Evaluating The Validity Of The PPP Hypothesis A Time Series Analysis Of The US-UK Exchange Rate - Dissertation Example

?Evaluating the validity of the PPP hypothesis – A time series analysis of the US-UK exchange rate This paper investigates the validity of the purchasing power parity hypothesis in predicting the US-UK exchange rate over the period of 1975-2012. We utilize a time series approach to evaluate stationarity properties of the real exchange rate. Augmented Dickey Fuller unit root test methodology is applied. Additionally, an OLS estimation of a log-linear specification of the nominal exchange rate with the price level differential between the two countries as explanatory variable is also carried out. Although significant coefficients in this regression seems to indicate that variations in the price level differential lead to changes in the exchange rate, deeper inspection of the stationarity properties of the relevant series establishes that we actually fail to find any evidence to support that PPP holds for the two countries under question. However since the time period covered is only of a short duration of 37 years, we conclude that this evidence should not be taken to be conclusive. It could still be the case that PPP holds in the long run but what has been examined in this paper covers only the short run and during this period the exchange rate is at a perturbed state. Introduction The exchange rate is one of the most important macro variables that have significant implications for policy of any open economy. It is therefore of primary importance to identify what determines the long run real exchange rate between two currencies for either of the countries involved. Additionally, given the state of other macro variables what should be expected of the medium and long term dynamics of the exchange rate for any given economy? That is, should it be expected to appreciate or depreciate over time? How does the nominal exchange rate affect inflation? These are all critical questions can be answered using the Purchasing Power Parity (PPP) theory. It is therefore critical to evaluate its empirical validity. The vital notion of the PPP hypothesis is that the real rate of exchange between the currencies of any two countries is determined essentially by the ratio of the price levels of the countries in question. Intuitively this follows the very simple concept that if the price of the same good in two countries is denominated into the same currency, the prices should be the same. This is essentially the implication of the law of one price which postulates that the same good should sell at the same price in all markets because if different prices are charged then arbitrage will arise until the prices are equalized. Alternatively, the theory suggests that changes in real exchange rates are essentially driven by relative price level changes (Froot and Rogoff, 1995). Now, there are absolute, relative and weak versions of the hypothesis and these are distinguished as follows. When the exchange rate is simply equal to the relative price level ratio absolute or strong PPP is said to prevail. If the variability of the exchange rate is caused by variations in the relative price levels, then we say that relative PPP holds. And finally, weak PPP is known to hold whenever changes in the relative price levels significantly affect the exchange rate. The reason that this theory has motivated a large number of studies and keeps on motivating new pursuits of empirically evaluating the PPP theory lies in the strong potential of the theory to have strong bearing on various policy aspects. For instance, an economy which has newly become independent can utilize this theory to ascertain its exchange rate. Forecasting macro-dynamics is critical for effective policy and this theory can be utilized to forecast the medium and long term exchange rates if it is found to be a valid determinant of the exchange rate. With this as the basic premise, in the present paper, we shall evaluate the validity of the PPP hypothesis as in its capacity of predicting real exchange rates. In particular, we want to evaluate whether the PPP hypothesis is a good theory of the exchange rate or not. We shall look at UK and the US as our two countries for the purpose. We shall look at the dynamics of the nominal and real exchange rates between the Pound Sterling and the US Dollar and try and evaluate whether the implications of the PPP hypothesis seem to hold indeed for these two countries. The analysis will be carried out using time series data. Our sample ranges from 1975 to 2012. Therefore, the paper will examine how the exchange rate or price of one currency in terms of the other for the UK and the US have moved about and whether such movements can be predicted by the tenets of the PPP hypothesis. The paper is structured in the following manner: the next section reviews the relevant literature that has developed. We trace the origins of PPP theory and identify where it stands empirically in present times in this section. The conceptual evolution of the hypothesis from belief of its continuous maintenance to its current belief of holding only in the long run equilibrium is presented to set the context and present the research in the paper. Also, the evolution of the empirical approaches to testing the theory is presented in this section as a premise of the empirical strategy that we adopt in the paper. Naturally, the immediately following section is the section on empirical strategy adopted in the paper to evaluate the veracity of the PPP hypothesis for the US-UK pair. In this section the basic empirical strategy is laid out. We start off the section by formalizing the notion of the law of one price and defining the notions of absolute and relative or weak PPP theory. Then, we show what the implication of PPP holding are and how these can be tested using stationarity analysis or empirically testing specifications using OLS. We develop two alternative procedures of testing the PPP hypothesis in this section. In the next section we present the results of running the empirical tests. The implications of these results are discussed in detail in the section as well. Finally, in the section following the discussion of the results, we conclude by summarizing the main findings of the paper. Literature Review After the 2nd World War, the Bretton Woods agreement was signed, whereby, the U.S. dollar was tied to the price of gold, and currencies of all other nations were “pegged,” (tied) to the U.S. dollar. But, in 1971, President Nixon devalued the US dollar relative to the value of gold and at the same time terminated the convertibility of the currency into gold. Various attempts to restore a modified Bretton Woods agreement failed, and the majority of the currencies of the world started “floating” against one another from the month of March in 1973. The monetarist approach was the dominating approach to exchange rate determination theories at that time. This school of thought assumed that the purchasing power parity or PPP exchange rate persisted continuously (Frenkel, 1976; Frankel and Rose, 1995; Taylor, 1995). The central argument of advocates of this school of thought was that the exchange rate by definition was the relative price of the money of one nation in terms of the money of another nation. Therefore, this relative price, like relative prices of other goods should be decided by the demand and supply balances in the respective market for each money in the fashion of equilibrium in the market for assets. The precise mechanism of how a small change in the supply of one money relative to the other, with other things remaining unchanged, transmitted into an equivalent change in the exchange rate was not apparent. The basic explanation was based on the argument of arbitrage: alternations in the relative supply of moneys alter the relative prices which include relative prices of goods that are traded and this in turn leads to goods arbitraging across economies. This arbitraging continues until equilibrium is restored via changes in the exchange rate. In the late 1970s there was a proliferation of empirical examinations of this theory of continuous holding of the PPP exchange rate and other aspects that were implied by the monetarist exchange rate theory. Initial studies found encouraging results. Evidence in research carried out by authors like Frenkel and Johnson (1978), seemed to support the continuous holding of the PPP exchange rate. These studies however were carried out using data that did not cover a long enough run of the data to be conclusive. Additionally, after the beginning of the floating exchange rate regime, after an immediate period of volatility, the exchange rate stabilized for a period of a couple of years. However, by the late 1970s, the inter-temporal length of the data improved and the volatility of the exchange rate was also substantial. Subsequent research found convincing evidence to reject both the continuous sustenance of the PPP exchange rate as well as the monetary theory of the exchange rate. As Frenkel (1981) points out, no econometric analysis was necessary to perceive the invalidity of the PPP theory. Simply observing the dynamics of the real exchange rate was enough. The exchange rate in real terms is simply the product of the nominal exchange rate, i.e., price of foreign currency in terms of the domestic currency and the ratio of the domestic to foreign price levels. This expression then will reflect the purchasing capacity of a single unit of a foreign currency in its local market relative to the purchasing capacity of the domestic currency in the domestic market. PPP essentially thus would imply a real exchange rate of unity. It is critical to note that the exchange rate that is reported in financial journals or economic news sections of daily newspapers is the nominal exchange rate. And this nominal exchange rate under the PPP can differ from unity. But a central difficulty in identifying the exact duration when PPP holds true is that we work with aggregate CPIs which take base periods arbitrarily so that when the measured real exchange rate normalized to equal unity may be unclear. There of course will be some value of the real exchange rate as measured that corresponds to PPP holding. More importantly, all variations observed in the measured real exchange rate must reflect underlying deviations from the PPP level. The specific explanation is best summed up in Taylor and Taylor (2004): “The real trade-weighted value of the U.S. dollar didn’t change too much from 1973 to 1976, thus lending a degree of plausibility to the continuous PPP argument. But the real value of the dollar dropped sharply starting in 1977, and from then on it became increasingly clear that continuous PPP could not hold as nominal exchange rates were patently far more volatile than relative national price levels.” Thus, the initial encouragement dampened and gradually it dawned upon theorists that continuous PPP could not hold. The idea was that there was a long run equilibrium convergence to the PPP exchange rate, but in the short run there were deviations. So, theoretical pursuits now moved in the direction of explaining the short run deviations. One of the most interesting explanations was developed by Dornbusch (1976). His theory was that due to sticky prices short run deviations can occur which can often lead to the exchange rate overshooting the long run equilibrium level. But over the long run the exchange rate gradually converges to the PPP which is the long run equilibrium. The empirical search for support to the theory of long run convergence to PPP in this period did not prove fruitful. As mentioned earlier, most empirical studies have looked at the real exchange rate. One particular approach tested for proof of mean reversion in the real exchange root series. Long run convergence to PPP can be thought of as a tendency of mean reversion of the real exchange rate series, with the mean being the real exchange rate consistent with the PPP level. Thus, mean reversion by itself can be thought of as a necessary condition for long run PPP. If the mean towards which the series reverts is found to be the PPP exchange rate indeed, this can be taken as evidence of long run absolute PPP. A number of empirical studies have tested the hypothesis that such mean reversion does not hold. The popular null hypothesis has been that of a random walk against the alternative of a mean reverting series (Roll, 1979; Adler and Lehman, 1983). The choice of the random walk series as the anti-thesis of the mean reversion was justified by arguments positing that random walk features in the real exchange rate series would result from efficient inter-economy markets since exchange rates and prices would reflect all information available which in turn would lead to all arbitrage opportunities being exploited quickly. Roll’s (1979) argument for instance was that changes in the real exchange rate which effectively measures the one-period return (real) from inter-economy arbitrage should be expected to have a value of zero because of the efficiency of markets. A number of authors later pointed out that this school of empirical scholarship suffered from certain econometric and theoretical drawbacks. As Taylor and Sarno (2004) point out these authors failed to realize that financing goods arbitrage was associated with a real cost and thus they did not adjust the expected return accordingly. If the adjustment is made for this arbitrage, efficiency would imply that the expected value of the change in real exchange rate will be equal to the expected value of the differential of interest rates. If this differential is found to be stationary, that long run PPP holds will be established. Obstfeld and Taylor (2004) provide evidence of such stationarity in the series of the differentials of interest rate. Empirical evidence supporting the random walk hypothesis was ambiguous and it was found that the results were not robust to the employed criteria (Cumby and Obstfeld, 1984). Gradually the econometric techniques became more sophisticated but these advanced tests suffered from low power as did the earlier ones. As a result of the low powers researchers may have not rejected the random walk null although it was false. In the later part of the 1980s, the “unit root tests” line of examination of the long run PPP theory started to develop. The core idea was that if a time series variable results from a unit root process, then although changes in the values of the variable may be predictable to a certain extent, the variable is unlikely to ever converge to any particular level even in the largest of time horizons. Thus, if the real exchange rate is found to be non-stationary or be the realization of a unit root process then it cannot be mean reverting. Thus, long run PPP must be rejected in that case. Taylor (1988), Mark (1990) etc are instances of research that followed this approach. In this paper, we shall utilize the basic unit root testing methodology as well. Empirical Framework Recall that the objective of the present study is to evaluate the veracity of the PPP theory for the UK and US economies. The basic empirical framework we shall utilize is that of testing the stationarity of the UK to US real exchange rate. Essentially this is a test of the relative version of the PPP theory. The basic idea of the PPP theory is that once we have accounted for the exchange rate, each good should sell at the same price across all economies. This is basically the law of one price. If be the price of good in the US and if be the price of the same good in the UK, and finally if be the US Dollar per Great Britain Pound nominal exchange rate then the Law of one price holds that: The equation above simply asserts that after the exchange rate is applied, good I should have the same price in both US and UK. However, because of tariff and non-tariff barriers, transport and transaction costs, and the presence of non-traded inputs, the price of all goods may not be equal in the US and UK. But in general if we consider the price level or an aggregate of all prices, then the relationship should be expected to hold. That is, for all goods the following should hold: This is the absolute version of the purchasing power parity theory. Typically, a consumer price index for each country is used for the purpose. However, this absolute PPP theory also has certain problems. For example, with a variety of price indices available, which one should be used? Often there are problems in base year mismatches across similar indices. Additionally, for the theory to hold the same basket of goods have to be included in the price indices of both countries. To circumvent these problems, we commonly use the relative version of PPP: Here the additional subscripts t and t-1 represent the respective time periods. So the above equation states that the change in the price level from period t-1 to period t in the US will equal the exchange rate adjusted price level change in the UK over the same period. So, relative PPP holds if the price growth differential across the two countries can be accounted for by the rate of growth of the exchange rate over the relevant time period. For testing the validity of these claims econometrically, it is convenient to use log transformations. So, we define the following: Then strong or absolute PPP requires: (1) Or in logs: (2) Relative PPP requires: (3) So, in logs, relative PPP requires: (4) Where is stationary. Equation (4) is basically the model estimated by Frankel (1978). However it suffers from the drawback that if is not stationary then standard hypothesis testing will be invalid. Additionally, as pointed out by Krugman (1978) under a floating regime, both the exchange rate as well as the price levels is endogenously determined. But if this endogeity is not explicitly addressed then the estimate of the coefficient will be biased downwards. Because of these difficulties, apart from estimating the coefficient, we shall also test the stationarity of the real exchange rate. This approach is based on the idea that we intend to test whether the long run real exchange rate is a random walk or it is actually a mean reverting process. Taking the former as the null hypothesis and the latter as the alternative, rejection of the null would imply empirical validity for the PPP hypothesis in the long run. Based on the suggestions of many authors (Frankel, 1986; Edison, 1987; Huizinga, 1987; Meese and Rogoff, 1988) for this approach we shall first construct the following identity for the real exchange rate: The formal test for the PPP hypothesis is then to test the null hypothesis that is non-stationary (integrated of the 1st order to be specific) against the alternative that it is stationary. Finally, we shall also look at the possibility of cointegration between the nominal exchange rate and the UK US CPI differential. If these two series individually are non-stationary, they can in principle make the real exchange rate series stationary if they are cointegrated. Notationally, if and are cointegrated, then can be stationary. So as an alternative to running a direct stationarity test on the real exchange rate, for robustness, we shall also evaluate whether and are cointegrated. Data Description For the purpose of the present study we are using time series data of monthly frequency. The range is February 1975 to March 2012. Our exchange rate series is US Dollars per GBP. The price level of US is represented by the United States all items CPI index. The price level of UK is represented by the United Kingdom all items consumer prices index series. Table 1 presents the descriptive statistics below. Table 1: Summary Statistics Statistic CPI_UK CPI_US Nominal Exchange Rate Mean 70.63092 72.82937 1.706026 Median 73.6033 74.2 1.6464 Maximum 125.433 117.5 2.4375 Minimum 16.0958 26.9 1.082 Std. Dev. 29.4393 25.56743 0.239456 Skewness -0.09438 -0.08584 0.757407 Kurtosis 1.951051 1.943825 3.618825 Jarque-Bera 21.1093 21.27752 49.7588 Probability 0.000026 0.000024 0 Sum 31501.39 32481.9 760.8874 Sum Sq. Dev. 385669.1 290893.6 25.51595 Observations 446 446 446 Figure 1: Movements in UK and US price levels over time The time plot of the two CPI series (figure 1) shows very similar trends. Although the UK series starts below the US series, it rises at a faster pace and by 1990, has caught up. Both series move almost inseparably together until the financial crisis. In the very recent years however there has been a marked departure. The UK CPI has continued its pace of growth but the US CPI growth has slowed down leading to the divergence we observe in the last few years. Figure 2: Time plot of the nominal exchange rate (USD per GBP) Figure 2 above presents the inter-temporal dynamics of the nominal UK pound – US dollar exchange rate. We find substantial volatility in the series. It starts from a high in 1977, declines and then recovers again in 1980. But by 1985 the series hits its lowest in the period concerned. Since then, the series has recovered gradually but not to the initial high values. Table 2: Correlation Matrix Correlation CPI_UK CPI_US Nominal Exchange Rate CPI_UK 1   CPI_US 0.998668 1   Nominal Exchange Rate -0.29024 -0.28507 1 Table 2 presents the correlation matrix among the nominal variables of interest. We find that UK and US consumer price indices are very highly positively correlated. This corroborates the trend found in figure 1. The Nominal exchange rate is negatively correlated with both price levels. Results and Discussion This section presents the results of the analysis and discussion of the results. We start off by exploring the stationarity of the real exchange rate. Figure 3: The real exchange rate over time The graph above seems to indicate that the real exchange rate has not been stationary. Stationarity requires frequent mean reversion and no persistence. But the graph above shows steady departures and significant persistence in the series. This seems to indicate that real exchange rate has not been stationary. Additionally, we should also note that the series has a positive intercept. Formal proof of non-stationarity of the series is given below: Table 3: Testing Startionarity of real exchange rate   ADF-statistic under null hypothesis Series level 1st differences   Real Exchange Rate -2.37 -19.52***   We fail to reject the null hypothesis that the real exchange rate has a unit root. The computed t-statistic under the null hypothesis of a unit root is -2.37 which is lower (in absolute terms) than even the 10% critical value of -2.57. Thus, the test fails to reject the null hypothesis that there is a unit root in the levels of the real exchange rate series. Now to identify the order of integration of the series we take first differences. Figure 4: The real exchange rate in first differences over time The time plot above shows the real exchange rate in first differences. This series visually at least satisfies all properties of stationarity. We find evidence of frequent mean reversion. Also no hints of persistence, i.e., short term continuation of trends is visible. Therefore, this visual inspection seems to indicate that the real interest rate is integrated of the 1st order. From the 2nd column of table 3, we find that the null hypothesis that the series of 1st differences of the real exchange rate has a unit root is strongly rejected. Thus, the table above shows that the real exchange rate in first differences is stationary. Therefore, we find evidence of the real exchange rate following an I(1) process. Therefore, this can be taken as evidence of a rejection of the PPP hypothesis. Now, for robustness we estimate equation (4) by OLS as the primary alternative method to evaluate the validity of the PPP hypothesis. The results are presented in table 4 below. Table 4 Regression Results     Specification     (a) (b) Price differential   0.488*** -     (-0.052)   Constant   0.49*** -0.35*     (0.006) (0.206) UK price level   - -1.130**       (0.164) US Price level   - 1.31***       (-0.208) R2   0.163 0.193         Adj. R2   0.16 0.19         F-statistic   86.82 53.26     [0.000] [0.000]         Notes: ( ) denotes standard errors. F-statistic: Test for the overall validity of the model; Any other test: give a brief explanation as done for the F-statistic. [ ] denote p-values. ***, **, and * denote significance at 1%, 5%, and 10% respectively. Column (a) presents the results of running an OLS regression specification denoted by equation (4). We find that the coefficient of the price level differential is positive and significant. Thus, the nominal exchange rate is influenced by movements in the relative price level differential. This would support the relative (weak) PPP hypothesis. But this also contradicts our results found in the earlier tests for the unit root in the real exchange rate. In column (b), we have the results from a regression where both the US and UK price levels were included individually as explanatory variables for the nominal exchange rate. The motivation for this specification was to evaluate what the individual impacts of movements in US and UK price levels were on the exchange rate. Observe that the constant turns out insignificant while the coefficient on the domestic price level is positive and significant while that on the foreign price level is negative and significant. Additionally, both coefficients are pretty close in terms of absolute values. This seems to reflect that the magnitude of impacts of movements in either the UK price level or the US price level have on the exchange rate are very close. This seems to support the relative PPP hypothesis even more strongly. Figure 5: Comparing actual vs fitted models However, in spite obtaining such significant coefficients, from the graph of actual versus the fitted model in figure 5, we find that our model(s) do a bad job of explaining the data. The series of residuals is almost identical to the series of actual data implying most of the variation is not explained by the model. Looking at the R2 values this is confirmed. The specification presented in column (a) of the table explains only 16 percent of the total variation in the exchange rate series, while specification (b) explains only about 19 percent of the variability of the exchange rate series. At this point the final exercise we carry out to evaluate the validity of the inferences is to look at the stationarity of the nominal exchange rate and the price level differential series. If these are found to be non-stationary then our standard hypothesis tests will be invalid. Table 6: Testing the stationarity of nominal exchange rate and the price differential series Series ADF-statistic under null hypothesis Nominal Exchange Rate -3.1**   (0.02) Price Level Differential -2.94   (0.15) Note: The numbers in parenthesis represent the p-value Table 6 presents the results of running Augmented Dickey Fuller tests on the nominal exchange rate and the price level differential series. Observe from the first row of table 6 that the ADF test rejects the null hypothesis that the nominal exchange rate is non-stationary. So, the series is stationary or integrated of order zero. However, from the 2nd row we find that the price differential series is non-stationary. Or rather, we fail to find evidence to state otherwise. That is we fail to find any statistical evidence that the series is stationary. Thus, this also implies that these variables follow different orders of integration. Therefore, these variables cannot be cointegrated either. Although we could formally try and establish that these variables are not cointegrated using a Johansen co-integration test we are not proceeding in that direction since it is already clear that these variables cannot be cointegrated. Two variables are cointegrated if they are individually non-stationary but a linear combination of these variables is. Howeever a prerequisite is that the variables follow a similar order of integration. This condition fails in this case which lets us conclude that the variables cannot be cointegrated. The diagram below also indicates this. Although both paths follow similar trends, evidently the nominal exchange rate is largely cyclical while the price level differential is not. Figure 6: The Nominal exchange rate and the US-UK CPI ratio over time Thus, we conclude that our inferences from the OLS estimation are likely to be flawed. We therefore retain the conclusions obtained from the primary unit root tests. Thus, we find evidence to support studies which conclude that relative PPP does not hold between the US and the UK. Conclusions This paper attempted to evaluate the validity of the PPP hypothesis. We searched for evidence of the theory taking the UK and the US as our two countries of which the exchange rate we were attempting to predict. Our methodology was that of estimating an OLS specification of the nominal exchange rate as well as evaluating the stationarity of the real exchange rate series. The tests of stationarity of the real exchange rate series revealed that the series follows an I(1) process. It is stationary in first differences but not in levels. Therefore, this implies that the relative PPP does not hold. The OLS estimation initially yielded results that seemed to indicate on the contrary that relative PPP does hold. We found evidence that strongly supported the idea that variability in the relative price levels did significantly affect the variability of the exchange rate. Additionally, we also explored the individual impacts that variability in the UK and US price levels had on the exchange rate. These were also found to be significant and of close magnitudes. The signs of the estimated coefficients were however opposite. Although these seemed to indicate that relative PPP does indeed hold, such conclusions were found to be unfounded. The nominal exchange rate series was found to be stationary but the differential of price level series was found to be non-stationary. Consequently the validity of the OLS results was compromised. Thus, from the research carried out in this study, we failed to find evidence to support the PPP hypothesis. However, before taking this as conclusive evidence that PPP does not hold between UK and US, it should be noted that we are using data that covers a time period of only 37 years: 1975 to 2012. Thus, we could have just examined a time interval over which the exchange rate did not converge to its long run equilibrium. That is, over the time horizon we inspected, the market was in a state of deviation. If we had access to data covering a truly long run horizon, then, conclusive comments could be made. But, to summarize our findings, we failed to find any evidence of the exchange rate being predictable in accordance to the PPP hypothesis for the US-UK economy pair over a time span covering 1975 to 2012. References Adler, M and Lehmann. B (1983) ‘Deviations from Purchasing Power Parity in the Long Run,’ Journal of Finance, 38:5, pp. 1471–1487. 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AK (1996) ‘A Panel Project on Purchasing Power Parity: Mean Reversion Within and Between Countries,’ Journal of International Economics, 40:1-2, pp. 209 –224 Frenkel, J A (1976) ‘A Monetary Approach to the Exchange Rate: Doctrinal Aspects and Empirical Evidence,’ Scandinavian Journal of Economics, 78:2, pp. 200 –224 Frenkel, J A and Johnson, HG (1976) ‘The Monetary Approach to the Balance of Payments’, London:Allen & Unwin Frenkel, J A (1981) ‘The Collapse of Purchasing Power Parity During the 1970s,’ European Economic Review, 16:1, pp. 145– 165 Mark, N (1990) ‘Real and Nominal Ex-change Rates in the Long Run: An Empirical Investigation,’ Journal of International Economics, 28:1-2, pp. 115–136 Obstfeld, M and Taylor. 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