Research On Stable Distributions And ARMA-GARCH Model In Stock Market Returns

Since the kurtosis of the distribution of the stock market return series is higher than the kurtosis of the normal distribution,this indicates that investment in financial products has an isotropic effect for more people,i.e.,more people have gains when the market has gains and more people will have a loss effect when the market has losses.The distribution is characterized by spikes and thick tails,so it is necessary to study in more detail under what distribution the returns obey,the fitted model works better.Also the volatility of the return series will change over time,and subsequently a volatility fitting model is needed,while investors react asymmetrically in the face of gains and losses,and the asymmetry of positive and negative shocks to the return series requires a more realistic model.The study of stock market returns in this paper is divided into two aspects,a theoretical research part and an empirical research part.In the theoretical study,firstly,the definition of stable distribution under different perspectives and its properties are given,and the focus is on the quantile estimation method to estimate the parameters of stable distribution and the goodness-of-fit test method KS test.The definition of the truncated stable distribution and its properties are then studied,and its second-order,fourth-order,sixth-order,and eighth-order moments are derived from the studied eigenfunctions,and its kurtosis expressions are also derived.Finally,it is shown that the kurtosis of the truncated stable distribution decreases as the characteristic index ? increases.In terms of empirical research,this paper first performs parameter estimation and fitting analysis of stable distributions for the SSE and CSI 300 indices,and conducts descriptive statistical analysis and normality tests on the return series.Subsequently,under the assumption that the yield series is stably distributed,its parameters are estimated by the quantile estimation method,and a series of 1000 values is randomly generated with its parameters and KS test is performed,and it is found that the original hypothesis cannot be rejected,while the images of the stable and normal distributions fitted to the yield series are drawn with its parameters,and both results show that the stable distribution fits the yield series better than the normal distribution.The time series of CSI 300 index return series for all trading days from January 1,2010 to December 31,2021 are selected for time series modeling and tested for smoothness and white noise,and the relative best-fit model ARMA(3,2)is established according to the AIC criterion,and the forecasting effect is found to be poor using the ARMA model.The residual series are then tested for ARCH effects and the volatility is modeled with a GARCH-like model,and comparing the AIC values revealed that the ARMA(2,2)-GARCH(1,1)model is the best in order,while the ARMA(2,2)-GARCH(1,1)model is found to fit the model best under the generalized error distribution.In order to respond to the asymmetry of positive and negative shocks in the return series,ARMA(2,2)-EGARCH(1,1)and ARMA(2,2)-TGARCH(1,1)modeling analyses are conducted respectively,and parameter estimates are given under three different distributions,and the results show that there is indeed a leverage effect in the logarithmic return series of the CSI 300 index.Finally,the ARMA(2,2)-GARCH(1,1)model is analyzed for the Va R value of the risk measure under normal distribution,and the forecasts are rolled backward starting from the first 2800 data with an interval of 1 day and a step size of 117 days,and for each forecast,a model is regenerated to continue the rolled backward forecasts,and the Kupiec failure rate test is performed,and it is found that the actual number of breakout days is not significantly different from the theoretical breakout days,i.e.,the estimated Va R value is valid.