faculty of physical science · study are purely time series data based on the fact that are indexed...

DETERMINATION OF VOLATILITY CLUSTERGARCH FAMILY MODEL USING R

NWOYE HENRY CHUKWUNONSO

PG/M.Sc/11/59547

DETERMINATION OF VOLATILITY CLUSTERGARCH FAMILY MODEL USING R

FACULTY OF PHYSICAL SCIENCE

DEPARTMENT OF STATISTICS

Azuka Ijomah

Digitally Signed by: Content manager’s

DN : CN = Webmaster’s name

O= University of Nigeria, Nsu

OU = Innovation Centre

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DETERMINATION OF VOLATILITY CLUSTER ING IN

PHYSICAL SCIENCE

DEPARTMENT OF STATISTICS

: Content manager’s Name

Webmaster’s name

a, Nsukka

2

DETERMINATION OF VOLATILITY CLUSTERING IN GARCH FAMILY MODELS USING R


PG/M.Sc/11/59547

A THESIS SUBMITTED TO THE DEPARTMENT OF STATISTICS UNIVERSITY OF NIGERIA, NSUKKA IN FULLFFILMENT OF TH E

REQUIREMENT FOR THE AWARD OF THE DEGREE OF MASTERS OF SCIENCE (M.Sc) OF THE UNIVERSITY OF

NIGERIA, NSUKKA, NIGERIA

SUPERVISOR: PROF F.I. UGWUOWO

SEPTEMBER, 2015

i

TITLE PAGE

DETERMINATION OF VOLATILITY CLUSTERING IN GARCH FAM ILY MODELS

USING R

ii

CERTIFICATION

I hereby certify that this project was carried out by me.

----------------------------------------

Nwoye, Henry Chukwunonso

15th September, 2015

iii

APPROVAL PAGE

This project has been read and approved as having met the requirements of the

Department of Statistics and the School of Post Graduate Studies, University of Nigeria,

Nsukka, for the Award of a Masters of Science Degree in Statistics.

---------------------------------- ---------------------------------------- Supervisor (Date) Prof. F.I. Ugwuowo Department of Statistics University of Nigeria, Nsukka,

--------------------------------- ----------------------------------------- Head of Department (Date) Prof. F.I. Ugwuowo Department of Statistics University of Nigeria, Nsukka,

---------------------------------- --------------------------------------(External Examiner) (Date)

iv

DEDICATION

I dedicate this work to my father and mother, the God Almighty from whom all good things come.

v

ABSTRACT

This work investigated the volatility clustering of exchange rate of Nigeria Naira against the United States

of America Dollar. The data used in the present study consist of the monthly exchange rates of the Naira

to Dollar from January 1999 to December 2012 obtained from the Central Bank of Nigeria. The main

focus is to provide a proper understanding of the theory and empirical working of GARCH family models

and to determine volatility clustering. The EGARCH(2,2) model was selected as the best model from

point of view of the Mean Absolute Error. The estimated parameters, plots of returns series and plot of

conditional variances were used in determining the volatility clustering. There were clear evidences of

volatility clustering in Exchange rate of Nigeria naira against United States dollar.

vi

ACKNOWLEDGEMENTS

It was learnt in the course of life that, gratitude is like a secret password that opens

hidden doors to unseen opportunities. I will therefore not hesitate to show it to all the

wonderful persons that contributed to the completion of this work.

I will ever remain grateful to God who stood by me during the course of my Post-

graduate studies. He was faithful to His promise, “I will never leave you nor forsake

you”. Several times the devil sought an avenue to do me mischief but God rebuked him.

That this study was consummated on record time, I owe it to you my supervisor,

Professor F. I. Ugwuowo. Your meticulous supervision and your insistence that the right

thing must be done, is reflected in the richness and near perfectness of this work. Against

your very tight schedule you always took ample time to give my work what you called,

“thorough bleeding”. How those ‘bleedings’ always sent me back to my typist and to the

drawing board! But it was always for the best. I cannot thank you enough for the

friendship you extended to me during the course of this study. You even went further to

break down the strict barrier between student and teacher. This provided the enabling

environment for me to remedy the constructive criticisms which you made on my work

My spiritual mentor, Rev. Dr. Father Paul Obayi and his effort remain fresh in my

memory. I cannot forget those times during this programme when he will sit me down in

his office to counsel and encourage me. I cherish dearly those moments. It turned out

exactly as was declared.

vii

Mr. Ben Agubuzu, you know I can not quantify your contributions both to this

work, the programme and to my life. When I was still contemplating to come for this

programme, you took it upon yourself to ensure it came to reality.

Your presence was everywhere- in my business, in my assignments and in my spiritual

life. I thank God for everything.

My bosom friend, Mr. Mondayokoh and wife , you are special. Your contributions

towards this programme is highly recognize. I thank u so much for your effort.

I thank God for my parents, Mr. and Mrs. Sylvester Nwoyewho stood by me both

morally and financially all through these years. I say a ‘Big Thank You’ for your support.

Your constant calls and enquiry over my health, was like a tonic that provided the needed

strength to forge ahead.

This acknowledgement will be incomplete without the mention of all the members

of the Graduate Students Fellowship, UNN, I say thank you. You provided home away

from home.

Dr. and Mrs. Maxwell Ekechukwu, I say thank you. My first encounter with you gave

me the inspiration that with God and diligence, I could reach the top most high. This

work is a proof that I am still committed to that promise.

Finally, my gratitude to all the professors and senior lecturers in statistics

department.


viii

TABLE OF CONTENTS

Title page - - - - - - - - - - -i

Certification - - - - - - - - - - -ii

Approval Page- - - - - - - - - - -iii

Dedication - - - - - - - - - - -iv

Abstract- - - - - - - - - - - -v

Acknowledgement - - - - - - - - - -vi

Table of contents - - - - - - - - - -viii

List of tables - - - - - - - - - - -xi

List of figures- - - - - - - - - - -xii

CHAPTER ONE: INTRODUCTION

1.1 Background of the Study - - - - - - - -1

1.2 Statement of Problem - - - - - - - -3

1.3 Objectives of Study - - - - - - - - -3

1.4 Significance of study- - - - - - - - -3

CHAPTER TWO: LITERATURE REVIEW

2.1 Volatility and Forecasting of Exchange rate - - - - -4

ix

2.2 Comparing other models with GARCH family models - - -6

2.3 Empirical Evidences in modeling Exchange rate volatility using

GARCH models- - - - - - - - - -8

CHAPTER THREE - METHODOLOGY

3.1 Source of Data- - - - - - - - - -12

3.2. Volatility Clustering:- - - - - - - - -12

3.3 Estimation of volatility (conditional variance):- - - - - -13

3.4 GARCH Model:- - - - - - - - -14

3.5 Standard GARCH model (sGARCH):- - - - - - -15

3.6 NGARCH (Non linear GARCH):- - - - - - - -16

3.7TGARCH Model: - - - - - - - - - -16

3.8 GJR-GARCH Model:- - - - - - - - 17

3.9AVGARCH Model:- - - - - - - - - -18

3.10NAGARCH Model:- - - - - - - - -18

3.11Asymmetric Power ARCH (APARCH) Model:- - - - - -18

3.12ALLGARCH/FGARCH Model:- - - - - - - -19

3.13EGARCH Model:- - - - - - - - -- -19

3.14 ARCH TEST - - - - - - - - - -19

x

3.15Adjusted Pearson goodness of fit test: - - - - - - -20

3.16Normal quantile: - - - - - - - - - -20

CHAPTER FOUR - ANALYSIS AND DISCUSSION

4.1 Exploratory Data Analysis- - - - - - - 21

4.2 Stationarity of Data- - - - - - - - -25

4.3Fitting the Garch Models- - - - - - - - -26

4.4Determination of Volatility Clustering - - - - - - -47

4.5Estimated Values of Conditional Variance (Volatility)- - - - -48

4.6Plot of Volatility- - - - - - - - - -49

4.7 Forecasting- - - - - - - - - - -55

4.8 Forecast Evaluation- - - - - - - - - -57

4.9Results and Interpretation- - - - - - - - -59

CHAPTER FIVE:

SUMMARY, CONCLUSION RECOMMENDATIONS

5.1 Summary-- - - - - - - - - 60

5.2 Conclusion- - - - -- - - - - 60

5.3 Recommendations- - - - - - - - 61

References

Appendix

xi

LIST OF TABLES

Table 1: Results of ADF and PP tests on Exchange rate and Echange rate

returns series - - - - - - - - -25

Table 2: ARCH LM-test: on residual of ARMA(0,1)Model.

Null hypothesis: no ARCH effects - - - - - -26

Table 3: Iterations for the GARCH (p,q) family models - - - -27

Table 4: Iterations for selection of 5 Best GARCH (p,q) family models - -29

Table 5: models and Estimate of the constant terms (omega) with

P-values in bracket- - - - - - - - -47

Table 6: Estimated values of Conditional variance(volatility)- - - -48

Table 7:Multi Step ahead forecast of the various models for the

conditional mean of the exchange rate returns series - - -56

Table 8: Conditional Standard Deviation of the Multi Step ahead forecasts

of the various models for the conditional mean of the Exchange rate return series --57

Table 9: Measurements of forecast Accuracy of the models - -- - - -59

xii

LIST OF FIGURES

Figure 1: Time series plot of Monthly Exchange rate series of Naira to Dollar - - -22

Figure 2: Time series plot of Monthly Returns of Exchange rate of Naira to Dollar - 23

Figure 3: ACF and PACF of Exchange rate and Exchange rate Returns series - - 24

Figure 4: Plot of volatility of EGARCH(2,1) - - - - - - - 50



Figure 7: Plot of volatility of AVGARCH(1,1) - - - - - - 53

Figure 8: Plot of volatility of GJR-GARCH(1,2) - - - - - - 54

1

CHAPTER ONE

INTRODUCTION

1.1 Background of the study

For almost five decades, exchange rates movements and fluctuations have become an

important topic of macroeconomic analysis and policy makers, particularly after the collapse of

the Bretton Woods agreement in 1973 (Omojimite and Akpokodje 2010), of fixed exchange rates

among major industrial countries. This volatility in exchange rate affects security valuation,

investment analysis, profitability and risk management. This is because it exposes exchange rates

to financial uncertainties. The volatility in the exchange rate may cause a significant impact on

macroeconomic analysis such as prices, wages, unemployment and the level of output.

Exchange rate means the exchange of one currency for another price for which the

currency of a country (Nigeria) can be exchanged for another country’s currency as in US

(dollar). A correct exchange rate do have important factors for the economic growth for most

developed countries whereas a high volatility has been a major problem to the economy of some

of the African countries such as Nigeria. Some factors which definitely affect exchange rate are

interest rate, inflation rate, trade balance, general state of economy, money supply and other

similar macro-economic giants’ variables.

A steady exchange rate can help financial institutions to tackle and to monitor the

performance of investments, financing, hedging and as well as reducing their operational risks.

Volatility can be represented by variance or standard deviation which is unconditional

and does not recognize that there are interesting patterns in asset volatility e.g. time-varying and

2

clustering properties. The financial analysts started to model and explain the characteristics of

exchange rate returns and volatility using time series econometric models. The popularly known

and steadily used models for capturing such volatility clustering is the Autoregressive

conditional Heteroscedasticity (ARCH) model advanced by Engle (1982) and Generalized

Autoregressive conditional Heteroscedasticity (GARCH) model developed independently by

Bollerslev (1986). After the Research work of Engle (1982) and Bollerslev (1986), many

GARCH models have been developed to model volatility. Some of the models include

integrated Generalized Autoregressive Heteroscedasticity (IGARCH) model originally proposed

by Engle and Bollerslev (1986), Generalized Autoregressive conditional Heteroscedasticity in

mean (GARCH-M) model introduced by Engle, Lilien and Robins (1987), Exponential

Generalized Autoregressive conditional Heteroscedasticity (EGARCH) model proposed by

Nelson (1991), Threshold Autoregressive conditional Heteroscedasticity (TARCH) and

Threshold Generalized Autoregressive conditional Heteroscedasticity (TGARCH) models were

introduced by Zakoian (1994) and Power Generalized Autoregressive conditional

Heteroscedasticity (PGARCH) model introduced by Ding, Granger and Engle (1993) and many

others.

On the other hand, many empirical studies have risen significantly in recent years to

investigate the behaviour of exchange rate volatility in the field of time series analysis of

financial returns such as leverage effect and volatility clustering. Recent works include, Musa et

al (2014), Ramzan et al (2012), Zakaria and Abdalla (2012), Kamal et al (2012), Fallman and

Wirf (2010),

The topic data were obtained and put together, for the purpose of this research work,

from central Bank of Nigeria Publication. The data range from 1999-2012 for a total of 168

3

observations. The monthly average official exchange rates of the Naira to Dollar used in this

study are purely time series data based on the fact that are indexed by time and that is in months.

The different models used, include GARCH, EGARCH, GJR-GARCH, AVGARCH, TGARCH,

NGARCH, NAGARCH,APARCH, ALLGARCH and sGARCH models.

1.2 Problem Statement

The exchange rate volatility clustering is a critical issue in Nigeria. An understanding of

its causes and manifestations is very important because it enhance economic predictions on

volatility. However, the study of exchange rate volatility clustering using GARCH family models

have not been exhausted and therefore the study investigate the appropriate model to determine

adequately exchange rate volatility clustering in Nigeria.

1.3 Objective of the Study

i. To fit the best GARCH family models to the Exchange Rate of the Naira to the Dollar.

ii. To determine the volatility clustering.

iii. To make Forecasts using the identified Models

1.4 Significance of Study

Determining of volatility clustering in Nigeria’s exchange rate is important to an understanding

of the causes of clustering in monetary policy fluctuations.

The study will therefore be relevant to financial policies and investment decisions.

4

CHAPTER TWO

EMPIRICAL LITERATURE REVIEW

2.1 Volatility and Forecasting of Exchange Rate

Volatility and the forecasting of exchange rate have been of much concern to researchers

over the years. Most models used in examining this trend were drawn from other modeling

families. Nonetheless, recent research interests are focused on the GARCH family models.

Muhammad (2010), investigated the volatility and forecasting on the Karachi Inter Bank

Offering Rate (Kibor).The data used was daily observations for the period of one month, six

month’ and one year with sample period ranging from January 2006 to May 2008, making a

total of 693 observations. First, they checked if the data was stationary through graph analysis of

exchange rate series that showed an upward trend, suggesting that the exchange rate series was

not stationary. They calculated the returns by taking first difference logs of two consecutive

month end bid rates. Therefore, they modeled ARMA (1, 1), GARCH (1, 1) and EGARCH (1,

1). They further, forecast the data by using mean Absolute Error (MAE), Root Mean Square

Error (RMSE), Mean Absolute Percentage Error (MAPE) and Theil – U inequality. How their

methodology did not include Phillips Perron (pp) tests. Muhammad (2010), shows that GARCH

(1, 1) was found to be the best to remove the persistence in volatility while EGARCH (1, 1)

successfully overcame the leverage effect in the distribution of returns.

In another study, Modelling Naira/Dollar Exchange Rate Volatility: Application of Garch

and AssymetricModels,Olowe and Ayodeji (2009), modeled the exchange rate volatility of

Nigeria against the U.S. Dollar. The data used in their work was average monthly exchange rates

of Nigeria from January 1970 to December 2007.They transformed the data to monthly exchange

5

rate return. The augmented Dickey – fuller (ADF) Unit root test was used to check the stationary

trend of time series and the volatility models. Their study presented full sample results and

separated the results of volatility in a fixed exchange rate regime from floating exchange rate

regime and managed floating rate regime. The GARCH family models used in their analysis

include: GARCH (1,1), IGARCH (1,1), TS – GARCH (1,1), GJR – GARCH (1,1), EGARCH

(1,1) and APARCH (1,1). Their finding shows that the APARCH and GJR-GARCH models

indicated the existence of significant asymmetric effect. Again, they found that the TSGARCH

and the APARCH models were the best among the models applied. Nevertheless, they did not

forecast the models. Summarily, Olowe and Ayodeji (2009), in their result, the hypothesis of

leverage effect was rejected by all asymmetry models, though all the coefficients of the variance

equations were significant, the TS-GARCH and APARCH models were found to be the best

models.

To measure the exchange rate volatility, Kamal et al (2012), in their study, ‘Modelling the

exchange rate volatility’, using generalized autoregressive conditionalheteroscedasticity

(GARCH) type models: Evidence from Pakistan, investigated the volatility of exchange rate of

Pakistan rupee against U.S. Dollar. The models used include EGARCH, TARCH and GARCH.

The data was in the form of daily observations for the period, ranging from January, 2001 to

December, 2009 with 2005 observations and monthly data for the same period with 108

observations. The Augmented Dickey – fuller (ADF) unit root test was used to check the

stationarity of the time series. A symmetric GARCH – M (1, 1) with other two asymmetric

models EGARCH (1, 1) and TARCH (1, 1) were used to analyze the daily and monthly

exchange rates of Pakistan. In their result, the GARCH – M (1, 1) model shows that the first

order autoregressive process supports the previous day exchange rate which affects the current

6

day exchange rate. However, their study did not check for ARCH effect before employing

GARCH models and also they did not forecast the model. For instance, in the variance equation,

ARCH (1) and GARCH (1) both remained significant at 1% for the daily and monthly exchange

rate returns. The TARCH showed that no presence of Autoregressive behaviour in the daily

exchange rate returns but the monthly exchange rate returns shows the presence of

Autoregressive behaviour. In summary, Kamal et al (2012), from their results, shows that among

all the models, EGARCH proved to be best to explain the behaviour of exchange rates on daily

and monthly data

2.2 Comparing Other Models with GARCH Family Models

Recent research focus has tended to compare the performance of other modeling groups

such as the realized volatility model against the GARCH family model. For instance, Fallman

and Wirf (2010), in their study ‘Forecasting foreign exchange volatility for value at risk’,

examined the practical task of forecasting one day ahead foreign exchange volatility of Euro

against the U.S. Dollar, Japanese Yen, Great British pound and Swedish Krona over the period

of 1st January 2009 to 29th October, 2010 resulting in a total of 476 observations. They used daily

returns and ten minute evenly spaced intervals. The models used include ARCH (1), GARCH (1,

1) and EGARCH (1, 1). Again, they used two approaches, the ARCH Frame work and Realized

Volatility. The value at Risk was calculated, price data was plotted and the volatility of the

exchange rate series was observable. Three standard forecast error measurements which includes

Mean Absolute Error (MAE) were used, Mean Absolute Percentage Error (MAPE) and Root

Mean Squared Error (RMSE). However, they did not treat Theil-U inequality in their forecast.

Thus, Fallman and Wirf (2010), shows that Realized Volatility based models were consistently

7

produced superior forecasts than the ARCH models which forecast accurate improvement of

about 50 percent

The findings of Hansen and Lunde (2005), has significant bearing to this present study

with regard to comparing volatility of models and the GARCH (1,1). Their study, A forecast

comparison of volatility models: Does anything beat a GARCH (1, 1)?, modeled the DM - $ spot

exchange rate data, the estimation sample spans from October 1, 1987 through September 30,

1992 resulting in a total of 1254 observations and the out of sample evaluation sample spans

from the period of October 1, 1992 through September 30, 1993 in a total of 260 observations.

The second data set consisted of IBM stock returns with the period from January 2, 1990 through

May 28, 1999 in a total of 2378 observations and the evaluation period spanned from June 1,

1999 through May 31, 2000 in a total of 254 observations. They used daily DM - $ exchange rate

data and IBM returns. Also, they compared 330 GARCH type models in terms of their ability to

forecast the one-day ahead conditional variance. The models were evaluated out of several

samples using six different loss functions, where the realized variance was substituted for the

latent conditional variance. Moreover, they tested for superior predictive ability, reality check

and divided the observations into an estimation period and an evaluation period. Their result

shows that the RC lacks power to an extent that makes it unable to distinguish ‘good’ and ‘bad’

models in the analysis. However, they did not check for ARCH effect before employing GARCH

models and also they did not test the stationarity of the data. Hansen and Lunde (2005), in their

analysis, found no evidence that a GARCH (1, 1) is out performed by so many models in the

behaviour of exchange rates but GARCH (1, 1) is purely inferior to models that can

accommodate a leverage effect in the analysis of IBM returns and reality check (RC) has less

power than the superior predictive ability (SPA) test.

8

2.3 Empirical Evidences in Modelling Exchange Rate Volatility using GARCH Models

To provide empirical evidence on using the GARCH family models in measuring

exchange rate volatility, Zakaria and Abdalla (2012), modeled the exchange rate volatility in a

panel of nineteen of the Arab Countries. The currencies were the United Arab Emirates Dirham,

Bahraini Dinar, Djiboutian Franc, Algerian Dinar, Egyptian Pound, Iraqi Dinar, Jordanian

Kuwaiti Dinar, Lebanese Pound, Libyan Dinar, Moroccan Dirham, Mauritanian Ouguiya, Omani

Rial, Qatari riyal, Saudi Arabian Riyal, Somali shilling, Syrian Pound, Tunisian Dinar and

Yemeni Rial, all against the U.S. Dollar. The data were daily returns of exchange rates from 1st

January 2000 to 19th November 2011 resulting in a total of 4341 observations. They transformed

the data to daily exchange rate return which the first difference of the natural logarithm of

exchange rate. The models used include GARCH (1, 1) and EGARCH (1, 1) and there are two

distinct specifications, the first for the conditional mean and the other for conditional variance.

Also, they investigated whether the daily returns are stationary by applying Augmented Dickey

fuller (ADF) test. The models are estimated using maximum likelihood method under the

assumption of Gaussian normal error distribution. The log likelihood function is maximized

using marquardt numerical iterative algorithm to search for optimal parameters. Their findings

indicated evidence of leverage effect for a good number of currencies, showing that negative

shocks imply a higher next period volatility than positive shocks. Forecasting the model, which

they did not do, may have shown a more robust outcome. To sum it up, Zakaria and Abdalla

(2012) have shown that the asymmetric EGARCH (1,1) provides evidence of leverage effects for

many currencies except for the Jordanian Dinar which indicates that negative shocks imply a

higher next period volatility than positive shocks which shows that the exchange rates volatility

can be adequately modeled by the family of GARCH models.

9

Kaur (2004), investigated the time varying volatility in the Indian stock market against U.S.

Market. The data is in daily stock form and the period ranges from 1993 to 2003. The daily stock

price has been converted to daily returns. They tested the stationarity of sensex and Nifty return

series by conducting Dickey – fuller and Philip-peron tests which shows that the series are

stationary. Moreso, they applied the Autocorrelation function and partial Autocorrelation

Function of return series and it shows random walk behaviour. Their analysis indicated the

presence of ARCH effect which is computed by lagrange multiplier (LM). In their study, high

level of volatility occurred in the 1999 – 2000 period while the period 1995 – 1998 was

relatively calm. It shows asymmetrical GARCH models, EGARCH (1, 1) to sensex and TARCH

(1, 1) to Nifty returns. Also, it shows that day of the week and the January effects were not

present while the return and volatility shows intra week and intra year seasonality. Like most of

modeling researches, they did not forecast the model, which as literature show, always predicts a

differential scenario. Thus, Kaur (2004) findings, shows mixed evidence of return and volatility

spillover between the U.S. and the Indian markets.

The dynamics in exchange rate volatility was modeled and forecasted by Ramzan et al

(2012), who investigated the modeling and forecasting of exchange rates of Pakistan against

U.S. Dollar. The data used in their study consisted of monthly average foreign exchange rates of

Pakistan with sample period ranging from July 1981 to May 2010 resulting in a total of 347

observations. Further, they examined the stationarity of exchange rate series using Graphical

analysis, correlogram and unit root test which showed the series as non stationary. The data was

transformed to monthly returns to achieved stationarity. The models used in their study were

ARMA, ARCH, GARCH, IGARCH and EGARCH. The forecast performance is measured

through different measures, which include Mean Absolute Error (MAE), Root Mean Squared

10

Error (RMSE), Mean Absolute Percentage Error (MAPE) and Theil – U inequality. Also, they

tested the presence of ARCH effect by using the Lagrange Multiplier (LM) test.Ramzan et al

(2012) have shown that GARCH (1,2) was found to be best to remove the persistence in

volatility while EGARCH successfully overcame the leverage effect in the exchange rate returns

and provides a model with a good forecasting performance.

Ning et al (2009), investigated the Modeling Asymmetric volatility clusters using copulas

and high frequency data. The data used was daily realized volatilities of the individual company

of stock and foreign exchange rate markets. They did not use monthly data. Thus, Ning et al

(2009), findings shows asymmetric pattern of volatility clusters maintain to be visible upon the

changes over time, and volatility clusters remain persistent after one month period.

Cont (2005), investigated on volatility clustering in financial markets: empirical facts and

agent-based models. They used several economic tools which show there was low and high

activity regimes with heavy tailed durations. Furthermore, Cont (2005), discuss a simple agent-

based model which are able to detect a link between volatility clustering and investors inertia,

thus, providing a useful complement to econometric analysis.

Chen and Zhu (2007), investigated on volatility clustering within industries: an empirical

investigation. They compared the short –run responses of stock returns to arrival of

macroeconomic news across general industry, banking, and real estate trusts. Summarily, Chen

and Zhu (2007), test the hypothesis by sample intraday stock price data of ten firms from three

industries. They further, conducted the brown – forsythe –modified levene tests. Thus, Chen and

Zhu (2007), findings shows that there exist different degrees of responses on macroeconomic

news, consequently return volatility clustering.

11

In another study, A Time Series Analysis of the Shanghai and New York Stock Price

indices, Chow and Lawler (2003), compared weekly rates of return. Moreso, the rates return was

calculated. Nevertheless, the rates of returns of both markets were serially uncorrelated while the

Autoregressions are not stable. Again, the volatility of Shanghai market was higher than New

York stock price. Thus, Chow and Lawler (2003), shows that the analysis has implications for

use of Autoregressions, Granger causality tests, and the interpretation of spurious correlation.

In conclusion, the findings of the reviewed literatures indicate that some of the

researchers used small range of period while others used few models to investigate their work. A

most significant outcome of their study is GARCH family model is a predictor of leverage in

exchange rate dynamics. However, this current study investigates the volatility clustering in

exchange rate of Nigeria against U.S. Dollar.

12

CHAPTER THREE

METHODOLOGY

3.1 Source of Data

The exchange rate data used in this research work consisted monthly average foreign

exchange rate of Nigeria (Naira per U.S. Dollar). The data is obtained from Central Bank of

Nigeria with sample period ranging from January 1999 to December 2012. The exchange rate of

Nigeria was transformed to returns. The models used are GARCH, EGARCH, GJR –GARCH,

AVGARCH, TGARCH, NGARCH, NAGARCH, APARCH, ALLGARCH and sGARCH

models.

In this study, the return on monthly exchange rate is given by

�� = ��

�

Where �� means Naira/dollar exchange rate at time t and �� means Naira exchange rate at time

t-1.

The ��ofabove equation represent the volatility of exchange rate of Naira to Dollar over the

period 1999 – 2012.

3.2. Volatility Clustering:

The volatility clustering is when large and small values in a time series tend to occur in clusters.

Moreso, volatility clustering refers to the observation, that “ large changes tend to be followed by

large changes, of either sign, and small changes tend to be followed by small changes “

13

(Mandelbrot, (1963). It is a stylized fact that a downward movement (depreciation) is always

followed by higher volatility. Volatility is higher after negative shocks than after positive shocks

of the same magnitude. This characteristic exhibited by percentage changes in financial data is

termed leverage effects. This feature was first suggested by Black (1976) for shock returns. In

finanacial time series such as, exchange rates, stock returns and other financial series are known

to exhibit certain behaviours like fat tails, leverage effect, volatility clustering and asymmetric

effect. When exchange rate returns is compared with normal distribution, fatter tails are

observed. This observation is also known as excess kurtosis. The standardized fourth moment for

a normal distribution is 3, whereas for many financial time series a value well above is observed,

Fama (1965). Summarily, volatility clustering is nothing but accumulation or clustering of

information.

3.3 Estimation of volatility (conditional variance):

The condition variance equation is used in estimating the conditional variance otherwise refers as volatility.

tetY εσµ +=

21

2110

2−− ++= tjtt BY σαασ

Where 21−tY and 2

1−tσ squared residuals and conditional variance of the previous period. The residuals of a return at time, t, may be given as

tetY εσµ +=−

Volatility of the returns can be obtained as follows:

211

2110

2−− ++= ttt Bσεαασ

But LVγα =0 where γ is a weight assigned to the long run average rate

14

LV .

Since weight must sum to 1,

1111 11 βαγβαγ −−=⇒=++

This implies that

11

0

1 βαα

−−=LV

Koima et al (2015)

This means that as the lag increases the variance forecast converges to unconditional variance given by

equation above.

3.4 GARCH Model: Bollerslev (1986) usefully generalized the simple ARCH model with the parsimonious and

frequently used Generalized ARCH (GARCH) model, which models current conditional

variance with geometrically declining weights on lagged squared residuals. The GARCH (p, q)

model can be expressed as

ttt zσε =

ℎ� = �� + �� + ��ℎ� �

Where;

��measures the extent to which a volatility shock today feed through into the next period’s

volatility. (�� + ��) Measures the rate at which this effect lies over time. ℎ� �is the volatility at

t-1.

15

3.5 Standard GARCH model (sGARCH):

The standard GARCH model (Bollerslev (1986)) may be written as:

,1

2

1

2

1

2 jjjjvjtp

jt

q

jjt

m

jt −++

+= ∑∑∑

==−

=

σβεαςωσ

With 2tσ denoting the conditional variance, ω the intercept and 2tε the residuals from the mean

filtration process. The GARCH order is defined by (q,p) (ARCH, GARCH), with possibly m

external regressorsjv which are passed pre-lagged. If variance targeting is used, then ω is

replaced by,

�� ∑=

−

−

−

m

j

vjjP1

^

1 ς

where ��is the unconditional variance of 2ε which is consistently estimated by its sample

counterpart at every iteration of the solver following the mean equation filtration, and

�� represents the sample means of the jth external regressors in the variance equation (assuming

stationarity), and ∧P is the persistence and defined below. If a numeric value was provided to the

variance. Targeting option in the specification (instead of logical), this will be used instead of

2σ for the calculation. One of the key features of the observed behavior of financial data which

GARCH models capture is volatility clustering which may be quantified in the persistence

parameter∧P . For the s’GARCH’ model this may be calculated as,

∑ ∑= =

∧+=

q

j

p

j

jjP1 1

βα

16

3.6 NGARCH (Non linear GARCH):

The NGARCH (p,q) model proposed by Higgins and Bera (1992) parameterizes the

conditional standard deviation raised to the power � as a function of the lagged conditional

standard deviations and the lagged absolute innovations raised to the same power,

�� = � + � ��|��

!

�"�|� + � ��

$

�"��

This formulation obviously reduces to the standard GARCH (p,q) model for � = 2 (see

GARCH). The NGARCH model is also sometimes referred to as Power ARCH or power

GARCH model, or PARCH or PGARCH model.

With most financial rates of returns, the estimates for � are found to be less than two, although

not always significantly so (see also APARCH and TS-GARCH).

3.7 TGARCH Model:

Threshold GARCH (TGARCH) model was proposed by Zakoian (1994), which has the

following form:

ℎ� = �� + � ��

$

�"�� + � &�

$

�"�'� �� + � �(

!

("�ℎ� (

Where:

)� � = *1 ,- �� < 00 ,-�� ≥ 0 1

17

&�=leverage effects coefficient. (if&�>0 it indicates the presence of leverage effect). That is

depending on whether �� is above or below the threshold value of zero, �� has different

effects on conditional variance ht: when �� is positive, the total effects are given by �� and

when �� is negative, the total effects are given by (�� + &�)�� . So one would expect &� to be

positive for bad news to have larger impacts.

3.8 GJR-GARCH Model:

The GJR-GARCH model is another volatility model that allows asymmetric effects. Glosten –

Jagannathan – Runkle Generalized autoregressive conditional Heteroscedasticity model was

introduced by Glosten et al (1993). The general specification of this model is of the form:

�� = 2 + �(��

!

�"�� + &�'� � �� ) + � �(

$

("�� (�

where'� � is a dummy variable which takes the value of 1 when &� is negative and 0 when &� is

positive. In this GJR-GARCH model, it is supposed that the impact of �� on the conditional

variance �� differs when �� is positive or negative. A nice aspect of the GJR-GARCH model is

that it is easy to test the null hypothesis of no leverage effects. Infact, &�=…= &!= 0 means that

the news impact curve is symmetric, i.e. past negative shocks have the same impact on today’s

volatility as positive shocks.

18

3.9 AVGARCH Model:

The absolute value generalized autoregressive conditional heteroscedastic (AVGARCH)

model was introduced by Taylor (1986), is specified as

�� = ��5�; �� = � + � ��$

�"�(|�� + 7| − 9(�� + 7))� + � �(

!

("�� (�

3.10 NAGARCH Model:

Nonlinear Asymmetric GARCH (NAGARCH) was introduced by Engle and Ng in 1993.

( )

0;0,

122

112

>≥

+−+= −−−

w

tttt

βα

βσθσεαωσ

For stock returns, parameter Ө is usually estimated to be positive; in this case, it reflects the

leverage effect, signifying that negative returns increase future volatility by a larger amount than

positive returns of the same magnitude.

3.11 Asymmetric Power ARCH (APARCH) Model:

Asymmetric Power ARCH (APARCH) model, introduced by Ding et al (1993). This model is

able to accommodate asymmetric effects and power transformations of the variance. Its

specification for the conditional variance is the following:

�� = �/5� + � ��

!

�"�(|;� �| − &�;� �) + � �(

$

( �� (�

19

Where:

�� ≡ √ℎ>, the parameter � (assumed positive but typically ranging between 1 and 2) performs a

Box-Cox transformation and & captures the asymmetric effects.

3.12 ALLGARCH/FGARCH Model:

Family Generalized autoregressive conditional heteroscedasticity model or Hentschel’s

FGARCH model is an omnibus model that nests a variety of other popular symmetric and

asymmetric Garch models. This model was introduced by Hentschel (1995).

3.13 EGARCH Model:

The Exponential GARCH (EGARCH) model was proposed by Nelson (1991). The model

has the following representation:

log ℎ� = �� + � ��

$

�"�

|�� | + &��

ℎ� �B

+ � 7(

!

("�log ℎ� (

Where, &�=1 leverage effect coefficient. (if&� > 0 it indicates the presence of leverage effect).

Note that when �� is positive or there is “good news”, the total effect of �� ,' (1 + &�)|�� |;

in contrast, when �� is negative or there is “bad news” the total effect of �� ,' (1 − &�)|�� |.

Bad news can have a large impact on volatility, and the value of &� would be expected to be

positive.

3.14 ARCH TEST -An uncorrelated time series can still be serially dependent due to a

dynamic conditional variance process. A time series exhibiting conditional heteroscedasticity or

20

autocorrelation in the squared series is said to have autoregressive conditional heteroscedastic

(ARCH) effects. Engle’s ARCH test is a lagrange multiplier test to assess the significance of

ARCH effects.

3.15 Adjusted Pearson goodness of fit test: It is a very common and useful test for

several purposes. It can help determine whether a set of claimed proportions is likely, or whether

a pair of categorical variables are independent.

3.16 Normal quantile: A Q-Q plot is a plot of the quantiles of two distributions against

each other, or a plot based on estimates of the quantiles.

21

CHAPTER FOUR

ANALYSIS AND DISCUSSION

4.1 EXPLORATORY DATA ANALYSIS

Let tR and tX be the monthly returns and exchange rate series of the Naira to Dollars at

time t for t=1,2,…,T. To investigate the presence of trend in the original exchange rate series of

the Naira to Dollar, a time series plot of the original exchange rate series is given in Figure1.

The plot shows the presence of an upward trend from the beginning of 1999 till the end of 2003

and a downward trend from 2004 till 2009 where it began another upward trending that lasted till

the end of the series used for this research. This plot indicates that the exchange rate series is non

stationary and hence need some transformation before it can be used for time series analysis.

Thus, we calculated the exchange rate return series as:

22

Figure1: Time Series Plot of Monthly Exchange Rate Series of Naira to Dollar

Figure 2, presents the time series plot of the monthly returns of the exchange rate of the Naira to

Dollar. This plot shows the return series is stationary and it also reveals the presence of volatility

clustering during the first three years and the last two years in the series of observations for this

study and relatively stable volatility from 2004 till the end of 2008.

23

Figure 2: Time series Plot of Monthly Returns of Exchange Rate of Naira to Dollar

The slow decay of the autocorrelation function plot of the exchange rate series presented

in Figure 3 further stresses the non stationarity of the exchange rate series this means that there

is a strong dependence of the Monthly exchange rate series on time for all the lags with

autocorrelation coefficients greater than the 95% limits. The partial autocorrelation plot of the

Exchange rate series on Figure 3 indicates that this non stationarity of this series could be

eliminated by differencing the exchange rate series just once [see Brockwell and Davis (1996),

chapter9].

The cut off at lag one of the autocorrelation plot of the return series and the exponential

decay of the partial autocorrelation function though alternating in sign, suggest that the return

series is an moving average of order one i.e. MA(1) where the current value of the return series is

a function of the immediate past innovation (error).

24

Figure 3: ACF and PACF of Exchange Rate and Exchange Rate Returns Series

25

4.2 Stationarity of Data

The major assumption for application of any time series models such as the Box and

Jenkin’s Autoregressive integrated Moving Average Models (ARIMA) and the Generalized

Autoregressive Conditional Hetroscedasticity (GARCH) family models is the stationarity of the

data. Apart from the graphical method used earlier in this study, we will also use the Augmented

Dickey-Fuller (ADF) test and Phillips-Perron (PP) test to further investigate the stationarity of

the both series.The results of the Augmented Dickey-Fuller (ADF) test and Phillips-Perron test

on the exchange rate series and exchange rate returns series is presented in Table 1 below.

Table 1: Results of ADF and PP test on Exchange rate and Returns series

Augmented Dickey-Fuller Test Phillips-Perron Test

SERIES t-statistic Lag p-value Adj t-statistic Lag p-value

Exchange Rate -2.077 5 0.5443 -8.394 4 0.633

Exchange Rate Returns -4.4947 5 <0.01 -89.5569 4 <0.01

From Table 1 above, with a t-statistic of (-2.077) and p-value of (0.5443) and with

adjusted t-statistic of (-8.394) and a p-value of (0.633) for the ADF and PP test on exchange rate

series respectively, the exchange rate is confirmed to be non-stationary hence the need to

transform the exchange rate series into a stationary series. From Table 1 also, with t-statistic of

(-4.4947) and adjusted t-statistic of (-89.5569) for the ADF and PP test respectively and with

both p-values (<0.01), it reveals that the exchange rate returns series is stationary compactable

for the application of our various models.

26

The ARCH family models require the presence of ARCH effect which we use the

Lagrange Multiplier (LM) test for the residuals of the MA (1) model as suggested by Engle

(1982 pp 987-1007). The results of the Lagrange Multiplier test on the residuals of the MA (1)

model are presented in Table 2.

Table 2: ARCH LM-test; on residual of ARMA (0,1) Model. Null hypothesis: no ARCH effects.

Chi-squared df p-value 30.0335 1 4.246e-08 34.3906 4 6.196e-07 33.6022 8 4.795e-05

From the results of the Lagrange Test presented above, the p-value of the test at different lags

indicates the presence of heterogeneity (ARCH effect). So therefore we reject the null hypothesis

of absence of ARCH effect for level of significance as low as 1%. This result leads us to the use

of GARCH family models to determine the conditional variance of the Exchange Rate returns

series.

4.3 FITTING THE GARCH MODELS

Haven established the presence of conditional heteroscedasticity, we then go further to

solve this problem by applying the GARCH family models to the returns series. The best models

among the GARCH (p,q) family models will be selected based on the Akaike information,

Schwarz information criteria, the Shibata information criteria and the Hannan-Quinn information

criteria. The models which minimize these criteria are selected as the best models and their

parameters estimated. This results of the iterations for the selection of the best GARCH (p,q)

models are presented in Table 3. In Table 3, the best 10 models are kept in bold. To make the

27

results obtained in Table 3 more realistic and manageable; the best 5 models are ordered and

presented in Table 4. This ordering is based on how much they were able to minimize these

information criteria.

Table 3: Iterations for the GARCH (p,q) family models

MODEL AKAIKE BAYES SHIBATA HANNAN QUINN

SGARCH(1,1) -5.8812 -5.7825 -5.8831 -5.8410

GARCH(1,1) -5.8811 -5.7825 -5.8831 -5.8410

NGARCH(1,1) -5.8649 -5.7466 -5.8678 -5.8169

TGARCH(1,1) -5.3711 -5.2554 -5.3766 -5.3256

GJRGARCH(1,1) -5.4163 -5.2979 -5.4191 -5.3682

AVGARCH(1,1) -5.9123 -5.7742 -5.9162 -5.8562

NAGARCH(1,1) -5.5508 -5.4325 -5.5537 -5.5027

APARCH(1,1) -5.8822 -5.7441 -5.8861 -5.8261

ALLGARCH(1,1) -5.8250 -5.6673 -5.8301 -5.7609

EGARCH(1,2) -5.9588 -5.8208 -5.9627 -5.9028

SGARCH(1,2) -5.8793 -5.7610 -5.8822 -5.8313

GARCH(1,2) -5.8793 -5.7610 -5.8822 -5.8313

NGARCH(1,2) -5.8721 -5.7341 -5.8760 -5.8160

TGARCH(1,2) -5.8644 -5.7264 -5.8683 -5.8084

GJRGARCH(1,2) -5.9064 -5.7684 5.9104 -5.8504

AVGARCH(1,2) -5.8472 -5.6894 -5.8522 -5.7831

NAGARCH(1,2) -5.9019 -5.7639 -5.9058 -5.8458

28

ALLGARCH(1,2) -5.8714 -5.6939 -5.8777 -5.7993

EGARCH(2,1) -6.4969 -6.3392 -6.5020 -6.4328

SGARCH(2,1) -5.8793 -5.7610 -5.8822 -5.8313

GARCH(2,1) -5.4909 -5.3726 -5.4938 -5.4428

NGARCH(2,1) -5.8656 -5.7275 -5.8695 -5.8095

GJRGARCH(2,1) -5.5407 -5.3829 -5.5457 -5.4766

ALLGARCH(2,1) -5.5867 -5.3698 -5.5960 -5.4986

EGARCH(2,2) -6.3327 -6.1552 -6.3390 -6.2606

SGARCH(2,2) -5.8663 -5.7283 -5.8702 -5.8103

GARCH(2,2) -5.8663 -5.7283 -5.8702 -5.8103

NGARCH(2,2) -5.8590 -5.7013 -5.8641 -5.7950

TGARCH(2,2) -5.8374 -5.6599 -5.8437 -5.7653

GJRGARCH(2,2) -5.5327 -5.3552 -5.5390 -5.4606

AVGARCH(2,2) -5.8142 -5.5973 -5.8235 -5.7261

NAGARCH(2,2) -5.6628 -5.4853 -5.6691 -5.5907

APARCH(2,2) -5.8393 -5.6421 -5.8471 -5.7592

29

Table 4: Selected 5 Best GARCH (p,q) Family Models MODEL AKAIKE BAYES SHIBATA HANNAN QUINN

EGARCH(2,1) -6.4969 -6.3392 -6.5020 -6.4328

EGARCH(2,2) -6.3327 -6.1552 -6.3390 -6.2606

EGARCH(1,2) -5.9588 -5.8208 -5.9627 -5.9028

AVGARCH(1,1) -5.9123 -5.7742 -5.9162 -5.8562

GJRGARCH(1,2) -5.9064 -5.7684 5.9104 -5.8504

Having presented the best five models, we go into the estimation of the parameters of the various

models.

The general GARCH model proposed by Bollerslev and Taylor (1986) represented by GARCH

(p,q) where p and q represents the GARCH and ARCH order of the model respectively is given

by:

2222

211

21|

22|11

21| qtqttptptptttt rrr −−−−−−−−− +++++++= ααασβσβωσ LL

In terms of the backward shift notation B, the model can be expressed as

21

21|1 )()1( t

qqtt

pp rBBBB ααωσββ LL ++=−−− −

Because conditional variances must be nonnegative, the coefficients in a GARCH model are

often constrained to be nonnegative. However, the nonnegative parameter constraints are not

30

necessary for a GARCH model to have nonnegative conditional variances with probability 1; see

Nelson and Cao (1992) and Tsai and Chan (2006).

The likelihood function of a GARCH model can be readily derived for the case of normal

innovations. We illustrate the computation for the case of a stationary GARCH(1,1) model.

Extension to the general case is straightforward. Given the parameters ω, α, and β, the

conditional variances can be computed recursively by the formula

−=−

−

− 21|2

1|

11| 2exp

2

1),,(

tt

t

tt

ttrrtf σπσ

L

Iterating this last formula and taking logs gives the following formula for the log-likelihood function:

∑= −

−−

+−=n

i tt

ttt

rnL

12

1|

22

2|1 )log(2

1)2log(

2),,(

σσπβαω

There is no closed-form solution for the maximum likelihood estimators of ω, α, and β, but they

can be computed by maximizing the log-likelihood function numerically. The maximum

likelihood estimators can be shown to be approximately normally distributed with the true

parameter values as their means.

Their covariances may be collected into a matrix denoted by Λ, which can be obtained as

follows. Let

31

=βαω

θ

be the vector of parameters. write the ith component of θ as θiso that θ1 = ω, θ2 = α, and θ3 = β.

The diagonal elements of Λ are the approximate variances of the estimators, whereas the off-

diagonal elements are their approximate covariances. So, the first diagonal element of Λ is the

approximate variance ofω̂ , the (1,2)th element of Λ is the approximate covariance between ω̂

andα̂ , and so forth. other parameters of higher ordered models can be computed similarly.

The rugarch package in the R statistical analysis software has function that implements the

maximization of this log-likelihood function above. We shall then obtain the parameter estimates

using the rugarch package. Recall that in Table 4, we identified some models to be the best

models, the model fit of these models are reported as follows

----------------------------------- GARCH Model : eGARCH(2,1)

Mean Model : ARFIMA(0,0,1)

Distribution : norm

Optimal Parameters

------------------------------------

Estimate Std. Error t value Pr(>|t|)

mu 0.002196 0.000001 1618.9 0.00 ma1 0.142463 0.000117 1219.5 0.00 omega -1.050754 0.000559 -1880.9 0.00 alpha1 0.384687 0.000224 1718.0 0.00 alpha2 -0.199219 0.000119 -1680.1 0.00

32

beta1 0.895579 0.000402 2225.1 0.00 gamma1 0.294625 0.000238 1237.9 0.00 gamma2 -0.844076 0.000512 -1648.5 0.00 LogLikelihood : 508.2626

Information Criteria ------------------------------------ Akaike -6.4969 Bayes -6.3392 Shibata -6.5020 Hannan-Quinn -6.4328

Weighted Ljung-Box Test on Standardized Squared Residuals ------------------------------------ statistic p-value Lag[1] 0.473 0.4916 Lag[8] 3.603 0.5756 Lag[14] 4.661 0.8102

H0 : No serial correlation

Sign Bias Test ------------------------------------ t-value prob sig Sign Bias 1.1408 0.2558 Negative Sign Bias 0.1261 0.8998 Positive Sign Bias 0.5388 0.5908 Joint Effect 1.6633 0.6451

Adjusted Pearson Goodness-of-Fit Test: ------------------------------------ group statistic p-value(g-1) 1 20 88.34 6.527e-11 2 30 93.01 1.243e-08 3 40 104.18 7.559e-08 4 50 121.97 3.725e-08

34

The ACF plot of the squared residuals of the model shows no significant autocorrelation across

the different lags which shows that the model has taken adequate care of the problem of

heteroscedasticity and further confirms the ARCH LM test of the residuals of the model whose

p-value shows that the model is adequate.

The deviations in the qq plot shows the presence of tail in the distribution of the residuals which

means that the residuals are asymmetric.

35

----------------------------------- GARCH Model : eGARCH(2,2) Mean Model : ARFIMA(0,0,1) Distribution : norm Optimal Parameters ------------------------------------ Estimate Std. Error t value Pr(>|t|) mu 0.001949 0.000002 1021.99 0.00

ma1 0.126403 0.000525 240.97 0.00

omega -0.736879 0.000440 -1674.34 0.00

alpha1 0.594831 0.000416 1429.21 0.00

alpha2 -0.468560 0.000336 -1392.67 0.00

beta1 0.876152 0.000407 2151.55 0.00

beta2 0.051987 0.000032 1625.42 0.00

gamma1 0.335808 0.000632 531.39 0.00

gamma2 -0.724933 0.000661 -1097.37 0.00

LogLikelihood : 496.6161



36

Sign Bias Test ------------------------------------ t-value prob sig Sign Bias 0.7874 0.4323 Negative Sign Bias 0.4684 0.6402 Positive Sign Bias 0.6287 0.5305 Joint Effect 0.7920 0.8514


37







38

---------------------------------- GARCH Model : eGARCH(1,2) Mean Model : ARFIMA(0,0,1) Distribution : norm Optimal Parameters ------------------------------------ Estimate Std. Error t value Pr(>|t|) mu 0.002444 0.001047 2.3336 0.019617

ma1 0.183312 0.056451 3.2473 0.001165

omega -9.999998 3.007586 -3.3249 0.000884

alpha1 0.330885 0.139997 2.3635 0.018102

beta1 0.091154 0.086469 1.0542 0.291796

beta2 -0.231219 0.192416 -1.2017 0.229494

gamma1 0.519903 0.166224 3.1277 0.001762



Weighted Ljung-Box Test on Standardized Residuals ------------------------------------ statistic p-value Lag[1] 2.539 0.11106 Lag[2] 2.601 0.07558 Lag[5] 3.234 0.37469 H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals

------------------------------------

statistic p-value Lag[1] 0.04941 0.8241

Lag[8] 0.41442 0.9979

39

Lag[14] 1.07321 0.9995

Sign Bias Test ------------------------------------ t-value prob sig Sign Bias 0.3083 0.7583 Negative Sign Bias 0.1036 0.9176 Positive Sign Bias 0.3016 0.7634 Joint Effect 0.1689 0.9825 Adjusted Pearson Goodness-of-Fit Test: ------------------------------------ group statistic p-value(g-1) 1 20 278.7 4.476e-48 2 30 332.2 3.199e-53 3 40 325.5 6.992e-47 4 50 309.6 1.566e-39

40







41

----------------------------------- GARCH Model : fGARCH(1,1) fGARCH Sub-Model : AVGARCH Mean Model : ARFIMA(0,0,1) Distribution : norm Optimal Parameters ------------------------------------ Estimate Std. Error t value Pr(>|t|) mu 0.002898 0.000912 3.17684 0.001489

ma1 0.291596 0.104876 2.78039 0.005429

omega 0.010442 0.001687 6.18942 0.000000

alpha1 0.275313 0.104861 2.62552 0.008652

beta1 0.000000 0.155878 0.00000 1.000000

eta11 -1.000000 0.432904 -2.30998 0.020889

eta21 -0.014496 0.067885 -0.21354 0.830904




42

Sign Bias Test ------------------------------------ t-value prob sig Sign Bias 0.03241 0.9742 Negative Sign Bias 0.03544 0.9718 Positive Sign Bias 0.14158 0.8876 Joint Effect 0.03900 0.9980 Adjusted Pearson Goodness-of-Fit Test: ------------------------------------ group statistic p-value(g-1) 1 20 231.2 1.941e-38 2 30 314.3 1.185e-49 3 40 341.6 5.409e-50 4 50 407.7 4.928e-58

43







44

----------------------------------- GARCH Model : fGARCH(1,2) fGARCH Sub-Model : GJRGARCH Mean Model : ARFIMA(0,0,1) Distribution : norm Optimal Parameters ------------------------------------ Estimate Std. Error t value Pr(>|t|) mu 0.002901 0.001276 2.273573 0.022992 ma1 0.297539 0.080354 3.702838 0.000213 omega 0.000114 0.000015 7.791639 0.000000 alpha1 0.190343 0.185851 1.024169 0.305755 beta1 0.000000 0.000080 0.000055 0.999956 beta2 0.000000 0.000004 0.000062 0.999950 eta11 -0.998596 0.847336 -1.178513 0.238592 LogLikelihood : 461.7992 Information Criteria ------------------------------------ Akaike -5.9065 Bayes -5.7684 Shibata -5.9104 Hannan-Quinn -5.8504 Weighted Ljung-Box Test on Standardized Squared Residuals ------------------------------------ statistic p-value Lag[1] 0.2166 0.6417 Lag[8] 0.7375 0.9887 Lag[14] 1.4544 0.9978 Sign Bias Test ------------------------------------ t-value prob sig Sign Bias 0.35814 0.7207 Negative Sign Bias 0.03222 0.9743 Positive Sign Bias 0.19291 0.8473 Joint Effect 0.16158 0.9835

45


46







47

4.4 DETERMINATION OF VOLATILITY CLUSTERING

The exchange rate of Nigeria naira on a united state dollar is a typical financial series.

This is because its exhibited most features of financial series. One of these features is the fact

that the conditional variance is not significantly zero. The conditional variance is volatility. The

plot of returns series of exchange rate in Figure 2, exhibited volatility clustering. In a closer

investigation of the plot, volatility clustering is noticed from the first three years and last two

years in the series of observations for this study and became stable from 2004 till end of 2008.

However, in other to determine the volatility clustering in exchange rate, Generalized

Autoregressive Conditional Hetroscedasticity (GARCH) family models were used. The major

determinant of volatility clustering is constant term (omega) in the conditional variance equation.

According to Musa et al (2014), the significant of this constant term means there is presence of

volatility clustering.

Table 5: Models and Estimate of the Constant Terms (Omega) with p-values in Brackets

Model Omega, ω

eGARCH(2,1) -1.050754(0,00)

eGARCH(2,2) -0.736879(0.00)

eGARCH(1,2) -9.999998(0.00)

AVGARCH(1,1) 0.010442(0.00)

GJRGARCH(1,2) 0.000114(0.00)

48

The constant term of selected five models and their P-values are shown in the above

Table 5.From the table, all constant term are significant. This shows that the coefficient of ω ( a

determinant of presence of volatility clustering, is statistically significant in all the models which

means that there is presence of volatility clustering in the exchange rate returns series.

4.5 Estimated Values of Conditional Variance (Volatility)

The estimated volatility from the fitted models are shown in the tables below.

Table 6: Estimated conditional variance

EGARCH(2,1) EGARCH(2,2) EGARCH(1,2) AVGARCH(1,1) GJRGARCH(1,2)

0.016054052 0.0164570692 0.015678249 0.007600085 0.01521091

0.016054052 0.0164570692 0.015678249 0.009686276 0.01521091

0.020743874 0.0212751442 0.012120364 0.014532028 0.01347140

. . . . .

. . . . .

. . . . .

0.013378623 0.0122651959 0.014877813 0.015910181 0.01458809

0.006268873 0.0052982981 0.011464666 0.010487229 0.01138927

0.015233829 0.0159907241 0.017094683 0.018280685 0.02038530

49

The extract of volatility as shown in the above table indicates that volatility at most points are

close to each other at different interval of time showing volatility clustering. The volatility plot

will show it better.

4.6 Plot of Volatility

The volatility plots of the five models are shown below.

50

Figure 4: Plot of volatility of EGARCH(2,1)

2000 2002 2004 2006 2008 2010 2012

0.00

00.

001

0.00

20.

003

0.00

40.

005

dates

sigm

a(eg

arch

21)̂

2

51


2000 2002 2004 2006 2008 2010 2012

0.00

000.

0005

0.00

100.

0015

0.00

200.

0025

dates

sigm

a(eg

arch

22)̂

2

52


2000 2002 2004 2006 2008 2010 2012

0.00

00.

005

0.01

00.

015

0.02

0

dates

sigm

a(eg

arch

12)̂

2

53

Figure 7: Plot of volatility of AVGARCH(1,1)

2000 2002 2004 2006 2008 2010 2012

0.00

000.

0010

0.00

200.

0030

dates

sigm

a(av

garc

h11)̂

2

54

Figure 8: Plot of volatility of GJR-GARCH(1,2)

The figures above presents time plot of conditional variance from five selected models in the

whole sample period ranging from 1999-2012. The plots show that the conditional variances or

volatility are clustered at beginning of first three years and last two years. Also, the plot of

2000 2002 2004 2006 2008 2010 2012

0.00

00.

002

0.00

40.

006

0.00

80.

010

0.01

2

dates

sigm

a(gj

rgar

ch12

)̂2

55

volatility of the best model (EGARCH(2,2)), showed a clearer display of clustering than all other

models.

4.7 Forecasting

Forecasting is an important application of time series analysis. As stated in the aims and

objectives of the research work and in line with every research work in time series analysis, the

primary objective of building a model for a time series is to be able forecast the future values of

that series. Haven properly taken care of the problem Heteroscedasticity, and adequately

modeled the conditional variance or volatility of the exchange rate returns series, we go further

then to forecast the conditional mean of the exchange rate return series. The method used in

obtaining the forecast for this work is the Minimum Mean Square Error Forecasting. We call

timet the forecast origin and l the lead time for the forecast, and denote the forecast itself as )(ˆ lrt

. The minimum mean square error forecast is such that:

2)(min)](ˆ[ gXElXXE ltg

tlt −−−−≤≤≤≤−−−− ++++++++

whereg is a function of the information available at time t (inclusive). We referred to )(ˆ lX t as the

l-step ahead forecast of tX at the forecast origin t.

The various steps ahead forecasts of the various models are presented in Table 7 alongside the

actual values of the exchange rate return series for the period of forecast. The conditional

standard deviations of the forecast series which can be used to obtain the confidence intervals for

the forecasts are presented in Table 7.

56

Table 7: Multi Step Ahead Forecast of the various models for the Conditional Mean of the Exchange Rate Returns Series

LEAD TIME

ACTUAL AVGARCH (1,1)

GJRGARCH (1,2)

EGARCH

(1,2)

EGARCH

(2,1)

EGARCH

(2,2)

Jan 012 0.0011 0.0050 0.0050 0.0043 0.0038 0.0034

Feb 012 -0.0032 0.0029 0.0029 0.0024 0.0022 0.0019

Mar 012 -0.0018 0.0029 0.0029 0.0024 0.0022 0.0019

Apr 012 -0.0017 0.0029 0.0029 0.0024 0.0022 0.0019

May 012 -0.0003 0.0029 0.0029 0.0024 0.0022 0.0019

Jun 012 0.0010 0.0029 0.0029 0.0024 0.0022 0.0019

Jul 012 -0.0000 0.0029 0.0029 0.0024 0.0022 0.0019

Aug 012 -0.0010 0.0029 0.0029 0.0024 0.0022 0.0019

Sep 012 0.0004 0.0029 0.0029 0.0024 0.0022 0.0019

Oct 012 -0.0002 0.0029 0.0029 0.0024 0.0022 0.0019

Nov 012 0.0000 0.0029 0.0029 0.0024 0.0022 0.0019

Dec 012 0.0000 0.0029 0.0029 0.0024 0.0022 0.0019

57

Table 8: Conditional Standard Deviation of the Multi step Ahead Forecasts of the Various Models for the Conditional mean of the Exchange Rate Return Series LEAD TIME

AVGARCH (1,1)

GJRGARCH (1,2)

EGARCH

(1,2)

EGARCH

(2,1)

EGARCH

(2,2)

Jan 012 0.0145 0.0123 0.0135 0.0066 0.0059

Feb 012 0.0137 0.0131 0.0115 0.0060 0.0062

Mar 012 0.0135 0.0133 0.0121 0.0060 0.0062

Apr 012 0.0135 0.0135 0.0127 0.0061 0.0061

May 012 0.0135 0.0135 0.0126 0.0061 0.0061

Jun 012 0.0135 0.0136 0.0124 0.0062 0.0061

Jul 012 0.0135 0.0136 0.0125 0.0062 0.0061

Aug 012 0.0135 0.0136 0.0125 0.0062 0.0061

Sep 012 0.0135 0.0136 0.0125 0.0062 0.0061

Oct 012 0.0135 0.0136 0.0125 0.0063 0.0061

Nov 012 0.0135 0.0136 0.0125 0.0063 0.0061

Dec 012 0.0135 0.0136 0.0125 0.0063 0.0061

4.8 FORECAST EVALUATION

The performance of volatility forecasts are crucial to many financial applications, which is why

the overall objective is to evaluate which of these selected models predict volatility with the

highest accuracy over a certain horizon of 12 months using five best GARCH family models in

predicting the volatility of the exchange rate returns series.

58

The performance of these various models were evaluated using two standard forecast error

measurements namely Mean Absolute Error (MAE) given by:

∑=

−=n

itt rf

nMAE

1

1

And Root Mean Square Error (RMSE) given by:

∑=

−=n

itt rf

nRMSE

1

2)(1

where: tr is true (observed) value of the exchange rate returns series, tf is the forecast value

and n is the number of fitted points.

The models are evaluated by assessing the returns of their in-sample modeling and their out-

sample forecasting. The one with the lowest error measurement is judged the best. The results of

the performance of these models are presented in Table 9.

59

Table 9: Measurements of Forecast Accuracy of the Models MODEL MAE RMSE

AVGARCH (1,1) 0.0035394 0.003706622

GJRGARCH (1,2) 0.003542399 0.00370946

EGARCH (1,2) 0.003064412 0.003254687

EGARCH (2,1) 0.002795525 0.003002927

EGARCH (2,2) 0.002766381 0.002539101

4.9 RESULTS AND INTERPRETATION

From the result presented in table 9 above, the point of view of the MAE, the best model

was the EGACH (2,2) model with an error of 0.002766381, followed the EGARCH (2,1) model

with an error of 0.002795525, then the EGARCH (1,2) model with an error of 0.003064412, then

AVGARCH (1,1) model with and error of 0.0035394 and GJRGARCH(1,2) model with an error

of 0.003542399.

The same order of ranking for the forecast error was for the case of the RMSE thereby

confirming the results of the MAE to be true.

The forecasts of all the models indicate that the series is going to be a little bit calm with the

values of the predicted volatilities declining and finally taking a constant value for some certain

period of time which might signal the beginning of a crash in price of the exchange rate series.

60

CHAPTER FIVE

5.1 SUMMARY

The main purpose of this research work is to investigate the volatility clustering of

Nigeria’s exchange rates against U.S. Dollar. The monthly average exchange rate of Naira for

the period ranging from January 1999 to December, 2012 with a total of 168 observations.

The stationarity of the exchange rate is examined using graphical analysis which showed the

series as non-stationary, using also the correlogram and unit root test on the series proved the

series to be non-stationary.

To make exchange rate stationary, the exchange rate are transformed to exchange rate

returns. The different models used include GARCH, EGARCH, GJR – GARCH, AVGARCH,

TGARCH, NGARCH, NAGARCH, APARCH, SGARCH and ALLGARCH. Also, Q-statistics

on standardized residuals, Arch test, sign bias test, adjusted pearson goodness of fit test, normal

quantiles, were used to examine the adequacy of these models.

5.2 CONCLUSION

This work investigated the volatility clustering of exchange rate of Nigeria Naira against

United States of America Dollar. It was observed that the selected GARCH family of models

determined volatility clustering of the exchange rate returns series. The work employs three

methods in determining volatility clustering.The results show that the coefficient of ω (a

determination of the presence of volatility clustering is statistically significant in all the models;

this shows the presence of volatility clustering in the exchange rate returns series. Moreso, the

plots of conditional variance of five best models shows that there is presence of volatility

clustering at the beginning of first three years and last two years of this study.

61

In all results the EGARCH models proved to be the best model that could determine the

volatility clustering, leverage effect and asymmetric effect present in the exchange rate returns

series followed by the AVGARCH model and then the GJRGARCH model. Finally, the

EGARCH (2,2) model of plot in conditional variances showed a clearer display of volatility

clustering.

5.3 RECOMMENDATIONS

This Research showed the presence of heteroscedasticity in the Naira to Dollar Exchange rate

series which is the case with most financial time series data.

Further research work needs to be done using other volatility models and higher frequency data

since exchange rate market is a volatile market.

Therefore, i recommend the use of EGARCH models in determining the volatility clustering in

exchange rate returns series because from point of view of the Mean absolute error and Root

mean square error proved it.

Findings from this study have shown that there exists an appreciable level of exchange rate

volatility clustering in the economy and this has its attendant negative effects among which

include; negative balance of payment from foreign trading, unstable, and inadequate flow of

foreign direct investment which leads to weakening of economic growth.

However, the findings will guide the government, through monetary policy makers, to effect

policies which will ensure a calmer economic environment better suited for Nigeria economy.

62

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65

APPENDIX: A

Exchange rate, represent in terms of U.S Dollar, consist of monthly average foreign exchange rate Nigeria currency against

U.S Dollar. The period of the data is ranging from January, 1999 to December, 2012 with a total point of 168.

Table D.3.1: Monthly Average Official Exchange Rate of the Naira (N/US$1.00)

Year January February March April May June July August September October November December Average

1999 86.00 86.00 86.97 90.00 94.88 94.88 94.88 94.88 94.88 94.90 96.45 97.60 92.69

2000 98.78 99.91 100.93 100.38 101.15 101.83 105.33 102.88 102.36 102.48 102.52 106.71 102.11

2001 110.50 110.70 110.66 113.70 113.57 112.48 111.85 111.70 111.60 111.60 111.99 112.99 111.94

2002 113.96 114.28 116.04 116.13 116.55 118.49 123.72 125.75 126.45 126.56 126.83 126.88 120.97

2003 127.07 127.32 127.16 127.37 127.67 127.83 127.77 127.90 128.58 129.79 136.61 137.22 129.36

2004 136.08 135.16 134.43 133.51 133.01 132.75 132.80 132.83 132.84 132.86 132.87 132.86 133.50

2005 132.86 132.85 132.85 132.85 132.82 132.87 132.87 133.23 130.81 130.84 130.63 130.29 132.15

2006 130.29 129.59 128.70 128.47 128.45 128.45 128.38 128.33 128.29 128.28 128.29 128.30 128.65

2007 128.28 128.27 128.15 127.98 127.56 127.41 127.19 126.68 125.88 124.27 120.12 118.21 125.83

2008 117.98 118.21 117.92 117.87 117.83 117.81 117.77 117.74 117.73 117.72 117.74 126.48 118.57

2009 145.78 147.14 147.72 147.23 147.84 148.20 148.59 151.86 152.30 149.36 150.85 149.95 148.90

2010 149.78 150.22 149.83 149.89 150.31 150.19 150.10 150.27 151.03 151.25 150.22 150.48 150.30

2011 151.55 151.94 152.51 153.97 154.80 154.50 151.86 152.72 155.26 153.26 155.77 158.21 153.86

2012 158.38 157.87 157.59 157.33 157.28 157.44 157.43 157.28 157.34 157.31 157.31 157.32 157.49

66

APPENDIX B:R CODE

library(zoo)

library(tseries)

library(timeDate)

library(timeSeries)

library(FinTS)

library(forecast)

library(rugarch)

X=read.csv(“C:/exchange.csv”)

Exchange=ts(exchange,start=c(1999,1),freq=12)

returns=(diff(log(exchange)))

plot(exchangee,type="l")

plot(returns,type="l")

par(mfrow=c(2,2))

par(mfrow=c(2,2))

acf(EXRate,50,main="ACF of Exchange Rate Series")

pacf(EXRate,50,main="PACF of Exchange Rate Series")

acf(returns,50,main="ACF of Exchange Rate Return Series")

pacf(returns,50,main="PACF of Exchange Rate Return Series")

adf.test(EXRate)

adf.test(returns)

pp.test(EXRate)

pp.test(returns)

m1=arima(returns,order=c(0,0,1))

ArchTest(residuals(m1),lags=1, demean = FALSE)



67

spec=ugarchspec(variance.model = list(model = "eGARCH", garchOrder = c(1,2)),mean.model = list(armaOrder = c(0,1), include.mean = TRUE),distribution.model = "norm")

egarch12=ugarchfit(spec,data=returns,out.sample=12)

egarch12



egarch22



egarch12

spec = ugarchspec(variance.model = list(model = "fGARCH",submodel='AVGARCH', garchOrder = c(1,1)),mean.model = list(armaOrder = c(0,1), include.mean = TRUE),distribution.model = "norm")

avgarch11=ugarchfit(spec,data=returns,out.sample=12)

avgarch11

spec = ugarchspec(variance.model = list(model = "fGARCH",submodel='GJRGARCH', garchOrder = c(1,2)),mean.model = list(armaOrder = c(0,1), include.mean = TRUE),distribution.model = "norm")

gjrgarch12=ugarchfit(spec,data=returns,out.sample=12)

gjrgarch12

sigma=signature(object = "uGARCHfilter")

sigma(egarch12)

sigma(egarch21)

sigma(egarch22)

sigma(avgarch11)

sigma(gjrgarch12)

dates <- seq(as.Date("01/01/1999", format = "%d/%m/%Y"), by = "months", length = length(sigma(egarch12)^2))

plot(dates,sigma(egarch12)^2,type="l")


68


plot(dates,sigma(avgarch11)^2,type="l")

plot(dates,sigma(gjrgarch12)^2,type="l")

igarch11for=ugarchforecast(egarch12, data = TRUE, n.ahead = 12)

egarch11for=ugarchforecast(egarch12, data = TRUE, n.ahead = 12)


egarch12for


egarch22for




egarch21


egarch21for

avgarch11for=ugarchforecast(avgarch11, data = TRUE, n.ahead = 12)

avgarch11for

gjrgarch12for=ugarchforecast(gjrgarch12, data = TRUE, n.ahead = 12)

gjrgarch12for

accuracy(returns[155:166],fitted(egarch12for))



accuracy(returns[155:166],fitted(avgarch11for))

accuracy(returns[155:166],fitted(gjrgarch12for))

Estimated Conditional variance(volatility) Appendix C.

> sigma(egarch12)

69

[1] 0.015678249 0.015678249 0.012120364 0.025857023 0.019979714 0.009549588

[7] 0.009346308 0.010590142 0.010553967 0.010352632 0.017393779 0.011935860

[13] 0.012487454 0.012641981 0.012113659 0.010836616 0.012727208 0.011118553

[19] 0.010777574 0.016918236 0.010883743 0.009633771 0.010364184 0.044089915

[25] 0.012907623 0.008721579 0.010115578 0.027786762 0.010598857 0.010028690

[31] 0.010927939 0.010523816 0.010428274 0.010372257 0.010689924 0.012786215

[37] 0.011829216 0.009921679 0.016729439 0.010538243 0.010174648 0.017332638

[43] 0.024622298 0.010655105 0.009641963 0.010363607 0.010258464 0.010410789

[49] 0.010243809 0.010196549 0.010630841 0.010201136 0.010128390 0.010326323

[55] 0.010525119 0.010278050 0.011382748 0.012713163 0.044594448 0.010104202

[61] 0.009327776 0.011259197 0.011043195 0.011047079 0.010615848 0.010496571

[67] 0.010297699 0.010380972 0.010414629 0.010391688 0.010398546 0.010418280

[73] 0.010404741 0.010413654 0.010404104 0.010404999 0.010433881 0.010354334

[79] 0.010411821 0.010397376 0.013005634 0.010595496 0.010345505 0.010659980

[85] 0.010362917 0.011067398 0.011112132 0.010390643 0.010314095 0.010409284

[91] 0.010486188 0.010439849 0.010425980 0.010404667 0.010392671 0.010407988

[97] 0.010416876 0.010413244 0.010518693 0.010545894 0.010769067 0.010455999

[103] 0.010554497 0.010860153 0.010920339 0.011996639 0.013822155 0.011537421

[109] 0.009925100 0.010033032 0.010846898 0.010443575 0.010387608 0.010414424

[115] 0.010447753 0.010428233 0.010405685 0.010412054 0.010383116 0.145591698

[121] 0.014552969 0.008000133 0.011858450 0.011271312 0.011001013 0.010097574

[127] 0.010168982 0.021658104 0.010384163 0.011823035 0.014792738 0.011046733

[133] 0.009744049 0.010238975 0.010896695 0.010311099 0.010293775 0.010579250

[139] 0.010464604 0.010234993 0.011278054 0.010342903 0.011179013 0.010463179

[145] 0.012124934 0.010260380 0.010541150 0.012889018 0.010836409 0.010433969

[151] 0.012610175 0.012543565 0.014877813 0.011464666 0.017094683

70

> sigma(egarch21)

[1] 0.016054052 0.016054052 0.020743874 0.027349976 0.023752614 0.009855612

[7] 0.010636731 0.011387914 0.012147481 0.012965564 0.020204256 0.014358308

[13] 0.015770885 0.014977681 0.014767767 0.011598043 0.014891850 0.013721650

[19] 0.012138143 0.018905393 0.010747405 0.011720100 0.012616301 0.037223463

[25] 0.010751053 0.007721130 0.008282571 0.025161870 0.005587537 0.004526130

[31] 0.003780534 0.003714640 0.003770919 0.003997707 0.005177063 0.008120372

[37] 0.006427481 0.005400168 0.013938750 0.004534714 0.005935547 0.013300680

[43] 0.012173867 0.003993727 0.003795395 0.003350493 0.003996560 0.004406733

[49] 0.005025489 0.005849613 0.006130657 0.006905164 0.007979500 0.008913088

[55] 0.009523809 0.010401536 0.012717261 0.014625910 0.036810131 0.006986172

[61] 0.006180434 0.005355425 0.004782990 0.003876931 0.003397749 0.003199553

[67] 0.003409259 0.003663435 0.003911844 0.004203869 0.004519309 0.004840001

[73] 0.005206256 0.005597901 0.006039010 0.006521454 0.006999544 0.007623026

[79] 0.008233316 0.009433038 0.007138117 0.007480301 0.007825705 0.007905007

[85] 0.008460763 0.008138409 0.007432452 0.007556568 0.008094444 0.008721145

[91] 0.009278423 0.009876645 0.010520607 0.011233611 0.012003900 0.012781323

[97] 0.013555201 0.014336629 0.014963324 0.015496325 0.015664455 0.016168948

[103] 0.016628016 0.016649967 0.016420479 0.014694914 0.010475202 0.007801500

[109] 0.007830039 0.008820422 0.009084857 0.009612821 0.010246451 0.010931360

[115] 0.011611445 0.012320458 0.013079459 0.013854695 0.014685388 0.075915213

[121] 0.011850319 0.005009880 0.006617840 0.005552450 0.007159983 0.006917364

[127] 0.008062239 0.020347386 0.006463307 0.004128054 0.009166218 0.002724378

[133] 0.003057447 0.003899013 0.003589816 0.003834346 0.004714996 0.004856890

[139] 0.005088763 0.005692203 0.007719194 0.006978261 0.006364362 0.007064163

[145] 0.010335942 0.008144161 0.009811167 0.012777030 0.010979849 0.010329458

71

[151] 0.008176806 0.010648946 0.013378623 0.006268873 0.015233829

> sigma(egarch22)

[1] 0.0164570692 0.0164570692 0.0212751442 0.0236442888 0.0203950198

[6] 0.0081001455 0.0116144748 0.0120204528 0.0133653945 0.0145421667

[11] 0.0208139779 0.0143013496 0.0168466656 0.0147413510 0.0150516361

[16] 0.0111036457 0.0157962301 0.0130896802 0.0115023685 0.0183135168

[21] 0.0089181997 0.0124762135 0.0125637849 0.0311947836 0.0079391571

[26] 0.0078866914 0.0090500712 0.0213447980 0.0046763477 0.0043042414

[31] 0.0033098008 0.0036616913 0.0039911699 0.0047033413 0.0066192664

[36] 0.0077857818 0.0065874553 0.0058084405 0.0124237637 0.0037308523

[41] 0.0077633543 0.0092063077 0.0128351859 0.0021830970 0.0045015883

[46] 0.0021002211 0.0040909547 0.0032989352 0.0048324871 0.0054263909

[51] 0.0053124390 0.0068809655 0.0077170914 0.0083231500 0.0091528801

[56] 0.0105877436 0.0130782138 0.0136616189 0.0284486991 0.0046568553

[61] 0.0048871983 0.0035416665 0.0028857139 0.0016697904 0.0011046740

[66] 0.0007911423 0.0010449727 0.0012302483 0.0014340403 0.0017159278

[71] 0.0020318238 0.0023639779 0.0027922311 0.0032635348 0.0038396713

[76] 0.0044913973 0.0051443662 0.0060894144 0.0069570557 0.0087737829

[81] 0.0044766557 0.0060975767 0.0059076392 0.0061161236 0.0070655838

[86] 0.0063597166 0.0053864517 0.0058278565 0.0066732824 0.0076065768

[91] 0.0084146196 0.0093429652 0.0103277012 0.0114240568 0.0125893567

[96] 0.0137413310 0.0148786683 0.0160163997 0.0168671180 0.0176156795

[101] 0.0177500099 0.0186120701 0.0191868990 0.0190825733 0.0187166359

[106] 0.0159306521 0.0097680257 0.0065083497 0.0072968924 0.0088770155

[111] 0.0083307348 0.0095281910 0.0104392686 0.0115286163 0.0125554234

[116] 0.0136408659 0.0147824485 0.0159174916 0.0171271356 0.0520138805

72

[121] 0.0111929760 0.0054935857 0.0091921794 0.0064754901 0.0096542592

[126] 0.0083473454 0.0101754169 0.0172675477 0.0066235078 0.0038956975

[131] 0.0093250049 0.0019796040 0.0039159463 0.0046596879 0.0035605379

[136] 0.0046971235 0.0062460270 0.0054810513 0.0064883740 0.0077003406

[141] 0.0099822394 0.0086639647 0.0080081585 0.0097211137 0.0120049346

[146] 0.0097879272 0.0124865193 0.0138055155 0.0122752189 0.0114874738

[151] 0.0082609715 0.0117510037 0.0122651959 0.0052982981 0.0159907241

> sigma(avgarch11)

[1] 0.007600085 0.009686276 0.014532028 0.022684375 0.028682226 0.009896751

[7] 0.011644201 0.009985209 0.010323550 0.010285276 0.016457943 0.013775720

[13] 0.014515666 0.014043716 0.013680960 0.009894635 0.014001007 0.012178814

[19] 0.009811864 0.016700102 0.009986850 0.011788468 0.010115317 0.025722530

[25] 0.021126308 0.009745628 0.010590310 0.020580295 0.009820878 0.009935411

[31] 0.009691368 0.010045452 0.009893536 0.010276682 0.011550717 0.013436078

[37] 0.013046205 0.010972312 0.016097977 0.009598504 0.011948216 0.016276696

[43] 0.025929662 0.014163252 0.012330300 0.010372836 0.011106901 0.010242984

[49] 0.010836520 0.010886766 0.009661315 0.010982116 0.010989465 0.010615999

[55] 0.010014614 0.010670319 0.012189228 0.013530549 0.029473429 0.009671346

[61] 0.009828398 0.009808733 0.009696540 0.009876614 0.009579629 0.009612244

[67] 0.010449507 0.010256108 0.010262101 0.010282607 0.010250522 0.010198626

[73] 0.010234818 0.010197939 0.010234323 0.010226958 0.010142660 0.010387877

[79] 0.010199759 0.011281163 0.010756688 0.012555739 0.009612489 0.009562880

[85] 0.010325637 0.009820664 0.009814161 0.010082763 0.010176941 0.010231870

[91] 0.010016783 0.010113874 0.010119647 0.010212427 0.010257501 0.010223736

[97] 0.010200993 0.010203397 0.009871956 0.009775500 0.009654085 0.010038034

[103] 0.009573180 0.009698326 0.009735357 0.010345409 0.011335983 0.010203439

73

[109] 0.010757539 0.010891239 0.009622622 0.010316003 0.010060767 0.010194528

[115] 0.010097471 0.010151125 0.010203967 0.010197415 0.010298588 0.037771483

[121] 0.060386187 0.010083097 0.014708378 0.009802493 0.012530874 0.010772758

[127] 0.011240648 0.018461736 0.010037864 0.010742881 0.016215271 0.010152634

[133] 0.011838602 0.011092166 0.009631066 0.010655740 0.011213024 0.009777603

[139] 0.010110095 0.010666565 0.012094345 0.010501647 0.009909885 0.011629436

[145] 0.012892414 0.010710456 0.011719424 0.013470109 0.011951516 0.009616962

[151] 0.010596627 0.014399133 0.015910181 0.010487229 0.018280685

> sigma(gjrgarch12)

[1] 0.01521091 0.01521091 0.01347140 0.02853775 0.03969821 0.01101759

[7] 0.01047267 0.01034397 0.01031922 0.01032235 0.01665885 0.01125888

[13] 0.01262326 0.01185288 0.01152809 0.01060903 0.01267886 0.01041724

[19] 0.01051318 0.01728176 0.01077410 0.01051854 0.01033794 0.03693638

[25] 0.02166612 0.01046843 0.01031278 0.02509056 0.01070562 0.01055383

[31] 0.01040787 0.01032716 0.01033609 0.01032058 0.01038799 0.01162838

[37] 0.01099157 0.01031436 0.01582114 0.01041679 0.01061377 0.01586098

[43] 0.03540127 0.01056186 0.01049752 0.01032922 0.01031226 0.01032758

[49] 0.01031112 0.01031145 0.01035456 0.01031111 0.01031085 0.01031582

[55] 0.01033396 0.01031255 0.01065072 0.01163490 0.04453386 0.01078605

[61] 0.01047255 0.01047418 0.01040755 0.01051592 0.01035910 0.01034716

[67] 0.01031535 0.01032378 0.01032256 0.01032200 0.01032310 0.01032470

[73] 0.01032323 0.01032466 0.01032324 0.01032367 0.01032659 0.01031849

[79] 0.01032536 0.01033092 0.01163639 0.01096254 0.01040485 0.01035102

[85] 0.01031820 0.01048894 0.01047411 0.01032630 0.01032480 0.01032320

[91] 0.01033162 0.01032655 0.01032689 0.01032361 0.01032257 0.01032388

[97] 0.01032449 0.01032430 0.01033797 0.01034041 0.01038898 0.01032729

74

[103] 0.01035308 0.01040703 0.01042720 0.01097484 0.01267188 0.01083657

[109] 0.01031167 0.01031134 0.01038740 0.01031829 0.01033045 0.01032387

[115] 0.01032817 0.01032561 0.01032406 0.01032452 0.01032122 0.06519843

[121] 0.10995402 0.01273161 0.01399093 0.01056696 0.01103234 0.01031776

[127] 0.01031796 0.02037369 0.01040668 0.01155005 0.01620148 0.01094966

[133] 0.01053721 0.01031109 0.01039269 0.01031202 0.01031540 0.01035039

[139] 0.01032535 0.01031284 0.01059697 0.01032321 0.01055524 0.01043962

[145] 0.01106673 0.01032006 0.01042644 0.01163825 0.01037610 0.01039654

[151] 0.01131298 0.01315366 0.01458809 0.01138927 0.02038530

faculty of physical science · study are purely time series data based on the fact that are indexed...

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