The Stochastic Unit Root Test Of Time Series Models

In order to explore the generation of data processes (DGP), we must try to model the DGP in historical data and forecast it. By the classical analysis of unit-root tests, whether ADF test or P-P test, they consider the unit-root tests of the models, whose errors are weak stationary or strong mixed processes. On the other hand, the models used to test the unit-root of time series data are assumed that the essence of the unit-root is not need to study in the DGP. In fact, the unit-root of real DGP in finance and economic doesn't exist certainly. Since Dickey and Fuller (1976) proposed the DF test and extend to ADF test in 1979, the essence of the unit-root of non-stationary time series data doesn't need to study in our mind, but the experience of the financial and economic data shows we should study the essence of the unit-root. Granger and Swanson (1994)[1] study the stochastic unit-root at first. They show the essence of the unit-root of time series should be studied. We can't be declare the unit-root weather exist or not in mind. The unit-root of DGP exists at some time, and does not exist at some time. This characteristic is called stochastic unit root, STUR. McCabe and Tremayne (1995)[2] consider the LBI(locally best invariant)test for difference stationary against STUR under local heteroscedastic integration. Granger and Swanson(1997)[3] study a unit root test against the alternative of a stochastic unit root under normal error in detail. Park WingFong and Wai KeungLi (2003)[4] study the tests on time series with randomized unit root and randomized seasonal unit root under the normal distribution error. Studying the papers about STUR processes, we find that the models are based on the assumption of the normal error. However, we can't determine the time series data, especially financial and economic data, follows the normal distribution. In fact that our practical experience show the real financial and economic data are fat tail distribution, it is unfit to assume that the real data are normality. In order to fit the real DGP, We allow for the general error distribution (GED), estimate the parameter by A-MLE (Approximate MLE), and show the limiting distribution of the statistic. When the parameter of GDE is equal to 2, it is normality. While the parameter of the GED follows 0