Asymptotic Distributions of the Least-Squares Estimators and Test Statistics in the Near Unit Root Model with Non-Zero Initial Value and Local Drift and Trend
研究了非零初始值且含漂移和趋势的模型中Dickey-Fuller检验的分布，通过随机积分表示和Fredholm方法推导特征函数并列表渐近分布，结果显示了检验对初始值的依赖。
This paper considers the distribution of the Dickey-Fuller test in a model with non-zero initial value and drift and trend. We show how stochastic integral representations for the limiting distribution can be derived either from the local to unity approach with local drift and trend or from the continuous record asymptotic results of Sørensen [29]. We also show how the stochastic integral representations can be utilized as the basis for finding the corresponding characteristic functions via the Fredholm approach of Nabeya and Tanaka [16,17], This “link” between those two approaches may be of general interest. We further tabulate the asymptotic distribution by inverting the characteristic function. Using the same methods, we also find the characteristic function for the asymptotic distribution for the Schmidt-Phillips [26] unit root test. Our results show very clearly the dependence of the various tests on the initial value of the time series.