Bayesian Estimation For Threshold Autoregressive Model With Two Threshold Variables

In the time series analysis,the research of the Threshold Autoregressive model has attracted much attention,but most studies only contain one threshold variable,which will limit its applicability in the actual data analysis.In practical problems,the autoregressive model with two or more threshold variables is required for analysis.In the analysis of the Hang Seng Index,Chen et al.(2012)used the two threshold variables threshold autoregressive(TTV-AR)model and studied the model's estimation and testing through classical statistical method.However,delay parameters were not considered in the TTV-AR model of Chen et al.(2012).Therefore,the TTV-AR model of Chen et al.(2012)is not perfect.From the perspective of Bayesian statistics,the TTV-AR model deserves further study.Therefore,for the TTV-AR model,we use Bayesian method to make statistical inference for the TTV-AR model.Using Markov chain Monte Carlo(MCMC)method,the required threshold value,delay parameter value and other parameter values were sampled from the marginal posterior density of each parameter to estimate,and the posterior distribution of parameters was deduced.Then using the TTV-AR model and parameter Bayesian estimation method to analyze the actual problem.Specific practices are as follows:Firstly,the Markov chain Monte Carlo MCMC method is briefly introduced,and the statistical structure analysis of TTV-AR model is introduced.Secondly,the selection of the prior distribution of parameters of TTV-AR model,the posterior distribution of parameters and MCMC sampling method are discussed;Then,through the simulation experiments verify the validity of the sampling method,the four models are simulated and analyzed respectively.The results show that our Bayesian estimation method can effectively estimate the delay parameters,threshold values and regression coefficients under various regimes,and the effect is very good.Finally,the two threshold variables threshold autoregressive(TTV-AR)model is used for empirical analysis.In the empirical analysis,we respectively illustrate three different types of data.1.Hong Kong Hang Seng Index Analysis: Using the past information of prices and market turnover as two threshold variables,we examined the daily yield series of the Hang Seng Index from 2013 to 2017.In the analysis,the model is compared by setting two different regimes models,and the optimal autoregressive TTV-AR model is selected according to the information criterion.2.The RMB exchange rate analysis: We use the two variables,that is,the difference between the day opening price and the closing price of the previous day and the amplitude of the exchange rate to construct threshold variables.studying the characteristics of nonlinear dynamics after the RMB exchange rate reform(taking USD/CNY as an example).Moreover,we analyze the impact of random shocks on the RMB exchange rate under the management of floating system,and reveal the characteristics of "reciprocity" between the exchange rate fluctuations and fluctuations.3.Gross National Product analysis: Modeling and analyzing the dynamic path of GNP growth rate in recent 40 years.Constructing AR(3)model and four-regime TTV-AR model,and the four regimes TTV-AR model is selected to describe the growth rate of GNP through the information criterion.The results shows that the piecewise linear nature of the TTV-AR model can capture the asymmetric behavior of GNP during recession and expansion.