Does Purchasing Power Parity (ppp) Hold?

Introduction

The theory of Purchasing Power Parity (PPP) is not new to economics. It was first introduced by Cassel in 1918, but it was presented by himself three years before using an equivalent term “theoretical rate of exchange”(Economics Discussion, 2015). Since then much research has been done using this model targeting different countries to determine its genuity throughout the globe. In this paper, we analyzed this theory based on U.S.A, Canada, Denmark and Norway from 1978 until 2015. The reason why we choose these countries because these are the countries that ring the Arctic Ocean. Throughout the paper, Purchasing Power Parity will be tested by using the United States as the domestic country, and Canada, Denmark and Norway as foreign countries. To specify, the producer price index for manufacturing (PPI) and consumer price index for all items (CPI) will be compared amongst the countries, after converting each country's currency to the US dollar. Investopedia.com explains, consumer price index is measured by a weighted average of all items related to consumer goods and services. It is determined as a basket of goods containing things such as food, medical care, and transportation (investopedia.com). In comparison, producer price index is “a family of indexes that measures the average change in selling prices received by domestic producers of goods and services over time” (investopedia.com).

Literature Review

According to Krugman, “The Law of One Price (LOP) states that in competitive markets free of transportation costs and official barriers to trade, identical goods sold in different countries must sell for the same price when their prices are expressed in terms of the same currency” (Krugman, P. 112).  Likewise, the Purchasing Power Parity (PPP) theory exists when a basket of goods in one country is worth the same as an identical or similar basket of goods in another country (Krugman, P. 113).  Notably, Purchasing Power Parity prices are compared after considering the exchange rates between the countries.

In terms of statistics, unit root tests are used to check whether or not the time series variables are non-stationary by employing augmented Dicky-Fuller test. “The Dickey–Fuller test is a popular unit root test used to assess the time series property of economic and financial data” (Tam, 2013, p.3495). While conducting the ADF test, an issue which people come across is the choice of autoregression used in the testing equation. For this paper we have determined Lag 1, Lag 4, Lag 8 and Lag 12. We chose various lags in order to increase our certainty. We chose these 4 specific lags because they vary in equal increments, as well it tests the data quarterly in addition to monthly.

In conducting the ADF test, an issue that arises is the choice of the order of the autoregression used in the testing equation. The existence of unit root verifies the H0, which means the time series variables are non-stationary. Regarding of the subject of the feasibility of Law of One Price, non-stationary series variables prove that Law of One Price is not possibly being implemented to the entire world. “A process is said to be stationary if it’s mean, variance, and covariance do not change over time” (Stadnytska, 2010, p.1). Additionally, several other features can justify the variables to be non-stationary as well. Specifically, if there is a trend that showed by those variables, then it can be implied that statistics are non-stationary due to the trend. In economic time series, trend is usually generated by slowly evolving preferences, technologies and demographics. (Fedc.wiwi.hu-berlin.de, 2015).

“Cointegration actually presents the long-run equilibrium relationship of different time series, which is a key basic thought and theory in the current econometric field and is also an important theoretical cornerstone in current researches on combination forecasting launched by time series” (Jiang et al., 2014, p.1). This is useful because it can tell one if one set of data is related to another. Practically in the world people could use a co-integration test to find out if a stock price is related to another, and if so, they could make more profitable investments.

Data and Data Analysis

In order to keep the data consistent, all data will be collected from St. Louis Federal Reserve of Economic Data (FRED). Moreover, the data will be compared in monthly non-seasonally adjusted intervals from January 1st, 1978 to March 1st, 2015. To enumerate, January 1st, 1978 was chosen because it was the longest time series available for the given data. The producer price index for manufacturing (PPI) and consumer price index for all items (CPI) will be compared amongst the countries, after converting each country's currency to the US dollar. They will be converted by using the nominal exchange rates between the US and each foreign country.

Purchasing Power Parity will be tested in econometric software called Eviews. The software will be used to run various unit root tests, specifically Augmented Dickey-Fuller, as well as various Cointegration tests. Firstly, unit root tests will be performed on CPI and PPI for each country. Specifically, we ran unit root tests on all data represented in the same currency given the formula Pˆ=P*E.

The Augmented Dickey-Fuller unit root test will allow one to identify whether the specific data they tested is stationary or non-stationary (has a unit root), using various lags. Both stationary and non-stationary can be determined by t-values in comparison to critical values, as well as the p-values. Secondly, assuming the data has a unit root, CPI and PPI data for each country will be expressed using logarithms in Eviews. By logging the variables, it aids in moving skewed variables to be more normal, as well as helps simplify the information. The logged variables are then tested using co-integration, comparing domestic prices to foreign prices. Various outcomes can be determined from co-integration tests, and will be further explained in the co-integration section of this paper. Finally, the co-integration test residuals will be saved, then tested using Augmented Dickey-Fuller unit root test. This will again test for stationarity. After running numerous unit root and Cointegration tests, the values will be analyzed to determine whether the researched data is stationary or non-stationary if there is a long run relationship between the data, and ultimately if Purchasing Power Parity is present given the CPI and PPI, comparing the United States to Canada, Denmark, and Norway.

Empirical Results

       Regarding of CPI, discussing the US’s in lag 1, the T value -2.854 are larger than the critical value -3.978(1%), -3.420(5%),-3.133(10%) which indicates the time series variables are non-stationary. When it is lag 4, T value -2.147 is bigger than -3.978(1%), -3.420(5%),-3.133(10%), suggesting time series variables are non-stationary. In terms of lag 8, T value of US is -1.992 which is greater than -3.978(1%), -3.420(5%),-3.133(10%), proving the time series variables are non-stationary. When it is lag 12, T value -2.217 is bigger than -3.978(1%), -3.420(5%),-3.133(10%), suggesting time series variables are non-stationary

       According to the above table, for lag 1 Canada, the T value -1.83 is larger than the critical value  -3.978(1%), -3.420(5%),-3.133(10%), thus the time series variables are non-stationary. In regard of lag 4. Still for Canada, the T-value -2.157 is bigger than critical value, so we can see that the set of time series variables are non-stationary. When lag 8, Canada has -1.72 for T value that is bigger than critical value -3.978(1%), -3.420(5%),-3.133(10%), thus the time series variables are non-stationary When lag 12, Canada has -1.758 for T value that is bigger than critical value -3.978(1%), -3.420(5%),-3.133(10%).In conclusion for Canada, the time series variables are considered to be non-stationary.

The same principle can be employed to Norway, the T value -2.882 from lag 1 is larger than the critical value -3.978(1%), -3.420(5%), -3.133(10%), thus the time series variables are non-stationary. When using lag 4, Norway is -2.763 for T value is bigger than the critical value for -3.978(1%), -3.420(5%),-3.133(10%). verifying the null hypothesis of not being non-stationary. When evaluating lag 8, Norway has -2.403 is bigger than critical value -3.978(1%), -3.420(5%),-3.133(10%), thus the time series variables are non-stationary. Furthermore, in lag 12, The T-value is 1.902 is bigger than critical value -3.978(1%), -3.420(5%),-3.133(10%), the set of time series variables are non-stationary.

For Denmark, using lag 1 values, the T value -2.532 is larger than the critical value  -3.978(1%), -3.420(5%),-3.133(10%), therefore, the time series variables are non-stationary. When analyzing lag 4, the T value -2.58 is bigger than -3.978(1%), -3.420(5%),-3.133(10%), which means this variety of time series is nonstationary.  In terms of lag 8, the T value -2.409 is bigger than the critical value -3.978(1%), -3.420(5%),-3.133(10%), showing this variety of time series is nonstationary. Likewise, in lag 12, the T value -2.222 is bigger than the critical value -3.978(1%), -3.420(5%),-3.133(10%), showing this variety of time series is nonstationary.

Second, to explore the unit roots for PPI, we have utilized three different countries, including Canada, Norway, and Denmark for three significance levels, 1%, 5% and 10% levels, respectively to compare the critical values with calculated T value.

       When viewing PPI for the United States, the lag 1 T value -2.357 is larger than the critical value -3.978(1%), -3.420(5%),-3.133(10%), which indicates the time series variables are non-stationary. When the test was run using lag 4 , the T value was -2.411, which is larger than-3.978(1%), -3.420(5%),-3.133(10%), suggesting time series variables are non-stationary. In terms of lag 8, T value of US is -2.119 which is greater than -3.978(1%), -3.420(5%), -3.133(10%), proving the time series variables are non-stationary. Again, using lag 12, T value of US is -2.158 which is greater than -3.978(1%), -3.420(5%),-3.133(10%), means the time series variables are non-stationary.

Based on the above table created by augmented Dicky-Fuller test. For Canada, the T value -2.069 (lag 1) is larger than the critical value -3.978(1%), -3.420(5%),-3.133(10%), thus the time series variables are nonstationary. Still for Canada, the T-value -2.312 (lag 4) is bigger than critical value -3.978(1%), -3.420(5%),-3.133(10%), so we can see that the set of time series variables are nonstationary. Regarding lag 8, Canada has -1.851 for T value that is bigger than critical value -3.978(1%), -3.420(5%),-3.133(10%), so the time series variables are non-stationary. Considering lag 12, Canada has -1.918 for T value that is bigger than critical value -3.978(1%), -3.420(5%),-3.133(10%), so the time series variables are non-stationary.

The same principle can be employed to Norway, the T value -2.554(lag 1) is bigger than the critical value -3.978(1%), -3.420(5%),-3.133(10%), thus the time series variables are non-stationary. Then with lag 4, Norway is -2.524 for T value is bigger than the critical value for -3.978(1%), -3.420(5%),-3.133(10%) verifying the null hypothesis of not being stationary. Likewise lag 8, Norway has -2.241 is bigger than critical value -3.978(1%), -3.420(5%),-3.133(10%), the set of time series variables are non-stationary. The T value -1.94 (lag 12) is bigger than the critical value -3.978(1%), -3.420(5%),-3.133(10%), thus the time series variables are non-stationary.

For Denmark, the T value -2.39 (lag 1) is bigger than the critical value -3.978(1%), -3.420(5%), -3.133(10%), therefore, the time series variables are non-stationary. When running the augmented dickey-fuller using lag 4, the T value -2.559 is bigger than -3.978(1%), -3.420(5%),-3.133(10%), which means this variety of time series is nonstationary. In term of lag 8, the T value -2.368 is bigger than the critical value -3.978(1%), -3.420(5%),-3.133(10%), showing this variety of time series is nonstationary. In term of lag 12, the T value -2.15 is bigger than the critical value -3.978(1%), -3.420(5%),-3.133(10%), showing this variety of time series is nonstationary.

After running a unit-root test the next step in figuring out if purchase power parity (PPP) is true and if the law of one price (LOP) holds is a co-integration test. We made sure that before the Cointegration tests were ran that the data had been logged, this was in order to avoid problems of scaling with our data. Although when logging the chance of the data showing stationary becomes higher. The formula used for logging was p = α + βp̂.

Cointegration formula

p = α + βp̂t + Et.

The method used for co-integration:

Eg: ln_cpi_us c ln_cdn_cpi_us

The (ln) is the log

The (cpi) is the consumer price index

The (us) is the country which the (cpi) belongs to.

For the other labeling:

ln_cdn_cpi_us

It follows the similar method, however the (cpi) and country (cdn, in this case) have switched.

The (us) at the end of the name represents that the following data sets have been altered by the united states exchange rate. This change in labeling is done in order to avoid confusion well entering in the two data sets.

We used the United States as the domestic country for doing the co-integration. The other data sets were entered in a similar way as to the one explained above.

Ln_cpi_us c ln_nor_cpi_us

Ln_cpi_us c ln_den_cpi_us

The same method was followed for each of the countries purchase price indexes (PPI) as well.

Eg: Ln_ppi_us c ln_nor_ppi_us

From our table our D.W. stat is less than 2 and is very low, this means that there is a positive autocorrelation within our regression. The next value that should be noted on this table is the Beta symbol (β). This is included in our table because the β represents the risk of type II errors according to Ellis, P (2010). Type II errors are false negatives. Beta’s above 1 are more likely to show potential risk according to Investopedia (2003).

Once the values have been noted, the next step is to run an augmented Dickey Fuller Test (ADF) on the residual of the integration regression. The t statistic that comes up is noted on table two, for the ADF value. The use of an ADF instead of the normal Dickey Fuller test is because we are testing for long run sample. From the T statistic in the ADF one can tell if the value is stationary or non-stationary. The values on table 2 were all non-stationary after the test, as none of the values given were less than the critical values, at 10%, 5% and 1% levels.

Non-stationary was found within the data and lags, which then conflicts with the theory of PPP and LOP, meaning that both of these theories do not hold in the selected countries (Norway, Denmark and Canada) and that there is no significant relation between the countries and their prices with one another, in the time series selected (1978 – 2015).

Conclusion

According to Alan and Mark, PPP should consider manufacturing tradable instead of CPI, “since PPP is based on traded goods, it might be more usefully tested with producer price indices that tend to contain the prices of more manufactured tradable, rather than consumer price indices, which tend to reflect the prices of relatively more non-tradable, such as many services” (2004, p.137).

The PPP theory assumes that all goods of a country are traded internationally, but some goods and services used in the indices are not traded. This can lead to price differences between countries. For example: If we take some homogeneous goods or the medical services sector into consideration, it’s not traded internationally. As soon as the price of these non-traded goods change the price indices change along with it. But this won’t affect the exchange rate as those services are not traded internationally. “The application of non-traded goods to real exchange rates is direct” (Backus & Smith, 1993, p.298). So this won’t change the exchange rates as expected under the PPP theory. change in the nominal interest rates can also affect the PPP as stated by Backus and Smith, “if nominal interest differentials reflect expected rates of depreciation, then real interest differentials, measured with price indexes, will reflect rates of change of deviations from PPP” (1993, p.299).

As said by Krugman, the Purchasing Power Parity (PPP) theory exists when a basket of goods in one country is worth the same as an identical or similar basket of goods in another country. The basket of goods could differ if people in different countries consume different types goods. “It is often difficult to determine whether literally the same basket of goods is available in two different countries” (Taylor and Taylor, 2004, p.137). The basket of goods may not be comparable in this case as the goods used in the other country will be different all together. The PPP theory also assumes the non-existence of transaction costs between countries. But if in case they are present, the PPP won’t react the way it should. On the other hand, the international trade transactions are also subject to time lags between trade transactions. “the presence of transactions costs—perhaps arising from transport costs, taxes, tariffs and duties and nontariff barriers—would induce a violation of the Law of One Price” (Taylor and Taylor, 2004, p.137).

           When testing unit roots using the Augmented Dickey-Fuller method, initially we tested whether CPI and PPI for each country was stationary or non-stationary given lags 1, 4, 8, and 12. Additionally, to increase the certainty, t-values were tested against all critical values specifically, 1%, 5%, and 10%. Our results allowed us to identify that all samples had a unit root meaning our results were unanimous, out data is non-stationary.

           Once we determined that our results were non-stationary we were able to run a co-integration test in order to test if there is a long run relationship between the domestic CPI and PPI with the foreign CPI’s and PPI’s. By analyzing the Durbin Watson value, we were able to determine it is less than 2, which is considered low. Above all, this means that there is a positive autocorrelation within our regression. The residuals of the co-integration were then tested using an Augmented Dickey-Fuller unit root test. The results determined there was no long run relationship. No long run relationship proves both the Law of One Price (LOP) and Purchase Power Parity (PPP) theory do not hold.