Modeling and forecasting volatility of Shanghai Stock Exchange with GARCH
Supervisor: Lars Forsberg
Master Thesis in Statistics May 2011
Modeling and forecasting volatility of Shanghai Stock Exchange with GARCH
family models
Author: Yang Han Supervisor: Lars Forsberg
Master Thesis in Statistics May 2011 Department of Statistics
Uppsala University Sweden
Modeling and forecasting volatility of
Shanghai Stock Exchange with GARCH
Modeling and forecasting volatility of Shanghai Stock Exchange with GARCH family models
June 4, 2011
Abstract
This paper discusses the performance of modeling and forecasting volatility of daily stock returns of A-shares in Shanghai Stock Exchange. The volatility is mod- eled by GARCH family models which are GARCH, EGARCH and GJR-GARCH models with three distributions, namely Gaussian distribution, student-t distribu- tion and generalized error distribution (GED). In order to determine the perfor- mance of forecasting volatility, we compare the models by using the Root Mean Squared Error (RMSE). The results show that the EGARCH models work so well in most of daily stock returns and the symmetric GARCH models are better than asymmetric GARCH models in this paper.
Key words: Volatility forecasting; GARCH family models; RMSE.
Contents
1 Introduction 4
1.1 Background . . . 5
1.2 Shanghai Stock Exchange . . . 6
1.3 Literature Reviews . . . 7
1.4 Aim of the paper . . . 8
2 Methodology 9 2.1 ARCH model . . . 9
2.2 Generalized-ARCH (GARCH) model. . . 10
2.3 Exponential GARCH (EGARCH) model . . . 10
2.4 GJR-GARCH model. . . 11
2.5 Distribution of the error term . . . 11
2.5.1 Gaussian (Normal) distribution. . . 12
2.5.2 Student-t distribution . . . 12
2.5.3 Generalized error distribution (GED) . . . 12
2.6 Root Mean Square Error (RMSE) . . . 13
3 Data 13 3.1 Data description . . . 13
3.2 Data analysis. . . 14
4 Results 15 4.1 Summary the results of GARCH family models. . . 15
4.2 Out-of-sample Forecasts . . . 18
5 Conclusion 21
6 Reference 23
7 Appendix 26
1 Introduction
The publication of Robert Engle (1982) introduced the Autoregressive Conditional Heteroscedasticity (ARCH) model into the world. It is used to characterize and model observed time series in econometrics. In application, there have been some other methods on modeling and forecasting financial volatility, such as the Generalized ARCH (GARCH) model which has been proposed by Bollerslev (1986), and the Ex- ponential GARCH (EGARCH) model that introduced by Nelson (1991), and the GJR- GARCH model from Glosten, Jagannathan and Runkle (1993) and so on. These mod- els treat heteroskedasticity as a variance to be modeled. And the purpose of these models is to provide a tool to measure the financial volatility that can be used in finan- cial decisions which are concerned by risk analysis, portfolio selection and derivative pricing.
Since these methods being published, there are many studies model and forecast fi- nancial volatility on stock markets, especially primarily relies on the developed mar- kets, such as US and Europe. Therefore, in this paper, we focus on the Chinese Stock Exchange; specifically we pay attention on the A-shares of Shanghai Stock Exchange.
We will try to use the method to measure the volatility of the Chinese stock market and examine the predictability of the Chinese stock market returns.
In this paper, we will use the GARCH, EGARGH and GJR-GARCH models to model and forecast the volatility by using the daily stock market returns of the Shanghai Stock Exchange. Then we will compare these models to find which one provides the best forecast. It should be stressed that the GARCH family models do not fully cap- ture for the thick tails property of high frequency time series. In this case, we can try to apply non-normal distribution function for the error term to solve this problem. we
will also check the different distributions (Normal distribution, Student-t distribution and Generalized error distribution) of the error term in order to get the better estima- tion for the volatility of the stock market returns.
The paper is organized as follow. In the first section, we briefly introduce the back- ground of the financial market and the volatility, and also some information of the Chinese Stock Market. The second section, we will describe the methods which are used in this paper. Then the third section, we will carry out the data and analysis re- sults of these models. Finally, we will provide the summary of the paper.
1.1 Background
Today, the financial market, especially financial market volatility, is concerned by more and more people. In finance, volatility is a measure for variation of price of a finan- cial instrument over time (Lin C, 1996). It is common for discussions to talk about the volatility of a security’s price, even while it is the returns’ volatility that is being mea- sured. It is used to quantify the risk of the financial instrument over the specified time period. Because of the financial market volatility is a very important factor of setting up strategies related to portfolio optimization, option pricing, risk management and market regulation in stock markets (Poon and Granger, 2003). And also the financial volatility can be used to measure the risk of financial market and stock market stabil- ity by both investors and econometricians. As a result, a lot of researchers are working on modeling and forecasting the financial volatility.
Econometricians are being asked to forecast and analyze the size of the errors of the model. Another way of saying is all the questions are about volatility. They are consid-
ered with modeling volatility in asset returns. In fact, observations of financial asset returns observed at daily are not independent, while the observations in these series are uncorrelated or nearly uncorrelated (Timo T, 2006). Models of ARCH become the most popular way to do with this dependence.
1.2 Shanghai Stock Exchange
There are two stock exchanges operating independently in the People’s Republic of China, one is Shanghai Stock Exchange that is based in the city of Shanghai, China and the other is Shenzhen Stock Exchange which is based in the city of Shenzhen, China. Both of them contain the A-shares and B-shares. We will introduce the dif- ference between them later. The Shanghai Stock Exchange established on December 19th, 1990 and the Shenzhen Stock Exchange established on July 2nd, 1991.
Shanghai Stock Exchange is the world’s 5th largest stock market. In December 2009, there are 870 companies listed in Shanghai Stock Market, and the total market capital- ization had reached around 18465.5 billion Chinese Yuan. The A-shares in Shanghai Stock Market employ Chinese Yuan as the currency and the B-shares allows the US dollar to trade. Like many developing countries, in order to sustain the domestic control of local companies, the Chinese government still does not allow the Shanghai Stock Exchange entirely open to foreign investors.
The paper studies companies which are listed in A-shares of Shanghai Stock Exchange, so we do not care about other stock exchanges only A-shares of Shanghai Stock Ex- change. We can get visual impression of A-shares of Shanghai Stock Exchange by Figure 1.
Figure 1: The main index of A-shares of Shanghai Stock Exchange
1.3 Literature Reviews
Considering the time-varying behavior of volatility, ARCH (Auto Regressive Condi- tional Heteroscedasticity) models developed by Engle (1982), and then it was devel- oped into the GARCH (Generalized Auto Regressive Conditional Heteroscedasticity) models by Bollerslev (1986). A number of extensions of the basic GARCH model that are especially suited to estimating the conditional volatility of financial time series have been developed. Hence, these models became the fundamental of the financial volatility models (Alexander and Lazar, 2006). The analysis of ARCH and GARCH models and their many extensions provides a statistical stage on which many theories of asset pricing and portfolio analysis can be exhibited and tested (Engle, 2001). An interesting feature of asset prices is that bad news seems to have a more pronounced
effect on volatility than goods news. This interesting feature shows that there is an asymmetric volatility phenomenon in the relation. The asymmetric volatility phe- nomenon is a market dynamic that shows that there are higher market volatility lev- els in market downswings than in market upswings (Patrick D, Stewart M and Chris S, 2006). Volatility models which allow for such an asymmetric property not only improve the ability to describe return dynamics (Engle and Ng, 1993) but also pro- vide more accurate option prices (Heston and Nandi, 2000 and Christoffersen and Jacobs,2004). The main reason which causes this phenomenon is leverage effect. The tendency for volatility to rise more following a large price fall then following a price rise of the same magnitude is called the leverage effect (Brooks, 2002). For many stocks, there is a strong negative correlation between the current return and the fu- ture volatility (De Gooijer and Hyndman, 2006). Among these models that take into account asymmetric and leverage effect, we have Exponential-GARCH model (Nel- son,1991) and GJR-GARCH model (Glosten, Jagannathan and Runkle, 1993).
1.4 Aim of the paper
The aim of this paper is that we will use the GARCH, EGARCH and GJR-GARCH models to model and forecast the volatility of A-shares of Shanghai Stock Exchange.
And we will check whether they suit for the stock of companies of Shanghai Stock Market. Then we will compare the performance among these models and we also in- troduce some different density functions for the error term.
2 Methodology
In order to let you easily know the ARCH families, we start with ARCH model and then extend it into GARCH, EGARCH and GJR-GARCH models. After that, we will introduce the distributions of the error term which are used in this paper.
2.1 ARCH model
The autoregressive conditional heteroskedasticity (ARCH) model is the first model of conditional heteroskedasticity (Engle, 1982). Let the εtdenote the error term which has a mean and variance conditionally on the information set τt−1(the σ− f ield generated by εt−j, j ≥ 1). Then we can get the following properties of εt. The first thing is E(εt|τt−1) = 0. And second, conditional variance σt2 = E(ε2t|τt−1) is a positive- valued parametric function of τt−1. Then we can get
εt=yt−µt(yt),
where yt is observed value, and µt(yt) = E(yt|τt−1) is the conditional mean of yt. Then we can get the equation of εt, that is
εt=σtzt,
where zt is a sequence of independent identically distributed random variables with zero mean and unit variance (i.e ztiid∼N(0, 1)) and where the series σ2modeled by
σt2=α0+α1ε2t−1+...αqε2t−q=α0+
∑
q i=1αiε2t−i,
where α0>0 and αi ≥0, i>0, α0and αi are coefficients. This is the ARCH model.
2.2 Generalized-ARCH (GARCH) model
In applications, the ARCH model has been replaced by the generalized ARCH (GARCH) model (Bollerslev, 1986). In this case, the GARCH (p, q) model is given by
σt2 =c+α1ε2t−1+...+αqε2t−q+β1σt2−1+...+βpσt2−p= c+
∑
q i=1αiε2t−i+
∑
p i=1βiσt2−i,
where p is the order of the GARCH terms σ2and q is the order of the ARCH terms ε2. The most popular GARCH model is GARCH(1,1) which is q=p=1. A sufficient condition for the conditional variance to be positive with probability one is c > 0, αi ≥0 (Timo T, 2006) and βi ≥0, i>0, and c, αi, βi are coefficients. The necessary and sufficient conditions for positivity of the conditional variance in higher-order GARCH models are more complicated than the sufficient conditions just mentioned (Nelson and Cao, 1992). We use the GARCH model as GARCH(1, 1) in this paper. Then the model which use in this model is given by
σt2 =c+α1ε2t−1+β1σt2−1
It is a one-period previous estimate for the variance calculated on any past informa- tion thought relevant.
Just like we mentioned it before, there is some limitations in GARCH(1, 1). The non- negative conditions maybe limit this estimate method because the coefficients of mode could be negative and the basic GARCH model is symmetric and does not capture the asymmetry; it means that cannot account for leverage effect. For these reasons, we will introduce the EGARCH model in next section.
2.3 Exponential GARCH (EGARCH) model
In the EGARCH model, the nature logarithm of the conditional variance is allowed to vary over time as a function of the lagged error term (Nelson, 1991). The EGARCH(q,
p) is given by
log σt2 =c+
∑
p k=1g(Zt−k) +
∑
q k=1βklog σt2−k,
where, g(Zt) = γZt+α(|Zt| −E(|Zt|)), σi2 is the conditional variance, c, α, β and γ are coefficients. And we can also get that: Zt−k = εσt−k
t−k. And it is written to log σt2=c+
∑
p k=1βklog σt2−k+
∑
q k=1αk(
εt−k σt−k
−E(
εt−k σt−k
)) +
∑
q k=1γkεt−k σt−k
The Ztmay be a standard normal variable or come from student-t distribution or gen- eralized error distribution. In this paper, we also use the model EGARCH (1, 1). So the model could be written as
log σt2= c+β1log σt2−1+α1(
εt−1 σt−1
−E(
εt−1 σt−1
)) +γ1εt−1 σt−1
2.4 GJR-GARCH model
The GJR-GARCH model also models asymmetry in the ARCH process (Glosten, Ja- gannathan and Runkle, 1993). We have εt = σtzt, where zt is a sequence of indepen- dent identically distributed random variables. The model is given by
σt2 =c+βσt2−1+αε2t−1+φε2t−1It−1,
where It−1 =0 if εt−1≥0, and It−1=1 if εt−1<0.
2.5 Distribution of the error term
The distribution of the error term plays an important role in estimating the GARCH family models. When we use the GARCH family models to model and forecast the fi- nancial volatility, the most common application is assumed that the distribution of the error term is conditionally normally distributed. In this situation, the GARCH family models cannot fully explain the cluster of volatility phenomenon that appears in the
Chinese data. For modeling such fat-tailed distributions, I take student-t distribution or generalized error distribution to fix it.
2.5.1 Gaussian (Normal) distribution
The probability density function of the Gaussian distribution is given by
f(zt) = √ 1 2πσ2e−
(zt−µ)2 2σ2 ,
where parameter µ is the mean and σ2 is the variance. When the µ = 1 and σt2 = 0, this distribution is called standard normal distribution.
2.5.2 Student-t distribution
The probability density function of the student-t distribution is given by
f(zt) = Γ(v+21)
p(v−2)πΓ(v2)(1+ z
2t
v−2)−12(v+1),
where v is the number of degrees of freedom, 2< v ≤ ∞, and Γ is the Gamma func- tion. When v→∞ the student-t distribution approaches to standard normal distribu- tion, so that the lower v the fatter the tails.
2.5.3 Generalized error distribution (GED)
The probability density function of the generalized error distribution is given by
f(zt) = v exp(−12 zt
λv
) λ2(1+1/
v)Γ(1/v) ,
where 2 < v ≤ ∞, λ = (2
−2 v Γ(1/
v)
Γ(3/ v)
) Note that, when v=2 and constant λ = 1, the generalized error distribution is also standard normal distribution.
2.6 Root Mean Square Error (RMSE)
The root mean squared error is used for measuring the difference between the values forecasted by an estimator or a model and the true values from the series which gener- ate the model. The criterion is the smaller value of the RMSE, the better the forecasting ability of the model. And in this paper, I use this method to check the forecasting per- formance of GARCH families. It is defined like this
RMSE= v u u u t
∑
n i=1[r2i −
∧
σi2]
2
n ,
where n is number of out-of-sample observations.
3 Data
3.1 Data description
The data of this paper is composed by the daily closing prices of 10 different stocks which are in the A-shares of Shanghai Stock Exchange. I chose them because of these ten stocks are from different forums of Chinese stock market and they play a very important role in their own field. I will denote them dataset 1 to dataset 10. The length of these ten datasets is different because of releasing of these shares at different times. Basically, I will use the daily closing price of the stocks in the last ten years, however, some of them do not exist for ten years, and I will use it from the issue date to the present. The data was taken from the website of Shanghai Stock Exchange and National Bureau of Statistics of China. Then I will give the details of these ten stocks in Table 3. It includes the name of companies, the forums of the stocks and the length of datasets.
3.2 Data analysis
In this paper, I will use the returns of stocks to model and forecast the financial volatil- ity. Primarily, I convert the closing daily prices of stocks to returns as follow:
ret =ln pt−ln pt−1
Where, retis return for datasets and ptis the closing daily price at time t.
We will summary these returns and get features of these datasets, such as: mean re- turns, standard deviation, minimum returns and maximum returns. They will be shown in Table 1. Then we can get some informations from these characteristic di- rectly. From this table, we can find that mean equity market returns are positive of all datasets except dataset 10 which mean return is negative. And we all know that the higher standard deviation of return, the greater risk of the stock. The highest daily return of 0.421100 and the lowest daily return of -0.557000 are both reported for the dataset 6.
Then we inspected the returns series to examine how they evolved during the same period. We present graph for return series of datasets of Shanghai Stock Exchange in Figure 2. It gives the visual inspection to show that volatility changes over time and it tends to cluster with periods with low volatility and periods with high volatility.
In this part, we still have to test whether the error terms of the returns follow the normal distribution. Earlier, we get the method to calculate the error term, that is:
εt=yt−µt(yt)
And then we will provide two methods to test the hypothesis. Firstly, we present QQ- plot for the error term in Figure 3. And with this intuitive approach, we find that the
error term maybe not a normal distribution.
Secondly we will do the Jarque-Bera test. Because of this test is a goodness of fit measure of departure from normality, based on sample kurtosis and skewness. The results of it will be shown in Table 2. And we can find the values of X-squared of all error terms are too large and P-values are too small. So we reject the null hypothesis, and all of the error terms are not normal distribution.
4 Results
We used the GARCH, EGARCH, GJR-GARCH models which include the normal dis- tribution, student-t distribution and generalized error distribution of the error term to model and forecast the financial volatility with the daily stock returns which come from the Shanghai Stock Exchange. We will summary the results of GARCH models.
Then I compare them to find which model forecasting the volatility better.
4.1 Summary the results of GARCH family models
In this section, when we used the daily stock returns to fit the GARCH family models, we can get the results like this: the coefficients, the P-values based on t-test, AIC and log likelihood values of the models. I will summary these results of models for all datasets and show them in Table 4 to 13.
For re dataset1, in Table 4, we can see that the coefficients γ in EGARCH models and φ in GJR-GARCH models are not significant. That means the EGARCH and GJR- GARCH models are not asymmetric models. And the EGARCH model with student-t distribution has the smallest value of AIC, that means it will be the best in-sample
estimation among all the models.
For re dataset2, in Table 5, the coefficients γ in EGARCH models and φ in GJR-GARCH models are not significant either. For this reason, EGARCH and GJR-GARCH models are not asymmetric models too. Specifically the coefficient β of EGARCH model with student-t distribution is not significant and the constant of it is abnormal compared to the EGARCH models with Gaussian distribution and GED, it also has the largest value of AIC among all the models of re dataset2. The EGARCH model with student-t distribution is the worst model of in-sample estimation of this dataset. The smallest value of AIC are GJR-GARCH model with student-t distribution, this model is the best.
For re dataset3, in Table 6, and in these models, only the coefficients γ in EGARCH models with student-t distribution and GED are not significant. The EGARCH model with Gaussian distribution and GJR-GARCH models are asymmetric models. The EGARCH model with student-t distribution has the smallest value of AIC, it has the best performance of in-sample estimations.
For re dataset4, in Table 7, we find the coefficients γ in EGARCH models and φ in GJR-GARCH models with Gaussian distribution are not significant, these models are not asymmetry. We also find the EGARCH model with the student-t distribution has the smallest value of AIC in re dataset4.
For re dataset5, in Table 8, the coefficients γ in EGARCH models with student-t dis- tribution, GED and φ in GJR-GARCH models with student-t distribution and GED are not significan, they are not asymmetric models. We also find the EGARCH model with the GED has minimum value of AIC, it is the best model of in-sample estimation
of this dataset.
For re dataset6, in Table 9, the coefficients γ in EGARCH models and φ in GJR-GARCH models with Gaussian distribution and GED are not significant. And the EGARCH model with the student-t distribution has minimum value of AIC.
For re dataset7, in Table 10, only the coefficient γ in EGARCH models with student- t distribution is not significant, it is the only symmetric model of all EGARCH and GJR-GARCH models and the constant of EARCH model with student-t distribution is abnormal compared to the EGARCH models with Gaussian distribution and GED, and the coefficient β of EGARCH model with student-t distribution is not significant,it also has the largest value of AIC. It it the worst model of in-sample estimations. In this dataset, we find the coefficents c in GARCH model with student-t distribution and GED are not significant. It can not reject the hypothesis, so the constant would be zero. We have mentioned it before, the constant of GARCH model can not be zero, so the GARCH models with student-t distribution and GED are not suitable for forecast- ing volatility of this dataset. We also find the EGARCH model with the GED is the best model of in-sample estimation among all the models.
For re dataset8, in Table 11, we can see that the coefficients γ in EGARCH models and φin GJR-GARCH models with student-t distribution and GED are not significant. We also find the EGARCH model with the student-t distribution has the minimum value of AIC.
For re dataset9, in Table 12, the coefficients γ in EGARCH models with GED and φ in GJR-GARCH models with student-t distribution and GED are not significant. The constant of EGARCH model with student-t distribution is abnormal compared to the
EGARCH models with Gaussian distribution and GED. The coefficient β of EGARCH model with student-t distribution is not significant, it does not suit to forecast the volatility of returns in re dataset9. The constant of GARCH models are all not sig- nificant, it would be useless in forecasting volatility of this dataset. We also find the GJR-GARCH model with the GED has the smallest value of AIC, it has the best per- formance of forecasting volatility of returns in this dataset.
For re dataset10, in Table 13, we can see that the coefficients γ in EGARCH models with student-t distribution and GED, also φ in GJR-GARCH models with student-t distribution and GED all are not significant. Specifically the coefficient β of EGARCH model with student-t distribution is not significant either, it would not suit for fore- casting volatility of this dataset. The constant of EGARCH model with student-t dis- tribution is abnormal compared to the EGARCH models with Gaussian distribution and GED. The constant of GARCH model with student-t distribution are not signifi- cant, it would be the worst model for forecasting. We also find the EGARCH model with the GED has the smallest value of AIC.
4.2 Out-of-sample Forecasts
There are many ways to compare the merits of models, such like AIC, log likelihood value and so on. However, shown by the results, the AICs of the models do not pro- vide a clearly decision. In this paper, in order to contrast the performance of these models in forecasting the stock volatility, we will determine it with calculating the out-of-sample forecasts. The procedure of the out-of-sample forecast comparisons are shown as follow. we will take the dataset 1 as an example to explain this procedure, and all the datasets are done like this. In dataset 1, there are 1418 observations, and we
reserve the last 100 observations. For evaluation we take the next 1180 observations, then we abandon the first 138 observations, because, we want to use the observations throughout the year to be the window. Firstly, we use the window which has the fix length and contains the observations from 1 to 1180 to estimate the models. Then we can get the forecasting value of the stock volatility. Secondly, we repeat the first step except the window changes the observations from 2 to 1181. Such a process provides 100 one-step ahead forecasts; we can use the values from this method to calculate RM- SEs of the models. Then we compare the forecasting performance by using the RMSE which is mentioned previously. The results of RMSE are also shown in Table 14.
From the values of RMSE of re dataset1, we can find that the performance of EGARCH model with Gaussian distribution, student-t distribution and GED is better than GARCH and GJR-GARCH. And the values of RMSE of Gaussian distribution in GARCH and GJR-GARCH models show it better than the other two. And the EARCH model with GED has the smallest value of RMSE, it indicates that EARCH model with GED gives the best forecast of volatility among all the models.
The values of RMSE of re dataset2 show that the models use GED are better than the models using the other two distributions in forecasting volatility. The smallest value of RMSE in this dataset also belongs to the EARCH model with GED, it means that EARCH model with GED has the best performance.
For re dataset3, we can get such results from the values of RMSE. The Gaussian dis- tribution is the best one in forecasting conditional variance of stock returns, and per- formance of the models using student-t distribution is the worst. The GARCH model with Gaussian distribution has the smallest value of RMSE, it gives the best forecast.
The results of re dataset4 can be drawn from these values of RMSE. The EGARCH models are better than the other two. And the best forecast of volatility is the models with Gaussian distribution. The smallest value of RMSE is given by EGARCH with Gaussian distribution. That means it is the best one.
In re dataset5, the values of RMSE of Gaussian distribution in GARCH and GJR- GARCH models are better than the other two distributions. And the EARCH model with student-t distribution has the smallest value of RMSE. The main conclusion of dataset 5 is that the returns can be usefully forecast by EGARCH model using student- t distribution.
The result of re dataset6 is given out by these values of RMSE. The Gaussian distribu- tion is the best one in forecasting conditional variance of stock returns. The EGARCH model with Gaussian distribution has the smallest value of RMSE, it does so well in forecasting financial volatility. However, the GJR-GARCH model with GED is the worst performing model.
The conclusions of re dataset7 and re dataset8are very similar; we know it from the values of RMSE of these two datasets. In both of datasets, the models with student-t distribution are the best among all the models, but the models with Gaussian distri- bution give the worst forecast of stock returns. The EGARCH model with student-t distribution is the owner of the smallest value of RMSE, that means it gives the most accurate forecast.
The values of RMSE of models with student-t distribution are smaller than the mod- els with GED and Gaussian distribution. That means student-t distribution is better than the other two distributions in this dataset. And the GJR-GARCH with student-t
distribution has the smallest value of RMSE, it is the best one for forecasting financial volatility. These results are revealed by the values of RMSE of re dataset9.
In re dataset10, the values of RMSE of student-t distribution in GARCH, EGARCH and GJR-GARCH models show it better than the other two. And also, the EGARCH model with student-t distribution has the smallest value of RMSE. This result implies that it is the most suitable model for forecasting conditional variance of returns.
According to the summary of all datasets, we get the best model for every dataset which provide the forecast of stock market volatility of stock returns. And we can also use the figure to show the outcome of these models. In Figure 4, it shows the compar- ison between residual and actual value.
5 Conclusion
This paper has considered the models of stock returns in Shanghai Stock Exchange during the last decade. And ten stocks’ return data of Shanghai Stock Exchange have been examined to compare the performance of forecasting among GARCH, EGARCH and GJR-GARCH models under three distributional assumptions: Gaussian distri- bution, student-t distribution and generalized error distribution. The results of this paper show that EGARCH model is very useful for forecasting purposes in most of datasets. But the simple GARCH models and GJR-GARCH models are unsatisfactory in this task. Because there is only one GJR-GARCH model has the best performance in modeling stock returns in dataset 6, and also only one GARCH model with Gaussian distribution has the smallest value of RMSE. In these ten models which have the best forecasting performance, the coefficients γ in EGARCH models and φ in GJR-GARCH
model are all not significant. That means they are all symmetric models. These results show that the symmetric GARCH models may be still useful for forecasting purposes.
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7 Appendix
Table 1: Summary the returns of datasets for GARCH family models
Mean return Standard deviation Minimum return Maximum return
re dataset1 0.000912 0.036129 -0.120200 0.330500
re dataset2 0.000381 0.032039 -0.119100 0.105900
re dataset3 0.000872 0.027487 -0.147300 0.145000
re dataset4 0.000249 0.037505 -0.137900 0.270800
re dataset5 0.001718 0.032397 -0.120800 0.215000
re dataset6 0.002539 0.060741 -0.557000 0.421100
re dataset7 0.000309 0.030101 -0.097340 0.091460
re dataset8 0.001243 0.031988 -0.105900 0.380200
re dataset9 0.000159 0.032746 -0.107900 0.098080
re dataset10 -0.001176 0.031292 -0.111700 0.100800
Table 2: Results of Jarque-Bera test
Error1 Error2 Error3 Error4 Error5 X-squared 5669.086 390.7862 1178.095 1095.239 690.8457
Df 2 2 2 2 2
P-value p¡0.001 p¡0.001 p¡0.001 p¡0.001 p¡0.001 Error6 Error7 Error8 Error9 Error10 X-squared 10742.51 20.6189 47844.18 43.7327 83.8792
Df 2 2 2 2 2
P-value p¡0.001 p¡0.001 p¡0.001 p¡0.001 p¡0.001
Table 3: The details of the datasets
The name of the stocks The forum of the stocks
The duration of the data
Number of ob- servations Dataset 1 LUJIAZUI Finance & Trade
zone development CO., Ltd
Real Estate Forum 2005/03/08–
2011/03/04
T=1418
Dataset 2 Chongqing Three Gorges Wa- ter Conservancy and Electric Power CO., Ltd
Electricity Forum 2001/03/07–
2011/03/04
T=2348
Dataset 3 Shanghai Jahwa United Com- pany Ltd.
Chemical Manufac- turing Forum
2001/03/15–
2010/12/03
T=2302
Dataset 4 Beijing Capital Tourism Co., Ltd
Travel and Leisure Forum
2001/03/07–
2011/03/04
T=2357
Dataset 5 Zhejiang Supor Co., Ltd Appliance Forum 2004/08/17–
2011/03/03
T=1492
Dataset 6 Wangfujing Department Store(Group) Co., Ltd
Retail Forum 2005/03/08–
2011/03/04
T=1406
Dataset 7 Industrial Bank Co., Ltd Banking Forum 2007/02/05–
2011/03/04
T=978
Dataset 8 Inner Mongolia Yili Industrial Group Co., Ltd
Food Forum 2005/03/08–
2011/03/04
T=1403
Dataset 9 Ping An Insurance (Group) Company of China, Ltd
Finance Forum 2007/03/01–
2011/03/04
T=917
Dataset 10 Shenhua Group Co., Ltd Energy Forum 2007/10/09–
2011/03/04
T=827
Table 4: The results of dataset 1
GARCH(1,1) EGARCH(1,1) GJR-GARCH
Gaussian Student-t GED Gaussian Student-t GED Gaussian Student-t GED
c 1.83e-05 1.63e-05 1.73e-05 -0.218897 -0.249166 -0.248314 1.79e-05 1.58e-05 1.71e-05 (0.0000) (0.0278) (0.0120) (0.0000) (0.0001) (0.0000) (0.0000) (0.0320) (0.0136)
β
0.939744 0.920490 0.929173 0.980133 0.981729 0.979374 0.92106 0.921062 0.929315 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
α
0.048139 0.071256 0.059371 0.114882 0.168040 0.145283 0.044780 0.059516 0.052968 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0002) (0.0005)
γ
0.008246 -0.003346 0.003594 (0.4617) (0.8564) (0.8482)
φ
0.005865 0.024138 0.013215 (0.6345) (0.2868) (0.5379) AIC -3.890193 -3.987369 -3.970103 -3.894892 -3.989460 -3.971851 -3.888897 -3.986782 -3.968987 Log-
likelihood
2760.202 2830.051 2817.818 2764.531 2832.532 2820.057 2760.283 2830.635 2818.027
TXT: c, β, α, are coefficients of GARCH, EGARCH and GJR-GARCH models, and γ is the coefficient only for the EGARCH model, the φ is the coefficient for GJR-GARCH model. AIC stands for Akaike information criterion for all
models. The log-likelihood is the value which stands for the maximum log-likelihood value for each model.
Table 5: The results of dataset 2
GARCH(1,1) EGARCH(1,1) GJR-GARCH
Gaussian Student-t GED Gaussian Student-t GED Gaussian Student-t GED
c 3.21e-05 4.05e-05 3.71e-05 -0.392918 -7.010294 -0.429430 3.27e-05 3.96e-05 3.58e-05 (0.0000) (0.0001) (0.0001) (0.0000) (0.0000) (0.0000) (0.0000) (0.0001) (0.0001)
β
0.868134 0.840533 0.851737 0.966746 0.042532 0.963288 0.865960 0.841629 0.854259 (0.0000) (0.0000) (0.0000) (0.0000) (0.5582) (0.0000) (0.0000) (0.0000) (0.0000)
α
0.103714 0.130211 0.115199 0.212575 0.380029 0.227345 0.090830 0.105178 0.095390 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
γ
-0.016016 -0.006834 -0.017857 (0.1048) (0.7929) (0.2609)
φ
0.018735 0.033391 0.022855 (0.2233) (0.2347) (0.3614) AIC -4.198591 -4.254423 -4.251244 -4.202514 -4.141375 -4.253123 -4.198168 -4.254148 -4.250748 Log-
likelihood
4931.046 4997.565 4993.524 4936.650 4865.904 4997.040 4932.414 4998.243 4994.252
TXT: Just like the Table 4
Table 6: The results of dataset 3
GARCH(1,1) EGARCH(1,1) GJR-GARCH
Gaussian Student-t GED Gaussian Student-t GED Gaussian Student-t GED
c 1.37e-05 1.67e-05 1.54e-05 -0.242592 -0.288056 -0.269547 1.14e-05 1.41e-05 1.29e-05 (0.0000) (0.0023) (0.0014) (0.0000) (0.0000) (0.0000) (0.0000) (0.0034) (0.0042)
β
0.902268 0.890722 0.893248 0.983928 0.979830 0.981550 0.904190 0.891864 0.895226 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
α
0.083631 0.094990 0.090204 0.168712 0.025964 0.180707 0.068791 0.075887 0.072772 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
γ
-0.020889 -0.025242 -0.021875 (0.0036) (0.0918) (0.1071)
φ
0.037747 0.052435 0.045322 (0.0007) (0.0331) (0.0376) AIC -4.485753 -4.560345 -4.556767 -4.508420 -4.574754 -4.571267 -4.487492 -4.561753 -4.557896 Log-
likelihood
5164.859 5251.677 5247.560 5191.937 5269.254 5265.242 5167.859 5254.297 5249.860
TXT: Just like the Table 4
Table 7: The results of dataset 4
GARCH(1,1) EGARCH(1,1) GJR-GARCH
Gaussian Student-t GED Gaussian Student-t GED Gaussian Student-t GED
c 3.34E-05 1.92E-05 2.35E-05 -0.418733 -0.270152 -0.322354 3.22E-05 1.53E-05 2.29E-05 (0.0000) (0.0014) (0.0004) (0.0000) (0.0000) (0.0000) (0.0000) (0.0034) (0.0005)
β
0.88698 0.908887 0.902539 0.961201 0.980577 0.973496 0.888625 0.919615 0.902835 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
α
0.092465 0.085703 0.08414 0.213479 0.193707 0.194966 0.082076 0.049228 0.064056 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
γ
-0.006817 -0.026101 -0.017286 (0.4512) (0.0699) (0.2361)
φ
0.020396 0.060895 0.0443 (0.0898) (0.0018) (0.0314) AIC -3.883731 -3.949743 -3.954461 -3.897677 -3.958426 -3.962272 -3.883592 -3.953719 -3.956037 Log-
likelihood
4579.035 4657.797 4663.355 4596.463 4669.026 4673.557 4579.871 4663.481 4666.211
TXT: Just like the Table 4
Table 8: The results of dataset 5
GARCH(1,1) EGARCH(1,1) GJR-GARCH
Gaussian Student-t GED Gaussian Student-t GED Gaussian Student-t GED
c 4.92E-05 5.75E-05 5.62E-05 -0.443166 -0.544171 -0.530298 4.21E-05 5.80E-05 5.67E-05 (0.0000) (0.0016) (0.0031) (0.0000) (0.0000) (0.0000) (0.0000) (0.0017) (0.0030)
β
0.865700 0.834646 0.842134 0.955198 0.94668 0.946825 0.880705 0.833184 0.841032 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
α
0.089126 0.119666 0.107769 0.177285 0.236744 0.215128 0.10158 0.115769 0.106165 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0001) (0.0004)
γ
0.047659 0.030713 0.030815 (0.0004) (0.2211) (0.2339)
φ
-0.043611 0.01089 0.005145 (0.0099) (0.7867) (0.8969) AIC -4.105098 -4.181014 -4.187157 -4.121488 -4.187955 -4.194559 -4.105827 -4.179719 -4.185826 Log-
likelihood
3064.350 3121.946 3126.525 3077.569 3128.121 3133.044 3065.894 3121.980 3126.533
TXT: Just like the Table 4
Table 9: The results of dataset 6
GARCH(1,1) EGARCH(1,1) GJR-GARCH
Gaussian Student-t GED Gaussian Student-t GED Gaussian Student-t GED
c 1.30E-05 1.47E-05 1.50E-05 -0.177253 -0.139693 -0.149568 1.32E-05 1.13E-05 1.39E-05 (0.0016) (0.0137) (0.0234) (0.0000) (0.0000) (0.0000) (0.0013) (0.0284) (0.0337)
β
0.89809 0.914612 0.909406 0.99432 0.994739 0.994382 0.89878 0.927868 0.913071 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
α
0.107996 0.079213 0.086123 0.190287 0.140269 0.150510 0.114778 0.039867 0.062836 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
γ
0.004917 -0.017237 -0.012668 (0.6700) (0.2371) (0.4400)
φ
-0.017088 0.060825 0.045532 (0.3123) (0.0074) (0.1177) AIC -3.486554 -3.617521 -3.599594 -3.512821 -3.62612 -3.610816 -3.485505 -3.62075 -3.600116 Log-
likelihood
2453.304 2546.308 2533.750 2472.757 2553.349 2542.598 2453.567 2549.577 2535.081
TXT: Just like the Table 4
Table 10: The results of dataset 7
GARCH(1,1) EGARCH(1,1) GJR-GARCH
Gaussian Student-t GED Gaussian Student-t GED Gaussian Student-t GED
c 1.27E-05 1.02E-05 1.07E-05 -0.190407 -7.003536 -0.184157 1.52E-05 1.39E-05 1.39E-05 (0.0408) (0.1487) (0.1569) (0.0009) (0.0008) (0.0113) (0.0209) (0.0647) (0.0855)
β
0.945522 0.948650 0.948238 0.981524 0.01960 0.982502 0.946129 0.947809 0.947591 (0.0000) (0.0000) (0.0000) (0.0000) (0.9465) (0.0000) (0.0000) (0.0000) (0.0000)
α
0.039337 0.039747 0.039093 0.075924 0.138540 0.077046 0.015819 0.014214 0.015108 (0.0001) (0.0014) (0.0027) (0.0002) (0.0395) (0.0043) (0.0787) (0.2259) (0.2126)
γ
-0.035309 -0.085379 -0.037076 (0.0016) (0.0603) (0.0129)
φ
0.040236 0.045105 0.042844 (0.0025) (0.0116) (0.0191) AIC -4.216937 -4.226248 -4.235718 -4.226370 -4.180040 -4.243068 -4.222277 -4.231132 -4.239934 Log-
likelihood
2063.974 2069.522 2074.148 2069.582 2047.949 2078.739 2067.582 2072.908 2077.208
TXT: Just like the Table 4
Table 11: The results of dataset 8
GARCH(1,1) EGARCH(1,1) GJR-GARCH
Gaussian Student-t GED Gaussian Student-t GED Gaussian Student-t GED
c 2.31E-05 3.16E-05 3.00E-05 -0.411726 -0.340019 -0.374186 1.94E-05 3.16E-05 3.01E-05 (0.0005) (0.0063) (0.0186) (0.0000) (0.0003) (0.0009) (0.0009) (0.0054) (0.0181)
β
0.869557 0.879917 0.877092 0.965466 0.970454 0.967412 0.877917 0.882943 0.877173 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
α
0.128057 0.090839 0.099417 0.230101 0.175829 0.193700 0.159087 0.076950 0.097505 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0004) (0.0003)
γ
0.018473 0.006406 0.007228 (0.6045) (0.7372) (0.7413)
φ
-0.073478 0.021630 0.003556 (0.0010) (0.2804) (0.6906) AIC -4.122101 -4.260434 -4.242892 -4.172628 -4.266443 -4.255416 -4.125753 -4.259411 -4.241474 Log-
likelihood
2893.593 2991.564 2979.267 2930.012 2996.777 2989.047 2897.153 2991.847 2979.273
TXT: Just like the Table 4
