Buying power para ( PPP ) states that in the absence of dealing costs and barriers to merchandise, the nominal exchange rate between two states should compare the aggregative monetary value degrees of the several states. Since its formal debut in 1918 by Gustav Cassel as a agency of bracing the exchange rates of the major states after WW1, PPP theory has been extensively scrutinised and investigated by research workers to find its relevancy as a practical theory in exchange rate finding. Get downing with the testing of the most basic relationship of PPP in the 1970s boulder clay the recent usage of more advanced econometric techniques like Co-integration or Fractional Integration, the consequences gathered from the legion literature have non been consistent. But even with the assorted consequences, there was ever belief by academe or interested parties that the PPP holds the possible to be the basis for the finding of exchange rates. As a consequence this paper attempts to fall in the plentiful literature on PPP, by look intoing the convergence of long tally PPP for two of most advanced economic systems in the universe, United States of America and United Kingdom severally. Co-integration and Johansen trial have been correspondingly employed to the information set that spans from 1968 to 2010. The findings from both trials were contrasting where Co-integration advocating that PPP does non keep in the long tally while Johansen Co-integration trial shows the being of one co-integrating relationship. However, this consequence is consistent with bing literature where different econometric theoretical accounts have produced different consequences, even on similar dataset.

## 1. Introduction

Buying Power Parity ( PPP ) points out that in the absence of dealing costs and barriers to merchandise, the nominal exchange rate between two states should be equal to the aggregative monetary value degrees of the several states. Although the term Buying Power Parity ( PPP ) was seemingly first proposed by Gustav Cassel in 1918 as a agency of bracing the overly-inflated exchange rates of major industrialized states after World War 1, the inception of the underlying thought has a history dating back to the Spanish bookmans of 15th & A ; 16th centuries. To the bookmans, PPP serve as dependable step to grok the interaction between the trade between the Spanish and other economic systems and the pecuniary impact on the exchange rate. But to current users of this theory, PPP offers an easy understood economic theory for the finding of exchange rate and the relation of the states ‘ monetary value degree to interchange rate.

Since its debut, PPP theory has been extensively scrutinised and investigated but consequences have non been consistent. Even with the contrasting literature, there was ever belief that the PPP theory holds the possible to the theoretical solution to interchange rate finding. Dornbusch and Krugman ( 1976 ) mentioned that “ Under the tegument of any international economic expert lies a deep-rooted belief in some discrepancy of the PPP theory of the exchange rate ” . It was farther pointed out by Rogoff ( 1996 ) , that even with the assorted consequences from the assorted literatures, there was “ a surprising grade of consensus on a twosome of footing facts ” First, the existent exchange rate converges towards PPP in the long tally although the velocity is really slow. In add-on, the perturbation from the short tally equilibrium is usually big and volatile which prevent the finding of exchange rate in the short term. As a consequence of the divergence from PPP in the short tally, the “ Purchasing Power Parity Puzzle ” is accordingly presented to research workers and interested parties likewise.

They have searched and investigated different facet of PPP and improved their statistical attacks in the hope of explicating the divergences in the short tally and how it could be reconciled with the long tally exchange rates. In the earlier surveies like Gailliot ( 1970 ) , Lee ( 1976 ) and Friedman ( 1980 ) , they have used a simple method of utilizing the basic PPP relationship as the void hypothesis to prove for the relevancy of PPP. As expected due to the stringent standards, most surveies failed to happen the cogency of PPP in the long tally. Surprisingly, Frenkel ( 1978 ) found that the PPP theory holds for states undergoing hyper-inflation stage but was less successful for states with stable economic conditions. With the unsuccessful efforts in the early yearss, research workers started to revisit the issue from another position by proving the void hypothesis that the existent exchange rate does non return to its mean but alternatively follows a random walk ( Roll 1979 ; Darby 1983 ; Edison 1987 etc ) . Some techniques to prove for unit root included the Dickey-Fuller and Augmented Dickey-Fuller trial and Variance ratios. These techniques did non look to better the chance of rejecting the void hypothesis of random walk and that PPP holds in the long tally. More late, methodological analysiss like co-integration and fractional integrating & A ; co-integration have entered the image. Corbar and Ouliaris ( 1988 ) , Taylor ( 1988 ) , Kim ( 1990 ) applied co-integration techniques but most found that the void hypothesis of non co-integration can non be rejected in most instances while Rogoff and Froot ( 1994 ) compared the 3 techniques used to find PPP and concluded that although the co-integration trial have been more successful in rejecting the void hypothesis, it is still ill-defined whether this technique produces a benefit over the simple PPP hypothesis or random walk trial. On the other manus, Cheung and Lai ( 1993 ) , Masih and Masih ( 1995 ) and Soofi ( 1998 ) employed the fractional integrating attack and found comparatively favorable consequences. Regardless of the assorted consequences from different methods and econometric techniques, buying power para theory remains an of import facet in exchange rate finding as it is simple to grok and intuitive.

The aim of this paper is to look into the relevancy of the PPP theory on the exchange rate of two advanced economic systems, U.S and UK severally. This analysis will be conducted utilizing co-integration trial, a common clip series methodological analysis in the PPP literature. Furthermore to guarantee the hardiness of the consequences, Johansen trial will besides be applied.

The remainder of this paper is organised as follows ; Section 2 presents the debut of buying power para and jurisprudence of one monetary value, discourse the argument of PPP and the trial of unit root and random walk every bit good as explaining of some improved techniques for the testing of PPP in last decennary. Section 3 describes the information and empirical model. Section 4 will demo the findings from the analysis while subdivision 5 will reason the paper.

## 2. Literature reappraisal

## 2.1 Buying Power Parity and Law of One Monetary value

Buying Power Parity states that in the absence of dealing costs and barriers to merchandise, a trade good should monetary value the same regardless of its location. If the monetary values of similar goods are different between states, the exchange rate will set to be equal to the ratio of the monetary value degrees of the states, so as to countervail any possible arbitrage chances. If this scenario were to be, three economic reactions will be later observed ( Arnold 2008 ) ;

Bing comparatively expensive, the demand of the peculiar trade good will fall, coercing monetary value down. ( Domestic )

Bing comparatively cheaper, the demand of the peculiar trade good will increase, increasing the monetary value. ( Foreign )

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

Finally, the exchange rate and the monetary value degrees of the two states will set until they converge at equilibrium once more and PPP will keep once more.

Before researching farther into PPP, it will be appropriate to advert the jurisprudence of one monetary value theory as it forms the basis for Buying Power Parity ( PPP ) theory, which explains relationship between nominal exchange rates and monetary value degrees in the long tally.

Similar to PPP, the jurisprudence of one monetary value provinces that in the absence of dealing cost and barriers to merchandise, indistinguishable goods in 2 states should sell for the same monetary value. The jurisprudence of one monetary value applies to single trade goods while buying power para is relevant to a basket of goods usually measured as CPI or WPI. Consequently, if the jurisprudence of one monetary value holds true for all the trade goods in the state, Buying Power Parity theory should besides keep. The equation below shows that the monetary value of a similar good sold in different counties when expressed in a similar currency.

Where is monetary value of good I in dollars, is the nominal exchange rate and is monetary value of good I in British Pounds. Therefore, the exchange rate between the two states equates the comparative ratios of the states ‘ monetary value degrees. ( Shown below )

From the equation above, it can be seen that when there is any alterations to the buying power of the states, it will finally take to a relative grasp or depreciation of the nominal exchange rate of the states. ( Krugman and Obstfeld, 2006 )

## 2.2 Relative & A ; Absolute PPP

The absolute PPP provinces that exchange rate is equal to the national monetary value degrees in two states and the buying power would be the same in the two states if expressed in a common currency. Alternatively, Relative PPP looks at the per centum alterations in the monetary value degrees and the exchange rate and stipulate that any alterations in exchange rate should be matched by a relative alteration in the monetary value degrees so as to maintain the ratio invariable.

where is the % alteration in the nominal exchange rate and denotes the rising prices rate ( % alteration in monetary value degrees ) .

Froot and Rogoff ( 1994 ) pointed out that treatments associating to re-establishing exchange rate to equilibriums utilizing PPP, referred entirely to relative PPP. They further argued that dazes caused by existent factors can make alterations in the comparative monetary values of trade goods, particularly so for low rising prices economic systems. Furthermore, there is non much groundss to back up that the comparative PPP will exhibits a greater hardiness than the absolute PPP.

## 2.3 Argument of PPP and Trials for Unit root and Random walk.

In July of 1944, delegates from 44 states met in Bretton Woods, where they drafted and signed the Articles of the International Monetary Fund ( IMF ) . The aim of this coming together was to suggest an international pecuniary system that will guarantee internal ( full employment & A ; rising prices ) and external balances of the states. The Bretton Woods understanding called for fixed exchange rates against the US dollar whilst the US dollar was tied to the monetary value of gold. Subsequently, the “ pecuniary attack ” was normally used during that period of clip to find exchange rates. This attack advocates that exchange rate between states should be determined by the comparative monetary value of two monies, which is further influenced by the demand and supply factors in their several money market. More significantly, this attack is based on the premise that buying power para does keep continuously.

From 1970s, there have been legion surveies to find the empirical relevancy of buying power para theory. But due to the deficiency of theoretical and advanced statistical theoretical accounts, the consequences were mostly dissatisfactory. Early surveies of PPP have largely based on the simple empirical theoretical account shown below.

Where is the logarithm of the domestic currency monetary value of good I, is the logarithm of the foreign-currency monetary value and is the logarithm of the domestic-currency monetary value of foreign exchange. They test the void hypothesis that = 1 and and if the nothing can non be rejected, it meant the exchange rates and comparative monetary values follows the PPP theory. On the other manus when, Buying power para does non keep.

Frenkel and Johnson ( 1978 ) found some comparative success for the PPP theory when proving with states with hyper-inflation but this consequence was non unforeseen due to the important pecuniary dazes evidenced in these hyper-inflation environments. He carried out the similar trial for states with stable state of affairss and found that PPP theory is non valid, a consequence consistent with other probe of PPP during that period. Dornbusch ( 1976 ) argued that one possible ground for the failure of PPP in the short tally might hold been due to interchange rate over-shooting as a consequence the presence of gluey monetary values in the short tally.

One of the major drawbacks for this simple PPP theoretical account described above was that there were no trial for stationary belongings of the clip series of exchange rates and comparative monetary values severally. Lutkepohl and Lratzig ( 2004 ) argued that stationarity belongings is time-invariant ; does non depend on clip, in the first and 2nd minute of a stochastic procedure. This implied that the clip series is stationary if ;

for all t T and

for all t T and all whole numbers h such that t – H Thymine

Without finding whether the several clip series are stationary, the variables might look to be related even when they are non as a consequence of specious arrested development. Hence, any appraisal consequences ( i.e Ordinary Least Square ) will be biased and misdirecting when the clip series is found to be non-stationary and exhibits specious arrested development.

Testing for relevancy of PPP in the long tally involves analyzing whether the existent exchange rate revert to its ain mean, a status necessary to guarantee that long tally PPP holds. In the face of a series of dissatisfactory consequences, research workers began to follow another attack to look intoing the PPP theory. Alternatively, they test the void hypothesis that the existent exchange rate does non return to its mean but alternatively follows a random walk ( Roll ( 1979 ) , Darby ( 1983 ) and Edison ( 1985 ) ) . Some writers argued that the being of random walk in existent exchange rate signifies a well-functioning international market where all available information is reflected in the monetary values and exchange rate while arbitrage chances are expeditiously taken advantage of. In all, there are several methods utilised to prove if the existent exchange rate follows a random walk ;

First, the Dickey Fuller and Augmented Dickey Fuller ( ADF ) trials were used to find whether a peculiar clip series contains a unit root and later follow a random walk. Furthermore, the ADF trial has been able to separate from 3 possible scenarios ; has unit root, has unit root plus impetus and has unit root plus impetus and clip tendency.

where is a existent exchange rate at clip T, is a pth order multinomial map of slowdown operator L with coefficients: , , aˆ¦ , and is a white noise perturbation. The void hypothesis will non be rejected if, and where the procedure has a unit root. But if the void hypothesis is rejected ( ) , this implies that PPP theory holds. Surveies from Hakkio ( 1984 ) , MacDonald ( 1985 ) , Edison ( 1987 ) , Meese and Rogoff ( 1988 ) , Taylor ( 1988 ) , Roll ( 1979 ) , Mark ( 1990 ) have consecutively used unit root trial for their analysis but were coherent in their decision where they failed to reject the unit root hypothesis. Their failure to turn out the cogency of PPP could hold been due to the low power of the ADF trials. This meant that even when we do non reject the void hypothesis of a random walk ; it might non connote that the clip series is in existent fact non-stationary.

Another attack used to look into the presence of random walk is the discrepancy ratios method.

Where is a existent exchange rate at clip T, T is the sample size, one = 2, 3 aˆ¦ T-1. If the void hypothesis of random walk holds, the discrepancy of the existent exchange rate should turn linearly and k ( I ) should be the same for all i. Conversely, if k ( I ) tends to zero when I additions, the series will alternatively follow a random walk. Glen ( 1988 ) found that contrasting consequences when proving the discrepancy ratios on the monthly and one-year sample. He could reject the random walk for monthly informations but failed to happen similar groundss for average reversion while happening cogent evidence of average reversion and non able to reject the random walk hypothesis for the one-year information. Huizinga ( 1987 ) employed this attack and found a positive auto-correlation in existent exchange rate of the US Dollar for clip up to two old ages.

Frankel ( 1986, 1990 ) argues that even when the void hypothesis of a random walk is non rejected at a certain significance degree ; it might non intend that the consequences should be accepted. This is because of the deficiency of power of the statistical trials particularly for surveies covering less than 15 old ages or since 1973. To turn out his point, he used longer informations sets and was successful in rejecting the void hypothesis for the Dollar/Pound exchange rate during the period 1869-1984. Besides, he found the half life for PPP divergences to be around 4.6 old ages. Niso Abuaf and Phillipe Jorion ( 1990 ) besides rejected the random walk hypothesis by utilizing informations from 1901-1972 for eight currencies. They found half life of perturbations from PPP of approximately 3.3 old ages. Using about two centuries of informations for the Dollar/ Pound and Franc/ Pound, James R. Lothian and Mark P. Taylor ( 1996 ) found groundss of average reversion in both exchange rates with half life of 4.7 and 2.5 old ages correspondingly. Although these surveies gathered comparative success in the testing for PPP, their methodological analysis has been criticised by others. Michael Mussa ( 1986 ) and Qian and Strauss ( 2001 ) argued that existent exchange rates tend to be more volatile under drifting than fixed exchange rates and analysis inclusive of periods of drifting and fixed might finally ensue in econometric deductions.

In add-on, the power of the trial could be farther improved through the inclusion of more states into the analysis. Frankel and Rose ( 1995 ) found average reversion on a cross-sectional analysis of drifting exchange rate. They examined one-year informations from 1948-1992 for a sum of 150 states and were able to reject the random walk hypothesis utilizing merely post-1973 informations. Abuaf and Jorion ( 1990 ) found similar consequences when they analysed 10 first-order autoregressive arrested developments for existent dollar exchange rate utilizing station 1973 information. On the other manus, Taylor and Sarno ( 1998 ) criticised that the research workers are incorrect to reason that all the rates are average returning when void hypothesis of random walk is rejected. Alternatively, the rejection of the nothing would merely connote that at least one of the rates is average reversion. Therefore, they suggested carry oning an alternate hypothesis to prove that least one of existent exchange rate is non-mean backsliding.

## 2.4 Structural Deviations from PPP

One of the most known theoretical model presented to explicate the divergences from PPP in the long tally, is the Balassa-Samuelson theoretical account. ( Balassa 1964 and Samuelson 1964 ) . Their empirical consequences show that in general richer states will hold higher consumer monetary value index ( CPI ) than poorer states. They argued that wealthier states are comparatively more productive in the traded good sector than the poorer states. As they have greater technological high quality in the traded goods, this will ensue in a rise in productiveness which will in bend addition the degree of rewards for the sector. But as the monetary value of tradable goods is restricted by the fixed exchange rate and universe monetary values, it will non increase. Contrastingly for the non-traded sector, they will hold to increase their monetary value as they could non fit the productiveness addition of the tradable sector. In the terminal with the addition in the monetary value of non-traded goods while changeless monetary value of tradable goods, the overall monetary value degree of the state have to increase. But if the nontradable goods sector is able to fit the productiveness additions of the tradable sector, the overall monetary value degree and existent exchange rate will remain changeless. This economic principle can be applied to both fixed and drifting exchange governments. In hindsight, their findings and economic principle highlighted the differentiation between traded and nontraded goods and the impact on the monetary value degree and existent exchange rates in the states. Froot and Rogoff ( 1994 ) has mentioned that if the non-traded goods sector is more labour intensive, a even balanced growing in both sectors will ensue in an grasp of comparative tradable monetary value. Empirical consequences from Ragoff ( 1996 ) besides supported this claim as they found a positive relationship between income and monetary value degrees, with the richer states holding the greatest differentiation.

Other factors that might impact the PPP from keeping includes ; differences in exchange rate government, import and export limitations, travel costs, perishable goods and location amid others. These will impact the monetary value of similar goods sold in different states every bit good as the general monetary value degree in the several states, ensuing in the misdemeanor of the jurisprudence of one monetary value and the convergence to PPP in the long and long tally.

## 2.5 Inventions in the trial of PPP

Most early surveies on PPP based on the simple PPP hypothesis or trial for being of unit root and random walk have produced dissatisfactory consequences that does non back up the relevancy of the PPP theory. Subsequently, research workers started to implement more sophisticated econometric techniques, in hope of acquiring favorable consequences. Two of the more noteworthy techniques employed by them are the Co-integration trial and Fractional integrating.

## 2.5.1 Co-integration

Engle and Granger ( 1987 ) concluded that if two non-stationary variables are integrated of the same order and set into a arrested development, the two variables will possess a relationship, if the remainder is stationary. Therefore, Co-integration trial could be used to observe the presence of a weak relationship for time-series that are otherwise rejected under strong PPP standard. Test for “ strong ” or rigorous signifier PPP trial could be determined through the values for of I± and I? under a arrested development. For a “ strong ” Palatopharyngoplasty to keep, the value for I± and I? should be 0 and 1. But in co-integration, even if the variables does non conform to these conditions, there might still be a relationship in a weaker signifier if the remainder of two non-stationary variables, is found to be stationary.

Testing for co-integration fundamentally involves a three measure process. First, is to utilize the Dickey-Fuller trial to find whether the several clip series ( Exchange rates and monetary value degrees for the states ) have unit roots and whether they can be differenced to the same whole number order. Following on, the “ co-integrating ” equation is regressed utilizing OLS and residuary is determined. The last measure involves subjecting the OLS remainders to the Dickey-Fuller unit root trial so as to find its stationarity. Hence, the variables will be co-integrated when the remainder is stationary. Corbar and Ouliaris ( 1988 ) , Taylor ( 1988 ) , Kim ( 1990 ) applied co-integration techniques but most found that the void hypothesis of non co-integration can non be rejected in most instances. Rogoff and Froot ( 1994 ) compared the 3 techniques used to find PPP and concluded that although the co-integration trial have been more successful in rejecting the void hypothesis, it is ill-defined whether this technique produces a benefit over the simple PPP hypothesis or random walk trial. Though the consequences have non been satisfactory, these surveies have revealed common features of the trial and information. First, rejection of the void hypothesis of no-cointegration occurs more frequently under fixed than drifting exchange rate governments. Besides, informations based on CPI are less likely to reject than dataset for WPI. Last, the rejection of no-cointegration for drifting exchange rates under the station Bretton-Woods period occur more frequently for trivariate systems than for bivariate systems and remainders appear to be more stationary by the weakening of the proportionality and symmetricalness limitations.

To farther increase the power of co-integration techniques, research workers have suggested the usage of panel cointegration methods to bring forth more robust findings. { Banerjee ( 1999 ) , Pedroni ( 2000, 2001b ) and Qian and Strass ( 2001 ) } . This improved technique allows the research worker to selectively pool long tally information in the panel while leting for short tally heterogeneousness among the different members.

## 2.5.2 Fractional Integration

A more recent tendency that is used by current research workers of PPP is fractional integrating. This methodological analysis was implemented due to the jobs built-in in the unit root and co-integration trials. Cuestas and Gil-Alana ( 2009 ) suggested that the possibility of Type II mistake happening in unit root trial for Dickey-Fuller additions as the sample size expands. Therefore making structural alterations with the addition of informations sample and if non taken into consideration will hold inauspicious effects on power of the trial. In add-on, Dumas ( 1994 ) , Michael et Al. ( 1997 ) , Sarno et Al. ( 2004 ) , Juvenal and Taylor ( 2008 ) , and Cuestas ( 2009 ) among many others, found that larger the divergence from long tally equilibrium exchange rate, the faster the gait of reversion to the mean. Further, Gil-Alana ( 200 ) besides found that if the velocity of the average reversion is slow, the consequences from Dickey-Fuller might be deceptive and fail to place the being of unit root. As a consequence, fractional integrating was seen as an alternate solution to the PPP mystifier. With this method, the existent exchange rate will hold to be estimated as a fractionally incorporate procedure.

## ,

Where and are multinomial slowdown operators with roots outside the unit circle. is a white noise operator. If parameter vitamin D = 1 & A ; peers to, the void hypothesis of random walk can non be rejected. On the contrary, If 500 = 0 Null hypothesis rejected and PPP holds in the long tally.

Baum, Barkoulas and Caglayan ( 1999 ) used fractional integrating on 29 states and suggested that rejection of absolute PPP in the long tally. Cheung and Lai ( 1993 ) argues that the average reversion of PPP exist and can be categorised by a fractionally incorporate procedure in 3 out of 5 states they tested. Masih and Masih ( 1995 ) concluded that the fractional cointegration attack “ captures a much wider category of para or mean-reversion behavior ” than simple cointegration method. On the other manus, Soofi ( 1998 ) found that although the variables are fractionally cointegrated, the PPP theoretical account for all states does non hold a mean-reversion belongings. Alves, Cati and Fava ( 2001 ) infer that the grounds does non look to back up the absolute PPP hypothesis but the comparative PPP holds in the long tally when they tested the PPP theory for Brazil.

## 2.6 Reasons for taking U.S and UK

Before lucubrating into the empirical subdivision, it would be appropriate to explicate the principle for choice of the states for the probe, U.S and UK. Both these states are considered the most advanced in the universe and portion eldritch similarity in assorted facets ; talk the same linguistic communication, economic fundamental law of the state, life styles, outlooks, civilization, faith and etc. The “ developed ” position of both the states entails that they exist in an unfastened economic environment where the flow of information is efficient and unrestricted. More frequently, consumers in these states have no limitation on their picks of similar goods from non merely the local market but besides from foreign states. As mentioned in old subdivision, Buying Power Parity ( PPP ) theory provinces that “ in the absence of dealing costs and barriers to merchandise, the nominal exchange rate between two states should be equal to the aggregative monetary value degrees of the several states. ” As such, this pre-requisite status for PPP theory is “ best ” fulfilled in these two states and as a consequence makes these states excellent topics for the elaborate scrutiny of PPP in the long tally.

## 3. Research Design and Sample

## 3.1 Descriptive Statisticss of Datas

The sample informations 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 informations of U.S and UK exchange rate and several consumer monetary value index ( CPI ) .

Figure 1 illiustrate the clip series of nominal USD/ GBP exchange rate and the comparative CPI of U.S over UK. It will be good to advert that this dataset besides includes the period under Bretton -woods understanding where the Pounds was fixed to the Dollars. It was finally abolished and allowed to drift in the 1971.

From the figure, we can see that after the abolition of the Bretton-Woods understanding in 1971, the motion of the exchange rate ( $ / ? ) has been fickle and volatile. The exchange rate reduced significantly and on a downward tendency 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 set. As the consequence of the recent fiscal crisis, the lb 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/ grasp of the currency degree should be matched by a relative alterations to the differences between the domestic and foreign rising prices rates. As a consequence, the USD/GBP exchange rate and comparative monetary value degrees of both U.S and UK should travel proportionately and in the same way. Traveling back to Figure 1, it has been have observed that the motion of the exchange rate ( $ /? ) and the comparative monetary value degrees does non travel in the same way in most periods. Furthermore, the magnitude of the alteration between the two clip series varies significantly where the exchange rate motion is larger and more volatile than the comparative monetary value ratio. This explicitly implies that the comparative PPP does non keep and there are important divergences in the short tally.

## 3.2 Empirical Model

This subdivision will depict the empirical design that this survey employ to look into the convergence of the exchange rates to the PPP theory. Our analysis will analyze the undermentioned hypothesis ;

H0: Palatopharyngoplasty does non keep in the long tally

H1: PPP holds in the long tally

The nominal exchange rate, Et is calculated as ;

( 1 )

Where Et is the exchange rate between Dollars and Pounds, PtUS is CPI for U.S and PtUK is CPI for UK. By taking logs for Equation ( 1 ) , the nominal exchange rate will be later reduced to a additive equation of the domestic ( UK ) and foreign ( U.S ) monetary value degrees, as shown in equation ( 2 ) below.

( 2 )

As a consequence, the estimated theoretical account will be ;

( 3 )

Where I± is intercept, I? is incline and Iµt is the remainders.

## 3.2.1 ” Strong ” Palatopharyngoplasty

In this subdivision, I will seek to look into whether the “ strong signifier ” of PPP exist for the U.S and the UK. There are two chief methodological analysis to analyze whether the most direct relationship of PPP holds for the capable states.

Ordinary Least Square Regression

This method involves executing a arrested development on equation ( 3 ) and proving the void hypothesis that and If the void hypothesis can non be rejected, it meant that PPP holds in the long tally but alternatively the void hypothesis is rejected, PPP theory does non keep. From the descriptive statistic ( Figure 1 ) , it is seen that the exchange rate and comparative monetary value of U.S and UK are trended. This ascertained fact meant that Ordinary Least Square ( OLS ) calculator might non be suited for the analysis as non-stationary time-series variables tend to bring forth specious arrested development due to jobs of hetereoskedasticity and consecutive auto-correlation, which will finally take to deceptive and colored appraisal consequences.

Stationarity of Real Exchange rate

Alternatively, to analyze the basic relationship of PPP between U.S and UK, I have opted to prove for unit root in the existent exchange rate to find the cogency of PPP in the long tally. To accomplish this, the existent exchange rate will foremost be determined, as shown in equation ( 4 ) .

( 4 )

Where denotes the existent exchange rate, denotes logarithm of nominal exchange rate whereas and denotes logarithm of CPI for U.S and UK severally. Follow on, I will prove the void hypothesis that the clip series of existent exchange rate remain stationary and does non follow a random walk. If this occurs, it will be concluded that PPP holds in the long tally. Alternatively, if the existent exchange rate does non follow a random walk, On the contrary, if existent exchange rate does follow a random walk, PPP does non keep. Subsequently, this could be explained some dazes or factors i.e utmost autumn in production or productiveness, in the economic system due to unanticipated fortunes. Linear unit root trial could be used to observe the being of unit root in time-series. More specific, Augmented Dickey-Fuller ( ADF ) , which is a trial of unit root, will be performed ;

( 5 )

Where I”yt is the first difference of log ( Et ) ; are the optional exogenic regressors which might include a changeless or a changeless and tendency, I± and I? represents the parametric quantities that are to be estimated, represents the remainders.

## 3.2.2 Co-integration Trial

Co-integration trial involves proving the void hypothesis that the variables are co-integrated and alternate hypothesis that they are non co-integrated as shown below.

: And are cointegrated

: Andare non cointegrated

Before get downing the co-integration analysis, it will hold to find whether the two clip series are stationary or does non hold a unit root. This involves proving the void hypothesis of non-stationary in the variables. To prove the premise of non-stationarity of variables, Augmented Dickey-Fuller trial will be used.

Once settled on the position of stationary of the variables, the equal figure of slowdowns ( P ) that should be integrated in the ADF arrested development will next determined, so as to avoid the job of consecutive correlativity in the remainders. This could be determined by taking the sum of slowdowns that minimizes the standard information standard ( Schwarz Information Criterion ) . So, if both the variables are non-stationary and integrated to the same order, it will be deemed possible to transport on utilizing co-integration to prove the long tally relationship of the variables and convergence to PPP. On the contrary, if the variables can non be integrated to the same order or that either one or both of the variables are stationary ; it would non feasible to transport on with the testing.

Engle and Granger ( 1987 ) mentioned that two non-stationary variables when integrated to the same order, will be co-integrated if their remainder is stationary. Puting into application, if two non-stationary clip series and that are both integrated to the same order, produces a stationary variable after the arrested development, it can be inferred that the variables are co-integrated.

The trial of co-integration involves two chief processs. First, the undermentioned arrested development equation ( 3 ) will be estimated, so as to acquire the estimated mistake term or residuary ( equation 6 )

( 6 )

Following, Augmented Dickey-Fuller will be used to prove the void hypothesis that the estimated mistake term has a unit root or is non-stationary. By comparing the t-stats with the Dickey-Fuller t-Statistics ( specious co-integrating arrested development ) , the variables are co-integrated if we can non reject the void hypothesis that the remainder has a unit root or non-stationary. But if we reject the void hypothesis of unit root, the variables will be co-integrated.

H0: Palatopharyngoplasty does non keep in the long tally

H1: PPP keep in the long tally

Finally based on our empirical hypothesis, if the variables are found to be co-integrated, the nothing that PPP does non keep in the long tally will be rejected and I can reason there be an equilibrium relationship between the exchange rate and comparative monetary value degrees in the long tally. Alternatively, the void hypothesis can non be rejected if the variables are non co-integrated.

## 3.3 Robustness Trial

To guarantee that the consequences from the co-integration trial is robust, I will use another methodological analysis for the analyzing the long tally equilibrium relationships of the variables and whether PPP holds. The Johansen maximal likeliness ( ML ) trial is proposed to transport out the analysis.

Johansen ( 1988, 1991 ) came out with the maximal likeliness ( ML ) method for gauging the equilibrium relationships in the long tally or co-integrating vectors. He derives a likeliness ratio ( LR ) trial for co-integration in a Gaussian vector mistake rectification theoretical account.

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

Where are the smallest characteristic root of a square matrixs of S21S11-1 S12 with regard to S22.

Gonzalo ( 1989 ) argued that for the finite sample belongingss of the ML calculator are really consistent with the asymptotic consequences. Stock and Watson ( 1991 ) used the monte carlo to confirm the indifferent belongings of the ML calculator. Cheung and Lai ( 1993 ) besides concluded that the Johansen trial has important power high quality over the residual-based trial like co-integration. Furthermore, as the Johansen trial merely involves one measure every bit compared to three stairss under co-integration, there will be less chance for mistakes in the application procedure. Therefore, Johansen trial will be utilized for hardiness.

## 4. Empirical Consequences

## 4.1 ” Strong ” signifier PPP Results

Table 1 shows the consequences from the unit root trial for existent exchange rate. The ADF trial on the existent exchange rate shows that the void hypothesis of unit root can non be rejected at the 95 % assurance degree. As expected, the consequences show that existent exchange rate is non-stationary and there is no statistical grounds to back up the being of a convergence to PPP in the long tally for the “ strong ” signifier.

## 4.2 Co-integration Consequences

As mentioned antecedently, the initial showing for co-integration trials involves finding whether the variables are stationary. Augmented Dickey-Fuller unit root trial was performed and it was found that nominal exchange rate, CPI for both U. S and UK and difference between the CPI for both states are non-stationary or has unit root at the 95 % assurance degree. ( See table 2 )

After finding that the variables are non-stationary and follows a random walk, I will stipulate the sum of slowdown that will be included into the ADF arrested development to forestall the job of consecutive correlativity in our several variables. The ADF consequences which includes the minimal Schwarz information standard is presented in table 3. The consequences show that both variables are made stationary when integrated to the order of I ( 1 ) , with the inclusion of the slowdowns with minimal Schwarz information for each variable.

It will be executable to transport on proving for co-integration when the variables are found to be non-stationary and could be integrated to the same order. However, it will non be possible to transport on proving for co-integration if variables are integrated to different order or when either one of the variables ( Independent and dependent ) or both exhibits stationary clip series in the unit root trial. The consequences till now have shown the void hypothesis of non-stationary or unit root can non be rejected for both the variables at 95 % assurance degree. Furthermore, both variables could be integrated to the order of one, I ( 1 ) . As a consequence, I will continue with the consequences from co-integration.

The co-integration trial has carried out executing a arrested development on the undermentioned co-integrating arrested development ( 3 ) ;

( 3 )

The consequences for the co-integration arrested development are as follows:

Furthermore, the R2 and Adjusted R2 are 0.581843 and 0.581014 severally. This consequence shows that the coefficient for the independent regressor ( difference in CPI ) is positive and important. Following on, the remainder of the co-integrated arrested development will be determined and subjected to the ADF trial. This will prove the void hypothesis of unit root and stationary for the residuary. If the void hypothesis of unit root can non be rejected, I will reason that the variables are non co-integrated and PPP does non keep in the long tally. Conversely if void hypothesis is rejected, it will reason that the grounds seems to deduce that the variables are co-integrated and PPP theory holds for U.S and UK in the long tally equilibrium. The ADF trial on the remainder has given us a trial statistic of -3.039267. Comparing this t-statistics with the critical value taken from the Dickey- Fuller t-stats tabular array ( applied to remainders from specious co-integrating arrested development ) for Case 3 of -3.42, it can be inferred that the remainder has a unit root and is non-stationary. ( See table 5 )

H0: Palatopharyngoplasty does non keep in the long tally

H1: PPP keep in the long tally

Consequently, this implied that the variables are non co-integrated and the void hypothesis that PPP does non keep in the long tally equilibrium can non be rejected.

## 4.3 Johansen Test Results

Table 6a and 6b shows the Johansen trial consequences for the Trace and Maximum Eigenvalue trial severally. Each trial examines two hypotheses ;

H0: No Co-Integration

H1: One Co-Integrating Relationship

## & A ;

H0: One Co-Integrating Relationship

H1: Two Co-Integrating Relationship

The consequences from both trials show groundss that they reject the void hypothesis of no co-integration and can non reject the void hypothesis of one co-integrating relationship. Therefore, this meant there is presence of one co-integrating relationship within the variables.

The Johansen trial has shown that there exits a long-term equilibrium relationship between the exchange rate and the comparative monetary value degrees of U.S and UK. But in the short tally, there might be disequilibrium. In order to accommodate and explicate the passage the short-term to long tally equilibrium, I will use the usage of the mistake rectification mechanism or Vector Error Correction ( VEC ) . This procedure enables a proportion of the disequilibrium in one period to be corrected in the following period and aid to explicate the alteration in exchange rate to the alteration in comparative monetary value degrees and old period ‘s disequilibrium.

Table 7 nowadayss the consequences from the Vector Error Correction ( VEC ) estimates. The partial short tally accommodation coefficient shows how much the disequilibrium is “ corrected ” per month as the variables change. The accommodation coefficient for exchange rates is negative and undistinguished, at 0.5 % per month while the accommodation coefficient for the comparative monetary value degrees of the states is besides negative and stands at 1.25 % per month. This meant that all the “ rectification ” of the disequilibrium in the short tally is done by comparative monetary value degrees alternatively of exchange rate.

## 5. Decision

Buying Power Parity ( PPP ) has generated considerable sum of involvement from academe and interested parties ( Internet Explorer. Government, Central Bankss etc ) , since being officially introduced by Gustav Cassel in 1918 as it offers a basic theoretical theoretical account to interchange rate finding which is non merely simple to understand and besides intuitive for application. Since 1970s, academe has explored the PPP issue from changing positions and with the usage of increasingly advanced econometric techniques, but the consequences has non been consistent. In the thick of the assorted consequences, there is still belief that this theory holds the key to being an of import manual for explicating the exchange rate differences between states. This has lead to Dornbusch and Krugman ( 1976 p.540 ) declaring that “ Under the tegument of any international economic expert lies a deep-rooted belief in some discrepancy of the PPP theory of the exchange rate ” .

This paper joins the huge sum of bing literature by look intoing the convergence of the PPP in the long tally for two advanced states, U.S and UK. These states have been chosen as they “ best ” fulfilled the status for the Purchasing Power Parity theory. This analysis was achieved utilizing co-integration techniques and farther supported for hardiness by Johansen trial. The findings for both the trials ( Co-integration and Johansen trial ) have generated contrastive consequences. Co-integration trial consequences show that the variables are non co-integrated and PPP does non keep in the long tally equilibrium while Johansen trial has identified the presence of one co-integrating relationship within the variables and reject the void hypothesis that PPP does non keep in the long tally. Building on the consequences from the Johansen trial, an mistake rectification theoretical account or Vector Error Correction ( VEC ) was performed to further understand the short tally disequilibrium and their passage into long tally equilibrium. It was found that all the rectification or accommodation to equilibrium was done by the comparative monetary value index alternatively of exchange rate and at a rate of about 1.25 % per month. Although this paper was non able to supply any concrete decision on the relevancy of the PPP theory on these two advanced economic systems, it has given findings that are consistent with the bing literature on Buying Power Parity. The fact that different econometric theoretical accounts performed on similar dataset, have produced distinguishable consequences and decisions.