Modeling and predicting of different stock markets with GARCH model

Transcription

1 Modeling and predicting of different stock markets with GARCH model Author:Wei Jiang Supervisor: Lars Forsberg 2012/6/1 Master Thesis in Statistics Department of Statistics Uppsala University Sweden

2 Modeling and predicting of different stock markets with GARCH model June, 2012 Abstract This paper is mainly talking about several volatility models and its ability to predict and capture the distinctive characteristics of conditional variance about the empirical financial data. In my paper, I choose basic GARCH model and two important models of the GARCH family which are E-GARCH model and GJR-GARCH model to estimate. At the same time, in order to acquire the forecasting performance, I consider to use two different distributions on error term: normal distribution and student-t distribution. Finally, for each set of empirical stock price, I could get the best model to predict the conditional variance of the stock return based on comparing the Root Mean Square Error (RMSE) s values of different models. Here, I select several main global stock markets indexes: NASDAQ s daily index (America), Standard and Poor s 500 daily index (America), FTSE100 daily index (UK), HANG SENG daily index (Hong Kong) and NIKKEI daily index (Japan). Key words: conditional variance; GARCH and GARCH family models; error distribution; Root Mean Square Error (RMSE) 1

3 Contents 1 Introduction Background Literature review 4 2 Methodology ARCH model Generalized-ARCH model (GARCH) Exponential GARCH (EGARCH) model GJR-GARCH model Distribution of the error term Normal distribution Student-t distribution Root Mean Square Error (RMSE) 10 3 Data Data description NASDAQ Stock Market Daily Closing Price Index Standard & Poor 500 Stock Market Daily Closing Price Index FTSE100 Stock Market Daily Closing Price Index HANG SENG Stock Market Daily Closing Price Index NIKKEI Stock Market Daily Closing Price Index Data analysis NASDAQ analysis Standard & Poor 500 analysis FTSE100 analysis HANG SENG analysis NIKKEI analysis 15 4 Results Application in NASDAQ daily return Selection of ARMA (p, q) model Result of GARCH model and GARCH family model for NASDAQ Application in Standard &Poor 500 daily return Selection of ARMA (p, q) model Result of GARCH model and GARCH family model for Standard & Poor Application in FTSE100 daily return Selection of ARMA (p, q) model Result of GARCH model and GARCH family model for FTSE Application in NIKKEI daily return Selection of ARMA (p, q) model 25 2

4 4.4.2 Result of GARCH model and GARCH family model for NIKKEI Application in HANG SENG daily return Selection of ARMA (p, q) model Result of GARCH model and GARCH family model for HANG SENG ARCH-LM Test Out-of-sample Forecast 30 5 Conclusion 32 6 Reference article resource websites resource 34 7 Appendix Appendix A Appendix B 36 3

5 1. Introduction This article is an application about the GARCH and extension GARCH model. So it s mainly focused on the selection of the appropriate model to estimate the financial volatility data and using this model to forecast the conditional variance of the stock return. Full text is organized as follows. In the section 1, it includes background and the literature review. Section2 introduces the classic ARCH/GARCH model and the extension GARCH model, error distribution and the method of Root Mean Square Error. The data description and analysis are presented in Section 3. Section 4 considers the result of all models, ARCH-LM test and the out-of sample forecast, and Section 5 points out the conclusion. The reference and the appendix could be found at the end. 1.1 Background Econometricians are always committed to fitting models to various kinds of data, both cross-sectional data and time series data, also including some financial volatility data, such as stock price, exchange-rate, and gold price and so on. After the detailed research, I am aware of some special characteristics of financial volatility data: long memory, fat tails and excess kurtosis, clustering volatility, leverage effect and spillover effect, these features are introduced by Bollerslev and Mikkelsen (1996). Historically, they often focused their attention on modeling conditional first moments; see e.g. Bollerslev, Engle and Nelson (1994). They set an underlying assumption for these analyses and assure this assumption to be valid, so the assumption is a variation in the error terms is constant for a data set at any time and location. It means the variation in the error terms will not change along with the changes of the time. So this kind of variance is known to be homoscedastic. But this assumption is not always valid in all cases. The error variance might be larger for some special intervals, and smaller for others. Especially for the financial data, this situation is obvious. According to this particular phenomenon, the error term is then said to suffer from heteroscedasticity, which is proposed by Bollerslev (1987). After constantly exploring, some advanced ARCH, GARCH and extension GARCH models are utilized to analysis the financial volatility data including these features. Enocksson and Skoog (2012). 1.2 Literature review To large extent, Economists have already captured the changes in financial data over a long time. From the paper of Franses and Mcaleer (2002), it can be seen many financial economists are very concerned about how to estimate the volatility of assets returns better. They also do much try and explore many researches which made a possible is that almost every price series exhibits the same characteristics, so we have to find some approximate volatility models to fit these features. This was pointed out early by French, Schwert and Stambaugh (1987) and Bollerslev (1987), and is especially clear in some of the surveys of empirical work from Engle s paper (2002). 4

6 And next, I will simply summary what they have studied about the volatility models and some achievement which they have got until now. This first model is Autoregressive Conditional Heteroskedasticity (ARCH) which was early introduced in the Engle s paper (1982), it aimed to capture the conditional variance that is why it became the most popular way of describing the unique feature. Later on, for making this model better, Bollerslev (1986) and Taylor (1986) put forward, independently of each other, a generalization of this model, called Generalized ARCH (GARCH). And this model have been certificated not only to catch volatility clustering but also to contain fat tails from the volatility data. These are common features about the financial data. Even though the GARCH model is already the extension of the ARCH model, it still has some drawbacks. The main point is that the GARCH model is symmetric, so it has a poor performance in reflecting the asymmetry. Because a fact on an interesting feature of financial volatility data is that bad news seems to have a more significant effect on the fluctuation compared to good ones. In other words, positive and negative information generate different degrees of influence to the changes of financial data. So this asymmetric phenomenon is leverage effect. Considering the stock data, it always exist a strong negative correlation between the current return and the future conditional variance. That is why some advanced GARCH model will be introduced later. Such as exponential-garch model, Nelson (1991) and GJR-GARCH model, Glosten, Jangannathan and Runkle (1993), are proposed. Except these models, there still have many other extension GARCH models, such as TGARCH model threshold ARCH attributed to Rabemananjara and Zakoian (1993) and Glosten, Jaganathan and Runkle (1993), FIGARCH model introduced by Baillie, Bollerslev and Mikkelsen (1996) IGARCH model proposed by Engle and Bollerslev (1986) and so on. 2. Methodology 2.1 ARCH model ARCH (Auto-regressive Conditional Heteoskedastic Model) is the first and the basic model in stochastic variance modeling and is proposed by Engle (1982). The key point of this model is that it already changes the assumption of the variation in the error terms from constant Var(ε ) = σ to be a random sequence which depended on the past residuals ({ε ε }). That is to say, this model has changed the restriction from homoscedastic to be heteroscedasticity. This breakthrough is explained by Baillie and Bollerslev (1989). And this is an accurate change to reflect the volatility data s features. Let ε as a random variable that has a mean and a variance conditionally on the information set Ι, The ARCH model of ε has the following properties. Come from Teräsvirta (2006). First, E(ε Ι ) = 0. And second, conditional variance 5

7 σ = E(ε Ι ), is a positive valued parametric function of Ι. The sequence {ε } may be observed directly, or it may be got from the following formula. In the latter case, I can get ε = y μ (y ) Where y is observed value, and μ (y ) = E(y Ι ) is the conditional mean of y given Ι, Engle s (1982) application was of this type. In what follows, the ε could be expressed as another way on parametric forms of σ. So, here ε is assumed as follows: ε = z σ Where {z } is a sequence of independent, identically distributed (iid) random variables with zero mean and unit variance. This implied: ε ~D(0, σ ), So the ARCH model of order q is like this: σ = α + α ε α ε = α + α ε (1) Where α > 0, and α 0, i > 0. To assure {σ } is asymptotically stationary random sequence, I can assume that α α < 1. This is the ARCH model. With the generation of ARCH model, it already can explain many problems in many fields, for instance, interest rates, exchange rates and trade option and stock index returns. Bollerslev, Chou and Kroner (1992) already used these models to achieve a variety of applications in their survey. It s different between forecasting the conditional variance of these series and forecasting the conditional mean of them because the conditional variance cannot be observed. So how to measure the conditional variance should be considered from Andersen and Bollerslev (1998). 2.2 Generalized-ARCH model (GARCH) Because of some drawbacks and limitation on ARCH model, it has been substituted by the so-called generalized ARCH (GARCH) model that Bollerslev (1986) and Taylor (1986) proposed independently of each other. Based on the ARCH model has been raised, it adds the lagged conditional variance term (σ ) as a new term in the GARCH model. The improved ARCH model (GARCH model) also reduces the number of estimated parameters. In this model, the conditional variance is still a linear function of its own lags and error terms, it has the following form: 6

8 σ = α + α ε + β σ Here, I need to explain this function, q represent the order of ε, and the p represents the order of the σ, in order to acquire the positive value, a sufficient condition for the conditional variance is α > 0; α 0; j = 1,...,q; β 0; i = 1,...,p, The GARCH(p, q) process is weakly stationary if and only if α + β < 1, and the GARCH model keeps not only all the characteristics of the ARCH model but also a linear function of lagged conditional variance. So the GARCH model is an extension of ARCH model. In my paper, I just use the most basic GARCH (1, 1), a sufficient condition of GARCH (1, 1) model for the conditional variance to be positive with probability one is α > 0; α 0 β 0. The model which I need to use in the paper is given by Alexander and Lazar (2006): σ = α + α ε + β σ (2) The more complicated higher-order GARCH models are mentioned in the paper of Nelson and Cao (1992). In addition to this, this paper also describes the necessary and sufficient conditions for positive value of the conditional variance in higher-order GARCH models. The GARCH (2, 2) case has been studied in detail by He and Teräsvirta (1999). GARCH model has greater applications in some areas, but it also has some limitations in estimating the volatility asset pricing. From the article of Enocksson and Skoog (2012), the GARCH model generally has two limitations. First, it cannot measure the leverage effect. The GARCH model treats the influence which comes from positive and negative information in a series equally, but it s not reasonable in many cases. The negative information of stock price always has pronounced effect on the fluctuation than the positive information, thus the symmetric GARCH model does not capture this kind asymmetry performance, see Patrick, Stewart and Chris (2006). Second, it s also difficult to achieve all the parameters are assumed larger than zero in GARCH models. In order to solve these series of problems, the GARCH model has been improved further. For measuring the negative impact of leverage effect in the volatility models, Nelson (1991) proposed the EGARCH model. Glosten, Jagannathan and Runkel (1993) proposed GJR-GARCH model. And next, I will introduce the EGARCH model first. 2.3 Exponential GARCH (EGARCH) model The nature logarithm of the conditional variance is assumed as a linear function of its own lagged term and allowed to vary over time. See Nelson (1991). The EGARCH (p, q) is given by logσ = c + g(z ) + β log σ, 7

9 Where, I simplify the g(z ) = γ Z + α Z E( Z ), g(z ) is a function of both the magnitude and sign of Z, γ Z is sign effect, α Z E( Z ) is magnitude effect. σ is the conditional variance, c, γ, α, β are the coefficients, and I define Z = and the function should be written as: log(σ ) = c + β log σ + α ε E ε + σ σ ε γ σ π 2, when Z is normal distribution E( Z ) = νγ[0.5(ν 1)], when Z is student t distribution πγ(0.5ν) Here, Z have two different kinds of distributions, normal distribution and student-t distribution. And c, γ, α, β are the parameters, the α parameter represents a magnitude effect or the symmetric effect of the model, the same role as the ARCH effect. The parameter β measures conditional variance. The same role as the GARCH model. If β is quite large, it will consume a long time to die out under a market crisis and vice versa. See Alexander (2004). The parameter γ reflects the asymmetric performance or leverage effect. So the parameter γ is an outstanding extension from GARCH model to EGARCH model. If γ = 0, which means the model doesn t exist asymmetric, when γ < 0, which means the negative news generate pronounced effect than the positive news. When γ > 0, it implies the opposite situation that is positive information is more significant than negative information. see Patrick, Stewart and Chris (2006). Here, in the article of Wang, Fawson, Barrett and Mcdonald (2001), they point out E( Z ) is constant for all i when Z is normal distribution or else when Z is student-t distribution. And in my paper, I just use the EGARCH (1, 1) model which is simplified by log(σ ) = c + γ Z + α Z α E Z + β log(σ ) (3) Another GARCH family s model is GJR-GARCH model. 2.4 GJR-GARCH model The GJR-GARCH model, Glosten, Jagannathan and Runkle (1993) proposed, is another model could measure the asymmetry in the GARCH family models. The same, I define the ε = z σ, where z ~D(0, 1), ε ~D(0, σ ) so the GJR-GARCH model is written by 8

10 σ = c + β σ + α ε + γ ε I (ε < 0) Where I is an indicator variable taking the value one if the residual is smaller than zero and the value zero if the residual is not smaller than zero. I = 1, if ε < 0 0, otherwise, From this model, it also captures the asymmetric impacts by the sign of the indicator term to reflect different influence between good news and bad news. This is another expression different from the EGARCH model. And the more detail could be acquired from Patrick, Stewart and Chris (2006). The most common GJR-GARCH model is simplified as follows: σ = c + β σ 2.5 Distribution of the error term + α ε + γ ε I (ε < 0) (4) The distribution of error term also plays an important role in estimating the volatility model. And in my paper, I mainly introduce two common distributions. One is normal distribution, the other is student-t distribution. The most common application is assumed as standard normal distribution. In some situations, this normal distribution is not good; maybe the error term is fat tail, so normal distribution cannot capture this feature. That is why I choose the student-t distribution as my second choice. So I choose another distribution, student-t distribution, which maybe explains the fat-tailed distribution better. Actually, it still has many other assumptions, but in my paper I just introduce these two distributions as examples Normal distribution The probability density function of Z is given as normal distribution, f( Z ) = exp (μ ) (5) where μ is mean and σ is standard deviation Student t-distribution The Conditional density function of Z is student t-distribution and the density function is given by: 9

11 f( Z ) = ν (1 + ν (ν) ν ) (ν) (6) Where ν is the number of degree of freedom, 2 < ν, and the Γ is gamma function. When ν the student-t distribution nearly equals to the normal distribution. The lower the ν, the fatter the tails. So the student-t distribution maybe reflects the fat-tail of the volatility index more exactly. 2.6 Root Mean Square Error (RMSE) Many methods could be chose to test model is better or not, such as MSE, RMSE from Lars and Eric (2007). This paper uses the RMSE as a test to measure GARCH models. The Root Mean Square Error (RMSE) (also called the root mean square deviation, RMSD) is a frequently measure of the difference between estimated values predicted by a model and the true values actually observed. These individual differences are also called residuals, and the RMSE aims to aggregate all these residuals together as a standard of predictive power. The criterion is the smaller value of the RMSE, the better the predicting ability of the model. And in my thesis, I use this method to decide which model has the best forecasting performance from GARCH and GARCH family models. The RMSE of a model prediction with respect to the estimated variable r is defined as the square root of the mean squared error ( RMSE = (7) where r is observed values and σ is the predicted value of conditional variance at time i, n is the number of forecasts. 3. Data 3.1 Data description NASDAQ Stock Market Daily Closing Price Index The first data I used is NASDAQ daily index. The NASDAQ Stock Market, also known as simply the NASDAQ, it s an American stock exchange. "NASDAQ" originally represented National Association of Securities Dealers Automated Quotations". Except the New York Stock Exchange, it already becomes the second-largest stock exchange in the world s stock market. In addition to this, the NASDAQ has large trading volume than any other electronic stock exchange market, because NASDAQ daily closing price index is very significant and representative, so that is why I choose the NASDAQ daily closing price index as my data. The data I chose from Yahoo Finance between and ( 10

12 3.1.2 Standard & Poor 500 Stock Market Daily Closing Price Index The S&P 500 stands for Standard & Poor 500 and is a free-float capitalization-iighted index published since 1957 and it includes 500 large-cap common stocks actively traded in the United States. Comparing with the Dow Jones Industrial Average, S&P500 has more companies, so the risk is more dispersed and it could reflect the changes in the market broader. Based on these features, S&P 500 is generally considered as the standard of ideal stock index future contracts. I choose Standard & Poor 500 daily close index from Yahoo Finance between and ( FTSE100 Stock Market Daily Closing Price Index The FTSE 100 Index, also known as London's FTSE 100 index (the Financial Times Stock Exchange 100 Index), FTSE, is a share index of the stocks of the largest 100 companies listed on the London Stock Exchange. This index is a barometer of the UK economy and one of the most important stock indexes in Europe, including the FTSE 250 index and the combination of the FTSE350 index. It is the most widely used of the FTSE Group's indices, and is frequently reported (e.g. on UK news bulletins) as a measure of business prosperity. I choose FTSE100 daily close index from Yahoo Finance between and ( HANG SENG Stock Market Daily Closing Price Index The Hang Seng Index (abbreviated: HSI, Chinese) is a free float-adjusted market capitalization-iighted stock market index in Hong Kong. The Hang Seng Index is an important indicator of the Hong Kong stock market price index calculated by the market value of the number of constituent stocks (blue chips), representing 70% of all listed companies on the Hong Kong Stock Exchange. It is used to record and monitor daily changes of the largest companies of the Hong Kong stock market and is a most influential stock index that reflects the increase trend of the Hong Kong stock market price. I choose HANG SENG daily close index from Yahoo Finance between and ( NIKKEI Stock Market Daily Closing Price Index The Nikkei (Nikkei heikin kabuki, Nikkei 225), more commonly called the Nikkei, the Nikkei index, or the Nikkei Stock Average, is a stock market index for the Tokyo Stock Exchange (TSE). It has been calculated daily by the Nihon Keizai Shimbun (Nikkei), Tokyo Stock Exchange 225 varieties of the stock index. This index with longer duration and good comparability has already become the most common and reliable indicators to study the changes in the Japanese s stock market. NIKKEI is the widest index which has been quoted as the daily index by media. I choose NIKKEI 225 daily close index from Yahoo Finance between and

13 ( 3.2 Data analysis In my paper, I will use the returns of these two kind stock prices to estimate and predict the financial volatility. So I need to acquire the stock price s returns first, using the following function: r = 100(ln(p ) ln(p ) ), (8) where, r is return for each of stock index and p is the closing daily price for each stock index at time t. After getting the returns of the stock price, I need to summary and list the features of these data, including sample size, mean, standard deviation, minimum, maximum, skewness, kurtosis and Jarque-Bera test NASDAQ analysis I describe the NASDAQ stock market firstly, it includes about 1260 observations from the to in NASDAQ market. With the formula (8), I get the 1259 return data. 12 The returns of NASDAQ daily close price index 8 NASDAQ's returns DATE 12

14 3.2.2 Standard & Poor 500 analysis I use the Standard & Poor 500 Stock Market Daily Closing Price Index as my second application, and it includes 1260 observations from the to in Standard & Poor 500 Stock Market. And then, using the formula (8), I could get the 1259 returns data as follows: 12 The returns of S&P500 daily close price index 8 S&P500's returns DATE FTSE100 analysis Next, I describe the FTSE100 stock market data, which includes about 1263 observations from the to in FTSE100 market. Based on the formula (8), I could get the 1262 returns data as follows: 13

15 The returns of FTSE100 daily close price index FTSE100's returns DATE HANG SENG analysis I describe the HANG SENG stock market data which includes 1261 observations from the to in HANG SENG market. Using the formula (8) mentioned, the 1260 returns data are as follows: 14

16 The returns of HANG SENG daily close price index HANG SENG's returns DATE NIKKEI analysis NIKKEI stock market data includes about 1221 observations I used in the paper, from the to in NIKKEI market. By the formula (8), I acquire the 1220 returns data as follows: 15

17 The returns of NIKKEI daliy close price index NIKKEI's returns DATE The descriptions of different data NASDAQ S&P500 FTSE100 HANG SENG NIKKEI Sample size Min Max Mean SD Skewness kurtosis Jarque-Bera From above table, the kurtosis of the NASDAQ daily return is which is higher than the value of normal distribution (kurtosis=3). This value clearly shows the financial time series data has the fat-tail characteristic. The skewness is , not zero which means it s not symmetric. At the same time, the Jarque-Bera Test could tell us another feature, the higher value of Jarque-Bera Test represents the non-normality of the series data. And here, the value ( ) is enough large, so the distribution of NASDAQ daily return is not normal. In the same way, I could acquire the other results about the other stock return value, the kurtosis of the Standard & Poor 500 daily return is which is higher than the value of normal distribution (kurtosis=3). This value clearly shows the financial time series data has the fat-tail characteristic. Based on the skewness s value, the distribution of S&P500 s 16

18 return value is asymmetric. At the same time, the Jarque-Bera Test could tell us it s non-normality of the series data, because the value ( ) is enough large, so the distribution of Standard & Poor 500 daily return is not normal. And the kurtosis of another three daily return values (FTSE100, HANG SENG and NIKKEI) are , and which are all higher than the value of normal distribution (kurtosis=3). This value clearly shows the financial time series data has the fat-tail characteristic. And these return values are also non-normality of the series data. Because of the Jarque-Bera s values ( , and ) are enough large, so the distributions of daily return about FTSE100, HANG SENG and NIKKEI are not normal. After describing all of five stock market returns, I do the following Box-Ljung test, this test could help us to check if the ARCH effect is existed in the returns or not, the null hypothesis is that the returns data doesn t exist the ARCH effect, while the alternative hypothesis is opposite. See Forsberg and Bollerslev (2002). Box-Ljung test (for returns) The Box-Ljung test results of different market s returns NASDAQ S&P500 FTSE100 HANG SENG NIKKEI test value p-value The p-values of different markets returns have been show form this table. By the assumption of 5% significance level, all of the results are significant. So the null hypothesis has to be rejected, which means all different markets returns exist the ARCH effect. Next I use the ARCH/GARCH model to analysis the different markets returns. After modeling with ARCH/GARCH model, the ARCH effect should be eliminated if the ARCH/GARCH model is good to estimate the returns. Finally, the ARCH effect after estimation should be tested again. At that time, the ARCH effect should not be existed any more. 4. Result In this part, I will separately utilize GARCH model and GARCH family models with different distributions of error term to estimate and forecast the financial volatility using the daily stock return from NASDAQ market index, Standard & Poor 500 daily index, FTSE100 daily index, HANG SENG daily index and NIKKEI daily index. And then compare the results of all the models and choose which model has the best performance to forecast the financial volatility by calculating the RMSE. 4.1 Application in NASDAQ daily return Selection of ARMA (p, q) model First step is selection of suitable ARMA (p, q) model for NASDAQ daily return. For choosing the exactly parameter p and q to fit the ARMA model, I could roughly get them by observing autocorrelation and partial autocorrelation. According to comparing the value of 17

19 AIC and BIC, the best one could be picked up. Many models have been tried, such as AR (1), MA (1), ARMA (1, 1) and so on. Finally, I choose p=0, q=1 because the MA (1) has the smallest AIC and BIC of all the models. So I select the MA (1) model finally. From the following function: r = θ + θ e + e The estimated parameters are θ = 1e 04, θ = , And then, calculate and take e = r (θ + θ e ), after that estimate GARCH model on e Result of GARCH model and GARCH family model for NASDAQ Estimating MA (1)-GARCH (1, 1) model, MA (1)-EGARCH (1, 1) model and MA (1)-GJR-GARCH (1, 1) model with normal distribution and student-t distribution, In Table 1.1, all the results of the models estimated parameters for different distributions have been shown in the following table. Table 1.1: The results of all volatility models for NASDAQ GARCH(1,1) EGARCH(1,1) GJR-GARCH(1,1) Normal Student-t Normal Student-t Normal Student-t Ω (s.e.) (0.0124) (0.0129) (0.0065) (0.0056) (0.0121) (0.0128) (p-value) (0.0125) (0.1027) (0.0139) (0.2327) (0.0019) (0.0124) α (s.e.) (0.0163) (0.0201) (0.0204) (0.0251) (0.0146) (0.0178) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (1.0000) (1.0000) γ (s.e.) (0.0240) (0.0294) (0.0319) (0.0393) (p-value) (0.0000) (0.0000) (0.0000) (0.0004) β (s.e.) (0.0174) (0.0181) (0.0059) (0.0065) (0.0180) (0.0187) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) ν (s.e.) (1.7931) (2.3078) (2.3369) (p-value) (0.0001) (0.0003) (0.0004) (This data is return of NASDAQ daily close price index, from to , a total of 1259 observations. this data comes from yahoo finance. This table points out all of the coefficients under all of the GARCH family models.) Assumed the significance level is 5%, in GARCH model, all of the estimated parameters are significant except Ω in student-t distribution. In EGARCH model, the same as GARCH model, almost all of the estimated parameters of EGARCH model are significant under the different error distributions except Ω under student-t distribution. While, in GJR-GARCH model, the 18

20 estimation results of α are very bad, α is the coefficient of ε term, and here the p-value of α is very large, which represents the coefficient of ε is not significant and the term of ε (ARCH term) has no effect in interpretation of σ. To sum up, both GARCH (1, 1) model and EGARCH (1, 1) model with normal distribution could explain the daily stock return better than GJR-GARCH (1, 1) model with two distributions. Because of the poor performance on GJR-GARCH (1, 1) model, I attempt to the high-order GJR-GARCH model, GJR-GARCH (1, 2) model, GJR-GARCH (2, 1) model and GJR-GARCH (2, 2) model with normal distribution and student-t distribution, and the results are shown in the Table 1.2. Table 1.2: The results of high-order GJR-GARCH model for NASDAQ GJR-GARCH (2, 1) threshold=2 GJR-GARCH (1, 2) threshold=2 GJR-GARCH (2, 2) threshold=2 Normal Student-t Normal Student-t Normal Student-t Ω (s.e.) (0.0107) (0.0117) (0.0156) (0.0176) (0.0034) (0.0043) (p-value) (0.0000) (0.0013) (0.0000) (0.0007) (0.0000) (0.0047) α (s.e.) (0.0139) (0.0185) (0.0201) (0.0251) (0.0258) (0.0220) (p-value) (0.0000) (0.0000) (0.1180) (0.1997) (0.0000) (0.0000) α (s.e.) (0.0268) (0.0364) (0.0259) (0.0260) (p-value) (0.0007) (0.0032) (0.0000) (0.0000) γ (s.e.) (0.0358) (0.0446) (0.0446) (0.0531) (0.0380) (0.0416) (p-value) (0.0048) (0.0054) (0.0844) (0.0748) (0.0000) (0.0000) γ (s.e.) (0.0443) (0.0551) (0.0431) (0.0531) (0.0397) (0.0461) (p-value) (0.0969) (0.3411) (0.0000) (0.0000) (0.0036) (0.0000) β (s.e.) (0.0199) (0.0228) (0.1961) (0.2323) (0.0073) (0.0753) (p-value) (0.0000) (0.0000) (0.0969) (0.1761) (0.0000) (0.0000) β (s.e.) (0.1800) (0.2116) (0.0015) (0.0645) (p-value) (0.0038) (0.0127) (0.0000) (0.0000) ν (s.e.) (2.8029) (2.8137) (2.7858) (p-value) (0.0003) (0.0004) (0.0002) (This data is return of NASDAQ daily close price index, from to , a total of 1259 observations. this data comes from yahoo finance. This table points out all of the coefficients under all of the higher-order GJR-GARCH models.) From this table, assumed 5% significance level, I find the estimated parameters of 19

21 GJR-GARCH (2, 2) model under both distributions are significant. So the GJR-GARCH (2, 2) model is better than GJR-GARCH (1, 1) model. 4.2 Application in Standard & Poor 500 daily return Selection of ARMA (p, q) model First step is selection of suitable ARMA (p, q) model for S&P500 daily return. For choosing the exactly parameter p and q to fit the ARMA model, I could roughly get them by autocorrelation and partial autocorrelation. According to comparing the value of AIC and BIC, the best one could be picked up. Many models have been tried, such as AR (1), MA (1), ARMA (1, 1) and so on. Finally, I choose p=1, q=1 because the ARMA (1, 1) has the smallest AIC and BIC of all the models. So I select the ARMA (1, 1) model finally. From the following function: r = θ + θ e + θ r + e The estimated parameters are θ = , θ = , θ = , And then, calculate and take e = r (θ + θ e + θ r ), after that estimate GARCH model on e Result of GARCH model and GARCH family model for Standard & Poor 500 Estimating ARMA (1, 1)-GARCH (1, 1) model, ARMA (1, 1)-EGARCH (1, 1) model and ARMA (1, 1)-GJR-GARCH (1, 1) model with normal distribution and student-t distribution, In Table 2.1, all the results of the models estimated parameters for different distributions have been shown in the following table. 20

22 Table 2.1: The results of all volatility models for S&P500 GARCH(1,1) EGARCH(1,1) GJR-GARCH(1,1) Normal Student-t Normal Student-t Normal Student-t Ω (s.e.) (0.0108) (0.0121) (0.0059) (0.0070) (0.0093) (0.0096) (p-value) (0.0018) (0.0781) (0.0231) (0.3789) (0.0010) (0.0212) α (s.e.) (0.0166) (0.0199) (0.0189) (0.0243) (0.0181) (0.0214) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (1.0000) (0.9999) γ (s.e.) (0.0235) (0.0302) (0.0287) (0.0386) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) β (s.e.) (0.0176) (0.0181) (0.0053) (0.0062) (0.0186) (0.0177) (p-value) (0.0000) ) (0.0000) (0.0000) (0.0000) (0.0000) ν (s.e.) (1.1064) (1.2529) (1.2337) (p-value) (0.0000) (0.0000) (0.0000) (This data is return of S&P500 daily close price index, from to , a total of 1259 observations. this data comes from yahoo finance. This table points out all of the coefficients under all of the GARCH family models) Assumed the significance level is 5%, in GARCH model, all of the estimated parameters are significant except Ω in student-t distribution. In EGARCH model, the same as GARCH model, almost all of the estimated parameters of EGARCH model are significant under the different error distributions except Ω under student-t distribution. While, in GJR-GARCH model, the estimation results of α are very bad, α is the coefficient of ε term, and here the p-value of α is very large, which represents the coefficient of ε is not significant and the term of ε has no effect in interpretation of σ. To sum up, both GARCH (1, 1) model and EGARCH (1, 1) model with normal distribution could explain the daily stock return better than GJR-GARCH (1, 1) model with two distributions. Because of the poor performance on GJR-GARCH (1, 1) model, I attempt to the high-order GJR-GARCH model, GJR-GARCH (1, 2) model, GJR-GARCH (2, 1) model and GJR-GARCH (2, 2) model with normal distribution and student-t distribution, and the results are shown in the Table

23 Table 2.2: The results of high-order GJR-GARCH model for S&P500 GJR-GARCH (2, 1) threshold=2 GJR-GARCH (1, 2) threshold=2 GJR-GARCH (2, 2) threshold=2 Normal Student-t Normal Student-t Normal Student-t Ω (s.e.) (0.0059) (0.0070) (0.0083) (0.0091) (0.0085) (0.0094) (p-value) (0.0000) (0.0009) (0.0000) (0.0008) (0.0000) (0.0044) α (s.e.) (0.0132) (0.0182) (0.0187) (0.0243) (0.0156) (0.0188) (p-value) (0.0000) (0.0000) (0.0056) (0.0242) (0.0000) (0.0000) α (s.e.) (0.0203) (0.0331) (0.0249) (0.0359) (p-value) (0.0021) (0.0089) (0.0528) (0.0250) γ (s.e.) (0.0153) (0.0377) (0.0346) (0.0444) (0.0280) (0.0374) (p-value) (0.0048) (0.0174) (0.0351) (0.0779) (0.0107) (0.0156) γ (s.e.) (0.0349) (0.0495) (0.0364) (0.0488) (0.0451) (0.0667) (p-value) (0.0003) (0.0334) (0.0000) (0.0000) (0.0000) (0.0330) β (s.e.) (0.0166) (0.0007) (0.1243) (0.1908) (0.1903) (0.2948) (p-value) (0.0000) (0.0000) (0.0255) (0.0560) (0.0085) (0.0202) β (s.e.) (0.1145) (0.1759) (0.1774) (0.2717) (p-value) (0.0000) (0.0038) (0.0407) (0.4873) ν (s.e.) (1.3690) (1.4516) (1.3793) (p-value) (0.0000) (0.0000) (0.0000) (This data is return of S&P500 daily close price index, from to , a total of 1259 observations. this data comes from yahoo finance. This table points out all of the coefficients under all of the higher-order GJR-GARCH models.) From this table, assumed 5% significance level, I find the estimated parameters of GJR-GARCH (2, 1) model under both distributions are significant. So the GJR-GARCH (2, 1) model is better than GJR-GARCH (1, 1) model. 4.3 Application in FTSE100 daily return Selection of ARMA (p, q) model First step is selection of suitable ARMA (p, q) model for FTSE100 daily return. For choosing the exactly parameter p and q to fit the ARMA model, I could roughly get them by autocorrelation and partial autocorrelation. According to comparing the value of AIC and BIC, the best one could be picked up. 22

24 Many models have been tried, such as AR (1), MA (1), ARMA (1, 1) and so on. Finally, I choose p=1, q=1 because the ARMA (1, 1) has the smallest AIC and BIC of all the models. So I select the ARMA (1, 1) model finally. From the following function: r = θ + θ e + θ r +e The estimated parameters are θ = 0.01, θ = , θ = And then, calculate and take e = r (θ + θ e + θ r ), after that estimate GARCH model on e Result of GARCH model and GARCH family model for FTSE100 Estimating ARMA (1, 1)-GARCH (1, 1) model, ARMA (1, 1)-EGARCH (1, 1) model and ARMA (1, 1)-GJR-GARCH (1, 1) model with normal distribution and student-t distribution, In Table 3.1, all the results of the models estimated parameters for different distributions have been shown in the following table. Table 3.1: The results of all volatility models for FTSE100 GARCH(1,1) EGARCH(1,1) GJR-GARCH(1,1) Normal Student-t Normal Student-t Normal Student-t Ω (s.e.) (0.0135) (0.0158) (0.0071) (0.0073) (0.0127) (0.0138) (p-value) (0.0073) (0.0209) (0.0231) (0.2239) (0.0002) (0.0014) α (s.e.) (0.0222) (0.0248) (0.0192) (0.0222) (0.0169) (0.0200) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (1.0000) (1.0000) γ (s.e.) (0.0304) (0.0319) (0.0335) (0.0378) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) β (s.e.) (0.0215) (0.0236) (0.0075) (0.0080) (0.0202) (0.0217) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) ν (s.e.) (2.0368) (3.5244) (3.0625) (p-value) (0.0001) (0.0025) (0.0012) (This data is return of FTSE100 daily close price index, from to , a total of 1262 observations. this data comes from yahoo finance. This table points out all of the coefficients under all of the GARCH family models.) Assumed the significance level is 5%, in GARCH model, all of the estimated parameters are significant except Ω in student-t distribution. In EGARCH model, the same as GARCH model, almost all of the estimated parameters of EGARCH model are significant under the different error distributions except Ω under student-t distribution. While, in GJR-GARCH model, the estimation results of α are very bad, α is the coefficient of ε term, and here the 23

25 p-value of α is very large, which represents the coefficient of ε is not significant and the term of ε has no effect in interpretation of σ. To sum up, both GARCH (1, 1) model and EGARCH (1, 1) model with normal distribution could explain the daily stock return better than GJR-GARCH (1, 1) model with two distributions. Because of the poor performance on GJR-GARCH (1, 1) model, I attempt to the high-order GJR-GARCH model, GJR-GARCH (1, 2) model, GJR-GARCH (2, 1) model and GJR-GARCH (2, 2) model with normal distribution and student-t distribution, and the results are shown in the Table 3.2. Table 3.2: The results of high-order GJR-GARCH model for FTSE100 GJR-GARCH (2, 1) threshold=2 GJR-GARCH (1, 2) threshold=2 GJR-GARCH (2, 2) threshold=2 Normal Student-t Normal Student-t Normal Student-t Ω (s.e.) (0.0087) (0.0105) (0.0130) (0.0148) (0.0063) (0.0005) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.1575) α (s.e.) (0.0117) (0.0132) (0.0194) (0.0227) (0.0314) (0.0147) (p-value) (0.0000) (0.0000) (0.0057) (0.0052) (0.0091) (0.0000) α (s.e.) (0.0198) (0.0222) (0.0310) (0.0147) (p-value) (0.0007) (0.0031) (0.0124) (0.0000) γ (s.e.) (0.0493) (0.0558) (0.0405) (0.0442) (0.0502) (0.0499) (p-value) (0.0000) (0.0001) (0.0000) (0.0000) (0.0000) (0.0000) γ (s.e.) (0.0516) (0.0585) (0.0387) (0.0442) (0.0634) (0.0481) (p-value) (0.8242) (0.8064) (0.0000) (0.0000) (0.0152) (0.0000) β (s.e.) (0.0188) (0.0214) (0.1650) (0.1469) (0.0688) (0.0363) (p-value) (0.0000) (0.0000) (0.2855) (0.3395) (0.0000) (0.0000) β (s.e.) (0.1478) (0.1317) (0.0626) (0.0354) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) ν (s.e.) (1.3690) (4.6137) ( ) (p-value) (0.0000) (0.0063) (0.0785) (This data is return of FTSE100 daily close price index, from to , a total of 1262 observations. this data comes from yahoo finance. This table points out all of the coefficients under all of the higher-order GJR-GARCH models.) From this table, assumed 5% significance level, I find the estimated parameters of GJR-GARCH (2, 2) model under normal distribution are significant. So the GJR-GARCH (2, 2) 24

26 model with normal distribution is better than GJR-GARCH (1, 1) model. 4.4 Application in NIKKEI daily return Selection of ARMA (p, q) model First step is selection of suitable ARMA (p, q) model for NIKKEI daily return. For choosing the exactly parameter p and q to fit the ARMA model, I could roughly get them by autocorrelation and partial autocorrelation. According to comparing the value of AIC and BIC, the best one could be picked up. Many models have been tried, such as AR (1), MA (1), ARMA (1, 1) and so on. Finally, I choose p=1, q=1 because the ARMA (1, 1) has the smallest AIC and BIC of all the models. So I select the ARMA (1, 1) model finally. From the following function: r = θ + θ e + θ r + e The estimated parameters are θ = , θ = , θ = And then, calculate and take e = r (θ + θ e + θ r ), after that estimate GARCH model on e Result of GARCH model and GARCH family model for NIKKEI Estimating ARMA (1, 1)-GARCH (1, 1) model, ARMA (1, 1)-EGARCH (1, 1) model and ARMA (1, 1)-GJR-GARCH (1, 1) model with normal distribution and student-t distribution, In Table 4.1, all the results of the models estimated parameters for different distributions have been shown in the following table. 25

27 Table 4.1: The results of all volatility models for NIKKEI GARCH(1,1) EGARCH(1,1) GJR-GARCH(1,1) Normal Student-t Normal Student-t Normal Student-t Ω (s.e.) (0.0239) (0.0238) (0.0081) (0.0083) (0.0193) (0.0197) (p-value) (0.0090) (0.0352) (0.0122) (0.0416) (0.0045) (0.0104) α (s.e.) (0.0231) (0.0235) (0.0165) (0.0177) (0.0175) (0.0183) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (0.3091) (0.3516) γ (s.e.) (0.0341) (0.0342) (0.0291) (0.0306) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) β (s.e.) (0.0239) (0.0245) (0.0069) (0.0069) (0.0209) (0.0209) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) ν (s.e.) (6.0712) ( ) ( ) (p-value) (0.0255) (0.1919) (0.2597) (This data is return of NIKKEI daily close price index, from to , a total of 1220 observations. this data comes from yahoo finance. This table points out all of the coefficients under all of the GARCH family models.) Assumed the significance level is 5%, in GARCH model, all of the estimated parameters are significant except Ω in student-t distribution. In EGARCH model, the same as GARCH model, almost all of the estimated parameters of EGARCH model are significant under the different error distributions except Ω under student-t distribution. While, in GJR-GARCH model, the estimation results of α are very bad, α is the coefficient of ε term, and here the p-value of α is very large, which represents the coefficient of ε is not significant and the term of ε has no effect in interpretation of σ. To sum up, both GARCH (1, 1) model and EGARCH (1, 1) model with normal distribution could explain the daily stock return better than GJR-GARCH (1, 1) model with two distributions. Because of the poor performance on GJR-GARCH (1, 1) model, I attempt to the high-order GJR-GARCH model, GJR-GARCH (1, 2) model, GJR-GARCH (2, 1) model and GJR-GARCH (2, 2) model with normal distribution and student-t distribution, and the results are shown in the Table

28 Table 4.2: The result of high-order GJR-GARCH model for NIKKEI GJR-GARCH (2, 1) threshold=2 GJR-GARCH (1, 2) threshold=2 GJR-GARCH (2, 2) threshold=2 Normal Student-t Normal Student-t Normal Student-t Ω (s.e.) (0.0180) (0.0198) (0.0402) (0.0475) (0.0230) (0.0306) (p-value) (0.0000) (0.0004) (0.5273) (0.6087) (0.0025) (0.0077) α (s.e.) (0.0287) (0.0314) (0.0178) (0.0213) (0.0286) (0.0317) (p-value) (0.0011) (0.0011) (0.5324) (0.6088) (0.0009) (0.0010) α (s.e.) (0.0285) (0.0329) (0.0293) (0.0338) (p-value) (0.0001) (0.0002) (0.0001) (0.0002) γ (s.e.) (0.0428) (0.0481) (0.0382) (0.0445) (0.0427) (0.0480) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) γ (s.e.) (0.0522) (0.0564) (0.1078) (0.1290) (0.0874) (0.0868) (p-value) (0.0485) (0.1337) (0.1502) (0.2950) (0.5301) (0.5490) β (s.e.) (0.0222) (0.0235) (0.5603) (0.7040) (0.3317) (0.3471) (p-value) (0.0000) (0.0000) (0.0081) (0.0338) (0.0747) (0.0665) β (s.e.) (0.4807) (0.6082) (0.2895) (0.3053) (p-value) (0.2641) (0.3700) (0.4273) (0.5211) ν (s.e.) ( ) ( ) ( ) (p-value) (0.0915) (0.1189) (0.0939) (This data is return of NIKKEI daily close price index, from to , a total of 1220 observations. this data comes from yahoo finance. This table points out all of the coefficients under all of the higher-order GJR-GARCH models.) From this table, assumed 5% significance level, I find the estimated parameters of GJR-GARCH (2, 1) model under normal distribution are significant. So the GJR-GARCH (2, 1) model with normal distribution is better than GJR-GARCH (1, 1) model. 4.5 Application in HANG SENG daily return Selection of ARMA (p, q) model First step is selection of suitable ARMA (p, q) model for HANG SENG daily return. For choosing the exactly parameter p and q to fit the ARMA model, I could roughly get them by autocorrelation and partial autocorrelation. According to comparing the value of AIC and BIC, the best one could be picked up. 27

29 Many models have been tried, such as AR (1), MA (1), ARMA (1,1) and so on. Finally, I choose p=1, q=0 because the AR (1) has the smallest AIC and BIC of all the models. So I select the AR (1) model finally. From the following function: r = θ + θ r + e The estimated parameters are θ = , θ = And then, calculate and take e = r (θ + θ r ), after that estimate GARCH model on e Result of GARCH model and GARCH family model for HANG SENG Estimating AR (1)-GARCH (1, 1) model, AR (1)-EGARCH (1, 1) model and AR (1)-GJR-GARCH (1, 1) model with normal distribution and student-t distribution, In Table 5, all the results of the models estimated parameters for different distributions have been shown in the following table. Table 5: The results of all volatility models GARCH(1,1) EGARCH(1,1) GJR-GARCH(1,1) Normal Student-t Normal Student-t Normal Student-t Ω (s.e.) (0.0184) (0.0187) (0.0087) (0.0090) (0.0209) (0.0213) (p-value) (0.0405) (0.0655) (0.0061) (0.0185) (0.0100) (0.0154) α (s.e.) (0.0186) (0.0197) (0.0174) (0.0190) (0.0159) (0.0170) (p-value) (0.0000) (0.0000) (0.0000) (0.0001) (0.0053) (0.0083) γ (s.e.) (0.0277) (0.0294) (0.0298) (0.0324) (p-value) (0.0000) (0.0000) (0.0001) (0.0005) β (s.e.) (0.0191) (0.0200) (0.0064) (0.0067) (0.0197) (0.0205) (p-value) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) ν (s.e.) (6.4354) ( ) ( ) (p-value) (0.0243) (0.0119) (0.0206) (This data is return of HANG SENG daily close price index, from to , a total of 1260 observations. this data comes from yahoo finance. This table points out all of the coefficients under all of the GARCH family models.) Assumed the significance level is 5%, in GARCH model, all of the estimated parameters are significant except Ω in student-t distribution. In EGARCH model, all of the estimated parameters of EGARCH model are significant under the different error distributions. At the same time, in GJR-GARCH model, all of the estimation results are significant. To sum up, all of the models are good except the GARCH (1, 1) model under student-t 28