Unit Root Test Based On STAR Model

In recent years,as one of the main models of nonlinear time series,the smooth threshold autoregressive model(STAR)have recently gained popularity.When the nonlinear STAR model is used to model the data,some classical economic theories often involve.For example,the purchasing power parity theory(PPP)and the study of the real exchange rate are inseparable.Purchasing power parity means that the real exchange rate is stable.However,once the real exchange rate is close to the long-term equilibrium value,its behavior will be shown as a random walk process.Unit root test has become a very effective tool to test the purchasing power parity theory.On the other hand,if there exists a unit root in the data,the analysis method based on stationary data will no longer be suitable(for example,the t statistic does not follow the t distribution).Therefore,it is necessary to carry out unit root test before modeling the time series.Although the STAR model fits the data so well and gives a good explanation of e-conomics,but for this nonlinear model,using ADF test will lead to excessive acceptance of non-stationary time series hypothesis.Unlike conventional DF test,Kapetanios et al.(2003)propose KSS test statistics based on the auxiliary equation of the Taylor ex-pansion.After that,a number of unit root tests of STAR model have been of concern.However,the existed research of testing a unit root based on the STAR model has the following deficiencies:Firstly,in order to discuss the effect of nonlinear characteristics on unit root test,many studies have focused on the nonlinear characteristics of condi-tional mean of time series,restrict the local linear part and ignore the effects of local parameters on the test;Secondly,for the STAR process with the trend term,there is no difference between the two forms with the trend term.Different from the AR model,the two forms of STAR model with the trend term is not consistent,hence it is necessary to further study and analyze the trend model;Thirdly,there are few studies on the condition that the error term is heteroscedastic,especially the GARCH model with high order.Therefore,on the basis of the existing literature,this paper makes an in-depth study on the unit root test of STAR model.The main research work of this paper can be explained from the following four aspects:Firstly,considering that the local system may be stationary or non-stationary,this paper studies the unit root test of the STAR model in the case of unknown local area.When the local area was unknown(stationary or non-stationary),the Wald test statistic is present and the asymptotic distribution of test statistics is derived.Compared with the KSS test,the statistics of test level and test effect are investigated through Monte Carlo simulation.The simulation results show that the test statistic is more effective than the conventional test statistic KSS.Secondly,the unit root test of STAR model with deterministic trend is considered to distinguish two similar data generating process,which are the unit root process with drift and the STAR stationary process with deterministic trend term.Different from the linear AR model,the two forms of the nonlinear STAR model with trend term are not consistent.With regard to two different forms of STAR model with trend term,this paper intends to use direct test method and OLS de-trend method for the unit root test.Based on these two methods,this paper gives the test statistic and its asymptotic distribution,and compares these two forms,so as to obtain a more reliable test method to distinguish the STAR model with the trend from the unit root process with drift.Thirdly,from the perspective of financial time series,the local-explosive process(bubble process)occurs,but in this case,the unit root test based on STAR model is rarely mentioned in the literature.The unit root process of the STAR model is considered in the case of the local-non-stationary,especially the local-explosive process.The modified Wald type test statistic is put forward and the limiting distribution of the test statistic is derived in the paper.Through Carlo Monte simulation,this paper makes the critical value of the modified Wald statistic,and verifies the test results of the statistic.The results show that the test statistic has a very good effect.Fourthly,most of the nonlinear characteristics of financial time series not only display in the conditional mean of the sequence,but also the conditional variance.In the actual research,the fluctuation of the economic variable usually has the time variation and the fluctuation cluster in different time,hence it is necessary to study the non-linear STAR-GARCH model.In this paper,the unit root test of ESTAR-GARCH(p,q)model is discussed in detail.The asymptotic distribution of the t test statistic is derived based on the OLS method and the QML method.By Monte Carlo simulation,the paper compares the effect of the test statistic based on the different methods.It is found that the t test statistic has a better effect than the conventional DF statistic and the test statistic which based on QML method is a more effective test than OLS method.