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
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NWOYE HENRY CHUKWUNONSO
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
NWOYE HENRY CHUKWUNONSO
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.
NWOYE HENRY CHUKWUNONSO
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 5: Plot of volatility of EGARCH(2,2) - - - - - - - 51
Figure 6: Plot of volatility of EGARCH(1,2) - - - - - - - 52
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
33
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
Information Criteria ------------------------------------ Akaike -6.3327 Bayes -6.1552 Shibata -6.3390 Hannan-Quinn -6.2606
Weighted Ljung-Box Test on Standardized Squared Residuals ------------------------------------ statistic p-value Lag[1] 0.07347 0.7863 Lag[11] 1.89059 0.9670 Lag[19] 3.40688 0.9903
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
Adjusted Pearson Goodness-of-Fit Test: ------------------------------------ group statistic p-value(g-1) 1 20 113.3 1.981e-15 2 30 125.7 5.177e-14 3 40 135.9 1.208e-12 4 50 157.7 2.414e-13
37
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.
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
LogLikelihood : 465.8308
Information Criteria ------------------------------------ Akaike -5.9588 Bayes -5.8208 Shibata -5.9627 Hannan-Quinn -5.9028
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
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.
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
LogLikelihood : 462.2443
Information Criteria ------------------------------------ Akaike -5.9123 Bayes -5.7742 Shibata -5.9162 Hannan-Quinn -5.8562
Weighted Ljung-Box Test on Standardized Squared Residuals ------------------------------------ statistic p-value Lag[1] 1.023 0.3119 Lag[5] 1.264 0.7976 Lag[9] 1.554 0.9510
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
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.
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
Adjusted Pearson Goodness-of-Fit Test: ------------------------------------ group statistic p-value(g-1) 1 20 239.8 3.630e-40 2 30 280.0 7.024e-43 3 40 342.6 3.402e-50 4 50 348.6 8.621e-47
46
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.
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
Figure 5: Plot of volatility of EGARCH(2,2)
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
Figure 6: Plot of volatility of EGARCH(1,2)
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)
ArchTest(residuals(m1),lags=4, demean = FALSE)
ArchTest(residuals(m1),lags=8, 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
spec=ugarchspec(variance.model = list(model = "eGARCH", garchOrder = c(2,2)),mean.model = list(armaOrder = c(0,1), include.mean = TRUE),distribution.model = "norm")
egarch22=ugarchfit(spec,data=returns,out.sample=12)
egarch22
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
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")
plot(dates,sigma(egarch21)^2,type="l")
68
plot(dates,sigma(egarch22)^2,type="l")
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=ugarchforecast(egarch12, data = TRUE, n.ahead = 12)
egarch12for
egarch22for=ugarchforecast(egarch22, data = TRUE, n.ahead = 12)
egarch22for
egarch21for=ugarchforecast(egarch21, data = TRUE, n.ahead = 12)
spec=ugarchspec(variance.model = list(model = "eGARCH", garchOrder = c(2,1)),mean.model = list(armaOrder = c(0,1), include.mean = TRUE),distribution.model = "norm")
egarch21=ugarchfit(spec,data=returns,out.sample=12)
egarch21
egarch21for=ugarchforecast(egarch21, data = TRUE, n.ahead = 12)
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(egarch22for))
accuracy(returns[155:166],fitted(egarch21for))
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