英文文献：Type I and Type II Errors in the Unit Root Determination of a Fractional Brownian Motion-分形布朗运动单位根确定中的第一类和第二类误差
英文文献作者：Wongsasutthikul, Paitoon,Turvey, Calum G.,Power, Gabriel J.
英文文献摘要：
Economists who deal with time-series data usually take the unit root test as the ‘prerequisite’ test for a Brownian motion. It is typical for any researchers to apply a battery of well-known unit root tests to their models to confirm stationarity in the model specification. Nonetheless, often times, we see a conclusion that fail to reject the null in favor of the existence of unit root even though the model specification is such that the lag coefficients of an AR(q) process do not sum up to unity. In this study, we show that having the sum of the lag coefficients equals to unity is indeed a necessary and sufficient condition for the existence of a unit root. Hence, the aforementioned incident will lead to a type II error in the unit root determination. On the other hands, type I error results when we reject the null that there exists a unit root when in fact the null is true. The fractional Brownian motion (fBm) process which has stationary but not necessarily independent increments is used to convey the findings of this study. We use Hurst exponent as a gauge for persistency in the data and show that a fBm process is a legitimate stochastic process with unit root even though it exhibits a degree of persistency in time.

处理时间序列数据的经济学家通常将单位根检验作为布朗运动的“先决条件”检验。典型的是，任何研究人员都在他们的模型中应用一组著名的单位根检验来确认模型规范中的平稳性。尽管如此，很多时候，我们看到一个结论不能拒绝零，而支持单位根的存在，即使模型规范是这样的，一个AR(q)过程的滞后系数总和不是统一的。在本文中，我们证明了滞后系数之和等于一确实是存在单位根的充要条件。因此，上述事件将导致在确定单位根时出现第二类错误。另一方面，当我们拒绝存在单位根的空值，而实际上空值为真时，类型I错误就会产生。分数布朗运动(fBm)过程具有平稳但不一定独立的增量，用来表达本研究的结果。利用Hurst指数作为数据持久性的度量，证明了fBm过程是一个具有单位根的合法随机过程，尽管它在时间上表现出一定的持久性。