# Relevance of PPP Theory to UK-US Exchange Rate

Purchasing power parity (PPP) states that in the absence of transaction costs and barriers to trade, the nominal exchange rate between two countries should equate the aggregate price levels of the respective countries. Since its formal introduction in 1918 by Gustav Cassel as a means of stabilising the exchange rates of the major countries after WW1, PPP theory has been extensively scrutinised and investigated by researchers to determine its relevance as a practical theory in exchange rate determination. Starting with the testing of the most basic relationship of PPP in the 1970s till the recent use of more advanced econometric techniques like Co-integration or Fractional Integration, the results gathered from the numerous literature have not been consistent. But even with the mixed results, there was always belief by academia or interested parties that the PPP holds the potential to be the cornerstone for the determination of exchange rates. As a result this paper attempts to join the plentiful literature on PPP, by investigating the convergence of long run PPP for two of most advanced economies in the world, United States of America and United Kingdom respectively. Co-integration and Johansen test have been correspondingly employed to the data set that spans from 1968 to 2010. The findings from both tests were contrasting where Co-integration advocating that PPP does not hold in the long run while Johansen Co-integration test shows the existence of one co-integrating relationship. Nevertheless, this result is consistent with existing literature where different econometric models have produced different results, even on similar dataset.

## 1. Introduction

Purchasing Power Parity (PPP) points out that in the absence of transaction costs and barriers to trade, the nominal exchange rate between two countries should be equal to the aggregate price levels of the respective countries. Although the term Purchasing Power Parity (PPP) was apparently first proposed by Gustav Cassel in 1918 as a means of stabilising the overly-inflated exchange rates of major industrialised countries after World War 1, the origination of the underlying idea has a history dating back to the Spanish scholars of 15th & 16th centuries. To the scholars, PPP serve as reliable measure to comprehend the interaction between the trade between the Spanish and other economies and the monetary impact on the exchange rate. But to current users of this theory, PPP offers an easily understood economic theory for the determination of exchange rate and the relation of the countries' price level to exchange rate.

Since its introduction, PPP theory has been extensively scrutinised and investigated but results have not been consistent. Even with the contrasting literature, there was always belief that the PPP theory holds the potential to the theoretical solution to exchange rate determination. Dornbusch and Krugman (1976 p.540) stated that “Under the skin of any international economist lies a deep-seated belief in some variant of the PPP theory of the exchange rate”. It was further pointed out by Rogoff (1996), that even with the mixed results from the various literatures, there was “a surprising degree of consensus on a couple of basis facts” First, real exchange rate will tend to purchasing power parity in the long run but the speed of convergence is extremely slow. In addition, the deviation in the short run is large and volatile which eliminates its role for exchange rate determination in short term. It is this significant deviation of the exchange rate form PPP in the short run that presents the “Purchasing Power Parity Puzzle” to researchers and interested parties alike.

They have searched and investigated different aspect of PPP and improved their statistical approaches in the hope of explaining the deviations in the short run and how it could be reconciled with the long run exchange rates. In the earlier studies like Gailliot (1970), Lee (1976) and Friedman (1980), they have used a simple method of using the basic PPP relationship as the null hypothesis to test for the relevance of PPP. As expected due to the stringent criteria, most studies failed to find the validity of PPP in the long run. Surprisingly, Frenkel (1978) found that the PPP theory holds for countries undergoing hyper-inflation phase but was less successful for countries with stable economic conditions. With the unsuccessful attempts in the early days, researchers started to revisit the issue from another perspective by testing the null hypothesis that the real exchange rate does not revert to its mean but instead follows a random walk (Roll 1979; Darby 1983; Edison 1987 etc). Some techniques to test for unit root included the Dickey-Fuller and Augmented Dickey-Fuller test and Variance ratios. These techniques did not seem to improve the probability of rejecting the null hypothesis of random walk and that PPP holds in the long run. More recently, methodologies like co-integration and fractional integration & co-integration have entered the picture. Corbar and Ouliaris (1988), Taylor (1988), Kim (1990) applied co-integration techniques but most found that the null hypothesis of non co-integration cannot be rejected in most cases while Rogoff and Froot (1994) compared the 3 techniques used to determine PPP and concluded that although the co-integration test have been more successful in rejecting the null hypothesis, it is still unclear whether this technique produces a benefit over the simple PPP hypothesis or random walk test. On the other hand, Cheung and Lai (1993), Masih and Masih (1995) and Soofi (1998) employed the fractional integration approach and found relatively favourable results. Regardless of the mixed results from different methods and econometric techniques, purchasing power parity theory remains an important aspect in exchange rate determination as it is simple to comprehend and intuitive.

The objective of this paper is to investigate the relevance of the PPP theory on the exchange rate of two advanced economies, U.S and UK respectively. This analysis will be conducted using co-integration test, a common time series methodology in the PPP literature. Furthermore to ensure the robustness of the results, Johansen test will also be applied.

The rest of this paper is organised as follows; Section 2 presents the introduction of purchasing power parity and law of one price, discuss the debate of PPP and the test of unit root and random walk as well as explaining of some improved techniques for the testing of PPP in last decade. Section 3 describes the data and empirical framework. Section 4 will show the findings from the analysis while section 5 will conclude the paper.

## 2. Literature review

## 2.1 Purchasing Power Parity and Law of One Price

Purchasing Power Parity states that in the absence of transaction costs and barriers to trade, a commodity should cost the same regardless of the location of its purchase. If the prices of similar goods are different between countries, the exchange rate will adjust to be equal to the ratio of the price levels of the countries, so as to offset any possible arbitrage opportunities. If this scenario were to exist, three economic reactions will be subsequently observed (Arnold 2008);

Being relatively expensive, the demand of the particular commodity will fall, forcing price down. (Domestic)

Being relatively cheaper, the demand of the particular commodity will increase, increasing the price. (Foreign)

The demand for the foreign currency relative to domestic currency would increase

Eventually, the exchange rate and the price levels of the two countries will adjust until they converge at equilibrium again and PPP will hold again.

Before exploring further into PPP, it will be beneficial to mention the law of one price theory as it forms the foundation for Purchasing Power Parity (PPP) theory, which explains the long-run equilibrium relationship between nominal exchange rates and price levels.

The law of one price states that in the absence of transaction cost and barriers to trade, identical goods in 2 countries should sell for the same price at the same time. The law of one price applies to individual commodities while purchasing power parity is relevant to a basket of goods (ie. CPI or WPI). Consequently, if the law of one price holds true for all the commodities, PPP theory will hold too.

Where is price of good I in dollar , is the nominal exchange rate and is price of good i in British Pounds. Thus, the exchange rate between the two countries equates the relative ratios of the countries' price levels. (Shown below)

Therefore, an increase in the domestic purchasing power will be associated with a proportional appreciation of the currency in the foreign exchange market, vice versa. Moreover, PPP is implying that all countries' price levels are similar when compared in similar currency. (Krugman and Obstfeld, 2006)

## 2.2 Relative & Absolute PPP

The absolute PPP states that exchange rate is equal to the national price levels in two countries and the purchasing power would be the same in the two countries if expressed in a common currency. On the other hand, Relative PPP focuses on percentage changes in the price levels and the exchange rate and assets that the relative change in exchange rate must be matched by the same change in price levels in order to keep the ratio constant.

Where is the percentage change in the nominal exchange rate and denotes the inflation rate.

Froot and Rogoff (1994) pointed out that discussions relating to re-establishing exchange rate to equilibriums using PPP, referred exclusively to relative PPP. They further argued that real shocks can lead to changes in the relative prices of different commodities, especially so for low inflation economies. Moreover, there is not much evidences to support that the relative PPP will exhibits a greater robustness than the absolute PPP.

## 2.3 Debate of PPP and Tests for Unit root and Random walk.

In July of 1944, delegates from 44 countries met in Bretton Woods, where they drafted and signed the Articles of the International Monetary Fund (IMF). The objective of this coming together was to propose an international monetary system that will ensure internal (full employment & inflation) and external balances of the countries. The Bretton Woods agreement called for fixed exchange rates against the US dollar whilst the US dollar was tied to the price of gold. Subsequently, the “monetary approach” was commonly used during that period of time to determine exchange rates. This approach advocates that exchange rate between countries should be determined by the relative price of two monies, which is further influenced by the demand and supply factors in their respective money market. More importantly, this approach is based on the assumption that purchasing power parity does hold continuously.

From 1970s, there have been numerous studies to determine the empirical relevance of purchasing power parity theory. But due to the lack of theoretical and advanced statistical models, the results were largely disappointing. Early studies of PPP have mostly based on the simple empirical model shown below.

Where is the logarithm of the domestic currency price of good i, is the logarithm of the foreign-currency price and is the logarithm of the domestic-currency price of foreign exchange. They test the null hypothesis that = 1 and and if the null cannot be rejected, it meant the exchange rates and relative prices follows the PPP theory. On the other hand when, Purchasing power parity does not hold.

Frenkel and Johnson (1978) found some relative success for the PPP theory when testing with countries with hyper-inflation but this result was not unanticipated due to the significant monetary shocks evidenced in these hyper-inflation environments. He carried out the similar test for countries with stable situations and found that PPP theory is not valid, a result consistent with other investigation of PPP during that period. Dornbusch (1976) argued that one possible reason for the failure of PPP in the short run might have been due to exchange rate over-shooting as a result the presence of sticky prices in the short run.

One of the major drawbacks for this simple PPP model described above was that there were no test for stationary property of the time series of exchange rates and relative prices respectively. Lutkepohl and Lratzig (2004) argued that stationarity property is time-invariant; does not depend on time, in the first and second moment of a stochastic process. This implied that the time series is stationary if;

for all t T and

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

Without determining whether the respective time series are stationary, the variables might appear to be related even when they are not as a result of spurious regression. Hence, any estimation results (i.e Ordinary Least Square) will be biased and misleading when the time series is found to be non-stationary and exhibits spurious regression.

Testing for relevance of PPP in the long run involves examining whether the real exchange rate revert to its own mean, a condition necessary to ensure that long run PPP holds. In the face of a series of disappointing results, researchers began to adopt another approach to investigating the PPP theory. Instead, they test the null hypothesis that the real exchange rate does not revert to its mean but instead follows a random walk (Roll (1979), Darby (1983) and Edison (1985)). Some authors argued that the existence of random walk in real exchange rate signifies a well-functioning international market where all available information is reflected in the prices and exchange rate while arbitrage opportunities are efficiently taken advantage of. In all, there are few main methods utilised to test if the real exchange rate follows a random walk;

Firstly, the Dickey Fuller and Augmented Dickey Fuller (ADF) tests were used to determine whether a particular time series contains a unit root and subsequently follow a random walk. Moreover, the ADF test is able to distinguish between 3 possible scenarios; has unit root, has unit root plus drift and has unit root plus drift and time trend.

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 disturbance. The null hypothesis will be not be rejected if, and where the process has a unit root. But if the null hypothesis is rejected (), this implies that PPP theory holds. Studies from Hakkio (1984), MacDonald (1985), Edison (1987), Meese and Rogoff (1988), Taylor (1988), Roll (1979), Mark (1990) have consecutively used unit root test for their analysis but were coherent in their conclusion where they failed to reject the unit root hypothesis. Their failure to prove the validity of PPP could be due to the low power of the ADF tests. This meant that even when we do not reject the null hypothesis of a random walk; it might not imply that the time series is in actual fact non-stationary.

Another approach used to investigate the presence of random walk is the variance ratios method.

Where is a real exchange rate at time t, T is the sample size, i = 2, 3 … T-1. If the null hypothesis of random walk holds, the variance of the real exchange rate should grow linearly and k(i) should be the same for all i. Conversely, if k(i) tends to zero when i increases , the series will be a stationary process. Glen (1988) found that contrasting results when testing the variance ratios on the monthly and annual sample. He could reject the random walk for monthly data but failed to find similar evidences for mean reversion while finding proof of mean reversion and not able to reject the random walk hypothesis for the annual data. Huizinga (1987) employed this approach and found a positive auto-correlation in real exchange rate of the US Dollar for time up to two years.

Frankel (1986, 1990) argues that even when the null hypothesis of a random walk is not rejected at a certain significance level; it might not mean that the results should be accepted. This is because of the lack of power of the statistical tests especially for studies covering less than 15 years or since 1973. To prove his point, he used longer data sets and was successful in rejecting the null hypothesis for the Dollar/Pound exchange rate during the period 1869-1984. Also, he concluded that PPP deviations generally have half-life of 4.6 years. Niso Abuaf and Phillipe Jorion (1990) also rejected the random walk hypothesis by using data from 1901-1972 for eight currencies. They found half-life of disturbances from PPP of about 3.3 years. Using nearly two centuries of data for the Dollar/ Pound and Franc/ Pound, James R. Lothian and Mark P. Taylor (1996) found evidences of mean reversion in both exchange rates with half-life of 4.7 and 2.5 years correspondingly. Although these studies gathered relative success in the testing for PPP, their methodology has been criticised by others. Michael Mussa (1986) and Qian and Strauss (2001) argued that real exchange rates tend to be more volatile under floating than fixed exchange rates and analysis inclusive of periods of floating and fixed might eventually result in econometric implications.

In addition, the power of the test could be further improved through the inclusion of more countries into the analysis. Frankel and Rose (1995) found mean reversion on a cross-sectional analysis of floating exchange rate. They examined annual data from 1948-1992 for a total of 150 countries and were able to reject the random walk hypothesis using only post-1973 data. Abuaf and Jorion (1990) found similar results when they analysed 10 first-order autoregressive regressions for real dollar exchange rate using post 1973 data. On the other hand, Taylor and Sarno (1998) criticised that the researchers are wrong to conclude that all the rates are mean reverting when null hypothesis of random walk is rejected. Instead, the rejection of the null would only imply that at least one of the rates is mean reversion. Therefore, they suggested conducting an alternative hypothesis to test that least one of real exchange rate is non-mean reverting.

## 2.4 Structural Deviations from PPP

One of the most known theories presented to explain the deviations from PPP in the long run, is the Balassa-Samuelson model. (Balassa 1964 and Samuelson 1964). Their empirical results show that in general richer countries will have higher consumer price index (CPI) than poorer countries. They argued that wealthier countries are relatively more productive in the traded good sector than the poorer countries. As they have greater technological superiority in the traded goods, this will result in a rise in productivity which will in turn increase the level of wages for the sector. But as the price of tradable goods is restricted by the fixed exchange rate and world prices, it will not increase. Contrastingly for the non-traded sector, they will have to increase their price as they could not match the productivity increase of the tradable sector. In the end with the increase in the price of non-traded goods while constant price of tradable goods, the overall price level of the country have to increase. But if the nontradable goods sector is able to match the productivity gains of the tradable sector, the overall price level and real exchange rate will stay constant. This economic rationale can be applied to both fixed and floating exchange regimes. In hindsight, their findings and economic rationale highlighted the distinction between traded and nontraded goods and the impact on the price level and real exchange rates in the countries. Froot and Rogoff (1994) has mentioned that if the non-traded goods sector is more labour intensive, a even balanced growth in both sectors will result in an appreciation of relative tradable price. Empirical results from Ragoff (1996) also supported this claim as they found a positive relationship between income and price levels, with the richer countries having the greatest distinction.

Other factors that might impact the PPP from holding includes; differences in exchange rate regime, import and export restrictions, travel costs, perishable goods and location amid others. These will impact the price of similar goods sold in different countries as well as the general price level in the respective countries, resulting in the violation of the law of one price and the convergence to PPP in the long and long run.

## 2.5 Innovations in the test of PPP

Most early studies on PPP based on the simple PPP hypothesis or test for existence of unit root and random walk have produced disappointing results that does not support the relevance of the PPP theory. Subsequently, researchers started to implement more sophisticated econometric techniques, in hope of getting favourable results. Two of the more notable techniques employed by them are the Co-integration test and Fractional integration.

## 2.5.1 Co-integration

According to Engle and Granger (1987), if two non-stationary variables are integrated of the same order and put into a regression, the two variables will possess a relationship, if the residual is stationary. Therefore, Co-integration test can used to find the existence of weak PPP relationship for time-series that are otherwise rejected under strong PPP criterion. Test for strong PPP test depends on the value for of α and β. For a strong PPP, value for α and β should be 0 and 1 respectively, if both of the variables do not meet the criteria there might still exist a possibility of relationship but in a weaker form.

Testing for co-integration basically involves a three step procedure. First, is to use the Dickey-Fuller test to determine whether the respective time series (Exchange rates and price levels for the countries) have unit roots and whether they can be differenced to the same integer order. Following on, the “co-integrating” equation is regressed using OLS and residual is determined. The last step involves testing the null hypothesis that α2=1 and using the OLS residuals to run the Dickey-Fuller regression with the omitted time trend. The variables will be co-integrated when the residual is stationary. Corbar and Ouliaris (1988), Taylor (1988), Kim (1990) applied co-integration techniques but most found that the null hypothesis of non co-integration cannot be rejected in most cases. Rogoff and Froot (1994) compared the 3 techniques used to determine PPP and concluded that although the co-integration test have been more successful in rejecting the null hypothesis, it is unclear whether this technique produces a benefit over the simple PPP hypothesis or random walk test. Though the results have not been satisfactory, these studies have revealed common characteristics of the test and data. First, rejection of the null hypothesis of no-cointegration occurs more often for fixed than floating exchange rate. Also, data based on CPI are less likely to reject than dataset for WPI. Lastly, the rejection of no-cointegration for floating exchange rates under the post Bretton-Woods period occur more often for trivariate systems than for bivariate systems and residuals appear to be more stationary by the weakening of the proportionality and symmetry restrictions.

To improve the power of the co-integration techniques, researchers have suggested the use of panel cointegration methods to produce more robust findings. {Banerjee (1999), Pedroni (2000, 2001b) and Qian and Strass (2001)}. This improved technique allows the researcher to selectively pool long run information in the panel while allowing for short run heterogeneity among the different members.

## 2.5.2 Fractional Integration

A more recent trend that is used by current researchers of PPP is fractional integration. This methodology was implemented due to the problems inherent in the unit root and co-integration tests. Cuestas and Gil-Alana (2009) suggested that the possibility of Type II error occurring in unit root test for Dickey-Fuller increases as the sample size expands. Thus creating structural changes with the increase of data sample and if not taken into consideration will have adverse effects on power of the test. In addition, Dumas (1994), Michael et al. (1997), Sarno et al. (2004), Juvenal and Taylor (2008), and Cuestas (2009) among many others, found that larger the deviation from long run equilibrium exchange rate, the faster the pace of reversion to the mean. Further, Gil-Alana (200) also found that if the speed of the mean reversion is slow, the results from Dickey-Fuller might be misleading and fail to identify the existence of unit root. As a result, fractional integration was seen as an alternative solution to the PPP puzzle. With this method, the real exchange rate will have to be estimated as a fractionally integrated process.

## ,

Where and are polynomial lag operators with roots outside the unit circle. is a white noise operator. If parameter d = 1 & equals to, the null hypothesis of random walk cannot be rejected. On the contrary, If d = 0 Null hypothesis rejected and PPP holds in the long run.

Baum, Barkoulas and Caglayan (1999) used fractional integration on 29 countries and suggested that rejection of absolute PPP in the long run. Cheung and Lai (1993) argues that the mean reversion of PPP exist and can be categorised by a fractionally integrated process in 3 out of 5 countries they tested. Masih and Masih (1995) concluded that the fractional cointegration approach “captures a much wider class of parity or mean-reversion behaviour” than simple cointegration method. On the other hand, Soofi (1998) found that although the variables are fractionally cointegrated, the PPP model for all countries does not have a mean-reversion property. Alves, Cati and Fava (2001) infer that the evidence does not seem to support the absolute PPP hypothesis but the relative PPP holds in the long run when they tested the PPP theory for Brazil.

## 2.6 Reasons for choosing U.S and UK

Before elaborating into the empirical section, it would be appropriate to explain the rationale for selection of the countries for the investigation, U.S and UK. Both these countries are considered the most advanced in the world and share uncanny similarity in various aspects; speak the same language, economic constitution of the country, lifestyles, expectations, culture, religion and etc. The “developed” status of both the countries entails that they exist in an open economic environment where the flow of information is efficient and unrestricted. More often, consumers in these countries have no restriction on their choices of similar goods from not just the local market but also from foreign countries. As mentioned in previous section, Purchasing Power Parity (PPP) theory states that “in the absence of transaction costs and barriers to trade, the nominal exchange rate between two countries should be equal to the aggregate price levels of the respective countries.” As such, this pre-requisite condition for PPP theory is “best” fulfilled in these two countries and as a result makes these countries excellent subjects for the detailed examination of PPP in the long run.

## 3. Research Design and Sample

## 3.1 Descriptive Statistics of Data

The sample data for our analysis comprises of 507 monthly observations for the period of 1968 to 2010. The information is extracted from Thomson Data-Stream which provides historical data of U.S and UK exchange rate and respective consumer price index (CPI).

Figure 1 illiustrate the time series of nominal USD/ GBP exchange rate and the relative CPI of U.S over UK. However the dataset includes the period under Bretton –woods agreement where the British pound was fixed to the US Dollars until it was abolished and allowed to float in the 1971.

From the figure, we can see that after the abolishment of the Bretton-Woods agreement in 1971, the movement of the exchange rate ($/ £) has been erratic and volatile. The exchange rate reduced significantly and on a downward trend since 1971 and hitting its lowest exchange rate of 1.09 in 1985. After 1985, the exchange rate became less volatile and fluctuated within the 1.50 to 1.80 band. As the result of the recent financial crisis, the pound The GBP collapsed at 2009 against to USD, falling from a high of $2.0736/ £ in Oct 2007 to $1.523/ £ in Feb 2010.

## Figure 1: USD/GBP nominal exchange rate and US/UK CPI Ratio (1968 to 2010)

The theory of Relative PPP states that the depreciation/ appreciation of the currency level should be matched by a proportional changes to the differences between the domestic and foreign inflation rates. As a result, the USD/GBP exchange rate and relative price levels of both U.S and UK should move proportionally and in the same direction. Going back to Figure 1, it has been have observed that the movement of the exchange rate ($/£) and the relative price levels does not move in the same direction in most periods. Moreover, the magnitude of the change between the two time series varies significantly where the exchange rate movement is larger and more volatile than the relative price ratio. This explicitly implies that the relative PPP does not hold and there are deviations in the short run.

## 3.2 Empirical Framework

This section will describe the empirical design that this study employ to investigate the convergence of the exchange rates to the PPP theory. Our analysis will test the following hypothesis;

H0: PPP does not hold in the long run

H1: PPP holds in the long run

The nominal exchange rate, Et will be calculated as

(1)

Where Et is the exchange rate between dollar and pound, PtUS is CPI for U.S and PtUK is CPI for UK. By taking logs for Equation (1), the nominal exchange rate will be subsequently reduced to a linear equation of the domestic (UK) and foreign (U.S) price levels, as shown in equation (2) below.

(2)

As a result, the estimated model will be;

(3)

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

## 3.2.1 “Strong” PPP

In this section, I will seek to investigate whether the “strong form” of PPP exist for the U.S and the UK. There are two main methodology to analyse whether the most direct relationship of PPP holds for the subject countries.

Ordinary Least Square Regression

This method involves performing a regression on equation (3) and testing the null hypothesis that and If the null hypothesis cannot be rejected, it meant that PPP holds in the long run but instead the null hypothesis is rejected, PPP theory does not hold. From the descriptive statistic (Figure 1), it is seen that the exchange rate and relative price of U.S and UK are trended. This observed fact meant that Ordinary Least Square (OLS) estimator might not be suitable for the analysis as non-stationary time-series variables tend to produce spurious regression due to problems of hetereoskedasticity and serial auto-correlation, which will eventually lead to misleading and biased estimation results.

Stationarity of Real Exchange rate

Instead, to examine the basic relationship of PPP between U.S and UK, I have opted to test for unit root in the real exchange rate to determine the validity of PPP in the long run. To achieve this, the real exchange rate will first be determined, as shown in equation (4).

(4)

Where denotes real exchange rate, denotes nominal exchange rate whereas and denotes logarithm of CPI for U.S and UK respectively. Next, I will test the null hypothesis that the real exchange rate remain stationary and does not follow a random walk. If this occurs, it will be concluded that PPP holds in the long run. On the contrary, if real exchange rate does not follow a random walk, the nominal exchange rate is not determined by just its relative price levels and PPP will not hold. If this happens, it meant there is some sort of trending behaviour in the data due to some jolt in the economy like rise in production or demand because of unforeseen conditions. Linear unit root test is the most common test used to detect the presence of unit root in the real exchange rate. More specific, Augmented Dickey-Fuller (ADF), which is a test of unit root, will be performed;

(5)

Where Δyt is the 1st difference of log(Et); are the optional exogenous regressors which might include a constant or a constant and trend, α and δ represents the parameters that are to be estimated, represents the residuals. Moreover, constant and trend variables are included if seen from observations.

## 3.2.2 Co-integration Test

Co-integration test involves testing the null hypothesis that the variables are co-integrated and alternative hypothesis that they are not co-integrated as shown below.

: And are cointegrated

: Andare not cointegrated

Before begining the co-integration analysis, it will have to determine whether the two time series are stationary or does not have a unit root. This involves testing the null hypothesis of non-stationary in the variables. To test the assumption of non-stationarity of variables, Augmented Dickey-Fuller test will be used.

Once settled on the status of stationary of the variables, the adequate number of lags (p) that should be integrated in the ADF regression will next determined, so as to avoid the problem of serial correlation in the residuals. This could be determined by choosing the amount of lags that minimizes the standard information criterion (Schwarz Information Criterion). So, if both the variables are non-stationary and integrated to the same order, it will be deemed possible to carry on using co-integration to test the long run relationship of the variables and convergence to PPP. On the contrary, if the variables cannot be integrated to the same order or that either one or both of the variables are stationary; it would not feasible to carry on with the testing.

According to Engle and Granger (1987), if two non-stationary variables are integrated of the same order and put into a regression, the two variables will possess a relationship or co-integrated if their residual is stationary. Putting into application, if two non-stationary time series and that are both integrated to the same order, produces a stationary variable after the regression, it can be inferred that the variables are co-integrated.

The test of co-integration involves two main procedures. First, the following regression equation (3) will be estimated, so as to get the estimated error term or residual (equation 6)

(6)

Next, Augmented Dickey-Fuller will be used to test the null hypothesis that the estimated error term has a unit root or is non-stationary. By comparing the t-stats with the Dickey-Fuller t-Statistics (spurious co-integrating regression), the variables are co-integrated if we cannot reject the null hypothesis that the residual has a unit root or non-stationary. But if we reject the null hypothesis of unit root, the variables will be co-integrated.

H0: PPP does not hold in the long run

H1: PPP hold in the long run

Finally based on our empirical hypothesis, if the variables are found to be co-integrated, I will reject the null that PPP does not hold in the long run and conclude there exist an equilibrium relationship between the exchange rate and relative price levels in the long run. Alternatively, the null hypothesis cannot be rejected if the variables are not co-integrated.

## 3.3 Robustness Test

To ensure that the results from the co-integration test is robust, I will employ another methodology for the examining the long run equilibrium relationships of the variables and whether PPP holds. The Johansen maximum likelihood (ML) test is proposed to carry out the analysis.

Johansen (1988, 1991) came out with the maximum likelihood (ML) method for estimating the equilibrium relationships in the long run or co-integrating vectors. He derives a likelihood ratio (LR) test for co-integration in a Gaussian vector error correction model.

The likelihood ratio (LR) test statistic for the hypothesis of at most r co-integrating vectors is;

Where are the smallest eigenvalues of S21S11-1 S12 with respect to S22.

Gonzalo (1989) argued that for the finite sample properties of the ML estimator are very consistent with the asymptotic results. Stock and Watson (1991) used the monte carlo to substantiate the unbiased property of the ML estimator. Cheung and Lai (1993) also concluded that the Johansen test has significant power superiority over the residual-based test like co-integration. Moreover, as the Johansen test only involves one step as compared to three steps under co-integration, there will be less probability for errors in the application process. Therefore, Johansen test will be utilized for robustness.

## 4. Empirical Results

## 4.1 “Strong” form PPP Results

Table 1 shows the results from the unit root test for real exchange rate. The ADF test on the real exchange rate shows that the null hypothesis of unit root cannot be rejected at the 95% confidence level. As expected, the results show that real exchange rate is non-stationary and there is no statistical evidence to support the existence of a convergence to PPP in the long run for the “strong” form.

## 4.2 Co-integration Results

As mentioned previously, the initial screening for co-integration tests involves determining whether the variables are stationary. Augmented Dickey-Fuller unit root test was performed and it was found that nominal exchange rate, CPI for both U. S and UK and difference between the CPI for both countries are non-stationary or has unit root at the 95% confidence level. (See table 2)

After determining that the variables are non-stationary and follows a random walk, I will specify the amount of lags that will be included into the ADF regression to prevent the problem of serial correlation in our respective variables. The ADF results which includes the minimum Schwarz information criterion is presented in table 3. The results show that both variables are made stationary when integrated to the order of I(1), with the inclusion of the lags with minimum Schwarz information for each variable.

It will be feasible to carry on testing for co-integration when the variables are found to be non-stationary and could be integrated to the same order. However, it will not be possible to carry on testing for co-integration if variables are integrated to different order or when either one of the variables (Independent and dependent) or both exhibits stationary time series in the unit root test. The results till now have shown the null hypothesis of non-stationary or unit root cannot be rejected for both the variables at 95% confidence level. Furthermore, both variables could be integrated to the order of one, I(1). As a result, I will proceed with the results from co-integration.

The co-integration test has carried out performing a regression on the following co-integrating regression (3);

(3)

The results for the co-integration regression are as follows:

Furthermore, the R2 and Adjusted R2 are 0.581843 and 0.581014 respectively. This result shows that the coefficient for the independent regressor (difference in CPI) is positive and significant. Following on, the residual of the co-integrated regression will be determined and subjected to the ADF test. This will test the null hypothesis of unit root and stationary for the residual. If the null hypothesis of unit root cannot be rejected, I will conclude that the variables are not co-integrated and PPP does not hold in the long run. Conversely if null hypothesis is rejected, it will conclude that the evidence seems to infer that the variables are co-integrated and PPP theory holds for U.S and UK in the long run equilibrium. The ADF test on the residual has given us a test statistic of -3.039267. Comparing this t-statistics with the critical value taken from the Dickey- Fuller t-stats table (applied to residuals from spurious co-integrating regression) for Case 3 of -3.42, it can be inferred that the residual has a unit root and is non-stationary. (See table 5)

H0: PPP does not hold in the long run

H1: PPP hold in the long run

Consequently, this implied that the variables are not co-integrated and the null hypothesis that PPP does not hold in the long run equilibrium cannot be rejected.

## 4.3 Johansen Test Results

Table 6a and 6b shows the Johansen test results for the Trace and Maximum Eigenvalue test respectively. Each test examines two hypotheses;

H0: No Co-Integration

H1: One Co-Integrating Relationship

## &

H0: One Co-Integrating Relationship

H1: Two Co-Integrating Relationship

The results from both tests show evidences that they reject the null hypothesis of no co-integration and cannot reject the null hypothesis of one co-integrating relationship. Therefore, this meant there is presence of one co-integrating relationship within the variables.

The Johansen test has shown that there exits a long-run equilibrium relationship between the exchange rate and the relative price levels of U.S and UK. But in the short run, there might be disequilibrium. In order to reconcile and explain the transition the short-run to long run equilibrium, I will employ the use of the error correction mechanism or Vector Error Correction (VEC). This process enables a proportion of the disequilibrium in one period to be corrected in the next period and help to explain the change in exchange rate to the change in relative price levels and previous period's disequilibrium.

Table 7 presents the results from the Vector Error Correction (VEC) estimates. The partial short run adjustment coefficient shows how much the disequilibrium is “corrected” per month as the variables change. The adjustment coefficient for exchange rates is negative and insignificant, at 0.5% per month while the adjustment coefficient for the relative price levels of the countries is also negative and stands at 1.25% per month. This meant that all the “correction” of the disequilibrium in the short run is done by relative price levels instead of exchange rate.

## 5. Conclusion

Purchasing Power Parity (PPP) has generated considerable amount of interest from academia and interested parties (ie. Government, Central banks etc), since being formally introduced by Gustav Cassel in 1918 as it offers a basic theoretical model to exchange rate determination which is not just simple to understand and also intuitive for application. Since 1970s, academia has explored the PPP issue from varying perspectives and with the use of progressively advanced econometric techniques, but the results has not been consistent. In the midst of the mixed results, there is still belief that this theory holds the key to being an important manual for explaining the exchange rate differences between countries. This has lead to Dornbusch and Krugman (1976 p.540) declaring that “Under the skin of any international economist lies a deep-seated belief in some variant of the PPP theory of the exchange rate”.

This paper joins the vast amount of existing literature by investigating the convergence of the PPP in the long run for two advanced countries, U.S and UK. These countries have been chosen as they “best” fulfilled the condition for the Purchasing Power Parity theory. This analysis was achieved using co-integration techniques and further supported for robustness by Johansen test. The findings for both the tests (Co-integration and Johansen test) have generated contrasting results. Co-integration test results show that the variables are not co-integrated and PPP does not hold in the long run equilibrium while Johansen test has identified the presence of one co-integrating relationship within the variables and reject the null hypothesis that PPP does not hold in the long run. Building on the results from the Johansen test, an error correction model or Vector Error Correction (VEC) was performed to further understand the short run disequilibrium and their transition into long run equilibrium. It was found that all the correction or adjustment to equilibrium was done by the relative price index instead of exchange rate and at a rate of approximately 1.25% per month. Although this paper was not able to provide any concrete conclusion on the relevance of the PPP theory on these two advanced economies, it has given findings that are consistent with the existing literature on Purchasing Power Parity. The fact that different econometric models performed on similar dataset, have produced distinct results and conclusions.