Test of asymmetric Garch models

Henri Högkulla s082880 Carl-Anton Karlsson s081760

Hanken School of Economics Department of Finance and Statistics Vaasa

November 2011

Contents
Abstract ...................................................................................................................................... 1 1 Introduction ............................................................................................................................. 2 2 Methodology ........................................................................................................................... 4 2.1 ARCH and GARCH ......................................................................................................... 4 2.2 EGARCH ......................................................................................................................... 4 2.3 GJR-GARCH ................................................................................................................... 5 2.4 Distributions ..................................................................................................................... 5 2.5 Information criterions ....................................................................................................... 6 3 Data ......................................................................................................................................... 7 3.1 Descriptive statistics ......................................................................................................... 7 4 Results ..................................................................................................................................... 9 4.1 Evaluation of the models ................................................................................................ 10 5 Conclusion ............................................................................................................................. 13 Appendix .................................................................................................................................. 14 References ................................................................................................................................ 15

Table 1 - Descriptives ................................................................................................................ 8 Table 2 - Nasdaq 100 Results ..................................................................................................... 9 Table 3 - Nasdaq Composite Results ....................................................................................... 10 Table 4 - Nasdaq 100 Information Criterias............................................................................. 11 Table 5 - Nasdaq Composite Information Criterias ................................................................. 11

Figure 1 - Return plots ............................................................................................................... 8 Figure 2 - Nasdaq Composite Distribution .............................................................................. 14 Figure 3 - Nasdaq 100 Distribution .......................................................................................... 14

Abstract
In this paper the GARCH, EGARCH and GJR-GARCH were used to model time series data. First we compared the models using a normal distribution and after that using the student-t distribution. We found that the EGARCH tend to fit better to the data in our empirical study. Also using a student-distribution can improve fitting of a model. The data used in the empirical study was daily logarithmic returns from the Nasdaq 100 and Nasdaq Composite indices. The time period was 26.9.2003 to 10.11.2011.

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1 Introduction
Risk management has always been the key word for fund managers and traders. Volatility, i.e. the measure of variation in price, is one of the most important measures of risk. Volatility is also used in the pricing of assets and is also traded nowadays using different derivatives. Therefore it has always been interesting to be able to model future volatility using historical data. There is a significant amount of literature which shows evidence of different anomalies on financial markets. These known anomalies can have an effect on how to construct a financial model for research. One is the volatility cluster effect which means that large changes in volatility are often followed by large changes and vice versa, Taylor (2005). Leptokurtosis can also be present in the material which means that the probability distribution has fatter tails i.e. not normally distributed; this is stated in Hall and Yao (2003). One other thing that has been present in other researches is the so called leverage effect. This is when volatility increases after a negative shock but does not increase correspondingly after a positive shock of the same magnitude as the negative shock. Wu (2001) found that both volatility feedback and the leverage effect are important determinants of asymmetric volatility. Due to these problems there have been developed several different statistical techniques to capture these characteristics in the data material if present. Since these anomalies are often present in financial time series one must be able to choose a fitting model which capture these effects in a proper way. Engle (1982) proposed the ARCH model that models time-varying conditional variances with lagged disturbances. He showed that using the model you can capture volatility clusters and leptokurtosis in time series. However the modeled uses symmetric distributions and therefore fails to capture so called leverage effects Black (1976). Nelson (1991) developed the EGARCH model which allows different effects of positive and negative shocks. The GJR-GARCH proposed by Glosten, Jagannathan and Runkle (1993) is another popular model which is used to capture the leverage effect. Previous studies, such as Alberg, Shalit and Yosef (2006), show that asymmetrical GARCH models tend to improve the forecasting performance in volatility. This study also claims that the GJR-GARCH model perform worse than the other asymmetric

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model. Shamiri and Hassan (2005) also studied different asymmetrical GARCH models and also several different distributions to see which one best model the Malaysian and Singaporean stock market. Their results were that both the GJR-GARCH and the EGARCH provided better results than the GARCH. The GJR was better on the Malaysian market and EGARCH was better on the Singaporean market. Their selection criteria were AIC, BIC and log-likelihood ratios. Malmsten (2004) also tested the EGARCH model against the GARCH model using Lagrange multiplier tests. Peters (2001) also tested the forecasting performance of four different GARCH models (GARCH, EGARCH, GJR and APARCH) with three different distributions (Normal, Student-t and skewed Student-t). He used 15 year daily data from the FTSE 100 and the DAX 30 indices. The findings were that asymmetric GARCH models improve the overall forecasting but any significant improvement cannot be observed using nonnormal distributions. It is also suggested that shorter time intervals could better represent “true” volatility. Rabemananjara and Zakoian (1993) tested for asymmetries in financial data using the EGARCH and the threshold GARCH(TARCH) models. They found that asymmetries do exist in the volatility on US stock markets. The conclusions drawn from the research were that volatility tends to be higher after a decrease than after an increase of equal size and that the positivity constraints on ARCH parameters can be violated. The reactions of volatility to past positive and negative return shocks can also be conversed between large and small values. For example small positive return values can have a higher effect on volatility than small negative ones of equal size. These findings also support the need for asymmetric volatility models. The data used for this study is daily logarithmic returns for the NASDAQ Composite and NASDAQ 100 over a period from 25.9.2003 to 1.11.2011 with 2102 observations. The purpose of this term paper is to determine which of the GJR and EGARCH best captures the properties of the data. We will also compare the different models with the GARCH model. First we will introduce the methodology which will be used in the empirical study and then we discuss the data material. In the chapter after that the findings of the study will be presented an at last we will discuss the conclusions and implications of the study.

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2 Methodology
The volatility models used in this empirical study are the GARCH, GJR-GARCH and the exponential GARCH. The chapter starts with a short introduction of the ARCH model proposed by Engle in 1982 and after that GARCH, EGARCH and GJR-GARCH will be presented.

2.1 ARCH and GARCH
Stock price volatility has been a subject of study for researchers frequently last decades. In 1982 Engle proposed a model for modeling time-varying conditional volatility. He assumed that the variance is a linear function of past innovations ( for example ∑ where is the variance of the time series and is the innovation at time t Bollerslev, like

Engle and Nelson (1994:2967-2970). However empirical evidence show that if you want to capture the dynamics of a conditional variance you usually need a high ARCH order. Bollerslev (1986) solved this by proposing a model based on the ARCH model which is called the GARCH model: ∑ where and ∑ ∑ is the impact of the previous shocks

is the conditional variance, ∑

is the impact of previous volatilities. For this to hold all the

coefficients in the model have to be positive and the sum of them not equal to 1 or higher than 1. (Bollerslev, Engle and Nelson 1994:2967-2970)

2.2 EGARCH
Both ARCH and GARCH assumes symmetric distributions. Nelson (1991) proposed a model which could fit better to asymmetrically distributed data. Consider a generalized model of Egarch (p,q) like

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∑ | Where σ is the conditional variance and time t-1. If our hypothesis is then

| |

∑

where

is the info at

< 0 we can “test the leverage effect”. (Ahlberg,

Shalit and Yosef 2006) Main differences from the GARCH model is that it allows volatility shocks to have a bigger impact on volatility and doesn´t require same impact on volatility with negative and positive shocks.

2.3 GJR-GARCH
The model was proposed by Glosten, Jagannathan, and Runkle (1993) and is usually written as: ∑

∑

In its generalized form, where on

is a dummy variable. In the model the impact of

is different depending on the error term being positive or negative. If error term is = 0 and vice versa. GJR-GARCH has the same conditions as the

positive, then

GARCH i.e. the sum of the coefficient must be below 1 and positive. An addition to these conditions is however that if < 0 then the model is still admissible if the sum of ≥ 0. Bollerslev, Engle and Nelson (1994:2967-2970)

2.4 Distributions
In the empirical study two different probability distributions are used. The first one is the Gaussian normal distribution which is the most common in empirical researches and the second is the student-t distribution. Reason for testing both is that the times series could show signs of excess kurtosis and skewness resulting in a non-normal distribution. If this is the case the test using student-t distribution could therefore “capture more information” and fit better to the data. The model parameters are estimated using maximum likelihood methodology. The method selects values that maximize the likelihood functions.

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The log likelihood function for the Gaussian normal distribution is equal to ∑ and ∑ for [ [ the student-t ] with T observations distribution [ ] [ ] [ ] being the

] with v=degrees of freedom and

gamma function. When v →∞ it results in a normal distribution. (Shamiri and Hassan 2005)

2.5 Information criterions
The information criterion measures estimates the goodness of fit of a statistical model on time series data. In the empirical study the AIC and BIC will be used. The BIC is sometimes also called the Schwarz´s criterion. The AIC measure was developed and proposed by Akaike in 1974 and is usually written as ( )

where LogL is the log of the likelihood function, k is the number of the parameters and T is the sample size. Bayesian information criterion or the Schwarz criterion was originally developed by Schwarz(1976) and Bayesian (1978). The measure based on the maximum likelihood estimation is ( )

where Log(L) is the log of the likelihood function, k is the number of the parameters and T is the sample size. The selection criterion is to choose the model that minimizes these model selection criterions. (Brooks. C.2008:232-236)

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3 Data
For the empirical study daily stock market returns from the Nasdaq index has been used. Both the Nasdaq Composite and the capitalization-weighted index Nasdaq 100 were used. The Nasdaq Composite is one of the largest indices in the world with over 3000 different securities and is often used as a benchmark index. It consists of both US and non-US based companies. Nasdaq 100 consists of the 100 largest companies, financial institution excluded. The timeframe used is from 26.9.2003 to 10.11.2011 for both of the indices with 2120 observations for each index. The data is extracted from the Thomson Datastream and can therefore be considered as reliable.

3.1 Descriptive statistics
In the data material a return series was given using logarithmic returns in order to obtain stationary series. This is done according to formula . Table 1 presents the descriptive statistics of our time series. The average returns for both indices are slightly positive with 0,000204 average for Nasdaq Composite and 0,000285 for Nasdaq 100. The standard deviations are rather similar but higher for the Nasdaq 100 with 0,014765 and 0,014547 for the Nasdaq Composite. The reason for this may be that Nasdaq 100 consists of fewer securities than the composite index. One interesting observation is that the both time series has a negative skewness but still a positive mean return. This may be caused by more negative outliners but still a large assembly of observations slightly positive. Both indices show signs of high excess kurtosis as three is considered the kurtosis of the normal distribution. This can also be seen in the figures 1 and 2 in the appendix. High excess kurtosis and skewness in the data gives high Jarque-Bera statistics which indicates the data material does not follow a normal distribution.

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Table 1 - Descriptives

Average

Min.

Max.

Std. Dev. Skewness Kurtosis

Jarque-Bera

Nasdaq Comp 0,000204 -0,09586 0,111592 0,014547 -0,21912 9,825572 4132,276 Nasdaq 100 0,000285 -0,11115 0,118491 0,014765 -0,12961 10,30668 4721,832

The graphs 2 and 3 show volatility for both time series included in the study. The left side displays volatility and the bottom displays the time period. The time plots of the logarithmic returns shows the typical volatility clustering present in financial time series. This can particularly be seen during the financial crisis in 2008. The high magnitude of volatility also continues after the financial crisis. In 2011 both Indices have experienced a time-period with higher volatility. Reason for this could be the European crisis involving the over debt countries of Greece and Italy. Times like crises often show noisy trading behavior with overreaction and under reactions due to different interpretation of the news.
Figure 1 - Return plots

Comparing the different indices you can see a higher amount of volatility in the Nasdaq 100 index. The Nasdaq 100 index has also larger extreme values which is reasonable because of the larger weights on different stocks.

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4 Results
Table 2 Nasdaq 100 shows the estimated parameters for the GARCH, EGARCH and GJR-GARCH tests. The first three columns represent the GARCH (1,1) results. The empirical results satisfies the restrictions because all the parameters are positive and 0,00000241 + 0,068771 + 0,917922 < 1 for the model where normal distribution was used and 0,00000189 + 0,066270 + 0,925081 < 1 for the model with student-t distribution. All the parameters are also heavily significant at 1 percent level. The γ variable is the largest on both tests with 0,917922 in the model using normal distribution and 0,925081 in the model using student-t distribution. This indicates that previous volatilities have a big impact on today’s volatility.
Table 2 - Nasdaq 100 Results

GJRGarch(1.1) Variables
C β γ

Egarch(1.1) Variables norm.distr.
-0,254917*** 0,100277*** -0,107798*** 0,979605***

Garch(1.1) t-distr.
0,090309*** -0,10927*** 0,986663***

norm.distr. t-distr.
2,41E-06*** 0,068771*** 0,917922***

Variables norm.distr.
2,79E-06*** -0,009879 0,123983*** 0,929633*** β w γ

t-distr.
2.06E-06*** -0,013143 0.123942*** 0.936391***

1,89E-06*** C 0,066270*** β 0,925081*** γ b

-0,188402*** C

***Significant at 1% level

Results for the EGARCH(1,1) show high significance levels for all the parameters. The γ is negative in both tests which imply that positive shocks have a bigger impact on future conditional volatility than same size negative shocks. This result is not coherent with the leverage effect theory discussed in the introduction. The test were student-t distribution was used also displays a negative third coefficient. The three last columns in the table displays the results of the GJR-GARCH (1,1). All the coefficients except the β are significant at a 1 percent level. Restrictions using this model was that all the coefficients has to be positive except w if the total sum of β and w is bigger or equal to 1. The estimated value for β is -0,009879 for the model using

normal distribution and -0,013143 for the model using student-t distributions. This is not coherent with the model presented in the methodology chapter. Also the sum of the coefficients is > 1 which is against the restrictions.

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Table 3 reports the results of the test made with the Nasdaq Composite data set. The results are similar to those shown in tests with Nasdaq 100. The first three columns present the results of the GARCH(1,1). All the coefficients are significant at a 1 percent level and positive. The sum of the coefficients is < 1 in both tests indicating that the result is coherent with the restrictions. The beta coefficients are a little bit bigger compared to the GARCH(1,1) with Nasdaq 100 which indicate that previous shocks have a little more impact.
Table 3 - Nasdaq Composite Results

Garch(1.1) Variables
C β γ

Egarch(1.1) norm.distr. t-distr.
2,01E-06***

GJR-Garch(1.1) Variables norm.distr.
C β w γ 2,19E-06*** -0,012533 0,128798*** 0,932082***

Variables norm.distr. t-distr.
-0,231609*** -0,173803*** 0,101641*** 0,090324***

t-distr.
1,58E-06*** -0,014818 0,126478*** 0,938152***

1,53E-06*** C

0,072625*** 0,071994*** β 0,915664*** 0,920880*** γ b ***Significant at 1% level

-0,104926*** -0,109102*** 0,982612*** 0,988513***

Results for the EGARCH(1,1) show high significance levels for all the parameters. The γ is negative in both tests which imply that positive shocks have a bigger impact on future conditional volatility than same size negative shocks. This result is not coherent with the leverage effect theory discussed in the introduction. The last column displays the results of the GJR-GARCH(1,1) tests. All the coefficients except β are significant at 1 percent level. β is negative in both tests and this is not according to the theory. The total sum of the coefficients is > 1 which is a violation of the restrictions.

4.1 Evaluation of the models
The purpose of this term paper was to determine which of the GJR and EGARCH best captures the properties of the data. We will also compare the different models with the GARCH model. This chapter presents the main findings of the empirical study made in this term paper. Since the GJR-GARCH convergence could not be reached the results from this model are not trustworthy. Therefore we will only compare the findings of the EGARCH and GARCH tests. Table 4 - Nasdaq 100 and Table 5 - Nasdaq Composite presents the AIC, BIC and Loglikelihood values for all the tests made in the empirical study. Both GARCH tests seem 10

to be capturing more information when student-t distribution was used. This can be confirmed by looking at for example the AIC, BIC and log-likelihood values presented in table 4. The AIC values decrease from -5.947749 to -5.973855 and the BIC from 5.937072 to -5.960508 when student-t distribution was used. Reason for this could be the excess skewness reported in the descriptive statistics chapter for both series. Same phenomenon is present in the EGARCH results where the AIC value decreases from 5.969276 to -5.994788 and the BIC value from -5.955928 to -5.978772 when the student-t distribution was used. Comparing the measurement values between the GARCH and the EGARCH you see that the EGARCH(1,1) has the lowest measurement values for both time series and could therefore be considered the best fit to the data set in this empirical study.
Table 4 - Nasdaq 100 Information Criterias

Garch(1,1) Variables AIC BIC Log-likelihood norm.distr. t-distr.
-5.947749 -5.937072 6308.614

Egarch (1,1) norm.distr. t-distr.

GJR-Garch (1.1) norm.distr. t-distr.
-5.995513 -5.979496 6361.243

-5.973855 -5.969276 -5.960508 -5.955928 6337.287 6332.432

-5.994788 -5.975679 -5.978772 -5.962332 6360.476 6339.220

Table 5 - Nasdaq Composite Information Criterias

Garch(1,1) Variables AIC BIC norm.distr. t-distr.
-6.042377 -6.031700

Egarch (1,1) norm.distr. t-distr.

GJR-Garch (1.1) norm.distr. t-distr.
-6.088194 -6.072178 6459.486

-6.063623 -6.062775 -6.050276 -6.049428 6432.441 6431.541

-6.085254 -6.073303 -6.069238 -6.059956 6456.370 6442.702

Log-likelihood 6408.920

These findings support the findings of Shamiri and Hassan (2005) that the asymmetric GARCH models, such as the EGARCH, can better estimate the time series. Also Peters (2001) report findings of the asymmetric GARCH models outperforming the symmetric GARCH model. Both researchers also find better results when they used student-t distributions and skewed-t distribution instead of the Gaussian normal distribution. This could indicate that non-normal distributions can improve the results as long as the time

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series are not entirely normally distributed. Also Chen and Kuan (2002) finds evidence of the usefulness of the EGARCH model when modeling financial time series but they also found that when using a EGARCH (1,1) it tends to underestimate the leverage effect and overestimate the magnitude effect. The “magnitude effect” being the earlier shocks in the model.

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5 Conclusion
The purpose of this term paper was to determine which of the GJR and EGARCH best captures the properties of the data. We will also compare the different models with the GARCH model. For the empirical study daily stock returns from the Nasdaq 100 and Nasdaq Composite was used. The empirical study shows that asymmetric GARCH models tend to capture more of the information in the data set than the symmetric GARCH model. Also the use of non-normal distributions can improve the results as long as the time series are not entirely normally distributed.

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Appendix
Figure 2 - Nasdaq Composite Distribution

Figure 3 - Nasdaq 100 Distribution

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References
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Rabemananjara R. and Zakoian J. M. (1993). Treshold Arch Models and Asymmetries in Volatility. Journal of Applied Econometrics. Vol. 8, No.1, 31–49. Shamiri, A. and Hassan, A. (2005). Modeling and Forecasting Volatility of the Malaysian and the Singaporean stock indices using Asymmetric GARCH models and Non-normal Densities. EconWPA, No. 0509015. Taylor W. J. (2005). Generating Volatility Forecasts from Value at Risk Estimates. Management Science, Vol. 51, No. 5, 712–725. Wu G. (2001). The Determinants of Asymmetric Volatility. The review of Financial Studies. Vol. 14, No. 3, 837–859.

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