Prediction of Popular Global Stock Indexes Volatility by Using

ARCH/GARCH Models
Nagendra Marisetty
Nagendra Marisetty, Faculty, REVA Business School (RBS), REVA University, Bangalore,
Karnataka, India. nagendra.marisetty@gmail.com.

Abstract
This study investigates the volatility dynamics of five major global financial indices—FTSE
100, Hang Seng, NIKKEI 225, NSE 50, and S&P 500—using a range of GARCH models over
a ten-year period from January 1, 2014, to December 31, 2023. The analysis involves
preprocessing the data to ensure stationarity, calculating log returns, and conducting
stationarity and ARCH effect LM tests. Various GARCH models, including GARCH (0,1),
GARCH (1,1), GARCH (1,2), and GARCH (2,2), are applied to capture and forecast volatility.
The study aims to determine the most effective model for accurately reflecting volatility
dynamics while accounting for significant market events such as the COVID-19 pandemic.
The findings reveal that the GARCH (1,1) model generally provides a robust balance between
model simplicity and statistical significance, effectively capturing the time-varying volatility
of the indices. Despite some complex models offering better fit measures according to the
Akaike Information Criterion (AIC) and Schwarz Criterion (SC), the GARCH (1,1) model
consistently demonstrates significant parameter estimates and reliable predictive performance,
as evidenced by consistent Root Mean Square Error (RMSE) and Mean Absolute Error (MAE)
values. This suggests that the GARCH (1,1) model is a preferred choice for volatility
forecasting due to its effectiveness and parsimony, although future research might explore more
advanced GARCH model variations for further refinement.
Key words: ARCH Model, GARCH Models, Global Stock Indexes, and Volatility,

1. Introduction
The prediction of financial market volatility has long been a focal point of research within the
field of financial econometrics, primarily driven by the need for effective risk management and
accurate forecasting. Volatility, a measure of the degree of variation in asset prices, serves as a
critical indicator of market uncertainty and risk. Understanding and predicting this volatility is
essential for investors, policymakers, and financial institutions alike. One of the most
influential and widely used methodologies for analysing financial market volatility is the
Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, introduced by
Bollerslev (1986). The foundational work on ARCH models by Engle (1982) paved the way
for Bollerslev’s GARCH model, which extends the ARCH framework by incorporating past
conditional variances into the model. This innovation allowed for a more comprehensive
capture of the persistent volatility clustering observed in financial time series data. Since its
inception, the GARCH model has evolved significantly, giving rise to a variety of GARCH-
type models that address specific characteristics of financial markets. These models include the
Exponential GARCH (EGARCH) by Nelson (1991), which accounts for the asymmetric effects
of shocks on volatility, and the Threshold GARCH (TGARCH) by Zakoian (1994), which
models threshold effects in volatility.

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Empirical studies have demonstrated the robustness and versatility of GARCH models across
different financial markets and economic contexts. For instance, Rossetti et al. (2017) utilized
the EGARCH model to analyze fixed income market volatility across multiple countries,
revealing its effectiveness in capturing volatility influenced by macroeconomic events.
Similarly, Chkili et al. (2021) employed a hybrid model combining FIAPARCH and artificial
neural networks (ANN) to study Islamic stock market volatility, illustrating the superior
forecasting accuracy of this approach during periods of significant financial events such as the
9/11 attacks and the 2008 financial crisis. The application of GARCH models extends beyond
traditional financial markets to include commodity markets and macroeconomic indicators.
Paolella et al. (2008) explored the implications of emission allowances on power and gas
markets, emphasizing the importance of understanding the statistical distributions of emission
trading returns for hedging strategies. Furthermore, Liang (2013) introduced GARCH models
in reliability forecasting, demonstrating their efficacy in predicting failure data for electronic
systems.
In the context of global financial crises and pandemics, GARCH models have proven
invaluable. Rehman et al. (2023) examined the impact of the 2008 Global Financial Crisis and
the COVID-19 pandemic on stock markets in GCC economies, highlighting the significant
changes in market behaviour during these periods. Onali (2020) and Mobin et al. (2022) further
underscored the importance of GARCH models in analysing the effects of COVID-19 on stock
and bond market volatility in the US and G7 countries, respectively. Bitcoin and other digital
assets have also been analysed using GARCH models to assess their potential as alternatives
to fiat currencies. Cermak (2017) applied a GARCH(1,1) model to examine Bitcoin's volatility
in relation to macroeconomic variables, providing insights into its suitability as a stable
financial asset. In recent years, the integration of advanced computational techniques with
GARCH models has led to enhanced predictive capabilities. Liu et al. (2024) developed a BTC
trading prediction model by combining DCC-GARCH with artificial neural networks,
showcasing the benefits of leveraging dynamic correlation and volatility data in trading
strategies.
The comprehensive body of research on GARCH models underscores their critical role in
understanding and forecasting volatility across various financial contexts. Whether it is stock
markets, commodity prices, or macroeconomic impacts, GARCH models provide a robust
framework for capturing the intricate dynamics of volatility. Their adaptability and predictive
power make them indispensable tools for financial analysts, economists, and policymakers
navigating the complexities of modern financial markets. This study aims to build on this
extensive literature by specifically focusing on the prediction of volatility in popular global
indexes using ARCH/GARCH models. By doing so, it seeks to provide deeper insights into the
effectiveness of these models in different economic environments and enhance the existing
body of knowledge on financial market volatility prediction.

2. Literature Review
In the expansive realm of financial modelling, the GARCH (Generalized Autoregressive
Conditional Heteroskedasticity) model and its variants have emerged as fundamental tools for
analysing and forecasting volatility across diverse economic landscapes. This narrative
synthesis explores a wealth of research studies, each elucidating the nuances and applications

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of GARCH models in various contexts, from stock markets to commodity prices, and from
macroeconomic impacts to unique industry-specific challenges.
The study of financial market volatility has been a cornerstone of financial econometrics,
particularly through the development and application of various autoregressive conditional
heteroskedasticity (ARCH) models. Bollerslev (1986) pioneered this area by introducing the
generalized autoregressive conditional heteroskedasticity (GARCH) model, extending the
basic ARCH framework to incorporate past conditional variances. This model set the stage for
a plethora of GARCH-type models that aim to capture the complex dynamics of financial time
series data. Among the early extensions of the GARCH model, Nelson (1991) introduced the
exponential GARCH (EGARCH) model, which accommodates the asymmetric effects of
shocks on volatility. The EGARCH model, along with others like the GJR model by Glosten et
al. (1993) and the NGARCH model by Bera and Higgins (1993), have been instrumental in
understanding financial volatility. These models address the need to capture volatility
asymmetry, a common feature in financial markets where negative shocks often have a larger
impact on volatility than positive ones.
Further advancements include the Asymmetric Power GARCH model by Ding et al. (1993)
and the Threshold GARCH model by Zakoian (1994), which refine the ability to model
asymmetries and threshold effects in volatility. Bollerslev and Ghysels (1996) introduced the
periodic GARCH (PGARCH) model, which accounts for seasonal volatility patterns in high-
frequency asset returns. These models have significantly improved our understanding of
volatility dynamics, providing better tools for risk management and forecasting. Empirical
studies across different markets and time periods have demonstrated the utility of these models.
Rossetti et al. (2017) examined fixed income market volatility in 11 countries, focusing on
interbank interest rates from January 2000 to December 2011. Their study employed an array
of models, with the EGARCH model emerging as particularly adept at capturing volatility
influenced more by internal macroeconomic events than external shocks. Similarly, Chkili et
al. (2021) investigated Islamic stock market volatility from 1999 to 2017, encompassing pivotal
events like the 9/11 attacks and the 2008 financial crisis. They introduced a hybrid model
combining FIAPARCH and artificial neural networks (ANN), which outperformed traditional
models in forecasting accuracy.
Paolella et al. (2008) delved into the implications of carbon and sulphur dioxide emission
allowances on global power and gas markets. Their focus on the U.S. Clean Air Act and the
EU Emissions Trading Scheme highlighted the crucial role of understanding statistical
distributions and forecast abilities of emission trading returns for optimal hedging and
purchasing strategies. On a different note, Evgenidis et al. (2016) explored the predictive power
of the yield spread on real economic growth by examining its relationship with interest rate
volatility through GARCH models and Markov regime switching. In the realm of reliability
forecasting, Liang (2013) introduced the GARCH model to analyze and forecast failure data
for repairable systems, specifically electronic systems from Chrysler suppliers. This innovative
application demonstrated the model's effectiveness in analysing failure data volatility and
predicting future failures. Arthur et al. (1996) presented a theory of asset pricing based on
heterogeneous agents who adapt their expectations to market dynamics. Through
computational experiments with an artificial stock market, they showcased how the evolution
of traders' expectations leads to various market regimes.

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Banking sector studies by Elyasiani (2004) utilized a multivariate GARCH model to analyze
bank stock returns and their volatilities in response to short-term and long-term interest rates
across three portfolios: money centre banks, large banks, and small banks. Yang et al. (2022)
evaluated financial market risk in the digital economy using a GARCH-VaR model tailored for
big data, underscoring the increasing complexity of modern financial markets. Rehman et al.
(2023) examined the effects of the 2008 Global Financial Crisis and the COVID-19 pandemic
on stock markets in six GCC economies using ARCH/GARCH models, revealing significant
impacts on market behaviour. Bitcoin's potential as an alternative to fiat currencies was
assessed by Cermak (2017), who analysed its volatility using a GARCH(1,1) model in relation
to macroeconomic variables. Handika et al. (2016) evaluated the accuracy of various volatility
models through a Value-at-Risk (VaR) approach, examining their impact on investment
performance in financialized commodity markets. Rastogi et al. (2024) explored the volatility
of agricultural commodity prices and their impact on inflation in India using BEKK GARCH
and DCC GARCH models, highlighting the intricate interplay between commodity markets
and macroeconomic indicators.
Lim et al. (2013) modelled the volatility of Malaysia's stock market using symmetric and
asymmetric GARCH models, comparing their performance across different time frames from
1990 to 2010. Arsalan et al. (2022) analysed stock market volatility and mean reversion across
various global stock exchanges using the GARCH(1,1) model, providing insights into
international market dynamics. Onali (2020) studied the impact of COVID-19 cases and deaths
on the US stock market using a GARCH(1,1) model, highlighting the pandemic's profound
effect on financial markets. In a comprehensive analysis, Mobin et al. (2022) investigated the
impact of COVID-19 on the risk dynamics of stock and bond markets in G7 countries using
GARCH models, revealing shifts in market risk profiles. Liu et al. (2024) developed a BTC
trading prediction model by integrating DCC-GARCH and artificial neural networks (ANN),
leveraging dynamic correlation and volatility data to enhance trading strategies. Abbas et al.
(2020) examined the interaction between macroeconomic uncertainty and stock market return
and volatility in China and the USA using GARCH models and a multivariate VAR model.
Guo (2005) investigated the economic significance of predicting foreign exchange rate
volatility using GARCH models versus implied volatility from currency options. Debasish
(2009) explored the impact of Nifty index futures on the volatility of Indian spot markets using
econometric models, providing insights into derivative markets. Al-Rjoub et al. (2012)
examined stock returns and volatility during financial crises in Jordan using the GARCH-M
model, shedding light on market behavior during turbulent periods. Mahmoud Sayed Agbo
(2023) forecasted prices of key Egyptian export crops using ARIMA and GARCH models,
emphasizing the importance of volatility modelling in agricultural markets. Hartz et al. (2012)
demonstrated that using non-Gaussian innovation distributions in GARCH models is more
effective for capturing volatility clustering and improving value-at-risk predictions compared
to outlier removal. Badaye et al. (2020) introduced a novel methodology using MC-GARCH
and copula models to forecast intraday VaR and ES for foreign currency portfolios.
Setiawan et al. (2021) examined the impact of the COVID-19 pandemic on stock market returns
and volatility in Indonesia and Hungary using a GARCH(1,1) model, highlighting the
pandemic's disparate effects on emerging and developed economies. Lee et al. (2021) proposed
an orthogonal ARMA-GARCH approach for generating economic scenarios to manage risks
in financial institutions, especially during turbulent periods like the COVID-19 pandemic. Xie

4
et al. (2023) investigated how mixed-frequency investor sentiment affects stock volatility in
the China A-shares market using the MIDAS-GARCH model, demonstrating its superior
explanatory power. Rajvanshi et al. (2019) evaluated the forecasting power of GARCH models
for the Nifty 50 index's return volatility using realized volatility as a proxy. Dixit et al. (2010)
evaluated the informational efficiency of S&P CNX Nifty index options in India, providing
insights into derivative market dynamics.
Sreenu et al. (2022) examined the impact of volatility on asset pricing and financial risk in the
Indian stock market using GARCH-M and E-GARCH-M models, highlighting the complex
relationship between volatility and asset pricing. Duppati et al. (2017) examined the ability of
intraday data to predict long-term memory in volatility for Asian equity indices using GARCH-
based models and realized volatility approaches. Flannery (2005) estimated a GARCH model
to analyze how daily equity returns and their volatility are influenced by macroeconomic
variables, providing a comprehensive view of market dynamics. Ugurlu et al. (2014) evaluated
GARCH-type models for stock market volatility in European emerging countries and Turkey,
using daily data to offer insights into regional market behavior. Kinateder et al. (2014)
integrated long memory with a GARCH(1,1) model and fat-tailed innovations to forecast
market risk over multiple periods, enhancing risk management strategies.
Chen (2015) investigated the changing risk-return relationship in Chinese stock markets,
focusing on differences between Shanghai and Shenzhen, varying data frequencies, and
comparing GARCH-M model specifications. Abrosimova et al. (2005) tested weak-form
efficiency in the Russian stock market using various data frequencies, providing insights into
market efficiency. Naidoo et al. (2023) examined how exchange rate volatility affects South
Africa's stock and real estate markets using GARCH(1,1) models, highlighting the
interconnectedness of financial markets. Handika et al. (2018) examined the empirical
performance of GARCH models in forecasting volatility across financialized commodity
markets, demonstrating the model's robustness. Mușetescu et al. (2022) used GARCH(1,1),
GARCH-M(1,1), and EGARCH(1,1) models to estimate and predict Brent Crude Oil return
volatility, emphasizing the importance of accurate volatility modelling in energy markets.
Rastogi et al. (2023) explored the volatility spillover effects of crude oil, gold, interest rates,
and exchange rates on inflation in India using BEKK-GARCH and DCC-GARCH models.
Amelot et al. (2020) evaluated time series models, artificial neural networks (ANNs), and
statistical topologies to forecast foreign exchange rates, providing a comparative analysis of
forecasting methodologies. Wang et al. (2022) used a GARCH model with structural breaks to
forecast stock volatility in the Chinese stock market, incorporating financial news sentiment
analysis to enhance predictive accuracy. Wu (2010) developed a threshold GARCH model to
analyze and predict long-term volatility, highlighting the model's sensitivity to different
volatility regimes. Xie et al. (2023) used the MIDAS-GARCH model to integrate mixed-
frequency investor sentiment into stock volatility forecasting, showcasing its superior
explanatory power over traditional models. In conclusion, the vast array of studies utilizing
GARCH models underscores their critical role in understanding and forecasting volatility
across diverse financial contexts. From stock markets to commodity prices, and from
macroeconomic impacts to specific industry challenges, GARCH models provide a robust
framework for capturing the intricate dynamics of volatility. These models' adaptability and
predictive power make them indispensable tools for financial analysts, economists, and
policymakers seeking to navigate the complexities of modern financial markets

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3. Methodology
The study will utilize daily closing prices of the FTSE 100, Hang Seng, NIKKEI 225, NSE 50,
and S&P 500 indexes. These indexes were selected due to their representation of major global
economies and their significance in financial markets. The data will be obtained from reliable
financial databases, specifically Yahoo Finance, covering a period of ten years, from January
1, 2014, to December 31, 2023. This extensive time span ensures a comprehensive analysis
that includes various market conditions, such as bull and bear markets, periods of economic
stability, and the effects of significant events like the COVID-19 pandemic. By capturing the
full spectrum of volatility dynamics, this study aims to provide a nuanced understanding of the
predictive capabilities of ARCH/GARCH models in the context of global financial indexes.
Before applying the GARCH models, it is essential to preprocess the collected data to ensure
its suitability for analysis. This preprocessing involves several key steps:
3.1. Log Returns Calculation
To facilitate effective volatility modelling, daily log returns will be computed from the closing
prices of the indexes. Log returns are preferred over simple returns because they tend to be
more stationary, which is crucial for accurate volatility estimation. The log return is calculated
using the following formula:
𝑝𝑡
rt = ln (𝑝 ), Where pt and pt-1 are the closing prices at time t and t-1
𝑡−1

3.2. Basic descriptive statistics

Mean, standard deviation, skewness, and kurtosis—will be calculated to gain insights into the
characteristics of the log returns. These statistics will help in understanding the central
tendency, dispersion, and distribution shape of the data.
3.3. Stationarity Check
The stationarity of the log returns will be assessed using three tests: Augmented Dickey-Fuller
(ADF) Test: This test checks for the presence of a unit root in the time series. If the test statistic
is less than the critical value at a given confidence level and the p-value is below the chosen
significance level, the null hypothesis of a unit root can be rejected, indicating that the series
is stationary. Augmented Dickey-Fuller Generalized Least Squares (ADF-GLS) Test: An
enhancement of the ADF test, this test also evaluates the presence of a unit root but includes
generalized least squares detrending to improve power. Kwiatkowski-Phillips-Schmidt-Shin
(KPSS) Test: This test examines the null hypothesis of stationarity against the alternative
hypothesis of non-stationarity. It determines whether the series is stationary around a trend
(trend-stationary) or around a mean (level-stationary).
3.4. ARCH Effect
The ARCH-LM (Autoregressive Conditional Heteroskedasticity - Lagrange Multiplier) test
will be employed to detect the presence of autoregressive conditional heteroskedasticity
(ARCH) effects in the time series. This test is crucial for identifying whether the variance of
the residuals is dependent on past error terms, which is a typical feature in financial time series.
The test statistic is compared with critical values from the chi-squared distribution; if the
statistic exceeds the critical value, the null hypothesis of no ARCH effects is rejected.

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3.5. Visual Analysis
Time series plots of the log returns will be generated to visually inspect the data for patterns,
anomalies, and the presence of ARCH effects. This visual inspection helps in identifying trends,
cycles, or irregularities that might not be apparent through statistical tests alone.
3.6. Model Specification
The study will utilize several GARCH models to forecast the volatility of the selected stock
indexes, incorporating the normal distribution. The analysis will encompass symmetric model,
including the GARCH (0,1), GARCH (1,1), GARCH (1,2) and GARCH (2,2).
3.7. ARCH Model
In traditional econometrics, it is commonly assumed that the variance of a random variable is
constant over time. However, financial time series often display heteroscedasticity, where
variance remains stable over the long term but fluctuates in the short term. To address this time-
varying volatility, Engle (1982) developed the Autoregressive Conditional Heteroskedasticity
(ARCH) model. This model is specifically designed to capture and model the evolving variance
and mean of time series data. The general representation of the ARCH model is:
yt = ϕ xt + μt -----------------------------------------------------------------------------(1)
σt2 = E(μt2∣μt-1, μt-2…...) = α0 + α1𝜇𝑡−1
2 2
+ ………… + αp𝜇𝑡−𝑝 = ∑𝑝𝑖=1 α𝑖 𝜇𝑡−𝑖
2
-----(2)

In the ARCH model, ϕ is a non-zero parameter that needs to be estimated, xt represents the
independent variable observed at time t, and ut is the random error term, which is typically
assumed to follow a normal distribution in the standard model. The core idea of the ARCH
model is that the variance of the residuals μt at time t depends on the squared error terms from
previous periods. Specifically, the model asserts that the variance of the error term at time t is
a linear function of the squared error terms from the preceding p periods. However, the ARCH
model assumes that both positive and negative shocks impact volatility equally, making it less
suitable for analysing time series data with asymmetric effects.
3.8. GARCH Model
Bollerslev (1986) introduced an important refinement to the ARCH model known as the
GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model. This model is
designed to better capture the volatility clustering commonly observed in financial time series.
Unlike the ARCH model, the GARCH model incorporates the conditional variance as a
GARCH process, allowing for more accurate estimation of time-varying volatility. The
defining equations of the GARCH model are as follows:
yt = ϕ xt + μt , μ~N(0, σt2) ------------------------------------------------------(3)

σt2 = ω + ∑𝑝𝑖=1 α𝑖 𝜇𝑡−𝑖

2
+ ∑𝑝𝑖=1 β𝑖 𝜎𝑡−𝑖
2
------------------------------------------(4)
2 2
In this model, 𝜇𝑡−𝑖 represents the ARCH parameter, while 𝜎𝑡−𝑖 is the GARCH parameter. The
coefficients associated with the ARCH and GARCH terms are indicated by α and β,
respectively, and p and q indicate the lag order of the model. Therefore, the ARCH model can
be seen as a specific case within the broader GARCH framework. In this study, primarily utilize
the GARCH(1,1) model, which includes one lag, to estimate the sample series. The strength of

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the GARCH model lies in its ability to reflect and interpret heteroscedasticity. However, it still
falls short in capturing asymmetry in financial time series.

3.9. Diagnostic Tests

To evaluate the adequacy and predictability of the GARCH models used in this study, several
diagnostic tests will be conducted. The Akaike Information Criterion (AIC) and the Schwarz
Criterion (SC), also known as the Bayesian Information Criterion (BIC), are utilized to assess
model fit while balancing complexity. The AIC helps in selecting models that achieve a good
balance between fit and parsimony by penalizing excessive complexity. Similarly, the SC also
accounts for the number of parameters but imposes a stricter penalty for model complexity as
the sample size increases. Both criteria are instrumental in identifying models that are both
accurate and efficient. Additionally, Root Mean Square Error (RMSE) and Mean Absolute
Error (MAE) will be used to evaluate the predictive accuracy of the models. RMSE measures
the average magnitude of prediction errors, providing insight into how well the model predicts
the observed data. MAE, on the other hand, calculates the average absolute differences between
observed and predicted values, offering a more robust measure of accuracy that is less
influenced by outliers. Together, these diagnostic metrics will guide the selection of the most
appropriate model for forecasting volatility, ensuring that the chosen model effectively captures
the dynamics of the financial time series data.

4. Results Discussion
The results analysis will focus on evaluating the performance of the various GARCH models
in forecasting volatility for the selected stock indexes. By comparing the models' predictive
accuracy and fit using metrics such as AIC, SC, RMSE, and MAE, the study aims to identify
the most effective approach for capturing volatility dynamics. This analysis will provide
insights into the relative strengths and weaknesses of each model, guiding future volatility
forecasting strategies.
Table 1: Descriptive statistics of Five Global Indexes during 2014 to 2023 period.

Variable FTSE 100 HANGSENG NIKKE I225 NSE 50 S&P 500

Mean 0.0053909 -0.012707 0.029454 0.05045 0.03768

Median 0.056513 0.028185 0.075226 0.07675 0.05972
Minimum -11.512 -6.5673 -8.2529 -13.904 -12.765
Maximum 8.6668 8.6928 7.7314 8.4003 8.9683
Std. Dev. 1.0004 1.3009 1.2585 1.0485 1.1212
C.V. 185.56 102.38 42.727 20.783 29.757
Skewness -0.86492 0.070791 -0.1421 -1.3743 -0.8079
Ex. kurtosis 12.828 3.2598 4.2747 20.453 16.041
5% Perc. -1.5223 -2.1453 -2.0566 -1.5115 -1.6827
95% Perc. 1.4742 2.001 1.9493 1.5095 1.544
IQ range 0.94588 1.3986 1.272 1.046 0.94678
(Source: Statistical calculations)

Table 1 provides a comprehensive overview of the descriptive statistics for five major global
indexes—FTSE 100, Hang Seng, NIKKEI 225, NSE 50, and S&P 500—over the period from

8
2014 to 2023. The mean returns of these indexes vary, with the NSE 50 showing the highest
average return at 0.05045, followed by the S&P 500 and NIKKEI 225. In contrast, the Hang
Seng index exhibits a negative mean return of -0.012707. The median returns across the indexes
are generally higher than their means, indicating skewness in the return distributions. For
instance, the FTSE 100 and NSE 50 have medians of 0.056513 and 0.07675, respectively,
compared to their mean returns, suggesting positive skewness in their return distributions.
The table also reveals significant variability in the indexes, with the Hang Seng index showing
the highest standard deviation of 1.3009, indicating greater volatility compared to the other
indexes. The coefficient of variation (C.V.), which reflects the relative dispersion of the returns,
is notably high for the FTSE 100 at 185.56, suggesting extreme variability in returns relative
to its mean. The indexes exhibit varying degrees of skewness and excess kurtosis, with the
FTSE 100 and NSE 50 showing pronounced negative skewness and high excess kurtosis,
indicating heavy tails and a higher likelihood of extreme return values. Overall, the descriptive
statistics highlight the diverse volatility and return characteristics of these global indexes,
underscoring the importance of tailored volatility modelling approaches for accurate
forecasting.
Table 2: Unit Root Test of Selected International Indices Returns

Indexes ADF Test (12 lag) ADF GLS Test (12 lag ) KPSS Test (8 lag ) ARCH LM Test (5 lag)

FTSE 100 -15.2617* (0.0000) -11.1424* (0.0000) 0.022667* (>0.1000) 429.598* (0.0000)

HANG SENG -49.3353* (0.0000) -49.2632* (0.0000) 0.022667* (>0.1000) 234.640* (0.0000)

NIKKEI 225 -28.7549* (0.0000) -17.6862* (0.0000) 0.020862* (>0.1000) 223.998* (0.0000)

NIFTY 50 -13.5031* (0.0000) -10.8833* (0.0000) 0.037368* (>0.1000) 431.221* (0.0000)

S&P 500 -15.7598* (0.0000) -9.17740* (0.0000) 0.024392* (>0.1000) 913.263* (0.0000)
(Source: Statistical calculations)(* 5 percent level of significance) (Probabilities in parenthesis)

Table 2 presents the results of unit root and ARCH-LM tests conducted on the returns of
selected international indices, including the FTSE 100, Hang Seng, NIKKEI 225, NSE 50, and
S&P 500. The results from the Augmented Dickey-Fuller (ADF) and ADF GLS tests, with 12
lags, show that all indices are stationary as indicated by their test statistics being significantly
negative and their p-values being well below the 5 percent significance level. This confirms
that the returns of these indices do not contain a unit root and are thus appropriate for further
analysis. The Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test results also support stationarity
for all indices, as the test statistics are below the critical value thresholds, confirming that the
time series are stationary around a mean.
The ARCH-LM test results reveal significant autoregressive conditional heteroskedasticity
(ARCH) effects in the returns of all indices, with high test statistics and p-values well below
0.05. This indicates that the volatility of these indices is not constant over time but instead
depends on past squared returns, which aligns with the common observation of volatility
clustering in financial time series. The large test statistics suggest that the variance of the
returns is significantly influenced by past error terms, justifying the use of GARCH models to
analyze, and forecast volatility for these indices. The combination of results from these
diagnostic tests confirms that the returns are stationary and exhibit ARCH effects, making them
suitable for further volatility modelling using GARCH models.
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Time Series Plot

Chart 1: FTSE 100 daily returns from 2014 to 2023

10

-5

-10

-15
2014 2016 2018 2020 2022 2024

(Source: Statistical calculations)

The chart 1 shows the daily returns of the FTSE 100 index from 2014 to 2023. The data exhibits
significant volatility, with periods of higher fluctuations around 2016 and a notable spike in
2020, likely corresponding to the COVID-19 pandemic's market impact. Post-2020, the
volatility seems to stabilize but remains elevated compared to the earlier part of the period.
This pattern highlights how major global events can significantly impact financial markets,
causing sharp fluctuations in daily returns. The overall trend does not show a clear directional
movement, indicating a mix of gains and losses throughout the period.
Chart 2: HANG SENG daily returns from 2014 to 2023
10

-2

-4

-6

-8
2014 2016 2018 2020 2022

(Source: Statistical calculations)

The daily returns of the Hang Seng Index from 2014 to 2023 (Chart 2) exhibit characteristics
typical of ARCH (Autoregressive Conditional Heteroskedasticity) effects, where periods of
high volatility are followed by more high volatility and periods of low volatility tend to follow
low volatility. The spikes in volatility around 2016 and the significant increase in 2020
highlight this clustering behavior. This clustering effect is indicative of the ARCH process,
where the magnitude of returns tends to be auto-correlated. Specifically, the heightened
volatility during the COVID-19 pandemic in 2020 and the elevated levels post-2020
demonstrate how past volatility can influence future volatility, resulting in persistent periods
of high and low fluctuations. This visual effect underscores the importance of using ARCH or

10
GARCH models to better understand and forecast future volatility based on past patterns in
financial time series data like the Hang Seng Index returns.
Chart 3: NIKKEI 225 daily returns from 2014 to 2023
8

-2

-4

-6

-8

-10
2014 2016 2018 2020 2022

(Source: Statistical calculations)

The chart 3 depicts the daily returns of the Nikkei 225 index from 2014 to 2023, showcasing
notable volatility patterns over the period. Observing the chart, there are periods of intense
fluctuation around 2016 and a significant spike in 2020, likely due to the global impact of the
COVID-19 pandemic. Post-2020, the data shows a continuation of heightened volatility,
although with some stabilization compared to the peak. The chart clearly exhibits ARCH
(Autoregressive Conditional Heteroskedasticity) effects, where clusters of high volatility
follow previous high volatility periods, and similarly, clusters of low volatility follow low
volatility periods. This visual clustering suggests that past volatility has a predictive influence
on future volatility, making ARCH models particularly useful for analyzing and forecasting the
volatility patterns seen in the Nikkei 225 daily returns.
Chart 4: NSE 50 daily returns from 2014 to 2023
10

-5

-10

-15
2014 2016 2018 2020 2022

(Source: Statistical calculations)

The daily returns (Chart 4) of the NSE 50 Index from 2014 to 2023 display typical
characteristics of ARCH (Autoregressive Conditional Heteroskedasticity) effects, where high
volatility periods are followed by more high volatility, and low volatility periods follow low
volatility. Notable spikes in volatility around 2016 and the significant surge in 2020 highlight
this clustering behavior. This pattern is indicative of the ARCH process, where the magnitude
of returns tends to be auto-correlated. The heightened volatility during the COVID-19

11
pandemic in 2020 and the elevated levels of volatility post-2020 demonstrate how past
volatility influences future volatility, resulting in persistent periods of fluctuation. This visual
effect emphasizes the importance of using ARCH or GARCH models to better understand and
forecast future volatility based on past patterns in financial time series data like the NSE 50
Index returns.
Chart 5: S&P 500 daily returns from 2014 to 2023
10

-5

-10

-15
2014 2016 2018 2020 2022 2024

(Source: Statistical calculations)

The daily returns of the S&P 500 Index from 2014 to 2023 (Chart 5) exhibit characteristics
typical of ARCH (Autoregressive Conditional Heteroskedasticity) effects, where periods of
high volatility are followed by more high volatility and periods of low volatility tend to follow
low volatility. The spikes in volatility around 2016 and the significant increase in 2020
highlight this clustering behavior. This clustering effect is indicative of the ARCH process,
where the magnitude of returns tends to be auto-correlated. Specifically, the heightened
volatility during the COVID-19 pandemic in 2020 and the elevated levels post-2020
demonstrate how past volatility can influence future volatility, resulting in persistent periods
of high and low fluctuations. This visual effect underscores the importance of using ARCH or
GARCH models to better understand and forecast future volatility based on past patterns in
financial time series data like the S&P 500 Index returns.
The analysis of the daily returns for five major indexes from 2014 to 2023—FTSE 100, Hang
Seng, Nikkei 225, NSE 50, and S&P 500—reveals common patterns of volatility influenced
by global events, particularly the COVID-19 pandemic in 2020. All five indexes exhibit
significant volatility spikes around 2016 and a pronounced surge in 2020, highlighting the
impact of global crises on financial markets. Post-2020, while some stabilization is observed,
volatility remains elevated compared to the earlier part of the period across all indexes. This
persistent volatility underscores the presence of ARCH (Autoregressive Conditional
Heteroskedasticity) effects, where periods of high volatility follow high volatility and low
volatility follows low volatility, indicating that past volatility influences future fluctuations.
This clustering behaviour is evident in the Hang Seng, Nikkei 225, NSE 50, and S&P 500
indexes, with notable periods of heightened volatility during and after the COVID-19
pandemic. The FTSE 100 also shows significant volatility but without a clear directional trend,
reflecting a mix of gains and losses. These observations underscore the importance of using
ARCH or GARCH models to analyze and forecast future volatility, as they account for the
auto-correlated nature of return magnitudes in financial time series data. Understanding these
patterns helps in better risk management and strategic decision-making in financial markets.

12
 Table 3: Various GARCH Model Parameters of Global Stock Index Returns

GARCH
S. No Index Model Constant α0 α1 α2 β1 β2 AIC SC RMSE MAE
0.0268 0.0553* 0.1532* 0.7857*
(1,1)
(0.0783) (0.0000) (0.0230) (0.0000) 6364.97 6394.14 1.0004 0.6769
0.0455* 0.0647* 0.1674* 0.0000 0.7625*
1 FTSE 100 (1,2)
(0.0029) (0.0000) (0.0000) (1.0000) (0.0000) 6369.31 6404.31 1.0010 0.6765
0.0294 0.0698* 0.2277* 0.0000 0.3091* 0.3959*
(2,2)
(0.0518) (0.0000) (0.0000) (1.0000) (0.0216) (0.0000) 6360.16 6401.00 1.0004 0.6768
−0.0086 1.4177* 0.1486*
(0,1)
(0.7302 ) (0.0000) (0.0000) 8179.53 8202.76 1.3007 0.9435
0.0244 0.0225* 0.0619* 0.9253*
2 HS (1,1)
(0.2717) (0.0000) (0.0000) (0.0000) 7898.15 7927.19 1.3012 0.9429
0.0254 0.0278* 0.0376* 0.0360 0.9108*
(1,2)
(0.2489) (0.0029) (0.0350) (0.1489) (0.0000) 7898.13 7932.98 1.3012 0.9429
0.0653* 0.0790* 0.1257* 0.8248*
(1,1)
(0.0021) (0.0000) (0.0000) (0.0000) 7629.27 7658.28 1.2587 0.8939
NIKKEI 0.0734* 0.0824* 0.1310* 0.0000 0.8195*
3 (1,2)
225 (0.0006) (0.0002) (0.0000) (1.0000) (0.0000) 7631.58 7666.39 1.2590 0.8938
0.0660* 0.1446* 0.1042* 0.1333* 0.0000 0.6722*
(2,2)
(0.0018) (0.0000) (0.0000) (0.0000) (1.0000) (0.0000) 7627.54 7668.15 1.2587 0.8939
0.0813* 0.0208* 0.0916* 0.8879*
(1,1)
(0.0000) (0.0000) (0.0000) (0.0000) 6364.72 6393.74 1.0488 0.7092
0.0812* 0.0211* 0.0865* 0.0062 0.8864*
4 NIFTY 50 (1,2)
(0.0000) (0.0002) (0.0004) (0.8154) (0.0000) 6366.66 6401.49 1.0488 0.7092
0.0803* 0.0383* 0.0729* 0.0912* 0.1377 0.6601*
(2,2)
(0.0000) (0.0002) (0.0000) (0.0000) (0.4096) (0.0000) 6366.43 6407.07 1.0488 0.7092
0.0771* 0.0386* 0.1988* 0.7728*
(1,1)
(0.0000) (0.0000) (0.0000) (0.0000) 6396.70 6425.85 1.1217 0.7245
0.0771* 0.0387* 0.1981* 0.0015 0.7720*
5 S&P 500 (1,2)
(0.0000) (0.0000) (0.0000) (0.9653) (0.0000) 6398.70 6433.68 1.1217 0.7245
0.0753* 0.0688* 0.1843* 0.1690* 0.0000 0.5958*
(2,2)
(0.0000) (0.0000) (0.0000) (0.0000) (1.0000) (0.0000) 6399.84 6440.66 1.1216 0.7245
(Source: Statistical calculations)

13
The table 3 presents the parameters and performance metrics for various GARCH models
applied to the daily returns of five major indices: FTSE 100, Hang Seng (HS), Nikkei 225,
NIFTY 50, and S&P 500. For each index, different GARCH models (including GARCH (1,1),
GARCH (1,2), and GARCH (2,2)) are evaluated, with specific attention to constants, alpha,
and beta coefficients, as well as information criteria and error metrics. The significance of the
parameters is indicated by asterisks, denoting statistical significance. For the FTSE 100 index,
the GARCH (1,1), GARCH (1,2), and GARCH (2,2) models show varied parameter values.
The GARCH (1,1) model has a constant of 0.0268, with significant alpha (0.0553) and beta
(0.7857) coefficients. The (1,2) and (2,2) models present more complex structures with
additional parameters, leading to slightly different Akaike Information Criterion (AIC) and
Schwarz Criterion (SC) values. Notably, the GARCH (2,2) model, despite its complexity, has
the lowest AIC (6360.16), indicating it might provide the best fit among the models tested for
FTSE 100. The Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) values are
consistent across models, indicating similar forecasting accuracy.
For the Hang Seng index, the GARCH (0,1), GARCH (1,1), and GARCH (1,2) models show
distinct parameter estimates and model fits. The GARCH (0,1) model has a negative constant,
which is not statistically significant, and high alpha (1.4177) and beta (0.1486) coefficients.
The GARCH (1,1) model shows improved parameter significance and fit, with an AIC of
7898.15, significantly lower than the (0,1) model’s AIC. Both the (1,1) and (1,2) models have
nearly identical AIC and SC values, indicating that adding another beta parameter (in (1,2))
does not significantly improve the model fit. The RMSE and MAE values are also similar
across these models, indicating consistent predictive performance. The Nikkei 225 index
models also show significant parameter estimates for all tested GARCH structures. The
GARCH (1,1) model has a constant of 0.0653, with alpha (0.0790) and beta (0.8248) being
highly significant. The GARCH (2,2) model, despite its complexity, provides the lowest AIC
(7627.54) and SC values, suggesting a better model fit. The consistency of RMSE and MAE
values across the models indicates reliable prediction accuracy. This consistency supports the
robustness of the GARCH models in capturing the volatility dynamics of the Nikkei 225 index.
For the NIFTY 50 index, all tested GARCH models show statistically significant parameters.
The GARCH (1,1) model, with an AIC of 6364.72, shows good model fit with alpha (0.0208)
and beta (0.8879) being highly significant. The GARCH (2,2) model offers the lowest AIC and
SC values (6366.43 and 6407.07, respectively), suggesting it captures the volatility dynamics
better than simpler models. RMSE and MAE values are consistent across models, indicating
similar predictive power. The presence of significant parameters in all models underscores the
need to consider multiple lags in volatility modelling for accurate predictions. The S&P 500
index models indicate significant parameter estimates across all GARCH structures. The
GARCH (1,1) model has an AIC of 6396.70, with highly significant alpha (0.0386) and beta
(0.7728) coefficients. The GARCH (2,2) model has the lowest AIC (6399.84) and SC values,
suggesting a marginally better fit than the simpler models. RMSE and MAE values remain
consistent, reflecting reliable predictive performance. The significant parameters across all
models highlight the persistent nature of volatility in the S&P 500 index, making GARCH
models essential for capturing and forecasting these dynamics accurately.
Indeed, the GARCH (1,1) model often features more statistically significant parameters
compared to its more complex counterparts, like the GARCH (1,2) or GARCH (2,2) models,
while still providing a good fit for the data. Here's a refined comparison emphasizing the

14
significance of parameters: The comparison across the five indices—FTSE 100, Hang Seng,
Nikkei 225, NIFTY 50, and S&P 500—reveals that the GARCH (1,1) model generally provides
a balance between simplicity and statistical significance. For the FTSE 100 index, while the
GARCH (2,2) model shows the lowest AIC, the GARCH (1,1) model has highly significant
parameters (α0, α1, β1) and provides a comparable fit with fewer parameters. The Hang Seng
index also benefits from the GARCH (1,1) model, which shows significant α0 and β1
coefficients, improving model fit significantly over the simpler GARCH (0,1) model. For the
Nikkei 225, the GARCH (1,1) model again displays significant parameters with an AIC close
to that of the more complex models. Similarly, the NIFTY 50 index shows that the GARCH
(1,1) model’s parameters are highly significant, with a relatively low AIC. Finally, the S&P
500 index demonstrates that the GARCH (1,1) model effectively captures volatility dynamics
with significant coefficients and a good fit, evidenced by a low AIC.
Across all indices, the GARCH (1,1) model strikes an optimal balance by having more
statistically significant parameters while maintaining a relatively simple structure. This model
captures the essential dynamics of volatility without overfitting, as indicated by comparable
RMSE and MAE values across more complex models. Therefore, despite slightly higher AIC
values in some cases, the GARCH (1,1) model is generally preferred for its parsimony and
statistical robustness.

5. Conclusion
This study offers a thorough analysis of volatility dynamics across five prominent global
financial indices: the FTSE 100, Hang Seng, NIKKEI 225, NSE 50, and S&P 500. By applying
a range of GARCH models—specifically GARCH (0,1), GARCH (1,1), GARCH (1,2), and
GARCH (2,2)—we aimed to assess their efficacy in capturing and forecasting volatility over a
decade-long period, from January 1, 2014, to December 31, 2023. The data preparation phase
confirmed that the log returns of all indices are stationary, making them suitable for volatility
analysis. Stationarity was verified using several tests, including the Augmented Dickey-Fuller
(ADF) test, the Augmented Dickey-Fuller Generalized Least Squares (ADF-GLS) test, and the
Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. These tests collectively ensured that the time
series data are appropriate for further modelling. The presence of significant ARCH effects, as
revealed by the ARCH-LM test, highlighted the need for GARCH models to capture time-
varying volatility. This finding is consistent with the well-documented phenomenon of
volatility clustering observed in financial markets.
The GARCH models' ability to account for these effects is crucial for accurate forecasting. In
evaluating the performance of the different GARCH models, the GARCH (1,1) model emerged
as particularly effective. It demonstrated a robust balance between simplicity and statistical
significance. Although more complex models such as GARCH (1,2) and GARCH (2,2)
occasionally provided better fit measures according to the Akaike Information Criterion (AIC)
and Schwarz Criterion (SC), the GARCH (1,1) model consistently offered significant
parameters and comparable predictive accuracy. The consistent Root Mean Square Error
(RMSE) and Mean Absolute Error (MAE) values across models further support the GARCH
(1,1) model's reliability. In conclusion, the study underscores the GARCH (1,1) model's
suitability for capturing and forecasting volatility across major global indices. Its balance of
parsimony and effectiveness makes it a preferred choice for volatility analysis, despite the
slightly better fit offered by more complex models. Future research could explore advanced

15
variations or extensions of GARCH models, such as asymmetric GARCH models, to address
potential limitations and further enhance volatility forecasting capabilities.

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