Dr. Mak, Billy S C
The Report of Modeling Stock Market Volatility in Four Asian Tigers
During Global Subprime Crisis Using ARMA-GARCH Approach
BY
MUNG Tin Hong
12012084
Finance Concentration
LI Yeung
12002127
Finance Concentration
An Honours Degree Project Submitted to the
School of Business in Partial Fulfillment
of the Graduation Requirement for the Degree of
Bachelor of Business Administration (Honours)
Hong Kong Baptist University
Hong Kong
April 2015
1 Contents
Abstract ......................................................................................................................................................... 3
1.
Introduction .......................................................................................................................................... 4
2.
Literature Review ................................................................................................................................. 6
3.
Methodology ........................................................................................................................................ 9
3.1
Data and descriptive statistics...................................................................................................... 9
3.2
Augmented Dickey-Fuller test................................................................................................... 14
3.3
Criterion for model selection ..................................................................................................... 15
3.4
Autoregressive Moving Average, ARMA model ...................................................................... 15
3.5
Autoregressive Conditional Heteroskedasticity Lagrange Multiplier, ARCH LM test ............. 17
3.6
ARMA-GARCH model ............................................................................................................. 17
4.
Empirical Results ............................................................................................................................... 19
5.
Conclusion.......................................................................................................................................... 31
2 Abstract
The Global Subprime Crisis in U.S. had caused global stock markets experiencing huge
fluctuation, including the Four Asian Tigers. The paper aims to model the volatility of stock
returns in Hong Kong, Singapore, South Korea and Taiwan by using ARMA-GARCH approach
and using the daily financial time series covering a period from July 17, 2002 to April 24, 2014.
The selection of the best fitting model is based on the Schwartz Information Criterion and it is
observed that the return and volatility of the stock market in Four Asian Tigers will change over
time. Our empirical results show that ARMA(0,1)-GARCH(2,1) models are best fitted for
during-crisis period. Changed market condition due to the emergence of Global Subprime Crisis
is proved by comparing different ARMA-GARCH models among pre-crisis, during-crisis and
post-crisis period in the Four Asian Tigers.
Keywords: Four Asian Tigers, Global Subprime Crisis, stock market, volatility, ARMA,
ARMA-GARCH
3 1. Introduction
The Global Subprime Crisis between year 2007 and 2008 caused by the United States is a recent
global financial crisis which affects and shocks different financial markets around the world.
This financial crisis is historically severe and is described by many economists and finance
experts as the worst financial crisis since the Great Depression in 1930s. The paper attempts to
study the volatility of stock market returns in Four Asian Tigers including Hong Kong,
Singapore, South Korea and Taiwan. Asian stock markets are gaining the importance among the
world economy and therefore more researches on the markets helps investors and people
understand Asian markets more through Four Asian Tigers in the paper.
Among different financial markets, the Subprime Mortgage Crisis has affected the global stock
markets around the world including Asia undoubtedly, which leads to the attention of different
investors to the stock market volatility. As the stock return is the leading indicator of the
economy and volatility reflects the unpredictable change in returns. Hence, the modeling of
volatility is helpful in description of previous stock market performance and forecasting the
future volatility.
Four Asian Tigers (Hong Kong, Singapore, South Korea and Taiwan) are famous for its rapid
economic and industrial growth since 1960s, Hong Kong and Singapore are developed into
leading international financial centers while South Korea and Taiwan are main manufacturers of
information technology and electronic products in the world. The large market capitalization and
economy size of the four regions could represent the high growth economies in Asia and use
them as a proxy for over viewing the Asia economies and stock markets. A study about the effect
of stock market volatility on Four Asian Tigers during the Global Subprime Crisis can be
beneficial for understanding more about stock markets and resilient economies of Four Asian
Tigers.
The paper examines the relationship between stock market volatility and the Global Subprime
Crisis in Four Asian Tigers through the model. Moreover, since the Global Subprime Crisis is a
4 recent and severe financial crisis which affect global financial markets, we would like to
understand more about the whole crisis including the causes, formation, duration and the related
effects towards economies, thus a comprehensive understanding can be developed towards the
economics of Hong Kong, Singapore, South Korea and Taiwan.
Many researchers have investigated into the volatility modeling on different stock markets
individually, especially mainly focusing on U.S. and Europe stock markets. Few researches have
been done on Asian stock markets, especially on the stock markets of Four Asian Tigers. One of
the purposes in our paper is to enrich the knowledge about stock volatility in Asia. Furthermore,
many studies about volatility are conducted to use a simple GARCH approach by the assumption
of constant mean of returns, our study apply ARMA-GARCH approach to construct the mean
equation with ARMA model and variance equation with GARCH model, which is different from
previous papers and our study attempts to fill the gap.
The remaining parts of our paper is organized as follows: Section 2 illustrates relevant literature on
stock market volatility with models our paper employed. Section 3 describes the data and
methodology which includes ADF test, SIC for model selection, ARMA, ARCH LM test and
ARMA-GARCH model. Section 4 covers the empirical results. Section 6 concludes our paper.
5 2. LiteratureReview
Volatility changes in stock market are the big consideration to regulators and risk-averse
investors, volatility in financial market is an important indicator which help set up their strategies
in asset pricing and allocation as well as risk management in both mature and emerging stock
markets, emphasized by Poon and Granger (2003). Modeling the volatility patterns though
standard deviation and variance are quite common for measuring risk and volatility. However,
the constant volatility assumption in earlier financial models is unrealistic since the variation in
volatility is widely recognized to change over time. Time series models are frequently employed
to investigate the dynamic nature of financial time series.
Autoregressive Integrated Moving Average (ARIMA) model proposed by Box-Jenkins (1976)
have been a standard model for different kinds of time series fluctuation. Leseps and Morell
(1977) found the exchange rate follows a long term trend accompanied with short term volatility.
Chen (1995) estimated S&P 500 index volatility by the ARMA model with a pre-differencing
transformation, which was outperformed than mean reversion model and GARCH model in his
empirical study. Dharmaratne (1995) used ARMA model to model and forecast the long stay
visitors in Barbados. Abdel-Aal and Al-Garni (1997) applied Box-Jenkins time series analysis to
monthly electric energy consumption in Saudi Arabia, they found the ARMA model requires less
data, has fewer coefficients but with higher accuracy. ARMA model applied to a nine-year air
quality record for forecasting maximum ozone concentration in Athens by Slini, et. al (2001),
which show the index of agreement is satisfactory but the forecasting alarms is the weakness.
Mahadevan (2002) discovered that ARMA has more accuracy in direction than the moving
average model when a static forecast in ten-year Indian government securities yield, however,
ARMA model was not outperformed the lagged moving averages in a dynamic forecasting.
Ruey S. Tasy (2002) found the financial time series possess some special characteristics which
make the modeling complicated. The presence of volatility clustering, leptokurtosis and leverage
effects is obvious in times series. Mandelbrot (1963) observed volatility clustering and
leptokurtosis are commonly found in financial time series. Volatility clustering occurs when big
shocks tend to be followed by big shocks and small shocks by small shocks while leptokurtosis is
6 the fat tailed distribution of the returns. On the other hand, Black (1976) pointed out that the
negative correlation between change in stock returns and volatility, which called “leverage
effect”. The effect implies the volatility after negative shocks is higher than after positive shocks
with the same magnitude.
Engle (1982) developed to model time varying conditional variance with the Autoregressive
Conditional Heteroskedasticity (ARCH) to capture the characteristic of volatility clustering,
leptokurtosis in financial time series by using past disturbances. In order to capture the dynamic
of the conditional variance, a high order of ARCH is required according to early empirical
evidence. In the history of finance literature about the ARCH model, different researchers used
ARCH model to measure the volatility of different financial markets, for example, C. Mustafa
(1992) used ARCH model for measuring volatility of option prices, G.W. Schwert (1990) for
volatility of future market and V. Akgiray (1989) for index returns. One of the main discoveries
about ARCH by F.C. Drost and T.E. Nijman (1993) is that ARCH effects are significant with
high frequency data such as daily and weekly data. Moreover, in the study of T.J. Brailsford and
R.W. Faff (1996), they conclude that volatility forecasting is not as easy as they think but ARCH
models and simple regression are the most applicable for volatility modeling.
ARCH model was further extended to Generalized ARCH model by Bollerslev (1986), based on
an infinite specification of ARCH model and the number of estimated parameters were
minimized from infinity to two. ARCH and GARCH model take the two characteristics into
account, but they fail to model the leverage effect because of the symmetric of their distributions.
Many extensions of the basic GARCH model have been developed to handle the problem,
especially those are suitable to estimate the conditional volatility of financial time series. In the
history of finance literature about the GARCH model. E. Barucci and R. Reno (2001) discovered
that GARCH models are more suitable for volatility forecasting if Fourier analysis is used. In a
study of National Stock Exchange of India, M.J. Rijo (2004) find that GARCH (1,1) model is the
best for measuring volatility. Moreover, in a study of the short term interest rate forecasting, S.
Radha and M. Thenmozhi (2006) find that GARCH provides better result. L.H. Erdington and W.
Guan (2005) compare the different volatility forecasting model and conclude that the GARCH
(1,1) model is better than exponentially weighted moving average model while Awartani and
Corradi (2005) find that GARCH(1,1) is the best compared to other GARCH model when
7 asymmetries are not allowed. Under the assumption that the innovations follow a normal
distribution, F.J. Magnus and O.E. Fosu (2006) give a positive comment to the GARCH (1,1)
model and reject the random walk hypothesis. Furthermore, in a study of Malaysian and
Singaporean stock indices, A. Shamiri and M.S.N. Abu Hassan (2007) estimate them using the
three GARCH(1,1) models. They discover that AR(1)-EGARCH best fit the estimation for
Singaporean stock market while AR(1)-GJR model is more applicable for estimating Malaysian
stock market. Apart from that, four non-period GARCH models are compared by M.N.
Haniffand W.C. Pok (2010) and they find that consistently superior results are produced from
EGARCH model. The traditional GARCH model is symmetric and it fails to capture the
asymmetry effect inherent in most market returns. The asymmetric effect means to the behavior
of financial time series tends to have greater fluctuation with “bad news” than “good news”. The
specific design of Exponential GARCH (EARCH) model and Threshold GARCH (TGARCH)
model is to solve the problem of capturing asymmetric shock to the conditional variance.
8 3. Methodology
This chapter is divided into six parts. The first part introduces the data and descriptive statistics.
The second part explains Augmented Dickey-Fuller test. The third part is model selection. The
fourth part describes the ARMA model as the mean equation. The fifth part shows the model of
ARCH LM test. The final part illustrates ARMA-GARCH model as the variance equation for
modeling the volatility.
3.1
Dataanddescriptive statistics
The study makes use of daily returns of Hang Seng Index, HSI (Hong Kong), Taiwan
Capitalization Weighted Stock Index, TAIEX (Taiwan), Straits Times Index, STI (Singapore)
and Korea Composite Stock Price Index, KOSPI (Korea) during three different periods. Stock
Return is an important indicator of any economy to study and estimate the volatility of stock
market. The coefficients in findings will illustrate the impact of the Global Subprime Crisis on
Four Asian Tigers. All the data of market indices are extracted from Bloomberg.
Periods are chosen after the recovery of 1997 Asian Financial Crisis in order to minimize its
effect and better capture the volatility caused by 2008 Global Subprime Crisis. To distinguish the
impact of Global Subprime Crisis, pre-crisis, crisis and post-crisis periods are chosen
individually. Although the exact dates of outbreak and end of crisis do not well specified by
public and there are a lot of disparities in different papers, most of the papers defined the crisis
period almost happened from Summer in 2007 to Spring in 2009. Hence, we select July 17 2007,
the date Bear Stearns announced its two hedge funds were nearly worthless, as the start date of
crisis. For the end date, we choose April 24 2009, the date International Monetary Fund claimed
it was the turning point for global economy. News and articles are well documented in the
appendix, as the supporting evidence.
The whole period is from July 17, 2002 to April 24, 2014, covering the financial crisis started at
the end of year 2007 and ended at the beginning of year 2009. Nearly 3000 observations are
9 included in each index. As mentioned, the crisis period was from July 17 2007 to April 24 2009.
The pre-crisis period started from July 17, 2002 to July 16 2007 while the post-crisis period
began from July 17, 2002 to July 16 2007. The pre- and post- crisis period cover a 5-year period
before and after the crisis period respectively, which makes the data observation symmetric and
is good for comparison of the change in volatility among four indices.
The daily return of indices could be obtained by using the adjusted close price as following:
log
/
(1)
is the daily returns of the indices during time t,
Where
at time t,
is
the daily close prices of the indices
is the daily close prices of the indices at time t-1.
Figure 1. Daily close prices of four indices from July 17, 2002 to April 24, 2014
Heng Seng Index - whole period
Korea Composite Stock Price Index - whole period
32,000
2,400
28,000
2,000
24,000
1,600
20,000
1,200
16,000
800
12,000
8,000
400
02
03
04
05
06
07
08
09
10
11
12
13 14
Straits Times Index - whole period
02
03
04
05
06
07
08
09
10
11
12
13 14
Taiwan Capitalization Weighted Stock Index - whole period
4,000
10,000
3,500
9,000
8,000
3,000
7,000
2,500
6,000
2,000
5,000
1,500
4,000
1,000
3,000
02
03
04
05
06
07
08
09
10
11
12
13 14
02
03
04
05
06
07
08
09
10
11
12
13 14
10 Stock returns of HSI (Hong Kong), TAIEX (Taiwan), STI (Singapore) and KOSPI (Korea) from
July 17, 2002 to April 24, 2014 are included on a daily basis in our study. The overall trends of
four indices are showed in Figure and they almost moved together with the same direction over
the period, regardless of their magnitudes. Generally, the four indices rise continuously until year
2007. After the outbreak of Global Subprime Crisis in summer 2007, all four stock markets had a
significant decline and the economic downturn prolonged to year 2009. In spring 2009, 2009 G20 London summit discussed the issue of Global Financial Crisis and reached an global
agreement to regulate hedge funds and credit-rating agencies. Later, IMF announced the
international economy was on the road of recovery which make the global stock market’s
rebound and the four indices showed their trends less volatile afterward.
Figure 2. Daily return series of four indices from July 17, 2002 to April 24, 2014
Daily return - Hang Seng Index
Daily return - Korea Composite Stock Price Index
.15
.12
.10
.08
.05
.04
.00
.00
-.05
-.04
-.10
-.08
-.15
-.12
02 03
04
05
06
07
08
09
10
11
12
13 14
Daily return - Straits Times Index
02 03
04
05
06
07
08
09
10
11
12
13 14
Daily return - Taiwan Capitalization Weighted Stock Index
.08
.08
.04
.04
.00
.00
-.04
-.04
-.08
-.12
-.08
02 03
04
05
06
07
08
09
10
11
12
13 14
02 03
04
05
06
07
08
09
10
11
12
13 14
Then we further transform the data by the steps of logarithm and differencing transformation,
stationary mean and variance of the time series data can be observed visually on Figure. The
11 return series of indices appear stationary and fluctuate around their means at zero during whole
period. We can see the return patterns of HSI, KOSPI and STI are alike while TWII has more
volatility over the period. Stock returns of these indices demonstrate volatility clustering which
means the previous level of volatility tends to have a positive correlation with the next level of
volatility. At the same time, the volatility after negative shocks tends to be larger than after
positive shocks even though they are in same magnitude, which called leverage effect.
Table 1. Descriptive statistics for return series of four indices from July 17, 2002 to April 24, 2014
Descriptive statistics
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Observations
HSI
0.000268
0.000584
0.134068
-0.135820
0.015436
0.040541
12.36263
10625.78 (0.000)
2909
TAIEX
0.000182
0.000638
0.065246
-0.069123
0.013250
-0.313372
6.043730
1176.959 (0.000)
2925
STI
0.000247
0.000655
0.075311
-0.086960
0.011847
-0.184238
8.269936
3439.660 (0.000)
2958
KOSPI
0.000324
0.000888
0.112844
-0.111720
0.014608
-0.477568
8.495551
3790.645 (0.000)
2924
In Table 1, the average returns of HSI, TAIEX, STI and KOSPI from period July 17 2002 to
April 24 2014 are 0.000268, 0.000182, 0.000247 and 0.000324 respectively, all mean of indices
were very close to zero and they had an positive mean returns during whole period. The standard
deviation of HSI, TAIEX, STI and KOSPI are 0.015436, 0.013250, 0.011847 and 0.014608
respectively, which measures the dispersion of return series. The standard deviation of HSI and
KOSPI are relatively higher than others, which means they are more volatile than TAIEX and
STI. The skewness of HSI, TAIEX, STI and KOSPI are 0.040541, -0.313372, -0.184238 and 0.477568 respectively, only HSI had positive skewness while other three indices obtained
negative skewness. The value of HSI is close to zero, nearly symmetrical distribution while other
three indices showed little left skewed distributions. The kurtosis of HSI, TAIEX, STI and
KOSPI are 12.36263, 6.043730, 8.269936 and 8.495551 respectively. All indices showed
leptokurtic distribution which is not a normal distribution, HSI had relatively larger kurtosis
which means a higher probability of extreme values. Jarque-Bera tests whether the series is a
normal distribution, assuming the null hypothesis of a normal distribution is true. All the findings
of indices showed zero value of probability and rejected the null hypothesis at 5% significant
level and concluded all stock returns of four indices are not normally distributed.
12 Table 2. Descriptive statistics for return series of four indices during pre-crisis period
Descriptive statistics
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Observations
HSI
0.000645
0.000637
0.040510
-0.041836
0.010021
-0.118944
4.234636
81.48306 (0.000)
1237
TAIEX
0.000460
0.000314
0.054845
-0.069123
0.012450
-0.259637
6.220900
550.8190(0.000)
1242
STI
0.000651
0.001118
0.040327
-0.039108
0.009728
-0.264029
4.505343
132.9712(0.000)
1254
KOSPI
0.000748
0.001616
0.048772
-0.059653
0.013901
-0.349808
4.427529
130.5773(0.000)
1240
During the pre-crisis period, the average returns of HSI, TAIEX, STI and KOSPI are very close
to zero with positive values, shown in Table 2. The standard deviation of HSI, TAIEX, STI and
KOSPI are0.010021, 0.012450, 0.009728 and 0.013901 respectively, HSI and STI are less
volatile than TAIEX and KOSPI before the crisis. The skewness of four indices are negative,
showing them have little skewed distributions to the left while KOSPI is the most left skewed
distribution among the group. All indices have leptokurtic distribution, compared to a normal
distribution with kurtosis value of 3. The stock returns of four Asian tigers are not normal
distributed since the p-values of Jarque-Bera is zero.
Table 3. Descriptive statistics for return series of four indices during crisis period
Descriptive statistics
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Observations
HSI
-0.000949
0.000565
0.134068
-0.135820
0.028818
0.190450
6.114575
178.4526 (0.000)
435
TAIEX
-0.001092
0.000778
0.060990
-0.067351
0.019771
-0.217169
3.647159
11.13687(0.000)
440
STI
-0.001467
-0.001334
0.075311
-0.086960
0.020349
0.016810
4.882566
65.88135(0.000)
446
KOSPI
-0.000812
0.000651
0.112844
-0.111720
0.022675
-0.381515
6.892027
286.4187(0.000)
437
From Table 3, Four Asian Tigers have negative but close to zero average stock returns during the
Global Subprime Crisis and experienced a high level of volatility from the value of standard
deviation. Most of their standard deviations approach and even exceed the value of 0.02 while
the values in other period are close to 0.01.TAIEX and KOSPI has left skewed distribution and
other two indices are skewed right. The kurtosis of TAIEX is 3.647159 which is relatively close
to normal distribution, but four indices still classified as leptokurtic distribution due to the excess
13 values. Due to the low p-values of Jarque-Bera, the normal distribution of four stock returns are
rejected.
Table 4. Descriptive statistics for return series of four indices during post-crisis period
Descriptive statistics
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Observations
HSI
0.000339
0.000429
0.055187
-0.058270
0.012944
-0.089539
4.843697
176.5683 (0.000)
1235
TAIEX
0.000362
0.000860
0.065246
-0.057422
0.010909
-0.205779
6.242300
552.7879(0.000)
1242
STI
0.000471
0.000621
0.054934
-0.037693
0.009373
0.072837
5.857499
428.4280(0.000)
1256
KOSPI
0.000321
0.000475
0.049000
-0.064202
0.011305
-0.436458
6.112964
542.6605(0.000)
1246
The average returns of HSI, TAIEX, STI and KOSPI after the crisis are positive again and still
very close to zero in Table 4. The stock markets are said to be less fluctuated after Global
Subprime Crisis by the observation of the decreasing values in standard deviations. All returns of
indices have little left skewed distributions except STI. Four Asian Tigers are distributed
leptokurtically, the excess skewness accounts for the distribution. The zero p-values of JarqueBera suggest the four indices are not normally distributed.
3.2
AugmentedDickey‐Fullertest
Before conducting the estimation of ARMA and ARMA-GARCH model, we have to examine
whether the return series have a unit root. If the series possess a unit root, it means they were
non-stationary. Stationarity of the series could be tested and detected with Augmented DickeyFuller test proposed by Dickey and Fuller(1979) or called unit root test. The null hypothesis of
the test is that the return series have a unit root and are not stationary. If the p-values of the series
during periods are lower 0.01, we could reject the null hypothesis and conclude the series are
stationary at all levels of significance. Since ARMA and ARMA-GARCH approach are only
applicable to stationary time series which does not vary over time. The ADF test is essential to
test the stationary nature of the time series before apply these models.
14 3.3
Criterionformodelselection
The optimal lag for AR and MA terms determines the most appropriate model to fit and capture
the memory in the residuals. The best model has to be found through trial and error listing all
reasonable combinations within 5 lag lengths. Akaike (1973) and Schwartz (1978) proposed
Akaike information criterion (AIC) and Schwartz Information Criterion (SIC)respectively, which
are the goodness of fit statistic to measure for an ARMA(p,q) model. The general form of SIC
can be written as:
AIC
SIC
2 ∙ ln
2 ∙ ln
2k
(2)
k ∙ ln
(3)
In equation 2 and 3, ln is the maximized log likelihood, k is the number of parameters and n is
the sample size in the model. Zahid Asghar and Irum Abid (2007) suggested SIC outperforms
than other criteria for sample size equal or larger than 240. Furthermore, SIC penalizes the model
more consistently in choosing model if the number of observations in samples are large enough.
Therefore, we employ SIC as the criterion for model selection in our paper and the best fit model
must have the minimum value of SIC.
3.4
AutoregressiveMovingAverage,ARMAmodel
Autoregressive Moving Average model are employed to model the return series in our paper. It
is useful to illustrate the dependency of data in financial time series based on its past values. The
ARMA model is used as mean equation for filtering the returns. The residuals are the white noise
which is the building black for complex time series models such as GARCH model in our paper.
They are essential to fit with and construct GARCH model. Finding the best ARMA model is our
next step to fit the daily returns of the series. ARMA(P,Q) model consists of two parts:
autoregressive model and moving average model, where P and Q is the lag length of the
autoregressive and moving average part respectively.
15 AR(P) model is defined as:
(4)
are model parameters, is a constant, and
is the residual term in equation 3. The AR term
has an infinite and long memory due to the correlation between the current value
previous values
and all
.
MA(Q) model is defined as:
(5)
In equation 4,
are the parameters in the model,
is a constant, and
is the residual term. The
MA term has a finite and short memory because there is no correlation between current
value and past values of .
ARMA(P,Q) model combines the properties of AR and MA model:
(6)
In equation 5,
is the conditional mean,
respectively, is a constant and
is
and
are the model parameters for AR and MA
the residual term.
16 3.5
AutoregressiveConditionalHeteroskedasticityLagrangeMultiplier,
ARCHLMtest
Lagrange Multiplier test proposed by Engle (1982) is to check the existence of any ARCH effect
in the residuals from the mean equation. The best fitted ARMA model has already been found by
lowest SIC selection. The model can be written as:
̂
Where is the residual term,
̂
(7)
is a constant and q is the lag length. The squares of in
ARMA model have a regression on α and q. The null hypothesis of the test is absence of the
Autoregressive Conditional Heteroscedastic (ARCH) effect. The test statistic of ARCH test is the
F statistic for the squared residuals regression. Values of F-statistic and R-squared observations
are found by the tests. We could reject the null hypothesis and conclude a strong existence of
ARCH effect if the p-values are less than 0.05. The existence of ARCH effect suggests the
appropriateness of using GARCH model to describe the conditional volatility.
3.6
ARMA‐GARCHmodel
Engle (1982) first introduced ARCH and then further generalized by Bollerslev (1986) to
GARCH model, which can illustrate the characteristics of heteroskedacisticity and volatility
clustering in time series data. It suggests two parameters to its current variance of returns, which
are for the past values of variance and squared residuals. Therefore, selecting a good GARCH
model can capture the time-varying characteristics of volatility, meanwhile it can identify the
impact of current and old news on volatility. The used lags are specified as GARCH(p,q), where
q (GARCH term) and p (ARCH term) denote as the lags of the conditional variance and squared
residuals respectively. Since ARMA(P,Q) model for conditional mean is perfectly compatible to
17 GARCH(p,q) model for conditional variance, which leads to the combination of two models and
forms our final volatility model: ARMA(P,Q)-GARCH(p,q) model.
ARMA(P,Q)-GARCH(p,q)model is represented as:
(6)
,
~
0,
(8)
(9)
The equation 5 and 8represented the conditional mean equation and conditional variance
equation of returns series respectively. In equation 7,
is the residual term and
is the
independent and identically distributed random variables of residuals with zero mean and the
variance. In equation 8,
is the conditional variance and ω is a constant.
is the ARCH
term representing the effect of shocks from the previous period on current volatility while
is
the GARCH term representing the effect of forecasted conditional variance from last period on
current volatility.
value of
and
and are the coefficients of ARCH and GARCH term respectively. The
indicates the extent ofeffect of shocks from the previous period on volatility
and the persistence of volatility. If the sum of ARCH and GARCH coefficients, i.e.
1,
the time series is said to have very persistent volatility shocks.
18 4. EmpiricalResults
First, we use unit root test to see whether our time series data is stationary or not. The unit root
test we use is Augmented Dickey-Fuller test and the daily return of our four time series data is
tested. The lag length is automatically selected by Eviews according to the SIC value. As showed
in the session of data and descriptive statistics, there is no obvious increasing trend or decreasing
trend of the daily return of our four time series data in different periods. Therefore, it is
reasonable to conduct the Augmented Dickey-Fuller test with intercept only. Table 5, 6, 7, 8
below show the unit root test results of our four time series data.
Table 5. Augmented Dickey-Fuller Test of HSI in different periods with intercept only
Index
Hong Kong
Hang Seng
Index (HSI)
Period
Exact Date
Overall
17-7-2002 to 24-4-2014
Pre-Crisis
17-7-2002 to 16-7-2007
During-Crisis
17-7-2007 to 24-4-2009
Post-Crisis
27-4-2009 to 24-4-2014
Note: ***represent 1% level of significance.
t-Statistic
-54.85
-34.12
-22.07
-34.36
P-value
0.0001***
0.0000***
0.0000***
0.0000***
Table 6. Augmented Dickey-Fuller Test of KOSPI in different periods with intercept only
Index
Period
Exact Date
Korea
Overall
16-7-2002 to 24-4-20141
Composite
Pre-Crisis
16-7-2002 to 16-7-2007
Stock Price
During-Crisis
18-7-2007 to 24-4-2009
Index (KOSPI)
Post-Crisis
27-4-2009 to 24-4-2014
Note: ***represent 1% level of significance.
t-Statistic
-52.84
-33.99
-20.64
-34.77
P-value
0.0001***
0.0000***
0.0000***
0.0000***
Table 7. Augmented Dickey-Fuller Test of STI in different periods with intercept only
Index
Singapore
Straits Times
Index (STI)
Period
Exact Date
Overall
17-7-2002 to 24-4-2014
Pre-Crisis
17-7-2002 to 16-7-2007
During-Crisis
17-7-2007 to 24-4-2009
Post-Crisis
27-4-2009 to 24-4-2014
Note: ***represent 1% level of significance.
t-Statistic
-53.25
-34.85
-21.59
-32.45
P-value
0.0001***
0.0000***
0.0000***
0.0000***
1
There is no stock transaction on 17-7-2007 in South Korea, therefore, 16-7-2002 will be the starting date for precrisis period and 18-7-2007 will be the starting date for during-crisis period.
19 Table 8. Augmented Dickey-Fuller Test of TAIEX in different periods with intercept only
Index
Period
Exact Date
Taiwan
Overall
17-7-2002 to 24-4-2014
Capitalization
Pre-Crisis
17-7-2002 to 16-7-2007
Weighted Stock During-Crisis
17-7-2007 to 24-4-2009
Index (TAIEX) Post-Crisis
27-4-2009 to 24-4-2014
Note: ***represent 1% level of significance.
t-Statistic
-51.35
-34.22
-20.31
-31.93
P-value
0.0001***
0.0000***
0.0000***
0.0000***
The results of the Augmented Dickey-Fuller tests on the daily return of the above four indices
are nearly the same. The t-Statistics of HSI, KOSPI, STI and TAIEX are around -50 for our
overall study period, around -30 for pre-crisis period and post-crisis period and around -20 for
during-crisis period.
From Table 5, 6, 7, 8 we can conclude that the time series of the four indices from different
periods are stationary. The p-values of the four indices are 0.0001 for the overall study period
and 0.000 for the three sub-periods. Therefore, the null hypothesis that all the series are nonstationary is rejected due to the low p-values and we can conclude that the above four indices in
different periods are stationary.
After testing the stationarity of the HSI, KOSPI, STI and TAIEX in different periods, we are
going to choose the best ARMA (Autoregressive Moving Average) model that fit the above time
series according to the lowest SIC value. In this empirical study, ARMA is used as a tool to
model the daily return of the four time series data in different periods. In other words, the mean
equation is modeled with an ARMA process. Moreover, as mentioned in the session of
methodology before, SIC value is chosen as a tool for our model selection due to the bigger
penalty than AIC value and a better model consistence. We have tested different ARMA models
from ARMA(0,1) to ARMA(5,5) since it is reasonable that the above four indices can adjust
within a week that has five transaction days. Below is the table that shows the best ARMA model
which has the lowest SIC values for the pre-crisis period, during-crisis period and post-crisis
period.
20 Table 9. The best fitting ARMA model of HSI, KOSPI, STI and TAIEX in different periods
according to the lowest SIC value.
Index
Hong Kong Hang Seng
Index (HSI)
Korea Composite Stock
Price Index (KOSPI)
Singapore Straits Times
Index (STI)
Taiwan Capitalization
Weighted Stock Index
(TAIEX)
Period
Pre-Crisis
During-Crisis
Post-Crisis
Pre-Crisis
During-Crisis
Post-Crisis
Pre-Crisis
During-Crisis
Post-Crisis
Pre-Crisis
During-Crisis
Post-Crisis
Best Model
ARMA(1,0)
ARMA(0,1)
ARMA(2,1)
ARMA(5,4)
ARMA(0,1)
ARMA(2,1)
ARMA(1,1)
ARMA(0,1)
ARMA(2,2)
ARMA(4,2)
ARMA(0,1)
ARMA(3,1)
From table 9, most probably there are only one memory or two memories for the return
adjustment for each period and this shows that normally the daily return of the stock market will
adjust within a week. But it can be seen that the ARMA(5,4) is fitted for Korea Composite Stock
Price Index and ARMA(4,2) is fitted for Taiwan Capitalization Weighted Stock Index. That
means lagged return four and five days ago can still affect the stock market. The reason is that
Taiwan stock market and South Korea stock market is more lively and dynamic than Hong Kong
and Singapore stock market in pre-crisis period. Therefore, Taiwan stock market and South
Korea stock market tend to have a longer memory in pre-crisis period.
One interesting result we have discovered is ARMA(0,1) model best suit all the four indices in
during-crisis period which is from 17-7-2007 to 24-4-2009. The reason can be attributed to the
severe volatility during the Global Subprime Crisis. ARMA(0,1) is basically equal to MA(1)
model. A MA(1) model without AR component means the daily return of the index is affected
only by the residual term, and not affected by its lagged return. The ARMA(0,1) best fit the
time series of Global Subprime Crisis period since the market is too volatile so that only residual
or innovation can affect the stock market. It is clearly observed that the Global Subprime Crisis
affect the daily return adjustment in Four Asian Tigers consistently.
21 Another interesting result we have found is that the ARMA model tend to be larger after Global
Subprime Crisis. The ARMA(0,1) model is only fitted for the four indices in during-crisis period
and it changed to ARMA(2,1) for Hong Kong Hang Seng Index, ARMA(2,1) for Korea
Composite Stock Price Index, ARMA(2,2) for Straits Times Index and ARMA(3,1) for Taiwan
Capitalization Weighted Stock Index after during-crisis period. A higher-order ARMA model
means the memory tend to be longer and more parameters are needed to model the dynamic
change of the daily return. The reason is that the Global Subprime Crisis has passed after 24-42009 and the stock market return to a normal condition. Therefore, the lagged return few days
ago can influence the daily return of the above four indices in a normal stock market. This can
explain why the memory become longer after the Global Subprime Crisis because the unstable
market condition has changed and it is reasonable that lagged return few days ago can affect the
daily return of the stock market in a natural condition.
After finding the ARMA model which best suit the daily return of HSI, KPOSI, STI and TAIEX
in different crisis periods, the next step is to conduct heteroskedasticity test to check whether
there is ARCH effect in the selected ARMA models since this research aims to apply ARMA and
ARCH/GARCH model for studying return and volatility. Heteroskedasticity test is conducted for
the selected ARMA model and we test the heteroskedasiticity at 4, 8 and 12 lags for each crisis
period since the result will be more thorough and accurate by adding more lags for
heteroskedasticity test. Table 10 shows the results.
22 Table 10. Heteroskedasticity test for HSI with selected ARMA models in different lags
Index
Periods
Hong Kong
Hang Seng
Index (HSI)
Pre-Crisis
Selected
ARMA Models
ARMA(1,0)
During-Crisis
ARMA(0,1)
Post-Crisis
ARMA(2,1)
Residual
Lags
4
8
12
4
8
12
4
8
12
F-statistic
Prob. F
5.09
5.19
4.99
26.22
14.53
10.82
20.17
14.06
10.40
0.0005***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
Note: ***represent 1% level of significance.
Table 11. Heteroskedasticity test for KOSPI with selected ARMA models in different lags
Index
Periods
Korea
Composite
Stock Price
Index (KOSPI)
Pre-Crisis
Selected
ARMA Models
ARMA(5,4)
During-Crisis
ARMA(0,1)
Post-Crisis
ARMA(2,1)
Residual
Lags
4
8
12
4
8
12
4
8
12
F-statistic
Prob. F
31.44
21.02
16.75
18.44
16.34
12.44
34.38
22.33
18.01
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
Note: ***represent 1% level of significance.
Table 12. Heteroskedasticity test for STI with selected ARMA models in different lags
Index
Periods
Singapore
Straits Times
Index (STI)
Pre-Crisis
Selected
ARMA Models
ARMA(1,1)
During-Crisis
ARMA(0,1)
Post-Crisis
ARMA(2,2)
Residual
Lags
4
8
12
4
8
12
4
8
12
F-statistic
Prob. F
10.30
8.71
8.16
22.53
12.01
10.86
38.49
27.98
18.52
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
Note: ***represent 1% level of significance.
23 Table 13. Heteroskedasticity test for TAIEX with selected ARMA models in different lags
Index
Periods
Taiwan
Capitalization
Weighted Stock
Index (TAIEX)
Pre-Crisis
Selected
ARMA Models
ARMA(4,2)
During-Crisis
ARMA(0,1)
Post-Crisis
ARMA(3,1)
Residual
Lags
4
8
12
4
8
12
4
8
12
F-statistic
Prob. F
17.55
13.56
9.90
2.63
4.50
4.04
13.52
10.29
9.33
0.0000***
0.0000***
0.0000***
0.0340**
0.0000***
0.0000***
0.0000***
0.0000***
0.0000***
Note: ***represent 1% level of significance.
Table 10, 11, 12, 13 show the heteroskedasticity test for the residuals in different lags from the
selected ARMA models. It is clearly observed that the nearly all the probability of F-statistics are
equal to 0.0000 which is smaller than 5% level of significance except the pre-crisis period of HSI
at fourth lags and during-crisis period of TAIEX at fourth lags. But both are still significant at 5
% level. Therefore, it can be concluded that there are ARCH effect in all of the ARMA models
selected and this means that GARCH model can be applied to describe the conditional volatility
process.
(9)
Above is the formula of the GARCH(p,q) model, the former part is the ARCH term and the later
part is the GARCH term. After finding the best-fitting ARAM model and conducting
heteroskedasticity test, the GARCH(p,q) process is applied to model the residuals. According to
(Bollerslevet al., 1992), it is enough to model volatilities in financial time series by using
GARCH(1,1), GARCH(1,2) and GARCH(2,1). Therefore, we are going to test the
ARCH/GARCH model ranging from (1,0) to (2,2) to model the residuals. Table 14 below shows
the best GARCH model based on the ARMA models we have selected before. We also use the
lowest SIC value as our model selection because maintaining a single model selection criteria
will have a better consistence for our study.
24 Table 14. Best fitting ARMA-GARCH model for HSI, KOSPI, STI and TAIEX in different periods
Index
Hong Kong Hang Seng
Index (HSI)
Korea Composite Stock
Price Index (KOSPI)
Singapore Straits Times
Index (STI)
Taiwan Capitalization
Weighted Stock Index
(TAIEX)
Period
Pre-Crisis
During-Crisis
Post-Crisis
Pre-Crisis
During-Crisis
Post-Crisis
Pre-Crisis
During-Crisis
Post-Crisis
Pre-Crisis
During-Crisis
Post-Crisis
Best Model
ARMA1,0 GARCH1,1
ARMA0,1 GARCH2,1
ARMA2,1 GARCH2,1
ARMA5,4 GARCH1,1
ARMA0,1 GARCH2,1
ARMA2,1 GARCH1,1
ARMA1,1 GARCH1,1
ARMA0,1 GARCH2,1
ARMA2,2 GARCH1,1
ARMA4,2 GARCH1,1
ARMA0,1 GARCH2,1
ARMA3,1 GARCH1,1
Following the lowest SIC value, we have found the best ARMA-GARCH model for modeling
the return and volatility of Hong Kong Hang Seng Index, Korea Composite Stock Price Index,
Straits Times Index and Taiwan Capitalization Weighted Stock Index for different crisis periods
as listed above. Only GARCH (1,1) or GARCH (2,1) model is the best for modeling the
residuals. This is consistent for what Bollerslevet's findings we have mentioned before.
As showed in Table 14, it can be discovered that ARMA-GARCH(2,1) model is fitted for the
during-crisis period of HSI, KOSPI, STI and TAIEX. Except for the post-crisis period of HSI, all
other periods of HSI, KOSPI, STI and TAIEX are fitted with ARMA-GARCH(1,1) only. The
reason can be attributed to the serve volatility during Global Subprime Crisis and a longer
memory is needed to model and adjust the residuals from the ARMA model. The coefficient α
(lagged squared residuals) in the GARCH model captures the influence of volatility or shocks
from the previous period and the coefficient β (lagged conditional variance) in the GARCH
model captures the forecasted conditional variance from previous period which measures the
persistence of volatility shocks. So basically it is observed that the ARMA-GARCH model
change with time and we can see that the Global Subprime Crisis do affect the change of the
volatility modeling. It can also be discovered that the lagged shock and volatility two days before
still have an impact for the stock market. As showed before, the standard deviation of duringcrisis period is larger than other periods and this is consistent with the changing ARMA-GARCH
models we have found for the during-crisis period.
25 After that, it will be more solid if we conduct the heteroskedasticity test again to check whether
there is still ARCH effect left in the ARMA-GARCH model we have selected. We also test the
heteroskedasticity at 4, 8 and 12 lags this time in order to obtain a more thorough results.
Table 15. Heteroskedasticity test for HSI with selected ARMA-GARCH models
Index
Periods
Selected
ARMA Models
ARMA1,0
GARCH1,1
Residual F-statistic
Lags
Hong Kong
Pre-Crisis
4
1.53
Hang Seng
8
0.92
Index (HSI)
12
0.91
During-Crisis
ARMA0,1
4
1.38
GARCH2,1
8
1.52
12
1.25
Post-Crisis
ARMA2,1
4
0.36
GARCH2,1
8
0.54
12
0.55
Note: None of the p-values are significant at 1%, 5% and 10% level.
Prob. F
0.1911
0.4961
0.5337
0.2390
0.1484
0.2440
0.8368
0.8273
0.8840
Table 16. Heteroskedasticity test for KOSPI with selected ARMA-GARCH models
Index
Periods
Selected
ARMA Models
ARMA5,4
GARCH1,1
Residual F-statistic
Lags
Korea
Pre-Crisis
4
0.37
Composite
8
0.30
Stock Price
12
0.66
Index (KOSPI)
During-Crisis
ARMA0,1
4
0.37
GARCH2,1
8
1.03
12
1.09
Post-Crisis
ARMA2,1
4
1.01
GARCH1,1
8
1.25
12
1.07
Note: None of the p-values are significant at 1%, 5% and 10% level.
Prob. F
0.8275
0.9643
0.7939
0.8284
0.4134
0.3704
0.3999
0.2684
0.3813
26 Table 17. Heteroskedasticity test for STI with selected ARMA-GARCH models
Index
Periods
Selected
ARMA Models
ARMA1,1
GARCH1,1
Residual F-statistic
Lags
Singapore
Pre-Crisis
4
1.02
Straits Times
8
0.63
Index (STI)
12
0.85
During-Crisis
ARMA0,1
4
1.56
GARCH2,1
8
1.07
12
1.09
Post-Crisis
ARMA2,2
4
0.51
GARCH1,1
8
1.25
12
1.17
Note: None of the p-values are significant at 1%, 5% and 10% level.
Prob. F
0.3979
0.7525
0.5964
0.1835
0.3800
0.3649
0.4762
0.2636
0.2988
Table 18. Heteroskedasticity test for TAIEX with selected ARMA-GARCH models
Index
Periods
Taiwan
Pre-Crisis
Capitalization
Weighted Stock
Index (TAIEX) During-Crisis
Post-Crisis
Selected
ARMA Models
ARMA4,2
GARCH1,1
ARMA0,1
GARCH2,1
ARMA3,1
GARCH1,1
Residual
Lags
4
8
12
4
8
12
4
8
12
F-statistic
Prob. F
1.21
1.42
1.17
0.36
1.53
1.43
0.27
0.38
0.72
0.1956
0.1835
0.2963
0.8399
0.1456
0.1473
0.8975
0.9329
0.7357
Note: None of the p-values are significant at 1%, 5% and 10% level.
It can be clearly observed from Table 15, 16, 17, 18 that all the p-value of the F-statistic is larger
than 1%, 5% and 10% level of significance. The null hypothesis of no ARCH effect for this
heteroskedasticity test cannot be rejected. That means there is no more ARCH effect left for our
selected ARMA-GARCH model and the selected ARMA-GARCH models are suitable for
modeling return and volatility.
After confirming the best ARMA-GRACH model, the estimated coefficients of the selected
models are be shown.
27 Table 19. Coefficients estimation of fitted ARMA-GARCH model for HSI
Hong Kong Hang Seng Index (HSI)
Selected Model
Pre-Crisis
During-Crisis
Post-Crisis
Parameter
ARMA1,0
ARMA0,1
ARMA2,1
GARCH1,1
GARCH2,1
GARCH2,1
C(1)
0.000708 (0.0075)***
-0.000371 (0.1336)
7.42E-05 (0.4608)
AR(1)
0.037601 (0.2524)**
0.734641 (0.0000)***
AR(2)
-0.038344 (0.0949)*
MA(1)
-0.026533 (0.0208)**
-0.704067 (0.0000)***
C(2)
1.28E-06 (0.0080)***
3.67E-05 (0.0381)**
2.40E-06 (0.0050)***
ARCH(1)
0.031478 (0.0000)***
0.106567 (0.0363)**
-0.013297 (0.3480)
ARCH(2)
0.092272 (0.2341)
0.070755 (0.0001)***
GARCH(1)
0.954344 (0.0000)***
0.762528 (0.0000)***
0.926028 (0.0000)***
Note: The values in parenthesis are the p-value of the coefficients. ***, ** and * represent 1%, 5% and
10% level of significance respectively.
Table 20. Coefficients estimation of fitted ARMA-GARCH model for KOSPI
Korea Composite Stock Price Index (KOSPI)
Selected Model
Pre-Crisis
During-Crisis
Post-Crisis
Parameter
ARMA5,4
ARMA0,1
ARMA2,1
GARCH1,1
GARCH2,1
GARCH1,1
C(1)
0.002142 (0.0823)*
-0.000493 (0.0553)*
0.000282 (0.1458)
AR(1)
-1.094222 (0.0045)***
0.324086 (0.0033)***
AR(2)
-0.107744 (0.0086)***
-0.011146 (0.0991)*
AR(3)
0.688402 (0.0064)***
AR(4)
0.122877 (0.0044)**
AR(5)
-0.075972(0.0244)**
MA(1)
1.143493 (0.0033)***
-0.020616 (0.0344)**
-0.336054 (0.0036)***
MA(2)
0.153944 (0.0094)***
MA(3)
-0.729666 (0.0039)***
MA(4)
-0.240308 (0.0834)*
C(2)
3.78E-06 (0.0026)***
1.18E-05 (0.0609)*
1.49E-06 (0.0054)***
ARCH(1)
0.083093 (0.0000)***
-0.026794 (0.0442)**
0.064783 (0.0000)***
ARCH(2)
0.176191 (0.0008)***
GARCH(1)
0.896456 (0.0000)***
0.832258 (0.0000)***
0.922212 (0.0000)***
Note: The values in parenthesis are the p-value of the coefficients. ***, ** and * represent 1%, 5% and
10% level of significance respectively.
28 Table 21. Coefficients estimation of fitted ARMA-GARCH model for STI
Singapore Straits Times Index (STI)
Selected Model
Pre-Crisis
During-Crisis
Post-Crisis
Parameter
ARMA1,1
ARMA0,1
ARMA2,2
GARCH1,1
GARCH2,1
GARCH1,1
C(1)
0.001454 (0.0015)***
-0.001039 (0.2018)
9.23E-05 (0.1047)
AR(1)
-0.979045 (0.0000)***
1.437863 (0.0000)***
AR(2)
-0.674865 (0.0000)***
MA(1)
0.999913 (0.0000)***
-0.001256 (0.0095)***
-1.440782 (0.0000)***
MA(2)
0.702866 (0.0000)***
C(2)
1.34E-06 (0.0022)***
1.17E-05 (0.0765)
1.11E-06 (0.0099)***
ARCH(1)
0.080305 (0.0000)***
0.006782 (0.0085)***
0.079737 (0.0000)***
ARCH(2)
0.132327 (0.0050)***
GARCH(1)
0.905478 (0.0000)***
0.837030 (0.0000)***
0.904459 (0.0000)***
Note: The values in parenthesis are the p-value of the coefficients. ***, ** and * represent 1%, 5% and
10% level of significance respectively.
Table 22. Coefficients estimation of fitted ARMA-GARCH model for TAIEX
Parameter
C(1)
AR(1)
AR(2)
AR(3)
AR(4)
MA(1)
MA(2)
C(2)
ARCH(1)
ARCH(2)
GARCH(1)
Taiwan Capitalization Weighted Stock Index (TAIEX)
Selected Model
Pre-Crisis
During-Crisis
Post-Crisis
ARMA4,2
ARMA0,1
ARMA3,1
GARCH1,1
GARCH2,1
GARCH1,1
4.82E-05 (0.1838)
-0.000768 (0.4038)
0.000229 (0.1206)
0.302967 (0.0152)**
0.587403 (0.0000)***
0.665114 (0.0000)***
-0.096056 (0.0069)***
0.005142 (0.0093)***
0.016586 (0.6032)
-0.030389 (0.3060)
-0.250194 (0.0438)**
0.022009 (0.0080)***
-0.505465 (0.0000)***
-0.704365 (0.0000)***
1.65E-06 (0.0034)***
2.15E-05 (0.0033)***
1.09E-06 (0.0064)***
0.053026 (0.0000)***
-0.037590 (0.0303)**
0.050636 (0.0000)***
0.147737 (0.0047)***
0.934544 (0.0000)***
0.837840 (0.0000)***
0.938351 (0.0000)***
Note: ***, ** and * represent 1%, 5% and 10% level of significance respectively. Note: The values in
parenthesis are the p-value of the coefficients. ***, ** and * represent 1%, 5% and 10% level of
significance respectively.
29 After showing the coefficients for the best fitting model chosen before, we discover that the
model for return and volatility will change over time. Although there is some coefficients which
are not significant, nearly all of the coefficients are significant at 1%, 5% and 10% level. Since
this is an empirical study, what we are trying to do is to set up a reasonable selection criterion
and find the most suitable model for this study.
From the above four tables, the sum of the GARCH parameters is approximately equal to one
and is close to unity for all the models, for example, α1 + β1 ≈1 and α1 +α2 + β1 ≈ 1. That means
the volatility is persistent in our daily return data for our four stock indices. In other words, the
shocks to the conditional variance is highly persistent.
30 5. Conclusion
This study aims to model the stock market volatility in the Four Asian Tigers during Global
Subprime Crisis caused by the devaluation of housing-related securities in the United States. We
have found related news to support the starting date and ending date of Global Subprime Crisis
which started on 17th July 2007 and ended on 24th April 2009. In order to know more about the
change in volatility during Global Subprime Crisis, we divide our time series data into three
periods which are pre-crisis period, during-crisis period and post-crisis period. Pre-crisis period
is five years before the Global Subprime Crisis period and the post-crisis period is five years
after the Global Subprime Crisis period.
We employ the Augmented Dickey-Fuller test on the daily returns of Hong Kong Hang Seng
Index (HSI), Korea Composite Stock Price Index (KOSPI), Singapore Straits Times Index (STI)
and Taiwan Capitalization Weighted Stock Index (TAIEX) and all of the four indices are
stationary.
We also employ ARMA model first to find the best ARMA according to the lowest SIC value
and we conduct the heteroskedasticity test to check whether the ARCH effect exists in the bestfitting ARMA model or not. We find that the application of GRACH model based on the ARMA
model we selected is better since there are ARCH effects in all of our indices in different periods.
After that, the best fitting ARMA-GARCH model is found for our indices in different period.
The heteroskedasticity test is employed again to check whether the ARCH effect still exists in
the best-fitting ARMA-GARCH model or not. Finally, there is no more ARCH effect in the
ARMA-GARCH model we have selected and that means the ARMA-GARCH model we have
found is suitable.
We discover that ARMA(0,1) and ARMA-GARCH(2,1) model best suit for all the four indices
in during-crisis period due to the change in the market condition caused by the Global Subprime
Crisis. It can be concluded that the volatility model changed over time. The ARMA process also
changed after Global Subprime Crisis since the stock markets in the Four Asian Tigers have
returned to a normal condition.
31 It can be concluded that although the Global Subprime Crisis is caused by the United States, the
stock markets in other countries are globally affected by this Global Subprime Crisis even in
Asian countries.
This empirical study is important since it is about the stock market volatility which affects lots of
investors in Asia since Hong Kong, South Korea, Singapore and Taiwan also play important
roles in Asian stock markets. Stock investment is very popular and it is a common investment
tool for investors because its mechanism is easier to be compared with options, futures and other
derivatives. Therefore, studying stock market volatility can help people to know more about the
market direction.
32