In this paper, we estimate (by maximum likelihood) the parameters of univariate fractionally integrated real exchange rate time series models, and test for autoregressive unit roots on the alternative of a covariance stationary long-memory process. We use quarterly dollar-based real exchange rates (since 1957) for seventeen OECD countries, and that the finding of unit autoregressive roots does not go away even with this more sophisticated alternative.Foreign exchange rates ; Time-series analysis

Apostolos Serletis

Michael Dueker

English

Research Papers in Economics

WORKING PAPER SERIES
Do Real Exchange Rates have Autoregressive 
Unit Roots?  A Test under the Alternative 
of Long Memory and Breaks
Michael Dueker
Apostolos Serletis
Working Paper 2000-016A
http://research.stlouisfed.org/wp/2000/2000-016.pdf
July 2000 
FEDERAL RESERVE BANK OF ST. LOUIS
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Photo courtesy of The Gateway Arch, St. Louis, MO.   www.gatewayarch.comDOREAL EXCHANGERATES HAVE AUTOREGRESSIVE UNIT
ROOTS? A TEST UNDER THEALTERNATIVE OFLONG
MEMORYANDBREAKS
June 2000
Abstract
In this paper, we estimate (by maximum likelihood) the parameters of univariate fractionally
integrated real exchange rate time series models, and test for autoregressive unit roots on the
alternative of a covariance stationary long-memory process. We use quarterly dollar-based real
exchange rates (since 1957) for seventeen OECD countries, and that the finding of unit
autoregressive roots does not go away even with this more sophisticated alternative.
KEYWORDS: Fractional integration, Long memory processes,Real exchange rates
JEL CLASSIFICATIONS: C22
Michael Dueker Apostolos Serletis
Federal ReserveBank ofSt. Louis Department ofEconomics
ResearchDepartment The University ofCalgary
P.O. Box 442 Calgary, AlbertaT2N 1N4
St. Louis, MO 63102 (403)220-4091
(314) 444-8594 (403)282-5262—fax
dueker@stls.frb.org serletis@acs.ucalgary.caI. INTRODUCTION
The theoryof purchasing powerparity (PPP) has attracted a greatdeal ofattention and
has beenexplored extensively in the recent literature using recent advances in the fieldof applied
econometrics. Based on the law ofone price, PPP asserts that relative goods prices are not
affected by exchange rates -- or, equivalently, that exchange rate changes will be proportional to
relative inflation. The relationship is important not only because it has beena cornerstone of
exchange rate models in international economics, but also because ofits policy implications -- it
provides a benchmark exchange rate and hence has some practical appeal for policymakers and
exchange rate arbitragers.
Although purchasing power parityhas been studied extensively, empirical studies
generally fail to find support for long-run PPP, especially during the recent floating exchange rate
period. In fact, the empirical consensus is that PPP does not hold over this period -- see for
example, Adler and Lehman (1983),Mark (1990), Grilli and Kaminski (1991), Flynn and
Boucher (1993), Serletis (1994), and Serletis and Zimonopoulos (1997). But there are also
studies covering different groups ofcountries [see Phylaktis and Kassimatis (1994)] as well as
studies covering periods oflong duration [see Lothian and Taylor (1996) and Perron and
Vogelsang (1992)] or country pairs experiencing large differentials in price movements [see
Frenkel (1980) and Taylor and McMahon (1988)] that reportevidence ofmean reversion towards
PPP.
A sufficient condition for a violation ofpurchasing power parityis that the realexchange
rate is characterized by a unit root. A number ofapproacheshave been developed to test forunit
roots. Nelson and Plosser (1982), using augmented Dickey-Fuller (ADF) type regressions [see-2-
Dickey and Fuller (1981)], argue that most macroeconomic time series (including real exchange
rates) have a unit root. Perron (1989), however, has shown that conventionalunit root tests are
biased againstrejecting a unit root where thereis a break in a trend stationary process. More
recently, Serletis and Zimondpoulos (1997), using the methodology suggested by Perron and
Vogelsang (1992) and quarterly (from 1957:1 to 1995:4) dollar-based and DM-based real
exchange rates forseventeen OECD countries, show that theunit root hypothesis cannot be
rejected even ifallowance is made for the possibility ofa one-time changein the mean ofthe
series at an unknown point in time.
Given that integration tests are sensitive to the class of modelsconsidered (and may be
misleadingbecause ofmisspecification), in this paper weconsider a more general model. We
test forfractional integration (a series is fractionally integrated if it is integrated oforder zero
only after fractional differencing), using thefractional ARIMA model. Fractionalintegration is a
popularway to parameterize long-memory processes (whose autocorrelation structure decays
slowly to zero or,equivalently, whose spectral density is highlyconcentrated at frequencies close
to zero).
Weapply thefractional ARIMA model to quarterly dollar-based real exchange rates,
covering the period from 1957:1 to 1995:4, forseventeen OECD countries. The countries
involved are Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland,
Italy,Japan, theNetherlands, Norway, Spain, Switzerland, and theUnited Kingdom -- see
Serletis and Zimonopoulos (1997) fordetails regarding thecalculation and time plots ofthe real
exchange rates.-3-
The remainder ofthe paper consists ofthree sections. The first provides a brief
discussion of themethodology, the second discusses estimation issues and presents the results,
and the last summarizes the paper.
II. LONG-MEMORY REAL EXCHANGE RATE MODELS
The general fractionally integrated time series model can be written as [see Sowell (1992)
orBaillie (1996) formore details]
~(L) (1 ~L)dy~ =
where ~ IIDN(0, 02), ~(L) = I - 4~L- ... -4~, Ii’, and 0(L) = 1 + 01 L + ... + ~ All the roots
of4(L) and 0(L) are assumed to lie outside theunit circle (thus satisfying the stationarity and
invertibility conditions) and the fractional differenceoperator, (1 - Ly1, can be expanded as a
Taylor series about L = 0 to give
F(j-d) L~
j=o 1’(-d) F(j+1)
= I - dL + d(d - 1) L2
- d(d -1) (d -2) L3
+
2! 3!
Hence, the fractional difference operator provides an infinite-order lag-operatorpolynomial with
slowly and monotonically declining weights, meaning that autocorrelations in the fractionally-4-
integrated time series model decay ata hyperbolic rate, ratherthan the geometric rateat which
ARMA autocorrelations decline.
Iflong-memory processes are estimated with the usual autoregressive moving-average
[ARMA(p,q)] model, without considering fractional orders ofintegration, the estimated
autoregressive process can exhibit spuriously high persistence close to a unit root. Since real
exchange rates might depart from theirmeans with long memory, we condition our tests for
autoregressive unit roots in realexchange rates on the alternative ofa covariancestationary long-
memory [fractionally integrated 1(d)] process, ratherthan the usual alternativeofthe series being
1(0). In this case, if we fail to reject an autoregressive unit root, we know it is not a spurious
finding due to neglectof the relevantalternative offractional integration and long memory.
Omitted breaks in the mean ofthe series represent a second source ofpossibly spurious
findings ofautoregressive unit roots. In fact, as Perron (1989) shows, conventional unit root tests
(such as, for example, theDickey-Fuller and Phillips-Perron tests) are biased against rejecting a
unit root where there is a break in a trend stationary process. For this reason, weparameterize
the long-memory alternative by assuming a fractionally integrated time series model
~(L) (I ~L)d (~~-~1) = 0(L)c~
but we allow fora break point in the mean, pa., ofthe real exchange rate series we study, by
replacing ji~by sub-sample means in the estimation, where the break points are taken from
Serletis and Zimonopoulos (1997).-5-
III. MAXIMUMLIKELIHOOD ESTIMATES
To facilitateestimation, note that in univariate models, such as ours, we are free to
reverse the ordering ofthe AR polynomial, ~(L), and thefractional integrationpolynomial, (1 -
Ly’. At each function evaluation, we can then quasi-difference thedata, using the latest values of
the ARparameters
= - 4~
1
yt_l - ... -4~y~.
This simplification is useful because the quasi-differenceddata have theautocovariance
structure ofa fractionalARIMA (0, d, q) process forwhich closed-form expressionsexist.
Sowell (1992) shows that the autocovariances forthe general fractional ARIMA(p. d, q) model
do not have a closed-form expression and involve infinite sums. From Sowell (1992), we repeat
the expression forthe autocovariances at lag s fora fractionalARIMA (0, d, q) process, where k
= 1 forunivariate time series
kk qq F(l -d —d)F(d +s+m—Q)
= ~ ~o ~ ~0. (m)0. (1) fl
n=i r=l nr 1=0 j,r F(d~)F( 1 — d)F( 1 — d~+ s + m —
forelement (i,j), where a is the variance matrix ofe, and 0(m) is the matrix in the moving-
average polynomial corresponding with lag m, and F is the gamma function.
Fractionally integrated processes are covariancestationary only if d 0.5 and weimposed
this restriction. To estimate the model ford> 0.5 it is necessary to difference the data, in which
case we would have lost all inferenceconcerning autoregressive unit roots. Thus, in practice, we
test for autoregressive unit roots under thealternative of“covariance-stationary” long memory.
Moreover, because these models are still fairly computationally intensive and the number of-6-
series in ourdata set is large, we did not conduct a search for optimal orders, (p,q), ofthe AR
and MA processes. Instead,we estimated a fractional AR1MA (2, d, 2) model in all cases,
knowing that in some cases themodel might be overparameterized. We felt, however, that it was
better to err on the side ofovçrparameterization than to find spurious evidence oflong memory
due to understating the order ofthe ARMA process.
The results of thefractional ARIMA (2, d, 2) model are presented in Tables 1-2. In Table
1 we present results without a break in the mean ofthe real exchange rates. InTable 2 we allow
for a break in themean and we choose the break point so asto minimize (ormaximize) Perron
and Vogelsang’s (1992) t~(IO, Tb,k)statistic -- see Serletis and Zimonopoulos (1997) for more
details. On average the standard errors are not particularly small due perhaps to the relatively
short post-1957 sampleperiod. Therefore, the long-memory parameterd is generallynot
significantly different from zero. Nevertheless, we generally find some degree oflong-memory
positive autocorrelation in the real exchange rates,but the autoregressive unit roots appearto be
presentanyway. Conditional on fractional integration parameters in the rangeof0.1 to 0.2, the
real exchange rates still have autoregressive roots large enough that a unitroot cannot be rejected
-- the results are very consistent with the sum ofthe two ARcoefficients being (generally) above
0.9.
Onefinal issue concerns the non-standard distribution ofthe t-statistics. The Dickey-
Fuller distribution and critical values will not hold for the testfor autoregressive unit roots in the
presence of long memory. To ourknowledge, no one has tabulated this distribution, sowe do
not have critical values forourt-tests. Nevertheless, given that our t-statistics would almost
uniformly fail to reject under the classical t distribution and critical values, they arecertainnot to
reject under a non-standard distribution. This reasoning parallelsthe fact that the Dickey-Fuller-7-
critical values arealways larger than theclassical ones, so failure to reject at the classicalcritical
values implies failure to reject under a fatter-tailed non-standard distribution. Clearly, the low t-
statistics we generate leave little room for ambiguity.
IV. CONCLUSION
We have estimated (by maximum likelihood) the parameters of univariatefractionally
integrated real exchange ratetime series models and tested forautoregressive unit root on the
alternative of a covariancestationary fractionally integrated process. Our main contribution is
that the previous tests for autoregressive unit roots have 1(0) as the alternative, whereas we have
a much more general alternative of1(d) fractionalorder ofintegration with d taking on any value
less than 0.5. We show that thefinding ofunit autoregressive roots does not go away even with
this more sophisticated alternative.
An areaforpotentially productive future research would be to assume conditional
heteroscedasticity in the disturbances. To date, however, no methodology exists forhandling
conditional heteroscedasticity in a long-memory model withoutcompromising the fractional
integration in an ad-hoc mannerby truncating the fractional differencing operator at the number
ofpoints in the sample. In this paper, it seemed preferable to accept the lossofefficiency from
not addressing conditional heteroscedasticity, ratherthan deviate from maximum-likelihood
estimation of the long-memory process.TABLE 1
FRACTIONAL ARIMA (2,d, 2) MODELS OF DOLLAR-BASED REAL EXCHANGE RATES WITH NO BREAKS IN THE MEAN
Log “classical”
Country Likelihood d p~ P2 02 P1 + P2 f-statistic
Austria -303.7 .067 (.25 1) .887 (.297) .036 (.266) .219 (.358) -.152 (.155) 19.5 (2.23) .923 1.22
Belgium -322.5 .096 (.205) .827 (.272) .147 (.262) .153 (.310) -.159 (.117) 24.9 (2.85) .974 .882
Canada -320.0 .216(186) .726(.235) .208C214) .204(.257) -.229(.116) 24.0(2.75) .934.. 1.36
Denmark -320.0 .112 (.199) .828 (.229) .141 (.271) .226 (.128) .199 (.128) 22.8 (2.61) .969 1.07
Finland -321.2 .063 (.153) .135 (.089) .773 (.089) 1.07 (.191) .198 (.164) 24.3 (2.77) .907 1.25
France -323.7 .108 (.205) .804 (.263) .152 (.248) .200 (.297) .181 (.123) 25.3 (2.89) .956 1.14
Germany -302.6 0.27 (.221) 0.99 (.643) -.056 (.592) 0.96 (.626) 0.43 (.128) 19.2 (2.20) .934 1.01
Greece -322.6 .082C198) .877C251) .094(.241) .164(.289) -.174(.119) 25.0 (2.86) .970 1.13
Ireland -293.4 .060 (.196) .817 (.256) .156 (.250) .211 (.316) .202 (.137) 17.0 (1.95) .972 1.28
Italy -340.8 .019 (.202) .910 (.343) .079 (.337) .130 (.376) .106 (.115) 31.6 (3.62) .988 0.60
Japan -169.2 .300 (.157) .425 (.396) .487 (.369) .334 (.408) .199 (.136) 3.38 (.382) .912 1,40
Netherlands -320.7 -.436 (.195) 1.76 (.101) -.758 (.100) -.199 (.200) .119 (.090) 24.1 (2.76) 1.00 0.80
Norway -300.7 .191 (.195) .901 (.140) .026(.123) .047(.214) .107(.094) 18.7 (2.14) .927 1.58
Spain -299.9 .196 (.193) .836 (.461) .086 (.424) -.103 (.478) -.054 (.1 18) 18.5 (2.19) .921 1.28
Switzerland -311.5 .122 (.227) .781 (.301) .188 (.290) .425 (.353) .110 (.175) 21.5 (2.46) .969 0.85
U.K. -299.9 .196 (.193) .836 (.461) .085 (.424) -.103 (.478) -.054 (.118) 18.5 (2.19) .921 1.28
NOTES: Sample period, quarterly data, 1957:1 - 1995:4. Numbers in parentheses are standard errors.TABLE2
FRACTIONAL ARIMA (2,d, 2) MODELS OF DOLLAR-BASED REAL EXCHANGE RATES
WITH THE SERLETIS AND ZIMONOPOULOS (1997) BREAK POINTS, BASED ON
THE PERRON AND VOGELSANG (1992) t~ (JO, Tb, k)STATISTIc
Log “classical”
Country Tb Likelihood d Pt P2 01 02 Pt + P2 f-statistic
Austria 1970:4 -307.9 .036 (.205) .816 (.334) .075 (.292) .272 (.376) -.123 (.152) 20.5 (2.35) .891 1.56
Belgium 1971:1 -347.6 .102(209) .774(378) .129(337) .116(.399) -.106(107) 34.6 (3.96) .903 1.36
Canada 1976:1 -327.4 .195 (.174) .583 (.221) .325 (.197) .328 (.240) -.243 (.122) 26.5 (3.03) .908 1.58
Denmark 1971:1 -345.7 .077 (.201) .751 (.324) .152 (.289) .226 (.352) .127 (.122) 33.7 (3.86) .903 1.51
Finland 1972:3 -325.7 .002 (.13 1) .095 (.087) .731 (.065) .949 (.050) .047 (.068) 25.6 (2.94) .826 3.19
France 1971:2 -336.5 .071 (.199) .701 (.302) .193 (.266) .027 (.319) .149 (.119) 29.9 (3.42) .893 1.51
Germany 1970:3 -306.4 .038 (.226) .965 (.848) -.054 (.764) .089 (.825) .032 (.141) 20.2 (2.31) .911 0.92
Greece 1991:1 -338.4 .032 (.203) .886 (.249) .075 (.237) .222 (.305) .184 (.140) 30.6 (3.50) .961 1.37
Ireland 1984:3 -313.1 .010 (.147) 1.07 (.446) -.099 (.428) .026 (.474) .102 (.118) 22.0 (2.52) .971 1.21
Italy 1984:4 -365.1 .026 (.200) .792 (.369) .165 (.354) .236 (.413) .116 (.133) 43.5 (4.97) .957 1.15
Japan 1971:1 -171.0 .317 (.146) .447 (.386) .458 (.357) .308 (.394) .198 (.130) 3.41 (.391) .905 1.53
Netherlands 1970:4 -392.7 .000 (.076) .972 (.600) .029 (.545) .045 (.486) .029 (.078) 62.3 (7.13) .943 0.97
Norway 1971:1 -311.0 .161 (.181) .864 (.158) .035 (.137) .055 (.219) .075 (.091) 21.4 (2.44) .899 1.93
Spain 1971:2 -300.6 .193 (.195) .836 (.493) .831 (.452) -.103 (.496) -.051 (.125) 18.7 (2.14) .919 1.25
Switzerland 1972:3 -325.0 .022 (.207) .635 (.329) .223 (.278) .316 (.340) .122 (.121) 25.7 (2.94) .859 1.48
U.K. 1976:2 -340.3 .265 (.134) .262 (.134) .596 (.114) .593 (.166) -.256 (.124) 31.1 (3.56) .859 1.87
NOTES: Sample period, quarterly data, 1957:1 - 1995:4. Numbers in parentheses are standard errors.REFERENCES
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-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Value of fractional integration parameter
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0
C
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0
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Ci)
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4
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II

Deviations from Purchasing PowerParity in the Long Run.”

Does the Real Exchange RateFollow a Random Walk? ThePacific Basin Perspective.”

Exchange Rates, Prices and Money: Lessons from the 1920s.”

Likelihood Ratio Statistics forAutoregressive TimeSeries with a Unit Root.”

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LongMemory Processes and Fractional Integration in

Maximum Likelihood Cointegration Tests of Purchasing Power Parity: Evidence from Seventeen

Maximum Likelihood Estimation ofStationary Univariate FractionallyIntegrated Time Series Models.”

Nominal Exchange Rate Regimes and the Real Exchange Rate: Evidence from the United States and Great Britain,

Nonstationarity and Level Shifts With an Application to Purchasing Power Parity.”

Real and Nominal Exchange Rates in theLong Run:

Real Exchange Rate Behavior: The Recent Float from the Perspective ofthePast Two Centuries.”

Tests of Long-Run Purchasing Power Parity Using Alternative Methodologies.”

The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis.”

Do real exchange rates have autoregressive unit roots? a test under the alternative of long memory and breaks

http://research.stlouisfed.org/wp/2000/2000-016.pdf

Do real exchange rates have autoregressive unit roots? a test under the alternative of long memory and breaks

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