Theory And Application Of Unit Root Test

Unit root test is the basis for further analysis of time series. Since DF unit root test is proposed by Dickey and Fuller (1979,1981), unit root test theory has got further development, which mainly focused on two aspects:the first is the appearance of new Parametric, Semiparametric and Nonparametric unit root test (stationarity test) methods; the second is unit root test's development in nonlinearity, such as Structure Break, Threshold Autoregression, Smooth Transition Autoregression and Markov-Switching, etc. High order trend is one characteristic of the macroeconomic variables, and the other is structure break. For this reason, the study of unit root test with high order trend and structure break not only is one of the frontier research contents, but also will provide theory premise for macroeconomic variables'stationarity analysis. Structure break may appear in the deterministic part of the data, may also appear in its stochastic part, which reflects in the autoregressive coefficient on the one hand, and in the variance of the error term on the other hand. This is the main starting point of this study, and in this paper, the unit root test theory with high order trend and structure break is discussed in detail, and Breitung nonparametric unit root test has been expanded and improved.For unit root test with high order trend, through the comparative analysis of (high order) trend stationary process and unit root process with (high order) trend, the characteristics of their respective trends are made clear. And take ADF unit root test with second order trend item for example; ADF unit root test with high order trend item is discussed in detail. A detailed analysis and deduction of unit root test statistics' construction, of asymptotic theory's derivation and of test statistics'asymptotic distribution theory under null hypothesis are given.For unit root test with high order trend and structure break, the ADF unit root test theory with structure break existing in the deterministic part, in the autoregressive coefficient, in the variance of the error term, in both the deterministic part and the autoregressive coefficient, in both the deterministic part and the variance of the error term, in both the autoregressive coefficient and the variance of the error term, in the deterministic part, the autoregressive coefficient and the variance of the error term is discussed in detail. And a detailed analysis and deduction of unit root test statistics' construction, of asymptotic theory's derivation and of test statistics'asymptotic distribution theory under null hypothesis are given.As for the expansion of Breitung's nonparametric unit root test, this paper, based-on Quadratic Spectral and Bartlett kernel function, using the consistent estimator of series' Long Run Variance under stationary hypothesis as the denominator of test's statistic, improves Breitung's test. In this paper, the expanded test statistic is introduced and its asymptotic distributions under null hypothesis are deduced, firstly. And then, the Finite Sample Properties of expanded and non-expanded nonparametric unit root test are compared by Monte Carlo Simulation. Three conclusions are as follows:first, the difference between the asymptotic distributions under null hypothesis of the expanded test and that of the non-expanded test is only a constant in the denominator. Second, compared with the non-expanded test, the expanded test obviously reduces the left-skewed extent of statistics'probability density. Third, when error term is autocorrelated, the expanded test based on Quadratic Spectral and Bartlett kernel function has some improvement in Breitung test's size distortion. Along with the increase of error term's autoregressive (moving average) coefficient and series'autoregressive coefficient, the empirical power of expanded test based on Quadratic Spectral and Bartlett kernel function becomes greater than that of non-expanded test gradually, and this phenomenon becomes more obvious when sample size is small.In Empirical Research, the sationarity of real GNP, real GDP, real GDP per capita, real fixed investment and real final consume is analyzed in this paper. Firstly, the characteristic of high order trend, of structure break and of trend in each stage of the above macroeconomic variables is determined primarily by graphics and difference. The determined trend characteristics of the above five macroeconomic variables are all linear trend in the first stage and quadratic trend in the second and third stage. And, the structure break points of the above macroeconomic variables are determined by the method of minimizing the sum of squared residuals proposed by Perron and Zhu (2005). The years corresponding with the two structure break points of real GNP, real GDP, real GDP per capita and real fixed investment are 1983 and 2001 respectively. While the years corresponding with the two structure break points of real final consume are 1981 and 1999 respectively. Secondly, the stationarity of the above macroeconomic variables is tested using the ADF unit root test in the case of both structure break factor is included and isn't included, and the test results are further determined using simulation data. The results show that, real GNP, real GDP, real GDP per capita and real fixed investment have structure break in 1983 and 2001 respectively, and are I(1) process with drift in the first stage and with linear trend in the second and third stage. Real final consume has structure break in 1981 and 1999 respectively, and is I(1) process with drift in the first stage and with linear trend in the second and third stage. However, when structural break factor is not included, real GNP, real GDP, real GDP per capita and real final consume would be 1(2) process with quadratic trend, and real fixed investment would be an 1(3) process with quadratic trend. So, in unit root test with high order trend, the "Perron Phenomenon" that treats stationary process with structure break as unit root process is also appeared.