# Testing the PPP Hypothesis between UK and the USA

Purchasing power parity (PPP) says that the exchange rate (EP*/P) between two currencies is defined by the change in the relative prices of those countries. If PPP holds, the real exchange rate fluctuates around an equilibrium value. Thus, shocks to the real exchange rate do not have permanent effects. The real exchange rate is expected to rise when it is driven below its equilibrium value and fall when it is above the equilibrium.

In 1970s the floating exchange rate regime was again introduced in the economy. Since then, Purchasing Power Parity has been seen as a long-run relationship in many international economic models. PPP is mostly accepted theory for explaining the long-run behaviour of exchange rates. Therefore, it has been tested many times but mixed results were found. Many researchers state that whether PPP is rejected or not depends on the length of the sample period, the choice of countries and the econometric techniques used.

This paper investigates whether PPP holds based on dataset which includes nominal exchange rates and Consumer Price Index (CPI) data for UK, US and Mexico. The sample period is of 42 years starting from 1st of January 1968 and ending in February 2010.

Augmented Dickey Fuller test and two cointegration analysis (Engle- Granger and Johansen methods) were used to test for stationarity of variables and for long run equilibrium relationship between them. These methods were applied to full sample which include both fixed and floating exchange rate regimes.

## Literature review and theoretical framework

## Law of one price

The law of one price states that in competitive markets free of transformational 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 and Obstfeld, 2006). The law of one price assumes that there are no transportation costs and no differential taxes applied between the two markets. Thus, there can be no tariffs on imports or other restrictions on trade.

The purchasing power parity is the application of the law of one price. The purchasing power parity is a simple proposition that, once converted to a common currency, national price levels should be equal. (Rogoff, 1996)

According to Froot and Rogoff (1994) law of one price states that for any good i,

Pta = ptb + et (1.1)

where Pta is the log of the domestic currency price of good i, ptb is the analogous foreign currency price, and et is the log of the domestic currency price of foreign exchange, ensuring identical prices of unfettered trade in goods. If law of one price holds for every individual good then it leads to the assumption that it must hold for any identical basket of goods.

## Absolute and Relative Purchasing Power Parity

There are two different types of classical Purchasing Power Parity: absolute PPP and relative PPP. Absolute PPP implies that a bundle of goods should cost the same in domestic and foreign countries once the exchange rate is taken into account. Absolute PPP relates the exchange rate between two currencies to the ratio of price levels in each country,

Et = PLt / PLt* (1.2)

where Et is the domestic price of foreign currency, PLt is the domestic price level, PLt* is the foreign price level.

Relative PPP illustrates differences in the rates of inflation between two countries. It states that the rate of appreciation of a currency is equal to the difference in inflation rates between the foreign and the domestic country. Therefore relative PPP between the UK and the USA can be written as:

, (1.3)

where ΔEt is the percentage change of the nominal exchange rate between USD and GBP and denotes the inflation rate – the percentage change in the price level. In countries where inflation rates differ dramatically, relative PPP equation will hold.

## Simple Purchasing Power Parity as the Null Hypothesis

Stage one empirical analyses were limited by the lack of statistical models and theoretical knowledge for differentiating between short and long run.

Frenkel (1978) used the data on a very high inflation counties and tested regressions such as:

, (1.4)

Frenkel found estimates of β quite close to one and believed that PPP should be the foundation of any model on exchange rates. However, during low inflation, stage – one tests strongly rejected PPP. Frenkel (1981) showed that PPP performed poorly, with β far from one when used data from 1970s for industrialized countries. He argued that PPP partly failed due to sticky real prices and temporary shocks.

The main problem with the stage one analysis is that there was no test for non-stationarity of prices and exchange rates. As the later research indicates, if observed variables are trended, this can produce spurious regression.

## Stationarity

Lutkepohl and Kratzig (2004) argued that stationarity is time-invariant in the first and second moments in a stochastic process therefore is stationary for:

all t T and

all t T and all integers h such that t – h T

Equation number one indicates that there is a constant mean for all members of a stationary stochastic process. The second equation indicates that the variances are also constant because, for h = 0, the variance = = is independent of t. Moreover, the covariances

are independent of t but on the distance in time h of the two members of the process.

One of the significant things to test, is if the exchange rate and the price level are stationery, or simpler, whether the series contain stochastic trend. Stochastic trends quite often create a problem of spurious regression. It means that two time series might appear related when they are not. When the series turn to be non-stationary, the OLS can give misleading results. That is why cointegrated time series are more reliable since the series contain a common stochastic trend, i.e. trend components of the two series is the same (Stock and Watson 2003).

## The Real Exchange Rate as a Random Walk and Tests for Random Walk

During the seventies and eighties one of the most significant concerns in testing the PPP theory was to distinguish the real exchange rate from random walk process. After disappointing stage one test, researchers found an alternative stage two tests which state the null hypothesis that the real exchange rate is stationary with the alternative hypothesis of holding PPP in the long run. Three main techniques were used to test if the real exchange rate is a random walk.

The first and most common is the Dickey-Fuller and augmented Dickey – Fuller tests. The regression of the real exchange rate was engaged:

(1.5)

where is a real exchange rate at time t, is a pth order polynomial function of lag operator L with coefficients: , ,… , and is a white noise.

The null hypothesis states that has a unit root when . The alternative hypothesis states that PPP holds when equals zero and .

The second technique uses variance ratios and implies the variance of the real exchange rate to grow linearly and k(i) should be one for all i under the null hypothesis of random walk.

The test statistic is:

, (1.6)

where is a real exchange rate at time t, T is the sample size, i = 2, 3 … T-1.

Under alternative hypothesis when k tends to zero as k increases, the series is a stationary process.

The third technique of distinguishing the real exchange rate from the random walk is using fractional integration:

, (1.7)

where and are polynomial lag operators with roots outside the unit circle and is a white noise operator. If d = 0 the process takes the form of stationary ARMA process and PPP holds in the long run. If d = 1 and = and equals 1 then the process turns to the random walk. This technique is more powerful than tests for unit root because it allows for fractional integration when 0 < d < 1.

Stage – two testes were implemented by many researchers including Meese and Rogoff (1988), Mark (1990), Hakkio (1984), Darby (1983), MacDonald (1985), Frankel (1986), Edison (1987), Roll (1979). Unfortunately, the evidence of holding PPP in the long run was not found. Low power of stage – two tests and the usage of the data sets of short or medium-term time series can explain the failure to prove PPP.

## Results for Post-Bretton-Woods Data - fixed and floating exchange rate regimes

Real exchange rates were very volatile between 1973 and 1986 as measured by the standard deviations. Enders (1988) states that the standard error of the estimate during the fixed exchange rate was much smaller than the standard error of the estimate during the flexible exchange rate.

Researchers like Meese and Rogoff (1988) and Mark (1990) were unable to reject the null of a unit root for currencies that float against each other. In contrast, Huizinga (1987) constructed variance ratios and found a positive autocorrelation in US dollar real exchange rates for time series running up to two years.

Mark (1990) tested fixed currency pairs between 1973 - 1988 and was very close to reject the random walk but only for the Belgium/Germany exchange rate at 5% confidence level. Chowdhury and Sdogati (1993) when looking at time period 1979-1990 strongly rejected the random walk for the European currencies against Deutsche mark. However, they were unable to reject it for Euro/Dollar data during floating exchange rate period.

## Tests using cross sections of currencies

In order to solve the problem of low power of the tests, Hakkio (1984) introduced the usage of cross-section data. He chose four exchange rates against dollar and engaged GLS to enable cross-exchange rate correlation in the residuals. Even though the power was improved, Hakkio failed to reject the random walk model.

Abuaf and Jorion (1990) run analogous test but they generated more power by using larger time series and larger cross-section data. They rejected the random walk but at its weakest form, at 10% significance level using one-sided tests.

More resent study performed by Cumby (1993) uses data from time period 1987-1993 for dollar price of McDonald’s Big Mac hamburgers in up to 25 countries. Cumby finds a relatively rapid rate of convergence comparing with other PPP studies. This can be well explained by three factors. First of all, the currency pairs in the sample were fixed against one another. Secondly, the sample includes countries like Argentina, Brazil, Mexico, Thailand so countries with a high inflation. The third factor is the pricing policy of the company which creates more rapid rate of convergence in prices.

## Tests using long-horizon data

Another approach to improve the power of tests is to use longer data sets. Frankel (1986) employed annual data for the Dollar/Pound exchange rate for the time period 1869-1984 and was able to reject the unit root at the 5% significance level. Frankel concluded that PPP deviations have a half-life of 4.6 years. Many researches agreed with Frankel and were also able to reject the unit root. Abuaf and Jorion (1990) easily rejected the null hypothesis based on 1901-1972 data for eight currencies and found a half-life for PPP deviations of 3.3 years. Diebold, Husted and Rush (1991) used the data from gold-standard period and were able to strongly reject the random walk model.

Some researchers used different methods of testing. Edison (1987) used an error-correction mechanism and employed Dollar/Pound exchange rate for the time period 1980 – 1978. Edison rejected the null and found that PPP deviations have a half-life of 7.3 years. Johnson (1990) employed similar method and used Canadian dollar/ US dollar exchange rate for 120 years. He rejected the null and found that PPP deviations have a half-life of 3.1 years. Glen (1992) applied variance ratios to test for mean reversion in the real exchange rate between 1987-1900. He found a strong evidence of mean reversion.

Lothian and Taylor (1994) looked closer at an interesting issue of combining very volatile post Bretton-Woods exchange rate data and low variance pre Bretton-Woods data. They used two centuries of data for the Dollar-Pound (1791-1990) and the Franc-Pound (1803-1990) exchange rates. When only post Bretton-Woods data is applied, the null cannot be rejected.

In case when entire sample is applied, the random walk is easily rejected. The researchers came to the conclusion that they cannot support the opinion that the addition of fixed-rate periods biases unit roots tests of the real exchange rate.

## Cointegration

Cointegration techniques raise a question whether a group of nonstationary variables can be joined to produce a stationary variable. The series are cointegrated if two or more series are non-stationary, but a linear combination of them is stationary. Cointegration is said to test the weaker form of PPP, as it requires only some linear combination of exchange rates and prices to be stationary. Stage two tests were looking whether the real exchange rate

was stationary. Cointegration – stage three is testing if is stationary for any constant and and therefore relaxes the symmetry and proportionality restrictions.

Cointegration methods to test PPP are based on three stage procedure. First, the Dickey-Fuller test is used to test the exchange rate and the two domestic price series for unit roots. The next step is to estimate the cointegrated regression using OLS. Cointegration of prices and exchange rates shows that the error term in the regression in stationary. The last stage is to test the hypothesis that α2=1 and to use the OLS residuals to run the Dickey-Fuller regression with the omitted time trend. Unfortunately the three-step method is very inefficient. However, the maximum likelihood method developed by Johansen [1988] and Johansen and Juselius [1990] is able to avoid this inefficiency. The technique is mostly used when there are more than two time series variables engaged as it can determine the number of cointegrating vectors. In Johansen’s test there is only one step engaged and therefore less error is involved.

There are a large number of studies that test PPP using cointegration methods.

For example: Corbae and Ouliaris (1988), Enders (1988), Kim (1990), Mark (1990), Fisher and Park (1991), Cheung and Lai (1993) and Kugler and Lenz (1993). These studies show interesting information about the data. Firstly, the rejection of null hypothesis (no-cointegration) happens more often for fixed currency pairs then for floating currency pairs. Secondly, tests based on WPIs price levels tend to be rejected more frequently than tests based on CPI price levels. The reason for this is that CPI price levels have higher nontraded goods factor than wholesale prices which have a tendency to weight manufactured goods more heavily. Thirdly, the rejection of the null for post-Bretton-Woods floating exchange rate happens more frequently for trivariate systems where p and p* enter independently, than for bivariate systems, where they enter as p-p*, or for stage-two tests where the coefficient on

p-p* is constrained to be one. The residuals appear more stationary by weakening the proportionality and symmetry restrictions.

Although there are many tests applying cointegration testing to PPP, it is still not certain if this technique gives superior results than stage two test. The problem with stage three tests is that the estimates of μ and μ* differ across the various studies and are often unreliable. Also, for the long sample periods the technique does not provide any new results. All the results are already known from the stage two tests.

## Structural models of deviations from PPP

There are many studies that try to explain empirically deviations from PPP by looking at factors such as: productivity, government spending and strategic pricing decisions made by companies.

The most famous model is by Balassa and Samuelson (1964) which attempts to explain long – term deviations in consumption-based PPP. According to the researchers CPIs in poor countries will be lower in comparison to rich countries and CPIs in fast-growing countries will rise relative to CPIs in slow-growing countries. Balassa and Samuelson claim that technological progress is faster in the tradable goods. Higher productivity in this sector will make the wages to go up in the whole economy. In non-tradable goods sector producers will be able to increase the wages only by raising the relative price of non-tradable goods.

Balassa and Samuelson model depends fully on supply factors. Nevertheless, there are few assumptions such as: a small size of the country and no impact on world interest rates, capital is mobile across countries, capital and labour move together across sectors internationally, there is no third factor in production such as land which is immobile.

Baumol and Bowen (1966) also tried to explain deviations from PPP. They stated that within a country, the service intensive goods such as education, health care, banking etc. tend to rise over time.

There are many researchers who provided evidence in favour of Balassa and Samuelson as well as Baumol and Bowen effects. Edison and Klovan (1987) looked at time series data in the time period of 1874 – 1971 and studied real exchange rate between the British Pound and the Norwegian krone. Based on this long – horizon data they found a significant evidence of a productivity differential effect so an evidence of Balassa and Samuelson effect.

Asea and Mendoza (1994) used a dynamic two-country general equilibrium model and calculated relative tradable goods prices in the time period 1975 – 1985 for fourteen OECD countries. They came to the conclusion that changes in non-tradable goods prices account only for a small and insignificant part of real exchange rate changes across the countries and therefore they found evidence of Baumol and Bowen effect.

## The data set and Empirical Framework

## 3.1 Descriptive Statistics of Data

The data used for this study comprises of 507 monthly observations. The first sample consists of two time series: nominal GBP/USD exchange rate and ratio between UK CPI and US CPI. The research will also separate an interesting subsample from 1992 until 2006. This is the period of small fluctuations in comparison to very volatile periods before 1992 and after 2006 when the economy experienced a global financial crisis. (Figure 2)

The second sample includes: nominal Mexican Peso /American Dollar exchange rate and ratio between Mexican CPI and US CPI. Both samples cover a 42 years period starting from 1st of January 1968 and ending in February 2010. However, the Mexican CPI data is not available from 1968 until 1977.

In case of UK and USA the sample includes the period of Bretton-Woods fixed exchange rate system, under which until 1971 the British pound and exchange rate were fixed to US Dollar. After 1971, the exchange rates became floating, large fluctuations in exchange rate and upward sloping trend can be observed. There is a strong depreciation of the Pound in 1980’s. In February 1985, GBP/USD exchange rate increases to its highest level of 0.95942. After 1986 there are a large number of fluctuations within the 0.7-0.5 Pound per Dollar band.

Figure 1. GBP /USD nominal exchange rate and UK/ US price level ratio from

1968 until 2010

Figure 2. GBP /USD nominal exchange rate and UK/ US price level ratio from

1992 until 2006

Figure 3. Mexican Peso /American Dollar nominal exchange rate and Mexico/ USA price level ratio from 1968 until 2010

The relative PPP theory states that the price levels’ ratio in the two countries and the exchange rate between them should move proportionally. From Figure 1 and Figure 3 can be seen that the series from 1968 until 2010 do not move proportionally therefore they are contrary to the relative PPP theory. Large fluctuations of the exchange rate are not followed by the ratio of price levels, thus changes in national price levels cannot tell us about the exchange rate movements.

## 3.2 Empirical Framework

The formula below shows how the nominal exchange rate is calculated

Et= PiUK / PiUS (2.1)

where E denotes the exchange rate between British Pound and US Dollar, whereas PiUK and PiUS denotes CPI of UK and US respectively.

By taking the logarithms of Equation 2.1 the nominal exchange rate becomes the linear combination of the UK and US price levels:

Log(Et) = log(PtUK) + log(PtUS) (2.2)

The real exchange rate is defined in terms of its nominal exchange rate and the price level components. It helps to understand the relationship between nominal and real exchange rates to the concept of purchasing power parity.

qt= log(Et) – log(PtUK) + log(PtUS) (2.3)

where qt is the log real exchange rate, Et is the log nominal exchange rate, the domestic currency price of a unit of the foreign currency.

If purchasing power parity held perfectly, qt would equal a constant q*, and equation (2.3) could be rewritten as:

log(PtUS) + log(Et) = q* + log(PtUK) (2.4)

Under fixed exchange rate regime, equation (2.4) becomes a relation linking the price levels in the two countries. Under floating exchange rate regime, equation (2.4) shows the relation between the exchange-rate adjusted price level in the one country and the actual price level in the other.

The estimated statistical model is given by:

log(Et) = α + β ( log(PtUK) – log(PtUS)) + εt (2.5)

where α is intercept, β is slope and εt is the vector of residuals.

The Augmented Dickey Fuller test is used to test for a unit root, for whether the series is stationary.

Augmented Dickey Fuller (ADF) test is based on the regression:

(2.6)

where Δyt denotes the 1st difference of log(Et), are the optional exogenous regressors which may consist of constant, or a constant and trend, α and δ are parameters to be estimated, are residuals. When constant and trend variables can be seen from observation they are included.

The next step is to specify the number of lagged difference terms to be added to the test regression in order to avoid serial correlation in the residuals. By looking at the Schwarz information criterion, the number of lags that minimizes this criterion should be chosen.

Co-integration is a statistical relationship where two time series that are both integrated of the same order. However, co-integration is not possible if variables are integrated in different orders and also it is not possible when dependant variable is stationary and independent variable is non-stationary and vice versa.

The Engle and Granger 2 – step approach can be used to test for co-integration. The first step is to show that the variables are integrated of the same order. Thus, if Et and PtUK/PtUS are both integrated of the same order, they are said to be co-integrated if put together the linear regression (2.5) produces stationary variable – error term.

The estimated error term - can be obtained:

(2.7)

The estimated residual series is then tested for stationarity using the augmented Dickey-Fuller test (ADF).

If estimated error term is stationary we can conclude that there is co-integration between two variables, if it is non-stationary it is impossible to reject null hypothesis of no co-integration. The tables of correct critical values taken from Hamilton are used, which are only applicable for residuals from spurious co-integration regression.

The Engle and Granger approach may be exposed to errors arising from the econometric techniques employed. Firstly, the 2-step co-integration procedure makes results sensitive to the ordering of variables therefore, the residuals may have different sets of statistical properties. Secondly, if the bivariate series are not cointegrated, the “cointegrating equation” results in spurious estimators (Lim and Martin, 1995). This would have the effect of making any mean reversion analysis of the residuals unreliable. To overcome these problems this study will also use Johansen test for cointegration which is based on a Vector-Error-Correction model (VECM).

In order to specify the long run relationships between the variables, VAR-based co-integration relationship using the methodology developed by Johansen can be estimated.

Vector autoregression (VAR) of order p is shown below:

yt = A1yt-1 + …+ Apyt-p + Bxt + εt (2.8)

where in this case yt is a (k x 1) vector of I (1) variables, xt is a d-vector of deterministic

variables such as constant term, linear trend, seasonal dummies, and crisis variables and εt is a

vector of innovations.

The VAR can be rewritten as:

Δyt = Πyt-1 + iΔyt-1 + βxt + εt (2.9)

where: Π = I – 1, and Гi = - j

The rank r is the number of cointegrating relations and each column of β is the cointegrating vector.

Then if the rank of Π is r < k, there exist r cointegrating relationships between the k variables included in the system. In our case, k equals to 3, then r can be either 2 or 1 which means two or one cointegrating relationship or it can be 0 which means no cointegrating relationship. If cointegration is found, then a Vector Error Correction (VEC) model can be estimated. Suppose there are only two variables, y1,t and y2,t and that the cointegrating equation is:

y2,t = β y1,t (2.10)

The corresponding VEC model is:

Δ y1,t = α1 (y2,t-1 - β y1,t-1 ) + ε1,t (2.11)

Δ y2,t = α2 (y2,t-1 - β y1,t-1 ) + ε2,t (2.12)

where (y2,t-1 - β y1,t-1 ) denotes error-correction term and α are known as the adjustment parameters which measure the speed of adjustment of particular variables with respect to a disturbance in the equilibrium relationship.

Based on above empirical framework, this research will test two hypotheses:

H0: PPP does not hold in the long run

H1: PPP holds in the long run

## Interpretation of results

## 4.1 UK – USA (Appendix 1)

In order to test for unit root the Augmented Dickey Fuller test is used. By looking at Schwarz criterion value the number of lags is chosen. The lowest possible value of Schwarz criterion is at lag 0 and equals -4.210100.

The ADF test results show that t-statistic equals -2.416476 the critical values at the 1%, 5% and 10% levels are smaller than the t-statistic value therefore the null is not rejected. Results from Augmented Dickey Fuller test show that variables are non-stationary at level and therefore simple OLS regression would give misleading results.

The two-step Engle-Granger approach is applied to test for cointegration between nominal exchange rate and prices. The first step is to test for unit root in nominal exchange rate. The lowest possible value of Schwarz criterion is chosen and equals -4.269804 which is lag 0. The ADF test results show that t-statistic is -1.986967 the critical values at the 1%, 5% and 10% levels are smaller than the t-statistic value therefore we do not reject the null. Nominal exchange rate has a unit root and therefore is non-stationary.

In the case where the null of unit root cannot be rejected, test for unit root in fist difference has to be applied. The ADF t-statistic equals to -19.94706, the critical values at the 1%, 5 % and 10% equal -3.443098, -2.867055 and -2.569769 respectively. The null hypothesis is rejected. Nominal exchange rate does not have a unit root and is stationary.

The next step is to test for unit root in relative prices. In case of relative prices Intercept is chosen. The lowest possible value of Schwarz criterion equals to -7.400924which is lag 14. The ADF t-statistic equals to -2.807389 the 1% critical value is -3.443442, 5% critical value is -2.867207 and 10% critical value equals to -2.569850. The null cannot be rejected at 1% and 5% levels.Test for unit root in first difference is applied. The results indicate that the null hypothesis is rejected, logrp becomes stationary.

The results confirm that all series are integrated of order 1 therefore we can proceed with cointegration analysis. The ADF test on residuals in performed. Neither an intercept nor a trend is included. The lowest possible value of Schwarz criterion equals to -4.264567which is lag 0. The ADF test results on residuals show t-statistic of -2.722710 which is higher than

-2.76 at 5% confidence level but lower than -2.45 critical value at 10% confidence level. Critical values are taken from Dickey Fuller t statistic table when applied to residuals from spurious cointegrating regression. (Appendix 4) This critical value is only valid for Case 1 (no intercept, no trend).

From obtained result we can reject null hypothesis for the sample at 10% confidence level and conclude that there is a long-run equilibrium relationship between the variables in full sample that includes data 1968-2010 period.

More powerful Johansen approach is applied. Johansen’s cointegration testing framework is used to determine the absence or the presence of the cointegrating relationship among all test variables. Case 3 is chosen where all series are cointegreted of order 1 and have a unit root.

Testing no cointegration versus one cointegration gives the trace statistic of 57.53087 which is larger than the 5% critical value of 29.68. Hence, the null of no cointegration is rejected. At least one cointegration exists. Testing one cointegrating relationship versus two cointegrating relationships gives the trace statistic of 14.03722 with 5% critical value of 15.41. The null cannot be rejected, therefore there is one cointegrating relationship.

## 4.2 UK – USA subsample from 1992-2006 (Appendix 2)

In case of UK and USA when the less volatile data from 1992-2006 is taken into consideration, the Augmented Dickey Fuller test is performed and as a result the null hypothesis cannot be rejected.

The Engle-Granger approach is employed. The test for unit root in nominal exchange rate was performed. The ADF test results show that t-statistic equals to -0.102046 the critical values at the 1%, 5% and 10% levels are smaller than the t-statistic value therefore the nominal exchange rate is non-stationary. The ADF test in first difference indicates that null hypothesis can be rejected. Nominal exchange rate does not have a unit root and is stationary.

The test for unit root in relative prices is performed. The ADF test results show that t-statistic is -3.128789, the 1% critical value is -3.469691, 5% critical value is -2.878723 and 10% critical value equals to -2.576010. The null cannot be rejected at 1% level. Test for unit root in first difference is applied. The null hypothesis can be rejected at 10% level.

The next step is to perform the ADF test on residuals. The ADF test results on residuals show t-statistic of -1.619062 which is higher than the critical values. The results indicate that the null hypothesis cannot be rejected.

More powerful Johansen approach is applied. Testing no cointegration versus one cointegration gives the trace statistic of 32.21313 which is larger than the 5% critical value of 29.79707. This result shows that at least one cointegration exists. Testing one cointegrating relationship versus two cointegrating relationships gives the trace statistic of 11.14775 with 5% critical value of 15.49471 therefore the null cannot be rejected. The Johansen test indicates one cointegrating relationship.

## 4.3 Mexico – USA (Appendix 3)

When the data of Mexico and the USA is considered the same tests are performed. The Augmented Dickey Fuller test is used to test for unit root. In case of Mexico and USA the null hypothesis cannot be rejected therefore variables are non-stationary at level.

The cointegration was examined using the Engle-Granger approach. Firstly, the test for unit root in nominal exchange rate was performed. The ADF test results show that t-statistic

is -0.837575 the critical values at the 1%, 5% and 10% levels are smaller than the t-statistic value therefore nominal exchange rate has a unit root and therefore is non-stationary.

Test for unit root in fist difference has to be applied. The ADF t-statistic equals to -14.47882, the critical values at the 1%, 5 % and 10% equal -3.443123, -2.867066 -2.569775 respectively thus, the null hypothesis is rejected. Nominal exchange rate becomes stationary.

Secondly, the test for unit root in relative prices is applied. The ADF t-statistic equals

to -1.830018 the 1% critical value is -3.447036, 5% critical value is -2.868790 and 10% critical value equals to -2.570698. The null cannot be rejected therefore the test for unit root in fist difference is performed. The results show that the null hypothesis is rejected at 1% and 5% levels and thus becomes stationary.

All series are integrated of order 1 and we can proceed with cointegration analysis. The ADF test on residuals in performed. The ADF test results on residuals show t-statistic equals

to -2.822860. Critical values are taken from Dickey Fuller t statistic table when applied to residuals from spurious cointegrating regression. (Appendix 4) This critical value is only valid for Case 1 (no intercept, no trend). The null hypothesis for the sample can be rejected at 10% confidence level. Consequently there is a long-run equilibrium relationship between the variables in full sample.

Johansen cointegration test is performed. Testing no cointegration versus one cointegration gives the trace statistic of 111.4652 which is larger than the 5% critical value of 29.79707. This result shows that at least one cointegration exists. The null hypothesis is also rejected when testing one cointegrating relationship versus two cointegrating relationships. Testing two cointegrating relationships versus three two cointegrating relationships gives the the trace statistic of 6.125404 and 5% critical value of 3.841466. This result indicates that three cointegrating relationships exist.

## Conclusion

The present paper tests the PPP theory using cointegration analysis for three countries: UK, USA and Mexico. Most studies suggest that there is more frequent rejection of null hypothesis in fixed exchange rates than in floating exchange regime. The sample used in this paper consists of both fixed and floating exchange rates.

The research was able to conclude that PPP holds at 10% confidence level between US and UK in the long run as it was proven on full sample which included period of 1968-2010. Also, Johansen methods proved that there is one cointegrating relationship

When shorter time period was examined (1992 – 2006) and less volatile data was used, the study showed mixed results. The Engle-Granger approach indicated that the null hypothesis cannot be rejected and PPP does not hold. However, the Johansen cointegration test showed that there is one cointegration. The discrepancy of results can be caused by the differences in both tests. The main difference between the tests is that with the Johansen test is possible to have more than one cointegrating relationship. Also, the Johansen test is based on maximum likelihood whereas the Engle-Granger on OLS.

In case of Mexico and USA for the time period of 1968-2010, the PPP holds at 10% confidence level therefore there is a long-run equilibrium relationship between the variables. Johansen technique showed that that three cointegrating relationships exist.

However, the researchers who claim that PPP does not hold, argue that the study contain data sets that employ at least some fixed rate data. The researchers state that more floating data has to be used and also more powerful econometric techniques need to be found.

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