The Structural Changes Of Time Series Models And Unit Root Test

With the transformation of economic structure and the reform of social development,most economic variables exhibit varying structural features.For instance,the macro time series presents obvious deterministic tendency shift for external shocks or institutional changes;the stock or other asset prices go up and down due to policy factors or excessive speculation.When we take those structural change factors into account,the traditional linear unit root test,such as ADF or PP test,inclines to draw wrong conclusions for the stationary feature of studied data process.Therefore,recent related studies on unit root process are mostly made in the framework of structural changes.On one hand,the incorporation of structural changes analysis make the conclusion of unit root testing more convincing;On the other hand,in such a framework,we could more clearly understand the data generating mechanism and its changing characters.For the time series yt = a + bt+ut,the structural changes could occur in the deterministic trend term a + bt,or the stochastic trend term ut.The literature on unit root test with breaks,represented by Perron,mainly focus on the changes in time sequence’s deterministic trends,such as breaks in the time trend or intercept,and relevant literature has formed a comparatively integrated system.The study on structural changes in random trends could be found in the analysis of price behavior in the assets price market,such as the popular SADF and BSADF test,which simulates the "rational bubbles" in the stock market through the path changing from unit root process to explosive process,and has been widely used in empirical study.While unit root test with breaks under deterministic or stochastic trend are both built on the basis of linear time series analysis,there are some other literatures that depict structure variation from the perspective of nonlinear modeling,and focus on the unit root test against the nonlinear stationary STAR model,such as the study on unit root testing in ESTAR framework by Kapetanios et al.(2003).On the basis of predecessors’ literature,this dissertation is from 3 aspects:the change of deterministic trend,the change of stochastic trend,and nonlinear STAR model to make systematic analysis and organization for the structural unit root test,and further study the related problem,mainly including the following parts:1,In the unit root testing with deterministic trending breaks,existing theories have proposed many different detection methods for the break locations from diversified points of view,we generalize and analyze the common popular unit root testing method with endogenous breaks.Then,under both the unit root null hypothesis and stationary alternative with breaks,the detecting powers of several common methods were compared and summarized through Monte Carlo simulations,which are expected to provide beneficial help for empirical workers.The identification for the number of breaks within deterministic trend is another important research hotspot.Misjudging the data process without breaks as with breaks or conversely,misjudging the data process with breaks as without breaks,could both lead to an eventual unit root testing error.So it is necessary to effectively identify the number of breaks before unit root testing.Traditional CUSUM and MOSUM testing is the classic method to detect the parameter stability,but both are based on the stable innovation error term.Under the settings that the information for the integration order of the error term is unknown,we study and derive the asymptotic property of CUSUM and MOSUM tests,then make further revision for them based on dynamic regression or difference regression to ensure they could effectively identify the features of structural breaks under null hypothesis of either unit root processes or stationary alternatives.Finally,take the revised MOSUM for instance,we make the simulations and the results show our revised strategy greatly improved the recognition power.Meanwhile,based on the revised MOSUM test,we could effectively identify the breaks locations neighborhood with short estimated intervals.2,For stochastic trend changes,we make related study under the framework of varying variance or auto-regression coefficients for the innovation.The former mainly relate to the study on unit root feature under time varying variance,and we organize and comment on the existing literature from the two perspectives of traditional statistic inference and bootstrap method.The latter mainly involves the transformation of unit root process and explosive process,which is the theoretical basis to test the asset market bubbles,and be focused in our analysis.With an unit root process mutated to an explosion process at some point as the alternative hypothesis,Phillips,Wu andYu(2009)propose the SADF bubble test,which is been widely used in reality modeling.However,the sup-type ADF tests are based on the null hypothesis of unit root process,without considering the possible cases of deterministic trend mutation,which means that the data process that the SADF tests are constructed on fail to effectively reflect the generating process of data,subsequent simulation also shows that the test will lead to the conclusion of "false bubble".Consequently,we introduce the deterministic trend shifts into the SADF bubble test,and make extended study to complete the testing.Simulations show that,our new testing strategy could more effectively to identify the trend breaks feature and foam characteristics for the data,thus ensuring the correction of the testing conclusion.To highlight the practical significance of our study,we carry on detailed analysis on the recent bubble phenomenon of China’s stock market,which shows some bubble zone estimated by BSADF test is just benign rise brought by deterministic time trend instead of real market bubble.3,The nonlinear smooth transition autoregressive model,which is based on the smooth transformation function,is another kind of time-series model to describe the data structure changes,and it has wide application background.However,the literatures on unit root test for such nonlinear modeling are few and lack of systematicness.In this paper,from the two aspects of using time as transition variables or using the data’s lag term as transition variables,we make a detailed description and organization on the nonlinear STAR model unit root testing.Then,in the ESTAR and LSTAR framework,we propose new F statistics to make unit root testing,which are based more flexible and common models settings compared with the existing literature.We derive the asymptotic nonstandard distribution of the F test and explore its finite sample properties,simulation results show that our F tests have greater power to identify the unit root null hypothesis than existing test statistics,such as ADF test,tKSS test provided by Kapetanios et al.(2003)and the t-type statistic by Liu Xueyan(2008).Finally,an application on the applicability of PPP hypothesis in Asian countries further underpins its practical meaning and superiority.