Variance Stability Test With Time-Varying Parameter Time Series Model

In this paper,based on the U-statistics of Jull and Xiao(2013)for checking the stability of moment conditions,a new generalized method for diagnosing the residual variance stability of regression models is proposed.This method can be applied to time-varying regression parameters.The situation can not only diagnose single or multiple structural breakpoints,but also diagnose smooth structural transitions.We give the asymptotic distribution properties of the new statistics under the original assumptions,and also examine their power under the assumption of opposition.Among them,DGPP1-4 is a single breakpoint,two breakpoints,four breakpoints and a symmetric "U"-shaped smooth transition.According to Monte Carlo's results,we can use the standard normal critical value as the standard and randomly simulate 2,000 times,taking into account the performance of the six cases,taking into account the bandwidth and auto-regressive parameters.We can find that in the case of a sample size of 100,the performance of size is not ideal and is numerically small,but as the number of samples increases,the size performance gradually improves,with Scenario 1,Scenario 3,and Scenario 6 in the sample.When the number reaches 300,size performs best.For situations where there are single structural changes and multiple structural changes,the number of samples is generally good.However,when the number of samples is further expanded,the results of size are individually large.The effect of the parameter change can be found,when we increase the bandwidth,the size of the general will be reduced,can also be found when we increase the auto-regression coefficient ?,the size of the results have been reduced.We also performed a Monte Carlo simulation analysis of the power performance of U statistics.Here we consider four kinds of data generation processes:a single structural breakpoint for the residual variance,two structural breakpoints,four structural assertions,and a smooth structural change.Here we examine the power performance of statistics in the presence of different forms of structural changes.Similarly,in each data generation process,the time-varying characteristics of the regression parameters are considered,namely the six situations described above.At the same time take into account the impact of changes in parameters and bandwidth on power.We can see that as the number of samples increases,the power under various data generation processes and under various circumstances gradually increases,and power decreases as the bandwidth increases.When the autoregressive coefficient y increases,power increases.Through Monte Carlo simulation,we can find that the size performance of statistics in different situations is different,but when the number of samples and the choice of bandwidth are more appropriate,there will be a good size under the original assumption.Similarly,considering the four data generation processes,the statistics power performance is good,indicating that the statistics can be applied to structural changes in different forms.The time-varying characteristics of the parameters in the regression model also have different effects on power.Finally,an empirical analysis was performed to verify the validity of the statistic.Through the use of the world's 12 major stock indices as samples,regression coefficient AR(1)is used to test whether the parameters are time-varying.Three different test statistics were used for testing(CUSUM,Chow,BSADF).Combining the results of the three test statistics,we can obtain the time-varying characteristics of the regression parameters,so we construct a variable-coefficient AR(1)regression model for the first-order moments.After considering whether the second-order moment is time-varying,the GARCH and GJR-GARCH models based on the variable coefficient AR(1)are constructed.Through the estimation of two GARCH models,it can be found that the residuals have heteroskedastic properties.Finally,the U statistic test proposed in this paper verifies the existence of time-varying variance.Apply the U statistic to the actual stock index and verify its validity.We find that the statistics not only have good size and power in statistics,but also have good performance in empirical tests.At the same time,based on the regression analysis of different stock indexes,due to the influence of time-varying parameters,the use of traditional fixed-moment first-order and second-order moment models is not sufficient to fully reflect the characteristics of the data.