Purchasing Power Parity (PPP) Between UK and India
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This Paper discuss about the Validity of the Purchasing Power Parity around the world. Therefore in order to test the validity of the Purchasing Power Parity (PPP), I have decided to take to two countries and have carried out some empirical tests to test the applicability of PPP. I have taken and tested the PPP between India and United Kingdom (UK) where one being domestic and another being foreign country respectively. As we have seen Indian Economy is growing in an accelerating pace where as IMF economic outlook (2010) believes advance economy countries will face challenges (which is true if we look at the European economy in the last 2years) some factors liable are due to the uncertainty of the financial sector, increased in funding cost, consumers and businesses lower confidence reducing the private consumption and investment. On the other hand, IMF believes India's economy growing at an accelerating pace and is highly likely to raise more near future. One of the main reasons is because of their domestic demand of their goods all over their country and so as the demand has increased all over the world which leads to increase in their exports and eventually results in economic growth. As we can see the from the recent IMF update 2010, the emerging and developing countries tends to perform better than the advanced economies where we can take UK as a best example. 
Therefore in my opinion I thought it would be interesting idea to select 2 countries, India and UK, one being developing country and another being developed countries respectively.
A very important point which need to noted that, in the foreign exchange rate market the India's central bank intervenes in order to stabilise the movements in the exchange rate, therefore what we can say the exchange rate is not only determined inflation differential between both the countries.
In addition India have got some specific factors which determine the exchange rate, some of the key factors specific to India has been discussed later in the paper.
2. Theoretical framework:-
The term Purchasing Power Parity is a very simple and easy to understand theory that strives to explain the changes in exchange rates of different countries in an extended time period. It was first stated by Cassel in 1918 and is coming to the forefront at an accelerating pace due to global progression. Ideas of PPP have been discussed before as well by scholars from the 15th and 16th centuries such as Officer (1982).
Many argue that PPP is not a valid model as it does not take many important factors into consideration and it does not hold true when observing exchange rates on a short time period however long run period tends to hold. Should it be considered as a random walk phenomenon? Previous studies have shown that shocks to the real exchange rate dwindle out very slowly. There is also a proof showing that fast developing countries usually experience real exchange rate appreciations while richer countries it tends to be higher. The “Balassa Samuelson” theory is very well known within the framework of PPP. Yet in industrializing countries, the results of the examination of empirical evidence in favour of Balassa Samuelson are weak.
Frankel (1986, 1990) argued that the reason for failure to reject the random walk model of real exchange rates was a lack of power.
In order to understand PPP we must take in to account the concept of law of one price as PPP is the application of the law. The law of one price states that in a competitive market excluding transaction cost and barriers to trade, identical goods in 2 countries should sell for the same price at the same time, when expressed in the same currency. The law of one price implies that the dollar price of good i is the same wherever it is sold.
PiUS =(E$/€)(PiE) (1)
E$/€ = PiUS/PE
Thus the exchage rate amongst 2 countries' currencies is equal to the ratios of the countrys' price level. This means that if there is a decline in the domestic purchasing power of a currency, it will cause a proportional currency depreciation in the foreign exchange market. This shows that PPP claims that when measured in the same currency, the price levels of all countries will be the same.
An easy way to understand the law of one price is to consider a practical example. if the dollar/pound exchange rate is $1.60 per pound a cell phone costing $100 in New York must not sell for more than £62.5 (Pus/ (E$/€) ) in United Kingdom. So in dollars the cell phone sold in London is $1.60 × £62.5 = $100. If there is any deviation from this price, then there would be an arbitrage opportunity.
Absolute and Relative PPP
The absolute PPP exchange rate considers the national price levels in two countries are equivalent if expressed in a common currency at that rate, so that the purchasing power of one unit of a currency would be the same in the two countries. 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, thus we can say depreciation of currency levels should be equal to the difference between domestic and foreign inflation rate.
Formula for Relative PPP
log(eRs/£) = log(PRs) - log(P£) + θ (2)
where θ is the short-term deviation from PPP and is equal to equation
θ = log(eRs/£) + log(P£) - log(PRs) (3)
The short run deviation from PPP, θ, is equal to log(rRs/£) as shown in equation (3) and when θ=0, PPP holds.
θ = log(rRs/£) = log(eRs/£) + log(P£) - log(PRs)
If we set: log(rRs/£) = rt, log(eRs/£) = et
log(P£) = Pt* log(PRs) = Pt
Then we obtain equation (4):
rt = et + Pt* - Pt
3. Previous studies on PPP
There were only a few empirical evidences that specially look at India's exchange rate relevance to PPP. One working paper (Renu Kohli 2002) during the floating period tries to study whether specific factors like structural changes in exchange rate regime, trade liberalisation and loosening of foreign exchange restrictions in India has an impact on the real exchange rate . The paper tested for stationarity of the real exchange rate using unit root tests. There result was a mean reversion in the real exchange rate series with the CPI as deflator. However, tests for stationarity of relative differential of tradable and non-tradable, real shocks may then be used to explain real exchange rate movements.
Froot and Rogoff (1995) differentiated among three steps of time series analysis of PPP. Because of its failure to consider the possible non-stationaries in the time variables used, stage one tests were not to perfection. On the other hand, stage-two tests concentrated more on the null hypothesis that real exchange rates tend to follow a random walk while the alternative said that PPP holds in the long run. But Lothian and Taylor (1997) along with many others have seen to yield very little due to deficiency of numbers of years used and so these tests were not able to differentiate between random-walk behaviour and extremely slow mean-reversions in the PPP-consistant level of the real exchange rate if not longer yearly data are used. The stage three-test was not able to improve the theory of understanding the real exchange rate behaviour as it made use of co-intergration methods but suffered from low power as well.
An interesting study was made by Leon and Najaran in 2003 in which they put a lot of attention to whether deviations from PPP are stable if there are nonlinear specifications present. It was an important study and enriched the already enormous data of PPP. This study included a time-varying threshold autoregressive model (TVTAR) that elaborates the diligence of variations from PPP. The methods involved calculating frequencies, size and durations of the variations of exchange rates from predicted changes in the exchange rates. Thus there was some level of support found for the stationary TVTAR against the hypothesis of the unit root. However, TVTAR was not dismissed for some developing countries.
Leon and Najarian's paper made some interesting claims regarding different countries reacting to the threshold mode with the more deprived countries using the threshold model. A notable distinction was noted between well-off nations and the developing countries where the former usually tend to be more obliged to over-depreciate. In addition the paper also indicated that the time-period of countries in Asia are higher for over depreciation but is the opposite for the more affluent nations.
One of the main results of this paper was assymetric adjustment. It depicted that the macroeconomic effects of real exchange rate misalignments change with the level and type of misalignment, over-appreciations or over depreciations (Leon & Najarian 2003). Trade oneness is probably connected to the time period of over-appreciation misalignments (Leon & Najarian, 2003). But dept ratios are influenced differently because they are connected to deviations of over-depreciations. And finally the country which is being considered is a dominating factor in influencing the exchange rate pattern, especially with regards to the economic development of that country.
A study similar to mine, a research was carried out by Jyn-Lin Wu, who have studied and investigated if there is long-run purchasing power parity in the case of Taiwan exchange rates. The paper does not impose a priori restriction on the cointegrating vectors due to risk of errors in measuring the price indices. The paper has carried out 3 tests- unit root, Engle-granger and Johansen test for the validity of the long run equilibrium of PPP. The author found out that Taylor (1988), Taylor and McMahon (1988) and Davutyan and Pippenger (1985) measurement errors is key factor for the testing the hypothesis of PPP. His paper also discusses that many researchers have found result in favour of PPP since they considered measurement errors in data. Thus Jyn-Lin wu has also considered the measurement errors into consideration and examined the hypothesis of PPP of Taiwan using the Johansen test. The author investigated the validity of PPP between Taiwan and major industrial countries like France, West Germany, Canada, the United States (US) and the United Kingdom (UK).
When the author started carrying out investigation for the hypothesis of non-stationary of real exchange rate for each country named above. The result obtained shows in the long-run PPP fails to hold except for UK. It requires a proportional restriction between the exchange rates and prices when unit root test is carried out for real exchange rate but once the error measurement are concerned for the data the limitation may not be consistent. Therefore due to restriction imposed in the unit root test of real exchange rate could be a cause for which the PPP might have failed to hold.
Similar result have been achieved when the author applied Engle-Granger test, the result obtained no cointegration between exchange rate and the relative price level but once again except for UK even when the measurements errors where considered.
Finally when Johansen test was applied, both the trace test and the maximum eigenvalue show that except for UK there is at most one cointegrating vector. But UK's trace tests shows presence of two cointegrating vectors, but maximum eigenvalue tell that there is no evidence of cointegration.
Inflations occurs when there is a rise in price level of goods and services in an economy over a period of time, when this happens real exchange rate between the two countries that is, in our case UK and India, there might be a effect in the PPP
The inflation rate for UK and India were 3.1% and 13.73% respectively therefore there is a possible chance of PPP being divergent.
The table below is the inflation rate for India from July 2008 till July 2010. 
United Kingdom inflation rate from July 2008 till July 2010 
Balassa and Samuelson (1964) argues that real exchange rate changes due to economic growth price variation between tradable and non-tradable goods, thus changes is real exchange rate causes divergence in PPP.
The PPP divergent was sustained due to the economic growth variation between tradable and non-tradable goods and in addition due to real exchange rate of the country being affected.
Machlup ( 1964 p27) findings argues that movements in exchange rate are effected by factors such as changes in capital movements, a country's having a high inflation rate, taste and productivity therefore causing a ppp being divergent.
Contrary to Balasssa & Samuelson Imed Drine & Christophe (2007) results concluded that PPP is accepted in countries with higher inflation.
India population being world's 2nd largest country after china for population of around 1.15 billion and study by Population Reference Bureau (PRB) predicted that by 2050 India's population will be exceeding china population having 1.6billion in 2050.
Recent evidence shows that population growth is one of the factor which has an effect on PPP since population growth affects the overall economy of a country. But there are very few literature and empirical evidence which talks about the population growth.
A very recent Paper by Salam and Hassan (2009), studied the link between the change in population growth and the changes in PPP exchange rate due to balassa-Samuelson using 80 countries and has obtained a robust result supporting the hypothesis. Been discussed earlier that PPP exchange rate is affected by changes in population growth which has an impact on wages and which leads to change in price level. The author has used panel data and carried panel unit-root and panel co-integration to examine the relationship long run equilibrium. In the paper they have also talked about the paper of Aloy and Gente (2005) where is argued and concluded that a country's population structure has an impact on the overall spending, savings, therefore affecting the exchange rate and the Purchasing Power Parity. In addition Alloy and Gente went further and stated that population growth has an overall affect in the nominal average wage in an economy thus having an impact in PPP. Since the paper Mentioned above Balassa and Samuelson argued that labour is not same for all countries, poor countries are likely to have lower labour that richer countries, change in wages which is followed by change is working class population thus having impact on the price of the tradable and non-tradable goods and therefore appreciating the real exchange rate and gives a support for the Aloy and Gente findings.
Black market exchange rate/ Exchange rate regime
Indian government started preparing to make rupee convertible in the February 1992 and in March 1993, they implemented the single floating exchange rate. However during 1975-1992 rupee was controlled by the government and pegged to a trade-weighted basket of currencies.
Before moving we should know what a black market exchange rate it normally takes place in a less developed countries where the exchange it is an unofficial but widely accepted which is fully market determined and is an illegal but commonly used accepted source of foreign currencies which is fully market determined.
A thought provoking study by Badhestani and Noer (1993) inferred a direct relationship between black market rates and the official rates from the RBI. Surprisingly, it was the reserve bank which was taking cues from the black exchange market. They concluded that the PPP holds in the long run but with short term deviations. This however is not conclusive as the case of Indian scenario is unique as it boils upon the fact of pre and post liberalization period which has to be taken in to consideration in order to understand PPP of India entirely.
4. DATA AND METHODOLOGY
In order for empirical evidence to find the evidence of the long run PPP, we have taken the data on rupee-pound exchange rates on a monthly frequency from January 1968 to February 2010 and CPI of both UK and India.
The main aim of the paper is to test the relative PPP holds or not, therefore few of the test is carried out in order to check presence of PPP.
Test for Unit Root
This is a test which is used to check if the real exchange rate has a unit root. Discussion to the above what we can say ourselves does the real exchange rate is a random walk.
real exchange rate
R = E x P_in/P_uk
Linear logarithmic relationship between nominal exchange rate and Price ratio of both countries.
ln R = ln P_in – Ln_uk + ln E (5)
A very important point to note is that not only price levels determine the country's nominal exchange rate but it can also be affected by a rise in production or demand due to circumstances which were unexpected and results in some sort of trending behaviour in data thus PPP will not hold.
Therefore Augmented Dickey-fuller Unit Root (ADF) test is used to check if there is a unit root in the real exchange rate. This is a Common test been used in most empirical paper.
The testing Procedure for the ADF test is as Follows
âˆ†y t= α+ßt + γyt-1 + δ1 âˆ†yt-1 + … + δpâˆ†yt-p + εt, (6)
where α is constant, ß is constant on a time trend and p is the lag order of the auto-regressive process.
Test for Co-integration
The basic meaning of co-integration is when there is a long run relationship between two or more variables. Our Paper investigates if there is long run relationship between nominal exchange rate and the relative price level both being a dependently variable and independent variable respectively. In order to test the long run relationship between these two variables that is if they are Cointegrated, thus the stationarity of the residual is tested in order to justify if the long run relationship exist.
The approach used is called Engle-Granger for testing the relationship between the nominal exchange rate and the price level.
The equation used to carry out this test is
Here, Pt = log-domestic (ind) price index, P*t=log-foreign (UK) price, St = log-nominal IND/UK exchange rate
And thus the equation should be stationary, where µ and µ* are not constrained equal to 1.
Therefore our aim is to test
Ho : Nominal exchange rate and relative price level are not Co-Integrated
Ha: Nominal exchange rate and relative price level are Co-Integrated
However in order to test the Co-Integration we first have to make the 3variables St, Pt, P*t stationary. Thus ADF test can be used to check if the variables need to be first differentiating to make them stationary. As to test the hypothesis we need to make the 3variables above as I(1).
Once they are bought to stationary we estimate the co-integration regression
St = α+ µ(Pt-P*t ) + µt (8)
where we put all the variables of same order into a regression series. Later we carry out a unit root test in the residuals. If we reject the null of a unit root, we conclude there is an evidence of co-integration between nominal exchange rate and prices, hence in favour of PPP.
As we know the test above that is the Engle and Granger's method is straightforward, however Sephton and Larsen (1991) call attention that method is subject to some following criticisms 1-lack of information about the asymptotic distribution of the Engle-Granger test, 2-rather than all possible cointegrating vector only the dominant cointegration vector is examined in the Engle-Granger test.
Therefore I have also checked also used a more powerful test of cointegration what is known as ML procedure by Johansen.
Johansen (1998, 1991) offer the highest probability method to explore cointegration among variables. In addition it is more preferable then the Engle-Granger test as it is robust to different departures from normality, it does not face problems associated with the normalization, and well-known about its asymptotic behaviour. Cheung and Lai (1993) in his test discussed about both the test power and have concluded that Johansen test is much more better then Engle-Granger test, and have also found evidence that using Johansen test they find a long-run relationship, where the auther conclude PPP holds. 
We have considered as the following Vector Auto Regression (VAR) of order
Where Yt is (k X 1 ) vector of I (1) variables, in our working paper Yt = ( St, Pt / P*t) ` and Xt being a ( d X 1 ) vector of deterministic variables which could trend or a intercept.
However for Johansen Test the VAR is rewritten as
To carry out Johansen test, we need to make a decision for the lag length (k) in the vector autoregressive (VAR) model in addition to inspect the appropriateness of including a time trend in the model.
Now what we have to look at is the rank of II is r < k, (where k= 3 in our case) thus existence of r co integrating relationship between k variables included in the system, Where r can be any value 2 or 1- thus both showing co integrating relationship, therefore PPP holds or it could be 0, zero indicates no co integrating relationship thus PPP fails to hold.
Therefore once the test is accomplished, two test of output are reported 1-Trace Statistics, 2- the maximum eigenvalue statistics,
Hence the Trace (max eigenvalue) statistics tests the following null hypothesis.
No cointegration vs. One cointegrating relationship : we reject the null hypothesis
One cointegrating relationship vs. Two cointegrating relationship : we do not reject the null
We conclude there is one cointegrating relationship
Vector Error Correction (VEC)
[VEC model can be estimated when we find a co-integration that is a long run relationship between variables. In our case we have two variables one being exchange rate and another relative price.
Let's consider take Y1,t and Y2,t as both the variables, so that the co-integrating equation would be
Then, the resultant VEC model is
Where term used for error correction, αi measures the adjustment time taken of the i-th endogenous variable to the equilibrium.
Many reader has found their result in favour of PPP when Johansen test was been applied, one of the paper discussed above from JYN-LIN WU (1996) found in favour of PPP between Taiwan and UK when he carried out his test without considering measurement errors in data.
Thus I had a strong feeling, my PPP would also hold when I would carry out a Johansen test between India and UK and yet it turned out to be right.
5. Testing and Results
In this section, the first I have looked at the graphs of both the countries CPI where both shows a trending behaviour. Later in this section I have looked at the results of the Unit root test, co integration test using Engle-Granger test, Johansen test, finally the ECM ( error correction model)
The graph below table 3 shows a trending behaviour of CPI for India when we plot the figures, the data used for plotting the graph was monthly data ranged from January 1968- February 2010.
The graph below in table 4a also shows a trending behaviour of CPI for UK when we plot the figures, the data used for plotting the graph was monthly data ranged from January 1968- February 2010.
Both the line shows a gap in between them, which in my point of view indicates that in the short run there is a possible divergent in PPP, the prices are not stable sometimes increases and sometimes it falls, since in order to PPP to hold there should be no gap, what that means is change in relative price should have the same change is the nominal exchange rate, in order to PPP to hold. We observe the graph carefully we can see that both looks to meet at a point in the long-run which gives a signal that, there is a possibility of PPP to hold in the long-run.
However by looking at the graph we cannot come to a conclusion, in order to come to a conclusion we have to do some stationarity test and co integration test. The result for stationarity and co integrating test is discussed below.
The first step is to find the stationarity of the statistics of the real exchange rate is by using unit root test known as ADF unit root test, the result achieved for the unit root is based on the monthly data from January 1968- February 2010. The ADF T-Statistics obtained at a lag of 0( as it shows the minimum Schwarz criterion when ADF test is carried out with lag 0 to 14) is -1.641210 at a confidence level of 95% whereas its critical value at 5% is -2.867078 therefore being -1.641 > -2.867 (shown in table 5a as Real RSs/£) thus we cannot reject the null hypothesis showing that the unit root exists and thus nominal exchange rate is non-stationary for the given period and also suggests that PPP does not hold for the given period.
Once the ADF unit root has been done and the result which has been found which showed that the there is no statistical evidence of the existence of the PPP.
The two tables below gives the reader all the result obtained when the test was carried out for Unit root test and co-integration.
The next step which has been done is checking the validity of the PPP in the long-run using co integration test .In order to carry out the test the variables shown in table 5a are checked whether they stationary when they are integrated to order of one, I(1). Thus each of the variables first difference are taken to bring those variables stationary as shown in table 5b which are told as I(0). As we can see that ADF value in table 5d-f of the India CPI, UK Cpi and the nominal exchnage rate is -1.5498, -0.3256, -0.187934 respectively which is lower than the 5% critical value of -3.148798, -3.418744, -2.867044, thus being ADF value< critical value which means the variables are not stationary.
But once the first differences is taken, now the ADF value is greater than the 5% critical value as we can see that ADF value in table 5g-i of the India CPI, UK Cpi and the nominal exchnage rate is -13.62674, -15.16459, -21.52906 respectively which is greater than the 5% critical value of -2.867089, - 2.860755,-2.867055 therefore the variables are now stationary.
Later by using the residual data of the regression ADF test is carried out .This test shows the stationarity of the residuals. The ADF t-statistics value obtained shown in table 7, when the ADF test was carried out is -2.252760 which is greater than the Hamilton critical value at 5% confidence level(Hamilton value is used since the ADF t-statistics should be compared to correct critical value and the critical values for the dickey-Fuller t statistics when applied to residuals from co-integrating regression are tabulated e.g. in Hamilton, Table 11) which is -2.252760 ( -2.252760>-3.42), thus we cannot reject the null hypothesis at 5% confidence level and the residual test for co-integration that nominal exchange rate and relative price level are not in favour of co-integrated , hence not favour of PPP, thus PPP does not hold in case of India.
After the test carrying out the test to find the existence of PPP in the short run and long run through unit root and Co-integration, I have used a Different method of test for cointegration known as Johansen as discussed earlier in the data methodology. The test shows interesting results compared to rest. What that means is the result shows a reverse affect compared to Unit root test and Engle-Granger test.
Now what we are interested in the bottom table 8 is the Trace statistics of none. As we can see the trace statistics has a value of 54.303 where as the 0.05 critical value is 29.797, thus being trace statistics>critical value, we cannot reject the null hypothesis, therefore can conclude that there is one co integrating relationship between the 2 variable, thus the result is in the favour of PPP.
Error correction Model
Since we have got the empirical finding that both the variable X and Y are cointegrated and relationship of purchasing power parity holds in the long-run. Thus we carry out vector error correction model. If we look at the table 9, both the log CPI UK and India are in favour of statistically significance This could be told by looking at the value 2.93028(UK) and -2.85970(India) which are greater than 1.96 and -1.96 respectively in absolute value. In addition there is no error correction mechanism since the difference of (1.088603-(-0.906487)) =1.99509 which is a positive value. We can only say there is a existence of error correction mechanism when the value of 1.99509 would have been negative.
The paper has discussed and has examined the long-run relationship and also has talked about country's some of the most crucial reasons like inflation, population, black Market exchange rate which has a possible impact on the validity of PPP. Long period monthly data of UK and India's CPI and the data on the rupee-pound exchange rates from January 1968 to February 2010 have been used to test the applicability of relative PPP between India and UK. When the unit root and the Co-integration test was applied the result did not provide a strong statistical support for both the relative and absolute PPP what is means is the results was not in favour of PPP. Earlier been discussed in the literature review many studies like Machlup (1964 p27) has also argued that there will be a divergence from PPP due to upward pressure on price which effects the real exchange rate. In case of India, it has strong GDP growth which will affect the consumption, spending in the same time affecting the productivity where there will be an upward pressure in the economy compared to UK.
However, when the Johansen Test was applied, the result seem to be opposite than the test which was carried out (unit root, co-integration). Johansen results show that is a long-run relationship since model provide strong evidence to reject the null hypothesis of non-cointegration in addition the Johansen's co-integration's trace statistics indicate evidence of cointegration between the relative price level and the nominal exchange rate, thus PPP exit in the long-run. Supportive of that when JYN-LIN WU (1996) carried out his test, the result obtained was also in favour of PPP between Taiwan and UK.
In sum, However, based on the interpretation and the results the PPP does not hold in the short run and in the long-run PPP may has hold but only when the Johansen test and not in the Engle-Granger based results, the PPP may have hold in the long-run due to a significant mean reversion of the real exchange rate. 
Therefore to conclude, India would is a interesting country to work on since it gives a knowledge of determination of the exchange rate , where one of the major reason is because of the pace of expansion n the international stage. An interesting reader could extend the PPP and could investigate further on PPP in the case of India and may work with the specific factors for which we believe has an impact on PPP, in addition India would is a interesting country to work on since it gives a knowledge of determination of the exchange rate, where one of the major reason is because of the pace of expansion n the international stage. On top of it, India in the near future will be getting stronger in terms of economy because of its technology and productivity, thus it will be a remarkable idea to revisit for the application of PPP for the country's exchange rate.
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