Statistical Inference Of Nonstationary Time Series

A large number of empirical studies in the field of macroeconomics and finance show that most of the data in practical application are non-stationary.Therefore,the analysis of non-stationary time series is of great significance.This paper studies three kinds of important problems in nonstationary time series analysis.Firstly,for the ADF regression,based on the quantile regression,we extend Koenker and Xiao(2004)and Ling and McAleer(2004)'s works from finite-variance innovations to infinite-variance innovations.A robust t-ratio statistic to test for unit-root and a resampling method to approximate the critical values of the t-ratio statistic are proposed in this paper.It is shown that the limit distribution of the statistic is a functional of stable processes and a Brownian bridge.The finite sample studies show that the proposed t-ratio test always performs significantly better than the conventional unit-root tests based on least squares procedure,such as the Augmented Dick Fuller(ADF)and Philliphs-Perron(PP)test,in the sense of power and size when infinite-variance disturbances exist.Also,quantile Kolmogorov-Smirnov(QKS)statistic and quantile Cramer-von Mises(QCM)statistic are considered.An application to the Consumer Price Index for nine countries is also presented.Secondly,when the truncation lag is divergent for the ADF regression,we propose a new method for selecting of the truncation lag and estimating of the "autoregressive" parameters when the observations come from an integrated linear process with infinite variance innovations.One distinctive feature of the new approach is that we can simultaneously select variables and estimate parameters.Also,we propose a new test statistic based on Mestimate loss function with L1 constraint to test for unit-root and a re-sampling method to approximate the critical values of the statistic.The proposed methodology is illustrated with simulated data sets.Lasso penalty,in the sense of the mean absolute percentage error,is compared with AIC criterion and BIC criterion to select the lag order.The finite sample studies show that when a unit root exists,the MAPE is smaller and that the newly proposed penalty estimate is effective.An application to the Consumer Price Index for eight countries is also presented.Thirdly,we extend Chan and Kutoyants(2012)'s work from the threshold autoregressive(TAR)to generalized threshold autoregressive models(GTAR).It is shown that under certain regular conditions,the maximum likelihood estimator(MLE)and Bayesian estimator(BE)of the threshold parameters are super consistent with convergence rate n.And the moments of the Bayesian estimator exist when the corresponding moments of the noise are finite and its limit distribution is a functional of integrated compound poisson processes.Furthermore,the estimators of the regression parameters are shown to be asymptotically normal with convergence rate(?).We consider two threshold autoregressive(TAR)models with one and two thresholds,respectively.It is shown that BE has smaller bias and sd than those of MLE,which is consistent with Chan and Kutoyants(2012).