Nonlinear Modeling And Forecasting Of SSE Composite Index Return Rates Serie

Since 1990s, chaos theory, fractal theory have been widely used in financial market issues. The financial market is essentially a non-linear dynamical system. Therefore, using non-linear theories and methods can reveal the essential characteristics of financial markets better, and has opened up a new horizon for the study of financial markets. Chaos and fractal, respectively, exploring the complex non-linear system from the perspective of dynamics and geometry. Many scholars at home and abroad have confirmed that these non-linear characteristics widely exist in the financial markets through a large number of empirical studies.Based on nonlinear theory, this paper employs standardized analysis and empirical research, combined with theoretical models and mathematical models and other statistical methods, to characterize and calculate the nonlinear structure of the stock market, such as fractal, continuity, predictability, and a series of other non-linear characteristics and the related statistical quantity.1.I use a large number of analytical methods, such as chaos theory, fractal theory, and so on, as comprehensively as possible on the Shanghai Composite to describe the nonlinear structure, and explore the internal laws of the market. To improve the accuracy of analysis on the fractal structure depends on the application of new methods. I look up a large number of domestic and foreign literatures and consider the advantages, disadvantages and applicability of a variety of methods of calculating fractal, and then I employ the latest two algorithms, which are CMA method and IRMD method, to calculate the H index of 0.5412 and 0.5442 respectively. Therefore, it confirms the existence of weak long-term memory, and the H index is about 0.54. Meanwhile, based on chaos theory, I use GP algorithm to calculate the correlation dimension of 5.2, which means Shanghai Composite Index has chaotic characteristics, and it need at least six variables to describe the basic dynamics characteristics of the data flow. The optimum time-delay is estimated by complex correlation method ofτ= 25, and the minimum embedding dimension is estimated by CAO method of m= 12.2. On the basis of the above-mentioned non-linear structure, I establish models to predict stock price trends and test the validity of the inherent law. It is feasible to makes predictions to certain degree using mathematical model. But due to the complexity of scaling exponents, it is very difficult to yield accurate predictions. The meaning of forecast lies in comparison. I employed the ARFIMA (2,0.147476,0)-FIGARCH (1, 0.45376,0) model which can capture long-term memory property and neural network with structure of 12-7-1 and the linear ARMA (2,2) model on this time series for comparison and found that predictive power of nonlinear model is far better than the linear model. The results strongly support the existence of nonlinear structure and memory characteristics in Shanghai Composite.The innovations in this paper are as follows:1. Through a multi-angle (non-linear test, fractal, chaos), multiple methods (BDS, CMA, IRMD, ARFIMA-FIGARCH, neural networks), I comprehensively study on the non-linear nature of the Shanghai Composite, and quantify the degree of non-linear of the stock market.2. Most domestic scholars use R/S analysis method to calculate the fractal H index when they study the fractal structure. I analyze the advantages, disadvantages and applicability of a variety of methods for calculating fractal index H, and then choose CMA algorithm and IRMD algorithm. CMA analysis method was proposed in 2005, and no domestic scholars have used it. IRMD analysis which is the latest method for calculating the fractal dimension was proposed by Sy-Sang Liaw, Feng-Yuan Chiu in April 2009.3. Domestic scholars often model on the stock market return series, but rarely forecast based on models. This article is based on multiple models to predict the return rates, and the results of our empirical study strongly support the existence of non-linear structures and memory characteristics of Shanghai Composite.