# 代写留学生论文 实际汇率的非平稳性

### 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

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