INTRODUCTION
Financial market volatility is an important indicator of the dynamic fluctuations
in stock prices (Raja and Selvam, 2011). An understanding
of volatility in stock markets is important for determining the cost of capital
and for assessing investment and leverage decisions as volatility is synonymous
with risk. Substantial changes in volatility of financial markets are capable
of having significant negative effects on riskaverse investors (Premaratne
and Balasubramanyan, 2003).
The global financial crisis which happened at the end of 2007 caused and is
still causing, a huge impact on financial markets and institutions around the
world. Questions regarding bank solvency, declines in credit availability and
damaged investor confidence had an impact on global stock markets, where securities
suffered huge losses during the late 2008 and early 2009 and Malaysia was no
exception (International Monetary Fund, 2009). The Kuala
Lumpur Composite Index (KLCI) which is the main index and market indicator in
Malaysia, dropped around 558.93 points in 2008 and this comes to around a 40%
drop in its value. Ever since the Asian Financial Crisis of 1997, this was the
biggest decline (Chin, 2009). So, how huge an impact
did the global financial crisis have on the Malaysian stock market volatility?
The main objective of this study is thus to investigate the volatility of the
KLCI with regards to the recent financial crisis of 2007/2008, after the Asian
financial crisis 1997.
Studies on investment and financial market volatility have widely made use
of ARCH models and the existence of ARCH effects is well documented by Hsieh
(1984), Akgiray (1989), Engle
(1990) and Engle and Mustafa (1992) and they used
these models for various types of markets. It has been shown that ARCH effects
are highly significant with daily and weekly data due to the amount or quality
of information reaching the markets or the time between information arrival
and processing by the participants in the market (Diebold,
1988; Drost and Nijman, 1993) but the effects actually
weaken when frequency of the data decreases (Diebold and
Nerlove, 1989).
With so many different types of models, the forecasting ability of the models
is important and several studies have documented this. Brailsford
and Faff (1996) showed that in volatility forecasting, ARCH and simple regression
models provide superior forecasting ability but are sensitive to the error statistic
used to assess the accuracy of the forecasts. However, Barucci
and Reno (2002) found that when Fourier analysis is used calculate the diffusion
process volatility, GARCH models have better forecasting properties. Erdington
and Guan (2004) found that the GARCH(1,1) model ‘generally yields better
forecasts than the historical standard deviation and exponentially weighted
moving average models..’ although it is still lacking in forecasting accuracy.
Awartani and Corradi (2005) also found that the GARCH
(1,1) model is superior when not allowing for asymmetries but when taking asymmetries
into consideration, this model is inferior to the asymmetric GARCH models.
Similar results were found for Asian markets. The ARCH/GARCH type models have
been shown to provide the best fit in volatility forecasting for studies done
on the Indian stock markets. For instance, Rijo (2004)
found that the GARCH (1,1) model gives the best fit according to all model selection
criteria for the National Stock Exchange (NSE) of India, while Radha
and Thenmozhi (2006) showed that GARCH based models are more appropriate
to forecast short term interest rates than the other models. Padhi
(2006) analysis revealed that the GARCH (1, 1) model is persistent for all
the five aggregate market indices of India and for the individual company.
For the Malaysian case, using asymmetric GARCH, Shamiri
and Abu Hassan (2007) showed that the AR (1)GJR model provides the best
outof sample forecast for the Malaysian stock market while AR (1)EGARCH provides
a better estimation for the Singaporean stock market which implies that Malaysian
stock market has asymmetric effects. However, Haniff and
Pok (2010) comparison of the four nonperiod GARCH models revealed that
the EGARCH produced consistently superior results compared to the other GARCH
models. Assis et al. (2010) compared various
univariate timeseries methods in the forecasting of coffee bean prices and
found that the ARIMA/GARCH models outperformed the others.
Guidi (2010) found that while some indices were better
forecasted using asymmetric GARCH models, the simple symmetric GARCH models
with the normal distribution actually performed better in volatility forecasting
of 5 Asian stock markets and were good enough to be used for forecasting purposes.
Fahimifard et al. (2009) found that while nonlinear
models outperformed liner ones, when comparing the linear models, the GARCH
model outperformed the ARIMA model. Most recently, in their analysis, Mukherjee
et al. (2011) found that the EGARCH model was a better model compared
to the TARCH model for the SENSEX because there was an indication that there
was considerable amount of asymmetry in the series. Yaziz
et al. (2011) found that while the ARIMA (1,2,1) model was able to
produce good forecasts of crude oil prices based on historical patterns, the
GARCH (1,1) was actually much better as it was able to capture the volatility
effect.
MATERIALS AND METHODS
Univariate models of conditional volatility: Financial time series usually
exhibit a set of characteristics. Stock market returns display “Avolatility
clustering ” where large changes in these returns tend to be followed by
large changes and small changes by small changes (Mandelbrot,
1963), leading to contiguous periods of volatility and stability.
The Autoregressive Conditional Heteroskedasticity models (ARCH) (Engle,
1982) have been used extensively to model volatility. The general form of
ARCH (q) process is as follows:
The value of α and β should be greater than zero since standard deviation
and variance cannot be negative and value of betas should be less than one in
order for the process to be stationary. A deficiency of ARCH (q) models is that
the conditional standard deviation process has high frequency oscillation with
high volatility coming in short bursts. Bollerslev (1986)
generalized the ARCH model by including lagged values of the conditional variance.
The GARCH models permit a wider range of behaviour, in particular, a more persistent
volatility. The general form of the GARCH (p, q) model is:
where, α_{i} ε^{2}_{t1} is an ARCH component
and β_{i} σ^{2}_{t1} is a GARCH component.
However, the GARCH (p, q) is symmetric and does not capture the asymmetry that
characterizes most financial time series and it is known as”leverage effect”.
It refers to the characteristic of time series on asset prices that “bad news”
tends to increase volatility more than good news. One of the primary restrictions
of the GARCH models is that they enforce a symmetric response of volatility
to positive and negative shocks (Brooks, 2002). In these
models, therefore, a big positive shock will have exactly the same effect on
the volatility of a series as a negative shock of the same magnitude (Asteriou
and Hall, 2007).
In order to capture the asymmetric shock to the conditional variance, Nelson
(1991) proposed the Exponential GARCH (EGARCH) model. In the EGARCH model
the natural logarithm of the conditional variance is allowed to vary over time
as a function of the lagged error terms rather than lagged squared errors. The
EGARCH (p, q) model can be written as follows:
The exponential nature of the EGARCH ensures that conditional variance is always
positive even if the parameter values are negative, thus there is no need for
parameter restrictions to impose nonnegativity. The impact is asymmetric if
γ_{i} is not equal to zero whereas the presence of the leverage
effect can be tested by hypothesis that γ_{i} is less than zero.
The Threshold GARCH (TGARCH) modifies the original GARCH specification using dummy variables. The main target of this model is to capture asymmetries in terms of negative and positive shocks by adding into a variance equation a multiplicative dummy variable to check whether there is statistically a significant difference when shocks are negative. The specification of conditional variance for TGARCH is as follows:
For ε^{2}_{t1}, l (.) = 1, or l (.) = 0 for ε^{2}_{tj}
>0. If γ_{i} coefficients have positive values, this indicates
a presence of leverage effect. The GJRGARCH model is a similar model to TGARCH.
The difference lies in the fact that we are dealing with conditional standard
deviation in the TGARCH model or conditional variance in GJRGARCH model.
Data and analysis: The Asian financial crisis of 1997 caused a huge
collapse of the stock markets in the South East Asian region. However, from
January 2000 onwards, stock prices had resumed their increasing trend until
the eve of outbreak of the global financial crisis. Malaysia had a good recovery
by the middle of 1999. There is no specific date of full economic recovery from
the Asian financial crisis, but by the middle of 2000, it was almost recovered.
Guidi (2010) showed a downward pattern in Asian stock
prices at the end of 2007 with signals of recovery from late 2008, indicating
the presence of the global financial crisis.
Thus, in order to capture the impact of the crisis on volatility and asymmetry of returns, two different periods are used to see the effect and both periods are selected after the recovery of Asian financial which was in the middle of year 2000, to make sure there is no effect of the 1997 Asian financial crisis in our analysis. This study uses secondary data collected from DataStream, covering a period of six and half years after the financial crisis of 1997 in East Asia and before the crisis of 2008.
The sample of data used in this study is the daily closing prices of Kuala Lumpur Composite Index (KLCI) from 1 June 2000 till the end of 2007 and also cover a period of 10 years from 1 June 2000 until the middle of March 2010 which includes the crisis. In the first analysis the crisis is excluded but it is included in the second analysis, so if there is any impact of the crisis, a significant change in the models can be detected. Daily closing price of the Kuala Lumpur Composite Index (KLCI) to analyze the volatility is transformed to daily returns as below:
Where:
R_{t} represents the daily returns of the KLCI 
P_{t} represents the daily prices of the KLCI 
The statistics for the KLCI returns series are shown in Table
1. Generally, there is a large difference between the maximum and minimum
return of the index. The standard deviation is also high with regards to the
number of observations, indicating a high level of fluctuation of the KLCI daily
returns. The mean is close to zero and positive as is expected for a time series
of returns.
There is also evidence of negative skewness, indicating an asymmetric tail
which exceeds more towards negative values rather than positive ones and an
indication that KLCI has nonsymmetric returns. KLCI returns are leptokurtic,
given its large kurtosis statistics in Table 1.
Table 1: 
Descriptive statistics of returns of KLCI 

Table 2: 
Unit root tests 

The test critical value at 1, 5 and 10% is 3.43 , 2.86 and
2.56, respectively. For both series, the number exceeds the critical values
at all levels, corresponding to zero pvalue 

Fig. 1: 
KLCI Returns from June 2000 to March 2010. Note: The xaxis
represents the year, i.e., year 2000 to 2010 while the yaxis represents
the returns of the KLCI 
The kurtosis exceeds the normal value of three indicating that the return distribution
is fattailed. Jarque and Bera (1980) test for normality
confirms the results based on skewness and kurtosis and both series are nonnormal
according to JarqueBera which rejects normality at the 1% level.
Table 2 below shows the result of unit root tests. The Augmented
DickeyFuller (ADF) test (Dickey and Fuller, 1979) is
applied to both series. Based on the test results, we reject the null hypothesis
that returns have unit roots. It shows that both series are stationary as the
mean is constant across time.
In Fig. 1, we present the KLCI returns from June 2000 to March 2010. Virtual inspection shows that volatility changes over time and it tends to cluster with periods of low volatility and periods of high volatility. The volatility is relatively consistent from 2001 to the year 2007 and seems to increase in the middle of 2007 till 2009. Next, we model this volatility in order to capture the effect of the crisis on the index returns.
Table 3: 
ARMA models 

γ_{tk } represents the lag values at lag k =
1,2,3,4 with their corresponding coefficients,
is the intercept. Values in parenthesis are the pvalues of the parameters 
Table 4: 
ARCH LM test 

The test is conducted at different numbers of lags. Values
in parenthesis indicate the pvalues. The zero pvalue at all lags strongly
indicates the presence of ARCH effect in both series. Obs*Rsquared is the
number of observations times the Rsquared value 
We first estimate simple ARMA models as our conditional mean and select the
best ARMA model that fits the return of the series. Different ARMA models are
examined at different lags based on the pvalues, residual of Qstatistic pvalues,
AIC values and adjusted Rsquared. Among the models, some are rejected due to
the stationarity condition since the sum of the absolute coefficients is greater
than unity and then some rejected due to the magnitude of their pvalues. For
both series, ARMA (4, 0) or AR (4) has been chosen as the best process for modelling
the conditional mean since the relevant AIC were at minimum and the model meets
all the criteria including white noise residuals (Table 3).
Next, we perform the ARCH LM test to see if there is any ARCH effect in the residuals. Table 4 presents the results of this test.
The LM test for both periods shows a significant presence of ARCH effect with low pvalue of 0.0000. So, we reject the null hypothesis of no ARCH effect and detect a strong presence of ARCH effect as expected for most financial time series.
As the return exhibits an ARCH effect, we use GARCHtype models. In most empirical
implementations, the values (p< = 2, q< = 2) are sufficient to model the
volatility which provides a sufficient tradeoff between flexibility and parsimony
(Knight and Satchell, 1998). Franses
and Van Dijk (1996) and Gokcan (2000) have also
shown that models with a small lag like GARCH (1, 1) are sufficient to cope
with the changing variance. According to Brooks (2002),
the lag order (1, 1) model is sufficient to capture all of the volatility clustering
that is present from the data.
We examine symmetric GARCH and nonlinear asymmetric EGARCH, GJRGARCH models at different (p< = 2, q< = 2) lags. We also find that the GARCH (1, 1), EGARCH(1,1), GJRGARCH (1,1) are the most successful models according to AIC as they have the smallest value while satisfying restrictions such as nonnegativity for symmetric GARCH. The models are estimated for both series using QuasiMaximum likelihood assuming the Gaussian normal distribution. The results are presented in Table 5. The coefficients of all models for both periods are significant at all levels implying the strong validity of the models.
Table 5: 
GARCH models 

Values in parenthesis are the pvalues. All coefficients are
significant at 1, 5 and 10% levels 
Table 6: 
GARCH models residual diagnostics 

All correlogram Qstatistics and correlogram squared residuals
pvalues are greater than 1, 5 and 10% level at lag 500, suggesting that
residuals are white noise. All pvalues for ARCH LM test are greater than
1 and 5% level at lag one, suggesting no presence of ARCH effect 
In order to test whether present models have adequately captured the persistence
in volatility and there is no ARCH effect left in the residual of models, the
ARCHLm test is conducted again. The pvalues of LM test, standardized residuals
and standardized residuals squared are shown in Table 6.
The results of the diagnostic tests show that the GARCH models are correctly specified. The Qstatistics for the standardized residuals and standardized squared residuals are insignificant with high pvalues for all models, suggesting the GARCH models are successful at modelling the serial correlation structure in conditional means and conditional variances.
We have found the AR (4)/GARCH (1, 1), AR (4)/EGARCH (1, 1) and AR (4)/GJRGARCH (1, 1) to be good models to describe the process for the first series which excludes the crisis and also for the second period which includes the crisis. According to the statistical tests and diagnostics, all models are significant and capture the ARCH effect and volatility clustering successfully. Table 7 presents the comparison of the models for both periods. The difference for the coefficients of each model are obtained and also expressed in percentage terms.
The difference is obtained by subtracting the first period values from the second period values and the percentage is obtained by dividing the difference with the first period values.
Forecasting performance: The models are also evaluated based on their
forecasting ability of the future returns. The out of sample period of 6 months
for each period is used to evaluate this. The sample for forecasting is from
1 January 2008 to 1 July 2008, to include the crisis and from 16 March 2010
to 16 September 2010.
Table 7: 
Model differences 

The difference is obtained by subtracting the first period
values from the second period values and the percentage is obtained by dividing
the difference with the first period values 
Table 8: 
Forecasting performance 

The best model for forecasting is determined by the highest
values for RMSE, MAE, MAPE and lowest value for TIC 
The common measures of forecast evaluation, i.e., Root Mean Squared Error
(RMSE), Mean Absolute Error (MAE), Mean absolute Percentage Error (MAPE) and
Theil Inequality Coefficient (TIC) are used. Results are presented in Table
8.
DISCUSSION
In Table 7, with regards to the symmetric GARCH model, the value of the beta which indicates the correlation between σ^{2}_{t} and σ^{2}_{t1}, shows that the conditional variance has decreased by 2.16%, implying that the persistency in conditional variance has decreased by 2.16%. On the other hand, the rate of change of conditional variance has increased by 24.5%. This is consistent with EGARCH and GJRGARCH results of an increase of 21.8 and 34.4% in the rate of change of conditional variance respectively. However, since these two models are extensions of the simple GARCH model to be used to capture asymmetries effects mostly and some complications are added to them for this purpose, we will use them for this purpose solely. Thus, with respect to the GARCH model which has explicit and simple coefficients of lagged squared error and conditional variance, the volatility has increased by 24.5% while the persistency in volatility has just decreased by 2.16% during the crisis period.
The asymmetric (leverage effect) is examined by the nonlinear asymmetric models EGARCH and GJRGARCH. The coefficient γ in the case of GJRGARCH is significantly different from zero implying that both series are not symmetric. This was also shown in the descriptive analysis earlier, where the series exhibited negative skewness. The positive value of the parameter indicates the presence of leverage effect. In the case of EGARCH, the presence of the leverage effect can be detected by the hypothesis γ<0 whereas the impact is asymmetric if γ is not equal to zero. For both series, the parameter is significantly different from zero, indicating the presence of asymmetry and is also less than zero suggesting leverage effects.
For both series under consideration the asymmetry exists. Both nonlinear asymmetric
EGARCH and GJRGARCH produce the same results in terms of asymmetry and also
leverage effects. These results are consistent with the findings of Shamiri
and Abu Hassan (2007) and Haniff and Pok (2010)
for the Malaysian market. The comparison between the two series asymmetric parameters,
show an increase in leverage effect in the market by 11.5 and 18.5% by EGARCH
and GJRGARCH respectively. Since the leverage effect refers to the characteristics
of time series on asset prices that “bad news” tend to increase volatility more
than “good news”, it is expected that the crisis will increase the impact of
the different kinds of news as the percentages in Table 5
suggest.
In Table 8, according to these measures and their criteria,
the symmetric AR (4)GARCH (1, 1) has outperformed both the EGARCH and GJRGARCH
models, indicating that the GARCH(1, 1) model is the most appropriate model
for modelling the volatility of both series, despite the presence of asymmetry
and leverage effect. However, Present findings are different from those of Shamiri
and Abu Hassan (2007) who found that the GJR model best suits the Malaysian
market and of Haniff and Pok (2010) who found that the
EGARCH model produced consistently superior results compared to the other GARCH
models.
CONCLUSION
This study examined different GARCH models to investigate and quantify the changes in volatility of the Malaysian stock market with respect to the global financial crisis 2007/2008. The KLCI was used as the main market indicator and the prices were transformed to log returns. Descriptive statistics showed that KLCI returns the presence of skewness in the series for both periods.
The unit root test was applied to check for stationarity and both series were found to be stationary. Conditional mean was then modelled using ARMA models and the AR (4) model was selected as the best model, which satisfied all criteria, had the lowest AIC as well as white noise residuals for both periods. Using the ARCHLM test at different lags, we detected a high presence of ARCH effect in the residuals and evidence of a clustering effect. The GARCH models were estimated for both series using QuasiMaximum likelihood assuming the Gaussian normal distribution. Different lags were examined for each model and the GARCH (1, 1), EGARCH (1, 1), GJRGARCH (1, 1) were found to be the most successful models, in line with previous literature. Rechecking using the ARCHLM test then showed no presence of ARCH effect. Standardized residuals and Standardized residuals squared were found to be white noise.
AR (4)/GARCH (1, 1), AR (4)/EGARCH (1, 1) and AR (4)/GJRGARCH (1, 1) were the final models to describe the process. Both asymmetric models produced the same results for both series which were found to be asymmetric and also suggested leverage effect. A comparison of the models for both series revealed significant increases in volatility and the presence of leverage effect with just a small drop in persistency due to the global financial crisis.
With respect to the simple GARCH (1,1) model which has explicit and simple
coefficients of lagged squared error and conditional variance, the volatility
has increased by 24.5% and at the persistency in volatility has just decreased
by 2.16% during the crisis period. Asymmetric GARCH models, which are extensions
of the simple GARCH model to capture asymmetries, are used for interpretation
of the asymmetric effect. A comparison of the two series asymmetric parameters,
show an increase in leverage effect in the market by 11.5% and 18.5% using EGARCH
and GJR GARCH, respectively. Since the leverage effect refers to the characteristics
of time series on asset prices that “ bad news” tend to increase volatility
more than “good news”, it is expected that the occurrence of the crisis
will increase this impact significantly.