This paper aims to examine the long-run validity of PPP for United Kingdom and Japan using data from January 1968 to January 2010. The paper involves applications of different types of econometric techniques. The techniques to test PPP theory have been estimated with the formal analysis, unit root test, Engle-Granger cointegration test and Johansen method: the results of the tests show us that the PPP vector does not exist between in the cointegration space and the nominal exchange rate and domestic and foreign prices do not move one by one as implied by the theoretical PPP. Empirical results confirm that the explanation (reasons) of PPP failure can be correct.
Introduction
Nowadays there are a lot of studies in the empirical international economic literature dedicated to the problem of the testing purchasing power parity (PPP). Attempts to understand the dynamics of exchange rate make a focus on a long-term equilibrium exchange rate. The PPP theorem provides the basic framework for explaining long-term exchange rate equilibrium conditions.
Various techniques are implied to test the validity of the PPP hypothesis, but the results of these studies can be different and even opposite. The objective of this study is to examine the different approaches of PPP test in the empirical literature and consequently shed some light on the validity of PPP between observed countries as a long-run equilibrium employing traditional methods of cointegration analysis and fractional cointegration Engle-Granger approach.
All data in this study was provided by the University. Consumer Price Index (CPI) [1] and monthly nominal exchange rates are used to construct PPP models. For convenience (and according to the requirements) all the series of variables are expressed in logarithm form.
This paper organized as follows: section II briefly explains the theory of PPP concept, section III is the literature review of PPP framework, data and different methodology, section IV is the test for PPP including theoretical part and empirical framework, results of each required test: the empirical framework follows after the theory of each test. In section V we make a conclusion of the study and all tests.
1. Theoretical framework: Purchasing Power Parity Doctrine
The PPP theory is one of the classical concepts in economics theory. Firstly, in the sixteenth century the School of Salamanca in Spain was interested in the idea that exchange rates are related to national price levels. Later in the nineteenth century David Ricardo mentioned the same theoretical problems, but it was not until 1918 that the Swedish economist Karl Gustav Cassel (1866-1945) coined the term purchasing power parity.
The theory of PPP argues that the changes of the exchange rate between two countries are determined by the changes in the price levels in these two countries, or in other words, the long-run equilibrium exchange rate between two currencies is equal to the currencies purchasing power. Also, exchange rate responds to any differences in inflation rate between two observed countries.
The concept of the Law of One Price (LOOP) [2] and PPP concept are very similar and even interrelated: in case of absence of transaction costs (and other possible effects that can affect the international trade), the prices on the identical goods are the same.
There are two alternative versions of PPP: absolute PPP and relative PPP. Absolute PPP predicts that purchasing power of different currencies should equalize the prices of national basket of goods and services between two countries, because (in case of different prices) market arbitrage opportunity will force the prices to be the same, i.e.
S = P/P* (1)
where S is the nominal exchange rate, P is the domestic price level, P* - foreign price level. So, according to the formula and previous explanation it seems that exchange rate is constant because exchange rate is equal to the ratio of the domestic to the foreign price of aggregate basket of commodities. Obviously, in practice absolute PPP does not hold (Big Mac Index or iPod index prove it) and the explanations of this failure is the following:
the existence of significant transaction costs, such as tariffs, taxes, transportation costs and other trade barriers [3] ;
the existence of the non-traded goods (for example electric power that produced and sold immediately) and services that preclude arbitrage opportunity.
Also, the fact that the real exchange rate is not constant in the short-run because the price of basket is sticky and the exchange rate is affected by money or asset market shocks. There is the same fact in the long-run because persistent shocks exist.
Other version is relative PPP and it implies that the ratio of the inflation rate between two countries is equal to the percentage depreciation or appreciation of the exchange rate, in other words, exchange rate between two countries need to be adjusted to the differences of the rate of inflation in each country. Formally, it can be written by the formula:
S = k P/P* (2)
where k is a constant parameter.
However, empirical tests of PPP have been conducted using the following form:
St = β0 + β1Pt + β2Pt * + εt (3)
where St is the nominal exchange rate, Pt is the domestic price, Pt * is the foreign price. All of them are naturally logged, and εt is the error (deviation) from parity and need to be stationary. If these three variables are cointegrated and implied a stationary error term, deviation from parity will be mean reverting. Also, PPP fulfills the symmetry (β1 = β2) and proportionality (β1=1=β2) restrictions.
PPP can be the version of exchange rate determination by looking on the relationship between prices (or inflation rate) in two countries. By the following some previous studies the absolute version of PPP can be written as:
et = α + β(Pt/Pt*) + εt (4)
where εt is the nominal exchange rate, Pt* and Pt* are price indexes on the home country and foreign, respectively. This equation implies that PPP holds when the estimated coefficient of price ratio is equal to unity (β = 1).
It was said above that there are some reasons (such as transaction costs) why the prices are different, also there are difficulties in counting PPP as well: firstly, it is practically impossible to measure the quality of all goods and services in different countries , secondly, PPP numbers can vary with the specific basket of commodities used, making it a rough estimate.
Nevertheless, PPP approach is widely used by many international organizations in their researches and statistical data - usually PPP calculations are often used to measure poverty rates.
2. Literature review
Existence of PPP theory caused a lot of new studies in the econometric techniques about this concept. The early empirical studies determined the following equation for testing PPP:
St = α + β1Pt + β2Pt * +μt (5)
St - is the nominal exchange rate, P - domestic prices, P* - foreign prices, μt is a disturbance term. β1 and β2 are restrictions for absolute PPP: β1 = 1, β2 = -1. On the other hand, in order to test relative PPP a test for the same restrictions need to equation above (5) with the variables in the first differences form. There are distinctions between the test that β1 and β2 are equal and have an opposite sign - the symmetric condition - and the test that they are equal to unity and minus unity respectively - the proportionary condition.
Ordinary least square model was applied in the early literature with mixed results. On the other hand, most researchers did not introduce any dynamic elements in the estimated equation in such a way as to determine the difference between short run and long run effects (even in case when it was recognized that PPP is expected to hold in the long-run term). Nevertheless, the empirical tests are based on the estimation of equations.
However, in a significant work, Frenkel (1978) found evidence in favor of PPP only for economies with high inflation; in the study the estimates β1 and β2 are found to be very close to plus and minus unity. Nevertheless, later this author found that there is no obvious evidence for countries with high inflation.
Moreover, some of the empirical literature has been based on the empirical examination of the real exchange rate. If the real exchange rate is to settle down at a level consistent with PPP, it has to posses some reversion towards its own mean. So, mean reversion is only a necessary condition for lung-run PPP. It can be explained as if the real exchange rate is non mean reverting then the long-run PPP would collapse. There are several early quantitative studies such as Darby (1979) or Adler and Lehmann (1983). They have tested the null hypothesis that the real exchange rate does not exhibit mean-reversion. Instead of this it follows a random walk, the non-mean reverting time series process where changes in each period are completely independent and random. What they discovered was less convincing support for long-run PPP over the recent floating regime. The results from the empirical test shoes a failure to find evidence to reject that the real exchange rates closely reflect random walks, implying that the persistence nature of shocks not to allow the deviations from parity to reverse. The same point of view we can find in the articles by Darby (1979) and Lothian (1987), where similar econometric approach is used. According to these studies the real exchange rate does not exhibit a unit root, contracting the PPP proposition.
The econometric methods described above caused the criticism: later empirical studies address the issue of nonstationarity seriously. The basis of the general approach is based on testing for non-stationary of the real exchange rates. Changes in exchange rates or prices (in case of these variables have unit roots) are to some extent predictable, even they may still never settle down at any special level, so it can collapse PPP. Since 1980s a standard approach has been to employ a variant of the Augmented Dickey-Fuller (ADF) test for a unit root in the process driving the real exchange rate. This is generally based on a general regression equation for the real exchange rate qt over time in a general form:
∆qt = γ0 + γ1t + γ2qt-1 + ∑â¿i=1∆qt-1 + et (6)
where t represents time trend, ∑â¿i=1 is included to soak up any serial correlation and e is white noise process. In case of γ2 = 1, the process generating the real exchange rate contains a unit root. Even in the long-run the level of the real exchange rate may not be predictable: because the change each period may be equal to a constant plus an unpredictable random element, but the long-run level is equal to the sum of the changes each period plus the sum of a big number of different elements. So, over time as these different elements get cumulated there is no way of telling in advance what they will add up to. Therefore, null hypothesis testing that γ2 = 1 (a unit root) is a test for whether the path of the real exchange rate over time does not return to any average level and thus that that long-run PPP does not hold.
Engle and Granger (1987) developed the cointegration approach used in the econometrics literature to test long-run PPP. The technique argues that any two non-stationary series which are found to be integrated of the same order are cointegrated if a linear combination of the two exists which is itself stationary. In this case, the nonstationarity of one series exactly offsets the nonstationarity of the other and long-run relationship is established between the two variables (in our case the exchange rate and prices). Thus the general null hypothesis of the test is that two exchange rates and prices are not cointegrated and if it possible not to reject this null hypothesis then there will be no relationship between the two variables, and therefore PPP does not hold. More about this test will be written further.
Ideally a test for a long-run PPP should include a proper modeling of the dynamics of economics variables and their equilibrium relationship, but at the same time permitting for significant deviations from equilibrium in the short-run. Cheung and Lai (1993) were among the researches who supported this idea and found a long-run relationship between domestic and foreign prices and nominal exchange rates. This implies that any other tests that meet this requirement should capture the cases where the deviation from equilibrium is prolonged and the equilibrium can be slowly achieved.
Ching and Lai (1993) proposed alternative approach to the traditional cointegration techniques: they argued that the fractional cointegration technique allows a wide range of mean-reversion characteristics than standard cointegration analysis. This benefit of flexibility in modeling mean-reverting dynamics seems to be significant in evaluation of long-run PPP. They proved that regarding to the validity of the long-run PPP completely different conclusions hypothesis could arise when the analysis is based n fractional cointegration. Both methods were used to examine the reality of the PPP theory between the US dollar and five other foreign countries (the USA was classified as the home or "domestic" country and Canada, France, Italy, Japan and UK were defined as foreign countries). The obtained result showed that the fractional cointegration techniques detected significant fractional cointegration in all countries except Italy, and conventional unit root test procedures rejected the null hypothesis of any cointegration relationship in all countries.
Froot and Rogoff (1995) gave a comprehensive survey of studies investigating the long-run determinants of purchasing power parity, but they discovered the limitations of the tests used in three successive stages in the time series literature on PPP. Some possible non-stationarities were overlooked. Researchers argued that deviations from long-run PPP have a half-life of about three years: they require no assumptions concerning erogeneity and they imply a sensible dynamic relationship among price levels and the exchange rate.
Significant research of quantitative techniques used in PPP test was done by Charles Engel - one of his article "Long-Run PPP May Not Hold After All" (1996) states that unit root tests are subject to such a size bias when they are applied to real exchange rates and this argument he derives from Balassa-Samuelson (Harrold-Balassa-Samuelson) framework [4] . In other words, he argued the extent to which departures from PPP are caused by the presence of non-traded goods versus deviations from the law of one price in traded goods. Because of size bias, the non-stationary component of the real exchange can't be detected in tests for the long-run PPP. In other words and more in detail, Engel divided the real exchange rate to the stationary and non-stationary components or processes (stationary process determined as relative price of traded goods, while non-stationary as a relative price of non-traded goods). This two-component character severely biases unit root tests in favor of rejecting non-stationarity of the real exchange rate and demonstrates that results supportive of PPP may be due merely to a misspecification of the data-generating process of real exchange rates. Furthermore, Engel states that coefficient related to the external traded goods can be related to the LOOP in the concept version of PPP where the real exchange rate need to be constant, but the coefficient which is related to the both traded and non-traded goods can be referred to the Balassa-Samuelson effect or any other test where the real exchange rate demonstrates a trend. It is important to say that according to the Engel, unit root tests of real exchange rates are biased in favor of rejecting nonstationarity.
Anyway, the empirical studies so far have provided us rather mixed conclusions on the long-run PPP hypothesis. The possible reason of this different point of views emerges in part from a problem associated with traditional testing the long-run PPP.
The results from cointegration studies emphasize some important characteristics of the data. It was noticed that it is more likely to find support for the PPP hypothesis if fixed exchange rate regimes prevail instead of flexible, even it is more likely to reject the null of co-integration. Moreover, Sarno and Taylor (2002) argued that it is easier to find evidence against PPP if we use Engle-Granger two-step procedures (triviate systems instead of biviriate ones).
This is no doubt that the fractional cointegration analysis gives us an extra dimension to study the long-run PPP hypothesis and even enable us to analyze the mean-reverting property of the exchange rates towards the long-run equilibrium.
Finally, it need to be mentioned, that apart from the technique of fractional cointegration, there are some other improved econometric methods that are used to study the PPP subject. For instance Sarno and Taylor (2002) tried to improve the test by using panel unit-root tests applied jointly to a number of real exchange rate over the recent float, while some other researches tried to find an application of non-linear techniques. In this work these techniques will not be used.
3. Econometric Methodology and empirical results
In order to test the validity of the PPP hypothesis, a several econometric techniques will be described in this study. Four tests will be implied in this section:
Formal analysis for PPP;
Unit root test;
Engle-Granger approach;
Johansen method;
Empirical and theoretical parts are combined here.
3.1. Formal analysis for PPP.
Formal analysis of PPP is also called as a first step of a formal analysis of PPP. In the empirical literature, the main result of this test was the statement that PPP doesn't hold in the short-run, analysis of the long-run term was beyond this test.
The test includes logged relation of CPI and logged relation of nominal exchange rates. So, after testing the PPP hypothesis between UK and Japan we obtain the following graph:
The graph above demonstrates obvious noise, so we can conclude that formal analysis for PPP proves that PPP doesn't hold.
3.2. Unit root test.
According to the first test, the PPP doesn't hold while the latter tests provided new approach: on the second step the real exchange rate was checked on stationary. The stationary of the real exchange rate proves that in the long-run "noise" can be dissapered.
The real exchange rate is then given by:
qt = st - pt + pt* (7)
where st denotes log-nominal UK/Jp exchange rate, pt denote log-domestic (UK) price index, and pt* is a log foreign (Japan) price index. Nevertheless, with a Augmented Dickey-Fuller (ADF) [5] test the null may not be rejected because of dynamic misspecification and measurement error.
We need to test the null that et Ì´ I(1) is equivalent to testing that PPP does not hold between UK and Japan. To employ this test it is necessary to use twelve lags, because this number corresponds to the number of months. Swartz Information Criterion is used to ensure that the existed/possible errors are white noise. The following table shows the results of the employed test:
T-statistic is |-2,26|, the null hypothesis of a unit root is not rejected because computed ADF test statistics is greater in absolute value than critical value (|-3,44|, |-2,87|, |-2,57|). So, according to this test PPP doesn't hold between UK and Japan.
3.3. Engle-Granger approach
Two approaches of testing PPP were described above. In 1987 Engle-Granger proposed new approach to test PPP: the third step which tests cointegration between nominal exchange rate and prices. This is a more general test since it amounts to test the null hypothesis that
st -μpt + μ*pt* is not stationary, where μ and μ* are constrained to be equal 1. Otherwise, there is at least one stationary linearly combination.
In the PPP empirical literature the following expression μ=μ*≠1 is called two-step approach (biviriate), while the μ ≠μ* is called three-step (triviate) approach. In triviate approach the existence of two stationary linearly combinations are possible, and it means that we can reject the null. Firstly, we need to test prices and exchange rate for stationary. In case when prices stationary but exchange rate not (or vice versa), then there is no cointegration between them. If all variables are described by processes I(1), then method of least squares is used to estimate the equation st = α + μ(pt - pt*) + μt. To employ the standard ADF test the residual μt capped needs to be checked on stationary. If the residual have white noise, then the hypothesis of non-stationarity can be rejected and we can reject the null hypothesis of no cointegration; a long-run relationship exists between exchange rate and relative prices. It is need to mention, that in this test we do not need to use a trend. In case when residual is stationary linear combination st -μpt + μ*pt* is stationary as well, while series cointegrated.
Firstly, we need to test the hypothesis that st, pt, pt* are all integrated in order one.
Testing unit root in level of series we receive the t-Statistic for st is |-1.31|.
So, |-1.31| < 3.15 [6] - and it can be interpret that there is unit root in the level of series.
Further in order to find whether this variable integrated in order one we compute two test for each variables: first for level, second for level one (first difference). Testing the same but with level 1 gives us that t-Statistic for st is -19.71. |19.71| ˃ Hamilton value. It means that st integrated in order one.
Testing unit root in level we receive that t-stat of pt and pt* is |5,43| and |7,15| relatively and both of them is higher than Hamilton value. So, it means that they are stationary.
So, as a result we have that not all of our variables (st, pt, pt*) are integrated in order one. On this step we can conclude that st, pt, pt* are not cointegrated, therefore we can not reject the null. So, according to this test PPP between UK and Japan doesn't hold.
3.4. Johansen method.
Although Engle-Granger approach is simple to employ, and one of main advantage of it is that it can estimate only one cointegration relationship between the exchange rates and prices. However, in the PPP study, we have three variables in the equation, so there could be potentially up to two linear independent cointegration relationships. Therefore it is clear that we require an alternative method which can be used to elicit more than one cointegration relationship. This leads us to Johansen cointegration method.
The Johansen test is computed the following way. First, like the residual based Engle-Granger two-step technique, we have to ensure that the exchange rate and domestic and foreign prices are of the same order of integration. Then it is necessary to indicate that all variables are integrated of order one. The next step is to apply the Johansen test to the three variables that we expected to be cointegrated. Since we have three variables of interest - exchange rates, domestic prices and foreign price level - the Johansen procedure involves the identification of rank of the 3 x 3 matrix П in the following specification:
∆Yt = α + ПYt-k + ∑ [k] -1i=1Г∆Yt-i + εt (8)
where Yt is a column vector of the three variables. The test is to detect whether П has zero rank. If П is of zero rank, then there is no stationary linear combination between the three variables, and there is no cointegration. On the other hand, if the rank is r, there will be r cointegrated vectors. There are two trace statistics for cointegration under Johansen method: the trace test and maximum Eigenvalue test. The previous test involves a joint test where the null is that the number of cointegrating vectors is less than or equal to r against an unspecified or general alternative that there are more than r. In the latter case, the null of exactly r cointegrating relationships is tested against an explicit alternative hypothesis, r=0, r=1, r=2 and further. Johansen argued that both tests should be carried out in order to confirm consistent conclusions concerning to the null hypothesis.
The general rule for both tests is that if the trace statistic is greater than the critical value, we reject the null hypothesis that there are r cointegrated vectors in favor of the alternative that there are r+1 (for the trace) and more than r (for the maximum Eigenvalue test).
In case when we have obtained the results of cointegration, the next step is to analyze the restrictions required for the PPP hypothesis to hold. Referring back to the equation (3), the restrictions imposed on the cointegrating vectors are (1; -1; 1). If the PPP vector is found to be cointegrated, then the nominal exchange rate will move one-by-one with the relative prices, and on the long-run PPP holds. In other words, the following conditions are held: the symmetry condition with β1 = β2 and the proportionality condition with β1 = β2 =1. In order to perform the test of these restrictions, we conduct the likelihood ratio (LR) test. So, if the computed LR statistic exceeds given critical values, the null hypothesis is rejected that PPP-vector is contained in the cointegration space and we conclude that PPP is violated. On the other hand, if the null is retained and thus the restriction requirements are met we conclude that PPP hypothesis hold.
So, using this test for our observed countries we can say the following: the result of the trace test tells us that at "none" trace statistics (30,52) is higher than 5% critical value (29,78) and at "at most 1" trace statistic is less (13,59) that 5% critical value (15,49) . Therefore for the first test we can reject the null hypothesis but for the second one it is not rejected. As a result it means we have one cointegration between the elements. The first line of the test is having no cointegration versus one cointegration and when we reject it it means we have rejected the null hypothesis which is having no cointegration, therefore it means we have one cointegration which is the second side of the hypothesis.
For the second line (at most one) the hypothesis is having one cointegration versus having more than one cointegration. As it is written in the results we have not rejected it and it means we are confirming again that we have just one cointegration. Therefore as the results there is one cointegration.
For the second test which is maximum Eigenvalue, for the fist line(none) we have not rejected the hypothesis therefore it means we have no cointegration with this test which is more effective than the trace one.
We can conclude about Johansen test that we have done, there is no cointegration between the elements and therefore PPP does not hold.
4. Conclusion
This paper investigates the long-run validity of PPP theory between UK and Japan from January 1968 to January 2010. Various advanced econometric techniques were applied to analyze the PPP theory: the results of the tests show us that the PPP vector does not exist between in the cointegration space and the nominal exchange rate and domestic and foreign prices do not move one by one as implied by the theoretical PPP. Empirical results confirm that the explanation (reasons) of PPP failure can be correct. Certainly, some mentioned above obvious reasons why practically PPP doesn't hold affected PPP between UK and Japan. Moreover, I think, for Japan, the most important reason why the movement of the three variables does not constitute a one by one relationship may be due to the deviations in productivity differentials. It is well known that Japan has been experiencing the fast growing rate over the past few decades. As a result, Balassa-Samuelson theory can have a place, which can entail a "noise" of PPP and fluctuation of the exchange rate.
So, empirical results clearly demonstrate a PPP failure between UK and Japan during observed period.
