Standard Deviation Abstract

International Review of Economics and Finance 20 (2011) 485–497

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International Review of Economics and Finance

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Predicting foreign exchange movements using historic deviations from PPP

Mei Qiu a, John F. Pinfold a,⁎, Lawrence C. Rose b

a School of Economics and Finance (Albany), Massey University, New Zealand b College of Business, Massey University, New Zealand

a r t i c l e i n f o

⁎ Corresponding author. School of Economics and F Tel.: +64 9 414 0800x9463; fax: +64 9 441 8177.

E-mail addresses: [email protected] (M. Qiu), J.F

1059-0560/$ – see front matter © 2010 Elsevier Inc. doi:10.1016/j.iref.2010.09.005

a b s t r a c t

Article history: Received 25 June 2009 Received in revised form 8 September 2010 Accepted 8 September 2010 Available online 7 October 2010

The ability to forecast FX rates from historical exchange rate movements is examined. An eight nation study shows a currency's deviation from the rate predicted by PPP over a four year period can predict the direction of its movement in the subsequent one to four years. We show short term exchange rate movements of freely floating currencies are large in comparison with changes in economic fundamentals and these movements accumulate to create pressure which results in a predictable pattern of reversal. The results are robust across currencies and relatively insensitive to the time parameters used in the estimation.

© 2010 Elsevier Inc. All rights reserved.

JEL classification: F31 G15

Keywords: Purchasing power parity Currency Exchange rate Foreign exchange

1. Introduction

Market based economic management, where market forces are allowed to prevail, is based on the notion that most economic variables are self adjusting and outcomes are not enhanced by government economic intervention. Many nations have adopted a freely floating exchange rate system which relies upon market mechanisms to adjust the value of their currencies and interventions are rare, and rarely successful when attempted. This implies that market forces influence exchange rates, as the international competitiveness of nations is inextricably linked to the relative exchange rates between nations. If the movements in freely floating exchange rates are purely random, international competitiveness will simply be a lottery, with winners and losers being determined by chance. Clearly this cannot be the case. We know exchange rates adjust over time and a floating exchange rate regime produces results which, in the long run, reflect underlying economic fundamentals. The lack of models capable of providing reliable predictions of future exchange rates is therefore puzzling.

Purchasing power parity, the concept that in the long run exchange rates are determined by the relationship between domestic and foreign prices, is intuitively the most appealing of all exchange rate models. We know it must apply, yet we also know an immense body of research has generally failed to show any statistically significant predictive power except over very long time periods. We find it hard to accept PPP is not a major driving force behind exchange rate movements, yet we are forced to accept that, even though we can forecast future inflation rates with some accuracy, we cannot derive economically useful information on future exchange rates from this knowledge. This paper takes a fresh perspective on the problem and finds that an alternative way of approaching the problem greatly enhances the predictive power of PPP and makes it an economically useful tool for exchange rate management.

inance (Albany), Massey University, Private Bag 102 904, North Shore MSC, Auckland 0745, New Zealand

[email protected] (J.F. Pinfold), [email protected] (L.C. Rose).

All rights reserved.

.

486 M. Qiu et al. / International Review of Economics and Finance 20 (2011) 485–497

In the paper we assume there is an effective exchange rate adjustment mechanism operating in market based economic systems which employ floating exchange rates, and PPP is part of this mechanism. What then is obscuring the effect of PPP? One possible answer is relative inflation rates betweenmost nations have generally been small since the late 1980s; hence the currency movements predicted by PPP are small. Exchange rates on the other hand are volatile and annual currency fluctuations are at least an order of magnitude greater than the changes predicted by PPP. High volatility is tied to an inability to predict future FX rates. If market participants can accurately predict future FX rates, deviations from predictions would generate profitable opportunities to speculate and competition for these profits would drive exchange rates towards their predicted values. The current inability to predict future FX rates allows high volatility and this volatility makes predictions difficult if one takes a forward looking approach. The Achilles' heel of current predictions methods is that the portion of the volatility in exchange rates unexplained by PPP is large enough to mask the effect inflation differentials have in determining FX movements.

The approach taken in this paper is different from previously reported research; it looks at the movement of exchange rates over a period of years and looks for a correction in the misalignments between currencies which have accumulated over this period. The result, as will be seen, is a clear pattern in exchange rate movements as exchange rates adjust to the economic pressures created by movements away from sustainable long run values. The mechanism is slow. The movement of a currency away from the value predicted by PPP which occurs over a four year period takes up to four years to reverse and only about 40% of the deviation is reversed in this period. However, predictions based on this mechanismwill be shown to be accurate enough to be economically useful, and have a high level of statistical significance.

The results are consistent with the predictions derived from previous studies which use shorter term data to make predictions, for example, Abuaf and Jorion (1990) find that deviations in exchange rates from PPP take about three years to reduce in half. The difference in this paper is the approach taken. Abuaf and Joroin use first order auto regression coefficient estimates taken from monthly and annual data to make their estimate. We look at long term historical movements and the subsequent corrections. We show for the first time, how long term cumulative deviations from PPP can be used to make economically useful predictions of future exchange rates for freely floating currencies. We also demonstrate why this method of predicting FX rates has far greater explanatory power than traditional methods employing PPP.

Our approach faces a number of obstacles in statistical methodology, which needed to be overcome before valid results could be obtained. The results may be controversial but the story is compelling. Once you have examined the evidence we produce you may think it is too good to be true, and dismiss it as a statistical anomaly created by flawedmethodology. However, given generally accepted economic principles, it would be surprising if a pattern like the one presented did not exist. It is a clear demonstration of market based economic principles at work.

2. The background to our model

2.1. The basis of our approach

Theory predicts historical deviations from PPP will be reversed in subsequent periods, and the pattern of this occurring should be evident. The reversal itself is well documented and simple graphical methods such as that presented inMacdonald (1995) show when an exchange rate is graphed alongside the exchange rate adjusted for relative prices, the two rates diverge and converge over a period of years. Cochran and DeFina (1996) and others who have tried have failed to show any simple and statistically significant pattern to this reversal which can be translated into economically useful predictions. Such a pattern of reversal is of course what we would expect. If a currency rises sharply in value, the nation's exporters can insulate themselves by forward hedging, and sustain periods of loss making, but in the long run many will go out of business if the currency does not return to profitable levels. If importers are advantaged at the expense of exporters for too long the economy will suffer, the balance of payments will deteriorate and this should impact on the value of the currency. Rather than looking for a complex equation to model this process, we look for a simple one which applies equally to all market based economies with freely floating exchange rates. While a complex model may be better at explaining historical data, the more parameters you put in your model the easier it is mimic historical patterns without providing useful predictive power. The literature abounds with such models which work well on historical data but fail to accurately predict in future periods.

The model we use is simple. Exchange rates should behave as predicted by PPP and long run deviations from this predicted behaviour will reverse themselves in subsequent periods. In particular we show the deviation of an exchange rate from the value predicted by PPP over a four or five year period is partially reversed in the following three or four years, and these predictions are robust enough to produce very useful profit opportunities. The problem is not in the model we employ, but in overcoming the statistical minefield standing in the way of proving it, given the limited period for which data is available for floating currency rates, and the long time frame of the pattern of currency correction.

2.2. The relevant research

No attempt will be made to comprehensively analyse the vast body of work on PPP and exchange rate forecasting. We will not analyse macroeconomic exchange rate determination models or those studies which attempt to establish equilibrium conditions for PPP except to say such approaches have, at best, a marginal ability to predict future exchange rate movements. Instead we will concentrate on the studies which are directly relevant to the central issues contained in this paper.

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One of the key strands of the literature is the use of the Dickey Fuller unit root test. A finding of no unit root is considered supportive of PPP and the absence the contrary. Evidence from unit root tests is mixed. Glen and Shibata (1992) found supportive evidence in nine bilateral exchange rates. Lothian (1997) and Lothian and Taylor (2000) found strong evidence in US dollar- sterling and French franc-sterling rates. The half life of the reversion to PPP in these studies varies from one to six years. Other studies failed to find unit roots, such as Cuddington and Liang (2000) and Patel (1990). The results of these studies which fail to find evidence of PPP have been questioned by Froot and Rogoff (1995) and Sarno and Taylor (2002), who argue unit root tests may fail to support PPP more often than they should due to a loss of test power when using short time series with close-to-unity coefficient estimates. The debate on univariate unit root tests still rages with Taylor (2002) virtually declaring the debate closed in favour of long run PPP after analysing a century of data for 20 countries. Lopez et al. (2005) disagreed; pointing out that Taylor used sub optimal lags for the unit root tests and only nine of the 16 countries show evidence in favour of PPP when “superior methods of lag selection” were used. The debate on the evidence from univariate unit root tests seems likely to continue.

The problemswith univariate root tests led to the use of panel unit root tests. By simultaneously testing for unit roots on panels of exchange rates, the test power is increased, particularly when using limited data from the short history of free floating exchange rates. Perhaps themost prominentwork using panel unit root tests is the study by Abuaf and Jorion (1990)which has already been mentioned. They translated the coefficients from the first order auto regression model on monthly and annual exchange rate data into half lives for exchange rate deviations from PPP. These ranged from three to five years. Coakley and Fuertes (1997) use a similar technique to find evidence in favour of PPP for the G10 currencies and the Swiss franc between 1973 and 1996, with a half life estimate of less than three years. Fleissig and Strauss (2000) while supporting PPP found the speed of exchange rate adjustment differed when different price indices and test procedures were adopted. Comprehensive studies by Choi (2004), Chiu (2002), and Cerrato and Sarantis (2008) support long run PPP in the majority of currencies studied.

While the majority of the panel unit root tests support PPP, some such as Chortareas and Driver (2001) have found mixed evidence. In addition Froot and Rogoff (1995) argue that the standard multivariate unit root test as used by Abuaf and Jorion (1990) has a null hypothesis that all the series under consideration are unit root processes, meaning the null hypothesis would be violated only if one of the exchange rate series in the panel is stationary while the rest of the series are not stationary. Thus while accepting the null hypothesis rejects PPP, rejecting the null does not prove PPP holds. Further, O'Connell (1998) argues the failure to control for cross-sectional dependence or survivorship bias can lead to mistakenly supporting PPP. The use of panel unit root tests has proven as controversial as the use of univariate unit root tests.

Another approach to testing PPP is to use the Engel and Granger cointegration test whereby evidence of cointegration between exchange rates and relative prices supports the existence of PPP. This approach has produced mixed results with Enders (1988) and Taylor (1988) rejecting PPP for developed countries in the period between the mid 1970s and mid 1980s. On the other hand, Cheung and Lai (1993) found evidence supporting PPP from 1914 to 1989. Johansen extended the Engel-Granger cointegration test in 1998 to encompass a multivariate approach by using themaximum likelihood estimation procedure, thereby increasing the power of the test. This procedure has more power than the Engel and Granger cointegration test and has yielded results more favourable to PPP. Here, we are not only interested in evidence supporting PPP, but also the speed with which convergence towards the value predicted by PPP occurs. The results of Macdonald (1995) showing a half life for deviations of three years is typical of the results obtained using the Johansen method.

PPP has also been modelled using non-symmetric, non-linear techniques. These studies support the statistical validity of PPP but provide little guidance on how this might be exploited for practical applications. This line of research will not be elaborated on here.

The point wewish tomake is that studies using a variety of techniques provide general support for a slow convergence towards the values predicted by PPP with about half the deviation corrected in a period of three to five years. Many explanations have been given for the slowness of this convergence such as sticky prices, transportation costs, tariff and non-tariff barriers, information costs and immobility of labour. If this slow convergence is occurring, it should provide us with an ability to predict the direction of future exchange ratemovements. This convergence towards PPP has been used as a component in a number of models designed to predict future exchange rates with varying success. None have used the technique applied here.

To summarise; the large body of evidence contained in the literature on PPP provides general, but not unequivocal, support for the existence of PPP. The attempts at applying PPP, or for that matter any other methodology, to predicting future exchange rates have been less promising, with predictability being marginal at best.

3. Testing the model

3.1. Objective and data

The objective of the research is to establish the relationship between long term movements in exchange rates away from the values predicted by PPP and exchange rate movements in future years. The studies detailed above indicate half of any divergence in a single monthly or twelve monthly periods will be reversed over the subsequent three to five years. They do not look at the cumulative effect of divergences over a number of years, but it is reasonable to assume if deviations persist and accumulate over a number of years the pressure for a reversal will, if anything, be greater than deviations over shorter periods. The starting place for this analysis is a simple ordinary least squares regression, which is subsequently modified to overcome the deficiencies of the available data.

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As the mechanism being examined relies upon the largely uninhibited actions of a market economy, the currencies studied must have free floating currencies with well developed financial markets and no restrictions on international currency movements. While in the past there have been periods of free floating currencies, these periods are not as ideal as the more recent period. We will concentrate on the period of free floating currencies following the breakdown of the Bretton Woods agreement. The gold exchange standard established at Bretton Woods in 1944 began to break down by 1973 when a number of countries floated their currencies, and was officially abandoned in the Jamaica Agreement in 1976. At this point some nations adopted a freely floating exchange rate, while others chose alternative mechanisms to manage their currencies.

As we wish to concentrate on currencies which are freely floating, our choices are limited. Few nations with open market economies, well developed capital markets, and few restrictions on the flow of goods and capital, have a long enough period in which their currencies have been freely floating to allowmeaningful analysis. Eight nations meet our criteria. The sample consists of data from January 1974 to December 2007 for Canada, Germany, Japan, Switzerland, the UK and the US. Two nations which floated their currencies later are included. Australian is included from January 1984 and New Zealand from January 1986.

It could be argued the UK and Germany should not be included as they have not been freely floating during the entire period. Germany, and to a lesser extent the UK, were part of the European Monetary System for part of the study period and Germany is now part of the Euro zone. To exclude themwould however leave a large part of the developed world unrepresented in the study. The inclusion of the Pound and the German Mark in the study, if anything, reduces the effectiveness of the predictive technique employed, hence their inclusion will not invalidate the results presented. We assume the European currencies were floating as a block and the German Mark represents this block. The German Mark is therefore replaced by the Euro in January 1999 at the official conversion rate of 1.95583 applied at the time. The UK Pound is also included as, while it became part of the European Monetary System exchange rate mechanism in 1990, it withdrew in 1992 after speculative attacks on its currency forced it to devalue.

The inflation rate used in this study is the CPI for most of the tests. An alternative measure, the Producer Price Index (PPI), is used in a robustness check for the key tests presented in Table 5. Monthly CPI figures and exchange rates of the seven currencies against the US dollar were obtained from Datastream. Cross rates for other currency pairs were calculated from their cross rates against the US dollar. New Zealand does not produce monthly CPI data so monthly data was linearly interpolated from quarterly data. PPI data was sourced from Datastream and the websites of the Reserve Banks of Australia and New Zealand.

3.2. Approach taken

The first step taken is to observe the relationship between long term historical movements away from the values predicted using PPP (the deviation period) and exchange rates in subsequent periods (the correction period). The 28 different currency pairs that can be formed between the currencies of Canada, Germany, Japan, Switzerland, the UK, the US, and New Zealand are examined using simple OLS regression. This procedure is done, not to prove the relationship is statistically significant, but to show the pattern that exists. Currency deviations from PPP for periods of between three and seven years are compared with the corrections of these deviations in the subsequent three to five years. As there are only thirty six years of data the number of non- overlapping data points for each currency is too small for valid statistical testing. However, we establish the nature of the pattern and that it is highly consistent across all possible pairings of currencies. We also show it is relatively insensitive to the deviation and correction periods chosen.

The objective of the second step is to show why the corrections of long term deviations from PPP show such a consistent pattern; yet using PPP to predict future exchange rates in the conventional way is of little value. This is done by breaking the deviation from PPP into its two components. The first component is simply the movement of an exchange rate away from its historical value. The second component is an adjustment of the historical exchange rate for the effects of PPP. The results show that the predictive power of our model comes largely from the reversal of historical movement in exchange rates, that is, exchange rates are mean reverting. The addition of an adjustment for PPP is shown to enhance the results, in other words exchange rates revert more consistently to the values corresponding to the historical exchange rate adjusted for PPP than they do to the historical exchange rate itself.

Once the pattern of exchange rate movements is established the task of showing that the relationships observed are statistically significant is undertaken. Themethodological challenges are discussed and pooled regressions which are corrected for overlapping data and serial correlation are employed to establish the statistical validity of the patterns observed. To make sure our results are robust to choice of different inflation measures, we re-run the pooled regressions using PPI instead of CPI when calculating inflation rates. Finally, the out of sample predictive power of PPP deviations is tested against a random walk exchange rate model.

3.3. Establishing the response to long term deviations from PPP

We start by establishing how freely floating currencies adjust to long term deviations from the values predicted by PPP. The deviations and subsequent corrections for the 28 possible currency pairings of the eight currencies are analysed using OLS regressions of monthly data.

These results do not, in themselves, produce statistically reliable results, as the series are not always co-integrated or stationary and contain overlapping data. Serial correlation is also present. While we adjust the data to compensate for the overlapping observations, the other problems are not resolved until later when multi-nation pooled data is analysed. The results are

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nevertheless included as they help to complete the picture and show an intriguing consistency. Simple regression tests cannot resolve the problem for individual currencies because the period which these currencies have been floating is too short. The 36 years of data allows for just eight non-overlapping data points for our tests based on the optimum prediction periods which we subsequently determine. There are however twenty eight unique currency pairs formed by the eight currencies, and as will be seen, the consistency of the results from such limited data is intriguing.

No attempt is made to establish an equilibrium PPP condition as establishing equilibrium conditions is problematic, and minor differences in assumptions lead to significantly different equilibrium values. A predictive method which is simple to calculate and unaffected by underlying assumptions has definite advantages. We simply assume long term historic movements in exchange rates will reverse themselves, and seek to discover how long it takes for a market economy to make this adjustment. We do this because exchange rate movements are large, and these movements are probably largely random, or if not random, are unexplainable in terms of underlying economic forces given our current state of knowledge. If FX movements are largely random they will often drive exchange rates away from economically sustainable values. The pattern which exists in FX rates is a combination of a volatile random walk and the correcting force created by unsustainable differences in relative prices, in other words PPP. In this scenario the corrective force of PPP will not primarily be generated by the differences in inflation between nations, but by the pricing problems created by random historic fluctuations in currency values away from historical price relativities.

We assume that economic forces will act to correct miss-pricing and this will result in historic movements in currency values being reversed and this reversal will be supplemented by pricing pressure caused by differences in inflation between nations. This hypothesis is represented in Eq. (1), where St is the spot rate at time t, ΔSt+n is the change in spot rate over the next n years, and SPt is the spot rate that would exist in year t if the currency had moved as predicted by PPP over the designated number of preceding years. We test to see if the coefficient β1 is different from zero.

Table 1 Exchang

Devia

Period

3 Yea

4 Yea

5 Yea

6 Yea

7 Yea

Notes: F stated p period) New Ze which o directio

ΔSt+n = α1 + β1* St−SPtð Þ + et : ð1Þ

Attempting to show this relationship in a simple regression is problematic as some of the individual data series are non- stationary, and some of these non-stationary series are not co-integrated. For the majority of currency pairs the regression results do not establish the statistical validity of the relationships shown even though the t-statistics after adjustment for overlapping data are high. We cannot prove the results of individual regressions are statistically significant because of the deficiencies in the data. Nevertheless, we will conduct the first series of tests on individual currency pairs, before moving on to more robust pooled testing. We do this because the patterns are more easily seen using individual currencies, and for practical applications it is important to know not just what works in aggregate, but also that it works consistently for each currency pair.

You may ask why we run tests where the results cannot reliably be distinguished from random currency fluctuations. If this logic is followed the clear patterns in the data never become evident. Currencymovements over periods of three to seven years are found to be partially reversed over the following three to five years in a very consistent pattern, just as PPP predicts. The average β1 coefficients are converted figures representing the percentage correction of the exchange rate in the subsequent period. These figures, and the number of currency pairs which exhibit the correct sign, are presented in Table 1. Only summary information is supplied as there is not room here to present the 28 tables of data required to show the full details.

e rate corrections to cumulative deviations from PPP across various time intervals — summary of uncorrected simple regression results.

tion Correction Period

3 Years 4 Years 5 Years

rs Correct direction predicted 25:3 27:1 27:1 Average Correction 26% 37% 44% Range of Corrections −18% to 62% −14% to 76% −3% to 83%

rs Correct direction predicted 25:3 26:2 24:4 Average Correction 30% 40% 40% Range of Corrections −11% to 61% −8% to 78% −5% to 79%

rs Correct direction predicted 24:4 22:6 23:5 Average Correction 33% 33% 36% Range of Corrections −12% to 71% −32 to 83% −30% to 84%

rs Correct direction predicted 22:6 23:5 22:6 Average Correction 30% 50% 31% Range of Corrections −17% to 75% −30% to 130% −49% to 104%

rs Correct direction predicted 22:6 21:7 20:8 Average Correction 24% 26% 23% Range of Corrections −26% to 66% −37% to 106% −63% to 152%

or each of the 28 currency combinations of the eight countries studied, the movement away from the value predicted by PPP in the exchange rate over the eriod of years (the deviation period) is regressed against the movement of exchange rate which occurred in the subsequent period of years (the correction . The period tested is January 1974 to December 2009 for currency combinations of the currencies of Canada, Germany, Japan, Switzerland, the UK, the US aland and Australia. For each combination of periods the results of the regressions are expressed as the percentage reversal of the exchange movemen ccurred in the deviation period. The range of corrections exhibited in the 28 currency pairs is also given, as is the number of regressions in which the n of movement is correctly predicted.

, t

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The currency movements demonstrated in the Table 1 are unlikely to have occurred randomly. The prediction of currency movement is correct in almost all cases and the results are robust over different deviation and correction periods. If each currency movement was an independent event a non-parametric test would have a very high statistical significance, similar to the probability of a coin toss yielding as many as 27 heads from 28 tosses. The problem is the currency pairs are part of an interdependent system. For example, for any three currencies only two exchange rates are required to derive the third by calculating the cross rate. Nevertheless, the pattern is clearly highly unlikely to be the result of random chance. This will be verified shortly when more robust tests on pooled data are conducted. Figures for correction periods of less than three years are not presented as, while in aggregate they provide useful predictive power, individual currency pairs are less likely to correctly predict the direction of future currency movements than those presented in the table.

3.4. Regressions using a two component model

Conventional methods which use PPP to predict future exchange rates do so by adjusting the current exchange rate for the expected difference in inflation rates between countries. This does not, in general, yield economically useful predictions. Here we are looking at historic deviations from the values predicted by PPP and applying them to predicting currency adjustments in future periods. This means themajority of the historic deviations from PPP are the result of unexplainedmovements in exchange rates, in other words a process of reverting to historic means is added to the movement required to adjust for PPP.

In order to explain why PPP is such a poor predictor, but the reversal of historic deviations from PPP is so much more informative; we break the exchange rate movements into two components. Eq. (1) can be broken down into its two constituent parts which are found in Eqs. (2) and (3). In these equations, St is the current spot rate at time t, ΔSt+n is the change in spot rate in the subsequent n years, St−m is the spot rate m years ago and SPt is the spot rate that would currently exist if the spot rate had moved according to PPP over that previous m years. By breaking our predictor into these two constituent parts we can illustrate the relative importance of the two components, namely the reversal of historic exchange rate movements, and the effect of inflation differentials between countries. As shown in Fig. 1, these forces can be acting in the same or opposite directions.

Fig. 1. D rate dur represe exchang

ΔSt+n = α2 + β2* St−St−mð Þ + et ð2Þ

Time

St-5

St+3

St

SPt

0-5 Year +3 Year

D2D1

D3

0 Time

St-5 St+3

SPt

-5 Year +3 Year

D1

D2

D3

St

iagrammatic representation of exchange rate correction process. Notes: Over a five year period the spot exchange rate moves from St-5 to St. The exchange ing this period canmove in the direction predicted by PPP as shown in the upper part of the diagram, or in the direction opposite to that predicted by PPP as nted in the lower part of the diagram. The deviation from PPP, represented by D1, is made up of two components, the deviation from the historica e rate represented by D2, and the movement predicted by PPP, represented by D3.

l

Table 2 Analysis of the components of corrections to cumulative deviations from PPP.

Base Currency A$ C$ DM NZ$ SF UK£ Yen US$ Ave.

β1 −0.14 −0.49 −0.35 −0.63 −0.38 −0.47 −0.38 −0.36 −0.40 β2 −0.29 −0.47 −0.31 −0.74 −0.36 −0.41 −0.33 −0.43 −0.42 β3 0.14 −0.29 −0.05 −0.50 −0.03 −0.48 −0.46 −0.02 −0.21 Adj-R21 0.08 0.42 0.23 0.51 0.26 0.32 0.19 0.22 0.28 Adj-R22 0.19 0.38 0.22 0.59 0.25 0.23 0.14 0.22 0.28 Adj-R23 0.05 0.07 0.06 0.04 0.06 0.11 0.13 0.06 0.07

Notes: For each of the eight term currencies, key statistics obtained from regression estimations of each of the seven commodity currencies are averaged and reported. The term (home) currencies are listed in the first row of the Table. Coefficient β1 measures the extent to which cumulative exchange rate deviations from PPP over a four-year period are corrected in the following four-year period. Coefficient β2 measures the correction to movements from historic exchange rates in the past four years. Coefficient β3 indicates the relationship between future exchange ratemovement and the change predicted by PPP. Adjusted R-square statistic for each of the three regression estimations are also reported here. PPP theoretical values are estimated from relative price changes in two countries over a four year period.

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s -

ΔSt + n = α3 + β3* St−m−SPtð Þ + et : ð3Þ

On average, 40% of the four year deviation is corrected in the subsequent four years. We use these time periods to carry out the regressions in Eqs. (1) and (2). The results of the regressions showing the effects of the two components are presented in Table 2. Once again, the properties of the data make it impossible to tell the statistical significance of the results as the series do not meet the tests for stationarity or cointegration. We present R2s for comparative purposes. Relative ranking of the R2s will provide a measure of the relative predictive ability of the three measures. The average result for each currency regressed against the other seven currencies is presented as well as the overall average. The purpose of this analysis is to showwhy traditional attempts to use PPP to predict future exchange rates are less powerful than the method employed in this study. The movement in the currency predicted by PPP is represented by D3 in Fig. 1, and β3 in Table 2. The movement of the currency from its historical value is represented by D2 and β2, and the combinedmovement by D1 and β1. We are trying to show howmuch of the predictability in our model comes from the simple historical movement in a currency being reversed, and how much comes from the additional correction force exerted by movements in PPP during this period. Comparing the results β1 and β2 in Table 2 shows most of the explanatory power of our model comes from the simple reversal of historic movement of currencies. Detailed results of individual regressions are not present due to space limitations. In our model of exchange rates the regression line should cross the intercept at zero. At this stage we do not force this to happen in our regressions. This gives R2s higher than would be the case if the intercept were forced through zero. The effect of forcing the intercept through zero will be illustrated later.

Our results show the exchange rate corrections are primarily driven by the simple reversal of historic movements in exchange rates. In most of the period studied, the cumulative differences in inflation rates over extended periods were small and cumulative differences in exchange rate movements were large. Using PPP in the traditional way, by looking at differences in inflation rates between nations and using this to predict future exchange rates is therefore ineffective. The real force of PPP is exerted in reversing historic deviations from PPP caused by the high volatility of exchange rates. While this volatility must result from economic forces, the causes of these exchange rate movements has not been adequately identified. It may be due to imbalances between the supply and demand for currencies, the actions of speculators or a variety of other causes. Whatever the cause of exchange rate volatility, the economic force created by the imbalance of prices between nations drives the currency towards the value predicted by PPP, as demonstrated by the data in Table 2.

The pattern of exchange rates movements we have seen so far could just be a statistical anomaly created by random chance, and due to the short period of data available and the long term nature of the pattern presented; statistical tests on individual currency pairs will not be able to resolve the issue. To overcome this we move to a methodology which can yield statistically meaningful results. We need to adjust for the overlapping data and overcome the problems with stationarity and lack of cointegration.

3.5. Pooled regressions

The statistical problems found in the previous results are now addressed. The first problem is created by the use of overlapping data which results in the sample size being artificially enlarged. This does not bias the coefficient estimates but it does overestimate the t-statistics. This problem was addressed in Phillips (1986), Valkanov (2003), Hjalmarsson (2006) and Hansen and Tuypens (2004). The solution they provide is to adjust the OLS t-statistics by dividing them by two-thirds of the square root of the forecast horizon, or more conservatively by the square root of the forecast horizon. Wewill adopt themore conservative of the two approaches.

The problem with stationarity and lack of cointegration in the data for individual currencies in the earlier regressions is attributed to the short period for which floating rate data is available and the long time frame of the prediction method. When we analyse the exchange rate movements of individual nations we have as few as five non-overlapping data points, so it is not surprising the results of regressions are not statistically valid. Twomethods of increasing the power of the tests are available. These

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are to use pooled data or to use panel data. Here we present pooled regressions rather than panel tests. While panel tests might be considered to be a more powerful tool, they have not been employed as we want to find a single model that operates in all nations over the entire time span of the data. Wewant to restrict ourselves to a single intercept and slope for all currency pairs. Once these restrictions are placed upon the analysis, panel testing provides no advantage over pooling. In addition the R2s of pooled tests give a measure of the usefulness of predictions for practical applications. The same cannot be claimed for panel tests.

As the nominal value of exchange rates vary between nations it is necessary to standardise them before pooling the data. To achieve this, the currencymovements were all calculated in percentage terms. The data for the 28 currency pairs possible from the eight nations in the study was then combined. In effect we are testing a single predictor variable and seeing how useful it is in predicting the movement of any currency pair selected at any starting date during the time frame examined. The pooled data for various combinations of estimation and prediction periods were tested and all were found to be stationary. As the relationship we are testing should cross the intercept at zero, this was forced for the regressions even though this was not done for the earlier regressions. The tests were also run without forcing the intercept through zero with comparable results.

The results of the pooled regressions are presented in Table 3. A period of four years is the optimum deviation period as this gives higher adjusted t-statistics and R2s than other periods. The slope on the regression line, which is the decimal fraction of the deviation corrected in the subsequent period, increases with the lengthening of the correction time, reaching a peak after about four years. The process of exchange rate correction is clearly evident, and is not particularly sensitive to the time parameters chosen. Observing the high adjusted R2s shows the high economic significance of the exchange movement explained by the predictor.

The adjustment made to the t-statistics for overlapping data is designed to adjust for the effects of serial correlation created by overlapping the data, but it does not correct for the serial correlation in the underlying data. In order to correct for the serial correlation found in exchange rate data, the tests were repeated using the Cochrane-Orcutt regression procedure. These results are presented in Table 4. Caution must be exercised in interpreting the slopes of the regressions and the R2s as the Cochrane- Orcutt procedure used to correct the t-statistics biases both the R2s and β coefficients, especially when data series are as highly correlated as these. The adjusted t-statistics are highest for the one-year correction period, but are significant at the one percent level for all results presented in the table. Normally correction for serial correlation leads to weakening in the relationship between variables, particularly where overlapping data is used. This applies particularly when dependent and independent variables are positively correlated. However, in this case the variables are negatively correlated. As the Table 4 results indicate, if currency movements are not serially correlated, the correction of historic movements away from PPP is more rapid. This is entirely consistent with what we would expect given earlier research which shows the effect of PPP when tested on monthly data is weak, in other words currencies movements have a tendency to continue in the historic direction (be serially correlated) rather than reversing as would be expected if PPP applied. The most important point to be derived from Table 4 is that the t-statistics are not upwardly biased by serial correlation in the time series currency data. We now seek to confirm this by using non-overlapping data.

3.6. Robustness check using non-overlapping samples

It is anticipated many readers will be sceptical of the robustness of the results presented so far. The use of overlapping samples is always problematic, and while the method used to correct for it has been robustly debated in the literature, it is not

Table 3 Exchange rate corrections to cumulative deviations from PPP: pooled estimations across various time intervals.

Statistics Three-yr Deviation Four-yr Deviation Five-yr Deviation Six-yr Deviation Seven-yr Deviation

One-year Correction Coef. −0.05 −0.07 −0.07 −0.08 −0.07 Adj. t-statistics (−2.63) (−4.39) (−4.89) (−5.36) (−4.92) Adj. R2 0.1% 2.6% 3.3% 4.1% 3.6%

Two-year Correction Coef. −0.15 −0.18 −0.20 −0.19 −0.14 Adj. t-statistics (−3.89) (−5.30) (−5.95) (−5.86) (−4.02) Adj. R2 4.0% 7.5% 9.6% 9.7% 5.0%

Three-year Correction Coef. −0.25 −0.29 −0.31 −0.25 −0.16 Adj. t-statistics (−4.49) (−5.72) (−6.05) (−4.70) (−2.82) Adj. R2 8.0% 12.8% 14.6% 9.8% 3.9%

Four-year Correction Coef. −0.37 −0.41 −0.37 −0.29 −0.19 Adj. t-statistics (−5.04) (−6.02) (−5.25) (−3.78) (−2.29) Adj. R2 13.2% 18.5% 15.3% 8.9% 3.6%

Five-year Correction Coef. −0.44 −0.42 −0.37 −0.28 −0.21 Adj. t-statistics (−4.92) (−4.87) (−3.99) (−2.88) (−2.03) Adj. R2 15.9% 16.2% 12.0% 6.9% 3.8%

Notes: Percentage changes of exchange rates over a one-, two-, three-, four- or five-year period are regressed against percentage exchange rate deviations from PPP accumulated over a period between three and seven years. Overlapping monthly observations between January 1974 and December 2009 are collected for each currency pair. We stack cross-sectional observations across all currency pairs to form pooled observations and conduct OLS regression tests. We repor estimates for slope coefficients, t-statistics adjusted for overlapping observation and adjusted R-squares. Adjusted t-statistics are obtained by dividing standard t- statistics by the square root of the observation period in the dependent variable, an approach suggested by Hjalmarsson (2006). Intercepts are forced to be zero for all regression estimations.

t

Table 4 Exchange rate corrections to cumulative deviations from PPP: Cochrane-Orcutt regression tests.

Statistics Three-yr Deviation Four-yr Deviation Five-yr Deviation Six-yr Deviation Seven-yr Deviation

One-year Correction Coef. −0.36 −0.34 −0.32 −0.32 −0.32 Adj. t-statistics (−11.09) (−11.70) (−10.89) (−10.97) (−10.48) Adj. R2 15.9% 15.9% 14.6% 15.3% 14.7%

Two-year Correction Coef. −0.39 −0.39 −0.39 −0.38 −0.35 Adj. t-statistics (−8.53) (−8.52) (−8.41) (−8.12) (−7.10) Adj. R2 16.8% 17.3% 17.5% 17.1% 14.2%

Three-year Correction Coef. −0.43 −0.44 −0.44 −0.41 −0.43 Adj. t-statistics (−7.34) (−7.59) (−7.30) (−6.37) (−6.49) Adj. R2 18.9% 20.6% 20.0% 16.7% 17.9%

Four-year Correction Coef. −0.44 −0.45 −0.41 −0.43 −0.40 Adj. t-statistics (−6.45) (−6.51) (−5.72) (−5.66) (−4.94) Adj. R2 20.0% 21.0% 17.7% 18.0% 15.0%

Five-year Correction Coef. −0.45 −0.43 −0.45 −0.42 −0.44 Adj. t-statistics (−5.79) (−5.42) (−5.41) (−4.65) (−4.57) Adj. R2 20.8% 19.4% 20.1% 16.3% 16.6%

Notes: Percentage changes of exchange rates over a one-, two-, three-, four- or five-year period are regressed against percentage exchange rate deviations from PPP accumulated over a period between three and seven years. Overlapping monthly observations between January 1974 and December 2009 are collected fo each currency pair. We stack cross-sectional observations across all currency pairs to form pooled observations and conduct Cochrane-Orcutt regression test which are robust to serial correlation. We report estimates for slope coefficients, t-statistics adjusted for overlapping observation and adjusted R-squares. Adjusted t-statistics are obtained by dividing standard t-statistics by the square root of the observation period in the dependent variable, an approach suggested by Hjalmarsson (2006). Intercepts are forced to be zero for all regression estimations.

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r s

universally accepted. For this reason the tests were repeated on non-overlapping samples of pooled data for the time periods exhibiting the best predictions, namely a four year deviation period and a correction period of one to four years. As results from time series data can be sensitive to the starting period chosen, the tests were repeated using as many starting dates as the data permitted.

For the one year correction period there are 12 different starting dates which can be used for the test, after the 12th month the data merely repeats itself, losing data points in the process. For the two year correction period there are 24 starting dates available, 36 are available for the three year correction period, and 48 for a four year correction period. Table 5 presents the results of these regressions. It gives the average for the non-overlapping regressions, and the range of values obtained in the individual regressions. It compares these with the pooled regression results. In this instance the intercepts of the regressions are not forced through zero in order to give the reader the opportunity to see the effect this has on the results. The intercepts are very close to zero but they are statistically different from zero. The effect on the slope of the regression is minimal. Comparing the pooled regressions in Table 5 with their equivalents in Table 3 shows the question of whether or not to force the intercept is not material to the conclusions. Durban-Watson statistics show the non-overlapping regressions have values close to 2, indicating serial correlation is not a problem. The Durban-Watson statistics of the overlapping samples show a very high serial correlation, which would have vastly overstated the t-statistics had they not been corrected. The results indicate correcting the t-statistics by dividing them by the square root of the number of overlaps was effective, and applying the Cochrane-Orcutt procedure on top of this, as was done in Table 4, results in overstating the t-statistics.

At this point one may ask why non-overlapping samples were not used throughout. Firstly, this would not have been possible on individual currency pairs as there would be too few data points, and establishing if the model works well on individual currency pairs as well as in aggregate is very important. Secondly, these non-overlapping pooled regressions have relatively few different dates represented in the data, and the longer the estimation and correction periods the worse the problem becomes. Thirdly, the use of overlapping samples is supported by Hansen and Hodrick (1980), Richardson and Smith (1991) and Torous et al. (2004) who demonstrate regressions using overlapping observations can increase the power of statistical tests when the number of observations available is insufficient for carrying out reliable statistical tests. In any event, the results of the non-overlapping regression tests indicate very little difference in the average results, from the ones using overlapping data.

3.7. Robustness check using PPI as an alternative price measure

The CPImeasures prices of both tradable and non-tradable goodswhile PPPworks through arbitrage of tradable goods between two nations. Using CPI as the inflation rate may introduce measurement error into the model. Although most of the studies on PPP use CPI as proxy for national price levels, Sarno and Chowdhury (2003) and Coakley et al. (2005) find that PPP works better when PPI is used as the proxy for price levels. Therefore, as a robustness check, we repeat the pooled regression tests done in the previous section using PPI in place of CPI. We report the results in Table 6.

The results in Table 6 suggest that substituting CPI for PPI does not change the conclusions drawn earlier. A negative relationship still exists between historic deviations from PPP and future exchange rate movements. However, the absolute values of regression slopes are steeper when PPI is used as the measure of inflation, indicating that exchange rates revert to PPP faster when PPI is used. This is driven by the PPI regression line being materially further from the intercept for PPI than CPI based data.

Table 5 Exchange rate corrections to cumulative deviations from PPP: Pooled estimates across various time intervals.

Statistics OLS regressions based on non-overlapping observations OLS regressions based on overlapping observations

Average for all regressions Range in values from different start dates

One-year Correction α1 Estimation 0.01 0–0.01 0.01 t-statistics 1.52 0.59–2.17 1.52 p-value 0.165 0.030–0.557 0.000

β1 Estimation −0.08 −0.09 to −0.05 −0.07 t-statistics −4.62 −5.45 to −3.02 −4.54 p-value 0.000 0.000–0.003 0.000

Adjusted R2 2.8% 1.1%–3.8% 2.8 Durban-Watson 2.00 1.82–2.16 0.17

Two-year Correction α1 Estimation 0.01 0–0.02 0.01 t-statistics 1.37 0.64–2.10 1.39 p-value 0.206 0.037–0.523 0.000

β1 Estimation −0.18 −0.24 to −0.11 −0.18 t-statistics −5.35 −7.44 to −3.45 −5.40 p-value 0.000 0.000–0.001 0.000

Adjusted R2 7.6% 3.0%–13.5% 7.7% Durban-Watson 2.00 1.76–2.30 0.08

Three-year Correction α1 Estimation 0.01 0–0.03 0.01 t-statistics 1.20 0.27–2.04 1.23 p-value 0.273 0.043–0.787 0.000

β1 Estimation −0.30 −0.48 to −0.19 −0.29 t-statistics −5.86 −7.96 to −3.86 −5.77 p-value 0.000 0.000–0.000 0.000

Adjusted R2 13.2% 6.3%–21.6% 13.0% Durban-Watson 2.00 1.76–2.22 0.07

Four-year Correction α1 Estimation 0.01 −0.01–0.03 0.01 t-statistics 0.85 −0.81–1.91 0.93 p-value 0.353 0.059–0.900 0.000

β1 Estimation −0.40 −0.54 to −0.19 −0.40 t-statistics −5.94 −8.88 to −2.95 −6.01 p-value 0.000 0.000–0.004 0.000

Adjusted R2 17.9% 4.0%–32.5% 18.4% Durban-Watson 1.92 1.75–2.03 0.05

Notes: Percentage changes of exchange rates over a one- two-, or three-, four year period are regressed against percentage exchange rate deviations from PPP accumulated over a period of four years. Monthly observations between January 1974 and December 2009 are collected for each currency pair. We stack cross- sectional observations across all currency pairs to form a pooled sample. The OLS regressions using non-overlapping data and are repeated 12, 24, 36 or 48 times using consecutive monthly start dates. We report estimates for slope coefficients, t-statistics and adjusted R-squares. T-statistics reported for overlapping regressions have been adjusted by dividing standard t-statistics by the square root of the observation period in the dependent variable, an approach suggested by Hjalmarsson (2006). Intercepts are not forced to be zero for any of the regression estimations.

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When the intercept of the regression is forced through zero, the difference between the two measures of inflation in not material. It would have been possible to present the data in Tables 1–4 using PPI as the measure of inflation without materially altering the results. However, when making predictions for individual currency pairs CPI gave slightly more predictable results and was therefore used in the predictive model as financial decisions are made based on individual currency pairs rather than on aggregate currency movements.

3.8. Out-of-sample predicting power of PPP deviations

Having shown that exchange rate deviations from PPP provide in-sample predictive power we will test our model's performance for out-of-sample predictions. In this section, we evaluate the out-of-sample predictive power of Eq. (1) against that of the randomwalk model using the procedure developed by Clark andWest (2006). In particular, we compute the mean squared prediction errors (MSPE) for both models for one-step-ahead predictions of exchange rate movements between each currency pair. We then compare the two sets of MSPEs obtained from the two models to determine which model predicts future exchange rate movements more accurately.

For each model, we use the first T observations of the (2T+1) observations in our complete sample to obtain the first estimateof coefficient β1 using OLS regression of Eq. (1) while forcing a zero intercept. We then use this estimate of β1 in Eq. (1) with the (T+1)th observation on exchange rate deviation from PPP as input to obtain the first out-of-sample prediction of exchange ratemovement for the following period. Next, we obtain the second estimate of coefficient β1 using OLS regression of Eq. (1) and the first (T+1) observations as inputs. We then use the second estimate for β1 and the (T+2)th observation of PPP deviation in Eq. (1) to obtain the second out-of-sample prediction of the exchange rate movement. We repeat this procedure recursively adding one more observation each time for the β1 estimation. This gives T out-of-sample predictions for exchange rate movements.

Table 6 Robustness check on pooled estimates across various time intervals using alternative price index PPI.

Statistics OLS regressions based on non-overlapping observations OLS regressions based on overlapping observations

Average for all regressions Range in values from different start dates

One-year Correction α1 Estimation −0.50 −0.78 to −0.29 −0.52 t-statistics −1.38 −2.15 to −0.75 −1.45 p-value 0.202 0.032–0.453 0.000

β1 Estimation −0.28 −0.31 to −0.23 −0.28 t-statistics −11.73 −13.25 to −9.29 −11.69 p-value 0.000 0.000–0.000 0.000

Adjusted R2 16.0% 10.6%–19.6% 16.0% Durban-Watson 1.67 1.46–1.94 0.12

Two-year Correction α1 Estimation −1.21 −1.79 to −0.56 −1.19 t-statistics −1.84 −2.75 to −0.77 −1.76 p-value 0.117 0.006–0.444 0.000

β1 Estimation −0.44 −0.54 to −0.15 −0.44 t-statistics −9.87 −12.79 to −2.80 −9.97 p-value 0.000 0.000–0.005 0.000

Adjusted R2 22.2% 1.9%–31.9% 22.4% Durban-Watson 1.51 1.00–1.88 0.07

Three-year Correction α1 Estimation −1.90 −2.54 to −1.29 −1.96 t-statistics −2.01 −2.67 to −1.43 −2.02 p-value 0.057 0.008–0.155 0.000

β1 Estimation −0.59 −0.89 to −0.12 −0.59 t-statistics −9.63 −18.11 to −1.68 −9.18 p-value 0.005 0.000–0.095 0.000

Adjusted R2 29.3% 0.8%–59.0% 27.6% Durban-Watson 1.46 1.02–1.86 0.04

Four-year Correction α1 Estimation −2.72 −3.75 to −1.70 −2.70 t-statistics −2.10 −2.69 to −1.59 −2.04 p-value 0.042 0.008–0.115 0.000

β1 Estimation −0.68 −0.96 to −0.27 −0.73 t-statistics −8.66 −23.45 to −2.35 −8.49 p-value 0.000 0.000–0.020 0.000

Adjusted R2 29.2% 2.8%–77.2% 31.3% Durban-Watson 1.17 0.90–1.44 0.03

Notes: Percentage changes of exchange rates over a one- two-, or three-, four year period are regressed against percentage exchange rate deviations from PPP accumulated over a period of four years. PPI is used to compute PPP rates. Monthly observations between January 1974 and December 2009 are collected for each currency pair. We stack cross-sectional observations across all currency pairs to form a pooled sample. The OLS regressions using non-overlapping data and are repeated 12, 24, 36 or 48 times using consecutive monthly start dates. We report estimates for slope coefficients, t-statistics and adjusted R-squares. t-statistics reported for overlapping regressions have been adjusted by dividing standard t-statistics by the square root of the observation period in the dependent variable, an approach suggested by Hjalmarsson (2006). Intercepts are not forced to be zero for any of the regression estimations.

Statistics

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Next, we use the method described in Clark and West (2006) to compute the adjusted squared prediction error of the predictive model Eq. (1), for the kth prediction P(ΔSk) using Eq. (4):

σ2k–adj: = △Sk−P △Skð Þ½ �2− P △Skð Þ½ �2 ð4Þ

We also compute the squared prediction error for the kth prediction from the random walk model using Eq. (5):

σ2k1 = △Skð Þ2: ð5Þ

According to Clark and West (2007), the difference between the two squared errors for the kth prediction obtained from our prediction model and the random walk model can be calculated as:

fk = σ 2 k1−σ

2 k–adj: = △Skð Þ2− △Sk−P △Skð Þ½ �2 + P △Skð Þ½ �2: ð6Þ

We repeat these computations on each of the T out-of-sample predictions produced by both models and obtain T observations of the differential MSPEs, that is, T observations of fk. Finally, we conduct a t-test on the fk series. The random walk model is rejected in favour of our predictable model in the form of Eq. (1) if fk is statistically significantly greater than zero. Conversely, deviation from PPP is deemed to have no predictive power for future exchange rate movements if the fk series is centred on or below zero. We apply the test described above using a four-year estimation period and predictive periods from one to four years. Results of the mean value between MSPEs of the two models, together with statistics from t-test are reported in Table 7 for each currency pair.

Table 7 Results of out-of-sample tests against the random walk model.

Currency Pair A$ C$ JPY GM NZ SF UKP US$

A$ 0.034 1E−5 0.015 0.002 0.028 0.119 0.143 (9.38) (7.34) (6.55) (3.60) (6.09) (5.42) (8.04) 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000***

C$ 0.031 1E−5 0.022 0.035 0.042 0.101 0.084 (8.99) (9.39) (10.46) (12.46) (10.01) (10.76) (10.56) 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000***

JPY 411.23 832.60 388.99 350.99 374.91 3187.71 786.05 (5.03) (9.19) (13.00) (12.96) (12.10) (10.98) (9.11) 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000***

GM 0.028 0.046 2E−5 0.020 0.349 0.009 0.213 (5.61) (11.14) (13.57) (10.38) (5.33) (13.21) (12.74) 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000***

NZ 0.004 0.119 3E−5 0.033 0.059 0.528 0.476 (3.10) (12.02) (11.20) (9.85) (9.30) (10.95) (11.24) 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000***

SF 0.024 0.037 8E−6 0.004 0.016 0.148 0.104 (4.88) (10.12) (12.44) (12.76) (9.69) (12.88) (12.26) 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000***

UKP 0.005 0.005 3E−6 0.006 0.006 0.011 0.015 (4.65) (9.86) (11.75) (13.44) (10.95) (11.25) (13.85) 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000***

US$ 0.024 0.029 5E−6 0.025 0.034 0.033 0.088 (6.55) (10.06) (8.083) (14.69) (13.50) (12.05) (13.46) 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000*** 0.000***

Notes: Differences between theMSPEs obtained from the predictable model of Eq. (1) are compared to those obtained from the randomwalkmodel as described in Clark andWest (2006, 2007). The t-statistics and p-values from standard t-tests are also reported. The t-tests have a null of a zero mean for the f series defined by Eq. (6). Asterisk *** denotes for statistical significance at the one percent level.

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The results of this robustness check show the predictive model Eq. (1) out-performed the random walk model for exchange rates for all 56 currency pairs. This indicates that historical deviations from PPP are useful for predicting future exchange rate movements.

4. Discussion of results

The evidence produced here provides a quite compelling picture of an exchange rate mechanismwhereby PPP has a significant influence in determining exchange rates. By basing our predictions on long term historical deviations, rather than the shorter periods traditionally employed, we have demonstrated purchasing power parity has a predictable and sizable influence on future exchange rates inmarket based economies with freely floating exchange rates. As volatility results in exchange rates moving away from traditional values, changes in relative pricing between nations provide a driving force which influences the pricing of currencies and drives them in the direction which helps restore the historic relationships of prices for goods and services. Such a relationship has clearly been noted in the past, but we develop this into a useful predictive tool. The underlyingmechanism shown here is a slow process, whereby pricing pressure resulting from currency movements accumulates over a period of years and results in a partial reversal of the historic exchange rate movement. The slopes of the regressions indicate the reversal process can be seen in the first year, but continues to correct itself over a period of up to four years. The best method of predicting the future direction of exchange rate movement is to look at the deviation of a currency over a four period from the rate predicted by PPP. Forty percent of this historic deviation will be progressively reversed over the subsequent four years.

No attempt is made to assert the evidence is incontrovertible as the history of testing PPP is one of argument and counterargument centred on the validity of the statistical methods used. Our results are expected to be received no differently. The evidence presented is, however, quite compelling. The relationship is one everyone believes must exist, yet practical application of this relationship to produce a useful predictive tool has been tantalisingly hard to achieve. If the relationship shown in this paper is merely the result of errors in analysis, then they are an extraordinary set of errors. The relationship is shown to exist in all the currencies tested, and it does not result from choosing tightly specified time frame parameters, which could lead to claims of data mining. The period in which the pressure for a currency reversal accumulates was found to be anything from three to six years or beyond, with the strongest effect at four years. The presence of such historic deviations from PPP provides a predictor of the direction of future exchange rate movement over periods of up to four years. The pattern of a four year deviation and a four year reversal means less than eight exchange rate cycles can be present in the 36 year history of floating exchange rates examined. It is therefore hardly surprising tests on individual currency pairs cannot be proven statistically.

It is quite clear the model we have produced only explains a portion of exchange rate movements. However in many applications even a small degree of explanatory power can be highly useful. While the chances of winning a gamble on a single large currency transaction using the reversal pattern we have shown may be low, when multiple transactions and multiple currencies are involved the degree of predictive power demonstrated can have real economic benefits. For exporters and

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importers who make frequent foreign exchange transactions over years or decades even a small edge can yield large benefits over long periods of timewhere a selective hedging strategy is employed. For international portfolio investors knowingwhich currency exposures to hedge can give significant benefits evenwhen amodel of modest predictive power is used. Themodel has been tested on both foreign currency hedging strategies and international portfolio selection and found to yield worthwhile benefits, although we do not report these results here. Other situations which have not been testedmay also benefit from the predictive power of this model. Foreign currency speculators may be able to profit from it. Firms who borrow in foreign currencies may be able to use it to decide when to reduce exposure using swaps and when to remain exposed to the potential of favorable currency movements.

The model presented here has deliberately been left simple and it has been assumed all economies adjust at the same rate. It has been assumed large deviations are no more likely to be reversed than small deviations, and indeed this seemed to be the case when it was analysed. The problemwith introducing a sophisticated set of rules and filters to enhance the predictive power of the model is that without a compelling theory for why these enhancements should improve predictability, there is no reason to assume they will prove effective when applied to time periods other than the one from which they were derived. On the other hand, the simple model we present just assumes deviations from PPP which accumulate over time are slowly corrected over time in a predictable manner, and all market based economies with free floating currencies behave the same. The reversal principle on which our model is based has been a central theme of the literature for decades. Wemerely point out the pattern which is present, and its usefulness in predicting future exchange rates, in other words we show what should have always been obvious, but apparently was not.

Acknowledgements

We thank the anonymous reviewer for the very helpful comments, especially the suggestions for additional robustness tests which have strengthened the paper.

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