Modeling S&P 500 STOCK INDEX using ARMA-ASYMMETRIC POWER ARCH models Jia Zhou Supervisor: Changli He Master thesis in statistics School of Economics and Social Science HÖGSKOLAN Dalarna, Sweden June, 2009 Contents Abstract 1. Introduction ……………………………………………………………………….1 2. Methodology ……………………………………………………………………...3 2.1 Conditional Mean Equation ……………………………………………………3 2.2 Variance Equation ……………………………………………………………...4 2.2.1 GARCH ……………………………………………………………………4 2.2.2 APARCH …………………………………………………………………..5 2.3 Conditional Distributions ………………………………………………………7 2.3.1 Normal Distribution ……………………………………………………….7 2.3.2 Student t Distribution ……………………………………………………...7 2.3.3 Skew-t Distribution ………………………………………………………..8 3. Data …………………………………………………………………………...……8 3.1 Original Data and Pretreatment ……………………………………………..…8 3.2 Stylized Facts of Asset Returns ……………………………………………….10 3.3 Tests for Normality and Unit Root ……………………………………………10 3.4 Tests for Serial Dependence in Asset Returns and ARCH-Effect of ε in the t Fitted ARMA (0, 2) Model ……………………………………………………12 4. Results …………………………………………………………………………….13 4.1 Order Selection of ARMA (p, q) ……………………………………………...13 4.2 Estimation Results …………………………………………………………….14 4.2.1 ARMA (0, 2)-GARCH (1, 1) ……………………………………………..14 4.2.2 ARMA (0, 2)-APARCH (1, 1) ……………………………………………17 5. Forecast……………………………………………………………………………20 5.1 Forecasting Conditional Mean……………………………………………...…20 5.2 Forecasting Conditional Variance…………………………………………..…20 6. Conclusions and Further Discussion...…………………………………………….22 Reference Abstract In this paper, the S&P 500 stock index is studied for its time varying volatility and stylized facts. The ARMA mean equation with asymmetric power ARCH errors is used to model the series correlations and the conditional heteroscadesticity in the asset returns. The conditional distributions of the standardized residuals are assumed to be the normal distribution, the t distribution or the skew-t distribution. Furthermore, to capture the asymmetry and fat tail of the returns, an ARMA (0, 2)-APARCH (1, 1) model with the skew-t distribution is found to fit the data better than other models discussed in this paper. Finally, we use ARMA (0, 2)-APARCH (1, 1) model with the skew-t distribution to do the 10-step-ahead forecasting compared with the ARMA (0, 2)-GARCH (1, 1) model with the normal distribution and get the empirical conclusion that the ARMA (0, 2)-APARCH (1, 1) with the skew-t distribution also gives a better result in the forecasting. Keywords: fat tail; asymmetry; GARCH; APARCH; non-Gaussian distribution 1. Introduction In finance, researchers always put a lot of interests in modeling and forecasting volatility of asset returns. The reason is that the volatility of asset returns can be seen as a measurement of the risk for investment and provides essential information for the investors to make the correct decisions. While the asset returns themselves are uncorrelated or nearly uncorrelated, however, there exists high order dependence within the return series. To model the time varying volatility (conditional heteroscedasticity), Autoregressive conditional heteroscedasticity (ARCH) introduced by Engle (1982) is used to model the series correlations in squared returns by allowing the conditional variance as a function of past errors and changing over time. Then generalized autoregressive conditional heteroscedasticity (GARCH) model introduced by Bollerslev (1986) extended the ARCH model to have longer memory and more flexible lag structure by adding lagged conditional variance to the model as well. Since then, GARCH model has been studied widely and proved a lot in the literature to be a competent model in fitting the financial time series, sometimes specify the mean equation with a low order of ARMA (p, q) process to capture the autocorrelation of the financial time series. The empirical probability distributions for financial asset returns always exhibit some characteristics which are called stylized facts (Tavares, et al., 2008): Firstly, the well-known volatility clustering or persistence: large changes tend to followed by large changes, and small changes tend to followed by small changes are observed in the asset returns. Secondly, fat tail exits in the probability distribution of the assets returns that the kurtosis exceeds the value of the normal distribution which is 3. This fat tail phenomenon is called excess kurtosis, and the returns time series which exhibit fat tail are often called leptokurtic. From (Bai, et al., 2003), we know the excess kurtosis can be resulted from two aspects: one is the volatility clustering, which is a type of heteroscedasticity, accounts for some of the excess kurtosis; the other is the presence of non-Gaussian asset returns distribution can also results in the fat tail. 1 Thirdly, asymmetric distribution of the assets returns, which means volatility increase more when the change is negative than the change is positive, which is also called leverage effect (Black, 1976). That’s why we always see the bad news give a greater impact on the volatility of the stock market than the good news. In the initial assumption in ARCH model and GARCH model, the conditional distribution of the innovations is Gaussian. But in most case, the unconditional distribution of the high frequency financial time series seems to have fat tail than the Gaussian distribution. Assuming the fourth order moment exists, Bollerslev (1986) showed that the kurtosis implied by a GARCH (1, 1) process with normal errors is greater than 3, so the unconditional distribution of error which follows GARCH (1, 1) process is leptokurtic (fat tail). However, it’s not adequate to capture the fairly high kurtosis present in the financial time series, sometimes GARCH model with a non-Gaussian errors are required to capture the observed fat-tailed behavior in asset returns. Then GARCH model with t distribution as conditional distribution was first introduced by Bollerslev (1987). Further more, due to the skewness observed in the asset returns, the GARCH model with normal conditional distribution cannot capture the asymmetric characteristic of the returns. To overcome this drawback in the model, there are two ways: one is allowing for asymmetric conditional distribution, i.e. skewed t distribution which takes leptokurtic into account as well, while another is modeling the asymmetric directly in the conditional variance equation as nonlinear GARCH model: The APARCH (p, q) model- Asymmetric power ARCH model introduced by Ding, et al. (1993). It changes the second order of the error term into a more flexible varying exponent with an asymmetric coefficient which takes the leverage effect into account. The APARCH (p, q) model is a general class of model which includes special cases as ARCH, GARCH, TS-GARCH (Taylor, 1986 cited in Ding et al., 1993), GJR-GARCH (Glosten et al.,1993 cited in Ding et al., 1993) and TARCH (Zakoian, 1991 cited by Ding et al., 1993) by given the different definition of the model parameters. In this paper, ARMA process with asymmetric power ARCH errors (including 2 GARCH, TS-GARCH and GJR-GARCH) is used to model the S&P 500 stock index returns with the t distribution and the skew-t distribution in order to see how well it captures the asymmetric and fat tail of the asset returns compared with the normal distribution. To measure the goodness of fit, we use maximum log-likelihood value, Akaike Information Criterion (AIC), Bayesian information criterion (BIC) to choose the better fitted model. Finally, we choose the ARMA (0, 2)-APARCH (1, 1) with skew-t distribution to be the best fitted model and do the 10-step-ahead forecasting. This paper is organized as follows: section 2 will introduce the basic model which contains the ARMA (p, q) as the mean equation with innovations following GARCH and APARCH process. Normal distribution, student t distribution and skew-t distribution as the conditional distribution of the standardized residuals will be also specified. Section 3 will discuss about the statistical properties of the S&P 500 stock index returns, test for unit root and ARCH effect and give some empirical analysis. Section 4 is going to give the estimation result of the ARMA-GARCH/APARCH model with t and skew-t distribution compared with Gaussian distribution. Section 5 is going to do the forecasting based on the estimation results and the selected model and compare it with the ARMA (0, 2)-GARCH (1, 1) model with normal distribution. The final section will draw conclusions and give further discussion about this paper. 2. Methodology 2.1Conditional Mean Equation We can describe the mean equation of a financial time series y as t y = E(y ψ )+ε (1) t t t−1 t WhereE(y ψ )is the conditional mean of y given ψ , ψ is the information t t−1 t t−1 t−1 set at timet−1. Sometimes in order to model the serial dependence and get the conditional mean 3 equation, ARMA (p, q) model is used to fit the data to remove this linear dependence and get the residualε which is uncorrelated (but not independent). t p q y =μ+∑φy +∑ϕε +ε (2) t i t−i j t−j t i=1 j=1 This ARMA (p, q) process is stationary when all the roots of φ(z)=1−φz−φz−....−φz =0 lie outside of the unit circle. 1 2 p To specify the order of the ARMA process, use Akaike information criterion (AIC) and the Bayesian Schwarz criterion (BIC) to choose the ARMA term which minimize the corresponding value of the criterions. 2.2 Variance Equation 2.2.1 GARCH The error term ε in the ARMA mean equation is defined by Engle(1982) as t autoregressive conditional heteroscedastic process which can be decomposed as follows: ε =σz z ~iid(0,1) (3) t t t t whereσ2 = E(ε2ψ ) is the conditional variance of the error andE(εψ )=0. t t t−1 t t−1 The ε is uncorrelated but its conditional variance σis changing over time as the t t function of the past errors defined in Engle(1982), and then generalized by Bollerslev (1986) who extended the ARCH model to have longer memory and more flexible lag structure by adding lagged conditional variance to the model: εψ ~ N(0,σ2 ) (4) t t−1 t q p σ2 =α +∑αε2 +∑βσ2 =α + A(L)ε2 +B(L)σ2 (5) t 0 i t−i i t−i 0 t t i=1 i=1 4 Where p≥0,q≥0,α >0,α ≥0,i=1,....,q, β ≥0,i =1,....,p 0 i i When p =0, the GARCH model is reduced to ARCH model. Bollerslev (1986) has proved that the GARCH (p, q) process as defined in (4) and (5) is wild-sense stationary withE(ε)=0, var(ε)=α(1−A(1)−B(1))−1andcov(ε,ε)=0for t ≠ sif t t 0 t s and only ifA(1)+B(1)<1. The GARCH (1, 1) model is parsimonious but the most effective to model the conditional volatility. It has been studied a lot in the literature and yields abundant results modeling the conditional heteroscedasticity of the financial time series successfully. εψ ~ N(0,σ2) t t−1 t σ2 =ϖ+αε2 +βσ2 (6) t t−1 t−1 Whereϖ>0,α≥0,β≥0, ψ is the information set available at timet−1,ε is t−1 t wide-stationary if and only ifα+β>1. 2.2.2 APARCH The APARCH (p, q)- Asymmetric power ARCH model was introduced by Ding, et al., (1993). It changes the second order of the error term into a more flexible varying exponent with an asymmetric coefficient takes the leverage effect into account. The variance equation of APARCH (p, q) can be written as ε = zσ , z ~ N(0,1) t t t t p q σδ =ω+∑α(ε −γε )δ +∑βσδ , t i t−i i t−i j t−j i=1 j=1 ω> 0,δ>0, α ≥0,−1<γ <1, i=1,....,p, β ≥0, j =1,....,q i i j The APARCH(p,q) process is stationary if 5 ω>0,∑αk +∑β <1,k = E(z −γz)δ i i j i i i j The APARCH (p, q) model is a general class of model which includes special cases as ARCH by Engle (1982), GARCH by Bollerslev(1986), TS-GARCH by Taylor and Schwert (1986 cited in Ding et al., 1993), GJR-GARCH by Glosten et al. (1993 cited in Ding et al., 1993), TARCH by Zakoian (1991 cited in Ding et al., 1993) and other two models by giving the different definition of the model parameter. In this paper, I only talk about TS-GARCH model, GJR-GARCH model and the complete APARCH model. a) TS (Taylor-Schwert) GARCH whenδ=1,γ =0 i p q σ =ω+∑αε +∑βσ (7) t i t−i j t−j i=1 j=1 ω> 0α ≥0i=1,....,p β ≥0, j =1,....,q i j b) GJR-GARCH whenδ=2 p q σ2 =ω+∑α(ε −γε )2 +∑βσ2 (8) t i t−i i t−i j t−j i=1 j=1 ω> 0, α ≥0,−1<γ <1, i=1,....,p, β ≥0, j =1,....,q i i j c) DGE (Ding, Granger and Engle) GARCH p q σδ =ω+∑α(ε −γε )δ +∑βσδ (9) t i t−i i t−i j t−j i=1 j=1 ω> 0,δ>0, α ≥0,−1<γ <1, i=1,....,p, β ≥0, j =1,....,q i i j 6 2.3 Conditional Distributions 2.3.1 Normal Distribution The standard GARCH (p, q) model introduced by Tim Bollerslev(1986) is with normal distributed errorε =σz ,z ~ iid(0,1). Use maximum log-likelihood method t t t t to estimate the parameter in the standard GARCH model, given the error following the Gaussian and we can get the log-likelihood function: 1 − εt2 1 −zt2 1 L(εtθ) = ln∏ 2πσ2 e 2σ2t = ln∏ 2πσ2 e 2 = −2∑[log(2π)+log(σ2t)+ zt2] t t t t t where θ is vector of the estimates parameters. 2.3.2 Student t Distribution Known fat tail in financial time series, it may be more appropriate to use a distribution which has fatter tail than the normal distribution. Bollerslev(1987) suggested fitting GARCH model using student t distribution for the standardized error to better capture the observed fat tails in the return series. If z here has student t distribution withvdegree of freedom, and its density function t is given by v+1 Γ( ) 2 z2 −v+1 f(z v)= (1+ t ) 2 t v v−2 (v−2)πΓ( ) 2 ε v+1 t Γ( ) 2 σ −v+1 1 f(ε v) = (1+ t ) 2 ⋅(− ) t v v−2 σ2 (v−2)πΓ( ) t 2 Where v>2is shape parameter, then the log-likelihood function of GARCH model with t distribution error can be built based on this probability density function which 7