Introduction
The motivation of why I choose purchasing power parity Theory which is I am very interesting about the reason of the same good can have huge different price in different countries. For example in US a BMW335i motor is only 42800 dollar. But in China it cost 626000 yuan, equivalent of 98366 dollar. Another way , the cucumber per Kg in China is 5.6 yuan , but in US it is cost 3.9 dollar per Kg. That means the same cost can have different earnings in different countries. Of course there are many factors lead to this result, exchange rate, tariff, the relationship between supply and demand, interest rate and so on. But how can we say it is a correct or reasonable price?
The purchasing power parity Theory is a wide application approach. It initially by an economist Thornton in United Kingdom in 1802 .With the passage of time, it become a part of the classical economic theory and finally an Sweden economists Gustav Cassel(1866-1945) developed and enriched this theory.
There are four hypothesis on the theory. First, international trade must be totally free. Seco
nd, the changes in commodity prices should be the same for all ranges. Third, price is the only factor affecting exchange rate. Finally, the amount of currency is the only factor affecting purchasing power parity. And the most important viewpoint of purchasing power parity theory is a currency that is calculated according to the price levels in different countries between the coefficient of equivalence, to enable fair comparison of the gross domestic product of countries, and there have a lot of gaps between this theory and actual rates of exchange. In other word, the reason of domestic need foreign currency or foreign need domestic currency is the two currencies in the issuing country has the purchasing power parity of commodity, the exchange rate between the two countries due to the ratio between their purchasing power parity. There are two senses in  purchasing power  parity theory hypothesis. Absolute Purchasing Power Theory and Relative Purchasing Power Theory. Absolute Purchasing Power Theory is also known as The Law of One price. The conditions of law of one price contains all the countries have adopted the same degree of currency convertibility, the information must be fully, there is no transaction costs and the tariffs should be zero. Then we can conclude that the equilibriu
m exchange rate between the domestic currency and foreign currency should be equal to the ratio of purchasing power or price levels between the domestic and foreign countries. But absolute is not perfect, so the economists use Relative purchasing power theory which is “the change of the exchange rate over a given period just offsets the difference in inflation rates in the countries concerned over the same period(Journal of Economic Perspectives pp137)”. Contrast to absolute purchasing power parity theory , it consider with inflation rate because sometimes inflation will reducing the purchasing power parity of currencies. The relationship between absolute and relative PPP is that if absolute PPP holds then the relative PPP must hold because the price index is the ratio between . But if relative PPP holds, the absolute PPP dose not necessarily hold, since “it is possible that common changes in nominal exchange rates are happening at different levels of purchasing power for the two currencies(Journal of Economic Perspectives pp137)”.
  My study aims to investigate the relationship between the purchasing power parity and exchange rate, inflation rate. I chose China and US to be my investigation objects. China is the biggest developing country and it kept 10% growth rate over the last 30 years. In 20
10, it overtook japan as the world’s second largest economy. At the same time, China is the largest exporter and second importer of goods in the world. US, the largest economy of the world, approximately a quarter of the global GDP.
Theoretical and empirical relationships
In order to investigate whether the purchasing power parity holds, I collected the CPI data absolute relativeof China and US from 1993-2012 and the data of the exchange between those two countries. First I used the approach which has been proposed by Frankel on 1978. This approach attempted to test whether the real exchange rate is stationary. In the other words, it tested whether the real exchange rate process follows a random walk. Then I try to test the following equation:
qt=st-pt+pt*
which st is the log of nominal exchange rate, pt is the log of CPI in China and pt* is the log of CPI in US, qt is the real exchange rate. This approach has two hypothesis:
              H0: qt ~ I(1)
              H1:qt ~ I(0)
The null hypothesis means that the equation contains at least one unit root and the series is not stationary. The other one means that there is no unit root.So if we can reject the null hypothesis means that the purchasing power parity does hold in the long-run.
  From eviews I got a graph of the real exchange rate between the CPI of China and US.
Visual inspection offers several insights. First, we may note that the variable qt has a positive intercept at about 2.05. Also it shows an erratic behaviour. It dose not move around its mean, it fluctuated violently from 1994-1995 and the hump is about 2.29, after that it show a sharp in the trend until to 1996.After that it increase slowly and erratically. From 2006 it show a decrease in the trend. Finally there is not an obvious trend.

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