estimating and forecasting volatility of stock indices …...ijbfmr 3 (2015) 19-34 issn 2053-1842...

IJBFMR 3 (2015) 19-34 ISSN 2053-1842

Estimating and forecasting volatility of stock indices using asymmetric GARCH models and Student-t

densities: Evidence from Chittagong Stock Exchange

Md. Qamruzzaman A. C. M. A.

School of Business Studies, Southeast University, Bangladesh. E-mail: [email protected].

Article History ABSTRACT Received 07 January, 2015

Received in revised form 01 February, 2015 Accepted 06 February, 2015 Keywords: Unit root test, Random Walk model, Stock indices, Chittagong Stock Exchange. Article Type: Full Length Research Article

This paper examined a wide variety of popular volatility models for stock index return, including Unit Root Test, Random Walk model, Autoregressive model, Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model, and extensive GARCH model, with Normal, and Student t-distribution assumption. I fit these models to Chittagong stock return index from 04 January 2004 to 14September 2014. There has been empirical evidence of volatility clustering, alike to findings in previous studies. Each market contains different GARCH models, which fit well. From the estimation, we find that the volatility of the return were significantly higher after 2009. The model introducing GARCH effect with normal and Student t-distribution assumption can better fit the volatility characteristics. We find that GARCH-z, EGARCH-z, IGARCH-z, GJR-GARCH-z and EGARCH-t. It is suggested that these five models can capture the main characteristics of Chittagong stock exchange (CSE).

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INTRODUCTION The ability to forecast financial market volatility is important for portfolio selection, valuation of stocks, asset management, predictability of risk premiums and designing optimal dynamic hedging strategies for options and Futures. While most researchers agree that volatility is predictable in many asset markets. Up and down movement in the daily prices of the securities can be considered as one of the consequences of the stochastic nature of the financial markets. In the face of usual up-down price movements, investors invest their funds in the financial markets particularly in the stocks or stock indices with the expectation of being compensated by risk-premium. The variation in the returns provided by the stocks due to changes in the daily price is generally termed as volatility which is measured by the standard deviation or the variance.

In such a volatile market it is difficult for companies to raise funds in the capital markets. Uncertainty causes loss of investor confidence which is important in stock trading particularly in making investment and leverage. This uncertainty can aggravate volatility further. Excess

volatility may even lead to crashes or crisis in financial markets. Volatility can either be historical volatility which is a measure based on past data, or implied volatility which is derived from the market price of a market traded derivative particularly an option. The historical volatility can be calculated in three ways namely; (1) simple volatility, (2) Exponentially Weighted Moving Average (EWMA) and (3) generalized autoregressive conditional heteroscedasticity (GARCH).

Among the financial time series non-linearity research literature, (Engle, 1982) proposed an autoregressive conditional Heteroscedasticity (ARCH) model and (Bollerslev, 1986) presented the GARCH model. These two models can determine the financial properties when the conditional variance is not a fixed parameter (Nelson, 1990). Looked at stock price changes and discovered that they have both positive and negative relationships with future stock price volatility. The GARCH model supposes a settled time conditional variance for the preceding issue of conditional variance and an error term square function. Therefore, the error term’s positive and

negative values do not respond to its influence on the conditional variance equation. The conditional variance only changes along with the error term’s value change, and cannot go along with the error term’s positive and negative changes. To improve this flaw (Nelson, 1991) presented an exponential GARCH model and (Glosten, 1993) developed a GJR-GARCH model. These are so-called models of asymmetric GARCH.

A large number of researcher’s used ARCH and GARCH in capturing the dynamic characteristics of stock market return across the countries, such as Islam (2013a), Elsheikh (2011) Islam (2013b), Engle (1987), Bae (2007), Wann-Jyi (2009), Tse (2010), Koutmos (1995), Bucevska (2012), Dima Alberga (2008), Ajab Al Freedi (2012) and many more. Few of empirical research findings are as follows.

Md. Ariful Islam (2014) has studied Stock market volatility: comparison between Dhaka stock exchange and Chittagong stock exchange considering Standard deviation, coefficient of Variation, F-test. Study results revealed that stock price at CSE is more volatile than DSE. Even the stock price of leading companies (top 20 and 30 companies of DSE and CSE) also varies from DSE to CSE and the volatility is much high than CSE30 of DSE20.

D.D.Tewari (2013) have studied existence and the nature of the volatility clustering phenomenon in the Johannesburg Stock Exchange (JSE) considering GARCH-type models. Study results revealed that an asymmetric effect of positive and negative shocks on conditional volatility could not be identified.

Suliman Zakaria (2012) have studied Stock market volatility in two African exchanges, Khartoum Stock Exchange, KSE (from Sudan) and Cairo and Alexandria Stock Exchange by employing different univariate specifications of the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. Study results revealed that the asymmetric GARCH models find a significant evidence for asymmetry in stock returns in the two markets, confirming the presence of leverage effect in the returns series.

Mehdi (2012) has studied Tehran Stock Market with GARCH Models in Forecasting Volatility. Study result revealed that the estimation and test results for models suggest that the leverage effect term, is significant in EGARCH model (even with a one-sided test). So there does appear to be an asymmetric effect in Tehran stock market. In addition Evaluation forecasting with MSE criteria indicate that GARCH models in this paper have a same forecasting power, but when Log- Likelihood is evaluation criteria, CGARCH has the best forecasting power.

Alberg (2008) estimate stock market volatility of Tel Aviv Stock Exchange indices, for the period 1992-2005. They report that the EGARCH model is the most successful in forecasting the TASE indices. Various time

Int. J. Bus. Financ. Manage. Res. 20 series methods are employed by Tudor (2008), including the simple GARCH model, the GARCH-in-Mean model and the exponential GARCH to investigate the Risk-Return Trade-off on the Romanian stock market. Results of the study confirm that E-GARCH is the best fitting model for the Bucharest Stock Exchange composite index volatility in terms of sample-fit.

Balaban (2005) used both symmetric and asymmetric ARCH-type models to derive volatility expectations. The outcome showed that there has a positive effect of expected volatility on weekly and monthly stock returns of both Philippines and Thailand markets according to ARCH model. The result is not clear if using other models such as GARCH, GJR-GARCH and EGARCH. For emerging African markets, Ogum (2005) investigate the market volatility using Nigeria and Kenya stock return series. Results of the exponential GARCH model indicate that asymmetric volatility found in the U.S. and other developed markets is also present in Nigerian stock exchange (NSE), but Kenya shows evidence of significant and positive asymmetric volatility. Also, they show that while the Nairobi Stock Exchange return series indicate negative and insignificant risk-premium parameters, the NSE return series exhibit a significant and positive time-varying risk premium. By using asymmetric GARCH models, Chowdhury (2004) and Chowdhury and Rahman (2004) have studied the relationship between the predicted volatility of DSE returns and that of selected macroeconomic variables of Bangladesh economy. They have calculated volatility from errors after using an autoregressive and seasonality adjusted forecasting model. The volatility series derived from such process has some limitations, which have been corrected in Generalized Conditional Auto Regressive Heteroscedasticity (GCARH) models developed by (Bollerslev, 1986).

Peters (2001) examined the forecasting performance of four GARCH (1, 1) models (GARCH, EGARCH, GJR and APARCH) used with three distributions (Normal, Student-t and Skewed Student-t). Study result revealed that overall estimation are achieved when asymmetric GARCH are used and when fat-tailed densities are taken into account in the conditional variance. Moreover, it is found that GJR and APARCH give better forecasts than symmetric GARCH. Finally increased performance of the forecasts is not clearly observed when using non-normal distributions.

But to the best of my knowledge, there are no such empirical studies for the Chittagong Stock Exchange (CSE). This study is useful for number of reasons. Firstly, to the best of our knowledge, this is the first study of this kind of modeling volatility of the Bangladesh stock exchange especially Chittagong stock exchange. Secondly, the results of this study will be of great interest to academician, policy maker and investor both domestically and internationally. The main objective of

Qamruzzaman 21 this study is to study volatility pattern of Chittagong Stock Markets and get some insight on volatility modeling using ARCH and GARCH kind of models. Thus, one of the contributions of this paper is to provide empirical evidence on the fit of conditional volatility models for the Chittagong stock exchange (CSE).

The main objective of this paper is to model stock returns volatility in Chittagong Stock Exchange by employing different univariate specifications of GARCH type models for daily observations on the index returns series of each market over the period of January 2004 to June 2014, as well as describing special features of the markets in terms of trading activity and index components and calculations. The volatility models employed in this paper include both symmetric and asymmetric GARCH models. METHODOLOGY The main focus of this study is to conduct a comprehensive analysis of the volatility characteristics of the Dhaka Stock Exchange DSE. A number of widely recognized volatility models are used in this regard, namely 1. Random walk model 2. Unit root test.3. ARMA model 4. GARCH model and 5. Expanded GARCH Model. Daily returns as asset time series Daily returns are identified as the difference in the natural logarithm of the closing index value for the two

consecutive trading days, i.e. .

Where, rt is the logarithmic daily return at time t and

and are daily price of an asset at two successive

days t-1 and t respectively. In order to do time series analysis, transformation of

original series is required depending upon the type of series when the data is in the level form. I have transformed the series of return by taking natural logarithm of the series. Some scholars have pointed out two advantages of this kind of transformation of the series. First, it eliminates the possible dependence of changes in stock price index on the price level of the index. Second, the change in the log of the stock price index yields continuously compounded series. Random walk model:

If is a random series with mean µ and constant

variance and is serially uncorrelated that the series

[ ] is said to be random walk if:

(1)

Where return series of stock price is [ ], random walk

model as , where u is the model parameter,

E( , Var ( = , and to follow a normal

distribution ARMA model The ARMA (the autoregressive Moving Average)

proposed by the Box and Jenkins (1970). If the series [ ]

stratifies ARMA (p, q), then [ ] can be described as

follows;

(2)

Where, , ,i=1, ….. Ƥ are parameters, E =0, Var ( =

, and a particular distribution

The ARCH Models Let us consider univariate time series yt. If Ψt-1 is the information set (i.e. all the information available) at time t-1, we can define its functional form as: yt = E[yt|Ψt-t] + εt (1) Where, E [yt|Ψt-t] denotes the conditional expectation operator and is the disturbance term (or unpredictable part), with E[εt] = 0 and E[εtεs] = 0; 8t 6= s. The εt term in Equation 1 is the innovation of the process. The conditional expectation is the expectation conditional to all past information available at time t-1. The Autoregressive Conditional Heteroscedastic (ARCH) process of Engle (1982) is any {εt} of the form

(3)

Where, is an independently and identically distributed

(i.i.d.) process, E ( ) =0, Var ( ) =1 and where is a

time-varying, positive and measurable function of the

information set at time t-1. By definition, is serially

uncorrelated with mean zero, but its conditional variance

equals , therefore, may change over time, contrary to

what is assumed in ordinary least squares (OLS) estimations. Specifically, the ARCH (q) model is given by:

(4)

GARCH model

The conditional variance of is assumed constant for

random walk and AR model, but in financial time series

the data are usually volatility clustering, such that the

conditional volatility changes over time. The stock price’s return series is [rt], sample GARCH (p, q) model can be describe as follows:

) =0, )=

(5

Using the lag or backshift operator L, the GARCH (p, q) model is

(6)

and

Based on Equation 5, it is straight forward to show that the GARCH model is based on an infinite ARCH

specification. If all the roots of the polynomial 1- =0

of Equation 5 lie outside the unit circle, we have

or equivalently

Which may be seen as an ARCH ( process since the

conditional variance linearly depends on all previous squared residuals. EGARCH Our first asymmetric GARCH model is the exponential GARCH model of Nelson (1991);

………7

Where, is the normalized series.

The value of g ( depends on several elements, Nelson

(1991) note that to accommodate the asymmetric relation between stock return and volatility changes, the value of

g ( must be a function of both magnitude and the sign

Int. J. Bus. Financ. Manage. Res. 22

of ( . That is why he suggest to express the function g

( as,

, (8)

Where, and

known as . Another advantage of this

specification is that it does not require any stationary

constraint. Thus, E depends on the assumption made

on the unconditional density. GJR-GARCH This popular model is proposed by Gloston, Jagannathan and Runkle (1993). It is generalized version is given by;

) (9)

Where is a dummy variable.

In this model, it is assumed that the impact of on the

conditional variance is difference when is positive

or negative. That is why the dummy variable takes the

value “0” (respectively “1”) when is positive (negative).

Note that model of Zakoian (1994) is very similar to the GJR but models the conditional standard deviation instead of the conditional variance. APARCH Ding, Granger, and Engle (1993) introduce the asymmetric Power ARCH (APARCH) model. The APARCH (p,q) model can be expressed as;

(10)

Where ,

(j=1………p),

This model is quite interesting since it couples the flexibility of a varying exponent with the asymmetry coefficient to take the “leverage effect” into account. RESULTS AND DISCUSSION Descriptive statistics This research considers daily closing prices index for Chittagong stock exchange namely CSCX index in

Qamruzzaman 23

Figure 1: graph of CSCX price series Figure 2: Conditional Volatility of CSCX

Price

0

4,000

8,000

12,000

16,000

20,000

04 05 06 07 08 09 10 11 12 13 14

CSCX price series

0

10

20

30

40

50

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

Conditional variance Figure 1. Graph of CSCX price series.

Bangladesh from 04th January 2004 to 15th September 2014. The analysis is undertaken using the econometric package Eviews-8 for volatility modeling. The return indices are obtained from CSE market for different indices. Daily returns are calculated by using the following formula; rt =100*dlog(pt) Where, rt represents daily return, pt represents daily closing stock market index. Here, we also try to examine the behavior of the return series using all three price index of Chittagong stock exchange. The graphs of Price Series, BSE SENSEX Series and Descriptive Statistics, Conditional Volatility and Trajectory of Volatility Returns are displayed in Figures 1 to 9.

It is observed from the graphs that there is volatility clustering with non-normal distribution for each case. In case of the trajectory of volatility of returns indicates less sensitive for return from CSCZ index while rest two indexes show greater sensitivity toward return volatility.

Some of the descriptive statistics of three different

index are displayed in Table 1. It is observed that average return of CSE30 is 0.078, CASPI is 0.083 and CSCX is 0.078 respectively which is very much closure to each other. Volatility (measured as a standard deviation) shows maximum value for CSCX index is 15.210 whereas rest two index shows almost same level of volatility (1.50). Return of CSE show Leptokurtic distribution in all cases which is indicating that distribution of return is sharper than a normal distribution, with values concentrated around the mean and thicker tails. This means high probability for extreme values. According to Jarque–Bera statistics normality is rejected for the return series by showing high non-normality in all three indices. However, it is obvious from Table 2 that Strong positive correlation exists among all index in Chittagong Stock exchange. Unit root test It is found that the variables RTCSE30, RTCSCX AND RTCASPI have trends in their level.Augmented Dickey-Fuller (ADF) t-tests and (PP) Phillips and Perron (1988) tests are used for each of the three time series-CSE30,


Figure 1: graph of CSCX price series Figure 2: Conditional Volatility of CSCX

Price

0

4,000

8,000

12,000

16,000

20,000

04 05 06 07 08 09 10 11 12 13 14

CSCX price series

0

10

20

30

40

50

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

Conditional variance

Figure 2. Conditional Volatility of CSCX Price.

-8

-4

0

4

8

12

16

04 05 06 07 08 09 10 11 12 13 14

RTCASPI

Figure 3. The trajectory of volatility of returns.

Qamruzzaman 25

Figure 4: Graph of CSE30 price series Figure 5: Conditional Volatility of CSE30

Price

0

4,000

8,000

12,000

16,000

20,000

24,000

04 05 06 07 08 09 10 11 12 13 14

CSE30

0

400,000

800,000

1,200,000

1,600,000

2,000,000

2,400,000

2,800,000

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

Conditional variance Figure 4. Graph of CSE30 price series.

Figure 4: Graph of CSE30 price series Figure 5: Conditional Volatility of CSE30

Price

0

4,000

8,000

12,000

16,000

20,000

24,000

04 05 06 07 08 09 10 11 12 13 14

CSE30

0

400,000

800,000

1,200,000

1,600,000

2,000,000

2,400,000

2,800,000

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

Conditional variance Figure 5. Conditional volatility of CSE30 price.


-16

-12

-8

-4

0

4

8

12

16

20

04 05 06 07 08 09 10 11 12 13 14

RTCSE30


Figure 7: graph of CSE30 price series Figure 8: Conditional Volatility of CSE30

Price

0

4,000

8,000

12,000

16,000

20,000

04 05 06 07 08 09 10 11 12 13 14

CSCX

0

20

40

60

80

100

120

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013


Figure 7. Graph of CSE30 price series.

Qamruzzaman 27

Figure 7: graph of CSE30 price series Figure 8: Conditional Volatility of CSE30

Price

0

4,000

8,000

12,000

16,000

20,000

04 05 06 07 08 09 10 11 12 13 14

CSCX

0

20

40

60

80

100

120

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013


Figure 8. Conditional volatility of CSE30.

Figure 9: The Trajectory of Volatility of Returns

-600

-400

-200

0

200

400

600

04 05 06 07 08 09 10 11 12 13 14

RTCSCZ



Table 1. Descriptive statistics for logarithm differences rt=100*dlog (pt).

Index Average Min Max SD Kurtosis Skewness Jarque–Bera stat.

CSCX 0.078 -542.9845 544.025 15.210 1265.62 0.0864 1712467

CSE30 0.078 -15.53021 17.9378 1.5935 18.0681 0.3785 24744.56

CASPI 0.083 -7.764455 13.07073 1.4226 9.52616 0.0755 4632.453

Table 2. Correlation matrix among different index in Chittagong stock exchange.

CSCX CSE30 CASPI

CSCX 1

CSE30 0.9557 1

CASPI 0.97765 0.99027 1

Table 3. Unit root test.

Variables

Augmented Dickey Fuller (ADF) Test-(t stat) Phillips and Perron (PP) test-(t stat)

No. of observations

1 2 3 No. of

observations 1 2 3

RTCSCX

Level 2568 (25.02)* (25.02) * (26.67) * 2569 (264.24) * (283.54) * (240.84) *

1st difference 2550.00 (19.92) * (19.92) * (19.93) * 2576.00 (4930.71) * (4929.70) * (4929.65) *

RTCSE30

Level 2608.00 (51.67) * (51.74) * (51.55) * 2608.00 (51.68) * (51.74) * (51.61) *

1st difference 2585.00 (19.84) * (19.84) * (19.84) * 2607.00 (675.46) * (675.30) * (675.75) *

RTCSCPI

Level 2608.00 (49.29) * (49.37) * (49.14) * 2608.00 (49.60) * (49.61) * (49.65) *

1st difference 2585.00 (19.11) * (19.11) * (19.11) * 2607.00 (732.51) * (732.23) * (732.84) *

Note: Superscripts * indicate rejection of null hypothesis at 5% level of significance. 1, No trend, no intercept; 2, only intercept; 3, trend and intercept.

CSCX and CASPI to test for the presence of a unit root. To ensure that the residuals were white noise the lag length for the ADF tests was selected.

The outcomes of the Augmented Dickey Fuller (ADF) test by Engle (1987) with and without trend and the Phillips and Perron (1988) test again with and without trend are reported in Table 3. It is obvious from Table 3 estimation that at level the variables are non-stationary in both ADF and PP tests which is indicate there is no unit root existing among tested variables. Autocorrelation test We test for autocorrelation in the raw returns and their squares using Ljung-Box (L-B) Q-statistics. For detecting autocorrelation look at Q-Statistics in Tables 1, 4 and 5 and its associated probability values. If the probability value is greater than 0.05, we accept the null hypothesis

(it suggests absence of autocorrelation). In a situation where probability value is less than 0.05, we reject the null hypothesis (it suggests presence of autocorrelation). The statistics given in Tables 4, 5 and 6 of Sample Autocorrelation-CSCX, CASPI and CSE30 respectively suggest the presence of autocorrelation in all lags of the series.

I computed Q-statistics up to 36 lags but reported the Q-statistics up to 15 lags for both raw returns and their square to test for ARCH effect. All the lags are statistically significant, and the squares of the lag values are larger, suggesting that ARCH type modeling is more appropriate. Random walk test Table 7 exhibits the results of Random walk model with their z statistics and associated probabilities for each

Qamruzzaman 29 Table 4. Sample autocorrelation –CSCE.

Raw return series Return square series

Lags AC Q-Stat Probability Lags AC Q-Stat Probability

1 -0.003 0.0199

1 0.000 0.0004 0.984

2 0.001 0.0237 0.878 2 0.000 0.0008 1

3 -0.001 0.0273 0.986 3 0.000 0.0012 1

4 0 0.0276 0.999 4 0.000 0.0016 1

5 0 0.0277 1 5 0.000 0.002 1

6 -0.002 0.0433 1 6 0.000 0.0024 1

7 -0.001 0.0487 1 7 0.000 0.0028 1

8 0 0.0494 1 8 0.000 0.0032 1

9 0 0.0496 1 9 0.000 0.0036 1

10 -0.002 0.0573 1 10 0.000 0.004 1

11 -0.003 0.0833 1 11 0.000 0.0044 1

12 0 0.0837 1 12 0.000 0.0048 1

13 0 0.0838 1 13 0.000 0.0052 1

14 0.002 0.0986 1 14 0.000 0.0056 1

15 -0.002 0.106 1 15 0.000 0.006 1

Table 5. Sample autocorrelation -CSE30.



1 0.02 1.0379

1 -0.001 0.0025 0.96

2 0.003 1.0555 0.304 2 0.006 0.0872 0.957

3 0.032 3.7437 0.154 3 0.024 1.6233 0.654

4 0.062 13.872 0.003 4 0.010 1.8987 0.754

5 0.059 22.901 0 5 0.016 2.5323 0.772

6 -0.02 23.963 0 6 0.018 3.3793 0.76

7 -0.008 24.134 0 7 0.017 4.1442 0.763

8 0.023 25.493 0.001 8 0.009 4.3749 0.822

9 0.031 27.949 0 9 0.009 4.5687 0.87

10 0.009 28.183 0.001 10 0.017 5.2883 0.871

11 -0.037 31.833 0 11 0.020 6.36 0.848

12 0.025 33.504 0 12 0.013 6.826 0.869

13 -0.011 33.848 0.001 13 0.006 6.93 0.906

14 0.019 34.792 0.001 14 0.006 7.0248 0.934

15 0.05 41.279 0 15 0.034 10.124 0.812

case. It is manifested that that z-statistics of all three index are lower than associated probability, which is explain the existence of autocorrelation.

Table 8 lists the estimated results of AR models and their t-statistics as well as log likelihood for three index of Chittagong stock exchange. It is observed that estimated mean of each model is insignificant and dummy variable is insignificant as well, but volatility is significant. AR model shows negative likelihood values in all three index

for each model which indicates that there is log likelihood exist among each model from AR (1) to AR (5). GARCH model Estimation results of the GARCH models including the t-statistic as well as likelihood value are listed in Table 9. The comparison of all log likelihood values of AR models


Table 6. Sample autocorrelation-CASPI.



1 0.023 1.3579

1 -0.031 2.5438 0.111

2 0.005 1.4306 0.232 2 0.079 18.881 0

3 0.023 2.7711 0.25 3 0.110 50.694 0

4 0.047 8.4914 0.037 4 0.043 55.54 0

5 0.043 13.343 0.01 5 0.100 81.657 0

6 0.001 13.344 0.02 6 0.065 92.656 0

7 0.001 13.349 0.038 7 0.091 114.19 0

8 0.025 14.938 0.037 8 0.060 123.64 0

9 0.034 17.961 0.022 9 0.056 131.78 0

10 0.006 18.045 0.035 10 0.065 142.82 0

11 -0.008 18.234 0.051 11 0.080 159.78 0

12 0.027 20.118 0.044 12 0.059 168.84 0

13 -0.009 20.351 0.061 13 0.070 181.54 0

14 0.04 24.549 0.026 14 0.047 187.3 0

15 0.039 28.473 0.012 15 0.095 210.83 0

Table 7. Estimation of RW model with normal distribution of market index.

index Joint Tests Value d.f Probability

CSCX

Max |z| (at period 2)*

1.4202 2577 0.4915

CSE30 6.8541 2608 0.0000

CASPI 9.8009 2608 0.0000

*Probability approximation using studentized maximum modulus with parameter value 4 and infinite degrees of freedom

Individual Tests

Period

Var. Ratio Std. Error z-Statistic Probability

2

CSCX INDEX 0.3344 0.4686 -1.4202 0.1555

CSE30 INDEX 0.5043 0.0723 -6.8541 0.0000

CASPI INDEX 0.5194 0.0490 -9.8009 0.0000

4

CSCX INDEX 0.1672 0.7221 -1.1530 0.2489

CSE30 INDEX 0.2361 0.1146 -6.6597 0.0000

CASPI INDEX 0.2485 0.0809 -9.2779 0.0000

8

CSCX INDEX 0.0840 0.8569 -1.0688 0.2851

CSE30 INDEX 0.1238 0.1441 -6.0764 0.0000

CASPI INDEX 0.1275 0.1093 -7.9749 0.0000

16

CSCX INDEX 0.0423 0.9252 -1.0350 0.3007

CSE30 INDEX 0.0620 0.1776 -5.2803 0.0000

CASPI INDEX 0.0642 0.1511 -6.1899 0.0000

shows that adding the GARCH effect significantly improve the in sample fit of the models. The log likelihood values increases from (9902) to (9890) for CASPI, from (4512) to (4505) for CSE30 and values decrease from (9983) to (10070) for CSCX respectively. GARCH effect is significant for CASPI, CSE30 but insignificant for

CXCZ. In order to further diagnosis whether there is any

residual effect remain in GARCH model. Table 10 exhibits estimate results of standardized squared residual indicating that values of Q-stat are significantly lower in every lags with having maximum probability such ensure

Qamruzzaman 31 Table 8. Parameter estimation for AR models with normal distribution.

µ Α α

1 β Log likelihood

AR (1)

CAXPI 0.0840 0.0289 2.9122 0.0036

(4618.19) 0.0350 0.0196 1.7896 0.0736

CSE3O 0.0792 0.0308 2.5668 0.0103

(4915.59) (0.0121) 0.0196 (0.6186) 0.5362

CSCZ 0.0787 0.1742 0.4519 0.6514

-10308.9300 -0.4951 0.0171 -28.9134 0.0000

AR (2)

CAXPI 0.0842 0.0278 3.0262 0.0025

(4618.4720) (0.0018) 0.0196 (0.0938) 0.9253

CSE3O 0.0794 0.0306 2.5939 0.0095

(4913.7890) (0.0204) 0.0196 (1.0424) 0.2973

CSCZ 0.0790 0.3001 0.2632 0.7924

(10667.5500) 0.0007 0.0197 0.0379 0.9698

AR (3)

CAXPI 0.0844 0.0283 2.9782 0.0029

(4616.8280) 0.0159 0.0196 0.8134 0.4161

CSE3O 0.0795 0.0316 2.5141 0.0120

(4912.7430) 0.0122 0.0196 0.6202 0.5352

CSCZ 0.0793 0.2999 0.2643 0.7916

(10663.9100) (0.0003) 0.0197 (0.0175) 0.9860

AR (4)

CAXPI 0.0847 0.0291 2.9078 0.0037

(4613.3950) 0.0428 0.0196 2.1882 0.0287

CSE3O 0.0799 0.0327 2.4422 0.0147

(4908.6840) 0.0457 0.0196 2.3355 0.0196

CSCZ 0.0794 0.3008 0.2640 0.7918

(10660.2600) 0.0022 0.0197 0.1101 0.9124

AR (5)

CAXPI 0.0850 0.0284 2.9930 0.0028

(4613.9430) 0.0173 0.0196 0.8835 0.3771

CSE3O 0.0803 0.0322 2.4941 0.0127

(4908.6670) 0.0294 0.0196 1.5026 0.1331

CSCZ 0.0796 0.3001 0.2651 0.7909

(10656.6300) (0.0003) 0.0197 (0.0158) 0.9874

that there is no residual effects in GARCH models.

Expanded GARCH models with normal distribution and non-normal distribution:

The estimated parameters in expanded GARCH models with z-distribution and t-distribution are listed in Table 11 including z-statistics and t-statistics with log likelihood values. The results shows that the means are insignificant exception may occur in AR (1) GJR-GARCH. The sum of GARCH estimates α

1+ β is less than 1which

shows that the volatility is limited and the data is non-stationary, explaining why model fit well.

RESEARCH FINDINGS

This paper compared the forecasting performance of

several GARCH-type models. The comparison was focused on two different aspects: the difference between symmetric and asymmetric GARCH (i.e., GARCH versus EGARCH, GJR and APARCH) and the difference between normal tailed symmetric, fat-tailed symmetric and fat-tailed asymmetric distributions (i.e. Normal versus Student-t and Skewed Student-t).

Study results revealed that return of CSE leptokurtic, significant skewness, and deviation from normality and the return series are volatility clustering. Conclusion about CSE returns are as follows: (1) AR model, which is added in lag, cannot improve performance and error of the model in contrast to random walk model. There is no significant different between two models. (2) Adding the GARCH effect based on random walk model can improve performance and error of the model to some extent. GARCH model which have leverage effect do a little help to improve the model performance. Moreover, it can


Table 9. Parameter estimates for GARCH (1, 1) models with normal distribution.

µ Α α

1 β Log likelihood

AR(1) GARCH (1,1)

CASPI (0.0967) 2.3380 (0.0414) 0.9670

(9902.421) (0.0373) 0.4089 (0.0912) 0.9273

CSE3O 0.0913 0.0238 3.8406 0.0001

(4511.830) 0.0666 0.0200 3.3285 0.0009

CSCX (0.1259) 2.1637 (0.0582) 0.9536

(9982.879) (0.2360) 0.2590 (0.9113) 0.3621

AR(2) GARCH (1,1)

CASPI (0.1034) 2.4873 (0.0416) 0.9668

(9899.478) 0.0014 0.9676 0.0015 0.9988

CSE3O 0.0884 0.0215 4.1086 0.0000

(4513.690) (0.0432) 0.0232 (1.8647) 0.0622

CSCX (0.1696) 3.7743 (0.0449) 0.9642

(10143.790) (0.0628) 0.9155 (0.0685) 0.9454

AR(3) GARCH (1,1)

CASPI (0.1040) 2.4285 (0.0428) 0.9658

(9896.977) (0.0032) 1.0798 (0.0030) 0.9976

CSE3O 0.0936 0.0237 3.9522 0.0001

(4512.799) 0.0460 0.0219 2.0981 0.0359

CSCX (0.1616) 3.1603 (0.0511) 0.9592

(10103.160) (0.0536) 0.4256 (0.1258) 0.8999

AR(4) GARCH(1,1)

CASPI (0.1073) 2.4433 (0.0439) 0.9650

(9891.760) 0.0045 1.0173 0.0044 0.9965

CSE3O 0.0933 0.0236 3.9532 0.0001

(4510.487) 0.0526 0.0220 2.3882 0.0169

CSCX (0.1515) 3.1413 (0.0482) 0.9615

(10023.350) (0.0056) 0.4311 (0.0130) 0.9897

AR(5) GARCH(1,1)

CASPI (0.1014) 2.4809 (0.0409) 0.9674

(9890.686) 0.0196 0.2397 0.0819 0.9347

CSE3O 0.0978 0.0239 4.1013 0.0000

(4505.767) 0.0574 0.0216 2.6633 0.0077

CSCX (0.1508) 3.3508 (0.0450) 0.9641

(10069.240) (0.0153) 0.3495 (0.0439) 0.9650

Table 10. Diagnostic test for standardized squared residuals.

Lags AC PAC Q-Stat Probability

1 -0.000 -0.000 0.0004 0.984

2 -0.000 -0.000 0.0008 1.000

3 -0.000 -0.000 0.0012 1.000

4 -0.000 -0.000 0.0016 1.000

5 -0.000 -0.000 0.0020 1.000

6 -0.000 -0.000 0.0024 1.000

7 -0.000 -0.000 0.0028 1.000

8 -0.000 -0.000 0.0032 1.000

9 -0.000 -0.000 0.0036 1.000

10 -0.000 -0.000 0.0040 1.000

11 -0.000 -0.000 0.0044 1.000

12 -0.000 -0.000 0.0048 1.000

13 -0.000 -0.000 0.0052 1.000

14 -0.000 -0.000 0.0056 1.000

15 -0.000 -0.000 0.0060 1.000

Qamruzzaman 33

Table 10. Diagnostic test for standardized squared residuals.

Lags AC PAC Q-Stat Probability

1 -0.000 -0.000 0.0004 0.984

2 -0.000 -0.000 0.0008 1.000

3 -0.000 -0.000 0.0012 1.000

4 -0.000 -0.000 0.0016 1.000

5 -0.000 -0.000 0.0020 1.000

6 -0.000 -0.000 0.0024 1.000

7 -0.000 -0.000 0.0028 1.000

8 -0.000 -0.000 0.0032 1.000

9 -0.000 -0.000 0.0036 1.000

10 -0.000 -0.000 0.0040 1.000

11 -0.000 -0.000 0.0044 1.000

12 -0.000 -0.000 0.0048 1.000

13 -0.000 -0.000 0.0052 1.000

14 -0.000 -0.000 0.0056 1.000

15 -0.000 -0.000 0.0060 1.000

Table 11. Parameter estimates for expanded GARCH models with normal distribution.

µ α α

1 β Log likelihood AIC SIC

AR(1) GJR- GARCH

CAXPI 0.233 3.633 0.064 0.949

(10235.63) 7.948 7.962 (0.009) 0.010 (0.857) 0.392

CSE3O 0.084 0.028 3.048 0.002

(4510.15) 3.463 3.476 0.067 0.020 3.346 0.001

CSCZ 0.233 3.633 0.064 0.949

(10235.63) 7.948 7.962 (0.009) 0.010 (0.857) 0.392

AR(1) EGARCH(1,1)

CAXPI (0.397) 0.271 (1.465) 0.143

(9603.81) 7.458 7.472 (0.020) 0.020 (1.008) 0.313

CSE3O 0.060 0.026 2.270 0.023

(4485.43) 3.444 3.457 0.067 0.020 3.453 0.001

CSCZ (0.397) 0.271 (1.465) 0.143

(9603.81) 7.458 7.471 (0.020) 0.020 (1.008) 0.313

AR(1) EGARCH(t)**

CAXPI 0.080 0.027 3.030 0.002

(4404.81) 3.424 3.440 0.209 0.016 13.205 0.000

CSE3O 0.068 0.024 2.835 0.005

(4337.74) 3.331 3.347 0.097 0.019 4.996 0.000

CSCZ 0.106 0.025 4.279 0.000

(4319.12) 3.357 3.373 0.136 0.017 8.097 0.000

**Estimated values of AR (1) EGARCH consider student’s t-distribution.

increase the specification error of the model such as EGARCH, IGARCH, and GJR-GARCH model. Therefore, it can be said that all four models may be best suited for capturing CSE return volatility.

Scope of further study The study examined stock return volatility focusing on three indexes available at CSE with application of ARCH,

GARC and Expanded GARCH Models. Thus, the study paves the avenue for following further research agenda: 1. Factors responsible for stock return clustering volatility in both DSE and CSE. REFERENCES

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