An empirical evaluation of the Purchasing Power Parity - Dissertation Example

An empirical evaluation of the Purchasing Power Parity Abstract This paper analyses the empirical validity of the purchasing power parity theory, i.e., the notion that the exchange rate between two countries is determined by the relative price levels in these two countries, using a time series error correction approach. Using monthly data on the exchange rate of the Japanese Yen and US Dollar and the respective Consumer Price Index values to represent the price levels over a period of 1960-2010 time series econometrical analysis is employed to investigate evidence of the theory. Although no support for absolute or relative purchasing power parity in their stronger versions are found, evidence does lend support to the weakest form of purchasing power parity. 1. Theoretical Foundations and Formulation of the econometric model The central notion of the purchasing power parity hypothesis is that the exchange rate between the currencies of two economies is determined by the relative price levels of the two countries. Alternatively one could also perceive this theory as saying that the changes in the exchange rate are driven by changes in the relative price levels (Froot and Rogoff, 1995). If we define  as the exchange rate between currencies of country A and country B, and  as the price level of country j, {j=A,B}, then purchasing power parity (PPP hereon) maintains the following: It should be noted however that this is the “strong” or absolute version of PPP. Relative PPP holds if the changes in the exchange rate are found to be determined by changes in the relative price levels: Finally, weak PPP is said to hold if changes in the exchange rate are found to be significantly influenced by changes in the relative price levels, i.e., if the following holds: Where satisfies requirements of stationarity. In order to examine which of these holds, it is convenient to transform the model into logarithmic form as follows. The log variant of equation (1) is: Where,  and  The objective of the present paper is to evaluate whether the PPP holds for US and Japan. We are going to estimate the following regression specification: where is the natural log of the Japanese Yen to US Dollar exchange rate at time t, is the natural log of the price level index of Japan and  is the natural log of the price level index of USA and  is an additive iid noise term. 3. Testable Hypothesis The hypotheses that we can test using this specification are as follows: i) Strong or absolute PPP:  ii) Relatively weaker form of PPP:  ⟹ iii) Weakest form of PPP:  ⟹ Note that for ii) and iii) to be valid specifications, the additive error term will have to be stationary. Otherwise estimates will be spurious. 4. Data In pursuit of investigating the empirical validity of the PPP theory, this study uses the monthly Japanese Yen to US Dollar exchange rate and the seasonally adjusted Japanese and US monthly Consumer Price Index series as the representative of the price levels as available from OECD main economic indicators 2010. Our data set covers a sample period ranging from 1st January1960 to 1st June 2010. The indexing of CPI for both the Japanese and US series is in accordance with assuming the price level of 2005=100. 5. Estimation This section presents the results of the estimations specified in section 2. We start of by presenting the statistical preliminaries and time series plots of the variables to engender a preconception of what can be expected from the estimated equations. a. Statistical preliminaries Table 1 presents the summary statistics for the variables of interest. Note that these are expressed in terms of natural logarithms of the levels. Table 1: Summary Statistics of the variables of interest Figure 1 below depicts the inter-temporal dynamics of the natural logarithm of the series of Japanese Yen to US Dollar Exchange rates. Observe that the series provides a clear visual evidence of a downward trend implying that over the period of 1970 to 2011, there has been a gradual decline in the exchange rate. Additionally, the series also appears to be non-stationary. Of course stationarity properties can only be conclusively verified after formal conduction of unit root tests. Figure 1: Inter-temporal dynamics of the (natural logarithm of) Yen-Dollar Exchange Rates Figure 3 presents the time plots of Japanese and US consumer price indices in logs. The US series exhibits a steady upward trend. However, the Japanese series seems to have become stationary since the middle of the 1990s after a gradually slowing climb before that. Around the middle of the first decade of the current millennium the cost of purchasing a given consumption basket (the CPI) in the US has overshot the cost of purchasing the very same basket in Japan reflecting a relative gain of purchasing power of the Japanese economy compared to the US economy. Figure 2: Time series of US and Japan consumer price indices b. OLS estimation The first model we estimate is an OLS specification. Recall that the specification we are testing is: The results of this regression are presented in table 2 below. The first and foremost point to is that the constant is positive and highly significant (t-value = 50.18, p-value=0.00). Therefore, the 1st condition required for the hypothesis, i.e., the value of the constant=0, of strong PPP to hold is not satisfied. Further, the estimated coefficients on the Japan and US CPI’s are also highly significant. However, apparently they lie quite far apart from one another although a formal test is required to ensure that they are indeed not identical and equal to unity in absolute value. A t-test is employed to test the null hypothesis that the coefficients are equal in absolute value to unity. The null is rejected in favour of the alternative of inequality at all standard levels of significance. Therefore, neither strong nor relatively weaker PPP are supported by the data. However, there is strong support for the weakest form of PPP. The coefficients have the expected signs. The coefficient on CPI of Japan is positive and approximately equal to 0.26 while that on the CPI of USA is negative, and approximately equal to -0.99. Table 2: OLS regression of equation (5) Apart from the fact that the t-statistics for the respective independent variables and F-statistic imply that the parameters are highly significant, the R-squared and adjusted R-squared values are also quite high. All this seems to imply that PPP, in its relatively weakest form does seem to hold in the context of the US and Japanese economies. Since there was a clear evidence of a downward trend in the time path of the Japanese Yen-US Dollar exchange rate, it may be instructive to account for a trend in our OLS specification. Table 3 presents the results of a specification identical to equation (5) in all but the addition of a trend. Table 3: OLS specification with trend Observe from table 3 that the inclusion of the trend leads to the Japanese CPI becoming insignificant in the determination of the exchange rate. The constant, trend and the US CPI are all significant determinants. Additionally note that the US CPI still retains the expected sign on its coefficient. Further, the R-squared and adjusted R-Squared values reflect a small improvement of fit. However, since the only requirement of the weakest form of PPP to hold is that neither of the coefficients equal unity in absolute value the estimated coefficients in the OLS specification with trend is not a violation of the weakest form of PPP. The implication is that only the US CPI drives the exchange rate. In fact the estimated model implies that a unit increase in the US CPI leads to a decline in the exchange rate by approximately 47%. However, a crucial point to recall here is that these models are valid estimates and inferences based on these can be made provided these regression specifications do not lead to spurious relations. To ascertain this, it is imperative to explore the stationarity properties of the variables of interest. c. Stationarity Analysis There is evidence in figure 1 that suggests that the series of logs of the Yen-Dollar exchange rate may not be stationary. To verify this hunch we conduct Augmented Dickey Fuller (ADF) and Philips-Perron (P-P) tests for unit roots on the series. The results of running the diagnostics are presented in tables 4 and 5. Table 4: ADF test for unit roots in the exchange rate series Observe in the table above the computed absolute value of the test statistic falls short of all of the critical values in absolute terms. Additionally the reported p-value is 21%. Thus, we fail to reject the null hypothesis of the presence of a unit root in the series. Table 5: P - P test for unit root in Exchange rate Table 5 confirms the non-stationarity of the exchange rate series as well. Thus, we fail to reject the possibility of a unit root in the series. Tables 6-9 present the results of ADF and P-P unit root tests on the series of US and Japanese CPI respectively. Table 6 shows that the ADF test fails to reject the null hypothesis of a unit root in the US CPI series and Table 7 confirms this using the P-P test. Table 6: ADF test for unit root in US CPI Table 7: P-P test for unit roots in US CPI Recall that that the US CPI series may not be stationary was predicted by examining its time plot (figure 2). However, additionally it was also inferred that the Japanese counterpart to the US CPI series may be stationary. In tables 8 and 9 the stationarity properties of Japan’s CPI series are explored. Table 8: ADF test for unit root in Japan's CPI series As table 8 shows, the null hypothesis of a unit root in the Japanese CPI series is convincingly rejected. Not only is the absolute value of the statistic larger than the absolute values of all the critical values of standard significance levels, MacKinnon’s approximation of the p-value equals 0.00. Table 9: P-P test for unit root in Japanese CPI series The rejection is further confirmed by the results of the P-P test portrayed above. Thus, we conclude that although the exchange rate and the US CPI series are non-stationary, the Japanese CPI series is stationary. Finally, to evaluate whether the estimates in the previous section were results of running spurious regressions, an approach that can be utilized is of looking at the fitted residual. If there is no evidence of the fitted residual being non-stationary, then we can rely on the estimates that these portray the possible long run dynamics of the model in the least. With this in mind, we first look at the time plot of the fitted residuals from our OLS specification. This is given in figure 3. Note that although there are apparent trends of persistence, over all the series seems to be stationary around a zero mean. Figure 3: Time series nature of fitted errors d. Correcting for Unit roots: estimating an Error Corrected Model (ECM) While the presence of unit roots can prove to be confounding for precision of estimation and even the validity of them if not specifically tended to, it can also reveal substantial nuances of the underlying data generating process that would remain unknown otherwise in certain cases. An instance of such a convenience is if we have co-integrated variables. In that case, by using error correction methodology it is possible to demarcate the short run and long run dynamics of the model. That is precisely the objective of this subsection. We shall use a closely related variant of the Engle and Granger (1988) two step error correction methodology. However, prior to that it is crucial to identify the order of integration of the non-stationary variables. The next step is checking for evidence of co-movements of the series. We start off by looking at the stationarity properties of 1st differences of the non-stationary series, namely the exchange rate series and the US CPI series. Figure 4 represents the cross-temporal variations of the 1st differenced exchange rate series. Figure 4: Time series of exchange rate in 1st differences There is a clear indication that the series is stationary around a zero mean. However, to formally verify this, we employ the ADF and P-P unit root tests (tables 10 and 11). Table 10: ADF test for unit root in 1st differences of exchange rate Table 11: P-P test for unit root in 1st differences of exchange rate series Observe that both tests reject the null hypothesis of a unit root in the series convincingly at all levels of significance. Thus, the 1st differenced series of exchange rates is integrated of order 1 or in the standard time series literature shorthand is {I(1)}. Figure 5 shows the temporal dynamics of the 1st differences of the US CPI. Note that 1st differences of the natural log of the CPI index, which is what is being used here represents the percentage growth of the price levels which is also known as inflation. Not only do we find that the series seems stationary, we also note that the dynamics are quite similar to that exhibited by the 1st differenced exchange rate series thereby indicating a strong possibility of co-integration. Figure 5: US CPI in first differences over time Formally, the fact that US inflation is indeed a stationary series is supported in tables 12 and 13 where existence of a unit root in the series is rejected strongly. Table 12: ADF test for unit root in US inflation Table 13: P-P test for unit roots in US inflation Thus, we find that the US inflation series is also integrated of order 1 or {I(1)}. Given the identical orders of integration and the similarities in the dynamics as evinced by the respective time plots there seems to be a high chance that the two series will be co-integrated. However, it is crucial that we formally test for such co-integration before we start utilizing the relationship to our convenience. The Engle and Granger (1988) 2 step error correction methodology essentially incorporates the co-integration test in its first stage. The first stage is simply running the OLS regression in levels of the non-stationary variables. The estimated relationship is valid and reflects the long run dynamics of the model as long as the series of estimated errors is stationary. If the fitted errors are found to be stationary, the non-stationary variables in levels are said to be co-integrated. The 2nd stage is to estimate the short run dynamics by regressing the model in the order of differences that made the series stationary on same order differenced series of the explanatory variables and the fitted residual with a lag order equal to the order of integration. Recall that we plotted the fitted residuals from the OLS in figure 3 which suggested that the series was indeed stationary. Table 14 below presents formal evidence of this in the form of an ADF test for unit roots in the series of residuals. However, it should be noted that the Dickey-Fuller critical values now are not valid as shown by MacKinnon. We need to use MacKinnon’s critical values derived via simulation. Table 14: ADF test for unit roots in residual from OLS regression The critical values suitable for the cointegration test given the characteristics of our specification at the 1, 5 and 10 percent levels of significance are: 3.90, 3.236 and 3.05. As is shown in the table the computed value of the ADF test statistic = -3.236 which implies it is just significant at the 5% level. Thus, the null hypothesis of a unit root in the series of fitted residuals can be rejected. Note that this also implies that the estimates obtained in tables2 and 3 reflect the long run dynamics of the model. Now we estimate 3 different specifications of error corrected models that exhibit the short run dynamics of the model. Table 1 presents the results for the 1st specification. It is a regression in differences with the lagged residual term included to correct for errors and utilize the co-integration of the exchange rate and the US inflation. Only the constant turns out to be significant at the 5% level. CPI Inflation of US and the lagged residual are significant at the 10% level. Note that inflation has a positive impact on the growth of the exchange rate. Table 15: ECM specification 1 Table 16 presents the results for the 2nd specification which adds in the levels of Japanese CPI as an explanatory variable since it was found to be stationary. Although the variable itself turns out to be insignificant, it leads to an increase in the significance of the US inflation and the lagged residual. The constant is no longer significant, but the significant variables retain the same signs as ECM specification 1. Table 16: ECM specification 2 Finally, table 17 presents the results of the final specification. The modification over specifications 1 and 2 is the incorporation of a trend. Table 17: ECM specification 3 As is evident from the table, the significance of the US inflation rises even more. The 1st row shows that a percentage rise in the variability of inflation causes a 70% increase in the rate of growth of the exchange rate. Additionally, the lagged residual, which is significant at the 10% level has an impact of dampening the rate of growth of the exchange rate by approximately 1%. Note that this explains the persistence in the decay of the exchange rate that we note from figure 1. As in the case of the previous specifications, neither the constant nor the consumer price inflation in Japan seems to affect the rate of change in the exchange rate. 6. Summary of findings and discussion Thus, what emerges from the results discussed above is that in the context of the US and economies Japan we find support for only the weakest form (iii) of PPP. Over the long run, the Japanese Yen to US Dollar exchange rate is significantly influenced by only the US price level. The constant is positive and significant and additionally we also find evidence of the existence of a significant but miniscule downward long run trend. Thus, our results refute the existence of either absolute or relative PPP. It may be worthwhile to consider the possible reasons. PPP being satisfied essentially implies that the same sets of goods were present and that too with identical weights in the representative consumption baskets of the two economies and that the law of one price held steady for all products in all markets. Recall that PPP in its strongest form in order to be satisfied requires (Feenstra and Kendall, 1997) i) that foreign goods have perfect substitutes in the domestic trade-able goods and ii) goods markets converge to perfection, i.e., they are characterized by negligibly small transaction costs and near perfect, symmetric information. Clearly, ii) does not hold realistically unless the economies are almost identical which is certainly not true in the case of US and Japan which are economically, culturally and politically entirely different. Further, because of the cultural differences, tastes and preferences are likely to differ which terminates the possibility of i) being satisfied. However, this raises the question of whether culturally and economically convergent economies observe convergence to PPP. This may be an interesting question for future research pursuits. We found that neither does relative PPP hold. Looking at the differences of the US and Japan economies the possible reason(s) may be any or all of existence of tariffs and transport costs, permanent movements of the terms of trade, disproportional movements of relative prices of non-tradables to tradables (Engel, 1999). All these provide ample opportunities for further research. Finally, there may be significant imperfections in our estimates arising out of a small sample bias. It can also be argued that the data does not encompass a broad enough time span to verify the validity of either absolute ore relative PPP in the long run. To conclude, in the context of the Japanese and US economies, we fail to find any evidence indicating PPP holds in either absolute or relative forms. We do find evidence that inflation in the price level of USA does substantially affect the exchange rate which can be loosely interpreted as satisfaction of the weakest form of PPP. However, paradoxically, the exchange rate seems to be unperturbed by variations in the Japanese consumer price index either in levels or in differences (inflation). References: Dittmann, I., (2002) “Residual Based Tests for Fractional Cointegration: A Monte Carlo Study”, Journal of Time Series Analysis, Volume 21 Issue 6, Pages 615 – 647. Dornbusch, Rudiger(1987) “Exchange Rates and Prices.” American Economic Review, March 1987, 77(1), pp. 93-106 Dickey, D.A. and W.A. Fuller. “Distribution of the Estimators for Autoregressive Time Series with a Unit Root,” Journal of the American Statistical Association, 74, p. 427–431,1979. Engel, C.,(1999) “Accounting for U.S. Real Exchange Rate Changes.” Journal of Political Economy, 107(3), pp. 507-38 Engel, R. and Granger, (1987) “Cointegration and error correction: Representation, estimation and testing”, Econometrica, 55: 251-276. Feenstra, Robert C. and Kendall, Jon D., (1997) “Pass-through of Exchange Rates and Purchasing Power Parity” Journal of International Economics, August 1997, 43(1/2), pp. 237-61. Froot, K. A. and Rogoff, K., (1995) “Perspectives on PPP and Long-Run Real Exchange Rates,” in Gene Grossman and Kenneth Rogoff, eds., Handbook of International Economics, Volume 3. Amsterdam: North Holland Press, 1995. Harris, R.I.D., (1995) “Using Cointegration analysis in Econometric Modeling”, FT Prentice Hall Publication MacKinnon, J.,(1991) “Critical Values for Cointegrstion Tests”, in R. Engel and C. Granger “Long-run Economic Relationships,” Oxford University Press. OECD (2010), "Main Economic Indicators - complete database" Main Economic Indicators (database),http://dx.doi.org/10.1787/data-00052-en (Accessed on 08/01/2011) Rogoff, K., (1996) “The Purchasing Power Parity Puzzle.” Journal of Economic Literature, 34(2) pp. 647-68 Read More