BSc Dissertation

Using GARCH-family
models to forecast stock
market volatility
Darja Kiseleva
489688
BSc Finance
Supervisor
Dr Konstantinos Vergos
March 2014

Using GARCH-family models to forecast stock market volatility 2014
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Statement of originality
This dissertation is submitted in partial fulfilment of the requirements for the degree of
BSc Finance. I declare that this dissertation is my own original work. Where I have
taken ideas and/or wording from another source this is explicitly referenced in text.
I give permission that this dissertation may be photocopied and made available
through the university library, in printed from and/or electronic form.
I provide a copy of the electronic source from which this dissertation was printed. I
give my permission for this dissertation, and electronic source, to be used in any
manner considered necessary to fulfil the requirements of the University of Portsmouth
Regulations, Procedures and Codes of Practice.
Word count: 10,107 (excluding acknowledgements, abstract, bibliography,
appendices, covers page, glossary, list of tables/figures and tables in the body part).
Signed Date

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Acknowledgements
First of all, I would like to thank my supervisor Dr Konstantinos Vergos for his
continuous help, guidance and knowledge throughout the duration of this dissertation,
as well as, thank you for the inspiration you gave me.
I would also like to thank my Mum and Grandparents for their help, support and
encouragement during my time at the University of Portsmouth.
A special thanks to the best housemates – Alex and Greg for their help and amazing
time together.
Finally, thank you for taking the time to read this dissertation.

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Abstract
Reliable and accurate volatility forecasts are vital for such financial field activities, as
risk management, portfolio pricing, hedging, options pricing and exercising and general
investments strategy. This dissertation looks at the abilities of GARCH family models to
forecast stock market volatility. Stock returns of the FTSE 100 covering 10 years period
from January 2003 to December 2013are examined in attempt to contribute to wide
range of studies made on GARCH model.
The empirical analysis is conducted by means of various GARCH models –
symmetric and asymmetric. Through the analysis, it was found that data set exhibits
ARCH effects, as well as, stylised volatility factors are present. The findings suggest
that asymmetric GARCH – EGARCH and TARCH are the most appropriate to model
FTSE 100 stock return volatility. The results provide evidence of the superiority of
EGARCH(1,1) model over TARCH(1,1) model, however, performance of models does
not differ significantly. Both models are superior to symmetric GARCH models, which
indicate that asymmetries play a major role in returns distribution. Normal error term
distribution has shown better performance than Student’s-t distribution, meaning that fat
tails are well captures by a normal distribution model. The conclusion is supported by
six different loss function measures and two information criterion that were used to
evaluate the forecasting accuracy, in order to present the clear distinction between the
best forecasting models.

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Table of contents
Statement of originality ............................................................................................ 2
Acknowledgements .................................................................................................. 3
Abstract ..................................................................................................................... 4
Table of contents ...................................................................................................... 5
List of tables and figures.......................................................................................... 8
Glossary..................................................................................................................... 9
1. INTRODUCTION.............................................................................................. 12
1.1. Background and motivation ....................................................................... 12
1.2. Outline of the dissertation.......................................................................... 13
2. THE FTSE 100 STOCK MARKET ................................................................... 15
3. STOCK PRICES VOLATILITY ........................................................................ 16
3.1. Stylised factors about volatility................................................................... 17
3.1.1. Volatility persistence ........................................................................... 17
3.1.2. Leverage effect and volatility asymmetry ............................................ 17
3.1.3. Long memory of shocks...................................................................... 18
3.1.4. Other factors that affect stock market volatility.................................... 18
3.2. Volatility forecasting................................................................................... 19
3.3. Frequency of observations ........................................................................ 19
4. AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTICITY MODEL...... 21
4.1. The basic ARCH-GARCH models ............................................................. 21

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4.2. GARCH model with conditional mean........................................................ 24
4.3. Asymmetric GARCH models ..................................................................... 25
4.4. “Long memory” GARCH model.................................................................. 27
4.5. GARCH with non-normal error terms distribution ...................................... 27
4.6. Literature review of GARCH models.......................................................... 28
5. METHODOLOGY............................................................................................. 37
5.1. Data........................................................................................................... 37
5.2. Statistical analysis of data ......................................................................... 37
5.3. Mean equation........................................................................................... 39
5.4. Testing for ARCH effects........................................................................... 40
5.4.1. Autocorrelation test ............................................................................. 40
5.4.2. Lagrange multiplier test....................................................................... 40
5.5. Model selection.......................................................................................... 41
5.6. Evaluation of estimated results.................................................................. 42
5.6.1. Loss function....................................................................................... 42
5.6.2. Information criterion ............................................................................ 44
6. EMPIRICAL RESULTS AND ANALYSIS........................................................ 45
6.1. Test for ARCH effects................................................................................ 45
6.1.1. LM test ................................................................................................ 45
6.1.2. Ljung-Box test ..................................................................................... 46
6.2. Model selection process ............................................................................ 46
6.3. Estimation of different GARCH models...................................................... 48

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6.4. Statistical measures of estimated models.................................................. 52
6.5. Estimating forecast results......................................................................... 56
6.5.1. Loss functions ..................................................................................... 56
6.5.2. Akaike information criteria and Bayesian information criteria.............. 59
6.6. Comparison of findings with other studies on GARCH model.................... 60
7. CONCLUSION................................................................................................. 64
7.1. Summary of findings.................................................................................. 64
7.2. Limitations ................................................................................................. 65
Bibliography............................................................................................................ 67
Appendices.............................................................................................................. 77
Appendix A............................................................................................................ 77
Appendix B............................................................................................................ 79
Appendix C............................................................................................................ 85

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List of tables and figures
Table 1 Various studies on GARCH-family models................................................... 29
Table 2 Statistical measure of daily returns of FTSE 100 ......................................... 38
Table 3 Descriptive statistic of residual series .......................................................... 46
Figure 1 ACF and PACF values................................................................................ 47
Table 4 Criteria values of GARCH(p,q)-family models.............................................. 48
Table 5 Estimates of GARCH(1,1)-family models ..................................................... 49
Table 6 Estimates of GARCH(1,1)-family models with Student’s-t distribution ......... 50
Table 9 Statistical measures of GARCH(1,1)-family models..................................... 52
Table 10 Ljung-Box Q test statistic of GARCH(1,1)-family models ........................... 54
Table 11 Loss function value of GARCH(1,1)-family models .................................... 57
Table 12 Loss function values of GARCH(1,1)-family models with Student’s-t
distribution..................................................................................................................... 58
Table 13 AIC and BIC values of models with normal error terms distribution ........... 59
Table 14 AIC and BIC values of models with Student’s-t error terms distribution ..... 59
Table 15 Overall findings of different studies on GARCH-family models .................. 61
Figure 2 FTSE 100 returns........................................................................................ 61

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Glossary
AGARCH – Absolute value GARCH model
AIC – Akaike information criterion
ANN-GARCH – Artificial Neutral Network GARCH model
APARCH – Asymmetric Power ARCH model
AR – Autoregressive model
ARCH – Autoregressive Conditional Heteroscedasticity model
BEKK – model of Baba, Engle, Kraft and Kroner (1990)
BIC – Bayesian Information criterion
CAPM – Capital Asset Pricing Model
CC-GARCH – Conditional Correlation GARCH model
CGARCH – Component GARCH model
COGARCH – Continuous time concept GARCH model
DAX – Deutscher Aktienindex
DCC – Dynamic Conditional Correlation model
DTGARCH – Double Threshold GARCH model
EGARCH – Exponential GARCH model
EWMA – Exponentially Weighted Moving Average
FGARCH – Factor GARCH model
FIGARCH – Fractionally Integrated GARCH model
FIREGARCH – Fractionally Integrated Range Exponential GARCH model
FTSE 100 – Financial Times Stock Exchange Index

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GARCH – General Autoregressive Conditional Heteroscedasticity model
GARCH-HT – GARCH Heavy-Tailed distribution model
GARCH-M – GARCH-in-Mean model
GARCH-SGT – GARCH Skewed Generalised t-distribution model
GARCH-t – GARCH t-distribution model
GED-GARCH – Generalized Error Distribution GARCH
GJR-GARCH – Glosten, Jagannathan and Runkle GARCH model
GOGARCH – Generalised Orthogonal GARCH model
IGARCH – Integrated GARCH model
MAE – Mean Absolute Error
MAPE – Mean Absolute Percentage Error
ME – Mean Error
MedSE – Median Square Error
MGARCH – Multivariate GARCH model
MSE – Mean Square Error
NAGARCH – Nonlinear Asymmetric GARCH model
NGARCH – Nonlinear GARCH model
NIKKEI – Japanese Stock Market Index
NM-GARCH – Normal Mixture GARCH model
NN – Neutral Networks model
OLS – Ordinary Least Squares
PGARCH – Periodic GARCH model

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QGARCH – Quadratic GARCH model
RMSE – Root Mean Square Error
RSGARCH – Regime Switching GARCH model
S&P 500 – Standard & Poor’s 500 Index
SAGARCH – Simple Asymmetric GARCH model
SSR – Sum of Squared Residuals
TARCH – Threshold ARCH model
TIC – Theil’s Inequality Coefficient
VaR – Value-at-Risk model
VAR – Vector Autoregression model
VC-GARCH – Varying Correlation GARCH model
VEC-GARCH – Vector GARCH model

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"Volatility forecasting is a little like predicting whether it will rain; you can be correct in
predicting the probability of rain, but still have no rain."
- Engle (1993)
1. INTRODUCTION
1.1. Background and motivation
The topic of volatility has always been in the centre of attention. Modelling and
forecasting stock market volatility has attracted a large number of researchers,
academics and regulators, due to its importance in several financial applications, like
the analysis of market timing decisions, understanding of better portfolio selection and
asset allocation process. Volatility is also vital in such financial filed activities, as risk
management assessment, portfolio management, as well as option pricing and trading
and value-at-risk models. Therefore, financial institutions are interested not only in
knowing the current level of volatility of the market or individually managed assets, but
they also should not underestimate the importance of future volatility predictions.
Mathematical and econometric modelling is a useful tool to build the relationship
between current values of financial indicators and their future expected values. Different
models are used to provide investors, researchers or various financial institutions with
estimates of a future market trends. Financial data volatility features such
characteristics, as leptokurtosis and clustering (Mandelbrot, 1963) thus forecasting
models have to capture these characteristics in order to produce reliable future
estimations and, therefore, provide better protection against risk that investors are
facing. ARCH model of Engle (1982) and GARCH model of Bollerslev (1986) are
successful in capturing the volatility factors and provide reliable estimations of time
varying volatility of different financial data. There are a lot of evidences that support
GARCH models (see Schwert et al, 1990, Brainsfold and Faff, 1996) in both its ability to
estimate and forecast volatility. However, in spite of its success, the GARCH model has

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been criticised for not being able to accommodate all dependencies that volatility has,
like asymmetry and fat tails. Nonetheless, these shortcomings have been overcome by
introduction of models like IGARCH, EGARCH, GARCH-M, TARCH, APARCH, and
others. However, there is no unified opinion on what model is superior, as some favour
simple GARCH, yet other give preference to more sophisticated variations of GARCH.
Therefore, this dissertation employs various models to study volatility.
The data is composed of FTSE 100 stock market returns covering 10 years period
from 1st
January 2003 to 31st
December 2013. The choice of stock market is based on
the fact that UK stock market is one of the largest and influential stock exchanges in the
world. Presence of heteroscedasticity in the data set indicates that GARCH models
have to be used.
The main aim of the paper is to estimate FTSE 100 stock market volatility and
indicate the most accurate model to forecast it. Statistical features of models, loss
functions and different information criteria help to determine the best model.
1.2. Outline of the dissertation
The dissertation has the following structure:
Section 1 is the introduction. It indicates the main objectives of dissertation and
covers the main aspects of volatility forecasting, as well as, topic’s importance.
Section 2 explains the choice of stock market and briefly covers some most vital
aspects of the FTSE 100 stock market.
Section 3 covers the topic of stock prices volatility. It shows why volatility forecasting
is important and discusses stylised volatility factors that provide a better understanding
of the subject.

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Section 4 focuses on GARCH-family models. It describes the econometric application
of models, their ability to forecast volatility and provides the literature review of various
studies.
Section 5 describes the methodology that is employed in the dissertation; shows
preliminary data tests and criteria to choose the best forecasting model.
Section 6 is the empirical research of the data set. Section provides with estimates
obtained when regressing different GARCH-family models. It also sums up the overall
results and indicates the best performing model.
Section 7 is the conclusion part which covers the most vital findings and limitations of
the study.

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2. THE FTSE 100 STOCK MARKET
The correct choice of the stock market and period of data set is one of the important
aspects after choosing the topics specification. Nowadays, the financial world is highly
interconnected between its different parts, as globalisation becomes more substantial.
When talking about financial institutions, like stock markets, it is important to look at the
broader picture and learn about relationship between stock markets of different
countries. However, under the scope of this research, one specific stock market will be
studied.
Even though it is believed that American stock markets are the biggest and most
developed in the world, the impact of the smaller European stock exchanges on the
world economy should not be underestimated. Being one of the biggest stock market in
Europe along with German DAX 30 and French CAC 40, FTSE 100 is the popular place
for investment decisions among traders and investors in the world.
The Financial Times Stock Exchange 100 index – the FTSE 100 has been launched
back in January 1984. It is being part of the FTSE UK series, which also include indices
like FTSE 250 and FTSE All Shares. The FTSE 100 is the index of the 100 largest listed
UK companies weighted according to their market capitalization.

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3. STOCK PRICES VOLATILITY
The importance of understanding future movements of the stock market cannot be
underestimated. Volatility of asset prices and, consequently, returns is one of the vital
components of modelling future behaviour of stocks. Therefore, a lot of attention has
been addressed to understanding and predicting volatility, as well as, many researches
were conducted to find the most suitable and efficient prediction model. In 2003 Poon
and Granger (p. 478) have estimated that around 93 reviews or working papers have
been published on the topic of volatility forecasting. It is highly probable to say that
amount of studies have probably risen by a couple of times since then.
Future volatility of the individual stock or portfolio of stocks is unpredictable.
Generally speaking, stock volatility is an up and down movements of the stock prices.
As from an academic point of view, volatility is a “measure of the uncertainty of the
return realised on an asset” (Hull, 2009, p. 792) and is expressed as a standard
deviation or a variance . Commonly used formula is as follows (Hull, 2009, p. 477):
∑ , where (3.1)
where indicates a mean return.
Since volatility is a vital factor on the stock market, different features of volatility that
have been observed have to be accounted for. Engle and Patton (2001) were among
many who have studied the subject of volatility and have summarised the stylised
factors about it. Some of the most important factors are presented in the next section.

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3.1. Stylised factors about volatility
3.1.1. Volatility persistence
Volatility is a stochastic parameter, i.e. it is not constant when looking at it over a time
horizon. For instance, Merton (1980, p. 354) said that “because the variance of the
market return changes significantly over time, estimators which use realized return time
series should be adjusted for heteroscedasticity”.
Moreover, findings of Mandelbrot (1963) and Fama (1965) have shown that volatility
exhibits such aspect as clustering. Hill et al (2012) explain volatility clustering as periods
when small changes are followed by further small changes and periods when large
changes are followed by further large changes. Moreover, Engle and Patton (2001) said
that presence of clustering means that volatility comes and goes and therefore, exhibits
mean reversion. That means, in long run volatility tends to converge to a normal level.
Another question that Fama (1965) and many other researchers have address is the
distribution of the stock returns. In his research, Fama (1965) suggests that stock
returns are not normally distributed; making an emphasis especially on the third and
fourth moments, i.e. mean and standard deviation. For instance, Hagerman (1978) has
also supported the fact that financial data series exhibit high kurtosis and is skewed. He
concluded that mixture of normal distributions and Student’s-t distribution might fit
financial data better.
3.1.2. Leverage effect and volatility asymmetry
A research by Engle and Ng (1993) has shown that news impact stock prices and
consequently volatility. It has been observed that volatility tends to react differently to
big stock price drops and big stock price increases. That means that good and bad
news affect volatility differently. This tendency is called a leverage effect (Tsay, 2005,

18
p.99). It is closely related to the problem of asymmetric volatility and is an important
characteristic of financial data volatility.
Wu (2001, p. 837) points out that “returns and conditional variance of next period’s
returns are negatively correlated”. In other words, negative (positive) returns are
generally associated with upward (downward) revision of the conditional volatility.
3.1.3. Long memory of shocks
Another feature of stock returns volatility is a so-called “long memory”. Granger et al
(1993) suggested that stock returns contain a very little degree of serial correlation.
However, many argue that the presence of clustering can be attributed to the fact that
today’s volatility shocks influence the future expected volatility.
Granger et al (1993) studied the long memory of stock returns. They concluded that
there is a high degree of autocorrelation in long lags in absolute returns. Long memory
has also been studied by Bollerslev and Mikkelsen (1996). Presence of “long memory
components in the volatility processes of asset returns have important implications for
many paradigms in modern financial economics” (Bollerslev and Mikkelsen, 1996, p.
181).
3.1.4. Other factors that affect stock market volatility
As well as special features that volatility exhibits, there are other “outside” factors that
influence it. It cannot be said that prices of assets change or behave independently and
have no relationship with other economic indicators and market conditions. Especially,
attention has to be paid on the correlation of the stock market with stock markets
elsewhere in the world and other financial markets, like bonds, derivatives and currency
markets.
For instance, Engle, Ng and Rothschild (1990) have provided an example of how an
equity market and its volatility shocks are connected and affect a bill market. As another

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example, Engle, Ito and Lin (1990) investigated country specific news on the conditional
volatility. As one of the conclusions, they said that news process cannot be ignorant of
terrestrial geography; and after examining the impact of news in one market on the
volatility of other, they have found a cross-market dynamic effect, especially in the short
run. Another study by Aggarwal et al (1999) gave a support to the importance of the
news and economic conditions when examining volatility. They argue that “large
changes in volatility seem to be related to important country-specific political, economic
and social events” (Aggarwal et al, 1999, p. 14).
3.2. Volatility forecasting
Since the establishment of the very first stock market, investors and traders started to
wonder about the subject of future movements of stock prices, i.e. volatility. As it has
been mentioned above, volatility is indeed a vital “ingredient” of the stock market. A
problem of volatility forecasting has always been present. However, some events, like
stock market crashes of 1987 and of 2008, have affected the global economy so hugely,
that a question of volatility forecasting became even more necessary and important.
Historically, many ways have been introduced to forecast future volatility. The
features of volatility and its distribution, which have been discussed, do play a vital role
in a process of building a model that will precisely forecast volatility. Those features
cannot be avoided and ignored, as if there is no attention paid to them the model may
be incorrect and lead to faulty results. So a good model to forecast volatility must be
able to capture those features mentioned above.
3.3. Frequency of observations
One of the main question to be asked prior to conducting research, is what type of
data frequency should be taken to obtain the most efficient and meaningful results.
Frequencies themselves may vary from those obtained every minute to monthly or even
yearly measurements. Figlewski (1997) argued that longer sample periods are used

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when forecasting for a long term, but low frequencies rather than high should be
chosen. At the same time, Andersen et al (1999) said that using high frequency data
may lead to improvement of the forecasts. Even though, it mainly depends on the
chosen model, it was been shown by many studies that the most popular choice is daily
returns.

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4. AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTICITY
MODEL
There is a vast variety of models developed that specifically address the problem of
volatility forecasting. For instance, Granger and Poon (2003) in their research studied
time series models, like Exponentially Weighted Moving Average [EWMA] model,
stochastic volatility [SV] model, and ARCH-GARCH family models. Under scope of this
research, attention will be paid to various ARCH models.
4.1. The basic ARCH-GARCH models
Issues, like stochastic volatility over time and volatility clustering have successfully
been addressed by Engle in 1982. He proposed an ARCH model – Autoregressive
Conditional Heteroscedasticity model specifically designed to deal with an implausible
assumption of constant one-period forecast variation (Engle, 1982, p. 987). ARCH can
be seen as a more sophisticated model comparing to other models that forecast
volatility. Engle (2001, p. 158) pointed out that the goal of ARCH model is to provide a
volatility measure that can be used in making financial decisions, for example, risk
analysis, portfolio selection or pricing derivatives on the market.
Assumption of conditional heteroscedasticity is reasonable because in the reality,
financial time series do not exhibit a constant error term. The simple regression model
has to be adjusted accordingly, so to produce the ARCH(q) model. According to Hill et
al (2008, p. 364) this adjustment is shown below:
(4.1)
(4.2)
∑ (4.3)

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Equations 4.2 and 4.3 are ARCH(q) class models. The error variance term is
varying over time – it is heteroscedastic. The distribution of the error term is assumed to
be conditionally normal , where represents the information
available at the time t - 1. And ht is a function of a constant term and the lagged error
squared . It is important for all coefficients to be positive in order
for variance to be positive as well. The coefficient must be less than 1; otherwise
the variance term will continue to increase over time. The model has a form of ARCH(1)
when variance term is a function of a constant term plus term . In this case
only one lag is included in the equation.
Hill et al (2008, p. 365) pointed out that the ARCH(q) model is an important
econometric model because “it is able to capture stylised features of real world
volatility”. Engle’s research in 1982 was based on studying inflation rates in the United
Kingdom, especially the fact that the future inflation rates are unpredictable (Engle,
1982). The ARCH(q) model is useful to predict future values, however, it has one
disadvantage – too many parameters are needed to be calculated when q is a large
number, in other words, when many lags are included in the equation. As the result, the
estimation loses accuracy.
In 1986 Bollerslev introduced the GARCH model – a Generalised ARCH. GARCH
model allows for more flexible structure of lagged values that are used in estimations
comparing to ARCH model (Bollerslev, 1986). According to Hill (2008, p. 372) the
structure of GARCH is presented below:
∑ ∑ (4.4)
where represents the number of lagged error terms, and represents the number
of lagged variance terms. It is also assumed by Hill (2008) that .
Rachev et al (2007, p. 284) said that GARCH is different to ARCH because it “allows

23
the conditional variance to be modelled by past values of itself in addition to the past
shocks”, which can be clearly seen from the formula 4.4. However, simple GARCH
model that allows for one lagged terms might be not enough to capture all volatility
factors. That leads to introduction of a model that allows for more than one lagged term.
 GARCH(p,q)
GARCH(p,q) model is valuable as it allows for several lagged terms to be included in
the mean equation. It can be seen as more flexible version of GARCH. The special case
of GARCH(p,q) – GARCH(1,1) has a following form of conditional variance:
(4.5)
GARCH(1,1) is a fairly popular specification of the generalised form – GARCH(p, q)
that includes only one lagged variance term and one lagged error term. As said by
Engle (2001), GARCH(1,1) is just the basic model introduced by Bollerslev. The number
of lagged terms may be extended to higher value, depending on the data used and fit of
the particular model.
 IGARCH
When GARCH model is estimated the sum of α and β is close to unity. IGARCH, or
Integrated GARCH puts a restriction that sum of terms and has to be equal to 1
within the formula 4.6 which represents the GARCH(1,1):
(4.6)
IGARCH model is very much similar to the basic GARCH model. Engle and
Bollerslev (1986, p. 27) describe IGARCH as a model belonging to a “wider class of
models with a property called “persistent variance” in which the current information
remains important” when forecasting a conditional variance of any horizon. Lamoureux

24
and Lastrapes (1990, p. 225) said that the potential problem of IGARCH model is a lack
of theoretical motivation to be applied.
Unfortunately, ARCH and GARCH models do not capture all stylized aspects of the
volatility. It led to introduction of different extensions and specifications of GARCH that
provide better forecasting results by capturing more patterns in the data. Nowadays,
there is a large number of different GARCH extensions, for instance, TARCH,
EGARCH, GARCH-M, A-PARCH and others, as well as, models with non-normal error
terms distribution, like Student’s-t. These models allow for different stylized volatility
factors to impact the forecast.
One of the issues that investor faces, is the understanding of risk-return relationship
behind different assets. Next section shows that GARCH model can show how return
can be explained by risk (Hill et al, 2012).
4.2. GARCH model with conditional mean
 GARCH-M
GARCH-M helps to understand the risk premium and conditional variance of returns
relationship. It is particularly suited to study asset markets and assumes that risk can be
measured by the variance of returns on asset (Enders, 2004, p. 128). ARCH-M or
ARCH-in-mean model was developed by Engle, Lilien and Robins in 1987. Because
ARCH-M allows the conditional variance to affect the risk premium, expected return is
also affected by changing conditional variances. The equation of GARCH-M is as
follows (Hill et al, 2012, p. 528):
(4.9)
(4.10)

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∑ ∑ (4.11)
Equation 4.9 is the mean equation; however, it allows the conditional variance to
affect the dependent variable by a factor .
However, there are some volatility features that GARCH, IGARCH and GARCH-M
models fail to account for. They are leverage and asymmetry effects. Both
characteristics are addressed in the next section.
4.3. Asymmetric GARCH models
Until some point in time, there has been no attention paid to the impact of different
price sensitive information on stock market volatility. Franses and Dijk (1996) said that
GARCH model is an effective tool to eliminate some features of financial data, but not
the asymmetry effect. The standard GARCH model assumes that news have symmetric
effect on volatility, however, it has been shown that not only news have a major effect
on volatility but the degree to which volatility is affected by news depends on whether
they are good or bad.
 TARCH
TARCH model is an example of when different news – positive and negative are
treated asymmetrically and it is a major improvement to the simple GARCH model.
Model was proposed by Rabemananjara and Zakoian (1993) and has received a lot of
support from different authors. Hill et al (2008, p. 373) show the generalised version of
TARCH as:
∑ ∑ (4.7)
{

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where is known as the asymmetry term. The positive news hit the stock market
volatility is affected by the term , but when news is negative volatility is affected by the
term . Negative news has a larger effect on volatility, assuming is significant
and positive.
 EGARCH
Another major improvement to GARCH model is EGARCH. The main idea of
EGARCH model is quiet similar to that one of TARCH. EGARCH, or Exponential
GARCH model (Nelson, 1991) takes the effect of news into account and therefore,
allows for asymmetric effect between positive and negative returns that take place due
to that potentially different effect of news on the stock market (Tsay, 2005, p. 124).
Engle and Ng (1993, p. 1752) gave the following formula interpretation of variance term
in EGARCH:
√
[
√
√ ] (4.8)
where , , , are constant terms. Asymmetry of EGARCH model comes from
the term as it can be either negative or positive, depending on yesterday’s shock that
hit the stock market.
Another volatility factor that has to be taken into account is the long memory of
shocks. It is connected with the problem of presence of autocorrelation in returns and,
therefore, their clustering.

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4.4. “Long memory” GARCH model
 A-PARCH
The asymmetric power ARCH or A-PARCH model that was developed by Ding et al
(1993) addresses the volatility clustering problem. The model particularly focuses on the
long-memory property of stock market returns; as it has been found by Taylor (1986)
that stock returns r exhibit little degree of serial correlation, but do possess serial
autocorrelation over long lags (Ding et al, 1993, p.83). A-PARCH model encompasses
seven different models in the literature – ARCH, GARCH, TS-GARCH, GJR-GARCH,
TARCH, NARCH, and log-ARCH. The conditional variance of general A-PARCH model
has the following form (Bollerslev, 2007, p.3):
∑ ∑ (4.12)
The model is believed to take into account all the factors of volatility that are featured
in seven different models that A-PARCH model is comprised of.
Different models were discussed in previous sections, however, all of them assumed
normal distribution of error terms. This assumption may be implausible and models
have to adjust for such property as fat tails and leptokurtosis of stock returns.
4.5. GARCH with non-normal error terms distribution
Under the general assumptions of ARCH and GARCH models, standard errors of
residuals are normally distributed. Back in 1965 in has been reported by Fama that
stock returns exhibit non-normal distribution – skewness or excess kurtosis. Financial
data has also been found to feature so-called fat tails. Bollerslev (1992) said that even
though ARCH generates some degree of excess kurtosis, it is not enough to fully
account for fat tails. In 1987 Bollerslev has detected the GARCH(1,1) with t-distribution

28
to be a good model to fit data with fat tails because it has a lower peak and is more
spread out than the normal distribution (Hill et al, 2012, p. 682).
 GARCH with Student’s-t distribution
Student’s-t distribution is the next step from normal distribution and it is believed to
provide better capture distribution features as it allows for fatter tails (Rachev, 2007,
p.300). Formulas 4.13 are adopted from Rachev et al (2007, p. 281), where is the
error term, and is the random variable which is assumed to be normally distributed
under ARCH(q).
√ (4.13)
The Student’s-t distribution with degrees of freedom is shown below (Angelidis
et al, 2010, p. 4):
( )
⁄ √
(4.14)
According to Angelidis et al (2010, p. 4), is the gamma function and
represents the thickness of the distribution tails.
4.6. Literature review of GARCH models
GARCH models have been widely used since their introduction. The research by
Engle (1982) was based on forecasting UK inflation rates. Later, Bollerslev et al (1988)
based their research on testing the CAPM – Capital Asset Pricing Model with GARCH,
as it has been shown that CAPM fails on some occasions. They specifically allowed for
covariance matrix set to vary over time, assuming that agents update estimated means
and covariance of returns using newly available information on returns. Bollerslev
(1988) found that model satisfied all specifications of financial data set.

29
As GARCH model became more developed many authors started to compare the
predictive abilities of different GARCH models. Table 1 gives an insight of some studies
that have been conducted.
Table 1 Various studies on GARCH models
Author Sample examined Period examined Method used Findings
Engle, R. (1982) UK inflation rates 1958 - 1977 ARCH
ARCH model improves performance of
OLS and provides superior forecasts.
Bollerslev, T. (1986) US GNP rate of inflation
1948 - 1983
(143 observations)
GARCH
GARCH model provides better
forecasts than ARCH and has a more
reasonable lag structure.
Bollerslev, T., Engle, R.
& Wooldridge, J.
(1988)
6-month Treasury bills, 20-
year Treasury bonds and
NYSE stocks returns
1959 - 1984
(102 observations)
GARCH(p,q)-M
In general, autocorrelation and
heteroscedasticity are present in data.
However, even better model can be
constructed to account for all data
specifications.
Akgiray, V. (1989) CRSP index
1963 - 1986
(6030 observations)
ARCH, GARCH(1,1)
Stock returns exhibit a significant level
of dependence. GARCH(1,1) model
shows the best fit and forecast
accuracy.
Engle, R., Ito, T. & Lin,
W. (1990)
¥/$ exchange rate 1985 - 1986
GARCH(1,1), GARCH(1,4),
GARCH(4,4)
News and volatility spillovers have to
be accounted for. Volatility clustering
has also been observed.
Engle, R., Ng, V. &
Rothschild, M. (1990)
Treasury bills, NYSE index
and AMSE stocks
1964 - 1985 FACTOR-ARCH
Forecsts obtained on Treasury bills
data support application of FACTOR-
ARCH model. The model is believed to
be used for forecasting purposes for
other asset classes.
Baillie, R. &
DeGennaro, R. (1990)
CRSP index
1970 - 1987
(4542 observations)
GARCH(p,q), GARCH-M(p,q)
GARCH-M model with Student's-t
distribution provides a good
description of relationship between
returns and volatilities.
Pagan, A. & Schwert,
W. (1990)
n/a 1835 - 1925
Two step, GARCH(1,2),
EGARCH(1,2), Markov
switching-regime,
Nonparametric kernel (1 lag),
Nonparametric Fourier (1 and
2 lags)
Nonparametric procedures give a
better explanation of the squared
returns. EGARCH has also performed
well.
Lamoureux, C. &
Lastrapes, W. (1990)
30 random stocks of
CRSP index, NYSE and
American Exchange stocks
1963 - 1979 GARCH(1,1)
Application of GARCH model to long
time series of stock returns yields a
high measure of persistence.
Engle, R. & Ng, V.
(1993)
Japanese TOPIX index
1980 - 1988
(2532 observations)
GARCH, EGARCH,
AGARCH, VGARCH,
NGARCH, GJR-GARCH,
PNP
Impact of news on the volatility has to
be appreciated. The best model is
GRJ-GARCH.
Bollerslev, T. & Engle,
R. (1993)
DM/$, £/$ exchange rate
1985 - 1985
(1245 observations)
GARCH(1,1), IGARCH(1,1)
Proposal of the idea of co-persistence
in variance.

30
Ding, Z., Granger, C. &
Engle, R. (1993)
S&P 500 index
1928 - 1991
(17,055 observations)
ARCH(q), GARCH(p,q),
APARCH(p,q)
New APARCH model is introduced
that encompasses seven other
models. It helps to better address the
long-memory property of returns.
Hentschel, L. (1995)
Dow Jones, S&P 500,
CRSP returns
1926 - 1962
EGARCH, TGARCH,
AGARCH, GARCH, NA-
GARCH, GJR-GARCH,
APARCH, NARCH
Development of a nested family of
asymmtric GARCH models. However,
no model is superior to others.
Frances, P. & Van Dijk,
D. (1996)
DAX, EOE, MAD, MIL,
VEC stock markets
1986 - 1994
(469 observations for
each of 5 markets)
GARCH, QGARCH, GJR-
GARCH, Random Walk
QGARCH provides superior forecasts
than other models. GJR-GARCH
shown the worst performance.
Brailsford, T & Faff, R.
(1996)
Statex-Actuaries
Accumulation index
(Australia)
1974 - 1993
(4900 observations)
Random walk, Historical
mean, Moving average,
Exponential smoothing,
EWMA, Simple regression,
GARCH, GJR-GARCH
No model is clearly superior. The
choice of model depends on the error
statistic that is used.
Bollerslev, T. &
Mikkelsen, H. (1996)
S&P 500 index n/a
GARCH, IGARCH,
FIGARCH, AR, AR-GARCH,
AR-IGARCH, AR-FIGARCH
The new FIGARCH model was
introduced. It is good at characterising
the long-run dependences in the stock
market volatility.
Choudry, T. (1996)
Stock markets in
Argentina, Greece, India,
Mexico, Thailand,
Zimbabwe
1976 - 1994 GARCH(1,1)-M
Presence of changes in GARCH-M
estimations, risk premia and volatility
persistence before and after 1987.
Fraser, P. (1996)
FTA All share index, Hoare-
Govett small company
index, 3-month Tresury bills
1970 - 1992
GARCH(p,q), GARCH-
M(p,q), GARCH-M(p,q)-t
The smaller company shares have
common characteristics with the whole
market. However, some risk-return
behaviour differences exist.
Henry, O. (1998)
Hang Seng index (Hong
Kong)
1990 - 1995
(1415 observations)
GARCH, EGARCH, GJR-
GARCH, GQARCH
GARCH(1,1) was found to produce
biased estimations. GQARCH shown
the best forecasting results.
Andersen, T. &
Bollerslev, T. (1998)
DM/$, ¥/$ spot exchange
rate
1987 - 1992 GARCH(1,1)
ARCH models provide a good volatility
forecasts. The (1,1) order was found to
be the most suitable.
Hansson, B. & Hordahl,
P. (1998)
Swedish stock market 1977 - 1990
Various CAPM and GARCH-
M
The Sharpe-Lintner-Mossin CAPM
model provides best explanation of
risk-return relationship.
Engle, R. & Lee, G.
(1999)
S&P 500 index, CRSP
index, Nikkei index, 14
individual stocks
1941 - 1991 (S&P 500)
1973 - 1991 (CRSP)
1971 - 1991 (Nikkei)
1973 - 1991 (individual
stocks)
GARCH(p,q), GARCH-M,
Component GARCH model
Risk premium is only related to long-
run movements of volatility. Leverage
effect has mainly a temporary
behaviour.
Aggarwal, R., Inclan, C.
& Leal, R. (1999)
19 stock market around the
world, including Nikkei,
DAX, FTSE 100, S&P 500
1985 - 1995 GARCH(p,q)
High volatility was present in the stock
markets during examined period. It is
associted with economic and political
event that took place.
McMillan, D., Speight,
A. & Gwilym, O. (2000)
UK FTA All share and
FTSE 100 indices
1984 - 1996 (FTSE 100)
1969 - 1996 (FTA All
share)
Historical mean, Random
walk, Moving average,
EWMA, Simple regression,
GARCH, TARCH, EGARCH,
CGARCH
Random walk model provides better
forecasting results than GARCH
models.

31
Bekaert, G. & Wu, G.
(2000)
Nikkei 225 1985 - 1994 CCAPM and GARCH-M
Asymmetries exist in the data and are
driven by the variance dynamics at the
firm level and not changes in leverage.
Tse, Y. & Tsui, A.
(2001)
DM/$ and ¥/$ exchange
rates
1990 - 1998
(2131 observations)
GARCH, M-GARCH, CC-
MGARCH, VC-MGARCH,
BEKK
New M-GARCH model with time
varying correlations is proposed. The
new model provides satisfactory
results and is favourable against
BEKK.
Yu, J. (2002) NZSE (New Zealand)
1980 - 1998
(4741 observations)
Random walk, Historical
average, Siple regression,
EWMA, ARCH, GARCH(p,q),
Stochastic volatility
Stochastic volatility models provide the
best results. But GARCH(3,2) performs
the best among all ARCH type models.
Xing, X. & Howe, J.
(2003)
UK and World stock
market indices
1973 - 1999
GARCH-M, EGARCH,
Bivariate GARCH-M,
Bivariate EGARCH,
EGARCH-M
Bivariate GARCH-M provides the best
forecasting results. It also detects the
positive relationship between stock
returns and variance of returns in the
UK after accounting for the covariance
between the UK and world stock
market.
Bauwens, L. & Laurent,
S. (2003)
£/$, ¥/$, €/$
1989 - 2001
(3066 observations)
GARCH(p,q), GJR-
GARCH(p,q), AR(p)-GARCH,
M-GARCH, VaR
A multivariate skew-Student density is
introduced to GARCH model. It
improves modelling and VaR
forecasts.
Awartani, B. & Corradi,
V. (2005)
S&P 500 index
1990 - 2001
(3065 odservations)
GARCH, EGARCH, GJR-
GARCH, QGARCH,
TGARCH, AGARCH,
IGARCH, RiskMetrics
exponential smoothing,
ABGARCH
Asymmetic GARCH model provides
better forecasts than GARCH that
does not allow for asymmetries.
However, other symmetric models
perfomed worse than simple GARCH.
Li, Q., Yang, J., Hsiao,
C., Chang, Y. (2005)
12 wrolds' largest stock
markets
1980 - 2001 EGARCH-M
Semiparametric specification of model
is more robust that parametric. It is
found that stock return volatility is
negatively correlated with stock
returns.
Hansen, P & Lunde, A.
(2005)
DM/$ exchange rate, IBM
stock returns
1987 - 1992
(1254 observations)
330 ARCH models
No evidence of poor performance of
simple GARCH(1,1) was found.
Alexander, C. & Lazar,
E. (2006)
£/$, ¥/$, €/$ exchange
rates
1989 - 2002
(3652 observations)
GARCH, GARCH-t, NM-
GARCH, NM-GARCH-t
NM-GARCH(1,1) model is the
preferred specification for this data
set.
Wilhelmsson, A. (2006) S&P 500 index 1996 - 2002
GARCH(1,1) and GJR-
GARCH(1,1) with nine
different distributions of error
terms
Models with Student's-t distribution
provide superior forecasting results.
Karanasos, M. & Kim,
J. (2006)
Stock markets in Korea,
Japan, Hong Kong,
Taiwan, Singapore
1980 - 1997
(4518 obserations)
APARCH(p,q)
APARCH(p,q) process can be
expressed as ARMA process.
Brandt, M. & Jones, C.
(2006)
S&P 500 index
1962 - 2004
EGARCH(p,q), FIEGARCH,
REGARCH(p,q),
FIREGARCH
Fractionally integrated ranged-based
models offer a comparable and
superior performance than two-factor
models. FIREGARCH model provides
reliable long horizon volatility forecasts.

32
Roh, T. (2007) KSE KOSPI 200 index 930 observations
NN, NN-EWMA, NN-GARCH,
NN-EGARCH
Hybrid model between the ANN and
financial time series models proposed.
ANN-models can enhance the
predictive power for the perspective of
deviation and direction accuracy.
Hybrid NN-EGARCH model can be
improved in forecasting volatility.
Alberg, D., Shalit, H. &
Yosef, R. (2008)
Tel Aviv Stock Index (TA 25
and TA 100 indices)
1992 - 2005 (TA 25)
1997 - 2005 (TA 100)
GARCH, GARCH-t,
EGARCH, EGARCH-t, GJR-
GARCH, GJR-GARCH-t,
APARCH, APARCH-t
Asymetric GARCH model with
allowance for fat-tails improves overall
performance. EGARCH model with
Student's-t distribution is the most
successful in forecasting.
Bildirici, M. & Ersin, O.
(2009)
Istanbul stock exchange
1987 - 2008 (5274
observations)
GARCH, EGARCH,
TGARCH, GJR-GARCH,
SAGARCH, PGARCH,
NGARCH, APGARCH,
NPGARCH, ANN-GARCH
family
ANN-GARCH models provide
significant improvement to the
forecasting results.
Wang, Y. (2009)
TAIFEX (Taiwan Futures
Exchange)
2005 - 2006
GARCH, GJR-GARCH, Grey-
GJR-GARCH
Grey-GJR-GARCH model achieves
better forecasting performance than
GARCH and GJR-GARCH.
Liu, H. & Hung, J.
(2010)
S&P 100 index 1997 - 2003
GARCH-N, GARCH-t,
GARCH-HT, GARCH-SGT,
EGARCH, GJR-GARCH
GJR-GARCH provides superior
forecasts to EGARCH. Models with
normal distribution of error terms are
also more preferable.
Chen, X., Ghysels, E. &
Wang, F. (2011)
S&P 500 futures 1982 - 2008
30 variations of GARCH,
including HYBRID GARCH
models
New HYBRID GARCH model is
presented. It can be used in multi-
period volatility forecasts in risk and
portfolio analysis.
Engle, R. & Sokalska,
M. (2012)
2721 companies from TAQ
database
April 2000 - June 2000 GARCH(p,q), GJR-GARCH
Stochastic intraday component within
the model improves forecasting
results. The new intraday volatility
forecasting model is proposed and it is
believed to become popular among
traders.
Orhan, M. & Koksal, B.
(2012)
4 stock markets in Brazil,
Germany, USA, Turkey
2006 - 2009
ARCH, GARCH, IGARCH,
SAGARCH, Taylo/Schwert
GARCH, TGARCH, GJR-
GARCH, GJR-PGARCH,
EGARCH, PGARCH,
NGARCH, AGARCH,
NGARCHK, APGARCH,
NPGARCH, NPGARCHK
ARCH model provides the best
forecasts. T-distribution aslo performs
better than normal distribution.
Hou, A. & Suardi, S.
(2012)
West Texas intermediate
(WTI) crude oil spot prices
1992 - 2010 (4845
observations)
GARCH, IGARCH,
RiskMetrics, GJR-GARCH,
EGARCH, APARCH,
FIGARCH, HYGARCH,
FIAPARCH
Nonparametric GARCH model is
superior in out-of-sample forecasts
and can be considered as a useful
alternative method of modelling crude
oil price return volatility.
Constantinides, A. &
Savel'ev, S. (2013)
S&P 500 index 16,000 observations GARCH(1,1), TLF-GARCH
TLF-GARCH model describes many
volatility factors of the stock market.
The model still has to be investigated
more.
Haas, M., Krause, J.,
Paolella, M. & Steude,
S. (2013)
DAX 30, S&P 500, DJIA
30, Nikkei 225, NASDAQ
COMPOSITE and $/€, ¥/€
exchange rates
1999 - 2009 (stock
markets) 2004 -
2009 (currency rates)
GARCH, ASYM-GARCH,
GJR-GARCH, EGARCH,
MixN-GARCH, MixN-GARCH-
ASYM, MixN-GARCH-GJR,
MixN-GARCH-LIK, MixN-
GARCH-LOG
MixN-GARCH-LIK delivers a clear-cut
superior out-of-sample performance
compared to all entertained models.

33
The “simple” GARCH model is present in all the researches because it is the base
model. However, it hasn’t been used a lot to forecast volatility because, as it has been
said, it is not capable to capture such volatility factors, like leverage effect, long-memory
property and fat tails. In spite of this, a few authors like, Engle and Lee (1999) have
specifically studied a particular variation of GARCH(p,q) model – GARCH(2,2). Akgiray
(1989), Hansen and Lunde (2005), Andersen and Bollerslev (1998) also said that
GARCH model provides good results and there is no particular reason to dismiss it. Yu
(2002) supported GARCH(3,2) application on the New Zealand stock market.
The use of asymmetries in models is supported by the majority of researchers
because they make a big impact on the volatility forecasting. Engle, Ito and Lin (1990)
and Engle and Ng (1993) agreed that asymmetry effects that is caused by nature of
news should be considered when forecasting volatility. Engle and Ng (1993) found that
asymmetric GARCH model performs better than other models when studying Japanese
stock market returns. EGARCH has become a popular choice of model to forecast
volatility. For example, Awartani and Corradi (2005) who looked at the role of
asymmetries in forecasting performance of GARCH model also conclude that a basic
Gilenko, E. &
Fedorova, E. (2014)
BRIC stock market indices
2003 - 2012 (2425
observations)
ARCH, GARCH, 4-
dimensional BEKK-GARCH
Internal and external spillover effects
are examined. External mean-to-mean
spillover effect were analysed.
Influence of the developed stock
market on the BRIC stock markets
decays over time.
Efimova, O. & Serletis,
A. (2014)
EIA prices on crude oil,
natural gas, electricity
2001 - 2013
GARCH, GARCH-M,
MAGARCH, BEKK, DCC,
VAR-GARCH, VEC-GARCH
Univariate and multivariate models
yield similar estimates, but univariate
models produce more accurate
forecasts. The MGARCH has an
advantage as it helps to investigate
interactions between several markets.
Bayraci, S. & Unal, G.
(2014)
Turkish Treasury bonds
2006 - 2010 (1286
observations)
GARCH(1,1),
COGARCH(1,1), DTGARCH,
COGARCH(1,1) provides excellent
results in modelling the interest rate
series, as they capture the features of
the volatility process and yield better
conditional volatility estimates.
Oueslati, A., Hammami,
Y. & Jilani, F. (2014)
Tunisian mutual fund
industry
2002 - 2010
Unconditional approach,
Conditional approach,
Bivariate GARCH, M-GARCH
M-GARCH model does not improve
the perception of the timing ability of
fund managers relative to other
models.

34
GARCH model was beaten by GARCH that allowed for asymmetries. Brandt and Jones
(2006) changed the standard EGARCH into FIREGARCH and gave the priority to it. But
also believe that news asymmetries have to be accounted for. On the other hand, Henry
(1998) applied various GARCH models on Hong Kong stock market, including
EGARCH. He found that QGARCH model is a better mechanism to forecast future
volatility rather than EGARCH. Frances and Van Dijk (1996) also supported use of
QGARCH and said that asymmetric models performed the worst. McMillan et al (2000)
have estimated daily, weekly and monthly returns on the UK FTA All Share and FTSE
100 stock market index. They used various specifications of ARCH model, like GARCH,
TARCH and EGARCH to estimate the forecasting ability. However, contradicting to
conclusions mentioned above, McMillan et al specified that “moving average and
exponential smoothing models provide marginally superior daily volatility forecasts”
(McMillan, 2000, p. 448) rather than giving a preference to any particular specification of
GARCH.
APARCH and GARCH-M are also a popular choice. Xing and Howe (2003) applied a
bivariate Generalised ARCH-M to the weekly returns on the UK stock market as well as
the world market. They argued that the world market should be taken into account when
studying a risk-return relationship in a partially integrated market (p. 344). They
suggested bivariate GARCH-M model as a good model to forecast UK stock market
returns. GARCH-M has been used by Choudry (1996) in 6 different stock markets. Ding
et al (1993) have used the A-PARCH to estimate daily returns on the S&P 500 index.
Karanasos and Kim (2006) have given a lot of attention to APARCH model in the
context of ARMA process.
Financial data volatility exhibits fat tails and non-normal error terms distribution may
help to account for this feature. So that Wilhelmsson (2006) found that non-normal error
term distribution GARCH and GJR-GARCH models provided superior forecasts of S&P
500 volatility. Baillie and DeGennaro (1990) have examined a relationship between
stock returns and volatility with GARCH-in-mean with t-Student distribution. They

35
pointed out “controlling for excess kurtosis by use of the student-t density is found to be
important” (p. 211). Study conducted by Alberg et al (2008) compared the forecasting
abilities of different GARCH models on Tel Aviv stock market. They used such models,
as EGARCH, GJR-GARCH, APARCH and their various errors distribution
specifications. However, their result suggests that EGARCH with t-Student distribution
of standard errors outperforms other models. On the other hand, Liu and Hung (2010)
accessed the forecasting ability of EGARCH, GJR-GARCH, TARCH and different error
distribution-type GARCH models (GARCH-N, GARCH-t, GARCH-HT and GARCH-
SGT). Scope of the research was covering daily S&P 100 data. Authors found that
GJR-GARCH achieved the most accurate forecasting results and did not give the
preference to models that incorporate non-normal errors distribution. Other authors who
did not give any support to Student’s-t distribution are Alexander and Lazar (2006).
Their study of modelling an exchange rate clearly concluded that even though GARCH
with t-density performed well in moment specification tests, it was found to be inferior to
normal mixture GARCH models for unconditional density (2006, p. 325). Even though
they did not favour the GARCH-t models, they still point out that the skewed model
improved on the symmetric GARCH-t according to the most of the criteria.
Nowadays, GARCH models become even more “exotic”. Alexander and Lazar (2006)
preferred NM-GARCH, Engle, Ng and Rothschild (1990) said that FGARCH provides
good forecasts and Chen et al (2011) introduced the HYBRID GARCH to forecast future
values. It is quiet probable that GARCH models will experience more development in
the future.
Despite usefulness of GARCH models, some authors have given the priority to other
models. Like, McMillan et al (2000) prioritised random walk model and Hansson and
Hordahl (1998) appreciated CAPM, but Brainsford and Faff (1996) experienced difficulty
of determining the superior model. A significant study was made by Poon and Granger
(2003) where they have looked at forecasting abilities of different models used in 93
different studies. They have come to the conclusion that when the GARCH class

36
models are used, asymmetric models perform better than simple GARCH; however,
overall preference has been given to more sophisticated variations like fractionally
integrated GARCH (FIGARCH) and regime switching GARCH (RSGARCH). However,
only 44% of 93 research papers that were covered have said that GARCH model
provides the “best” forecasting results. The historical volatility models have provided
better results in 56% of 93 papers. These models include “random walk, historical
averages of squared returns, or absolute returns, time series models based on historical
volatility using moving averages, exponential weights, autoregressive models, or even
fractionally integrated autoregressive absolute returns” (Poon and Granger, 2003, p.
506). This shows that historical volatility methods work well if not better than GARCH
models when forecasting volatility. This, therefore, raises the question of necessity of
sophisticated models like GARCH at all. However, this question is left to be answered
by more professional researchers in this particular field of study.
Concluding, there is clearly no uniformity of the choice of “best” model among all the
authors, as their studies vary in the choice of the stock market, chosen length of
analysed data and specifications of chosen forecasting models. This does raise the
issue of inability to precisely say which model is superior.

37
5. METHODOLOGY
5.1. Data
The main purpose of the dissertation is to find the best model to forecast FTSE 100
stock market volatility. Before applying any models, it is important to understand the
nature of examined data.
The empirical research is composed of applying different models on daily returns.
This frequency of data observations is chosen because it was found to be a popular
choice among researchers and is believed to provide a better estimation according to
Bollerslev et al (1999). In order to have more accurate results, data covers a period of
10 years from January 1st
2003 until December 31st
2013. FTSE 100 daily closing prices
have been downloaded from Yahoo Finance UK website (www.uk.finance.yahoo.com).
Overall number of observations is 2778. The formula below was applied in order to
obtain daily returns on the stock market index.
(5.1)
denotes a closing price at time t (t=1,…, 2778). This way of obtaining returns has
been used by a vast majority of authors, one of them, for example, Ding et al (1993)
where he explained daily returns as the continuously compounded returns.
5.2. Statistical analysis of data
Brooks (2008, p. 381) said that non-linear models were found to be efficient when
studying financial data. However, only if data exhibits features applicable to non-linear
process, ARCH model can be applied. Historically, it was reported by many
researchers, first among others were Mandelbrot (1963) and Fama (1965), that returns
on stocks are not normally distributed, i.e. do not follow white noise process. This is
closely related to so-called volatility stylized factors, as they are present due to non-

38
normality of returns. The features that returns exhibit are skewness, leptokurtosis and
fat tails. For instance, Mills (1995) have found those non-normal features to be present
in FTSE 100 index returns.
Therefore, this research pays attention on whether returns of FTSE 100 index of
chosen period are white noise or not.
Table 2 Statistical measure of daily returns of FTSE 100
Statistic Daily returns
№ of observations 2778
Mean 0.000187
Median 0.000555
Max 0.093842
Min -0.092646
Standard deviation 0.012127
Skewness -0.117811
Kurtosis 11.11457
Jarque-Bera test 7628.126
Probability 0.000000
The Table 2 represents a summary of the most important statistical measures that
can help analysing distribution of daily returns. According to Hill et al (2012, p. 33) the
standard normal distribution has a mean value of 0 and standard deviation value of 1.
From the table above it can be seen that mean and median values are positive and not
significantly different from zero, however, standard deviation is relatively low. Under the
normal distribution assumptions, the value of kurtosis should be 3 and there should be
zero skewness. The Jarque-Bera test statistic is closely related to values of kurtosis and
skewness. It is performed under the null hypothesis of normal distribution. Table 2
shows that null hypothesis is rejected; the probability value is less than 0.01 that is

39
implied by 1% significance level. The daily returns are negatively skewed, even though
the value is not too low. If skewness is zero, it means that returns are perfectly
symmetrically distributed around zero (Hill et al, 2012, p. 148). It case of FTSE 100,
negative skew indicates presence of long tails to the left of distribution plot (Hill et al,
2012, p. 658). The kurtosis value is large, which indicates a high peak of daily returns
distribution. High kurtosis, or in other words, leptokurtosis is also associated with fat
tails that are usually present in financial data series (DeCarlo, 1997, p. 294).
First of all, results in Table 2 are consistent with theory of non-normal stocks
distribution by Fama (1965). Moreover, the findings are consistent with those of Xing
and Howe (2003) where they found non-normal distribution of stock index returns of the
UK and those of Choudry (1996) where he observed non-normal distribution of stock
returns in six different stock markets.
5.3. Mean equation
Brainsford and Faff (1996) suggested that the emphasis of most researches has
moved away from the mean equation of stock market returns to the volatility of returns.
Mainly previous studies that focused on stock prices volatility forecasting measured it
with variance of the error terms. Therefore, the attention is primarily paid on the
unpredictable part of future returns, which is the error term. Error terms, in other words,
disturbances are obtained when conducting regression analysis.
Returns of stock market can be shown in a form of simple regression model – the
mean model (Hill et al, 2008, p. 364):

40
where returns can be explained by the constant term and the error term
which is assumed to be normally distributed; as well as . This model has a
simple from; however, more explanatory variables can be included along with constant
and error terms.
Different models, such as, GARCH, GARCH (1,1), TARCH, GARCH-M, EGARCH
and GARCH with Student’s-t distribution will be applied to FTSE 100 returns. These
models are found to be among the most widely used. As it has been mentioned above,
these models have an advantage of capturing various features of financial data.
5.4. Testing for ARCH effects
Following Engle (1982, p. 999) it is reasonable to mention that Ordinary Least
Squares [OLS] estimation can be an appropriate measure of financial data. It might be
suitable if the disturbances of returns are not conditionally heteroscedastic. Therefore, it
is necessary to test for the so-called ARCH effects before conducting an estimation of
forecast model.
5.4.1. Autocorrelation test
The procedure is well studied by Tsay (2005) where he described steps needed to be
taken to perform autocorrelation test. First of all, it is needed to obtain squared
residuals . According to Tsay (2005, p. 101), there are two different methods to test
for presence of ARCH effects within financial data series. The first test is Ljung-Box
Q(m) statistic which is applied on the squared residuals. The test assumes the null
hypothesis of zero autocorrelation of the series in the first m lags.
5.4.2. Lagrange multiplier test
The second test is Lagrange Multiplier [LM] test that has been used by Engle (1982).
Firstly, mean equation has to be estimated using Ordinary Least Squares procedure to

41
obtain squared values of residuals ̂ . Secondly, squared residuals have to be run on
their lagged values. Hill (2012, p. 523) shows the first-order ARCH with following
formula:
̂ ̂
where is a random term. Number of lagged terms depends on the order of ARCH.
If ARCH is not present in data, then and goodness of fit term R2
will have low
value. If ARCH is present it indicates that ̂ depends on lagged values; R2
is going to
be high. According to Hill (2012, p.523), the LM test is , where T indicates a
sample size, q is a number of lagged terms, R2
is the coefficient of determination.
Assuming the null hypothesis of , and if it true, then has a chi-
squared distribution. In case when , then null hypothesis
of is rejected and it can be said that ARCH effects are present within sample
data.
5.5. Model selection
Besides testing data set for presence of ARCH effects, the attention has to be drawn
on the selection of most appropriate length of lags, in other words, the order of model.
GARCH(1,1) is a rather popular model among many researchers. Its main feature is
that it only allows for one lagged term of variance and one lagged error term to be used
in the estimation. Brooks and Burke (2003, p. 558) have two explanations of why
GARCH(1,1) is so widely used. Firstly, it may be because this model specification is
enough to capture the entire volatility clustering problem, and so there is no need for
additional lags to be included. Secondly, many researchers find it difficult to determine
the most suitable length of lags and, therefore, focus on GARCH(1,1) for simplicity.

42
The ARCH model strongly relies on AR – autoregressive properties within financial
data sets. Enders (2004, p.69) said that even though additional lags will lead to
reduction of residuals sum of squares [SSR], it will require for extra coefficients to be
estimated and will, therefore, reduce degrees of freedom. However, presence of those
additional lags still may lead to better performance of the estimated model. Zivot (2008)
among many others paid his attention on determination of ARCH order p and GARCH
order q. He specifies two selection criterion models – Bayesian information criterion
(BIC) and Akaike information criterion (AIC) (Zivot, 2008, p.13). According to Zivot
(2008) those value have to be at the minimum when the lag length is the most
appropriate. Exact formulas for those criteria are given in section 5.6.2. The actual
results of AIC values of different GARCH(p,q) models are presented in section 6.2.
5.6. Evaluation of estimated results
The evaluation of performance of forecast models plays a major role, as it
consequently leads to the choice of the most accurate model. Therefore, the criteria to
evaluate them have to be chosen.
5.6.1. Loss function
Unfortunately, not a lot of researchers covered topic of evaluation criteria, and rather
discussed the obtained results. However, Bollerslev (1994) gave attention to model
selection. He pointed out the complexity of models choice by saying that “the usual
model selection difficulties are further complicated in ARCH models by the uncountable
infinity of functional forms allowed by variance equation and the choice of an
appropriate loss function” (Bollerslev, 1994, p. 3011). He also said that the loss
functions are usually used as a measure of forecast evaluation. The loss function itself
represents a ““loss” or “cost” associated with various pairs of forecasts and realisations”
(Diebold and Lopez, 1996, p. 12).

43
The most widely used loss functions are Mean Error (ME), Mean Squared Error
(MSE), Mean Absolute Percent Error (MAPE), Root Mean Squared Error (RMSE), Mean
Absolute Error (MAE) and Theil’s Inequality Coefficient (TIC). According to Diebold and
Lopez (1996) and Lopez (1999) the chosen loss functions are as follows:
∑( ̂ ) ∑ |
̂
|
∑ ̂
∑ ̂
∑
√ ∑( ̂ ) ∑| ̂ |
where {̂ } is the volatility forecast of one step ahead and is used as a
proxy because the actual conditional variance of next period is not observable.
In spite of the loss function that is used, rarely it will determine one superior model.
Diebold and Lopez (1996, p. 12) pointed out that usually different forecasts are
compared and combined. Brailsford and Faff (1996, p. 432) said that even use of all
loss functions does not reveal a dominant model and the way forecasting models were
ranked depends on the choice of loss function. For instance, Akgiray (1989) and
Brailsford and Faff (1996) used the statistics mentioned above to find the most suitable
forecasting model. On the other hand, Yu (2002) while forecasting volatility of the New
Zealand stock market also used RMSE and MAE along with other measures like Theil-U
statistic and LINEX loss function; however, LINEX is not covered in the scope of this
research.

44
5.6.2. Information criterion
According to Hull (2012, p. 236) another way to choose the model is based on the
Information criterion mentioned above in section 5.5. Those are Akaike (AIC) and
Bayesian (BIC) information criteria. Both formulas are presented below:
( ) ( )
where SSE is the sum of squared errors, K is the number of coefficients estimated,
and N is the number of observations. While using these criteria, the preference should
be given to the model with smallest AIC or BIC. For instance, Xing and Howe (2003)
have applied both criteria to select the most suitable GARCH model while studying UK
stock market returns.

45
6. EMPIRICAL RESULTS AND ANALYSIS
6.1. Test for ARCH effects
As it has been described in section 5, two tests for conditional heteroscedasticity are
Lagrange Multiplier test and Ljung-Box test (Tsay, 2005). Both tests were carried out in
this research with help of EViews 8.0 software. Tests are performed under the following
hypothesis:
H0 squared residuals do not depend on their lagged values
HA squared residuals depend on lagged values
If null hypothesis is rejected in favour of alternative hypothesis, it indicates presence
of autocorrelation in squared returns.
Firstly, mean equation was estimated with Ordinary Least Squares process to obtain
residuals set.
6.1.1. LM test
Following Brooks (2008, p. 389) the LM test has been performed at 5 lags of squared
residuals. LM-statistic is calculated as number of observations multiplied by coefficient
of determination R2
. Results of F-statistic and LM-statistic are shown below, while
Appendix A contains full test.
F-statistic 158.4258 Prob. F(5,2767) 0.0000
Obs*R-squared 617.1660 Prob. Chi-Square(5) 0.0000
F-statistic and LM-statistic are significant at 99% confidence level. Therefore, null
hypothesis can be rejected in favour of alternative hypothesis, indicating that ARCH
effects are present in returns of FTSE 100.

46
6.1.2. Ljung-Box test
The second test is the Ljung-Box test. It indicates presence of autocorrelation in
squared residuals. Following Tsay (2005) test is performed at lag 5, 10 and 20. Ljung-
Box Q-statistic was found to be high, as well as p-values equal to zero at all levels
indicating significance at 99% confidence level. It shows presence of strong correlation
between squared residuals and, consequently, is one of the indications of
heteroscedasticity. The table of serial correlation values of all lags can be found in the
Appendix A section. The Table 3 show some descriptive statistic values of residuals
series of the data set.
Table 3 Descriptive statistic of residual series
It was observed that FTSE 100 index features ARCH effect in returns and this finding
is consistent with general assumptions of financial data series, as discussed by many
authors such as Engle (1982) and Bollerslev (1992).
6.2. Model selection process
As mentioned in section 5.5, it is important to determine the length of lags so that the
model provides good forecasts. Following Enders (2004) and Tsay (2005), Akaike
information criterion (AIC) and Bayesian information criterion (BIC) help to determine
most suitable values of p term in ARCH (p) and q term in GARCH(p,q) model. As
described in section 5.5 and 6.1, it is important to study the autocorrelation between
residuals. Moreover, Enders (2004, p.118) said that looking at autocorrelation of
Skewness Kurtosis
Jarque-
Bera test
Q2
(5) Q2
(10) Q2
(20)
residuals
series of OLS
estimation
-0.117811 11.11457 7628.126* 1270.0 2022.5 3286.8
Note: Q(n) follows the Chi squared distribution. Critical values at 1% significance level for lag 5, 10
and 20 are 15.086, 23.209 and 37.556 respectively.

47
squared residuals rather than just residuals can help to find the order of GARCH(p,q)
model. The summary chart of first 10 lags can be seen below in Figure 1.
Figure 1 ACF and PACF values
It is observed from Figure 1 that autocorrelation is present in all 10 lags. Therefore, it
can be concluded that probably more than one lag of q term should be included in the
equation. However, it is not a very reliable observation, and therefore, it is desirable to
estimate different variations of models and then chose the most appropriate by looking
at the specific valuation criteria. The Table 4 represents values of AIC and BIC when
different terms p and q in GARCH(p,q) are used.
There is no universal length of lags that should be included in estimation of GARCH
model. Choice of lag length among many authors differs and this may be due to
differences between sets of data. According to the AIC values in the Table 4, the most
appropriate model is GARCH(2,1) as it has the lowest AIC value. However, when
looking at the BIC values, the GARCH(1,1) seems to perform better than other models.
This particular finding is in accordance with vast number of papers written on GARCH
models, as mainly they indicated good performance of GARCH(1,1).
-0.1
0
0.1
0.2
0.3
0.4
1 2 3 4 5 6 7 8 9 10
Correlationcoefficient
Lag number
ACF PACF

48
Table 4 Criteria values for different GARCH(p,q)
p=1
q=1
p=1
q=2
p=2
q=1
p=2
q=2
p=1
q=4
p=2
q=4
SSR 0.408820 0.408809 0.408805 0.408805 0.408807 0.408795
AIC -6.435011 -6.435269 -6.435412 -6.434696 -6.434302 -6.434757
BIC -6.426474 -6.42597 -6.424739 -6.421889 -6.419361 -6.417681
Since the results of AIC and BIC have not shown the same conclusion on the model,
the assumption of the most appropriate lag length cannot be made.
6.3. Estimation of different GARCH models
In this section the attention is paid on the actual estimates of models, rather than
theoretical background. The models under consideration are: GARCH(p,q), EGARCH,
TARCH, APARCH, GARCH-M, IGARCH. Both mean and variance have been modelled
simultaneously and estimations were made with EViews 8.0 software.
As there is no particular value of p and q terms in GARCH(p,q), models of different
orders were estimated. Moreover, it has been stated in section 4.5 that presence of
Student’s-t distribution rather than normal distribution might improve the forecasting
ability of models. Therefore, models were also estimated with both kinds of distribution,
to see whether there are any changes in forecasting capabilities of models when
distribution shifts from normal to non-normal.
As there are major differences in the main equations of GARCH model specification
(see section 4), the estimates of the variables within models produced different values
depending on the model used.

49
Firstly, all models have been estimated with only one lagged error and one lagged
variance term. Estimates are presented in Table 5. In the scope of this research, all
variables are desired to be significant at 99% confidence level.
Table 5 Estimates of GARCH(1,1), IGARCH(1,1), TARCH(1,1), EGARCH(1,1),
GARCH-M(1,1) and APARCH(1,1) models
Table 5 presents results from 6 different models. It has been observed that all
variables are significant at 99% level, expect of APARCH(1,1) model which has not
performed so well. It is especially important that variable γ is not significant, because it
is the variable that differentiates APARCH from GARCH. Hence, it is irrelevant to study
APARCH model on later stages. However, it is impossible to say which model is
superior by looking at the estimates values. Table 6 shows that presence of Student’s-t
distribution of error terms does not necessarily make any changes in the significance
levels of equation variables as well as any major difference in values of estimates. In
both cases of error terms distribution, GARCH(1,1) model has performed well, and all of
its variables are significant at 1% level. IGARCH(1,1) and GARCH-M(1,1) has provided
similar results with all variables being significant as well. Both EGARCH(1,1) and
TARCH(1,1) can be viewed in the same context, as they are two models that allow for
γ θ
GARCH(1,1)
0.00000134* 0.094908* 0.896556* 0.000583*
IGARCH(1,1)
0.070646* 0.929354* 0.00052*
TARCH(1,1)
0.000000533* 0.142911* 0.940431* -0.155348* 0.000925*
EGARCH(1,1)
-0.113236* 0.109044* 0.996425* 0.131131* 0.000992*
GARCH-M(1,1)
0.00000163* 0.10673* 0.882805* 0.001447* -12.16799*
APARCH(1,1)
1.24E-05 0.061938 0.939232* -0.999986 0.001011*
Variance equation Mean equation
Note: * denotes significance at 99% level.

50
asymmetries within data. Both models have also performed well and produced values
that are significant at 1% level.
GARCH-M(1,1) and APARCH(1,1) models with Student’s-t distribution
As higher order of GARCH model may be needed to capture all the volatility factors,
the next step was the inclusion of additional lagged terms in the variance equation. It is,
therefore, necessary that the additional terms are significant at 1% confidence level;
otherwise, their inclusion is not reasonable. Table 7 and Table 8 provide results of
estimates when additional error term lag – q term and when additional variance term lag
– p term are included in variance equation. Therefore, the models have a form of
GARCH(1,2) with extra q term, and GARCH(2,1) with extra p term.
When comparing results in Table 5 with those in Table 7 and Table 8, it is clearly
seen that additional p and q terms of GARCH(p,q) in most cases are not significant at
1% level. All models except of EGARCH(1,2) and GARCH-M(2,1) have performed
worse with additional lags in the equation. Even though EGARCH(1,2) and GARCH-
M(2,1) showed different results to other models, when taking the whole sample of
γ θ
GARCH(1,1)
0.00000154* 0.102781* 0.888112* 0.000598*
IGARCH(1,1)
0.076394* 0.923606* 0.000563*
TARCH(1,1)
0.00000062* 0.152471* 0.935604* -0.165408* 0.000886*
EGARCH(1,1)
-0.129755* 0.117754* 0.995311* 0.143517* 0.000984*
GARCH-M(1,1)
0.00000192* 0.116919* 0.871427* 0.001471* -12.35357*
APARCH(1,1)
1.51E-05 0.066382 0.935348* -0.999999 0.000965*
Variance equation with Student's-t distribution Mean equation

51
models into account, they do not produce enough justification of necessity of additional
lagged terms.
γ θ
GARCH(1,2)
1.59E-06* 0.064631* 0.041525 0.883611* 0.000576*
IGARCH(1,2)
0.064088* 0.007946 0.927965* 0.000515*
TARCH(1,2)
5.79E-07* 0.121656* 0.025093 0.936102* -0.155519* 0.000921*
EGARCH(1,2)
-0.137734* 0.004285 0.118466* 0.994839* 0.142130* 0.001030*
GARCH-M(1,2)
1.94E-06* 0.073582* 0.047007 0.866798* 0.001452* -12.29561*
APARCH(1,2)
9.65E-06 0.064152 -0.009858 0.945732* -0.955291 0.001011*
Variance equation Mean equation
γ θ
GARCH(2,1)
1.08E-06* 0.071467* 1.222581* -0.301028 0.000578*
IGARCH(2,1)
0.066215* 1.004242* -0.070457 0.000517*
TARCH(2,1)
4.68E-07* 0.121122* 1.120196* -0.172141 -0.129756* 0.000918*
EGARCH(2,1)
-0.126045* 0.118972* 0.917979* 0.077741 0.144248* 0.001033*
GARCH-M(2,1)
1.29E06* 0.078077* 1.250776* -0.337289* 0.001463* -12.42601*
APARCH(2,1)
1.06E-05 0.049285 1.88332* -0.236607 -0.994651 0.001005*
Mean equationVariance equation

52
From the conducted estimations, GARCH(p,q) model of higher order than (1,1) did
not perform better than the basic GARCH(1,1), as additional terms were found to be
insignificant at 99% confidence level. It can, therefore, be said that additional lags are
not needed to capture all features of volatility.
According to the obtained results, it is reasonable to focus further research on
models that allow for only one of each lagged terms within variance equation. However,
as it is impossible to provide any reasonable justification on supremacy of one model
over the other. Thus, next sections focus on different measures to differentiate between
forecasting abilities of models.
6.4. Statistical measures of estimated models
It has been shown in section 6.1 that ARCH effects are present in the data set. One
of the main aims of GARCH model is to eliminate presence of heteroscedasticity and
autocorrelation in squared residuals. Moreover, a good GARCH model or its
specification should account for both leptokurtosis (excess kurtosis) and skewness
(Wilhelmsson, 2006). These measures can also help in identifying the most suitable
model, as it accounts for those volatility factors better than others, i.e. brings the
kurtosis value to desirable, reduces skewness and eliminates autocorrelation.
Table 9 shows such values as skewness, kurtosis and Jarque-Bera test statistic that
were applied on standardised residuals series. As regression models with Student’s-t
distribution of error terms did not show poor performance in section 6.3, it is reasonable
to study statistical measures of those models alongside with models that allowed for
normal distribution of error terms.
Table 9 shows that two models that allowed for asymmetries – TARCH(1,1) and
EGARCH(1,1) clearly outperform GARCH(1,1), IGARCH(1,1) and GARCH-M(1,1).
TARCH(1,1) alongside with EGARCH(1,1) model have shown better values for kurtosis
and Jarque-Bera test statistic.

53
Table 9 Statistical measures of GARCH(1,1), IGARCH(1,1), TARCH(1,1),
EGARCH(1,1) and GARCH-M(1,1) models
Asymmetric models were able to reduces the value of kurtosis from 11.115 (see
Table 2) to values close to 3, the value that is in accordance with normal distribution
assumption (Enders, 2004). Jarque-Bera test – the test for normality (Hill, 2012, p. 148)
has also been significantly reduced from its initial value of 7628.126. All values of
Jarque-Bera test are significant at 1% level and, therefore, indicate that normal
distribution assumption is not reasonable at this level of significance. That means that
even though all of the models have provided much smaller values of Jarque-Bera test,
they still indicate that residuals are not normally distributed. However, reduction of the
test statistic value is a good indicator. Skewness demonstrates how symmetric residuals
are distributed around zero (Hill, 2012, p. 148), but neither of models has produced a
value of zero skewness, showing that their distribution is not perfectly symmetrical.
EGARCH(1,1) has shown the smaller value of skewness than other models, as well as
the value shifted to positive from its initial value of -0.117811. Yet, it is still not desirable,
and it can be said that models fail to fully account for skewness.
Furthermore, Enders (2004, p. 147) said that Ljung-Box test can indicate if there is
any serial correlation remained in the squared residuals. It also provides a better
Skewness Kurtosis
Jarque-
Bera test
Skewness Kurtosis
Jarque-
Bera test
GARCH(1,1)
0.197851 3.846175 101.0026* 0.205743 3.856506 104.5132*
IGARCH(1,1)
0.195375 3.898824 111.1861* 0.208817 3.913883 116.8612*
TARCH(1,1)
0.173193 3.678776 67.21829* 0.175853 3.702772 71.48548*
EGARCH(1,1)
0.161342 3.700664 68.87752* 0.163780 3.738889 75.61402*
GARCH-M(1,1)
0.212277 3.856424 105.7618* 0.220293 3.869005 109.8799*
Student's-t distributionNormal distribution

54
understanding of how well the data fit in the model. Ljung-Box Q statistic at lag, 5, 10
and 20 was applied on standardised residuals and standardised squared residuals
series. Results are presented in the Table 10.
Table 10 Ljung-Box Q test statistic of different GARCH(1,1) models with both
normal and student’s-t distributions
According to the tests based on the Ljung-Box Q statistic, all models are accounting
for dependences between residuals reasonably well. In comparison to Table 3, where
values of Q2
statistic are far outside the critical values, it is seen that all models have
majorly reduced autocorrelation of residuals. Table 10 shows that both sets of models
Q(5) Q(10) Q(20) Q2
(5) Q2
(10) Q2
(20)
GARCH(1,1)
13.948 15.410 28.114 8.854 16.358 28.755
IGARCH(1,1)
14.370 15.805 29.610 11.846 19.374 35.751
TARCH(1,1)
14.369 16.175 29.310 5.290 11.223 24.659
EGARCH(1,1)
15.399 17.446 30.556 8.245 15.961 28.413
GARCH-M(1,1)
15.263 16.432 25.194 5.740 12.632 24.800
Q(5) Q(10) Q(20) Q2
(5) Q2
(10) Q2
(20)
GARCH(1,1)
13.712 15.171 27.806 8.611 16.050 28.208
IGARCH(1,1)
14.181 15.617 29.309 10.752 18.118 35.834
TARCH(1,1)
14.110 16.014 28.861 5.045 10.766 24.408
EGARCH(1,1)
15.062 17.257 30.041 7.280 14.104 27.363
GARCH-M(1,1)
14.815 15.957 24.539 6.236 13.042 24.717
Normal distribution
Student's-t distribution
Note: Q(n) follows the Chi squared distribution. Critical values at 1% significance level for lag 5, 10
and 20 are 15.086, 23.209 and 37.556 respectively.

55
with normal and with student’s-t distributions did not perform very differently from each
other in terms of Q statistic values. Therefore, no preference can be given to a particular
distribution or error terms within the mean equation, as both sets of values are within
the critical value range and are close to each other. However, when looking at the
uniformity of results in standardised residuals and standardised squared residuals
preference is given to standardised residuals as being a more reliable source of
justification. Enders (2004, p.118) also appreciated squared residuals rather than non-
squared residuals and said that they provide more rationale. Results of standardised
residuals set are controversial, because GARCH(1,1) has performed better at lag 5 and
10, but lag 20 is given to GARCH-M(1,1) model. Nevertheless, all values are safely less
than respective critical values. On the other hand, the choice of the most suitable model
according to the squared standardized residuals is obvious as it is clearly seen that
TARCH(1,1) and EGARCH(1,1) have produced better results in terms of eliminating
autocorrelation and heteroscedasticity effects in residuals. In both variation of error term
distribution, TARCH(1,1) has slightly outperformed the EGARCH(1,1), however, the
differences are not fundamental.
The worst performing models are EGARCH(1,1) for standardised residuals and
IGARCH(1,1) for standardised squared residuals. The results are still in accordance
with expectations, i.e. they are smaller than critical values, however, Q statistic values
are the highest when comparing them to other models. Both Tables 8 and 9 do not
include APARCH(1,1) model as it has been found to have variables that are not
statistically significant when applying to given data set.
In this section, it has been found that 5 different GARCH(1,1) models are able to
eliminate ARCH effects well. To fulfil the main objective of this dissertation, the next
section focuses on predictive abilities of models. Even though some models did not
produce desirable statistical measures, their forecasting ability is still studied in the
section 6.5.

56
6.5. Estimating forecast results
It is not only important to examine how good does data set fit the chosen model and if
the equation variables are statistically significant, but also how well can the chosen
model produce future volatility forecasts. Lopez (2001) mentioned that even if the model
provides good performance of all estimates, its forecasting abilities may be not as good.
Relating to section 5.6.1, there are two ways of evaluate forecasting performance. They
are the Loss functions and Information criterion. These two measures help to identify
the model that produces the best forecast.
6.5.1. Loss functions
Different kinds of loss functions have been more widely discussed in section 5.6.1.
The evaluations were made for both mean and variance equations using OxMetrics 6.0
software and its G@RCH package. The last 30 observations out of the whole sample
were left specifically for forecasting purposes. As covered by Diebold and Lopez (1996)
the model with smallest values of loss functions indicates that difference between the
actual and predicted values is small and the model provides better predictions.
However, as loss function values can be both positive and negative, the values closer to
zero are preferred.
Alberg et al (2008, p. 1206) describe the advantage of using various loss functions in
“the robustness in choosing an optimal predictor model”. As it has been said by
Brailsford and Faff (1996) it is unlikely that one model will produce all values that are
seen as the “best”. Therefore, if the majority of values are desirable, the model is
concluded to provide the best forecast.

57
Table 11 Loss function value of different GARCH(1,1) models
According to the results shown in Table 11, it is seen that EGARCH(1,1) is the model
that has the majority – 8 out of 13 different loss function values more desirable than
those of other models. Table 11 shows that in some cases, like ME and RMSE both
mean and variance equations produce desirable values, but in case of MSE, for
instance, only variance equation is seen as the “best”. On the other hand, the desirable
values of loss functions that do not “belong” to EGARCH(1,1) model are spread out
between other models without any uniformity. TARCH(1,1) model has only two values of
loss functions – MedSE(1) and MAE(1) that are better than those of other models,
which is not in consistency with previous sections where TARCH(1,1) model has shown
very good performance when comparing to other GARCH(1,1) models.
GARCH(1,1) IGARCH(1,1) TGARCH(1,1) EGARCH(1,1) GARCH-M(1,1)
MSE(1) 3.548E-05 3.548E-05 3.543E-05 3.542E-05 3.537E-05
MSE(2) 2.064E-09 2.080E-09 2.578E-09 1.896E-09 2.072E-09
MedSE(1) 1.371E-05 1.368E-05 1.105E-05 1.216E-05 1.346E-05
MedSE(2) 1.956E-09 1.937E-09 2.643E-09 1.876E-09 1.950E-09
ME(1) -0.000246 0.0002422 0.0001346 -2.905E-05 -0.0001991
ME(2) -1.996E-05 -2.050E-05 -2.996E-05 -1.383E-05 -2.022E-05
MAE(1) 0.004757 0.004757 0.004732 0.004742 0.004747
MAE(2) 4.132E-05 4.141E-05 4.589E-05 3.974E-05 4.14E-05
RMSE(1) 0.005956 0.005956 0.005953 0.005951 0.005974
RMSE(2) 4.543E-05 4.561E-05 5.077E-05 4.354E-05 4.55E-05
MAPE(2) 83.06 82.5 102.9 77.6 83.21
TIC(1) 0.9192 0.9197 0.976 0.9502 0.9244
TIC(2) 0.4099 0.4092 0.4209 0.4168 0.4098
Note: (1) – mean equation, (2) – variance equation. Figures in bold indicate the best results for the
forecasting measures.

58
Table 12 Loss function values of different GARCH(1,1) models with Student’s-t
distribution
In comparison with Table 11, Table 12 does not provide the same uniformity of
indication of which model is superior. The only difference between Table 11 and 12 is
that models presented in Table 12 allowed for student’s-t distribution within error terms.
Desirable values in Table 12 are much more spread out between models and no trend
of superiority of a particular model can be indicated.
Regarding the worst performing models, GARCH(1,1), GARCH-M(1,1) and
IGARCH(1,1) were found to be the “worst” among others in Table 11. On the other
hand, Table 12 shows that GARCH(1,1) has shown relatively good results, however,
IGARCH(1,1) and GARCH-M(1,1) have shown poor performance. This can be a strong
indicator that models that allow for asymmetries have smaller loss function values, and
GARCH(1,1) IGARCH(1,1) TGARCH(1,1) EGARCH(1,1) GARCH-M(1,1)
MSE(1) 3.553E-05 3.553E-05 3.542E-05 3.542E-05 3.544E-05
MSE(2) 2.071E-09 2.116E-09 2.567E-09 3.335E+04 2.076E-09
MedSE(1) 1.370E-05 1.370E-05 1.209E-05 1.166E-05 1.391E-05
MedSE(2) 1.957E-09 1.929E-09 2.668E-09 3.822 1.952E-09
ME(1) -0.000338 -0.0003361 -2.025E-05 4.247E-05 -0.0002984
ME(2) -2.016E-05 -2.141E-05 -2.959E-05 -80.050 -2.030E-05
MAE(1) 0.004763 0.004763 0.004742 0.004738 0.004755
MAE(2) 4.139E-05 4.172E-05 4.577E-05 80.050 4.143E-05
RMSE(1) 0.005961 0.005961 0.005951 0.005951 0.005953
RMSE(2) 4.551E-05 4.600E-05 5.066E-05 182.6 4.556E-05
MAPE(2) 83.4 83.55 103.7 1.241E+07 83.43
TIC(1) 0.9076 0.9073 0.9516 0.9612 0.9114
TIC(2) 0.41 0.4093 0.4215 1 0.4099
Note: (1) – mean equation, (2) – variance equation. Figures in bold indicate the best results for the
forecasting measures.

59
therefore, provide the “best” performance when forecasting FTSE 100 stock market
volatility. This conclusion is in accordance with those made in section 6.4.
Use of various loss functions rather than just one is obvious, as it allows making an
unbiased conclusion. However, it is still reasonable to study other measures that can
indicate the superior model and it will provide more justification of the results, as
conclusion will be made taking a view from different angles. Therefore, next section
focuses on various information criteria.
6.5.2. Akaike information criteria and Bayesian information criteria
All models, both with Student’s-t and normal distribution of error terms have been
estimated to produce AIC and BIC results. EViews 8.0 statistical package provides
these values alongside with variables estimation. As mentioned in section 5.6.2, the
preference should be given to the model that produces smallest AIC and BIC values.
Table 13 AIC and BIC values of models with normal error terms distribution
Table 14 AIC and BIC values of models with Student’s-t error terms distribution
GARCH(1,1) IGARCH(1,1) TARCH(1,1) EGARCH(1,1) GARCH-M(1,1)
AIC -6.435011 -6.420551 -6.476049 -6.478759 -6.446834
BIC -6.426474 -6.416282 -6.465377 -6.468087 -6.436162
GARCH(1,1) IGARCH(1,1) TARCH(1,1) EGARCH(1,1) GARCH-M(1,1)
AIC -6.454758 -6.443339 -6.488105 -6.491093 -6.467526
BIC -6.444086 -6.436936 -6.475298 -6.478287 -6.454719

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