A Cointegration Analysis of Purchasing Power Parity: 1973-96

International Advances
in Economic Research

VOLUME 5
NUMBER 3

August 1999

Pages 369-385

This paper tests the purchasing power parity (PPP) hypothesis for five industrial countries using cointegration and error-correction modeling. The cointegration test indicated that for all countries the PPP hypothesis holds in the long run but not in the short run. Further, the error-correction models suggested that deviations of the actual exchange rate from its long-run PPP value were corrected in subsequent periods. Finally, the high frequency monthly data models did a better job of tracking the turning points of the actual data than the low-frequency quarterly and yearly models. (JEL F3, F4)

The theory of purchasing power parity (PPP) is relatively simple and straightforward, and amounts to nothing more than applying the law of one price to a comparable market basket of goods and services across countries. In its simplest form, it states that in the absence of government intervention and significant freight charges and tariffs, an internationally traded basket of similar goods should sell for the same effective price when converted into the same currency. Although simple in theory, the real world is characterized by a number of complications such as differentiated products, tastes, and costly information. These have created considerable problems for economists testing the theory empirically in the post-Bretton Woods era.

In determining the validity of PPP, the results from several empirical studies have been mixed. Few, if any, studies have found evidence for the theory in the short run while the results on PPP in the long run have been more varied. For example, Hakkio [1984] finds evidence supporting PPP in the 1970s using cross-country tests as does Dockery and Georgellis [1994] for the Greek economy during the 1980-92 period and Geppert [1997], in the presence of transaction costs. However, Krugman [1978], Dornbusch [1980], and Frenkel [1981a, 1981b] found evidence against long-run PPP. In view of the consensus on short-run results and the lack of consensus on long-run results, recent empirical work has focused on the validity of the theory in the long run.

The mixed results emerging from these long-run studies have forced investigators to apply a number of different methods and statistical techniques to obtain more conclusive evidence. For example, the data used for PPP models have been incorporated in all possible forms, ranging from high-frequency monthly data to annual data (see Johnson [1990]). With regard to constructing PPP ratios, a number of price indices have been used for this purpose such as the consumer price index (CPI), the world price index, and the cost of living index. Most of the tests have used data mainly from the major industrial nations of the world, with the U.S. dollar and price index acting as the common denominator in regressions.⁽²⁾ However, other currencies and price indices have also been used as the common denominator. Frenkel [1981a], for example, uses the German mark and world price index. Furthermore, economists have also analyzed the variability in real exchange rates to test PPP (see Roll [1979] and Huizinga [1987]).

Economists have relied on econometric analysis to determine the validity of PPP theory and, until the mid-1980s, tests were carried out using the ordinary and generalized least squares methods [Isard, 1995, p. 65].⁽³⁾ However, little reliability can be placed on results carried out using these two methods because these tests fail to incorporate the fact that the data used usually exhibit a time trend and, therefore, are nonstationary. According to Granger and Newbold [1974] and Engle and Granger [1987], regressions involving nonstationary variables using the classical methods are spurious.⁽⁴⁾

In the wake of these studies, this paper conducts a cointegration analysis of PPP for five industrial nations to determine the validity of the theory in the long run. The second section briefly describes the data and the rationale for selecting it as well as determining whether it exhibits an underlying trend. The third section uses the ordinary least squares (OLS) method to test the model in (1). The fourth section tests the data for unit roots (nonstationarity), thus determining their order of integration. The fifth section runs cointegration tests of the time series variables to determine whether a stable long-run relationship exists between the exchange rate and the price indices, ignoring for the moment short-run dynamics. The sixth section uses these results to construct an error-correction model that reconciles the short- and long-run dynamics of PPP.

The data required to test PPP were obtained from the online Haver Analytics database. Two time series variables were used for this purpose: the spot exchange rate and the CPI.⁽⁵⁾ The countries chosen to test PPP were Germany, the United Kingdom, Japan, Canada, and France (the U.S. dollar was used as the common currency in all exchange rates). The data is in a high-frequency monthly format and spans a period of 23 years from January 1973 to December 1996 for most countries. The exceptions are Canada and Japan for which the data begins in January 1978.⁽⁶⁾ Data is chosen from major industrial countries because they have had floating exchange rates since the inception of the floating exchange rate mechanism in 1973.⁽⁷⁾ Also, relative to developing nations, they exhibit a greater degree of economic and financial integration along with fewer commercial and financial restrictions. Finally, the results in this study, which encompass a longer time span, can be compared with previous studies of PPP by using a more limited data set from these countries.

It is also important to analyze the graphs of the exchange rates and price indices for the countries in question to determine whether the variables exhibit any time trend. The existence of a time trend might hint of nonstationarity of data, which can create serious problems in OLS estimation of PPP.

Because of space constraints, the key findings are summarized.⁽⁸⁾ In almost all cases, there is clear time trend involved. For example, for the exchange rates of Japan, Germany, and the United Kingdom, there is a negative time trend, whereas, for Canada's exchange rate, there is a positive time trend. This provides indirect evidence that the exchange rates for these countries are nonstationary. However, more precise tests will be conducted to determine the stationarity of the variables. The exchange rate for France is the exception because it exhibits a minimal positive time trend.

Similarly, the CPIs for the countries in question seem to possess a common trend. A positive time trend is involved for all the countries' price indices, except for the price index for Germany which exhibits a negative time trend. Due to the existence of a time trend in the price indices, nonstationarity of the data is likely to be present.

There are two main reasons for using the classical method of estimation. First, it allows comparing the results from this study with those of prior studies. One of the differences between this and previous studies is the length of the time series variables used. The data in this case spans a period of 23 years and is considerably greater than the data set used in prior research, mainly from the 1970s until the early 1980s (see Dornbusch [1980], Frenkel [1981a], and Enders [1988]). Second, it is important to determine whether the regressions in question are spurious.

The model used for this purpose is the absolute PPP model in standard logarithmic form:

where s_t is the logarithm of the actual exchange rate (foreign currency to domestic currency), and p^* and p are the foreign and domestic CPIs, respectively. For the PPP relationship to hold requires that coefficient

The OLS method was used for running the regressions for five different cases, all of which involved the U.S. dollar and CPI as the domestic currency and price index, respectively. The results from the regressions using (1) are shown in Table 1.

Notice that none of the estimates fulfill the requirement that

In the case of the PPP ratio, the estimated coefficients in four cases are of the wrong sign. The closest that any of the coefficients come to unity (and with the correct sign) is the coefficient for the PPP ratio of France. Although standard errors of the coefficients are included in Table 1, they should not be used to determine the significance of the coefficients since they may be misleading [Taylor, 1988, p. 1375]. Judging from the values of the coefficients (and ignoring their significance), the results indicate that PPP failed to hold in the long run for the major industrial nations, using data from 1973 to 1996.⁽⁹⁾

Having established that PPP fails to hold using the OLS method of estimation, it must be determined whether these results are spurious. Results from the regressions will not be meaningful if the exchange rate or price indices are nonstationary, that is, if they possess a time trend. Although formal tests will be run in the next section to determine whether these variables are nonstationary, it would be interesting to analyze the statistics produced from the regressions and to determine whether they are spurious. According to Granger and Newbold [1974] and Phillips [1988], spurious regressions will tend to have inflated statistics such as high t-statistics, R² values, and F-statistics.

The regressions reported in Table 2 exhibit this pattern. None of the adjusted R² values are less than 97 percent, which is very high considering that the coefficients have the wrong signs and the PPP relationship does not hold. Furthermore, the F-statistics are very high as well, leading to an initial conclusion that some (or all) of the variables involved in the regressions are nonstationary.

The OLS regressions from the previous section hint toward nonstationarity of the variables, however, a more precise procedure is required to make a firm determination. It is important to check whether the time series variables, exchange rate, and price indices are nonstationary, that is, whether they have means, variances, and covariances that are time dependent.

A formal test of the null hypothesis of nonstationarity can be conducted via the ADF. The ADF tests were run using four lags and no constant or time trend. The McKinnon critical value at the 5 percent level is -1.9407 and -1.9411 for 288 and 228 observations, respectively. The null hypothesis of a unit root is rejected if the computed ADF test statistic is greater in absolute value than the critical value. The results of running the ADF test on the CPI and spot rates in level form suggested that the null hypothesis cannot be rejected at the 5 percent level and that the time series variables are all nonstationary in level form. This is true in the case of both the logarithmic and nonlogarithmic transformations. The variables were therefore differenced and the ADF tests run again. The number of lags used in this case is still four, and no constant is included either.

Table 2 shows that in all cases except one, the null hypothesis of a unit root in the first difference of the CPI variables can be rejected at the 5 percent significance level. The only exception is the CPI for France where the null hypothesis cannot be rejected at the 5 percent level in both logarithmic and nonlogarithmic form. However, if the significance level is relaxed to 10 percent, the critical value becomes 1.6162, leading to a rejection of the null. For the purposes of this study, it will be assumed that the CPI for France is stationary in first difference form as well. The ADF tests in Table 2 indicate that the spot exchange rates in their first difference are all stationary in both logarithmic and nonlogarithmic form at the 5 percent significance level.

The results in Table 2 are consonant with those of prior studies that have found the exchange rate and price indices to be nonstationary for most of the countries in question. For the same countries, Corbae and Ouliaris [1988] find all relevant variables to be nonstationary except for the United Kingdom and Japan for which the price level and exchange rate, respectively, were stationary. The ADF tests by Cheung and Lai [1993] also confirm the results from this study.

To perform a cointegration test, it is necessary that the order of integration of all the variables in the long-run relationship be the same [Enders, 1995, p. 219].⁽¹⁰⁾ The order of integration can be defined as the number of times a time series variable must be differenced for it to become stationary. From the ADF tests above, all the variables are integrated of order one, or I(1), because the first difference of these variables is stationary, or I(0).

The ADF tests rule out directly using OLS estimation for testing PPP in the long run. However, given that the order of integration for all the countries' variables is I(1), cointegration analysis is warranted.⁽¹¹⁾ The advantage of using this procedure is that it can determine the existence of a stable long-run (equilibrium) relationship among the nonstationary time series variables. It also ignores the short-run dynamics that might cause the relationship not to hold in the short run.

At this stage, it is important to clarify what the cointegration test actually determines since there is still some confusion among the economists using it to test PPP. Some economists have used their cointegration results to determine whether there is a long-run equilibrium relationship between the relevant variables, that is, the exchange rate and the price levels. However, cointegration existence should not be interpreted as an established ex ante equilibrium relationship because the data is not sufficiently rich (detailed) enough to enable inferring whether economic agents' plans were realized in the market. Cointegration only shows the existence of a long-run ex post stable relationship between the variables in question [Enders, 1995, p. 359]. Also, when econometricians use the term "equilibrium" in cointegration analysis, it should not be interpreted in its strictly economic sense.

There are two alternative techniques for running cointegration tests: the Engle-Granger [1987] two-step test and the maximum likelihood method developed by Johansen [1988] and Johansen and Juselius [1990]. The latter test is preferred when there are more than two time series variables involved because it can determine the number of cointegrating vectors. Furthermore, less error is involved in the Johansen technique because only one step is involved rather than the two steps required in the Engle-Granger technique.

The results suggest at least one cointegrating equation at the 5 percent significance level for all five cases, except in the case of France which has two cointegrating equations.⁽¹²⁾ It can therefore be concluded that there is a stable long-run relationship between the exchange rate and price indices. This means that although the three time series variables may diverge from each other in the short run, they will stay close to each other and not drift far apart in the long run.

The results in this study differ from those of prior studies because a long-run relationship has been found in four of the five cases. Previous studies have found a cointegrating relationship for one or two cases at most. For example, Enders [1988] finds weak evidence of cointegration of the Canada price, the U.S. price, and the U.S. dollar/Canadian dollar exchange rate since 1973. However, for the five cases in this study where a cointegrating relationship was found, Corbae and Ouliaris [1988] do not find any evidence of cointegration. On the other hand, Cheung and Lai [1993] do find evidence of cointegration supporting long-run PPP by using the Johansen likelihood ratio test.

According to the Granger representation theorem, the existence of a stable long-run relationship between the exchange rate and price levels enables the investigator to estimate at least one error-correction model (see Ramanathan [1995, p. 574]). Error-correction models are useful because they reconcile the short- and long-run behavior of the variables involved. Obviously, ignoring either the short- or long-run properties of a model results in a misspecified relationship. The long-run relationship is incorporated by including the (lagged) cointegrating vector into the model, and the short-run dynamics are incorporated by including the variables in their differenced form. The simplest form of an error-correction model in this case would be:

where

is the first difference of the exchange rate (in logarithmic form),

is the first difference of the ratio of the foreign and domestic price levels, and

is the error-correction term.

Theory predicts that the error-correction term must be negative and significantly different from zero. The coefficient

is an estimate of the speed of adjustment back to the long-run equilibrium relationship. A negative

implies that in the event of a one-unit deviation between the exchange rate and the long-run PPP rate, there would be an adjustment back to the long-run (stable) relationship in subsequent periods to eliminate this discrepancy. On the other hand, the coefficient of

must be significantly different than zero in a positive direction.

In terms of the short-run dynamics, only the first difference of the ratio of price levels is included in (2). However, other short-run determinants of the exchange rate can also be included in the error-correction model. It is well known that a number of economic variables can affect the exchange rate in the short run. Such short-run variables include interest rates, output levels, and money supply values for the two countries in question (see Frenkel [1979]). Hence, in order to render the error-correction model complete, these variables were also included in the estimation process. The monthly data for such variables were obtained from the Haver Analytics database. All variables were represented in first difference form because ADF tests indicated that they were nonstationary and integrated of order I(1). The modified error-correction model incorporating these variables can be represented as:

where (m^*/m), (r^*/r), and (y^*/y) are the ratios of the foreign and domestic money, interest rate, and output values, respectively. The residuals from the PPP test were used as instruments in estimating the error-correction models for the five countries in question. The results are displayed in Table 3.

Given that the cointegration tests found only one cointegrating equation in all cases except France, only one error-correction term was included in the estimated models. Regarding short-run variables in the model (excluding the price ratio), the included lags were not greater than t - 12. This was done to capture their effects on the change in the exchange rate within one year, that is, the short run. On the other hand, since prices are sticky in the short run, their effect on the change in the exchange rate might not occur within a period of one year, hence, no upper limit was placed on the lag length (k) in

Only the first significant positively signed term was included in the error-correction model for the price ratio variable.

The coefficients of the error-correction terms in Table 3 can be interpreted as follows. In the event of a one-unit deviation from long-run PPP, there is a correction of approximately 3 percent in the subsequent time period for all five cases. However, the difference among the five cases is that the correction occurs at different time periods. For example, in the case of the United Kingdom, the correction occurs in the next month, whereas, in the case of Japan, the correction occurs after 28 months, perhaps reflecting the (historically) less open trading regime of that country compared with other industrialized nations (see Lawrence [1993]). In the case of Germany and France, a one-unit deviation from long-run PPP is corrected after one year. The

term is present in all five cases, however, the lag lengths (k) vary from model to model. The

term has an almost instantaneous effect on the change in the mark/dollar exchange rate (after one month), whereas it has a much slower effect on the change in the Canadian dollar/U.S. dollar exchange rate (after 31 months).

Turning to the short-run determinants of the exchange rate, the lagged interest rate had a significant effect on the exchange rate in all five cases. In addition, the coefficient on the interest rate term was negatively signed which is consistent with the monetary model of exchange rate determination. Also present in all five error-correction models is the

term. Although this variable is statistically significant in all regressions, unlike the interest rate variable, the signs were not all of the same direction. Coefficients with a negative sign conform with the traditional flow model of exchange rate determination, whereas the positively signed coefficients conform with the monetary model. The money supply variables for France, Germany, and Japan have the anticipated (positive) signs and are statistically significant. In the Canada and United Kingdom models, however, this variable had the expected sign but was not reported in the table because it was statistically insignificant.

In general, the error-correction models reported here are supportive of PPP but, given that the latter is essentially a long-run proposition, it is important to test models using quarterly and annual data. The high-frequency monthly data were therefore converted to low-frequency quarterly and annual data. The estimates of the error-correction models using quarterly and annual data are listed in Tables 4 and 5, respectively.

The quarterly data estimates reported in Table 4 are an improvement over the results obtained from the models using monthly data. First, there is a significant increase in the absolute value of the error-correction terms. In the event of a one-unit deviation from long-run PPP, there is a 12 percent correction in subsequent periods for the cases of France, Japan, and the United Kingdom. For Germany and Canada, there is a 9 and 10 percent correction in subsequent periods, respectively. The lag of the error-correction term is between two (United Kingdom) and five quarters (Canada and Japan). For France and Germany, the error-correction term is lagged four quarters.

As far as the short-run variables are concerned, the change in the interest rate ratio is no longer significant in four of the five cases, with Germany being the exception. This would suggest that the interest rate effects are better represented using high-frequency data since their effect on exchange rates is relatively quicker. The change in the logarithm of the price ratios continues to have an effect on the exchange rates in all cases except Canada. Furthermore, the coefficients of these terms, like those of the error-correction terms, have a larger effect on the change in the exchange rate.

To summarize, the error-correction terms using quarterly data suggest that on average, a proportionately greater deviation in the long-run PPP is corrected in subsequent time periods. Second, variables that are designed to capture short-term effects become economically and statistically insignificant. Not surprisingly, the models using annual data are even more supportive of PPP. First, there is an almost twofold increase in the R² values, with the lowest value at 0.43 for the United Kingdom and the largest at 0.74 for Canada. More importantly, there is a relatively significant increase in the absolute value of the coefficients of the error-correction terms. This means that current deviations of the actual exchange rate from its long-term PPP value are corrected by a larger amount in subsequent periods. On average for all five cases, a one-unit deviation from long-run PPP results in a correction of almost 40 percent in subsequent time periods. France exhibits the largest correction in PPP disequilibrium (65 percent) while, perhaps not surprisingly, Japan exhibits the slowest speed of adjustment back to long-run PPP (37 percent).

Table 5 also reveals that because of statistical insignificance, the interest rate and money variables have dropped out from all the models except Japan. This provides further evidence that the effects of the short-run variables are better captured using high-frequency data. The change in the output ratios is present in three of the five cases. Like the error-correction coefficients, the coefficients for all the other variables are larger than their respective counterparts in the models using quarterly data. The results from Table 5 suggest that the error-correction models using annual data are relatively better in explaining the long-term variation in the exchange rate than those using quarterly or monthly data.

To judge the forecasting ability of the various models, the models were used to track the historical exchange rate series from 1973 to 1996 (or from 1975 and 1978 in some cases). Figures 1 and 2 provide graphical evidence of the models' capacity to track the actual exchange rate in the case of France. In Figure 1 (annual data), the forecasted series (LOGFRAUF) tracks the actual exchange rate series (LOGFRAUS) loosely. There are instances when the actual exchange rate is rising, whereas the forecasted exchange rate is falling. Figure 2 (quarterly data) shows that the forecasted rate tracks the actual exchange rate, although the latter series constantly lags the former series.

This study tested the theory of PPP for five industrial countries using cointegration and error-correction modeling. The results are encouraging and provide empirical evidence for PPP holding in the long run. The diagnostic tests showed that all relevant variables were integrated of order I(1). The nonstationarity of the relevant variables in level form ruled out directly using OLS estimation techniques to test the theory of PPP. The Johansen maximum likelihood test was used to determine whether a long-run relationship is present between the exchange rate and the PPP ratio. The cointegration test indicated that the null hypothesis of no cointegration among the relevant variables could be rejected. This is one of the most important findings of this paper because it provides empirical support for the PPP hypothesis in the long run.

Error-correction modeling was employed to reconcile the short- and long-run dynamics of PPP. Once again, the results from the models were supportive of the theory of PPP. That is, they indicated that the actual exchange rate reverted back to its long-run PPP value in subsequent periods following one-unit deviation during the current period. The first set of models were generated using monthly data. However, since PPP is a long-run theory and prices tend to be sticky in the short run, the models were also estimated using quarterly and annual data. This is an area of error-correction modeling regarding PPP in which prior studies have not focused in much detail. The models generated with low-frequency data were relatively better than those with high-frequency monthly data. In addition, the high-frequency models were much better forecasters of the (historical) movement in the exchange rate than the low-frequency error-correction models. This is not an altogether surprising finding because the high-frequency models included a number of variables, such as the interest rate and the money supply, that explain short-term movements in the exchange rate.

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2. Data from industrial countries have been used for testing purposes because these countries have had floating exchange rates since 1973, therefore, providing a larger data sample. Additionally, the exchange rates of these countries have been allowed to float, though with a significant amount of interference by the monetary authorities.
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3. It should be noted that analysis of PPP studies shows that econometric analysis has been used only for verifying the long-run equilibrium relationship. To determine the existence of a short-run relationship, economists have relied on more informal methods such as simple statistical tests and graphical analysis (see, for example, Hakkio [1992]).
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4. Engle and Granger [1987] indicate that failure to incorporate the long-term properties of the model--the lagged residual of the cointegrating equation--results in a misspecified relationship.
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5. It is well known that there are several problems with using price indices to explain exchange rate changes. Including the prices of nontraded goods (such as housing) in constructing these indices represents one major problem. Changes in the prices of these goods do not translate into changes in international trade flows, therefore, they do not affect exchange rates. However, using these indices can be rationalized by arguing that changes in the prices of nontraded goods affect the price of traded goods indirectly through their impact on wage demands and the cost of living.
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6. A complete listing of all data is available from the authors upon request.
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7. Although these countries have had floating exchange rates, by no means have they been pure or freely floating exchange rates but, rather, managed floating rates.
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8. The relevant figures are available from the authors upon request
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9. It is interesting to compare the estimated coefficients of the PPP ratio before and after the Corbae-Ouliaris [1988] technique is used in removing serial correlation. The estimated

coefficients from the regressions without correcting for serial correlation are very close to unity in four of the five cases and would indicate that PPP relationship held (ignoring serial correlation).
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10. The bivariate case requires that the variables be of the same order of integration. However, for the multivariate case, this requirement need not hold, that is, cointegration tests can be run with variables of different orders of integration.
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11. Generally, cointegration attempts to find a linear combination of the nonstationary variables that is of an order less than the highest ordered variable [Enders, 1995, p. 361]. In this case, since the highest order of the variables is I(1), an order less than this would be I(0) which implies stationarity.
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