Theoretical Extensions And Applications Of Cointegration Tests With Structural Changes

Since the introduction of cointegration test in the 1980s,the cointegration theory has drawn extensive attention in the field of econometrics.A large amount of research has emerged,enriching and developing the cointegration theory from many aspects.As the most important analysis tool in the field of multivariate non-stationary time series research,cointegration test has an irreplaceable role in the empirical research.Considering that the deterministic trend or volatility parameters of many macroeconomic and financial time series may change over time,the cointegration test with structural change has become a cutting edge of econometrics.By insightfully interpreting of the traditional cointegration theory,we are committed to substantially expanding the cointegration theory and applying it to some practical economic issues.In multivariate time series analysis,the spurious regression phenomenon may arise among several independent non-stationary time series variables.In contrast,cointegration means that there is a long-term stable relationship among variables.In this thesis,through theoretical derivation and Monte Carlo simulations,we analyze the cause of statistical inference failure in the case of spurious regression and explain the super-consistency of the regression coefficients among cointegrated variables.The cointegration test provides a method for identifying long-term relationships among non-stationary variables.This thesis takes the EG two-step method,the Johansen test and the ECM test as examples to provide an introduction and in-depth interpretation of the idea of cointegration test,model estimation,as well as the construction and the limiting properties of test statistics.In the standard cointegration test,people assume that the parameters in the data generating process remain unchanged,while many key macroeconomic or financial variables are characterized by permanent volatility shifts.We relax the constant-variance assumption on the standard residual-based cointegration test,allowing a wide class of permanent variance changes in variable disturbance terms without the need to model specific form of variance.First,we extend the residual-based DF cointegration test to allow for time-varying variance and obtain its asymptotic null distribution under this circumstance.Since this distribution depends on the unknown form of time-varying variance,the standard test is no longer applicable.To deal with time-varying variance,we propose a wild bootstrap algorithm which is suitable for multivariable systems.By bootstrap resampling data,the corresponding test statistics is constructed to obtain the critical values of the residual-based DF test under time-varying variance.We prove that the bootstrap-based DF cointegration test achieves asymptotic validity in the presence of time-varying volatility.Furthermore,we extend the residual-based(5(5test to allow for autocorrelation and time-varying variance and derive the asymptotic properties of the test statistics.The corresponding wild bootstrap test is proposed,and its asymptotic validity is proved.Simulation results show that the two wild-bootstrap-based tests proposed in this thesis perform well with finite samples.Applying our proposed method to the Bitcoin and the China’s stock market,we test whether there was a cointegration relationship between the price of Bitcoin and the CSI 300 Index during a nine-month growth period before China banned the Bitcoin trading in September2017.The results of the tests provide evidence of the relative isolation of the Bitcoin market from China’s stock market.This thesis further studies the cointegration test for the variables with deterministic trends.For the residual-based cointegration test,the usual operation is to perform GLS detrending on the data,then use the detrended data to construct test statistics.We improve and extend the GLS detrended version of residual-based cointegration test from two aspects.First,we modify the choice of GLS detrending method under certain deterministic component setting.Second,we extend the GLS detrended version of residual-based cointegration test to allow for time-varying variance.The weak convergence of the partial sum processes constructed by the GLS-detrended data is proved under this circumstance.Theoretical studies indicate that the residual-based(5(5test with GLS detrended data is invalid since the time-varying volatility changes the asymptotic distribution of its test statistics.We propose a wild bootstrap algorithm suitable for multivariate GLS detrended data to obtain the finite sample critical values of the original test.The wild bootstrap test statistic has the same limiting distribution as the original test statistics,which confirms the validity of this algorithm.Monte Carlo simulations show our GLS detrended version of wild bootstrap procedure works well with finite samples.We apply our proposed method to test the long-term relationships among crude oil price,stock index and exchange rate of the world’s major oil-importing and oil-exporting countries.The results show that these three variables are cointegrated for most major oil-importing countries,while the patterns are different for most major oil-exporting countries.Furthermore,this thesis extends the cointegration test to allow for trend structure and variance structure changes.First,we study the residual-based cointegration tests when the trend structure and variance structure have a single break at the same time.We test the null of no-cointegration with a single break of variance against the alternative of cointegration with a single break of trend and variance at the same point.When the break point is known,we derive the asymptotic distribution of the usual residual-based cointegration test statistics,which depends on the break point and variance change ratios.When the break point is unknown,we propose to estimate the break point by quasi maximum likelihood method and calculate the variance change ratios.Based on this,we can construct the residual-based cointegration test statistics and obtain the critical values under the corresponding parameter setting.Monte Carlo simulations show that our proposed tests perform well with finite samples when the regression form is set correctly.With further relaxation of the assumption on variance of the error terms,we study the residual-based cointegration tests in the case where the trend structure has single or double breaks and the variance has more general form of time-variation.The null hypothesis is no-cointegration with time-varying variance,while the alternative is cointegration with time-varying variance and single or double breaks of trend structure.We propose two infimum cointegration tests based on the wild boostrap method and prove their asymptotic validity.Monte Carlo simulations show that our proposed tests have good finite sample size and power performance.