage of outliers (mixtures) are present in the data

and are more apparent in larger sample sizes. This results, when 5.3=c , the equation

50

(2.32) that derives 12JB and, when 3=c , the equation (2.33) deriving 16JB , both

outperform 1JB and 2JB in allowing normality. For both equations (2.32) and (2.33), the

JBs become as sensitive as 1JB and 2JB when c is higher. Here, to eliminate the

necessary outliers as well as detecting the non-normality of the data accurately, an

appropriate value of c should be set up in applying our new RJB to accept the normally

distributed data that are contaminated and to detect the non-normal symmetric data.

2.4.2 Other test statistics

• Shapiro-Wilk Test

The Shapiro-Wilk test (Shapiro and Wilk (1965)) calculates W-statistic that tests

whether a random sample, Tuuu ,,, 21 … comes from (specifically) a normal

distribution. The W-statistic is calculated as follows.

(2.38) ( )

( )∑

∑

=

=

−

⎥⎦

⎤⎢⎣

⎡

= T

tt

T

ttt

uu

uaW

1

2

2

1

where ( )tu are the ordered sample values ( ( )1u is the smallest value) and the ta are

constants generated from the means, variances and covariances of the order

statistics of a sample of size T from a normal distribution. The constants, ia , can be

computed as:

( )

( ) 2/11'

',,MVVM

VM11

1

−−

−

=Taa …

51

where, ( )',,1 TMM …=M are the expected values of order statistics of standard

normal distribution, and V is the covariance matrix of these order statistics. The

null hypothesis is rejected when W is small8.

• Lagrange Multiplier (LM) test (Autocorrelation test)

Another assumptions that must be verified is that the white noise are uncorrelated (i.e.

independent). The common tests used to assess autocorrelations are the Lagrange

multiplier (LM) and Ljung–Box (LB) tests. For the LM test, we let white noise, tu , be

modeled as a linear function of its lag:

ttMtMtt uuu ηηθθθ +=++++= −− θ'Utˆˆˆ 110 …

The hypothesis tested under the LM-test are:

0H : 0θ = (i.e. no autocorrelation) against 1H : 0θ ≠

The LM-test statistic is given by,

(2.39) ( )ηη'

ηU'UUUη' 1−

= Tξ

Under 0H , the test statistics is 2~ Mχξ , where M is the number of lags. If the

calculated value of ξ is larger than 2,Mαχ at α significant level, the null hypothesis

is rejected and thus implying that the fitted model (GARCH) is not sufficiently good

for the given time series data.

8 Royston, J. P. (1982) discussed extensively on the probability of W statistic.

52

• Ljung – Box autocorrelation test

The autocorrelation for estimated tu at lag s is defined as

(2.40) ∑

∑

+=

+=−

= T

stt

T

ststt

s

u

uur

1

2

1

ˆ

ˆˆˆ

Basically, the statistical significance of any autocorrelation, sr can be assess by its

standard error. Bartlett (1946) has shown that if a time series is purely random, that

is white noise, then the s -th sample autocorrelation coefficients, sr , are

approximately normally distributed,

( )TNrs /1,0~ˆ

where T denotes the sample size. The 95% confidence interval for any sr is

Trs /196.1ˆ ×±

We do not reject the null hypothesis of no autocorrelation if the ‘observed’ sr lies

within the interval, ( )TrTr ss /196.1ˆ,/196.1ˆ ×+×− and vice versa.

Instead of testing the statistical significance of individual autocorrelation

coefficient, we can also test the joint hypothesis that all the sr up to certain lags, say

S lags, are equal to zero. To test this, we initially calculate sr given in equation

(2.40). The Ljung – Box test statistic is then defined as,

53

(2.41) ( )∑= −

+=M

s

sS sT

rTTQ

1

2ˆ2ˆ

Under the null hypothesis of no autocorrelation, 2~ˆSSQ χ . Using this distribution,

the null hypothesis is rejected if the calculated 2,

ˆαχ SSQ > at α significant level;

implying that the model is insufficient or could perhaps be miss specified.

54

CHAPTER 3

Least Absolute Deviation (LAD) for GARCH Models

3.1 Introduction

Regression quantile developed by Koenker and Basset (1978), has been extended and

applied to time series modeling by Koenker and Zhao (1994), and Koenker and Zhao

(1996). In volatility modeling, they developed the properties of quantile regression in

ARCH model. Jiang et al (2001) extended Koenker and Zhao’s (1996) study by limiting the

scope into L1–ARCH modeling, while Park (2002a) extended this work to L1–GARCH.

In this chapter, we introduce briefly the properties of L1–(G)ARCH model using Bahadur

representation, which has been described by Koenker and Zhao (1996) and Jiang et al

(2001). We also describe briefly, the method of estimation of L1–(G)ARCH, which have

been described by Jiang et al (2001), Park (2002a) and Peng and Yao(2003). In addition,

we show extensively the detail algorithm of estimating LAD – GARCH( pq, ) of model

introduced by Peng and Yao (2003), that will be used in Chapter 4, next. The properties of

L1–ARCH model are important as we are using it in Chapter 5.

As the assumption innovation being general i.i.d with zero mean or median is allowed in

L1–ARCH model, we adopt the bootstrap standard error motivated by De Angelis, Hall and

Young (1993) in estimating the standard error of the ARCH parameters.

55

Finally, we review the goodness of fit method of the model via robust L1–residuals

autocorrelation and robust portmanteau test (see Jiang et al (2001)).

3.2 L1–ARCH

3.2.1 Model

The L1–ARCH model is the special case of quantile regression for linear ARCH introduced

by Koenker and Zhao (1996). Developed by Jiang et al (2001) , the L1–ARCH model is

given as:

(3.1) tty ε+=− bx'ψ t1t|

and

( )αz'ttt u=ε

where ( )',,,1 1 ktt yy −−= …tx , ( )',,, 10 kbbb …=b , ( )',,,1 1 qtt −−= εεtz and

( )',,, 10 qααα …=α . The innovations, tu , of L1–(G)ARCH (see equation (3.1)) is different

from that of ordinary (G)ARCH, with tu of L1–(G)ARCH being identical and

independently distributed (i.i.d) for Tt ,,1= , and ( ) 0=tumedian , ( ) 1=tumedian (see

Jiang et al (2001) and Park (2002a)).

3.2.2 Parameter Estimation

With the assumption that tu (in equation (3.1)) are i.i.d. with ( ) 0=tumedian and

( ) 1=tumedian , we have the following results:

56

From (3.1),

(3.2) ( ) bx'ψ t1t =−|tymedian

and

(3.3) ( ) ( )( ) ttt umedianmedian σε === αz'αz' tt

As the ARCH consists of two main parts: (i) the conditional expectation and (ii) conditional

scale, the parameter estimation are carried out as follows (Jiang et al (2001)). The estimate

of b , is obtained by minimizing the average absolute deviations over all possible b , i.e.

(3.4) ∑=

−=T

tty

1

minargˆ bx'b tb

The residuals is computed as

bx'tˆˆ −= tt yε ,

and the estimated conditional scale α~ , is that value of α , which minimizes the following

criterion:

(3.5) ∑=

−=T

tt

1

ˆˆminarg~ α'zα tα

ε

Let the initial conditional scale at time t be tσ~ be

(3.6) α'z t~ˆ~ =tσ

57

To improve the efficiency of parameter estimation for ARCH models, let

(3.7) ∑=

−−=T

ttt

1

1~ˆˆminargˆ σε α'zα tα

The improved conditional scale estimates tσ , are given by

(3.8) α'z t ˆˆˆ =tσ

3.2.3 Asymptotic normal distribution of parameters estimates

We will review the properties of quantile regression estimates for ARCH models developed

by Koenker and Zhao (1996).

Under mild conditions, Jiang et al (2001) showed that the L1–ARCH parameters are

asymptotically normal distributed. The mild conditions being:

1. ( ) 0=tumedian and ( ) 1=tumedian

2. +∞<+δ2tyE for some 0>δ and ty is strictly stationary

3. The density function of u , ( )uf is symmetric and continuous at 0=u and 1=u

with ( ) 00 >f and ( ) 01 >f

4. ( ) atttt Dx'xx'x == ∑=

T

t

at

at T

E1

1 σσ are positive definite for 1,0a = ;

( ) atttt Gz'zz'z == ∑=

T

t

at

at T

E1

1 σσ are positive definite for 2,1,0a = ;

5. tu is independent of 1tψ − .

58

For these models, the L1–(G)ARCH parameters, following Bahadur (1966) representation

are

(3.9) ( ) ( )[ ] ( ) ( )10/ˆ1

2/1p

T

tt ouTfT +=− ∑

=

−− ψt1

1 xDbb

with asymptotic normal for conditional median parameters:

(3.10) ( ) ( ) ⎥⎦

⎤⎢⎣

⎡→−

−−

04,0ˆ

2fNT

D 110

11 DDD

bb

For the ARCH parameters,

(3.11) ( ) ( ) ( )[ ] ( ) ( )1111ˆ1

2/11p

T

tt ouTffT +−+−=− ∑

=

−−− ψt1

1 zGαα

with asymptotic normal for conditional scale parameters:

(3.12) ( )( ) ( )[ ] ⎭

⎬⎫

⎩⎨⎧

+−→−

−−

2114,0ˆ

ffNT

D 110

11 GGG

αα

where, ∑=

=T

t

atT 1

1 σtta x'xD and ∑=

=T

t

atT 1

1 σtta z'zG

3.3 L1–GARCH

This model developed by Park (2002a), is an extension of Jiang et al (2001) ARCH model.

The L1–GARCH model follows equation 3.1, with modification as

(3.13) ( )γz'bx'bx'bx' tttt ttttt uuy +=+=+= σε

59

Here, we have ( )',,,,,,1 11 pttqtt −−−−= σσεε …tz and ( )',,,,,, 110 pq ββααα ……=γ and

tu is assumed i.i.d. with zero median and ( ) 1=tumedian . Park (2002a) applied the L1–

GARCH model to USD – JY and USD – BP exchange rates data from 2 January 1990 to 31

December 1999.

In estimating the parameters for (3.13), Park proposed an estimation method based on

Taylor expansion, that is, let consider ( )'γ'b'ξ ,= , then

(3.14) ∑=

−−=T

tty

1minargˆ γz'bx'ξ ttξ

To find ξ that minimizes (3.14) requires the first and second differentiations. However, this

objective function is not differentiable as it is not continuous at certain points. Therefore,

we require some special algorithm to allow the objective function to be continuous similar

to the L1–estimate of parameters. The smoothen continuous function of L1–estimates has

been discussed briefly by Park (2002a), and detailed by Hitomi and Kagihara (2001). By

replacing the function of u , Park adopted the method by Gallant and Tauchen

(unpublished manuscript, 2000), i.e.

( ) ( )( )( )⎩

⎨⎧

−≥+−

=otherwise/.cos1

2/. if/12/.SuS

uSSuSu

ππρ

whereas Hitomi and Kagihara use the nonlinear smoothed LAD (NSLAD) function, that is,

( ) 22 duu +=ρ , where 0>d is the distance from the origin, thus overcoming the

nonlinear model. Further, the BHHH algorithm is used to speed up the rate of convergence.

60

3.4 L1–(G)ARCH – Peng and Yao (2003)

Another alternative to the linear GARCH( pq, ) model is that of Peng and Yao (2003). The

approach is similar to Engles (1982) and Bollerslev (1986), but tu is distributed i.i.d and

need not be normally distributed. The model considered is given as follows

(3.15) tty ε+= bx't

with

ttt u σε =

and

γz't=++++++= −−−−22

1122

1102

ptotqtqtt σβσβεαεαασ ……

where ( )',,,,,,1 221

221 pttqtt −−−−= σσεε …tz and ( )',,,,,, 110 pq ββααα ……=γ . When tu are

i.i.d with zero median and ( ) 12 =tumedian , we have ( ) 0=tmedian ε and

( ) γz't== 22ttmedian σε . Thus, the conditional median of ty and 2

ty are

( ) bx'ψ t1t =−|tymedian

and

( ) γz'ψ t1t ==−22 | ttymedian σ .

If 0=p , the model becomes ARCH model. The asymptotic normal distribution of ty of

(3.15) follows (3.10). Peng and Yao (2003) claimed that γ is also asymptotically normal

distributed. Further work is needed in establishing the asymptotic normality of ( )γγ −ˆT

and we leave this for further work.

61

To estimate the parameters of the robust GARCH model, Peng and Yao introduced three

approaches:

(3.16a) ( )∑=

−=T

ttty

1

22 1/minarg γγγ

1 σ

(3.16b) ( ) ( )[ ]∑=

−=T

ttty

1

222 loglogminarg γγ

γσ

and

(3.16c) ( )∑=

−=T

ttty

1

22minarg γγγ

3 σ

3.5 Alternative estimations for L1–GARCH parameters

Engle (1982), Bollerslev (1986) and Fiorentini et al (1992) provide a nice procedure in

estimating ordinary (G)ARCH parameters. Jiang et al (2001) constructed the L1–ARCH( q )

estimate of parameters procedure, but details of L1–GARCH( pq, ) model procedure is not

available yet.

L1–GARCH parameter estimates procedure involves iteration process as well as GARCH

model. To solve this, we consider the model by incorporating conditional expectations of

Jiang et al (2001) and Park (2002a), and conditional variances of Peng and Yao (2002). The

modified algorithm will be used in simulation study that is shown in Chapter 4. The model

used is,

(3.17) tty ε+= bx't

where

62

ttt u σε =

and

∑∑=

−=

− ++=p

jjtj

q

iitit y

1

2

1

20

2 σβαασ

In estimating conditional expectation, we find the value of b which minimizes the sum of

absolute deviations, over all possible b (i.e. LAD),

(3.18) ∑=

−=T

tty

1

minargˆ bx'b tb

Using the estimates from (3.18), we compute residuals, tε as

bx'tˆˆ −= tt yε

For GARCH( pq, ) , we know that

(3.19a) ( ) γ'zψ t1t ˆˆ|ˆ1

2

1

20

22 =++== ∑∑=

−=

−−

p

jjtj

q

iitittE σβεαασε

where ( )',,,,,,1 221

221 pttqtt −−−−= σσεε …tz and ( )',,,,,, 110 pq ββααα ……=γ .

In estimating the conditional variance, 2ˆ tσ , the following algorithm is constructed:

1. Let the initial value of the parameter be ''β'αγ 000 ⎟⎠⎞

⎜⎝⎛= ,

2. The newly updated LAD parameters, ( )mγ is that value of γ which minimizes the

following criterion:

63

(3.19b) ( ) ( ) ( )∑=

−=T

tt

1

2 ˆˆminargˆ mmt

γ

m γ'zγ ε

3. At each iteration, ( )mtz and ( )mγ change and these changes continue until

convergence. Average absolute errors of the conditional variances given

(3.19c) ( )∑=

−T

t

mttT 1

22 ˆˆ1 σε

is calculated for each ( )mγ obtained, thus monitoring the convergence of the

estimates.

4. Repeat step (3) and (4) until convergence is reached, or stop once *mm = . Here,

the maximum number of iteration is set to *m (i.e. *1 mm ≤≤ ).

5. The stopping criterion used is when ( ) ( )

( ) 001.0ˆ

ˆˆmax <⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−

−

1m

1mm

γγγ . If the iteration

reaches its maximum number, *m , then γ is that value of possible γ which

minimizes the average absolute errors of conditional variances in (3.19c), i.e.

(3.19d) ( )*

11

22 ˆˆT1 min

m

m

T

t

mtt

==

⎟⎠

⎞⎜⎝

⎛−= ∑ σε

γγ

6. Next, estimate the new conditional variance 2ˆ tσ via γ'z t ˆˆˆ 2 =tσ .

The complete program of L1–GARCH(1,1) in SPLUS is available in Appendix 4.7 of the

thesis.

64

3.6 Bootstrap Standard Error

Since the asymptotic distribution of L1–estimators involve the unknown probability density

function of the innovation, ( )uf , we must consider a method of estimate the standard error,

in practice. In estimating standard error of ARCH parameters, Jiang et al (2001) used the

bootstrap approximation method, motivated by De Angelis, Hall and Young (1993), which

allows the innovations being distributed other than Normal distribution. Note that

standardized residuals, 1ˆˆˆ −= tttu σε , ( )Tt ,,1…= have behavior similar to the innovations

tu , they proposed the following bootstrapping innovation approach.

Let tu * , ( )Tt ,,1…= be i.i.d. from the empirical distribution. The bootstrap analogues *b

and *α of b and α , respectively, are computed by models equations (3.4) and (3.7) with

ty replaced by tty *ˆˆ* ε+= bx*'t , ( )Tt ,,1…= and tε replaced by

( ) tqttt u **ˆˆ*ˆˆˆ*ˆ 1110 −− +++= εαεααε … , ( )Tt ,,1…= .

We can estimate the distribution of bb −ˆ and αα −ˆ by the bootstrap distributions of

bb ˆ*ˆ − and αα ˆ*ˆ − respectively. However, the accuracy of the approximation requires

further exploration and one may consider smoothed bootstrap analogues.

3.7 Testing on the autocorrelations of the L1–(G)ARCH models – L1–residuals

autocorrelation and robust portmanteau test

In assessing the adequacy of the fitted model, we usually use residual autocorrelations in

order to identify the number of lags that is adequate for the model. The portmanteau test

which is based on the squared residual autocorrelations (see McLeod (1979)) is also used to

65

test for the adequacy of the model. McLeod and Li (1983)) suggested the use of squared

residual autocorrelations for checking the homogeneity of the variance over time. Li and

Mak (1994) considered the squared residual autocorrelations in order to assess the

appearance of conditional heteroscedastic process and derived their asymptotic

distributions. This result is useful for checking the adequacy of a conditional

heteroscedastic for time series model. However, this approach lacks robustness since the

appearance of sensitivity to outliers and misspecified error distributions. In situations where

there are minor departures from the underlying distribution, a robust measure for

autocorrelation is desirable. We re-visit robust standardized absolute L1–residual

autocorrelation coefficient to assess the adequacy of the model, in particular, the

specification of the conditional scale, tσ .

Let the empirical version of the robust lag s absolute innovation autocorrelation be:

(3.20) ( ) ( )

( )∑+=

−

−

−−=

T

st t

stts uu

uuuur

12ψψψ

where tttu σε /= , ( )tTt umedianu ≤≤= 1 and ( ) ( )02/1 <−= xIxψ . Since 1=u , by

assumption, the estimated robust autocorrelation can be written as

(3.21) ( ) ( )∑+=

− −−=T

ststts uuuu

Tr

1

4~ ψψ

Jiang et al (2001) showed that, if L1–norm fitted model is adequate, sr would be good

estimator of sr . Thus, it could be used to construct robust portmanteau statistic for

assessing the fitted model.

66

Let ( )',,1 Mrr …=r and ( )'ˆ,,ˆˆ 1 Mrr …=r for any given positive integer M . Under mild

assumption of the following:

1. ( ) 0=tumedian and ( ) 1=tumedian

2. +∞<+δ2tyE for a 0>δ and ty is strictly stationary

3. ( )uf is symmetric and continuous at 0=u and 1=u with ( ) 00 >f and

( ) 01 >f

4. ( ) atttt Dx'xx'x == ∑=

T

t

at

at T

E1

1 σσ are positive definite for 1.0=a ;

( ) atttt Gz'zz'z == ∑=

T

t

at

at T

E1

1 σσ are positive definite for 2,1.0=a .

5. tu is independent of 1tψ − ,

we have

( ) ( )( ) ( ) ( )1ˆ114ˆ ps oααU' +−+−−= TffrTrT ss

and

( ) ( )( ) ( ) ( )1ˆ114ˆ poααUrr +−+−−= TffTT

where ( )11 −= −−

stt uψσts zU and ( )'UUU M1 ,,…= .

The robust portmanteau test statistics is then constructed (see Jiang et al (2001)) such that,

(3.22) rV'r 1~ˆ~~ −= TQM

67

where, the MM × matrix of V is derived as 'UGUIV 12

ˆˆˆ4ˆ −−= , while 2G and sU are

derived such that ∑=

−=T

ttT 1

2ˆˆˆ1ˆ σtt2 'zzG and ( )∑+=

−− −=

T

ststt u

T 1

2 1ˆˆˆ1ˆ ψσts zU .

Asymptotically, under mild assumption of 1 to 5 above, MQ is chi-squared distributed with

M degrees of freedom. Here, in order to find the adequacy of the model, the statistic

rV'r 1~ˆ~~ −= TQM must be less than MQ at 10% significant level. This shows that, the

innovation disturbances are uncorrelated at M lags.

68

CHAPTER 4

Study on the Performance of Conditional Variances – L1–GARCH

and GARCH – a comparison.

In this chapter, we conduct simulation studies to provide better understanding of the

behavior of the estimated of L1–(G)ARCH parameters discussed in Chapter 3. In this study,

we also compare the performance of robust GARCH with ordinary GARCH when tu are

derived from alternatives to normal distribution.

4.1 Introduction

Several studies towards the simulation tests and observations of time series modeling

previously were based on the performance of parameter estimates (see Jiang et al (2001)

and Peng and Yao (2003)), performance of forecasting (see Park (2002a)), ARCH and

GARCH tests (see Dijk et al (1999)) as well as time series outlier detection (see Pena

(1990), Chen and Liu (1993), and Franses and Ghijsels (1999)). Other simulation studies

which concerned the speed of the iteration to parameter convergence of GARCH model had

been investigated by Fiorentini et al (1996) who imposed the method of estimation using

the matrix function as follows – Hessian matrix (H), Outer Product matrix (OP), estimated

Information matrix (I) and mixed gradient. Some of these examples will be discussed to

enhance our understanding.

69

To enhance our understanding on the behavior of the estimated GARCH parameters

produced by various method of estimation, several simulation studies have been conducted

by several authors. Amongst them are that of Jiang et al (2001) and Peng and Yao (2003)

while Park (2002a) focus on forecasting performance. Dijk et al (1999) and Frances and

Ghijsels(1999) on the other hand study on other aspects of ARCH and GARCH models

such as the ARCH/GARCH tests (see Dijk et al (1999)) and outlier detection (see Franses

and Ghijsels (1999)).

Several measures of performance of the estimating procedure are revisited and amongst

them are:

• Jiang et al (2001) carried out simulation studies to estimate parameters of ARCH

models using the robust ARCH method, namely L1–ARCH ( )q , such that

( )||3.0||4.015.05.0 21 −− +++=+= ttttt uy εεε , by allowing tu to take alternatives

(Students- t with 3 degrees of freedom and a mixture distribution,

( ) ( )25,01.01,09.0 NNH += ) to normal distribution. 400 replications had been made

and for each replication, the ARCH parameters were re-estimated. The average

estimated parameters showed that, the estimated L1–ARCH parameters were closed

to the true ones when imposing various innovations.

• Park (2002a) extends the work of Jiang et al (2001) to L1–GARCH ( )pq, . In

assessing the performance of L1–GARCH ( )pq, , Park compares L1–GARCH ( )pq,

to ordinary GARCH and E-GARCH and employs the mean squared error (MSE),

root MSE (RMSE) and mean absolute error (MAE) to measure the performance the

one–step–ahead–forecast errors. Unlike Jiang et al (2001) simulation studies,

contamination in the data enters through the mean part of the models in (4.1) as:

70

tT

tty ε+= bx*

where ⎩⎨⎧

==+

=0 if ,1 if ,

x*

tt

tttt x

xοοηο

For details, refer to Brooks (1997) and Franses and Ghijsels (1999).

• Peng and Yao (2003) assessed the performance of the proposed robust GARCH

procedure average absolute error (AAE) of the estimated parameter defined as

(4.1a) ( ) 2//ˆ/ˆ/ˆ/ˆ 02020101 αααααααα −+−

for ARCH(2), and

(4.1b) ( ) 2/ˆ/ˆ/ˆ 110101 ββαααα −+−

for GARCH(1,1).

71

We first compare the performance in estimating the conditional variances between the

ordinary GARCH and L1–GARCH model. To do this, we derive the relationship between

the GARCH and L1–GARCH model in the form of conditional variances. The algorithm for

estimating the parameters and the conditional variance of L1–GARCH(1,1) is presented in

Chapter 3. A simulation study including outlier diagnostic technique to produce the

additive outlier9 (AO) corrected (AO-corrected) data is carried out to estimate the

conditional variances for both models and the results are being presented in boxplot

presentation. It is carried out to further understand the behavior of the estimated parameters

of the conditional variance.

4.2 Measures of performance

Here, our aim is to assess the various performance measures and subsequently adopt for

future performance measure of a newly proposed robust GARCH procedure.

The measures of performance for the following estimators are considered as discussed in

section 4.1 including:

• Jiang et al (2001) 1L -ARCH( q ) model:

(4.2a) ( )22110110 −−− +++++= tttttt uybby εαεααε

• Park (2002a) 1L -GARCH( pq, ) model:

(4.2b) ( )11110110 −−− ++++= ttttt uybby σβεαα

9 Outliers can occur as additive and innovative outliers. Additive outlier is an outlier occurs in a single time and does not affect the whole series of data. Innovative outliers are the appearance of outliers in a certain period that affect the entire period. This could be seen when the major changes of policy that has been imposed by a constitution.

72

In (4.2a) and (4.2b), tu is a random variable, distributed as i.i.d. with zero median

and ( ) 1=tumedian .

• Peng and Yao (2003) robust GARCH(1,1)

(4.2c) ( )211

2210110 −−− ++++= ttttt uybby σβεαα

In (4.2c), tu is a random variable, distributed as i.i.d. with zero median and

( ) 12 =tumedian .

To ease explanation of the procedure, we let 1=p and 1=q . The simulation study is

carried out by setting 10 =b , 5.01 =b , 10 =α , 5.01 =α and 2.012 == βα in (4.2a) to

(4.2c). The values are chosen such that the condition 1<+ βα ; 0, >βα , is satisfied.

Following Jiang et al (2001) we simulate 500=K data sets (replications), each of size

400=T with the random variable tu distributed as follows:

a. Standard Normal distribution;

b. Student- t distribution with 3 degrees of freedom, 3t ;

c. Mixture distribution, ( ) ( )25,01.01,09.0 NNH +=

The results of simulation are presented in terms of the average estimated parameters. The

tables of average estimated parameter are given as follows.

73

Table 4.1a Average estimated parameters of (4.2a), L1–AR(1)–ARCH(2) model. Innovation distribution ( )tu

0b 1b 0α 1α 2α

Normal 1.03514 (0.36311)

0.48256 (0.10150)

1.13054 (0.23407)

0.47234 (0.07249)

0.19026 (0.06538)

Mixed Normal 1.04692 (0.39822)

0.48474 (0.10573)

1.12132 (0.23207)

0.46616 (0.07338)

0.19492 (0.06399)

Student’s t with 3 d.f. 1.06392 (0.52240)

0.47098 (0.15332)

1.18446 (0.30788)

0.47468 (0.0865)

0.19090 (0.07644)

Note: Numbers in parentheses indicate the standard error of parameter estimates

Table 4.1b Average estimated parameters of (4.2b), L1–AR(1)–GARCH(1,1) model. Innovation distribution ( )tu

0b 1b 0α 1α 1β

Normal 0.99437 (0.35250)

0.496542 (0.09392)

1.11987 (0.29244)

0.48078 (0.07833)

0.18177 (0.09870)

Mixed Normal 1.02646 (0.35621)

0.48939 (0.08925)

1.10800 (0.31715)

0.47284 (0.07281)

0.19452 (0.10522)

Student’s t with 3 d.f. 1.02296 (0.41773)

0.47886 (0.11231)

1.13615 (0.28287)

0.47397 (0.07423)

0.19520 (0.08535)

Note: Numbers in parentheses indicate the standard error of parameter estimates

Table 4.1c Average estimated parameters of (4,2c), L1–AR(1)–GARCH(1,1) model. Innovation distribution ( )tu

0b 1b 0α 1α 1β

Normal 1.02466 (0.40736)

0.48335 (0.13843)

1.21373 (0.46893)

0.47811 (0.09576)

0.19182 (0.08613)

Mixed Normal 1.01470 (0.38194)

0.48960 (0.14589)

1.25465 (0.45629)

0.46928 (0.08923)

0.18955 (0.07869)

Student’s t with 3 d.f. 1.07604 (0.97915)

0.46850 (0.19498)

1.90343 (2.48777)

0.50377 (0.10403)

0.17351 (0.06719)

Note: Numbers in parentheses indicate the standard error of parameter estimates

Tables 4.1a through 4.1c display the average estimates of parameter of the generated

distributions by respective models and assumptions. All results are adequate, except for the

case of Student’s t with 3 degrees of freedom using equation (4.2c).

74

Next, we employ AAE (from equation (4.1a) and (4.1b)) as a measure of performance.

With reference to equations (4.1a) and (4.1b), we now have

(4.3a) ( ) 3//ˆ/ˆ/ˆ/ˆ/ˆ/ˆ020201010101 αααααααα −+−+− bbbb

for ARCH(2), and

(4.3b) ( ) 3/ˆ/ˆ/ˆ/ˆ/ˆ1101010101 ββαααα −+−+− bbbb

for GARCH(1,1).

The details of the computing algorithms in SPLUS are shown in Appendices 4.5 through

4.7.

Figure 4.1 Boxplot of Average Absolute Errors of parameter estimates.

05

1015

20

JIANG-MIxedNorm

JIANG-Std

Norm

JIANG-t3

PARK-MixedNorm

PARK-Std

Norm

PARK-t3

PYAO-MixedNorm

PYAO-Std

Norm

PYAO-t3

Average Absolute Errors

75

The AAE from equation (4.3a) and (4.3b) is presented into boxplot to assess the

accurateness of parameter estimates of GARCH models (see equation (4.2a) to (4.2c)).

Here we see that there is a small spread in AAE when tu are distributed as standard normal

distribution. However, the spread increases when tu deviates from the normal distribution.

4.3 Relationship between 21tσ and 2

2tσ

Consider the following GARCH(1,1) model:

(4.4) 2,11

2110 ititiitititit yuuy −− ++== σβαασ

where 1=i represents ordinary GARCH(1,1) model while 2=i represents the L1–

GARCH(1,1) model. Under normality assumption, ( ) ( ) 021 == tt umedianumean ,

( ) 1var 1 =tu , and ( ) 122 =tumedian . It is interesting to derive the relationship between 1tu and

2tu . Here, we form the relationship between 1tu and 2tu as 12 tt auu = and thus

( ) ( ) 121

222 == tt uamedianumedian . This results 4862.1=a , and, in this case, 21 6745.0 tt uu = .

Subsequently, we have ( ) 4549.021 =tumedian . The conditional variance for the

GARCH(1,1) model of (4.4) is given by

(4.5) ( ) 1Tt11t θzψ =++== −−−

21,111

211101

221 | tttt yyE σβαασ

For L1–GARCH(1,1), 22tσ is obtained by replacing the expectation in (4.5) by median, thus

yielding:

(4.6) ( ) 2Tt21t θzψ =++== −−−

22,112

211202

222 | tttt yymedian σβαασ

76

While equation (4.5) is referred to conditional variance, equation (4.6) will be referred as

robust conditional variance.

It is interesting to observe the following relationship between 21tσ and 2

2tσ , that is,

(4.7) ( ) ( ) ( )2221

21

2 | ttttt uVaruVaryE σσ ==−1tψ

and finally, under the assumption of normality, the equation (4.7) becomes

(4.8) ( ) 222

21 ttt uVar σσ =

Thus, the ordinary GARCH conditional variance can be computed using L1 – GARCH(1,1)

as ( ) 222 ttuVar σ .

4.4 Outlier diagnostic procedures

In an attempt to improve volatility forecasting, Franses and Ghijsels (1999) employs the

method of Chen and Liu (1993) towards GARCH models. They considered returns series,

ty , defined as 1loglog −−= ttt ppy , where tp denotes stock price or stock market index.

By considering the GARCH(1,1) model of (4.4) and (4.5), the conditional volatility

(equation (4.5)) of the returns is frequently used for out-of-sample forecasting of volatility.

The GARCH(1,1) in equation (4.5) can be re-written as a linear function,

(4.9) ttttt yy δδσβαα +=+++= −− θz Tt

211

2110

2

where ( )T21

21 ,,1 −−= tty σtz , ( )T

110 ,, βαα=θ and tδ is the residual of the equation (4.9).

Equation (4.9) forms linear regression function indicating 2ty is response variable where as

77

21−ty and 2

1−tσ are explanatory variable. It states that volatility at time τ=t is influenced by

volatility at time 1−= τt . Thus, if an additive outlier (AO) occurs at τ=t in y-direction

(returns), that is 2τy , it will affects 2

1+τy through 12

12

10 ++++ τττ δσβαα y , in the

explanatory variable. As such, outlier detections procedure such as the jackknife residual,

Mahalanobis distance and hat matrix, should assist in providing reliable models for the

GARCH in the presence of outliers, both in the response 2ty as well as in tz variables.

4.5 Outlier detection and estimation of conditional variances

The DGP for the simulation study is carried out as follows:

1. Let ( ) 2,1 ,1,0~ =iNuti

2. Let the true parameters be 5.0 ,1 10 == ii αα and 2.01 =iβ ; ( )iii 110 ,, βαα=1θ

3. Generate the data as follows:

1211 =σ and 11111 σuy = ,

211

21

221 2.05.01 σσ ++= y and 21212 σuy =

:

:

2

1,12

121 2.05.01 −− ++= τττ σσ y and 11 τττ σuy =

Repeat until ty at Tt = is obtained. From the above step, we obtain the true values of

Tyyy ,...,, 21 as well as 21

221

211 ,...,, Tσσσ .

4. To allow for contamination in the data, let Hut ~ , where ( ) GH γγ +Φ−= 1 for a

small proportion, γ , and ( )25,0NG ≡ . We then proceed the above steps from 1 to

78

3 by imposing the contaminated tu to generate the contaminated conditional

variances.

To provide insight into the time series generated, we plot the time series of true conditional

variances (solid line) and that with contaminated values (dotted line) in Figure 4.2.

Figure 4.2 Conditional variances of GARCH(1,1) (solid lines) model and the contaminated conditional variances (dotted lines).

Time

0 50 100 150

510

1520

2530

True Conditional Variance - Non-contaminated and Contaminated

79

Our proposal is to estimate the conditional variance using L1–GARCH procedure by

incorporating diagnostic procedures as follows:-

Given a time series data of size T , Tyy ,,1 … , we carry out the following procedure in order

to estimate the conditional variances.

1. Using ordinary GARCH(1,1) and L1–GARCH(1,1), first obtain their conditional

variances, 1Tt1θz ~~~2

1 =tσ and 22

~tσ respectively. By considering equations (4.6) and

(4.7), for L1–GARCH(1,1), the conditional variances are computed as ( ) 222

~~var ttu σ ,

where, ( )T2,1

21

~,,1~itty −−= σtiz , ( )T110

~,~,~~iii βαα=iθ and

22

2 ~~

t

tt

yuσ

= .

2. Compute the hat matrix, such that ( ) i

1

iTi

Tii ZZZZH ˆˆˆˆ −

= , where,

( )Tˆˆˆˆ TTi

T2i

T1ii z,,z,zZ …= . Note that the diagonal of the hat matrix, itth , satisfy

(4.10) pqhT

titt ++=∑

=

11

,

3. Compute the scale parameter, is , such that, ( )( )pqT

ys

T

ttit

i ++−

−=∑=

1

~1

222 σ.

4. The jackknifed residuals are being obtained as

(4.11) ( )itti

titi hs

ytk

,

22

1

~

−

−=

σ

Outlier can be detected when ( ) 5.2>tki . By incorporating equation (4.11), the

corrected values tiy which is corrected by diagnostic outlier detection procedure, tiy * is

given as follows

80

( )( )( ) ( )⎪⎩

⎪⎨⎧

>−+≤

=5.21sgn5.2~5.2

*,

2

22

tkhstktky

yiittiiti

itti σ

and thus,

(4.15) ( )

( ) ( )( ) ( )⎪⎩

⎪⎨⎧

>−+

≤=

5.21sgn5.2~5.2

*,

2 tkhstkysign

tkyy

iittiitit

itti σ

5. The new conditional variance, 1Tt1θz ˆˆˆ 2

1 =tσ and 22ˆ tσ for both GARCH and L1–

GARCH respectively, are then re-estimated from tiy * obtained by Equation (4.15).

Again, the conditional variance for L1–GARCH(1,1) is ( ) 222 ˆˆvar ttu σ , where,

( )T2,1

21 ˆ,,1ˆ itty −−= σtiz , ( )T110

ˆ,ˆ,ˆˆiii βαα=iθ and

22

22

ˆ*ˆ

t

tt

yuσ

= .

4.6 Results

To measure the performance of the conditional variances, we employ the Average Error

and Average Absolute Error of conditional variances as follows:

Average error of 2tiσ are given as,

(4.16a) ( )

TAE

T

ttt

t

∑=

−= 1

21

2

21

σσσ

for GARCH(1,1), and

(4.16b) ( )( )

T

uVarAE

T

tttt

t

∑=

−= 1

22

22

2

22

ˆˆ σσσ

for L1–GARCH(1,1), where as, Average absolute error of 2tiσ are given as,

81

(4.16c) T

AAE

T

ttt

t

∑=

−= 1

21

2

21

σσσ

for GARCH(1,1) and

(4.16d) ( )

T

uVarAE

T

tttt

t

∑=

−= 1

22

22

2

22

ˆˆ σσσ

for L1–GARCH(1,1), in order the average distance from that of the true parameter values.

The procedure above are being repeated K times.

Finally, we plot boxplot for 2tAEσ and 2

tAAEσ above to observe visually, the performance

between GARCH and L1–GARCH estimator, with adopting diagnostic checking procedure.

Appendix 4.8 presents the algorithm of simulation study for assessing the performance of

estimating conditional variance for both ordinary GARCH(1,1) and L1–GARCH(1,1).

Five hundred simulated time series (replications), each of size T = 300, 500 and 1000 with

γ = 0%, 1% and 5% were generated. For each simulated time series, we compute 2tAEσ

and 2tAAEσ as measures of performance and the results are displayed in the form of

boxplots.

82

Figure 4.3 Boxplots for 2tAEσ and 2

tAAEσ for non contaminated data, 0=γ .

-20

24

GARCH-0300

GARCH-0500

GARCH-1000

L1-GARCH-

0300

L1-GARCH-

0500

L1-GARCH-

1000

No ContaminationAverage Errors

01

23

4

GARCH-0300

GARCH-0500

GARCH-1000

L1-GARCH-

0300

L1-GARCH-

0500

L1-GARCH-

1000

Average Absolute Errors

Figure 4.3 shows that GARCH(1,1) performs well in estimating conditional variances in

comparison to L1–GARCH(1,1). This is as expected as in theory when there is no

contamination (AO) in the given time series. However, the 2tAEσ and 2

tAAEσ of the L1–

GARCH(1,1) model are relatively close to that of GARCH(1,1) especially when the size of

data increases to 1000.

83

Figure 4.4 Boxplot for 2tAEσ and 2

tAAEσ for 1% contaminated data.

-10

-50

GARCH-0300

GARCH-0500

GARCH-1000

L1-GARCH-

0300

L1-GARCH-

0500

L1-GARCH-

1000

1% ContaminationAverage Errors

02

46

810

1214

GARCH-0300

GARCH-0500

GARCH-1000

L1-GARCH-

0300

L1-GARCH-

0500

L1-GARCH-

1000

Average Absolute Errors

Figure 4.5 Boxplot for 2

tAEσ and 2tAAEσ for 5% contaminated data.

-25

-20

-15

-10

-50

GARCH-0300

GARCH-0500

GARCH-1000

L1-GARCH-

0300

L1-GARCH-

0500

L1-GARCH-

1000

5% ContaminationAverage Errors

05

1015

2025

GARCH-0300

GARCH-0500

GARCH-1000

L1-GARCH-

0300

L1-GARCH-

0500

L1-GARCH-

1000

Average Absolute Errors

84

Table 4.2 Statistical summary for 2tiAEσ

Size of data GARCH(1,1) L1-GARCH(1,1) 300 -0.05395

(0.39280) 0.17146

(0.51162) 500 -0.02505

(0.30105) 0.23559

(0.43209)

No contaminated data

1000 -0.01081 (0.17936)

0.25610 (0.29499)

300 -0.78408 (1.24575)

-0.40994 (0.90161)

500 -0.77135 (1.07199)

-0.36988 (0.83933)

1% contaminated data

1000 -0.66643 (0.55667)

-0.26058 (0.46501)

300 -3.76946 (3.21766)

-3.34830 (3.25362)

500 -3.40443 (2.22523)

-3.23409 (2.76298)

5% contaminated data

1000 -3.42363 (1.42567)

-3.06711 (1.63676)

The results from the simulated study shows that when γ increases from 1% to 5% (i.e.

contaminated time series data), the L1–GARCH(1,1) is more reliable in estimating 2tσ (see

Figure 4.4 and Figure 4.5). The reliability in estimating 2tσ increases with increasing T

(less spread in box-plot). Thus, incorporating some exploratory analysis to identify outlier

does assist in yielding reliable estimates of the conditional variances, especially for smaller

T (<500). Table 4.2 summarize the average error of conditional variance for both

GARCH(1,1) and L1-GARCH(1,1) respectively. The non-significant values of 2tiAEσ

indicate that the estimated 2tiAEσ are significantly zero.

85

CHAPTER 5

Inflation Uncertainty and Economic Growth: Evidence from LAD

ARCH Model

5.1 Introduction

The relationship between inflation uncertainty and economic growth has received much

attention among economists in the recent years. These studies include Fischer (1993),

Tommassi (1994), Barro (1996), Judson and Orphanides (1996), Grier and Perry (2000),

Dotsey and Sarte (2000), Apergis (2005), to name just a few10. Unfortunately, the empirical

evidence on the relation from all these studies is far from conclusive. A negative effect of

inflation uncertainty on growth has been reported in Fischer (1993), Grier and Perry (2000)

and Tommassi (1994), among others. Recent studies on this issue by Grier et al (2004) and

Apergis (2005) appear to support the argument, that is, inflation uncertainty is an

impediment to growth11. Apergis utilized data set from 17 OECD countries for the period

that covers from 1969-1999 to show that inflation uncertainty has an adverse effect on

economic growth. In contrast, Dotsey and Sarte (2000) find a positive association between

the two series. Barro (1996) found a negative, but insignificant, relationship between them.

Sala-i-Martin (1997) and Bruno and Eastery (1998) found that the adverse effect of

10 In a related line of research, several studies have argued that inflation is detrimental to economic growth (see Sala-i-Martin (1991), De Gregorio (1992), Taylor (1996) Temple (2000), among others). 11 In Judson and Orphanides (1996), for example, the used a panel data approach over a long period to show that inflation uncertainty is negatively correlated to economic growth across level of inflation, time and structure of the economy.

86

inflation on growth to be fragile12. Fountas and Karanasos (2007) using data from G7

countries found mixed results. For the case of Africa, Little et al (1992) show that

macroeconomic stability is not sufficient to affect growth. Sauer and Bohara (1995) provide

empirical evidence on the relationship between the two variables for the US and Germany.

They found the link between the inflation uncertainty and output growth exists for the US

but not for Germany. In a recent article by Bredin and Fountas (2005), the authors

concluded that contrary to popular belief, uncertainty is not detrimental to growth but in

some cases it may also enhance growth. Together, all these findings demonstrate the

difficulty to document the exact impact of inflation uncertainty on the economy.

Recent empirical evidence argues that output growth is substantially affected by inflation.

Empirical studies that has provided support for this hypothesis includes De Gregorio

(1992), Gylfason and Herbertsson (2001) and Guerrero (2006), among others. The

mechanism through which inflation could have impact on growth are those of savings (Fry

(1995)), the structure of the tax system, such as depreciation allowances (Feldstein (1983)),

the effect of tax systems on instruments (De Long and Summers (1991)), the impact of

inflation on the activities of financial markets (Boyd et al (1995)), the impact of inflation on

macroeconomic like interest rates and exchange rates13 (Gylfason (1999)) and the effect of

inflation through the distribution of human capital across tasks (De Gregorio (1992)). For

more detail discussion on the relationship between inflation and growth (see Temple

(2000)).

12 A survey in Holland (1993) reported the mixed results. The unambiguous results is due to factors like the method used to measure inflation (nominal) uncertainty, the chosen econometric methodology, the country and sample period. 13 It is widely recognized that exchange rate stability push firms to invest in innovation and strengthen them to compete in the global market. Technological leadership and firms’ growth are closely connected to investment capacity. Countries with higher quality and larger quantity of investment spending, with technological leadership, such as Germany, Japan and the US had strong and stable currency.

87

Although the empirical evidence did not provide a definite result on the growth-uncertainty

nexus, theory predicts that inflation uncertainty generates in efficient allocation of

resources and hence has negative impact on output. Friedman (1977) postulates that a

higher variability of inflation is usually accompanied by higher inflation rates, and the

increase in inflation uncertainty tends to adversely affect real economic activities — the

Friedmen hypothesis. Additionally, uncertainty associated with inflation is expected to

contribute to lower productivity. Fischer (1993), for example, pointed out uncertainty

lowers the rate of productivity growth and thus influences the growth process14. Others

have argued that uncertainty about inflation affects the financial markets by rising long-

term interest rates. In what follow, business will invest less in plants and equipments and

consumers will invest less in housing and other durables good, and therefore lowers output.

Most of the literature on growth mainly focused on the relationship between inflation and

economic growth. Recent studies have provided evidence that inflation may negatively

affect growth. In this study, we look at the relationship between price stability, rather than

inflation.

The purpose of this study is to assess the effects of inflation and inflation uncertainty on

five ASEAN countries (ASEAN-5: Malaysia, Singapore, Thailand, Indonesia and the

Philippines). We choose these countries because little empirical evidence on the growth-

inflation uncertainty in the literature. Most of the countries were able to grow with low to

14Higher uncertainty also implies more frequent negotiations of nominal contracts, undermines economic agents’ task to distinguish between nominal and relative price changes, and thus may adversely affect real economic activity.

88

moderate inflation, except for several brief episodes in the mid 1980s and more recently in

the 1997 crisis and post-Iraq War. In this study, we provide further evidence on the impact

of inflation uncertainty and output growth in the emerging ASEAN countries. This chapter

fills the gap in the literature in two important ways. First, our focus is on price stability and

to this end we relied on conditional volatility (inflation uncertainty) that is modeled using

the robust L1–ARCH model proposed by Jiang et al (2001). Besides, the outcome from the

Bollerslev’s (1996) GARCH measures of inflation uncertainty is also presented for

comparison. Second, we relied on robust method of estimations and the OLS method as

well to highlight the relationship if any between growth and inflation uncertainty.

Specifically, the Least Trimmed Squares (LTS) method is adopted in this study to account

for potential outliers in the regression model. The rest of this chapter is structured as

follows. Section 5.2 presents the robust ARCH model and the data in section 5.3. In section

5.4 the empirical findings are reported while Section 5.5 concludes.

5.2 The Volatility Model: Robust L1–ARCH model

To model the time-varying volatility in financial and macroeconomic variables, a large

fraction of this literature has relied on the popular autoregressive conditional

heteroscedastic (Engle (1982), ARCH) and generalized ARCH (Bollerslev (1986),

GARCH) models (see Bredin and Fountas (2005), Grier and Perry (1988) and the

references contained therein). These models assume that current volatility is a function of

past volatility. For example, the volatility models were widely used to construct the

conditional volatility of inflation in high-income countries, especially the using the US

(Engle (1982), Greir and Perry (2000) and Apergis (2005))15. The framework for modeling

15 For recent application of the GARCH model to estimate the inflation uncertainty and the impact of inflation uncertainty on output growth, see Bredin and Fountas (2005) and Apergis (2005), among others.

89

inflation uncertainty in this study is based on the work of Jiang et al (2001). The technique

was also adopted by Jiang et al (2001) to measure the volatility in stock returns.

The robust L1–ARCH model involves estimating the following specifications:

(5.1) tttt uy σε +=+= bx'bx' tt

and αz't=tσ

where ty is the inflation uncertainty, ( )'xt ptt yy −−= ,,,1 1 … , ( )'z t qtt −−= εε ,,,1 1 … ;

( )'b kbbb ,,, 10 …= and ( )'α pααα ,,, 10 …= are the respective k and p vector of the

unknown parameters to be estimated with tε being the residual error term with mean zero

and ( ) ttmedian σε = .

The L1–ARCH estimates of b and α in (5.1), denoted by b and α is that value of which

minimizes the following:

∑=

−n

tty

1min bx'tb

Likewise, α , is that value of which minimizes:

∑=

−−n

ttt

1

1min σε αz'tα

An alternative method to compute the standard error for L1–ARCH parameters is obtained

by using bootstrap standard error. To do this, generate data set of size T using equation

90

(5.1), where tu is computed once b and α are obtained. The bootstrap data

Ttu t ,,1 ,* = , are then generated as discussed in section 3.6.

Since robust ARCH is used, the conventional residual-based diagnostic test may is no

longer be appropriate in determining the overall adequacy of the specification of the model

and hence, we revert to the robust portmanteau Q -test statistics as discussed in section 3.7.

5.3 Data

The panel data consists of annual data from the ASEAN-5 countries (Indonesia, Malaysia,

Singapore, Thailand and the Philippine). Consumer price index (CPI) from Indonesia

(1925–2003), Malaysia (1949–2003), Singapore (1960–2003), Thailand (1944–2003) and

the Philippines (1938–2003), are used to construct the inflation rates and conditional

variance for each of the countries under investigation. Output growth is measured by the

difference in log annual gross domestic product (GDP) from 1980-2003 to construct

economic growth in these countries. All the GDP and CPI figures were taken from the

International Monetary Fund’s International Financial Statistics. The above data for

ASEAN-5 are provided in Appendix 4 of this thesis. The inflation rates ( jt ,π ) were

constructed using natural log, ( )jtjtjt CPICPI ,1,, log −=π . We then compute the residuals of

the inflation rates as ( )jjtjt median ππε −= ,, .

Table 5.1 provides a statistical summary of the inflation rates for all the countries under

investigation. Indonesia recorded the highest average annual inflation rate (11%) followed

by Philippines (8%). Singapore exhibited the lowest inflation rate and as expected

displayed the lowest standard deviation. The descriptive statistic presented in Table 5.1

91

shows that the distribution of the inflation rates series is characterized by long-tailed

distribution with large skewness and kurtosis. In addition, Jarque-Bera and robust Jarque-

Bera tests revealed that the all data series are characterized by non-normal distribution. This

skewness and excess kurtosis is consistent with the fact that the distribution of inflation rate

movement is non-normal. Our proposed robust 12JB test statistic however suggests that

Indonesia inflation data has no ARCH effect, but the robust Gel and Gastwirth, 2JB

successfully rejects all countries, which is as consistent as the classical Jarque-Bera test,

1JB .

Table 5.2 shows the absolute residuals autocorrelations of inflation, obtained by computing

( )( ) ( )( )

( )( )∑

∑

=

+=−

−

−−= T

ttt

T

sttsttt

js

median

medianmedianr

1

2

1,

ˆˆ

ˆˆˆˆ~

εεψ

εεψεεψ. The autocorrelation functions are all

greater than jT

1/1

, suggesting past rates of volatility affect future inflation uncertainty.

These results allow us to proceed with the robust ARCH procedure as mention earlier.

Table 5.2 also indicates that lag 6 ( 6=k ) is sufficient for computing the L1–residual

autocorrelation, sr~16.

16 The measure of volatility of inflation involves ARCH modeling. To employ L1-ARCH model, we first need to calculate the residual of the inflation rate, ( )tjtjtj median ππε −=

92

Table 5.1 Summary statistics of inflation rates data for ASEAN-5 Indonesia Malaysia Philippines Singapore Thailand

Min -0.5866 -0.0726 -1.7789 -0.0184 -0.0879 Max 3.0136 0.2584 2.1355 0.2333 0.6263 Mean 0.1115 0.0295 0.0839 0.0276 0.0607

Median 0.0675 0.0258 0.0565 0.0183 0.0466 MAD 0.0652 0.0230 0.0550 0.0194 0.0441

Std Dev 0.3702 0.0477 0.3849 0.0469 0.0931 Skewness 6.7966 2.3178 0.7987 3.3681 4.0439 Kurtosis 54.1419 10.1232 20.9563 12.3494 23.3398

1JB 7811.068 [0.000]

211.800 [0.000]

925.967 [0.000]

256.469 [0.000]

1177.273 [0.000]

2JB 4388071 [0.0000]

1379.926 [0.0000]

215516.6 [0.0000]

7457.636 [0.0000]

16999.19 [0.0000]

12JB 4.8901 [0.0867]

17.7837 [0.0001]

16.8889 [0.0002]

72.0146 [0.0000]

6.4530 [0.0397]

Note: The figures in bracket are p-values for the classical Jarque-Bera ( 1JB ), Gel and Gastwirth, 2JB , and 12JB tests. The JB tests can be referred from equation (2.36), while equation (2.37a) and (2.37k) are used for computing robust 2JB and 12JB

Table 5.2 Standardized absolute L1–residual autocorrelations, sr~ Indonesia Malaysia Philippines Singapore Thailand

Lag 1 0.3288* 0.2037* 0.3281* 0.0465 0.3390* Lag 2 0.2329* 0.1481* 0.3750* -0.2093* 0.1864* Lag 3 0.1644* 0.1667* 0.1406* -0.1395 0.0339 Lag 4 0.1781* 0.2593* 0.2813* 0.2093* -0.0508 Lag 5 0.2192* 0.05556 0.1719* 0.0930 0.0678 Lag 6 0.2603* 0.1481* 0.1875* 0.0233 0.0169

Note: The asterisk (*) indicates that the standardized absolute L1–residual autocorrelations,

sjr~ are significantly autocorrelated, that is greater than critical values, jT/1 , for country j.

5.4 Empirical Results

First, we construct the GARCH(1,1) model to inflation uncertainties for ASEAN-5

countries. The GARCH(1,1) model is as follows,

(5.2) ( ) ( ) 2/12/12,1

2,110 jtj αz'tjjtjjtjjtjtjtjtj uuu =++== −− σβπαασπ

93

where, the white noise, tju , is Gaussian. The results for GARCH(1,1) is as shown in Table

5.3.

Table 5.3 Parameter estimates and tests for model: GARCH(1,1) model Indonesia Malaysia Philippines Singapore Thailand

0α 0.0957 (0.0658)

0.0003 (0.0002)

0.00635*** (0.0006)

0.0025*** (0.0002)

0.0005 (0.0003)

1α 0.0384 (0.1650)

1.5872*** (0.4270)

3.0892*** (0.1760)

1.1576 (0.7641)

1.9681*** (0.1862)

1β -0.2535 (0.8464)

0.0016 (0.0675)

-0.0015 (0.0155)

-0.7135*** (0.0551)

-0.0185 (0.0322)

AIC 88.3175 -184.9153 -1.5108 -136.5564 -138.1828 BIC 95.1888 -178.9483 4.9659 -131.2728 -131.9502

1JB 8525 [0.0000]

3.8740 [0.1441]

1198 [0.0000]

957.8 [0.0000]

318.2 [0.0000]

12JB 6.5610 [0.0376]

5.8229 [0.0544]

13.1211 [0.0014]

8.7531 [0.0126]

7.7427 [0.0208]

SW 0.3867 [0.0000]

0.9507 [0.0492]

0.6049 [0.0000]

0.5682 [0.0000]

0.8241 [0.0000]

12Q 4.217 [0.9792]

28.62 [0.0045]

5.701 [0.9304]

7.955 [0.7887]

4.346 [0.9763]

212Q 0.2623

[1.0000] 15.39

[0.2210] 0.8726

[1.0000] 0.8829

[1.0000] 1.16

[1.0000] LM 0.2573

[1.0000] 13.69

[0.3211] 0.707

[1.0000] 0.7824

[1.0000] 12.94

[0.3732] Note: The figures in parentheses indicate the standard error of parameter estimates. The figure in bracket show the p-values of estimated statistics

Table 5.3 shows that GARCH(1,1) model is insufficient for the ASEAN-5 inflation data as

the estimated parameters of GARCH(1,1) are not significant. The Shapiro-Wilk’s test

reveals that all inflation data for ASEAN-5 except Malaysia, are non-Gaussian. This result

is consistent with the classical JB . The robust JB , ( 12JB ) also supports that the estimated

white noise for all countries except Malaysia, are non-Gaussian. The result of 12Q and 212Q

suggest that the estimated innovations for most countries are not autocorrelated. In addition,

since the sum of the ARCH and GARCH parameters for Malaysia, Philippines, Singapore

94

and Thailand exceed 1, and the estimated GARCH parameter or Indonesia is negative, we

need an alternative model and test statistics to overcome these problems.

As an alternative fit to the model, we reconsider fitting a robust ARCH( p ) model to the

data which takes the form:

(5.3) ( )jptpjjtjjtjtjtjjtj uuM ,,110 −− +++=+= εαεαασπ

where, ( )tjj medianM π= , jtjtj M−= πε and j = 1 (Indonesia), 2 (Malaysia), 3

(Philippines), 4 (Singapore) and 5 (Thailand). Table 5.4 reports the robust estimates of

inflation uncertainty using the LAD-ARCH model as outlined in the previous section for

the ASEAN-5 countries to combat the sensitivity of the result due to the problem outliers.

Initially, we model the inflation series as an L1–ARCH (6) model. Based on the bootstrap

standard errors17, insignificant variables were eliminated and the selected model for each of

the country is displayed in Table 5.4. All in all, the model appears to fit the data of the

ASEAN countries, except for Singapore, that have not only recorded very low inflation rate

but also experience high growth rates for most part of the period considered in the analysis.

Note that both the robust standardized absolute L1–residual autocorrelations and robust

portmanteau tests are not statistically significantly at the usual significance levels for

Singapore. This means that the estimated innovation variables are not autocorrelated even

after the 10th lags. Again, the finding suggests that the robust modeling technique failed to

capture changing volatility of inflation data for Singapore. Malaysia’s inflation volatility

has been influenced at lag 4 and 6 while for other Asian countries, the ARCH (1) model

appears to adequately capture the effect of time-varying volatilities. Although L1–ARCH(1) 17 The procedure of estimating standard error via bootstrap standard error are discussed in Chapter 3 of this thesis

95

parameter estimates are well fitted for Indonesia, it does not when considering that the

robust portmanteau tests are significantly rejected at lags 5 and 10.

Table 5.4 L1–ARCH models for each inflation rates for ASEAN-5 countries.

Countries Indonesia Malaysia Philippines Singapore Thailand

0α 0.02025** (0.0094)

0.0095*** (0.002)

0.0212*** (0.008)

0.0105*** (0.002)

0.0120*** (0.004)

1α 0.2825* (0.152)

- 0.4388*** (0.157)

0.1561 (0.137)

0.4470*** 0.1458)

4α - 0.1279 (0.086)

- - -

6α - 0.1466* (0.084)

- - -

1~r 0.1096

(0.081) 0.0185 (0.136)

-0.0469 (0.095)

-0.0930 (0.117)

0.0000 (0.092)

2~r 0.1781

(0.113) -0.0370 (0.1348)

0.1563 (0.1212)

-0.3023 (0.1522)

0.0169 (0.1284)

3~r 0.3288

(0.117) -0.0185 (0.135)

-0.0781 (0.122)

-0.0465 (0.150)

-0.0678 (0.130)

4~r 0.1781

(0.1151) -0.0741 (0.1089)

0.0625 (0.1248)

0.2093 (0.1521)

-0.1525 (0.130)

5~r 0.0822

(0.114) 0.1667 (0.132)

0.07813 (0.122)

0.0930 (0.148)

-

6~r - 0.0741

(0.107) - - -

1~Q 1.8179 0.0186 0.2441 0.6310 0.0000

5~Q 23.1791 1.9467 2.7585 6.8805 1.7326

10~Q 40.9480 5.3741 7.9595 15.6537 4.7535

1JB 10974.73 [0.00]

1016.215 [0.000]

2400.567 [0.000]

259.8958 [0.0000]

1627.414 [0.000]

12JB 7.3698 [0.0251]

9.2355 [0.0099]

1.5952 [0.4504]

71.9154 [0.0000]

6.2633 [0.0436]

Note: L1–estimates with their bootstrap standard error (based on 1000 bootstrap replication); Sr~ (for s = 1,…,6) – the robust lag s standardized absolute L1–residual

autocorrelation estimator. α,~

MQ - robust portmanteau test statistic which tests the adequacy of the fit with the criteria values of chi-squared distribution at 1.0=α : 2.706 for 1=M , 9. 24 for 5=M and 15.99 for 10=M degrees of freedom

96

The robust Jarque-Bera ( RJB ) in Table 5.4 suggests that the model used for fitting the

inflation rates for all countries except Indonesia, is adequate. The yearly plots of

conditional scale from 1980 to 2003 for ASEAN-5 countries are shown in Appendix 4 of

this thesis.

Next, we address an important question: does inflation uncertainty affect economic growth?

Here, we relied on the robust estimates of the volatilities, αz'tj ˆˆ =tjσ in equation (3.8), to

measure inflation uncertainty in the ASEAN countries. Figure A4.1d, from Appendix 4

reveals that the inflation uncertainty for Singapore is relatively small compared to others.

To determine the impact of inflation uncertainty ( )tjσ on economic growth, we construct a

panel for the time series and estimate the following model:

(5.4) iii egrowth ++=Δ σθμ ˆ

For this purpose, we consider two types of method of estimations to obtain the parameters

of the model – the Ordinary Least Squares (OLS) and Least Trimmed Squares (LTS). Table

5.5 presents the estimates of parameters of the model. The coefficients of the conditional

scale (inflation uncertainty) in both fits, are negative and are statistically significant at the

one percent level. Comparatively, the absolute size of the coefficient of the inflation

uncertainty variable from the LTS (-1.49) appears to be larger than that of the LS estimates

(-1.02) and this difference can be captured in Figure 5.1.

97

Table 5.5 Parameter Estimates of the simplest model, 1980-2003 OLS LTS μ 0.0818***

(0.0056) 0.0972*** (0.0047)

θ -1.1017*** (0.1705)

-1.4900*** (0.1597)

%70γ 0.03080 0.0296

%80γ 0.0466 0.0445

%90γ 0.0697 0.0735

5Q 19.1404 [0.0018]

17.1719 [0.0042]

LM 9.8040 [0.0810]

7.49589 [0.186294]

1JB 47.9821 [0.0000]

46.1580 [0.0000]

12JB 34.3958 [0.0000]

38.0530 [0.0000]

SW 0.9348 [0.0000]

0.9287 [0.0000]

RESET 1.8039 [0.1504]

2.2671 [0.0848]

Note: All estimates are significant at the 1% level. Standard errors of parameter estimates are in the parentheses. %70γ , %80γ , %90γ : are measures where 70%, 80% and 90% out of the observed data lie within γ distance away from the fitted model. LM:- Lagrange Multiplier test for estimated residuals based on five lags, with p-values in brackets

98

Figure 5.1 Inflation Uncertainty and Economic Growth: LTS and OLS fits

Inflation Uncertainty

Gro

wth

Rat

e

0.05 0.10 0.15

-0.1

0-0

.05

0.0

0.05

0.10

0.15

The LM tests in Table 5.5 shows that residuals are not autocorrelated for both fits and this

follows Apergis (2005) and others argument that growth and inflation uncertainty are

negatively correlated. This result is also supportive of the Friedman hypothesis, that is,

uncertainty as measure by the conditional variance of the inflation exerts a negative impact

on economic growth the ASEAN countries. In addition due to the outliers triggered, the

OLS is no longer adopted to fit the relationship between growth and inflation uncertainty of

five East Asian countries. This can be observed by considering %70γ and %80γ of LTS are

smaller than those of OLS. By filtering thirteen outliers of data using LTS, further analysis

can be done via OLS.

Some researchers have also examined the relationship between inflation, inflation

uncertainty and economic growth (Judson and Orphanides (1996)). Using panel data

99

approach, Judson and Orphanides (1996) reported that high inflation is detrimental to

growth and volatility is related to lower growth at all levels of inflation. In this chapter, we

look at the effect of inflation ( )ijπ and inflation uncertainty ( )tjσ towards growth

( )ijgrowthΔ across countries. The cross relationship is formed as follows

(5.5) ijijijjjij eDgrowth ++++=Δ θπσβαα ˆ

where, j = 1 for Indonesia, 2 for Malaysia, and 3 for Thailand respectively, and jD are

their dummies. Here, Philippines is set as a benchmark, α representing the constant

parameter where as jα represent parameter shifters from α for different countries. As the

overall inflation uncertainties for Singapore are too small relatively (see Appendix 4), we

omit this country from our observation.

100

Table 5.6 The comparison of method of estimates between OLS and LTS, for the equation (5.5).

OLS LTS α 0.0827***

(0.0108) 0.0786*** (0.0074)

1α 0.0155 (0.0104)

0.0076 (0.0071)

2α 0.0133 (0.0117)

0.0203** (0.0081)

4α 0.0241** (0.0109)

0.0235*** (0.0077)

β -0.7665*** (0.1889)

-0.7703*** (0.1375)

θ -0.2349*** (0.0464)

-0.1637*** (0.0434)

R2 0.4694 0.405

5Q 29.2555 [0.0000]

27.7955 [0.0000]

LM 31.5456 [0.0000]

25.0237 [0.0001]

1JB 11.6300 [0.0030]

19.7866 [0.0001]

12JB 7.7586 [0.0207]

6.6958 [0.0352]

SW 0.9526 [0.0058]

0.9202 [0.0000]

Note: figure in parenthesis is standard errors. Asterisks ***, ** and * represents significance at 1, 5 and 10% respectively

The estimated parameters for both OLS and LTS are sufficiently significant. But, the

autocorrelation of estimated errors, ije and its normality tests are not showing better fit for

this model. The OLS however is better in explaining the behavior of the data than LTS as

the larger values of R2.

101

CHAPTER 6

Discussion

This thesis mainly deals with the development and applications of L1–GARCH model,

especially the robust version, in analyzing economic data where the heterogeneity of

variances is more prevalent. It also anticipates that, L1–GARCH model could be more

useful in modelling this problem, especially in estimating conditional variances, which

usually form the basis of many analyses in this area research – for example, estimating the

uncertainty of inflation data (see Apergis (2005), Fischer (1993), Grier and Perry (2000)

and Dotsey and Sarte (2000)), particularly when ordinary GARCH fail as the time series

data do not follow strict assumption of normality.

In this thesis, we show that GARCH(1,1) performs well if normality assumption holds

while L1–GARCH(1,1) performs better when violations of the assumption are present in the

data. For ordinary GARCH models, MLE is used to estimate conditional variances. The

MLE is non-robust, that is, it is subject to influence by outlying data points which occurs

when there are departures from the normality assumption. In such situations, the method of

LAD for GARCH, denoted as L1–GARCH, is a possible alternative in estimating the

appropriate conditional variances. The use of exploratory analysis and outlier detection

procedure prior to the fitting helps to further improve the estimation process.

102

In studying the performance of estimating conditional variance, a cut-off at 2.5 is used as

typically illustrated by Rousseeuw and Leroy ((1987) in flagging outliers. Under the

Gaussian distribution, this corresponds to less than 1 percentile. To enhance detection, it

would be interesting to consider various possible cut-off points from 2.0 to 5.0. In addition

to the above study, inclusion of other types of outliers such as additive, innovative and

transient outliers should also be incorporated for further studies.

In Chapter 2, we developed a robust version of Jarque-Bera ( RJB ) test which is commonly

used for detecting the normality of errors. The RJB provides alternatives when the classical

Jarque-Bera test fails in the presence of outliers in the data. This development is very much

related to economic research because nonnormality is more prevalent in economic data and

the presence of outliers in this type of data is rather a rule than exception. The RJB test

statistic which utilizes a filter function, is asymptotically χ2-distributed when its cut-off

point, tends to infinity as this will lead us back to the classical JB -test. While the proposed

RJBs are robust to the presence of a small percentage of outliers (or contaminants), it is

difficult to differentiate between outlyingness and deviation from the actual assumption.

However, the properties of RJB – by utilizing filter function that is bounded and

continuous – should also be derived to show theoretically the increasing of the cut-off point

value leads to convergence of l (see Chapter 2) value to 3. Furthermore, deriving the

properties of robust skewness and kurtosis would accurately compute the values of

asymptotic variance and thus to show that these statistics are not correlated. Finally, the

new developed RJB could be constructed and thus the study of power of rejection would

be improved. This leads us for future research.

103

Since estimating the parameters involve iterative procedure, the modified Newton-Raphson

procedure is employed. The computational complexity involves in the search for a global

minimum suggest that other efficient method such as the genetic algorithm or simulated

annealing may be alternatives that could be used to reduce the computation burden.

The bootstrap standard error is used to estimate the standard errors for individual parameter

estimates in the case when f (u) is unknown. If f (u) is known, we apply equations (3.10)

and (3.12) to evaluate the significance of the parameters. Since u ~ N(0,1) is allowed, we

can test this using skewness, kurtosis and JB -tests. The RJB tests which consist of robust

measures of skewness and kurtosis that are less sensitive to outliers are used when outlier is

suspected in the data. If f(u) is not known, computing f(u), at u = 0 in equations (3.10) and

(3.12) is demanding in terms of time consuming to find appropriate distribution of u , and

as such, one may resort to compute alternatives to the bootstrap standard errors.

In chapter 5, we adopt the L1–ARCH model to measure inflation uncertainty and to shed

light on the relationship between inflation uncertainty and economic growth in five ASEAN

countries. Essentially, we found that the volatility model fits adequately the data from the

ASEAN-5 countries. The advantage of the L1–ARCH model is the ability to handle extreme

observation(s). We find overwhelming statistical evidence supporting the hypothesis that

increased inflation uncertainty lead to fall in economic growth. This positive association

between inflation uncertainty and growth is consistent with Tommassi (1994), Grier and

Perry (2000), Grier et al (2004) and Apergis (2005), among others, conducted for the major

industrialized countries. This finding is in line with Friedman’s conjecture that uncertainty

concerning regime changes depresses real economic activity. The policy implication for

104

countries experiencing high inflation uncertainty (inflation) in the post Iraq war due to oil

price shocks is clear: Policy measures aimed at lowering inflation uncertainty will reduce

the negative consequences of uncertainty on real economic activities.

Finally, we take note that our studies on AR( k )–GARCH( q, p ) using Least absolute

deviation (LAD) method of estimation can be extended into ARMA( k,l )–GARCH( q, p )

in

studying the behaviour of the mean as well as variance of volatility data. We can also

observe the asymmetry / leverage of the distribution of GARCH model via Exponential

GARCH models using LAD. Such approach is usually employed in stock market data as

leverage can be an indicator to observe whether good or bad news does affect the volatility

of stocks. For this, we leave it as an open study.

105

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113

APPENDIX 1

Sample data of volatility

Table A1.1 Sample data of volatility discussed in Chapter 1 1 -0.023 32 -0.012 63 -0.036 94 -0.013 2 -0.018 33 0.041 64 -0.033 95 -0.001 3 0.016 34 -0.001 65 -0.005 96 -0.035 4 -0.002 35 0.049 66 0.008 97 -0.04 5 0.033 36 -0.043 67 0.076 98 0.024 6 -0.017 37 -0.004 68 0.004 99 -0.034 7 -0.01 38 0.021 69 0.014 100 -0.03 8 0.015 39 0.024 70 -0.044 101 -0.066 9 -0.001 40 0.025 71 -0.006 102 0.047 10 -0.001 41 0.043 72 -0.003 103 0.048 11 -0.003 42 0.001 73 0.001 104 -0.018 12 -0.004 43 0.035 74 -0.035 105 -0.016 13 0.027 44 -0.035 75 0.055 106 -0.06 14 0.005 45 0.026 76 0.02 107 -0.01 15 0.009 46 0.009 77 -0.025 108 0.003 16 0.043 47 -0.013 78 -0.012 109 0.053 17 0.036 48 0.046 79 -0.016 110 0.056 18 -0.015 49 0.045 80 0.026 111 0.012 19 0.007 50 0.025 81 -0.013 112 -0.029 20 0.01 51 0.054 82 0.028 113 0.009 21 0.005 52 0.071 83 0.057 114 -0.048 22 -0.008 53 -0.081 84 0.011 115 0.029 23 0.018 54 -0.054 85 0.028 116 -0.003 24 -0.018 55 -0.065 86 -0.031 117 0.027 25 -0.038 56 0.029 87 0.041 118 -0.019 26 0.038 57 0.053 88 0.014 119 -0.002 27 -0.011 58 -0.037 89 -0.002 120 0.013 28 0.011 59 -0.02 90 -0.017 121 -0.028 29 0.029 60 0.036 91 -0.038 122 0.006 30 0.009 61 -0.056 92 -0.006 123 0.059 31 0.014 62 -0.015 93 0.007 124 -0.006

114

Table A1.1 continues…

125 0.033 166 0.019 207 0.013 248 0.002 126 0.025 167 0.001 208 0.003 249 0.049 127 -0.027 168 0.025 209 0.02 250 -0.049 128 0.008 169 -0.005 210 -0.001 251 -0.136 129 -0.005 170 0.002 211 -0.008 252 -0.031 130 -0.036 171 -0.005 212 0.016 253 0.073 131 0.033 172 0.018 213 -0.024 254 -0.04 132 0.013 173 -0.002 214 0.021 255 -0.185 133 -0.019 174 0.003 215 0.003 256 -0.011 134 -0.006 175 -0.018 216 0.012 257 0.122 135 0.02 176 -0.005 217 0.005 258 -0.052 136 -0.017 177 -0.004 218 -0.017 259 0.013 137 -0.025 178 -0.009 219 -0.008 260 -0.028 138 0.003 179 -0.008 220 -0.024 261 0.025 139 -0.018 180 -0.005 221 0.006 262 -0.149 140 -0.025 181 0.02 222 -0.068 263 0.106 141 0.015 182 -0.01 223 0.007 264 0.037 142 0.026 183 -0.003 224 -0.027 265 0.02 143 -0.017 184 0.005 225 -0.01 266 0.245 144 -0.017 185 -0.011 226 0.013 267 -0.077 145 -0.02 186 -0.038 227 0.012 268 0.052 146 -0.002 187 0.015 228 -0.06 269 0.014 147 0.006 188 0.01 229 0.04 270 -0.086 148 -0.039 189 -0.003 230 0.026 271 0.024 149 -0.024 190 0.017 231 0.011 272 0.03 150 -0.008 191 -0.002 232 -0.037 273 -0.01 151 0.042 192 -0.013 233 0.02 274 -0.091 152 0.057 193 0.012 234 -0.026 275 0.014 153 -0.002 194 0.012 235 0.005 276 -0.067 154 0.012 195 -0.001 236 -0.053 277 0.016 155 -0.002 196 0.017 237 -0.012 278 -0.011 156 0.002 197 -0.01 238 0.043 279 -0.073 157 0.045 198 0.03 239 -0.038 280 -0.016 158 0.015 199 -0.005 240 -0.07 281 0.026 159 -0.01 200 -0.003 241 -0.018 282 -0.066 160 -0.008 201 0.005 242 0.001 283 -0.057 161 0.019 202 0.01 243 -0.113 284 -0.058 162 -0.016 203 0.02 244 0.033 285 0.002 163 0.028 204 0.005 245 0.044 286 -0.039 164 -0.003 205 -0.032 246 -0.076 287 0.071 165 0.002 206 -0.003 247 0.023 288 -0.097

115

Table A1.1 continues…

289 0.048 330 0.033 371 0.038 412 -0.028 290 -0.059 331 0.015 372 0.007 413 0.021 291 -0.032 332 0.095 373 -0.017 414 0.013 292 -0.094 333 -0.01 374 -0.06 415 -0.046 293 -0.093 334 -0.038 375 0.001 416 -0.003 294 0.004 335 -0.019 376 -0.014 417 -0.022 295 -0.056 336 0.041 377 0.039 418 -0.011 296 0.196 337 0.013 378 0.003 419 0.017 297 0.013 338 0.02 379 -0.027 420 0.025 298 0.064 339 0.043 380 -0.02 421 0.03 299 -0.021 340 0.001 381 -0.034 422 0.028 300 -0.068 341 -0.016 382 0.003 423 -0.02 301 0.027 342 -0.046 383 0.033 424 -0.016 302 0.063 343 -0.051 384 -0.014 425 -0.003 303 0.054 344 -0.087 385 0.042 426 -0.014 304 -0.042 345 0.103 386 -0.074 427 -0.011 305 0.11 346 -0.013 387 0.011 428 -0.033 306 0.017 347 0.003 388 -0.033 429 -0.001 307 -0.016 348 -0.026 389 -0.027 430 -0.031 308 0.071 349 -0.007 390 -0.025 431 -0.111 309 0.048 350 -0.034 391 0.031 432 0.017 310 0.003 351 -0.024 392 -0.014 433 -0.003 311 0.022 352 -0.017 393 0.046 434 0.024 312 -0.005 353 0.095 394 -0.031 435 -0.012 313 0.054 354 -0.034 395 -0.038 436 -0.007 314 0.001 355 0.025 396 0.024 437 -0.021 315 0.006 356 0.004 397 -0.015 438 0.015 316 0.027 357 -0.031 398 0.024 439 0.016 317 -0.052 358 0.002 399 -0.025 440 0.009 318 -0.059 359 0.017 400 -0.04 441 0.024 319 0.024 360 0.017 401 -0.033 442 -0.004 320 -0.01 361 -0.031 402 0.023 443 0.008 321 -0.041 362 0.027 403 -0.015 444 0.055 322 -0.045 363 0.045 404 -0.033 445 -0.014 323 0.029 364 0.003 405 -0.001 446 0.042 324 -0.026 365 0.034 406 0.06 447 -0.009 325 -0.024 366 0.003 407 0.038 448 0.017 326 0.082 367 0.122 408 0.013 449 -0.021 327 0.05 368 0.017 409 -0.043 450 0.011 328 0.05 369 -0.03 410 -0.011 451 0.039 329 0.07 370 0.018 411 -0.023 452 0.003

116

Table A1.1 continues…

453 0.011 494 -0.006 535 -0.006 576 -0.014 454 -0.082 495 -0.023 536 -0.002 577 0.008 455 -0.057 496 0.029 537 0.011 578 -0.009 456 0.018 497 -0.007 538 -0.013 579 -0.006 457 -0.008 498 -0.018 539 0.001 580 0.013 458 0.007 499 -0.011 540 0.006 581 0.042 459 0.009 500 0.005 541 0.01 582 0.017 460 -0.004 501 0.007 542 0.023 583 0.004 461 -0.025 502 0.02 543 0.03 584 -0.006 462 0.008 503 -0.01 544 0.012 585 0.02 463 0.06 504 -0.026 545 0.009 586 -0.014 464 0.005 505 -0.023 546 -0.015 587 -0.011 465 -0.005 506 -0.006 547 0.012 588 -0.006 466 0.035 507 -0.021 548 0.046 589 -0.025 467 0.001 508 -0.035 549 -0.007 590 0.018 468 -0.002 509 -0.001 550 -0.014 591 -0.044 469 0.038 510 -0.006 551 -0.011 592 -0.001 470 -0.002 511 0.027 552 0.017 593 -0.049 471 0.011 512 0.007 553 -0.003 594 0.002 472 -0.01 513 0.004 554 0.008 595 0.025 473 -0.01 514 -0.001 555 0.019 596 -0.006 474 0.035 515 -0.014 556 -0.003 597 0.029 475 -0.015 516 -0.013 557 0.016 598 -0.004 476 0.014 517 -0.014 558 -0.024 599 0.003 477 -0.019 518 -0.016 559 0.004 600 0.008 478 0.01 519 0.025 560 -0.004 601 0.028 479 0.04 520 0.01 561 -0.004 602 -0.016 480 0.002 521 0.02 562 0.066 603 -0.007 481 -0.002 522 -0.028 563 -0.017 604 -0.006 482 0.003 523 0.005 564 0.028 605 -0.017 483 0.006 524 0.054 565 0.013 606 -0.005 484 0.018 525 -0.006 566 -0.017 607 -0.008 485 0.026 526 -0.009 567 -0.021 608 0.018 486 0.001 527 -0.007 568 -0.011 609 0.025 487 -0.012 528 -0.008 569 -0.007 610 0.009 488 -0.01 529 -0.004 570 0.011 611 0.01 489 -0.002 530 -0.012 571 0.007 612 -0.008 490 -0.029 531 -0.017 572 -0.029 613 -0.003 491 -0.031 532 -0.01 573 0.015 614 0.008 492 0.021 533 0.008 574 0.012 615 -0.013 493 -0.008 534 0.006 575 0.046 616 0.003

117

Table A1.1 continues…

617 0.009 652 -0.006 687 -0.01 722 0.007 618 0.012 653 0.011 688 0.006 723 0.023 619 0.012 654 0.014 689 0.001 724 0.016 620 0.015 655 0.023 690 0.004 725 0.016 621 0.011 656 -0.006 691 0.001 726 0.015 622 0.007 657 0.005 692 0.018 727 0.014 623 -0.028 658 -0.009 693 -0.008 728 -0.013 624 0.01 659 -0.014 694 0.01 729 -0.012 625 0.004 660 -0.008 695 -0.001 730 0.014 626 -0.001 661 -0.01 696 0.012 731 0.02 627 0.01 662 0.011 697 0.004 732 -0.004 628 0.014 663 0.005 698 -0.025 733 0.024 629 -0.002 664 -0.001 699 -0.015 734 0.017 630 -0.015 665 0.007 700 0.002 735 0.03 631 -0.011 666 -0.003 701 -0.016 736 0.021 632 0.009 667 0.001 702 -0.023 737 0.012 633 -0.008 668 -0.018 703 0.018 738 0.012 634 -0.005 669 -0.004 704 0.011 739 -0.102 635 -0.02 670 0.01 705 0.013 740 0.024 636 0.02 671 -0.018 706 -0.014 741 -0.005 637 -0.009 672 0.005 707 0.013 742 0.046 638 -0.012 673 0.004 708 0.011 743 0.007 639 -0.008 674 -0.022 709 0.003 744 0.024 640 -0.013 675 0.017 710 0.004 745 0.02 641 0.014 676 -0.006 711 -0.002 746 0.002 642 0.005 677 -0.001 712 0.009 747 0.004 643 0.002 678 0.01 713 0.01 748 0.026 644 0.026 679 0.013 714 -0.006 749 -0.013 645 -0.013 680 -0.001 715 0.002 750 0.002 646 -0.01 681 -0.008 716 0.006 751 -0.016 647 -0.016 682 0.01 717 0.002 752 0.015 648 -0.003 683 0.015 718 0.003 753 -0.008 649 0.029 684 -0.01 719 0.012 754 0.006 650 0.008 685 0.007 720 -0.006 755 0.022 651 -0.001 686 -0.003 721 0.011 756 -0.03

118

APPENDIX 2

Asymptotic Normal Distribution for Median Statistic and Bahadur

Representation for L1–GARCH model.

In this appendix, we referred and thus, derive Hoaglin, Mosteller and Tukey (1983)

regarding to asymptotic normal distribution for median statistic. Given the variables,

nXX ,,1 … , the probability and cumulative distribution function is given by ( )xf X and

( )xFX respectively. Thus, the statistic of X , ( )XT is defined as median of X , denoted as

M . We may also define ( ) ( ) xX xF εδε +−1 as X is contaminated at the proportion of ε ,

( )10 << ε , and the contamination point is defined as xδ . The influence curve is given as

(A2.1) ( ) ( ) ( )[ ] ( )ε

εδε XTxFTTFxIC xX −+−

=1

,;

In this case, we consider the nominator of equation (A2.1), can be denoted as

(A2.2) ( ) ( )[ ] ( ) MMXTxFT xX −=−+− *1 εδε

Approximately, we may also define

(A2.3) ( ) ( ) ( ) ( )MfMMMFMFMXM XXX )*(**Pr −≈−=<<

If we define ε=−MM * , we might conclude that,

119

(A2.4) ( ) ( )[ ] ( )2

sgn0

2

02*lim εε

ε

MxMx

MxMFMF XXn

−=⎪⎩

⎪⎨

⎧

<−−

≥−=−

∞→

For clear understanding to the equation (A2.4), we run small simulation – we generate n

size of data assumed standard normal distribution, and thus, compute the median of the

data, denoted as M . From the generated data, we contaminate 10% of the data assumed to

be normal distribution with both mean and standard deviation being 5, then we estimate the

median of this data, denoted as *M . Since computing 2/ε can be obtained by both

( ) 2/* MM − and ( ) ( )MFMF XX −* we are to show that,

( )[ ] ( ) ( )[ ] 0*2/* ≈−−−= MFMFMML XX . We repeat K replications to compute K

values of L . We then increase the size of the generated data to asses the consistency of

estimating L . The SPLUS programming language for this simulation study is shown in

Appendix 3.9. The results is shown as follows

Table A2.1 The statistics of 1000 replicated data of ( )[ ] ( ) ( )[ ]MFMFMML XX −−−= *2/* . Size

n 30 50 100 300 1000

Mean 0.0829 0.0831 0.0823 0.0806 0.0799 Std Dev 0.9923 0.9283 1.0427 1.0167 0.9819

Skewness 0.6325 0.0863 0.2249 -0.0509 0.0276 Kurtosis 1.3438 -0.8564 0.9937 -0.4299 -0.0104

Further, we visualize the density plot for 1000 replicated data of computed L which sizes

1000 data.

120

Figure A2.1 Density plot for 1000 values of L which has been generated with 1000 size of data.

0.02 0.04 0.06 0.08 0.10 0.12 0.14

05

1015

2025

The results and figure above support the equation A2.4. Thus, by substituting (A2.2),

(A2.3) and (A2.4) into (A2.1), we have

(A2.5) ( ) ( )( )

( ) ( )( )Mf

MFMFMf

MXMMMTFxICXX εεε−

=<<

=−

=**Pr*,;

( ) ( ) ( ) ( )MfMxMx

Mf XX 21sgn

2sgn.1

−=⎥⎦⎤

⎢⎣⎡ −=

εε

The asymptotic variance can be defined as

(A2.6) ( ) ( )∫∞

∞−= dxxfICTFA 2,

Thus, for median statistic being the mean of asymptotic normal distribution, the asymptotic

variance of X can be derived as follows

(A2.7) ( ) ( ) ( ) ( )( )[ ]2

2

41

21sgn,

Mfdxxf

MfMxTFA

XX

=⎥⎦

⎤⎢⎣

⎡−= ∫

∞

∞−

121

Therefore, the asymptotic Normal distribution of X is

( )[ ] ⎭⎬⎫

⎩⎨⎧

241,~Mf

MNXX

or

(A2.8) ( )[ ] ⎭

⎬⎫

⎩⎨⎧

− 241,0~Mf

NMXX

Given the transformed variables, nXX ,,1 … , the probability and cumulative distribution

function is given by ( )xf X and ( )xF X respectively. By following the above procedure,

the asymptotic normal distribution of |X| is

(A2.9) ( )[ ] ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

− 241,0~

MfNMX

X

We might transform the conditional variable of X into X as follows

(A2.10) ( ) ( ) ( ) ( )MXMPdxdMXP

dxdMF

dxdMf XX <<−=<==

( ) ( )MFdxdMF

dxd

XX −−=

( ) ( ) ( ) ( ) ( )MfMfxdxdMfMf XXXX −+=−−−=

Thus, by substituting (A2.10) into (A2.9), the asymptotic Normal distribution of X is

(A2.11) ( ) ( )[ ] ⎭

⎬⎫

⎩⎨⎧

−+− 24

1,0~MfMf

NMXXX

122

These asymptotic normal distribution can be used to derive the asymptotic normal for L1–

estimate for ARCH model.

For further understanding to the asymptotic normality for L1–GARCH distribution, we

follow Bahadur (1966) representation, that is, by defining following model,

tttt ybby εε +=++= − bxTt110

and

(A2.12) θz Tt=++= −− 11110 ttt σβεαασ

We might also define, ( ) ( )uIu −= 2/1ψ , which forms ( )( ) 0=UE ψ and ( )( ) 4/1=UVar ψ .

Thus, the Bahadur representation for L1–GARCH model is defined as

(A2.13) ( ) ( )( ) ( )( ) ( )12/112/1

ˆ1

11 p

T

tut

uu

oFuTFf

T +−=− ∑=

−−

−

ψt1

1 xDbb

where, ∑=

=T

trtT 1

1σ

Ttt

rxx

D , for 1,0=r . From equation (A2.13), we might derive

( )[ ] ( ) ( )[ ] 0xDbb t

11 ==− ∑

=

− T

tt

u

uETf

TE1

10

ˆ ψ

and

( )[ ] ( ) ( )[ ] ( )

T

1

10

10

ˆ ⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

=−−

=

−

∑ TfuVar

TfTVar

u

T

tt

u

11T

tt

11 D

xxD

bb ψ

( ) ( )T1

21

41

01 1

1Ttt

11 DxxD −

=

−⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

T

tu Tf

Since TDD = ,

( )[ ]( )204

ˆuf

TVar1

101

1 DDDbb

−−

=−

123

These terms form

(A2.14) ( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡→−

−−

204,ˆ

ufNT

110

11 DDD

0bb

For the parameter of conditional volatility, the Bahadur representation forms

(A2.15) ( ) ( )( ) ( )( ) ( )12/112/1

ˆ1

11 p

T

tu

uu

oFuTFf

T +−=− ∑=

−−

−

ψt1

1 zGθθ

( ) ( ) ( )1111 1

p

T

tu

ouTf

+−= ∑=

−

ψt1

1 zG

By substituting (A2.10) into (A2.15), we may define

(A2.16) ( ) ( ) ( )[ ] ( ) ( )11111

ˆ1

p

T

tuu

ouTff

T +−−+

=− ∑=

−

ψt1

1 zGθθ

where, ∑=

=T

trtT 1

1σ

Ttt

rzz

G , for 1,0=r . From equation (A2.16), we might derive

( )[ ] ( ) ( )[ ] ( )[ ] 0zGθθ t

11 =−

−+=− ∑

=

− T

tuu

uETff

TE1

1111

ˆ ψ

and

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ]

T

1

111

1111

ˆ⎭⎬⎫

⎩⎨⎧

−+⎭⎬⎫

⎩⎨⎧

−⎭⎬⎫

⎩⎨⎧

−+=−

−

=

−

∑ TffuVar

TffTVar

uu

T

tuu

11T

tt

11 GzzGθθ ψ

( ) ( )[ ] ( )T1

21

41

111 1

1Ttt

11 GzzG −

=

− ⎟⎠

⎞⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧

−+= ∑

T

tuu Tff

Since TGG = ,

( )[ ]( ) ( )[ ]2114

ˆ−+

=−−−

uu ffTVar

110

11 GGG

θθ

124

These terms form

(A2.17) ( )( ) ( )[ ] ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

−+→−

−−

2114,ˆ

uu ffNT

110

11 GGG

0θθ

An exhaustive derivation of Bahadur representation has been done by Koenker and Zhao

(1996), and, we do not derive it detail in this thesis.

125

APPENDIX 3

Programming language in S–PLUS for simulation study

Appendix 3.1 – Estimating of Computing k values by setting c = 1.960 and 2.456.

#SIMULATION OF COMPUTING VALUES OF l BY SETTING c = 1.96 to 4. saiz_1000 #size to be considered in simulation are 50, 100, 200, 500, 1000, 10000 and

50000 rep_10000 #replications to be considered in simulation are 1000, 5000 and 10000 c1_1.96 c2_2.5 c3_3 c4_3.5 c5_4 kurt1_kurt2_kurt3_kurt4_kurt5_kurt6_kurt7_kurt8_kurt9_kurt10_NULL for(i in 1:rep){ MEAN_runif(1,-50,50) STD_runif(1,0.1,100) x_rnorm(saiz,MEAN,STD) n_length(x) s_mad(x) m_median(x) J_mean(abs(x-median(x)))*sqrt(pi/2) u_(x-mean(x))/stdev(x) U_abs(u) sgn_sign(u) w1_u w1[U>c1]_c1*sgn[U>c1] w2_u w2[U>c2]_c2*sgn[U>c2] w3_u w3[U>c3]_c3*sgn[U>c3] w4_u w4[U>c4]_c4*sgn[U>c4] w5_u

126

w5[U>c5]_c5*sgn[U>c5] kurt1_append(kurt1,mean((w1)^4)) kurt2_append(kurt2,mean((w2)^4)) kurt3_append(kurt3,mean((w3)^4)) kurt4_append(kurt4,mean((w4)^4)) kurt5_append(kurt5,mean((w5)^4)) u_(x-mean(x))/J U_abs(u) sgn_sign(u) w1_u w1[U>c1]_c1*sgn[U>c1] w2_u w2[U>c2]_c2*sgn[U>c2] w3_u w3[U>c3]_c3*sgn[U>c3] w4_u w4[U>c4]_c4*sgn[U>c4] w5_u w5[U>c5]_c5*sgn[U>c5] kurt6_append(kurt6,mean((w1)^4)) kurt7_append(kurt7,mean((w2)^4)) kurt8_append(kurt8,mean((w3)^4)) kurt9_append(kurt9,mean((w4)^4)) kurt10_append(kurt10,mean((w5)^4)) } c(mean(kurt1),median(kurt1)) c(mean(kurt2),median(kurt2)) c(mean(kurt3),median(kurt3)) c(mean(kurt4),median(kurt4)) c(mean(kurt5),median(kurt5)) c(mean(kurt6),median(kurt6)) c(mean(kurt7),median(kurt7)) c(mean(kurt8),median(kurt8)) c(mean(kurt9),median(kurt9)) c(mean(kurt10),median(kurt10)) hist(kurt1) hist(kurt2) hist(kurt3) hist(kurt4) hist(kurt5) hist(kurt6) hist(kurt7) hist(kurt8) hist(kurt9)

127

hist(kurt10) Table A3.1 Average and Median of values of 1l when 96.1=c .

96.1=c 50 100 200 500 1000 10000 Mean 1.974012 2.001004 2.018417 2.017295 2.018332 2.0209231000

Median 1.983183 2.008642 2.019907 2.017074 2.01878 2.02063Mean 1.979157 2.004906 2.012345 2.016513 2.018018 5000

Median 1.98506 2.012732 2.01497 2.017932 2.018809 Mean 1.981397 2.004623 2.011803 2.016263 2.018836 10000

Median 1.987031 2.00834 2.014532 2.016982 2.018998

Table A3.2 Average and Median of values of 1l when 50.2=c . 50.2=c 50 100 200 500 1000 10000

Mean 2.5151 2.587601 2.608227 2.622233 2.636669 2.6373291000 Median 2.504611 2.586412 2.602309 2.622791 2.636151 2.637269Mean 2.525419 2.585574 2.615646 2.626478 2.6328 5000

Median 2.515822 2.582672 2.612337 2.622115 2.631425 Mean 2.524936 2.58586 2.607276 2.628765 2.631483 10000

Median 2.512004 2.579194 2.60605 2.628479 2.631327

Table A3.3 Average and Median of values of 1l when 00.3=c . 00.3=c 50 100 200 500 1000 10000

Mean 2.71572 2.81812 2.835115 2.870192 2.888642 2.8911271000 Median 2.648491 2.800323 2.825109 2.862677 2.88214 2.88898Mean 2.7142 2.806102 2.852407 2.873423 2.884629 5000

Median 2.656527 2.783471 2.846611 2.869204 2.883372 Mean 2.715763 2.805064 2.842642 2.876688 2.881795 10000

Median 2.666547 2.785176 2.832332 2.872525 2.880127

Table A3.4 Average and Median of values of 1l when 50.3=c . 50.3=c 50 100 200 500 1000 10000

Mean 2.773802 2.879308 2.901112 2.950698 2.971517 2.9745781000 Median 2.648491 2.834585 2.874775 2.93503 2.963375 2.973654Mean 2.757603 2.869002 2.923273 2.95274 2.965814 5000

Median 2.657421 2.809338 2.90004 2.94839 2.959256 Mean 2.763721 2.864948 2.915045 2.956373 2.96293 10000

Median 2.66719 2.809273 2.889496 2.947908 2.958547

128

Table A3.5 Average and Median of values of 1l when 00.4=c . 00.4=c 50 100 200 500 1000 10000

Mean 2.785437 2.890296 2.91116 2.970243 2.993012 2.9947491000 Median 2.648491 2.834585 2.875661 2.949075 2.985007 2.992591Mean 2.763672 2.88074 2.938228 2.9717 2.984355 5000

Median 2.657421 2.809338 2.903095 2.959474 2.974352 Mean 2.77098 2.875815 2.93066 2.975064 2.98182 10000

Median 2.66719 2.809273 2.892517 2.959003 2.973754

Table A3.6 Average and Median of values of 2l when 96.1=c . 96.1=c 50 100 200 500 1000 10000

Mean 2.014575 2.02245 2.02571 2.021614 2.02358 2.0216761000 Median 2.01039 2.023862 2.021347 2.02013 2.022717 2.021675Mean 2.019586 2.027436 2.024245 2.020328 2.020362 5000

Median 2.015039 2.029174 2.024184 2.020905 2.018977 Mean 2.023573 2.026422 2.020389 2.021296 2.020345 10000

Median 2.016043 2.023096 2.016749 2.020254 2.020026

Table A3.7 Average and Median of values of 2l when 50.2=c . 50.2=c 50 100 200 500 1000 10000

Mean 2.612904 2.643291 2.629665 2.632894 2.647538 2.6387721000 Median 2.581369 2.628722 2.61586 2.631828 2.647844 2.639701Mean 2.627282 2.640012 2.644523 2.636266 2.638392 5000

Median 2.582222 2.629229 2.63444 2.634038 2.635474 Mean 2.629373 2.64021 2.630121 2.640556 2.635783 10000

Median 2.594509 2.622581 2.623178 2.637005 2.635232

Table A3.8 Average and Median of values of 2l when 00.3=c . 00.3=c 50 100 200 500 1000 10000

Mean 2.876375 2.909858 2.872307 2.887598 2.904325 2.8931131000 Median 2.74281 2.891633 2.855706 2.877538 2.8945 2.890786Mean 2.880012 2.892498 2.898727 2.889904 2.893518 5000

Median 2.743671 2.848699 2.87299 2.878706 2.88789 Mean 2.884764 2.892774 2.881023 2.895526 2.889229 10000

Median 2.771059 2.835475 2.856467 2.883461 2.88444

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Table A3.9 Average and Median of values of 2l when 50.3=c . 50.3=c 50 100 200 500 1000 10000

Mean 2.981897 2.994126 2.948814 2.973565 2.990503 2.9769071000 Median 2.74281 2.922686 2.907913 2.942711 2.981009 2.973449Mean 2.963944 2.977675 2.981741 2.973837 2.97709 5000

Median 2.744521 2.874159 2.935225 2.95668 2.966446 Mean 2.973732 2.974426 2.964404 2.979956 2.972552 10000

Median 2.771875 2.867004 2.914161 2.962548 2.96198

Table A3.10 Average and Median of values of 2l when 00.4=c . 00.4=c 50 100 200 500 1000 10000

Mean 3.014246 3.014624 2.962127 2.995745 3.013182 2.9972281000 Median 2.74281 2.922686 2.910967 2.958009 3.001618 2.993648Mean 2.983864 2.998818 3.001486 2.994919 2.996737 5000

Median 2.744521 2.874159 2.940488 2.971197 2.981951 Mean 2.99972 2.993271 2.98474 3.000995 2.992599 10000

Median 2.771875 2.867211 2.917017 2.974082 2.977642

130

Appendix 3.2 – Computing Asymptotic variance of Skewness and Kurtosis

#SIMULATION OF ESTIMATING ASYMPTOTIC VARIANCE OF ROBUST SKEWNESS AND KURTOSIS

saiz_100 rep_10000 c1_1.96 c2_2.5 c3_3 c4_3.5 c5_4 skew1_skew2_skew3_skew4_skew5_skew6_skew7_skew8_skew9_skew10_skew11_skew

12_skew13_skew14_skew15_ NULL kurt1_kurt2_kurt3_kurt4_kurt5_kurt6_kurt7_kurt8_kurt9_kurt10_kurt11_kurt12_kurt13_k

urt14_ NULL for(i in 1:rep){ MEAN_runif(1,-50,50) STD_runif(1,0.1,100) x_rnorm(saiz,MEAN,STD) n_length(x) s_mad(x) m_median(x) J_mean(abs(x-median(x)))*sqrt(pi/2) Q_matrix(,4,1) for(j in 1:4){ Q[j]_quantile(x, probs = j/4) } E_matrix(,8,1) for(j in 1:8){ E[j]_quantile(x, probs = j/8) } sk1_skewness(x,method="moment") sk2_mean((x-mean(x))^3)/(J^3) sk3_(Q[3]+Q[1]-2*Q[2])/(Q[3]-Q[1]) sk4_(mean(x)-Q[2])/mean(abs(x-Q[2])) sk5_(mean(x)-Q[2])/sqrt(var(x)) kr1_kurtosis(x,method="moment") kr2_mean((x-mean(x))^4)/(J^4)-3 kr3_((E[7]-E[5])+(E[3]-E[1]))/(E[6]-E[2])-1.23 kr4_(quantile(x, probs = .975)-quantile(x, probs = .025))/(quantile(x, probs = .75)-

quantile(x, probs = .25))-2.91 u_(x-mean(x))/stdev(x) U_abs(u)

131

sgn_sign(u) w1_u w1[U>c1]_c1*sgn[U>c1] w2_u w2[U>c2]_c2*sgn[U>c2] w3_u w3[U>c3]_c3*sgn[U>c3] w4_u w4[U>c4]_c4*sgn[U>c4] w5_u w5[U>c5]_c5*sgn[U>c5] sk6_mean(w1^3) sk7_mean(w2^3) sk8_mean(w3^3) sk9_mean(w4^3) sk10_mean(w5^3) kr5_mean((w1)^4)-2.02 kr6_mean((w2)^4)-2.64 kr7_mean((w3)^4)-2.89 kr8_mean((w4)^4)-2.97 kr9_mean((w5)^4)-2.99 u_(x-mean(x))/J U_abs(u) sgn_sign(u) w1_u w1[U>c1]_c1*sgn[U>c1] w2_u w2[U>c2]_c2*sgn[U>c2] w3_u w3[U>c3]_c3*sgn[U>c3] w4_u w4[U>c4]_c4*sgn[U>c4] w5_u w5[U>c5]_c5*sgn[U>c5] sk11_mean(w1^3) sk12_mean(w2^3) sk13_mean(w3^3) sk14_mean(w4^3) sk15_mean(w5^3) kr10_mean((w1)^4)-2.02 kr11_mean((w2)^4)-2.64 kr12_mean((w3)^4)-2.89 kr13_mean((w4)^4)-2.98 kr14_mean((w5)^4)-3 skew1_append(skew1,sqrt(saiz)*sk1)

132

skew2_append(skew2,sqrt(saiz)*sk2) skew3_append(skew3,sqrt(saiz)*sk3) skew4_append(skew4,sqrt(saiz)*sk4) skew5_append(skew5,sqrt(saiz)*sk5) skew6_append(skew6,sqrt(saiz)*sk6) skew7_append(skew7,sqrt(saiz)*sk7) skew8_append(skew8,sqrt(saiz)*sk8) skew9_append(skew9,sqrt(saiz)*sk9) skew10_append(skew10,sqrt(saiz)*sk10) skew11_append(skew11,sqrt(saiz)*sk11) skew12_append(skew12,sqrt(saiz)*sk12) skew13_append(skew13,sqrt(saiz)*sk13) skew14_append(skew14,sqrt(saiz)*sk14) skew15_append(skew15,sqrt(saiz)*sk15) kurt1_append(kurt1,sqrt(saiz)*kr1) kurt2_append(kurt2,sqrt(saiz)*kr2) kurt3_append(kurt3,sqrt(saiz)*kr3) kurt4_append(kurt4,sqrt(saiz)*kr4) kurt5_append(kurt5,sqrt(saiz)*kr5) kurt6_append(kurt6,sqrt(saiz)*kr6) kurt7_append(kurt7,sqrt(saiz)*kr7) kurt8_append(kurt8,sqrt(saiz)*kr8) kurt9_append(kurt9,sqrt(saiz)*kr9) kurt10_append(kurt10,sqrt(saiz)*kr10) kurt11_append(kurt11,sqrt(saiz)*kr11) kurt12_append(kurt12,sqrt(saiz)*kr12) kurt13_append(kurt13,sqrt(saiz)*kr13) kurt14_append(kurt14,sqrt(saiz)*kr14) } c(mean(skew1),mean(skew2),mean(skew3),mean(skew4),mean(skew5),mean(skew6),mean

(skew7),mean(skew8),mean(skew9),mean(skew10),mean(skew11),mean(skew12),mean(skew13),mean(skew14),mean(skew15))

c(mean(kurt1),mean(kurt2),mean(kurt3),mean(kurt4),mean(kurt5),mean(kurt6),mean(kurt7),mean(kurt8),mean(kurt9),mean(kurt10),mean(kurt11),mean(kurt12),mean(kurt13),mean(kurt14))

c(var(skew1),var(skew2),var(skew3),var(skew4),var(skew5),var(skew6),var(skew7),var(sk

ew8),var(skew9),var(skew10),var(skew11),var(skew12),var(skew13),var(skew14),var(skew15))

c(var(kurt1),var(kurt2),var(kurt3),var(kurt4),var(kurt5),var(kurt6),var(kurt7),var(kurt8),var(kurt9),var(kurt10),var(kurt11),var(kurt12),var(kurt13),var(kurt14))

c(cor(skew1,kurt1),cor(skew2,kurt2),cor(skew3,kurt3),cor(skew4,kurt3),cor(skew5,kurt3),c

or(skew3,kurt4),cor(skew4,kurt4),cor(skew5,kurt4),cor(skew6,kurt5),cor(skew7,kurt6),cor(skew8,kurt7),cor(skew9,kurt8),cor(skew10,kurt9),cor(skew11,kurt10),cor(skew12,kurt11),cor(skew13,kurt12),cor(skew14,kurt13),cor(skew15,kurt14))

133

Table A3.11 Estimated mean for skewness and kurtosis using 10,000 simulated standard Normal distribution.

Size, T 50 100 200 500 1000

1SKT 0.014 -0.011 -0.017 0.014 0.031

2SKT 0.015 -0.010 -0.017 0.015 0.032

3SKT 0.006 -0.008 0.006 0.018 0.013

4SKT -0.002 -0.008 0.004 0.008 0.017

5SKT -0.002 -0.007 0.003 0.006 0.013

6SKT 0.001 0.001 0.012 -0.013 0.020

7SKT 0.003 -0.009 -0.002 -0.016 0.032

8SKT 0.006 -0.010 -0.014 0.001 0.034

9SKT 0.013 -0.010 -0.015 0.010 0.033

10SKT 0.014 -0.011 -0.016 0.015 0.031

11SKT 0.000 0.002 0.012 -0.013 0.020

12SKT 0.002 -0.009 -0.001 -0.016 0.032

13SKT 0.005 -0.011 -0.013 0.001 0.035

14SKT 0.012 -0.009 -0.015 0.010 0.033

15SKT 0.016 -0.009 -0.016 0.015 0.032

1KRT -0.773 -0.588 -0.445 -0.285 -0.138

2KRT 0.138 0.088 -0.005 0.010 0.100

3KRT 0.082 0.090 0.064 0.076 0.110

4KRT -0.624 -0.503 -0.464 -0.383 -0.313

5KRT -0.249 -0.153 -0.121 -0.050 -0.051

6KRT -0.740 -0.538 -0.388 -0.295 -0.220

7KRT -1.156 -0.830 -0.585 -0.367 -0.172

8KRT -1.421 -1.000 -0.702 -0.376 -0.119

9KRT -1.516 -1.082 -0.766 -0.406 -0.138

10KRT 0.078 0.075 0.023 0.049 0.028

11KRT 0.056 0.019 -0.019 -0.052 -0.032

12KRT 0.106 0.066 0.017 0.024 0.129

13KRT 0.069 0.032 -0.083 -0.103 -0.057

14KRT 0.089 0.048 -0.076 -0.086 -0.041

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Table A3.12 Estimated correlation between skewness and kurtosis Size, T 50 100 200 500 1000

( )11, KRTKRTcor 0.010 0.006 -0.008 0.011 0.011

( )22 , KRTSKTcor 0.009 0.008 -0.003 0.009 0.014

( )33 , KRTSKTcor 0.016 0.003 -0.005 0.007 -0.003

( )34 , KRTSKTcor 0.019 0.003 0.004 -0.002 0.008

( )35 , KRTSKTcor 0.019 0.004 0.004 -0.002 0.008

( )43, KRTSKTcor 0.004 0.001 -0.011 0.002 0.004

( )44 , KRTSKTcor 0.009 0.000 0.000 -0.004 0.010

( )45 , KRTSKTcor 0.009 0.000 0.000 -0.004 0.010

( )56 , KRTSKTcor 0.010 -0.010 0.028 -0.010 0.012

( )67 , KRTSKTcor -0.011 0.000 -0.007 0.000 -0.002

( )78 , KRTSKTcor -0.004 0.009 -0.004 0.006 0.010

( )89 , KRTSKTcor 0.012 0.009 -0.006 0.008 0.010

( )910 , KRTSKTcor 0.015 0.005 -0.008 0.014 0.010

( )1011 , KRTSKTcor 0.003 -0.001 0.024 -0.002 0.018

( )1112 , KRTSKTcor -0.010 -0.002 0.000 0.002 0.007

( )1213 , KRTSKTcor -0.005 0.008 0.000 0.005 0.012

( )1314 , KRTSKTcor 0.005 0.014 0.000 0.007 0.012

( )1415 , KRTSKTcor 0.016 0.012 -0.002 0.011 0.014

135

Appendix 3.3 – Simulation of Setting up 5% critical region of normality test #SIMULATION OF SETTING UP CRITICAL REGION AT 5% rep_10000 c1_1.96 c2_2.5 c3_3 c4_3.5 c5_4 C1_6 C2_6 C3_1.8 C4_.9 C5_.6 C6_1.2 C7_3 C8_4.6 C9_5.5 C10_5.7 C11_1.2 C12_3.1 C13_4.7 C14_5.6 C15_5.9 D1_24 D2_60 D3_3.1 D4_15 D5_1.87 D6_5 D7_11.5 D8_18.5 D9_22 D10_3.8 D11_17 D12_35 D13_50 D14_56 critical_function(saiz){ JB1_JB2_JB3_JB4_JB5_JB6_JB7_JB8_JB9_JB10_JB11_JB12_JB13_JB14_JB15_JB16_J

B17_JB18_ NULL saiz=saiz for(i in 1:rep){ MEAN_runif(1,-50,50)

136

STD_runif(1,0.1,100) x_rnorm(saiz,MEAN,STD) n_length(x) s_mad(x) m_median(x) J_mean(abs(x-median(x)))*sqrt(pi/2) Q_matrix(,4,1) for(j in 1:4){ Q[j]_quantile(x, probs = j/4) } E_matrix(,8,1) for(j in 1:8){ E[j]_quantile(x, probs = j/8) } sk1_skewness(x,method="moment") sk2_mean((x-mean(x))^3)/(J^3) sk3_(Q[3]+Q[1]-2*Q[2])/(Q[3]-Q[1]) sk4_(mean(x)-Q[2])/mean(abs(x-Q[2])) sk5_(mean(x)-Q[2])/sqrt(var(x)) kr1_kurtosis(x,method="moment") kr2_mean((x-mean(x))^4)/(J^4)-3 kr3_((E[7]-E[5])+(E[3]-E[1]))/(E[6]-E[2])-1.23 kr4_(quantile(x, probs = .975)-quantile(x, probs = .025))/(quantile(x, probs = .75)-

quantile(x, probs = .25))-2.91 u_(x-mean(x))/stdev(x) U_abs(u) sgn_sign(u) w1_u w1[U>c1]_c1*sgn[U>c1] w2_u w2[U>c2]_c2*sgn[U>c2] w3_u w3[U>c3]_c3*sgn[U>c3] w4_u w4[U>c4]_c4*sgn[U>c4] w5_u w5[U>c5]_c5*sgn[U>c5] sk6_mean(w1^3) sk7_mean(w2^3) sk8_mean(w3^3) sk9_mean(w4^3) sk10_mean(w5^3) kr5_mean((w1)^4)-2.02 kr6_mean((w2)^4)-2.64 kr7_mean((w3)^4)-2.89

137

kr8_mean((w4)^4)-2.97 kr9_mean((w5)^4)-2.99 u_(x-mean(x))/J U_abs(u) sgn_sign(u) w1_u w1[U>c1]_c1*sgn[U>c1] w2_u w2[U>c2]_c2*sgn[U>c2] w3_u w3[U>c3]_c3*sgn[U>c3] w4_u w4[U>c4]_c4*sgn[U>c4] w5_u w5[U>c5]_c5*sgn[U>c5] sk11_mean(w1^3) sk12_mean(w2^3) sk13_mean(w3^3) sk14_mean(w4^3) sk15_mean(w5^3) kr10_mean((w1)^4)-2.02 kr11_mean((w2)^4)-2.64 kr12_mean((w3)^4)-2.89 kr13_mean((w4)^4)-2.98 kr14_mean((w5)^4)-3 jb1_n/C1*(sk1^2)+n/D1*(kr1^2) jb2_n/C2*(sk2^2)+n/D2*(kr2^2) jb3_n/C3*(sk3^2)+n/D3*(kr3^2) jb4_n/C4*(sk4^2)+n/D3*(kr3^2) jb5_n/C5*(sk5^2)+n/D3*(kr3^2) jb6_n/C3*(sk3^2)+n/D4*(kr4^2) jb7_n/C4*(sk4^2)+n/D4*(kr4^2) jb8_n/C5*(sk5^2)+n/D4*(kr4^2) jb9_n/C6*(sk6^2)+n/D5*(kr5^2) jb10_n/C7*(sk7^2)+n/D6*(kr6^2) jb11_n/C8*(sk8^2)+n/D7*(kr7^2) jb12_n/C9*(sk9^2)+n/D8*(kr8^2) jb13_n/C10*(sk10^2)+n/D9*(kr9^2) jb14_n/C11*(sk11^2)+n/D10*(kr10^2) jb15_n/C12*(sk12^2)+n/D11*(kr11^2) jb16_n/C13*(sk13^2)+n/D12*(kr12^2) jb17_n/C14*(sk14^2)+n/D13*(kr13^2) jb18_n/C15*(sk15^2)+n/D14*(kr14^2) JB1_append(JB1,jb1) JB2_append(JB2,jb2)

138

JB3_append(JB3,jb3) JB4_append(JB4,jb4) JB5_append(JB5,jb5) JB6_append(JB6,jb6) JB7_append(JB7,jb7) JB8_append(JB8,jb8) JB9_append(JB9,jb9) JB10_append(JB10,jb10) JB11_append(JB11,jb11) JB12_append(JB12,jb12) JB13_append(JB13,jb13) JB14_append(JB14,jb14) JB15_append(JB15,jb15) JB16_append(JB16,jb16) JB17_append(JB17,jb17) JB18_append(JB18,jb18) } JB1_matrix(sort(JB1),rep,1) JB2_matrix(sort(JB2),rep,1) JB3_matrix(sort(JB3),rep,1) JB4_matrix(sort(JB4),rep,1) JB5_matrix(sort(JB5),rep,1) JB6_matrix(sort(JB6),rep,1) JB7_matrix(sort(JB7),rep,1) JB8_matrix(sort(JB8),rep,1) JB9_matrix(sort(JB9),rep,1) JB10_matrix(sort(JB10),rep,1) JB11_matrix(sort(JB11),rep,1) JB12_matrix(sort(JB12),rep,1) JB13_matrix(sort(JB13),rep,1) JB14_matrix(sort(JB14),rep,1) JB15_matrix(sort(JB15),rep,1) JB16_matrix(sort(JB16),rep,1) JB17_matrix(sort(JB17),rep,1) JB18_matrix(sort(JB18),rep,1) list(crit=c(JB1[0.95*rep],JB2[0.95*rep],JB3[0.95*rep],JB4[0.95*rep],JB5[0.95*rep],JB

6[0.95*rep],JB7[0.95*rep],JB8[0.95*rep],JB9[0.95*rep],JB10[0.95*rep],JB11[0.95*rep],JB12[0.95*rep],JB13[0.95*rep],JB14[0.95*rep],JB15[0.95*rep],JB16[0.95*rep],JB17[0.95*rep],JB18[0.95*rep]))

} crit30_critical(saiz=30) crit50_critical(saiz=50) crit100_critical(saiz=100) crit200_critical(saiz=200) crit30$crit crit50$crit

139

crit100$crit crit200$crit

140

Appendix 3.4 – Power of Normality Rejection at 5% #SIMULATION OF COMPUTING POWER OF REJECTION AT 5% CRITICAL

REGION rep_10000 c1_1.96 c2_2.5 c3_3 c4_3.5 c5_4 c1_1.96 c2_2.5 c3_3 c4_3.5 c5_4 C1_6 C2_6 C3_1.8 C4_.9 C5_.6 C6_1.2 C7_3 C8_4.6 C9_5.5 C10_5.7 C11_1.2 C12_3.1 C13_4.7 C14_5.6 C15_5.9 D1_24 D2_60 D3_3.1 D4_15 D5_1.87 D6_5 D7_11.5 D8_18.5 D9_22 D10_3.8 D11_17 D12_35 D13_50 D14_56

141

alpha_.05 crit30_c(4.373734, 7.078235, 5.757927, 5.856534, 5.637705, 5.084975, 5.252559,

5.046551, 5.559391, 5.376404, 4.908059, 4.081077, 3.792516, 6.085139, 6.078918, 6.858621, 8.077855, 7.357841)

crit50_c(4.749200, 6.662342, 6.021109, 5.976240, 5.796864, 5.304477, 5.390281, 5.209217, 5.677982, 5.523863, 5.010710, 4.866857, 4.424410, 5.989329, 5.747696, 6.458764, 7.092843, 6.921381)

crit100_c(5.435817, 6.790764, 5.971975, 5.991599, 5.826622, 5.634947, 5.518572, 5.345708, 5.891736, 5.845516, 5.475450, 5.406109, 5.414771, 6.113067, 5.887316, 6.050292, 6.610210, 6.997644)

crit200_c(5.622800, 6.184989, 6.105504, 5.919023, 5.793940, 5.878129, 5.774319, 5.637825, 5.965581, 5.916134, 5.702797, 5.466362, 5.692495, 5.959128, 5.902721, 5.888013, 5.910931, 6.240791)

pow_function(saiz,crit){ JB1_JB2_JB3_JB4_JB5_JB6_JB7_JB8_JB9_JB10_JB11_JB12_JB13_JB14_JB15_JB

16_JB17_JB18_NULL saiz=saiz crit=crit for(i in 1:rep){ # x_rnorm(saiz,0,1) # x_rt(saiz,3) # x_rt(saiz,10) # x_rlogis(saiz,0,1) # x_rexp(100,rate=1/2) # x_rchisq(saiz,2) # x_rlnorm(saiz, meanlog=0, sdlog=1) # {x_rnorm(saiz,0,1) # v_runif(saiz,0,1) # for(j in 1:length(x)) if(v[j]<alpha)x[j]_rnorm(1,0,5) # } # {x_rnorm(saiz,0,1) # v_runif(saiz,0,1) # for(j in 1:length(x)) if(v[j]<alpha)x[j]_rexp(1,rate=1/2) # } # {x_rnorm(saiz,0,1) # v_runif(saiz,0,1) # for(j in 1:length(x)) if(v[j]<alpha)x[j]_rchisq(1,2) # } {x_rnorm(saiz,0,1) v_runif(saiz,0,1) for(j in 1:length(x)) if(v[j]<alpha)x[j]_rlnorm(1, meanlog=0, sdlog=1) }

142

n_length(x) s_mad(x) m_median(x) J_mean(abs(x-median(x)))*sqrt(pi/2) Q_matrix(,4,1) for(j in 1:4){ Q[j]_quantile(x, probs = j/4) } E_matrix(,8,1) for(j in 1:8){ E[j]_quantile(x, probs = j/8) } sk1_skewness(x,method="moment") sk2_mean((x-mean(x))^3)/(J^3) sk3_(Q[3]+Q[1]-2*Q[2])/(Q[3]-Q[1]) sk4_(mean(x)-Q[2])/mean(abs(x-Q[2])) sk5_(mean(x)-Q[2])/sqrt(var(x)) kr1_kurtosis(x,method="moment") kr2_mean((x-mean(x))^4)/(J^4)-3 kr3_((E[7]-E[5])+(E[3]-E[1]))/(E[6]-E[2])-1.23 kr4_(quantile(x, probs = .975)-quantile(x, probs = .025))/(quantile(x, probs = .75)-

quantile(x, probs = .25))-2.91 u_(x-mean(x))/stdev(x) U_abs(u) sgn_sign(u) w1_u w1[U>c1]_c1*sgn[U>c1] w2_u w2[U>c2]_c2*sgn[U>c2] w3_u w3[U>c3]_c3*sgn[U>c3] w4_u w4[U>c4]_c4*sgn[U>c4] w5_u w5[U>c5]_c5*sgn[U>c5] sk6_mean(w1^3) sk7_mean(w2^3) sk8_mean(w3^3) sk9_mean(w4^3) sk10_mean(w5^3) kr5_mean((w1)^4)-2.02 kr6_mean((w2)^4)-2.64 kr7_mean((w3)^4)-2.89 kr8_mean((w4)^4)-2.97 kr9_mean((w5)^4)-2.99

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u_(x-mean(x))/J U_abs(u) sgn_sign(u) w1_u w1[U>c1]_c1*sgn[U>c1] w2_u w2[U>c2]_c2*sgn[U>c2] w3_u w3[U>c3]_c3*sgn[U>c3] w4_u w4[U>c4]_c4*sgn[U>c4] w5_u w5[U>c5]_c5*sgn[U>c5] sk11_mean(w1^3) sk12_mean(w2^3) sk13_mean(w3^3) sk14_mean(w4^3) sk15_mean(w5^3) kr10_mean((w1)^4)-2.02 kr11_mean((w2)^4)-2.64 kr12_mean((w3)^4)-2.89 kr13_mean((w4)^4)-2.98 kr14_mean((w5)^4)-3 jb1_n/C1*(sk1^2)+n/D1*(kr1^2) jb2_n/C2*(sk2^2)+n/D2*(kr2^2) jb3_n/C3*(sk3^2)+n/D3*(kr3^2) jb4_n/C4*(sk4^2)+n/D3*(kr3^2) jb5_n/C5*(sk5^2)+n/D3*(kr3^2) jb6_n/C3*(sk3^2)+n/D4*(kr4^2) jb7_n/C4*(sk4^2)+n/D4*(kr4^2) jb8_n/C5*(sk5^2)+n/D4*(kr4^2) jb9_n/C6*(sk6^2)+n/D5*(kr5^2) jb10_n/C7*(sk7^2)+n/D6*(kr6^2) jb11_n/C8*(sk8^2)+n/D7*(kr7^2) jb12_n/C9*(sk9^2)+n/D8*(kr8^2) jb13_n/C10*(sk10^2)+n/D9*(kr9^2) jb14_n/C11*(sk11^2)+n/D10*(kr10^2) jb15_n/C12*(sk12^2)+n/D11*(kr11^2) jb16_n/C13*(sk13^2)+n/D12*(kr12^2) jb17_n/C14*(sk14^2)+n/D13*(kr13^2) jb18_n/C15*(sk15^2)+n/D14*(kr14^2) JB1_append(JB1,jb1) JB2_append(JB2,jb2) JB3_append(JB3,jb3) JB4_append(JB4,jb4)

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JB5_append(JB5,jb5) JB6_append(JB6,jb6) JB7_append(JB7,jb7) JB8_append(JB8,jb8) JB9_append(JB9,jb9) JB10_append(JB10,jb10) JB11_append(JB11,jb11) JB12_append(JB12,jb12) JB13_append(JB13,jb13) JB14_append(JB14,jb14) JB15_append(JB15,jb15) JB16_append(JB16,jb16) JB17_append(JB17,jb17) JB18_append(JB18,jb18) } JB1_matrix(JB1,rep,1) JB2_matrix(JB2,rep,1) JB3_matrix(JB3,rep,1) JB4_matrix(JB4,rep,1) JB5_matrix(JB5,rep,1) JB6_matrix(JB6,rep,1) JB7_matrix(JB7,rep,1) JB8_matrix(JB8,rep,1) JB9_matrix(JB9,rep,1) JB10_matrix(JB10,rep,1) JB11_matrix(JB11,rep,1) JB12_matrix(JB12,rep,1) JB13_matrix(JB13,rep,1) JB14_matrix(JB14,rep,1) JB15_matrix(JB15,rep,1) JB16_matrix(JB16,rep,1) JB17_matrix(JB17,rep,1) JB18_matrix(JB18,rep,1) power1_length(JB1[JB1>crit[1]])/rep power2_length(JB2[JB2>crit[2]])/rep power3_length(JB3[JB3>crit[3]])/rep power4_length(JB4[JB4>crit[4]])/rep power5_length(JB5[JB5>crit[5]])/rep power6_length(JB6[JB6>crit[6]])/rep power7_length(JB7[JB7>crit[7]])/rep power8_length(JB8[JB8>crit[8]])/rep power9_length(JB9[JB9>crit[9]])/rep power10_length(JB10[JB10>crit[10]])/rep power11_length(JB11[JB11>crit[11]])/rep power12_length(JB12[JB12>crit[12]])/rep power13_length(JB13[JB13>crit[13]])/rep power14_length(JB14[JB14>crit[14]])/rep power15_length(JB15[JB15>crit[15]])/rep

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power16_length(JB16[JB16>crit[16]])/rep power17_length(JB17[JB17>crit[17]])/rep power18_length(JB18[JB18>crit[18]])/rep list(power=c(power1,power2,power3,power4,power5,power6,power7,power8,power9,p

ower10,power11,power12,power13,power14,power15,power16,power17,power18)) } pow30_pow(saiz=30,crit=crit30) pow50_pow(saiz=50,crit=crit50) pow100_pow(saiz=100,crit=crit100) pow200_pow(saiz=200,crit=crit200) pow30$power pow50$power pow100$power pow200$power

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Appendix 3.5 – Estimating Parameters of LAD AR(1)–ARCH(2) model module(finmetrics) q_2 p_0 k_1 saiz_400 cont_0.00 rep_50 model_1 dist_1 iteration_50 coef.true_c(1,0.5,1,0.5,0.2) #Generating Robust-AR-GARCH data rgarch_function(dist,model,saiz,b0,b1,alpha0,alpha1,alpha2){ u_rnorm(saiz,0,1) #Normal(0,1) # u_rnorm(saiz,0,1) # if(model==1) u_u/median(abs(u)) else u_u/sqrt(median(u^2)) #if mix

normal with 5% distribution of N(0,5), and # u_rt(saiz,3) #if t-distribution with 3 degrees of freedom if(model==1) u_u/median(abs(u)) else u_u/sqrt(median(u^2)) h1_0 eps2_eps1_0 yold1_0 h_NULL ynew_NULL epss_NULL v_runif(length(u),0,1) for(i in 1:length(u)){ if(model==1){ h1_alpha0+alpha1*abs(eps1)+alpha2*abs(eps2) if(dist=1) epsnew_u[i]*h1 else if(v[i]<cont) epsnew_u[i]*h1+rnorm(1,0,5) } else { h1_alpha0+alpha1*eps1^2+alpha2*eps2^2 if(dist=1) epsnew_u[i]*sqrt(h1) else if(v[i]<cont) epsnew_u[i]*sqrt(h1)+rnorm(1,0,5) } y1_b0+b1*yold1+epsnew eps2_eps1 eps1_epsnew yold1_y1 h_append(h,h1) ynew_append(ynew,y1) epss_append(epss,epsnew)

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} list(ynew=ynew,epss=epss,h=h,u=u) } est.rgarch_function(model,y,k,q,p){ #Generating X matrix xmatrix_function(element,k,n){ x1_as.matrix(rep(1,n)) x2_matrix(0,n,k) for(i in 1:k)x2[(i+1):(n),i]_element[1:(n-i)] x.1_matrix(c(x1,x2),n,1+k) list(x=x.1) } #Generating Z matrix zmatrix_function(element,q,p,teta,n){ z1_as.matrix(rep(1,n)) z2_matrix(mean(element),n,q) for(i in 1:q)z2[(i+1):(n),i]_element[1:(n-i)] z3_matrix(mean(element),n,p) h.tilda_matrix(mean(element),n,1) for(i in (max(q,p)+1):n){ for(j in 1:p){ z3[i,j]_h.tilda[i-j] } h.tilda[i]_c(z1[i,],z2[i,],z3[i,])%*%teta } z.1_matrix(c(z1,z2,z3),n,1+q+p) list(z=z.1,h=h.tilda) } n_length(y) if(k!=0){ x_xmatrix(y,k,n)$x b_matrix(l1fit(x, y, intercept=F)$coefficients,1+k,1) } e_y-x%*%b if(model==1) element_abs(e) else element_e^2 z.tilda_xmatrix(element,q,n)$x a.tilda_matrix(l1fit(z.tilda, element, intercept=F)$coefficients,1+q,1) h.tilda_z.tilda%*%a.tilda z_matrix(,n,1+q) for(i in 1:(1+q)){ z[,i]_z.tilda[,i]/h.tilda }

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a_matrix(l1fit(z, element/h.tilda, intercept=F)$coefficients,1+q,1) list(coef=c(b,a),h=z%*%a,y=y,e=e,z=z) } teta_NULL for(i in 1:rep){ a_rgarch(dist,model,saiz+200,coef.true[1],coef.true[2],coef.true[3],coef.true[4],coef.tru

e[5]) y_a$ynew[201:(saiz+200)] result_est.rgarch(model,y,k,q,p) teta_append(teta,result$coef) } est.teta_matrix(teta,rep,2+k+q+p,byrow=T)

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Appendix 3.6 – Estimating Parameters of LAD AR(1)–GARCH(1,1) (Park (2002)) model module(finmetrics) q_1 p_1 k_1 saiz_400 cont_0.00 rep_500 model_1 dist_1 coef.true_c(1,0.5,1,.5,.2) iteration_50 #Generating Robust-AR-GARCH data rgarch_function(dist,model,saiz,b0,b1,alpha0,alpha1,beta1){ u_rnorm(saiz,0,1) #Normal(0,1) # u_rnorm(saiz,0,1) # if(model==1) u_u/median(abs(u)) else u_u/sqrt(median(u^2)) #if mix

normal with 5% distribution of N(0,5), and # u_rt(saiz,3) #if t-distribution with 3 degrees of freedom if(model==1) u_u/median(abs(u)) else u_u/sqrt(median(u^2)) h1_0 eps1_0 yold1_0 h_NULL ynew_NULL epss_NULL v_runif(length(u),0,1) for(i in 1:length(u)){ if(model==1){ h1_alpha0+alpha1*abs(eps1)+beta1*h1 if(dist=1) epsnew_u[i]*h1 else if(v[i]<cont) epsnew_u[i]*h1+rnorm(1,0,5) } else { h1_alpha0+alpha1*eps1^2+beta1*h1 if(dist=1) epsnew_u[i]*sqrt(h1) else if(v[i]<cont) epsnew_u[i]*sqrt(h1)+rnorm(1,0,5) } y1_b0+b1*yold1+epsnew eps1_epsnew yold1_y1 h_append(h,h1) ynew_append(ynew,y1) epss_append(epss,epsnew)

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} list(ynew=ynew,epss=epss,h=h,u=u) } est.rgarch_function(model,y,k,q,p){ #Generating X matrix xmatrix_function(element,k,n){ x1_as.matrix(rep(1,n)) x2_matrix(0,n,k) for(i in 1:k)x2[(i+1):(n),i]_element[1:(n-i)] x.1_matrix(c(x1,x2),n,1+k) list(x=x.1) } #Generating Z matrix zmatrix_function(element,q,p,teta,n){ z1_as.matrix(rep(1,n)) z2_matrix(mean(element),n,q) for(i in 1:q)z2[(i+1):(n),i]_element[1:(n-i)] z3_matrix(mean(element),n,p) h.tilda_matrix(mean(element),n,1) for(i in (max(q,p)+1):n){ for(j in 1:p){ z3[i,j]_h.tilda[i-j] } h.tilda[i]_c(z1[i,],z2[i,],z3[i,])%*%teta } z.1_matrix(c(z1,z2,z3),n,1+q+p) list(z=z.1,h=h.tilda) } n_length(y) if(k!=0){ x_xmatrix(y,k,n)$x b_matrix(l1fit(x, y, intercept=F)$coefficients,1+k,1) } e_y-x%*%b if(model==1) element_abs(e) else element_e^2 gama_matrix(,1+q+p,1) for(i in 1:100){ gama[2:(1+q+p)]_runif(q+p,0,1) if(sum(gama[2:(1+q+p)])<1) break } gama[1]_runif(1,0,3) zm_zmatrix(element,q,p,gama[1:(1+q+p)],n) z_zm$z

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h_zm$h out_NULL conv_NULL l_NULL for(kk in 1:iteration){ dia_1/h gama1<-

matrix(l1fit(diag(dia,n)%*%z,diag(dia,n)%*%element,F)$coefficients,1+q+p,1) if(any(gama1[2:(1+q+p)]>1)||gama1[1]<0||abs(sum(gama1[2:(1+q+p)]))>1){ gama_rep(NA,1+q+p) l_kk break } convergence_max(abs((gama1-gama)/gama)) gama_gama1 zm_zmatrix(element,q,p,gama[1:(1+q+p)],n) z_zm$z h_zm$h out_append(out,gama) conv_append(conv,convergence) l_kk print(kk) if(convergence<0.0001)break } out_matrix(out,l,1+q+p,byrow=T) list(iter=l,coef=c(b,gama),conv=conv,gama=out,z=z,h=h,res=e) } teta_NULL for(i in 1:rep){ a_rgarch(dist,model,saiz+200,coef.true[1],coef.true[2],coef.true[3],coef.true[4],coef.tru

e[5]) y_a$ynew[201:(saiz+200)] result_est.rgarch(model,y,k,q,p) teta_append(teta,result$coef) } est.teta_matrix(teta,rep,2+k+q+p,byrow=T)

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Appendix 3.7 – Estimating Parameters of LAD AR(1) – GARCH(1,1) (Peng and Yao (2003)) model module(finmetrics) q_1 p_1 k_1 saiz_400 cont_0.00 rep_500 model_2 dist_1 coef.true_c(1,0.5,1,.5,.2) iteration_50 #Generating Robust-AR-GARCH data rgarch_function(dist,model,saiz,b0,b1,alpha0,alpha1,beta1){ u_rnorm(saiz,0,1) #Normal(0,1) # u_rnorm(saiz,0,1) # if(model==1) u_u/median(abs(u)) else u_u/sqrt(median(u^2)) #if mix

normal with 5% distribution of N(0,5), and # u_rt(saiz,3) #if t-distribution with 3 degrees of freedom if(model==1) u_u/median(abs(u)) else u_u/sqrt(median(u^2)) h1_0 eps1_0 yold1_0 h_NULL ynew_NULL epss_NULL v_runif(length(u),0,1) for(i in 1:length(u)){ if(model==1){ h1_alpha0+alpha1*abs(eps1)+beta1*h1 if(dist=1) epsnew_u[i]*h1 else if(v[i]<cont) epsnew_u[i]*h1+rnorm(1,0,5) } else { h1_alpha0+alpha1*eps1^2+beta1*h1 if(dist=1) epsnew_u[i]*sqrt(h1) else if(v[i]<cont) epsnew_u[i]*sqrt(h1)+rnorm(1,0,5) } y1_b0+b1*yold1+epsnew eps1_epsnew yold1_y1 h_append(h,h1) ynew_append(ynew,y1) epss_append(epss,epsnew)

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} list(ynew=ynew,epss=epss,h=h,u=u) } est.rgarch_function(model,y,k,q,p){ #Generating X matrix xmatrix_function(element,k,n){ x1_as.matrix(rep(1,n)) x2_matrix(0,n,k) for(i in 1:k)x2[(i+1):(n),i]_element[1:(n-i)] x.1_matrix(c(x1,x2),n,1+k) list(x=x.1) } #Generating Z matrix zmatrix_function(element,q,p,teta,n){ z1_as.matrix(rep(1,n)) z2_matrix(mean(element),n,q) for(i in 1:q)z2[(i+1):(n),i]_element[1:(n-i)] z3_matrix(mean(element),n,p) h.tilda_matrix(mean(element),n,1) for(i in (max(q,p)+1):n){ for(j in 1:p){ z3[i,j]_h.tilda[i-j] } h.tilda[i]_c(z1[i,],z2[i,],z3[i,])%*%teta } z.1_matrix(c(z1,z2,z3),n,1+q+p) list(z=z.1,h=h.tilda) } n_length(y) if(k!=0){ x_xmatrix(y,k,n)$x b_matrix(l1fit(x, y, intercept=F)$coefficients,1+k,1) } e_y-x%*%b if(model==1) element_abs(e) else element_e^2 gama_matrix(,1+q+p,1) for(i in 1:100){ gama[2:(1+q+p)]_runif(q+p,0,1) if(sum(gama[2:(1+q+p)])<1) break } gama[1]_runif(1,0,3) zm_zmatrix(element,q,p,gama[1:(1+q+p)],n) z_zm$z

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h_zm$h out_NULL conv_NULL l_NULL for(kk in 1:iteration){ dia_1/h gama1<-

matrix(l1fit(diag(dia,n)%*%z,diag(dia,n)%*%element,F)$coefficients,1+q+p,1) if(any(gama1[2:(1+q+p)]>1)||gama1[1]<0||abs(sum(gama1[2:(1+q+p)]))>1){ gama_rep(NA,1+q+p) l_kk break } convergence_max(abs((gama1-gama)/gama)) gama_gama1 zm_zmatrix(element,q,p,gama[1:(1+q+p)],n) z_zm$z h_zm$h out_append(out,gama) conv_append(conv,convergence) l_kk print(kk) if(convergence<0.0001)break } out_matrix(out,l,1+q+p,byrow=T) list(iter=l,coef=c(b,gama),conv=conv,gama=out,z=z,h=h,res=e) } teta_NULL for(i in 1:rep){ a_rgarch(dist,model,saiz+200,coef.true[1],coef.true[2],coef.true[3],coef.true[4],coef.tru

e[5]) y_a$ynew[201:(saiz+200)] result_est.rgarch(model,y,k,q,p) teta_append(teta,result$coef) } est.teta_matrix(teta,rep,2+k+q+p,byrow=T)

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Appendix 3.8 – Simulation study on Assessing the Performance of Estimating Conditional Variance – Ordinary GARCH(1,1) Vs L1–GARCH(1,1) #par(mfrow=c(2,3)) module(finmetrics) q_1 p_1 k_0 saiz_300 cont_0.00 rep_500 model_2 dist_2 coef.true_c(0,0,1,.5,.2) iteration_30 #Generating Robust-AR-GARCH data rgarch_function(dist,model,saiz,b0,b1,alpha0,alpha1,beta1){ u_rnorm(saiz,0,1) #Normal(0,1) # if(model==1) u_u/median(abs(u)) else u_u/sqrt(median(u^2)) h1_0 eps1_0 yold1_0 h_NULL ynew_NULL epss_NULL for(i in 1:length(u)){ if(model==1){ h1_alpha0+alpha1*abs(eps1)+beta1*h1 epsnew_u[i]*h1 } else { h1_alpha0+alpha1*eps1^2+beta1*h1 epsnew_u[i]*sqrt(h1) } y1_b0+b1*yold1+epsnew eps1_epsnew yold1_y1 h_append(h,h1) ynew_append(ynew,y1) epss_append(epss,epsnew) } result1_list(ynew1=ynew,epss1=epss,htrue1=h,u1=u) h1_0 eps1_0 yold1_0 h_NULL

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ynew_NULL epss_NULL v_runif(length(u),0,1) for(i in 1:length(u)){ if(model==1){ h1_alpha0+alpha1*abs(eps1)+beta1*h1 if(dist==1) epsnew_u[i]*h1 else if(v[i]<cont) epsnew_u[i]*h1+rnorm(1,0,5) else epsnew_u[i]*h1 } else { h1_alpha0+alpha1*eps1^2+beta1*h1 if(dist==1) epsnew_u[i]*sqrt(h1) else if(v[i]<cont) epsnew_u[i]*sqrt(h1)+rnorm(1,0,5) else

epsnew_u[i]*sqrt(h1) } y1_b0+b1*yold1+epsnew eps1_epsnew yold1_y1 h_append(h,h1) ynew_append(ynew,y1) epss_append(epss,epsnew) } result2_list(ynew2=ynew,epss2=epss,htrue2=h,u2=u) list(result1=result1,result2=result2) } est.rgarch_function(model,y,k,q,p){ #Generating Z matrix zmatrix_function(element,q,p,teta,n){ z1_as.matrix(rep(1,n)) z2_matrix(mean(element),n,q) for(i in 1:q)z2[(i+1):(n),i]_element[1:(n-i)] z3_matrix(mean(element),n,p) h.tilda_matrix(mean(element),n,1) for(i in (max(q,p)+1):n){ for(j in 1:p){ z3[i,j]_h.tilda[i-j] } h.tilda[i]_c(z1[i,],z2[i,],z3[i,])%*%teta } z.1_matrix(c(z1,z2,z3),n,1+q+p) list(z=z.1,h=h.tilda) } n_length(y) e_y

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if(model==1) element_abs(e) else element_e^2 gama_matrix(,1+q+p,1) for(i in 1:100){ gama[2:(1+q+p)]_runif(q+p,0,1) if(sum(gama[2:(1+q+p)])<1) break } gama[1]_runif(1,0,3) zm_zmatrix(element,q,p,gama[1:(1+q+p)],n) z_zm$z h_zm$h out_NULL conv_NULL l_NULL error_NULL for(kk in 1:iteration){ dia_1/h gama1<-

matrix(l1fit(diag(dia,n)%*%z,diag(dia,n)%*%element,F)$coefficients,1+q+p,1) if(any(gama1[2:(1+q+p)]>1)||gama1[1]<0||abs(sum(gama1[2:(1+q+p)]))>1){ gama_rep(NA,1+q+p) l_kk break } convergence_max(abs((gama1-gama)/gama)) gama_gama1 zm_zmatrix(element,q,p,gama[1:(1+q+p)],n) z_zm$z h_zm$h error_append(error,mean(abs(element-h))) out_append(out,gama) conv_append(conv,convergence) l_kk print(kk) if(convergence<0.001)break } out_matrix(out,l,1+q+p,byrow=T) if(l==iteration){ for(ll in 1:l){ if(error[ll]==min(error)){ gama_out[ll,] zm_zmatrix(element,q,p,gama[1:(1+q+p)],n) z_zm$z h_zm$h } }

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} list(iter=l,coef=gama,conv=conv,gama=out,z=z,h=h,res=e,error=error) } aa_ab_ac_ad_ae_af_ag_ah_ai_aj_NULL for(j in 1:rep){ #GENERATE Y DATA

a_rgarch(dist,model,saiz+200,coef.true[1],coef.true[2],coef.true[3],coef.true[4],coef.true[5])

y_a$result2$ynew2[101:(saiz+100)] #TRUE CONDITIONAL VARIANCE #NON CONTAMINATED CONDITIONAL VARIANCE htrue1_a$result1$htrue1[101:(saiz+100)] #CONTAMINATED CONDITIONAL VARIANCE htrue2_a$result2$htrue2[101:(saiz+100)] #ESTIMATING PARAMETERS AND CONDITIONAL VARIANCE USING L1 -

GARCH PROCEDURE result_est.rgarch(model,y,k,q,p) gama1_result$coef res_result$res h_result$h z_result$z r1_y^2-h skel_sqrt(sum(r1^2)/(saiz-(1+q+p))) #ESTIMATING PARAMETERS AND CONDITIONAL VARIANCE USING

ORDINARY GARCH(1,1) b_garch(y~-1,~garch(1,1)) gama11_b$coef h3_b$sigma.t^2 r2_b$res^2-h3 z2_cbind(rep(1,saiz),b$res,h3) skel2_sqrt(sum(r2^2)/(saiz-(1+q+p))) #DETECTING OUTLIERS USING HAT MATRIX FOR L1 - GARCH hatt_diag(z%*%ginverse(t(z)%*%z)%*%t(z)) student1_sqrt(1-hatt)*skel student2_r1/student1 #ADJUSTING y TO NEW y1 y1_y for(i in 1:saiz){ if(abs(student2[i])>2.5) y1[i]_sign(y[i])*sqrt(h[i]+2.5*student1[i]*sign(r1[i])) } y1-y result_est.rgarch(model,y1,k,q,p) gama2_result$coef

159

if(all((gama2)=="NA")||(any(gama2[2:(1+q+p)]<0))||(any((result$h)<0))){ gama2_rep(NA,1+q+p) h2_rep(NA,saiz) error1_NA } else{ res_result$res h_result$h z_result$z if(model==1) u_res/h else u_res/sqrt(h) h2_h*rep(var(u),1) error1_htrue1-h2 } #DETECTING OUTLIERS USING HAT MATRIX FOR ORDINARY GARCH hatt2_diag(z2%*%ginverse(t(z2)%*%z2)%*%t(z2)) student1_sqrt(1-hatt2)*skel2 student2_r2/student1 #ADJUSTING y TO NEW y2 y2_y for(i in 1:saiz){ if(abs(student2[i])>2.5) y2[i]_sign(y[i])*sqrt(h3[i]+2.5*student1[i]*sign(r2[i])) } y-y2 c_garch(y2~-1,~garch(1,1)) gama21_c$coef h4_c$sigma.t^2 # ts.plot(h2,htrue1) # title("time series plot for l1-garch") # ts.plot(h3,htrue1) # title("time series plot for ordinary garch") #APPEND DATA FOR L1-GARCH(1,1) aa_append(aa,gama1) ab_append(ab,gama2) ac_append(ac,mean(error1)) ad_append(ad,mean(error1)^2) ae_append(ae,mean(abs(error1))) #APPEND DATA FOR ORDINARY GARCH af_append(af,gama11) ag_append(ag,gama21) ah_append(ah,mean(htrue1-h3)) ai_append(ai,mean(htrue1-h3)^2) aj_append(aj,mean(abs(htrue1-h3))) }

160

par(mfrow=c(1,1)) #OUTPUT OF SIMULATION FOR EVERY REPLICATION coef1_matrix(aa,rep,1+q+p,byrow=T) coef2_matrix(ab,rep,1+q+p,byrow=T) coef3_matrix(af,rep,1+q+p,byrow=T) coef4_matrix(ag,rep,1+q+p,byrow=T) for(i in 1:rep){ if(is.na(all(coef1[i,]))){ coef2[i,]_rep(NA,1+q+p) ac[i]_ad[i]_ae[i]_NA } } boxplot(coef1[,1],coef1[,2],coef1[,3]) boxplot(coef2[,1],coef2[,2],coef2[,3]) boxplot(coef3[,1],coef3[,2],coef3[,3]) boxplot(coef4[,1],coef4[,2],coef4[,3]) kump_c(rep(1,rep),rep(2,rep)) kump2_cbind(kump,c(ac,ah),c(ad,ai),c(ae,aj)) boxplot(split(kump2[,2],c(rep('l1garch',rep),rep('garch',rep)))) boxplot(split(kump2[,3],c(rep('l1garch',rep),rep('garch',rep)))) boxplot(split(kump2[,4],c(rep('l1garch',rep),rep('garch',rep))))

161

Appendix 3.9 – Small simulation study on confirming the equation A3.4 saiz_1000 #30 for T=30; 50 for T=50; 100 for T=100 and 300 for T=300 rep_1000 out_NULL for(i in 1:rep){ u_rnorm(saiz,0,1) v_u w_runif(saiz,0,1) for(i in 1:length(v)) if(w[i]<0.1)v[i]_rnorm(1,5,5) out_append(out,((median(v)-median(u))/2-(length(v[v<=median(u)])/saiz-.5))) } plot(density(out)) c(mean(out),sqrt(var(u)),skewness(u),kurtosis(u))

162

APPENDIX 4

Inflation Uncertainty and Economic Growth: Evidence from LAD

ARCH Model

Table A4.1 Consumer Price Index (CPI) for Asean–5 countries

Indonesia Malaysia Philippines Singapore Thailand 1925 0.037101 1926 0.036696 1927 0.036088 1928 0.035479 1929 0.035277 1930 0.034466 1931 0.030816 1932 0.025748 1933 0.02291 1934 0.020679 1935 0.020071 1936 0.018855 1937 0.019868 1938 0.020274 0.522475 1939 0.020274 0.526029 1940 0.0214 0.550909 1941 0.023371 0.561572 1942 - 0.90278 1943 - 2.182311 1944 - - 2.9030671945 - 3.909677 5.4304431946 - 2.932258 6.4038241947 0.475874 2.178757 7.5479741948 0.264687 2.043695 8.7775091949 0.25624 21.22438 1.92285 8.4701251950 0.298477 22.99308 1.983273 8.7775091951 0.501216 29.77309 2.146768 9.665506

163

Table A4.1 continues…

Indonesia Malaysia Philippines Singapore Thailand 1952 0.563164 30.36266 2.008152 10.7755 1953 0.678613 29.47831 1.940622 11.851341954 0.718034 27.41483 1.912188 12.431961955 0.777166 26.23569 1.894416 13.183341956 0.84193 26.53048 1.944176 13.866411957 1.027775 28.00439 1.976164 14.720261958 1.072828 27.70961 2.047249 14.122571959 1.230514 26.82526 2.025924 14.122571960 1.35441 26.82526 2.111226 30.89193 13.968881961 1.410726 26.82526 2.143214 30.99524 14.156721962 1.50928 27.12004 2.267613 31.16744 14.498261963 1.467043 27.70961 2.395566 31.85622 14.634871964 1.528991 27.70961 2.59105 32.33837 14.925181965 1.590939 27.70961 2.658581 32.44169 15.044721966 1.540254 28.00439 2.800751 33.09603 15.625331967 1.681045 29.18352 2.978463 34.16365 16.2401 1968 1.985154 29.18352 3.045994 34.40472 14.873951969 2.624345 28.88874 3.106416 34.3014 16.923171970 2.815821 29.47831 3.554252 34.43916 17.076871971 2.923295 30.00353 4.336188 35.07692 17.187041972 3.116748 30.92268 4.691613 35.79441 18.013341973 4.084015 34.20534 5.473548 45.2014 20.822761974 5.76061 40.17979 0.924106 55.32587 25.890731975 6.856846 41.95242 7.819355 56.76084 27.2679 1976 8.211019 43.06853 8.530205 55.72448 28.369631977 9.113801 45.10378 9.383226 57.47832 30.518011978 9.86612 47.33599 10.09408 60.18881 32.94182

164

Table A4.1 continues…

Indonesia Malaysia Philippines Singapore Thailand 1979 11.45674 49.04297 11.83566 62.66014 36.191941980 13.52024 52.32564 14.00375 68.0014 43.3532 1981 15.17534 57.38094 15.85196 73.58182 48.806781982 16.61549 60.72925 17.45138 76.45175 51.395861983 18.57152 62.96146 19.19296 77.32867 53.323891984 20.52755 65.45629 28.86053 79.32168 53.764581985 21.49481 65.65325 35.54252 79.72028 55.086661986 22.74151 66.11282 35.82686 78.6042 56.078221987 24.74053 66.30978 37.17748 79.0028 57.455391988 26.46012 68.01676 40.30522 80.1986 59.548681989 27.83578 69.85505 43.07754 82.03217 62.523361990 29.51238 71.56204 48.08903 84.82238 65.828561991 31.53289 74.45078 54.73548 87.53287 68.9685 1992 33.145 77.60214 57.89877 89.36643 71.227051993 35.23 79.9 60.6 91.2 73.1 1994 38.23 82.8 65.7 94 76.8 1995 41.84 85.7 70.9 95.6 81.3 1996 45.17 88.7 77.3 97 86 1997 48.21 91.1 81.8 98.9 90.8 1998 76.01 95.9 89.8 98.6 98.2 1999 91.45 98.5 95.8 98.7 98.5 2000 100 100 100 100 100 2001 112.55 101.4 106.1 101 101.7 2002 123.83 103.2 109.3 100.6 102.3 2003 126.78 104.3 112.5 101.1 104.1

165

Table A4.2 Growth of Gross Domestic Product (GDP) (%) for Asean-5 (1980 – 2003)

Year Indonesia Malaysia Philippines Singapore Thailand1980 0.099 0.074 0.051 0.07 0.058 1981 0.079 0.069 0.034 0.096 0.059 1982 0.022 0.059 0.036 0.069 0.054 1983 0.042 0.063 0.019 0.082 0.056 1984 0.07 0.078 -0.073 0.083 0.058 1985 0.025 -0.009 -0.073 -0.016 0.046 1986 0.059 0.012 0.034 0.023 0.055 1987 0.049 0.054 0.043 0.097 0.095 1988 0.058 0.099 0.068 0.116 0.133 1989 0.075 0.091 0.062 0.096 0.122 1990 0.072 0.09 0.03 0.09 0.116 1991 0.07 0.095 -0.006 0.073 0.081 1992 0.065 0.089 0.003 0.065 0.081 1993 0.073 0.099 0.021 0.127 0.083 1994 0.075 0.092 0.044 0.114 0.09 1995 0.082 0.098 0.047 0.08 0.092 1996 0.08 0.1 0.058 0.081 0.059 1997 0.045 0.073 0.052 0.086 -0.014 1998 -0.131 -0.074 -0.006 -0.009 -0.105 1999 0.008 0.061 0.034 0.069 0.044 2000 0.049 0.089 0.044 0.096 0.154 2001 0.038 0.003 0.018 -0.02 0.146 2002 0.044 0.041 0.043 0.032 0.03 2003 0.049 0.053 0.047 0.014 -0.027

166

Figure A4.1a L1–ARCH Estimates of Inflation Uncertainties - Indonesia Indonesia

0

0.1

0.2

0.3

0.4

0.5

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

Figure A4.1b L1–ARCH Estimates of Inflation Uncertainties - Malaysia Malaysia

0

0.1

0.2

0.3

0.4

0.5

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

167

Figure A4.1c L1–ARCH Estimates of Inflation Uncertainties - Philippines Philippines

0

0.1

0.2

0.3

0.4

0.5

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

Figure A4.1d L1–ARCH Estimates of Inflation Uncertainties - Singapore Singapore

0

0.1

0.2

0.3

0.4

0.5

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

168

Figure A4.1e L1–ARCH Estimates of Inflation Uncertainties - Thailand Thailand

0

0.1

0.2

0.3

0.4

0.5

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

169

APPENDIX 5

Tables of Distributions and their Properties

Table A5.1 Distributions and their Properties – Density function, Mean and Median Distribution Density function Mean Median

Normal

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−2

21exp

21

σμ

σπx

μ μ

kt -distribution ( )( )( )

⎟⎠⎞

⎜⎝⎛ +

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

Γ+Γ 2

12

12/2/1

k

kx

kkkπ

0 0

Logistic ( )[ ]( )[ ]{ }2/exp1

/expsx

sxμμ−−+−−

μ μ

Exponential ( )xλλ −exp 1−λ ( )λ2ln

170

Table A5.1 continues… 2kχ

( )2/12/

2/ 2/21 xk

k exk

−−

Γ

k Approximately, 3/2−k

Lognormal

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−2ln

21exp

21

σμ

πσx

x

( )2/exp 2σμ + ( )μexp

171

Table A5.2 Distributions and their Properties – Variance, Skewness and Kurtosis Distribution Variance Skewness Kurtosis

Normal 2σ 0 0

kt -distribution 2−k

k , for 2>k 0 for 3>k 4

6−k

, for 4>k

Logistic 22

3sπ

0 6/5

Exponential 2−λ 2 6

2kχ k2 k/8 k/12

Lognormal ( )[ ] ( )22 2exp1exp σμσ +− ( )[ ] ( ) 1exp2exp 22 −+ σσ ( )424

36

1

36422

222

−

−+−+ σσμ

σσσ

ee

eee