Analysis Of Shanghai Stock Index Based On Markov Switching Model

The stock market switches between bull and bear markets,with frequent fluctuations and complex changes.It is of great significance to study the fluctuation of Shanghai Composite Index for China's economic development and social stability.In the theoretical foundation part,the parameters of OU process model with Markov switching are derived from EM algorithm combined with composite likelihood function,and the convergence of EM algorithm is proved.In the case that the Markov chain has three states,assuming the mean and variance of the three states as well as the transfer rate matrix,(Q+,Q-)which is the Wiener-Hopf factorization for matrices of the Markov process is obtained.Using the diagonalization method,calculate matrix Q+ and matrix Q-corresponding left eigenvec-tor and right eigenvector respectively.According to the cross-product operation,we can get exp(Q+(l-x))and exp(Q-(x-l)).Finally,the average waiting time of bull market,bear market and shock city is obtained.In the part of empirical analysis,the Shanghai Composite Index from January 4,2011 to December 31,2020 is selected as the research object.The AR model and ARMA model about the daily return rate of Shanghai Composite Index were established successively through the white noise test and the stability test.The order of the model was determined according to the information criterion,and the model passed the significance test.Comparing AR model with ARMA model,it is found that ARMA model has better effect than AR model.Chow test was used to test ARMA model,and it was found that there were mutation points in the time series.So the Markov switching model is introduced.Firstly,the MSAR model is introduced.The number of states and order in the MSAR model is determined according to the information criterion.Then,the state transition matrix and model parameters are obtained by Eviews under the assumption that the residual term obeys normal distribution.LR test was used to compare AR model with MSAR model,and it was found that MSAR model was superior to AR model.Under the assumption that the residuals obey normal distribution and T distribution respectively,the state transition matrix and model parameters are obtained by MATLAB,and the filtering probability and smooth-ing probability graphs of different states are made.According to the filtered probability and smoothed probability,the analysis shows that in the nearly 500 days before December 31,2020,the probability of the market being in the bull market is greater than the probability of the bear market in most of the days,and in the nearly 100 days before December 31,2020,the probability of the market being in the bull market is much greater than the probability of the market being in the bear market.Then the OU process model with Markov switching is introduced,and the parameters and state transition matrix of the model are obtained by using MATLAB code according to EM algorithm.Comparing the OU process model with the MSAR model,the expected duration of different states is calculated by the state transition probability,and it is found that the expected duration of rising state is longer than that of falling state in these two models.Finally,according to the Geometric Brownian Motion model,the closing points of the Shanghai Composite Index in nearly 500 days are fitted.According to the model parameters,it is judged that the Shanghai Composite Index will rise in the near future,which is consistent with the results obtained from the filtering probability and smoothing probability.