predictability of the us dollar index

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Predictability of the U.S. Dollar Index using a U.S.

export and import price index-based relative PPP model

Axel Grossmann & Marc W. Simpson

# Springer Science + Business Media, LLC 2009

Abstract We use U.S. export and import price indexes to construct a relative

purchasing power parity-based model of the nominal U.S. Dollar Index. The model

is successful in predicting the future direction of change in the U.S. Dollar Index

over a six-month period up to 68% of the time. Finally, the model, in combinationwith a simple linear, recursive technique, is able to statistically significantly

outperform the random walk in predicting the value of the U.S. Dollar Index at

terms of less than four months for the period from 1996 to 2005. The paper provides

important implications for investors who are interested in the direction of change in

the Dollars value, forecasting the level of the U.S. Dollar Index, as well as the

extent of over- and undervaluation of the U.S. Dollar, in general.

Keywords Relative PPP. Equilibrium Exchange Rate .Nominal U.S. Dollar Index .

Half-Lives . Forecasting

JEL Classification F31 . F47

1 Introduction

Multinational corporations and international investors alike have long been

interested in either speculating with or hedging foreign currency exposures. While

A. Grossmann (*)

Department of Accounting, Finance and Business Law, Radford University, Radford, VA 24142, USA

e-mail: [email protected]

M. W. Simpson

Department of Finance, Northern Illinois University, DeKalb, IL 60115, USA

e-mail: [email protected]

J Econ Finan (2011) 35:417433

DOI 10.1007/s12197-009-9102-6

Published online: 3 September 2009


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many speculators or hedgers are interested in the specific bilateral exchange rates

between two currencies, others may be interested, broadly, in the general level of the

U.S. Dollar. The general level of the U.S. Dollar is also of interest to policymakers,

as it influences the overall level of economic activity, the competitiveness of U.S.

businesses, and the prices that U.S. consumers pay.All of these market participants and policymakers would find it useful to have a

guide to the general equilibrium level of the Dollar that perhaps could be used in

predicting future changes in the Dollars value. Although the last two and a half

decades have produced innumerable attempts to understand the underlying behavior

of exchange rates, the academic literature is replete with studies that demonstrate the

inability of economic models to outperform a random walk in forecasting exchange

rates (e.g. Meese and Rogoff 1983; Mark 1995; MacDonald and Marsh 1997; Kilian

1999; Mark and Sul 2001; Qi and Wu 2003; Xu 2003; Nucci 2003).

Morey and Simpson (2001a), however, demonstrate that a relative purchasingpower parity (PPP)-based model, first proposed by Hakkio (1992), could be useful in

predicting the direction of change in several bilateral exchange rates. Given the

importance of the U.S. Dollar Index for investors and policymakers the aim of this

paper is to: 1) construct a relative PPP-based equilibrium index, or equilibrium

exchange rate (EER) for the U.S. Dollar over the post-Bretton Woods period (1973

to 2005) that could allow agents to determine the extent to which the nominal U.S.

Dollar is over- or undervalued; 2) to investigate, using two different methodologies,

if deviations from the constructed EER provide any information about the future

path of the U.S. Dollar. First, following the approach by Hakkio (1992) and Moreyand Simpson (2001a), probabilities of convergence of the nominal U.S. Dollar Index

towards its constructed EER are calculated. Probabilities higher than 50% would

indicate that the constructed EER provides investors with some information about

the future change of the U.S. Dollar. Second, a linear recursive forecasting technique

is utilized to determine the horizon over which such a relative PPP-based model is

able to outperform a pure random walk in forecasting the nominal U.S. Dollar Index

out-of-sample.

The paper provides the following interesting results. First, the constructed EER

seems to perform well as a model of the equilibrium level of the U.S. Dollar index as

we find half-lives of less than 0.60 years for the early post-Plaza Accord period.

Second, the constructed relative PPP-based EER is successful in predicting the

direction of change in the nominal U.S. Dollar Index up to 68.4% of the time, which

indicates that the EER is able to provide investors with some information about the

future movement of the value of the U.S. Dollar. Finally, we provide evidence that a

simple, linear, recursive technique that uses the EER is able to statistically

significantly outperform the random walk in predicating the value of the U.S.

Dollar index over terms of less than four months.

2 Background

PPP has often been suggested by economists as a means to determine the

fundamental value of a currency. Yet, the empirical evidence of the efficacy of

PPP, especially with respect to its validity in the medium- and short-run is mixed at

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best (e.g. Breuer 1994; MacDonald 1995; Froot and Rogoff 1995; Xu 2003).1 Other

more complex models that have been put forward to construct EERs, particularly

relating to effective exchange rates, are: 1) the fundamental equilibrium exchange

rate, which is based on simultaneous attainment of external and internal balance

(Williamson 1994; Montiel 1999; Wren-Lewis 2003), and 2) the natural realexchange rate, which is based on simultaneous attainment of balance of payments

equilibrium and a natural rate of unemployment (Stein 1994; Allen 1995; Montiel

1999). Although those more complex equilibrium models provide promising

alternatives to PPP (Montiel 1999), they are based on fundamentals for which data

are only available at quarterly or annually frequencies; hence, they provide little help

if one wants to determine the aggregate value of the Dollar at shorter frequencies.

Furthermore, Montiel (1999) acknowledged that:

[the more complex EERs] have yet not been shown to deliver the robustness

and precision that would be required for a nonstructural approach. .

Thus, according to Montiel (1999), despite its limitations, the PPP-based model

appears to be the most promising approach to constructing an EER.

A common way to test the appropriateness of an EER is to demonstrate that

exchange rate reverts towards the constructed equilibrium over time. While many

academic researchers focus on real exchange rates, this paper uses an alternative

representation of the relative PPP equilibrium to construct an equilibrium value of

the nominal U.S. Dollar Index over the post-Bretton Woods period. This alternate

model was first developed by Hakkio (1992).

3 The U.S. Dollar equilibrium exchange rate

Hakkio (1992) uses an approach based on the concept of relative PPP to establish an

EER, and defines deviations from PPP as the difference between the actual spot

exchange rate and the constructed EER. The difference between the use of the real

exchange rate and the Hakkio style equilibrium rate is that: with the real exchange

rate, one is concerned about the deviation of the real exchange rate from its long-run

mean, while, with the Hakkio relative PPP EER, one is focusing on the deviation of

the actual exchange rate from the implied equilibrium rate.

While Hakkio (1992), who investigates bilateral exchange rates, uses consumer

price indexes to account for the relative inflation rates in two countries, we use in

this paper the U.S. export price index in place of a domestic goods price index, and

we use the U.S. import price index in place of an aggregate foreign goods price

index that accounts for the price levels of all trading partners. By focusing on the

prices of traded goods exported from and imported into the U.S., we are more

accurately capturing the spirit of the PPP hypothesis, which states that only prices of

those goods that are traded can be arbitraged and therefore affect exchange rates.

Along those lines, Hinkle and Nsengiyumva (1999) suggested the use of export price

indexes as proxies for traded goods price indexes, instead of aggregate goods price

1 For example Rogoff (1999) summarizes that the general consensus in the literature provides evidence

that the half-lives of deviation of real exchange rates from its long-run value are about 4 to 5 years.

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indexes. Furthermore, Xu (2003) illustrates that the forecasts of a PPP model based

on traded goods indices are more accurate than those of models based on either CPI

or WPI.

One of the problems with relative PPP, which postulates that the percent change

in the rate of exchange between two countries currencies should be equal to theinflation rate differentials between the two countries, is that it does not tell us the

level of the EER. If, however, one assumes that the spot exchange rate is in

equilibrium at a specific point in time, one could construct the EER in the period t +

1 as follows:

EERt1 ffi St 1 pt p*t

; 1

where EERt+1 is the equilibrium exchange rate at time t+1,St is the nominal

exchange rate, andpt

and p

*t are the percentage changes in inflation of the domesticand foreign country, respectively.2

Instead of assuming that the exchange rate is in equilibrium at a given point in

time, which obviously biases the level of the EER, it is more realistic to assume that

PPP holds, on average, over a longer time span. Hence, Hakkio (1992) constructed

36 EER series using each of the 36 monthly observations in the base period from

1980 to 1982 as a starting point.3 After these 36 different equilibrium exchange rates

series are created, an average across all 36 series at each observation represents the

EER (see Appendix A for a more detailed explanation on how the equilibrium

exchange rate is constructed).

Of course, any such constructed EER could be biased by the choice of the base

period. Morey and Simpson (2001a, b), Simpson (2003), and Grossmann et al.

(2009) all use a Hakkio-style relative PPP-based equilibrium exchange rate like the

one used in this study. These researchers carry out their analyses, each using a

number of different base periods, and find little qualitative differences in the results

of their studies. For robustness, this study reports its main results utilizing different

base periods as well as a rolling base period. For example, the EER, using a rolling

base period of the entire post-Bretton Woods period, is created by constructing the

first EER using the base period from 1973 to 1975 and then moving the base period

forward by 1 year. Following this approach 31 EERs are constructed over the entiresample period. Finally, the average of the 31 EERs represents the EER using a

rolling base period spanning the entire post-Bretton Woods period.

Figure 1 shows the constructed relative PPP-based EER of the nominal U.S.

Dollar Index using the rolling base period spanning the entire post-Bretton Woods

period. Additionally, Fig. 2 graphs the percent misalignments of the nominal U.S

Dollar Index from its constructed EER. Both figures demonstrate the wide initial

misalignment of the U.S. Dollar after the collapse of the Bretton Woods system, as

2 In our case, the nominal exchange rate is the major nominal U.S. Dollar Index obtained from the Federal

Reserve web-page, and the inflation rate differentials are calculated using import and export price indexes

from DataStream, where they are quoted (?I74F) and (?I75F), respectively. The index is constructed

in such a way that, as the U.S. Dollar appreciates, the index level rises.3 Hakkio (1992) claimed that the world economy was in equilibrium during that time. Furthermore, the

period coincides with the beginning of the Reagan administration, during which the U.S. initially did not

intervene in the foreign exchange rate market.

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well as, the success of the Plaza Accord at the end of 1985 in bringing the highlyovervalued U.S. Dollar back to its equilibrium.

4 Predictability of the nominal U.S. Dollar Index

Lagged U.S. export and import price indexes and a rolling base period spanning the

period from 1973 to 1985 are utilized in the subsequent predictions of the nominal

U.S. Dollar Index. This ensures that we only use information available at the time

forecasts are made. Consequently, predictions are made over the entire post-PlazaAccord period (1986 to 2005), as well as the two sub-periods (1986 to 1995 and

1996 to 2005). For robustness, the analysis is also done using the base period from

1973 to 1975, as well as the base period from 1980 to 1982.

4.1 Mean reversion of the U.S. Dollar Index

If one wants to analyze the predictability of the nominal U.S. Dollar Index based on

a constructed EER, one needs to show the appropriateness of such a perceived

equilibrium. The literature generally defines a proper EER by the reverting behavior

of the exchange rate towards the constructed equilibrium.

The most common approach in the literature to show that any exchange rate

converges towards its equilibrium is to calculate half-lives using the estimated roots

60

70

80

90100

110

120

130

140

150

74 77 80 83 86 89 92 96 99 02 05

U.S. Dollar Index Rolling PPP-based EER

Fig. 1 U.S. Dollar Index and rolling PPP-based equilibrium

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

74 77 80 83 86 89 92 96 99 02 05

Fig. 2 U.S. Dollar Index percent under-and overvaluation

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of constructed real exchange rates from stationary tests or EngleGranger

cointegration tests. The literature reports half-lives of around 2 to 5 years using

consumer price indexes (e.g. Rogoff1996; Xu 2003); while Xu (2003), using traded-

goods price indexes and the roots from a trivariate EngleGranger cointegration test,

finds half-lives of around 1 year for some of the countries in his analysis. 4 In thisstudy, we suggest an alternative way to calculate half-lives, which does not rely on

any stationarity test of the real exchange rate. That is to say, we use the obtained

misalignments to calculate the mean lifetimes it takes the nominal U.S. Dollar Index

to converge to its constructed EER. Second, half-lives are derived from the

calculated mean lifetimes based on Eq. (2):

T1=2 ln2t; 2

where T1/2 represents the half-lives and t the mean lifetime (see Appendix B for a

detailed explanation how mean lifetimes can be converted into half-lives). Table 1reports the half-lives of the nominal U.S. Dollar Index over the whole Bretton

Woods period, as well as the four sub-periods. Remarkably, the half-lives are less

than 2.2 years for the entire post-Plaza Accord Period and even less than 0.6 years

for the sub-period from 1986 to 1995, which provides strong evidence in favor of the

relative PPP-model utilized in this study.

4.2 Probabilities

Hakkio (1992) and Morey and Simpson (2001a) show how one can use constructedPPP-based EERs to calculate the probability that a nominal exchange rate will

converge towards its equilibrium over a given future time period. A probability of

50% or higher would suggest that the constructed EER provides useful information

about the future movement of the nominal exchange rate, while any probability of

less than 50% would leave investors with the suggestion that any guess of the future

exchange rate path provides the same predictive power as PPP. Based on the

approach proposed by Hakkio (1992) the probabilities are calculated as follows:

PROB NOHEERNOLEERNTO

100;NOHEER stk < st and st > EERt ;NOLEER stk > st and st < EERt ;

3

where NOHEER represents the number of observations where nominal U.S. Dollar

Index (St) is higher than the EER and moves towards the EER over a given future

horizon; NOLEER represents the number of observations where the nominal U.S.

Dollar Index (St) is lower than the EER and moves towards the EER over a given

future horizon; while NTO represents the number of total observations.

4 There are two problems that arise with estimating roots of constructed real exchange rates from

stationary tests or EngleGranger cointegration tests. First, these tests seem to provide low power over

short-time periods, which makes it difficult to determine if the real exchange rate is stationary (Froot and

Rogoff 1995). Second, the trivariate EngleGranger cointegration tests fail to incorporate the symmetry

and proportionality restrictions, which are implied if PPP holds (e.g. Breuer 1994). Therefore, one may

conclude that the commonly used methodologies suggested in the literature do not provide appropriate

tools to test if PPP holds, and especially not during the post-Bretton Woods period.

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Furthermore, since the literature points out that there might be a threshold band

around the EER in between which forces of PPP are rather weak (Hakkio 1992;

Sercu et al. 1995; MacDonald 2000), we also calculate probabilities using only large

deviations from the EER. In this case one can expect that forces of PPP are stronger

when the exchange rate goes beyond the threshold band; hence, the success rate to

predict the future nominal exchange rate should increase. We define large deviations

as those which are larger than the moving average of the absolute values of the

misalignments over the past 5 years.

Table 2 reports the success rates of correctly predicting the direction of the future

spot exchange rates over the entire post-Plaza Accord Period (Panel A), the period

from 1986 to 1995 (Panel B), and the period from 1996 to 2005 (Panel C), over

horizons of 1 month, 3 months, and 6 months, respectively. Table 2, column 2,

Table 1 Half-lives of the U.S. Dollar Index. This table reports the half-lives it takes for the nominal

effective U.S. Dollar exchange rate to converge to the constructed equilibrium exchange rate. The half-

lives are calculated based on mean lifetimes as shown in the appendix

Number of Observations Half-Lives

73-05 (whole sample period) 396 2.20

73-85 (pre-Plaza Accord) 156 1.75

86-05 (post-Plaza Accord) 540 2.56

86-95 (post-Plaza Accord; first subperiod) 120 3.95

96-05 (post-Plaza Accord; second subperiod) 120 0.57

Table 2 Probability that the nominal U.S. Dollar Index will converge towards its EER. This table reports

the probabilities that the nominal U.S. Dollar Index will revert towards the constructed equilibrium

exchange rate over a given future time horizon. A probability higher than 50 percent indicates that the

constructed equilibrium exchange rate provides useful information about the future movement of the U.S.

Dollar Index

Future Time Horizon Rolling Base Period Rolling Base Period Base Period Base Period19731984 19731984 Large 19731975 19801982

Panel A: 1986 to 2005

1 month 51.0% 50.0% 51.5% 54.8%

3 months 58.2% 61.2% 54.4% 61.2%

6 months 59.4% 62.1% 55.6% 63.2%

Panel B: 1986 to 1995

1 month 47.5% 50.0% 45.0% 51.7%

3 months 53.3% 65.5% 50.8% 60.8%

6 months 50.8% 67.2% 48.3% 61.7%

Panel C: 1996 to 2005

1 month 54.6% 50.0% 58.0% 58.0%

3 months 63.2% 55.0% 58.1% 61.5%

6 months 68.4% 54.1% 63.2% 64.9%

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shows that the success rates are all higher than 50% for all three sub-periods, with an

exception for the 1-month prediction horizon during the 1986 to 1995 period.

Moreover, during the second half of the post-Plaza Accord period (Panel C), the

success rate reaches over 68% for the 6-month prediction horizon. This means, for

example, that once the U.S. Dollar is undervalued one may be able to predict that thevalue of the Dollar will increase over the next 6 months. Focusing only on large

misalignments increases the success rate even further, with exception for the period

from 1996 to 2005 (column 3). Columns 4 and 5 indicate that the calculated

probabilities are robust with respect to the choice of different base periods, since the

success rates differ only slightly from those reported in column 2.

4.3 The recursive linear forecasting methodology

The forecasting methodology in this study encompasses two main steps. The firststep relates the change of the actual nominal U.S. Dollar Index to the misalignment

of the nominal U.S. Dollar Index in the previous period. The second step simply

models the misalignments as an AR(1) process.

For example, to find the one-month ahead forecast of the nominal U.S. Dollar

Index at any time, where t is any date in between and including December 1985

through November 2005, the following OLS regression is fit using all of the

available monthly observations from January 1973 to time t:

st st1 ab st1 eerPPP

t1 "t; 4where st is the log of the nominal U.S. Dollar Index at time t, and eer

PPPt is the log of

the relative PPP-based EER at time t. Given the coefficients from Eq. (4), the one-

month ahead forecast at time of the nominal U.S. Dollar Index, denoted Et[s

t+1], is:

Et st1 st bat bbt st eerPPPt h i; 5where ba

t is the intercept coefficient from estimating Eq. (4) using all available

observations from January 1973 up to, and including, those at time t, and bbt is theslope coefficient from estimating Eq. (4) using all available observations from

January 1973 up to, and including, those at time t.

To forecast the one-month ahead forecast at time t + 1, i.e., Et+1[st+2], Eq. (4) is

re-estimated using all available observations from January 1973 to time t + 1, and

the variables substituted into Eq. (5), with updated time subscripts.

The forecast of the n-month ahead nominal U.S. Dollar Index is done via an n-step

process where forecasts at later steps depend on forecasts generated in earlier steps. For

example, to forecast the two-month ahead U.S. Dollar Index at time t one must first

forecast the one-month ahead U.S. Dollar Index at time t. In addition to the forecastof the U.S. Dollar Index from earlier steps, the model also requires a forecast of the

nominal U.S. Dollar Index misalignments. Therefore, we fit the following OLS

regression using all available observations from January 1973 to time t:

st eerPPPt m g st1 eer

PPPt1

ht: 6

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Given the coefficients from Eq. (4), the two-month ahead forecast at time t,

denoted Et[s

t+2], is:

Et st2 Et st1 bat

bb

tEt st1 eer

PPPt1 h i

; 7

where the one-month ahead forecast of the nominal U.S. Dollar Index misalign-

ments, Et st1 eerPPPt1

, is found by using the coefficients from Eq. (6), as follows:

Et st1 eerPPPt1

bmt bgt st eerPPPt : 8

where bmt is the intercept coefficient from estimating Eq. (6) using all availableobservations from January 1973 up to, and including, those at time t, and bgt is theslope coefficient from estimating Eq. (6) using all available observations from

January 1973 up to, and including, those at time t.To forecast the nominal U.S. Dollar Index, three, or more, months into the future,

the process becomes one of recursively substituting the appropriate coefficient

estimates and previous forecasts into the following two formulas:

Et stn Et stn1 bat bbtEt stn1 eerPPPtn1 h i; for all n ! 3; 9and

Et

stn

eerPPPtn bmt bgtEt stn1 eerPPPtn1 ; for all n ! 3: 10

Given that the nominal U.S. Dollar Index has been converted into logs for the

estimation procedures, before calculating the forecast error, we first convert the

forecasts back into common numbers by taking the antilog of the forecast:

Et Stn eEt stn for all n 1; 2; 3; . . . 24: 11

Next, for each forecast, we calculate the n-month ahead squared error of the

relative PPP-based model:

SEPPPtn Stn Et Stn

2

for all n 1; 2; 3; . . . 24: 12

Finally, using all of the n-month ahead squared errors in each sample period we

compute the mean squared error (MSE):

MSEPPPtn Xxi1

SEPPPtn

i

xfor all n 1; 2; 3; . . . 24; 13

where SEPPPtn

iis the squared error of the ithn-month ahead forecast of the PPP-

based model in the sample period containing x such forecasts.

The forecast accuracy measured by the mean squared errors (MSE) of the aboverelative PPP-based model is compared to that of a nave random walk. If the nominal

U.S. Dollar Index follows a random walk, then one would expect the value of future

U.S. Dollar to be simply the value of the exchange rate today:

ERWt Stn St; for all n 1; 2; 3; . . . 24: 14

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The squared error MSE of the random walk model is given by Eqs. (15) and (16)

SERWtn Stn St 2; for all n 1; 2; 3; . . . 24; 15

MSERWtn Xxi1

SERWtn

i

x: 16

The comparisons of the forecast accuracy of the relative PPP-based model and the

random walk are structured in such a way that all of the out-of-sample tests have the

same number of observations. Thus, the estimations of the one-month forecasts

begin with the one-month forecast generated, given all of the data up to, and

including, December 1985 (thus, it is a forecast of the exchange rate in January

1986), while the two-month forecasts begin with the forecast generated, given all ofthe data up to, and including, November 1985 (thus, it also is a forecast of the

exchange rate in January 1985), and so forth.

Tables 3 and 4 report the percent difference between the MSE of the relative PPP-

based model against the random walk model, which is measured as follows:

%DMSE MSEPPP MSERW

MSERW; 17

whereMSEPPP represents the MSE of the forecast from the relative PPP-based model,

and MSE

RW

is the MSE of the random walk model for the specific time period andforecast horizon. By construction, a negative percent change MSE indicates that the

relative PPP-based model provides better forecast accuracy than the random walk

model. Tests of statistical significance of the percent change MSE are performed

using a one tailed t-test with the null hypothesis that the MSEs are equal for the

relative PPP-based model and the random walk model, against an alternate hypothesis

that the MSEs of the relative PPP-based model are less than the MSEs of the random

walk model. The forecasts are made over horizons of 1 month to 24 months.

Table 3 illustrates that for the period immediately following Plaza Accord (1986

to 1995) the relative PPP-based model is not able to outperform the random walk

model over any of the 24-month horizon. The results do not change if we use the

1973 to 1975 base period (column 3) or the1980 to1982 base period (column 4)

instead of the rolling base period spanning from 1973 to 1985 (column 2).

Table 4 reports the percent change MSE covering the period from 1996 to 2005,

which provides very encouraging results. Using the rolling base period (column 2), the

relative PPP-based model statistically significantly (at the 5% level) outperforms the

random walk model at terms of less than 4 months. Columns 3 and 4 of Table 4

demonstrate that the finding is fairly robust with respect to the choice of the base period.

5 Conclusion

Market participants and policymakers have long searched for a tool to determine the

equilibrium value of the U.S. Dollar. In this paper, we investigate a relative PPP-

based model using U.S. import and export price indexes. Moreover, while the literature

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on exchange rate forecasting has found that models based on pure macroeconomic

fundamentals are not very helpful in providing information for the path of exchange

rates, this paper reports promising results for the nominal U.S. Dollar Index.

The constructed EER provides strong evidence in favor of the relative PPP-model, as the half-lives deviations from the equilibrium are less than 2.2 years for

the entire post-Bretton Woods period. Notably, half-lives are less than 0.60 years for

the later post-Plaza Accord period.

This paper illustrates that the constructed relative PPP-based EER is able to

provide a rule of thumb for investors in predicting the future movements of the U.S.

Table 3 Percent difference in forecast errors for 1986 to 1995. This table reports the percent difference in

mean squared errors of the out-of sample forecasts of the export and import price index PPP-based model

and the mean squared errors of the random walk model. A negative number indicates that the PPP-based

model has a lower mean squared error by the given percentage. The tests for statistical significance come

from a one tailed T-test with a null hypothesis that the mean squared errors are equal for the PPP-based

model and for the random walk, against an alternate hypothesis that the mean squared errors of the PPP-based model are less than the mean squared errors of the random walk model

Forecast Horizon Rolling Base Period Base Period Base Period

19731983 19731975 19801982

19861995 19861995 19861995

1 month 0.86 0.73 4.54

2 months 1.16 1.04 6.59

3 months 1.36 1.19 8.71

4 months 1.51 1.34 11.13

5 months 1.74 1.54 13.68

6 months 2.02 1.68 16.73

7 months 2.86 2.32 21.04

8 months 4.07 3.32 26.00

9 months 5.10 4.16 30.68

10 months 7.70 6.47 38.94

11 months 10.04 8.62 47.17

12 months 11.11 9.68 52.32

13 months 11.67 10.33 56.21

14 months 12.11 10.83 59.62

15 months 12.69 11.46 64.09

16 months 13.26 12.10 68.66

17 months 13.75 12.70 72.83

18 months 14.32 13.42 77.76

19 months 14.91 14.21 82.07

20 months 15.33 14.85 85.87

21 months 15.78 15.51 89.31

22 months 16.40 16.27 92.78

23 months 16.91 16.88 96.18

24 months 17.28 17.35 99.89

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Dollar Index, as the rate of success in predicting the future direction of the nominal

U.S. Dollar Index is more than 68% for the 6-month term, during the 1995 to 2005

period. Furthermore, the paper provides evidence that a simple, linear, recursive

technique applied to the relative PPP-based model is able to statistically significantlyoutperform the random walk in predicting the value of the U.S. Dollar Index at

horizons of less than 4 months.

Consequently, this paper fills an important void in the literature with respect to the

misalignment of the U.S. Dollar Index, as well as predictability of its future path.

This information may be important for any participant in the foreign exchange rate

Table 4 Percent difference in forecast errors for 1996 to 2005. This table reports the percent difference in

mean squared errors of the out-of sample forecasts of the export and import price index PPP-based model

and the mean squared errors of the random walk model. A negative number indicates that the PPP-based

model has a lower mean squared error by the given percentage. The tests for statistical significance come

from a one tailed T-test with a null hypothesis that the mean squared errors are equal for the PPP-based

model and for the random walk, against an alternate hypothesis that the mean squared errors of the PPP-based model are less than the mean squared errors of the random walk model

Forecast Horizon Rolling Base Period Base Period Base Period

19731983 19731975 19801982

19962005 19962005 19962005

1 month 0.73 0.55 0.66

2 months 1.67 1.43 1.56

3 months 2.59a 2.22 2.44a

4 months 3.54b 3.07a 3.37a

5 months 4.55b 3.96b 4.41b

6 months 5.62c 4.84b 5.55c

7 months 6.56c 5.54c 6.56c

8 months 7.31c 6.06c 7.34c

9 months 7.97c 6.54c 8.02c

10 months 8.56c 7.04c 8.63c

11 months 9.17c 7.72c 9.30c

12 months 9.81c 8.46c 10.00c

13 months 10.34c 9.14c 10.61c

14 months 10.88c 9.74c 11.23c

15 months 11.51c 10.40c 11.98c

16 months 12.12c 11.04c 12.71c

17 months 12.59c 11.64c 13.28c

18 months 12.99c 12.18c 13.80c

19 months 13.28c 12.66c 14.18c

20 months 13.55c 13.12c 14.54c

21 months 13.87c 12.60c 14.97c

22 months 14.19c 14.06c 15.41c

23 months 14.41c 14.37c 15.75c

24 months 14.59c 14.62c 16.07c

a, b, c indicate statistical significance at the 10%, 5%, and 1% levels, respectively

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market (i.e. investors trading U.S. Dollar Index future contracts or exchange traded

funds based on these contracts) and to foreign and domestic policymakers.

Appendix A

Construction of the Hakkio based equilibrium exchange rate

To illustrate exactly how the relative PPP equilibrium rate is constructed, assume we

use the nominal U.S. Dollar Index in January 1980 of 94.750. Now further assume

the inflation for the U.S. based on the U.S. export price index was 3.5% and the

inflation for the foreign countries based on the U.S. import price index was 2.4%. In

this case, the implied change in the relative PPP equilibrium exchange rate between

January and February 1980 would be (3.5%2.4%) = 1.1 percent, meaning that the

relative PPP equilibrium exchange rate for the U.S. Dollar Index for February 1980

is 95.792 [i.e., 94.750 (1 + 0.0350.024) = 95.792]. One could then use the

percent changes in the U.S. export price index and the U.S. import price index to

construct the implied relative PPP equilibrium rate for each month in the entire

sample period. The next step would be to construct 36 such equilibrium rate series

using different actual exchange rates as the starting point. For example, the next

actual exchange rate to be used would be the U.S. Dollar Index in February 1980.

Once these 36 different equilibrium exchange rate series were created one could take

an average across all 36 series at each observation to come up with the equilibrium

exchange rate. Note that once the level of the equilibrium exchange rate is establishedin the base period, the only thing that causes a change in the equilibrium exchange rate

in periods subsequent to the 36-month base period is the difference in the percent

change of the U.S. export and import price index. That is, once the base period is used

to establish the level of the equilibrium exchange rate, any subsequent changes in the

level of the equilibrium exchange rate is driven entirely by relative inflation rates.

1 2 3 35 36 Average

Jan-73 / 36

Feb-74 / 36

/ 36

/ 36

Jan-80 94.750 93.314 91.663 / 36

Feb-80 95.792 94.375 92.706 / 36

Mar-80 100.250 98.941 97.191 / 36

99.818 / 36

118.527 117.624 / 36

Nov-82 117.435 116.541 / 36

Dec-82 116.568 115.686 / 36/ 36

/ 36

Nov-05 / 36

Dec-05 / 36

Fig. 3 Construction of the relative equilibrium exchange rate

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Appendix B

Calculating half-lives5

A general approach in the literature to calculate half-lives is to obtain the coefficientfrom an EngleGranger cointegration test. This can be done in two different ways.

One can test whether the real exchange rate follows a stationary process or if the

residuals of the regression of the log spot exchange rates on domestic and foreign

log price indexes are stationary. In either case, the following relationship is obtained:

M0=2

M0 bT; A:1

where Trepresents the half-life, andM0=2M0

is the ratio with which the deviations from

the exchange rate decay on average to a stationary process over the half-life T.Rearranging Eq. (A.1) allows one to derive the half-life from the obtained

coefficient as shown in Eq. (A.2):

T ln 0:5

ln b : A:2

For example, if is equal to 0.7071 the half-life (T) would be two time periods

(2t), if is equal to 0.5 the half-life (T) would be one time period (1t), and if is

equal to 0.25 the half-life (T) would be half a time period (1/2t), and so on.

The general decaying pattern can be graphed as shown in Fig. 4, where one canderive the following relationship:

M M02tT : A:3

Further, assuming that T = 2tand substituting it into Eq. (A.3) one gets after one

time period or half a half-life the following:

M

M0 2

12 0:7071: A:4

This means that after one-time period, or half a half-life, the deviations decay onaverage to 70.71% of their previous values, which is the same as obtaining a of

0.7071. This can easily be shown by substituting = 0.7071 and T = 2 into

Eq. (A.1).

bT M0=2

M0 0:70712 0:5 A:5

Moreover, the change in the decay of the exchange rate deviations from its

equilibrium in any given time interval t can be expressed as follows:

DM lMDt; A:6

5 Part of the presented material in this appendix, which follows the approach of calculating the decaying of

particles in chemistry, can be found at the following web-page:http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/meanlif.html

4 J Econ Finan (2011) 35:41743330
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where 1 is the decaying constant. Writing Eq. (A.6) in differential form one gets:

dM lM dt ordM

M ldt: A:7

Integrating Eq. (A.7) leads to lnM lt C and taking the exponent of bothsides one gets:

M eCelt A:8

with M

eC

M

0at time t = 0, Eq. (A.9) is derived:

M M0elt: A:9

Given Eqs. (A.3) and (A.9) one can write the following:

M

M0 2

tT elt: A:10

Taking the logarithm of Eq. (A.10) and rearranging it leads to the following

expression:

1l

Tln2 : A:11

Calculating the mean lifetime

Given Eq. (A.9) the probability of decay can be expressed as the following

distribution function:

fdecayt Melt: A:12

To normalize this distribution function:

Z10

fdecaytdt

Z10

Meltdt 1

lMelt j

1

0

M

l 1 thus; M l A:13

The probability of decay within a time period t is given by the integral of the

decay distribution function from 0 to t. To obtain the mean lifetime, however, one

0 1 2 3 4 5 6 T

Time as a multiple of the halflife T

AbsoluteMisalignment

M0

M0/2

M0/8

M0/16M0/32

M0/4

Fig. 4 Graph of decaying absolute misalignments

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needs to calculate the average time the deviation will exist without decaying. Hence,

the probability of no decay at time t can be expressed as follows:

Pt 1 Zt

0le

lt

dt 1 elt

j

t

0 elt

: 14

The average survival time is now the mean value (mean lifetime) of the above

probability:

t

Z10

teltdt: 15

Integrating by parts:

d (uv) = udv + vdu

u = t so that du = 1 and dv eltdt so that v 1l

elt

t

Z10

t eltdt 1

lelt j

1

0

Z10

1

leltdt

1

lelt j

1

0

1

l: A:16

Substituting Eq. (A.16) into Eq. (A.11) one can calculate mean lifetimes as shown

in Eq. (A.17):

t Tln2

: A:17

Rearranging Eq. (A.17) allows the calculation of the half-lives, once the mean

lifetime of the absolute misalignments is calculated:

T ln2t: A:18

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