1.1.1 Testing for non-stationarity of real exchange rate: Unit Root test, Variance Ratio test and Fractional Integration
When the test of PPP goes to the second stage, the main method is testing the non-stationarity of real exchange rate. At this time the researchers have noticed the existence of spurious regression problem using simple regression analysis, and tried to improve. From the middle of 1980s, they began to use the basic approach of augmented Dickey-Fuller tests (ADF tests) to test the existence of unit root in the real exchange rate changes. Consider the following general form of the regression:
(3-2)
Where is a white noise process.
Null hypothesis is , using the ADF test is equivalent to testing the existence of unit root in the generation process of . If the unit root exists, then there is no long-run equilibrium level.
The second approach to test nonstationarity of the real exchange rate is to the variance ratio test. A simple non-parametric z (k) can be used to test persistence of the real exchange rate. This method is originally proposed by Cochrane (1988):
(3-3)
Where k is positive, represents the variance. If the real exchange rate is a random walk process, then the variance ratio should be equal to 1, since changes of variance in k period should be k times of changes of variance in the first period. Conversely, if the real exchange rate has the property of mean reverting, and then z (k) should be in the range of 0 and 1.
The third approach is using Fractional Integration to test nonstationarity of time series in a wider context. The hypothesis used under this approach is different from traditional unit root tests. Typically, the process of real exchange rate is expressed as follows:
(3-4)
Which, and are lag operators L of the P-order polynomial, their roots are outside the unit circle. is a white noise process. In this approach, the parameter d can be continuous in the interval between 0 and 1. Fractional Integration process is more persistent than purely ARMA process, and more stable. If d = 0, then the real exchange rate would follow the ARMA process. On the other hand, if d, and are in the same units, then the real exchange rate is a random walk process.
After the late 1980s, some researches using the approaches described above to study the behavior of real exchange rate under floating exchange rate regime, found that the results cannot reject the hypothesis that the real exchange rates in the major industrial countries are all subject to random walk processes (Adler and Lehmann, 1983; Mark, 1990; Meese and Rogoff, 1988). This in turn led people to believe that the deviation of the real exchange rate from PPP is permanent. But later it was found that the above results do not support PPP since unit root test and integration test maybe inefficient (i.e. accept the wrong hypothesis).
1.1.1测试实际汇率的非平稳性：单位根检验，方差比检验和分数次积分
当PPP试验进行的第二阶段，主要的方法是测试实际汇率的非平稳性。在这个时候，研究者发现虚假回归问题，使用简单的回归分析的存在，并试图改进。从20世纪80年代中期，他们开始使用增广迪基-富勒测试的基本方法（ADF检验）检验单位根的实际汇率变动的存在。考虑以下的一般形式的回归：
（3-2）
白噪声过程。
零假设是，利用ADF检验是相当于测试产生过程中的单位根的存在。如果单位根的存在，就没有长期均衡水平。
测试的实际汇率的非平稳性的第二种方法是方差比检验。一个简单的非参数Z（K）可以用来测试实际汇率的持续性。这是由Cochrane最初提出的方法（1988）：
（3）
其中k是正的，表示的方差。如果实际汇率是一个随机游走过程，然后方差比应该等于1，因为在K周期方差变化应在第一期的方差变化的K倍。相反，如果实际汇率的均值回复特性，然后Z（K）应在0和1的范围内。
第三种方法是使用分数集成测试在更大范围内的时间序列的非平稳性。在使用这种方法的假设，不同于传统的单位根检验。通常情况下，实际汇率的过程表示如下：
（3-4）
其中，和滞后的p阶多项式算子L，其根源是单位圆外。是一个白噪声过程。在这种方法中，参数D可以在0和1之间的时间间隔连续。分数次积分过程比纯粹的ARMA过程更持久，更稳定。如果D= 0，则实际汇率也会跟着ARMA过程。另一方面，如果D，和在同一个单位，然后实际汇率是一个随机游走过程。
在20世纪80年代后期，利用上述研究浮动汇率制度下汇率行为的方法研究，结果发现不能拒绝的假设，在主要工业国家的实际汇率都是服从随机游走过程（艾德勒，1983；莱曼，1990；米斯和标记，罗戈夫，1988）。这反过来又导致人们认为，实际汇率与购买力平价的偏离是永久性的。但是，后来人们发现，上述结果不从单位根测试和集成测试也许低效支持PPP（即接受错误的假设）。
