37

Michael J. Jiucker

Michael J Dueker is an economist at the Federal ReserveBank of St. Louis. Richard 11 Jako provided researchassistance.

I Hypothesis Testing with Near- UnitRoots: The Case of Long-RunPurchasing-Power Parity

HE HYPOTHESIS THAT the purchasingpower of a given currency, like the dollar, willbe equal across countries has strong appeal. ifthe hypothesis is true, then inflation and ex-change rate movements will be such that a givencurrency will, over time, lose equal amounts ofits purchasing power in all countries. The se-quence of events by which deviations frompurchasing-power parity would be eliminatedcan best be illustrated by example: If the dollarcould put-chase more goods in other countriesthan in the United States, then U.S. consumerswould purchase more goods from abroad,which would raise the demand for foreign cur-rencies relative to the dollar and lead to adepreciation of the dollar and eventual equaliza.don of the dollar’s purchasing power acrosscountries. Despite the intuitive appeal of sucharguments for long-run purchasing-power pari-ty, statistical tests have been mixed. This paperargues that previous test results have conflictedbecause tests of purchasing-power parity haverelatively low power under both the null hypoth-esis that it holds, and the null that it fails. Hencethis paper contains tests of both null hypothesesand shows that frequently neither is rejectedfor monthly data from five major industrializedcountries. This result serves as a caution againsttesting only one null hypothesis, finding that thenull hypothesis cannot be rejected for a broad

set of countries and concluding that there isrobust evidence for or against the theory oflong-run purchasing-power parity.

The theory of long-run purchasing-power par-ity (PPP) implies that a currency’s purchasingpower is equal across countries in long-runequilibrium, but does not specify how long devi-ations from this equilibrium can last. Large andpersistent departures from PPP in the last 20years, however, have cast doubt on the validityof PPP. As we will discuss later, there is a liter-ature which tests whether long-run PPP holds,that is, whether departures from PPP are transi-

tory. This article aims to reconcile some of thedisparate results from previous studies by usinga long-memory model, which can do more thanclassify deviations from PPP as temporary orpermanent: it can provide specific measures oftheir persistence. Such measures are useful be-cause large, persistent differences in a curren-cy’s purchasing power across countries cangreatly affect trade flows and the allocation ofresources.

Empirically, long-run PPP holds if the real ex-change rate, which equals the nominal exchangerate multiplied by the ratio of the domestic andforeign price levels, is mean-reverting. This arti-cle will conform with the majority of the empir-ical PPP studies by using consumer price indexes

38

to calculate the real exchange rate. if price in-dexes measured the prices of identical basketsof goods across countries, absolute P1W wouldhold if r = S x P, ~vhere P is the domesticprice of the goods basket, P~is the foreignprice and S is the exchange rate in terms ofunits of foreign currency per unit of domesticcurrency. Because consumer price indexes domiot measure the cost of identical baskets ofgoods across countries, however, relative PPPmodifies the relationship to account for the ra-tio of the values of the two distinct baskets ofgoods: ps = kSP, where k is the ratio of thevalue of the foreign basket to the domesticbasket. The domestic country’s real exchangerate with the foreign counu’y is then i/k andequals 5p/p*i

The conventional approach to testing for long-run purchasing-power parity consists of testingfor a unit root in the real exchange rate: Long-run PPP holds if the real exchange rate is mean-reverting but not if it has a unit root. Previoustests have shown little po~•verto reject whicheverof the two null hypotheses is employed. Testswhose null hvpodesis is that the real exchangerate contains a unit root generally fail to reject,whereas tests whose null is that long-run PPPholds also often fail to reject. These disparatefindings are reconciled, however, if there islong memory in the real exchange tate, whichenables both acceptance and rejection of long-run PPP at conventional significance levels.2

This paper employs long-memory models toobtain estimates of the orders of integration ofreal exchange rates on a continuous scale. Theadvantage of estimating the order of integrationon a continuous scale is that we can confirmthat long-memory time series hehaviom’ in realexchange rates is a possible source of the dis-crepancies between previous tests of long-run

PPP. The finding of long memory in real ex-change rates also allows us to judge whetherthe real exchange rate reverts to its mnean withinan economically meaningful time frame.

WHY PURCHASIN&POWERPARITY MIGHT NOT HOLD

Before discussing statistical tests of PPP, it isworth repeating Engel’s (1992) list of possiblereasons for the empirical failure of PPP:

1. Barriers to trade such as tariffs and trans-portation costs.

2. Different consumption pm’efeitnces acrosscountries.

3. The presence of non-traded goods in priceindexes.

4. Prices which are sticky’ in terms of the cur-rency in which the good is consumned.

Barriers to trade, such as tariffs, are an obviousreason why the same goods do not sell at thesame price throughout the world. Different con-sumnption preferences, on the other hand, wouldlead consumers in each country to choosedifferent baskets of goods. Because price indexesare constructed for baskets of goods designedto represent a particular country’s consumption,an apparent failure of PPP could he due todifferent rates of price inflation across twocountr’s distinctive baskets of consumptiongoods, rather than different prices for the samegoods acm’oss countries. When included in priceindexes, non-traded goods can also muddle theinterpretation of the real exchange rate, becausenon-traded goods can be idiosyncratic amid arethus not directly comparahle across countries.Nevertheless consumer price indexes will heused in this paper, despite the presence of non-tradeables, because wholesale price indexes canfail to reflect the undem’lying rate of inflation ac-curately.~‘l’he fourth source of failure, sticky

‘Summers and Heston (1991) tabulate the costs of nearly-identical baskets of goods across countries, rather thanuse existing price indexes. They define the PPP nominalexchange rate to be V/P and use this implied exchangerate, rather than the market exchange rate, to make cross-country comparisons. The Summers and Heston measuresof the price levels could take the place of commonly usedconsumer price indexes when testing long-run PPP, ascould wholesale price indexes, The Summers and Hestondata, however, are only available on an annual basis andinclude data extrapolated between five year data collectionperiods. The analysis in this paper will be limited to theuse of consumer price indexes to facilitate comparison withprevious studies.

than zero. If the order of integration is greater than 0.5,the series is not covariance stationary and if the order ofintegration is greater than one, the series does not have amean, Long memory is not the same as an autoregressivenear-unit root, because a series with a near-unit au-toregressive root is still integrated of order zero, and is notconsidered a long-memory process.

3For example the wholesale price index for Japan suggeststhat Japan has had deflation on average from 1980 to thepresent, whereas the GDP deflator and CPI show moderateinflation.

2Long memory, as will be discussed later, means thatthe order of integration of a time series process is greater

39

prices, can best he explained by an example:Japanese autos sold in Japan and also exportedto the United States have sticky pr-ices in yenwhen sold in Japan and sticky prices in dollarswhen sold in the United States. Any exchangerate fluctuations would cause the yen (or riollat)price of the same model of car to differ acrossthe Pacific. Thus autos might contribute to thefailure of PPP in the tm’ue sense: the same goodbeing sold at different 1h~~(net of taxes)across coun tries.

PREVIOUS TESTS OFPURCHASIN&POWER PARITY

Tests of PPP in the literature can he classifiedaccording to many criteria. In this brief reviewof a large literature, three features will receiveattention: 20th century annual data vs. post-1973monthly data; the use of consumer price index-es versus wholesale price indexes in the calcula-tion of the real exchange rate; and whether ornot price levels are assumed to he measuredwith error.4 The aimn of this review is to illus-tm’ate the lack of consensus that has emergedfrom studies of long-run PPP and identify whichmodeling choices might have influenced the out-comes of those tests.

Coughlin and Koedijk (1990) conduct unit-roottests on real exchange rates using post-1973monthly data and consumer prices and find thatthe unit-root null hypothesis cannot he rejected.They also examine whether the real exchangerates are cointegrated with factors thought todetermine the real exchange rate. Cheung andLai (1993b) use post-1973 monthly data on con-sumer prices and allow for measurement errorin prices. ‘l’hey use a Johansen (1991) likelihoodratio test for cointegrating vectors, in whichlong-run PPP is the null hypothesis, and general-ly fail to reject long-run PPP.5 Thus the studiesof Coughlin and Koedijk (1990) and Cheung andLai (i993b) illustrate the importance of the nullhypothesis in testing long-run PPP. Edison andFisher (1991) and Fisher and Park (1991)represent another’ pair of studies that differ inthe null hypothesis employed and the generalconclusions about long-run PPP. Cheung and

4For a thorough introduction to what the authors call thepurchasing-power parity assumption, see Caves, Frankeland Jones (1990).

Provided that the hypothesis of zero cointegrating vectorshas been rejected, the null hypothesis for subsequenthypothesis tests using the Johansen procedure is thatthere is at least one cointegrating vector.

Lai (1993a) study 20th century annual data andfind evidence that the meal exchange rate haslong memory, hut not a unit root for mostcountries studied. Pippenger (1993) uses whole-sale price indexes and finds evidence that PPPbetween Switzerland and various countries ~l~-pears to hold in the long run.

Overall a lack of consensus emerges from em-pirical tests of long-run PPP. ‘The choice of nullhypothesis is one source of discrepancy, and itappears that results fm’om a long-memory modelcan reconcile the apparently conflicting resultsof several previous studies of long-run PPP, whichdiffer primarily in their choice of null hypothesis.Consequently, tests of each null hypothesis willbe highlighted in the estimation results below.

TESTING PURCHASING-POWERPARITY

Much research on whether PPP holds in thelong run consists of performing a unit-root teston the real exchange rate. It is well known,however, that unit-root tests, especially thosehaving a unit root as the null hypothesis, likeDickey-Fuller, have little power against long-memnorv alternatives. Such unit-root testing con-sists of classifying economic variables as eitherintegrated of order zero [1(0)] or one [I(1fl.G Incontrast, we use a parametric long-memorymodel in which data series, like the real ex-change rate, are modeled as integrated of orderd, denoted 1(d), where d does not have to be aninteger. Any series that is integrated of orderd<1 will return evemitually to its mean (or itsdeterministic trend), so shocks to the real ex-change m’ate are not permanent if the real ex-change rate is integrated of order d< 1.

This paper also provides information aboutthe sources of PPP failure by examining thecomponents of the real exchange rate. ‘l’he ratioof the price levels may have a higher order offractional integration than tile nominal exchangerate, or vice versa. If r is tile real exchangerate, s is the nominal exchange rate, p is thedomestic price level, and p~is the foreign pricelevel (all in natural logs), then r=s+(p.p*). If s isl(dl), (p~p*)is I(d2), then r will genem-ally he in-

°tntegrationof order one means that a variable’s first differ-ences are stationary, whereas its levels are not.

40

tegrated of order max{dl,dz}. If m’ is 1(b) whereb < max{dl,d2}, then the nominal exchangerate and the price ratio (p.p*) are fractionallycointegrated, that is, they share the samestochastic trend to some extent.7 The real ex-change rate does not have a unit root if b< 1.

We can also examine the point estimates of diand d2 and see which component appears tohave the strongest trend. This comparison an-swers critics of flexible exchange rates who ar-gue that floating rates actually have caused thereal exchange rate to be less stable than it wouldhave beemi under fixed nominal exchange rates.8

If di > d2, then shocks to the nominal exchangerate are more persistent than shocks to therelative price levels. The latter is somewhat cu-rious, because proponents of the switch to aflexible exchange rate regime envisioned flexibleexchange rates as sources of real exchange ratestability in a world in which countries mighthave persistent differences in inflation rates. Yetif shocks to the nominal exchange rate are morepersistent than shocks to the relative price levels,then the nominal exchange rate has persistenceabove what is potentially useful in reducing thevariance of the real exchange rate. In the em-pirical results that follow, the possibility of ex-cess persistence in the nominal exchange ratewill be examined.

BACKGROUND ON LONG-MEMORY MODELS

For mamly time series, autoregressive moving-average (ARMA) models serve as a parsimoniousway to summarize the autocovariance structureof the data. One limitation of such models is thatARMA processes are integrated of order zero,and the autocovariances die off relatively quickly,even when a root in the autoregressive polyno-mial is near one. Thus ARMA models can becalled short-memory models, because a shockaffects the level of the series for a relativelyshort time.

Long-memory models, in contrast, are suitablefor data that have slowly decayimlg coefficients

in their moving-average representations. Thefractional ARMA model can serve as a long-memory model, yet it adds only one parameterto a standard ARMA model. To illustrate, we be-gin with a simple ARMA(1,i) applied to the firstdifference of a data series y, where L is the lagoperator, is a mean-zero disturbance, p is theAR coefficient, and B is the MA coefficient:

(1) (1 —pL)(1 —fly = (1 + BL)

A fractional ARMA model is simply an ARMAmodel applied to fractionally differenced data:

~2)(1-pLhl _fldy = (1÷BL)

The fractional differencing operator’ is evaluatedby taking a Taylor series expansion around

(3)(i—LY’ = 1—dL + d(d—flL2—

2

d(d—ihd—2)L3

+

3!

Two characteristics of fractionally integrateddata are worth noting. First, a series that is in-tegrated of order d (1(d)) with d< I reverts toits mean (or at least to its deterministic trend).Second, if d< .5, the series is covariance station-ary. At first glance, it might seem counter-intui-tive that a mean-reverting series can fail to becovariance stationary. With long methory, how-ever, the departures from the mean can besufficiently persistent that the variance of theseries is infinite.

Furthermore, two commonly assumed data-generating processes fit within the subset offractional integration: trend and difference sta-tionarity. Fractional integration offers a bridgebetween the controversial assignment of a dataseries as either trend or difference stationary,so that questions about stationarity assumptions

7For reasons outlined below, the restriction that the coeffi-cients on s and (p~p*)equal one is relaxed, so that theorder of integration of a general linear (cointegrating) com-bination of s, p and p’ is assumed to be the order of in-tegration of the real exchange rate, The concept ofcoinfegration has been generalized [Granger (1986)] to in-clude cases in which series have stochastic trends thatonly partially offset each other, This is called fractionalcointegration. Originally, cointegration meant that a partic-

ular linear combination of two strongly trending series was(0).

8For example, Aliber (1993) notes that “the U.S. dollar ap-preciated from 1979 to 1986 even though the U.S. inflationrate was higher than the inflation rates in Germany andJapan

°Theconcept of fractional differencing was developed byGranger and Joyeux (1980) and Hosking (1981).

41

may he avoided. For example, if y is trend sta-tionary, we might model y as

(4)j’, =

In differences, equation (4) looks like

(5) (1—fly, =

If y is difference stationary, then

(6)(1—L)y, = +

Now suppose that y is fractionally integrated oforder d. The fim’st differences of y are thenequal to

(7) (1—fly, = + (1—fl’-’¼,

where j~is proportional to p. Clearly trend sta-tionarity (d=0) and diffem’ence stationarity (d= 1)are bridged by fractional integm’ation, which al-lows fom’ intermediate cases. An intuitive way tounderstand why fractional integration is an in-termediate case between trend and differencestationarity is to interpret each shock in adifference-stationary process to be a permanentshift away from any previous trend; a shock ina trend-stationary process is a short-lasting shiftfrom the trend; a shock in a fractionally inte-grated process is a long-lasting shift from thetrend. This paper uses estimates of the order offractional integration to discriminate, if possible,between long-lasting shifts from the mean realexchange rate and permanent shifts in the realexchange rate (the unit-root case).

ESTIMATES OF LONG MEMORY INREAL EXCHANGE RATES

The data used in this article consist of 234monthly observations of the nominal exchangerates and consumer price indexes for- the Unit-ed States, Great Britain, Germany and Japanfrom June 1973 to November 1992.10 Thus sixbilateral relationships will he examined.

Fractional ARMA models are used to estimatethe orders of integration of the nominal ex-change rates, the ratios of the pm’ice levels and

the real exchange rates. The general form ofthe fractional ARMA model is

(8) A(LH1 —fl”y,=B(fl ,

where y is 1(d) and is assumed to have zero

niean, no serial correlation and variance ~ A(L)is an autoregressive polynomial of order p andB(L) is a moving-average polynomial of order q:

(9) A(L) = I —p,L —p,L2— ...

(1W B(fl=1+B,L+O2L’+,.,+BL~

Estimation was carried out using the Fox amidTaqqu (1986) frequency-domain estimator offractional ARMA models. The estimator is basedon an approximation to the likelihood. Dahlhaus(1988, 1989) has analyzed the Fox and Taqqu es-timator and has shown that it shares the santeasymptotic efficiency as exact maximum-likelihood estimation. Further details regardingthe estimator appear in the appendix.

Before presenting estimation results, we mustdiscuss how the real exchange rate was calculat-ed. Any mismneasurement of the price levels canlead to spurious changes in the mean of thereal exchange rate and bias tests toward rejec-tion of long-run PPP. To minimize the possibilityof spurious rejections of long-run PPP, the realexchange rate was calculated by estimating afractionally cointegm’ating relationship betweenthe nominal exchange rate (s), the domesticprice level (p) and the fom-eign price level (p*):h1

(11) S,=a0—a,p,+aj3~+ ,

The residuals from equation (ii) were thentreated as the real exchange rate for unit-roottesting with the fractional ARMA model. Cheungand Lai (1993b) and Pippenger (1993) also esti-mate a general cointegrating relationship, ratherthan impose a =a, = 1. They both argue that,because of measurement en’or in price indexesand umiequal weights attached to the same goodin different indexes, it is umidesirahle to imposeunit coefficients on the pm-ice indexes whenstudying whether the real exchange rate ismean-reverting. The Phillips and Hansen (1990)

lcKoedijk and Schotman (1989) find that the real exchangerates between 15 industrialized countries are fairly wellspanned by the real exchange rates between the UnitedStates, Japan, Germany and Great Britain.

‘1Cheung and Lai (1993a) discuss the asymptotic theory be-hind estimating regressions where the residuals are frac-tionally integrated.

42

was used for equation (11). This method ac-counts for simultaneity in the determination ofleft- and right-hand side variables. Cheung andLai (1993h) suggest using estimation proceduresthat take into account interactions hetweemi left-and right-hand side vatiahles. in fact, thePhillips-Hansen estimates of a, amid a2 indicatethat the should miot he restricted tr) equal one.For example, for the nominal exchange mate be-tween Bm’itain and the United States, the estimatesof a, and a1 are 1.45 and 1.22, respectively.

The next point of focus is the null hypothesisto he tested: PPP holds as the null hypothesis ifthe hypothesis that b < 1, where b is the orderof integration of the real exchange rate, is notrejected; the altem’native null hypothesis that PPPfails is not rejected if the null that b ~ 1 is notrejected. Using the fractional ARMA model, it iseasy to test both null hypotheses and show howthe results depend on the choice of the null.

Table I contains the main results on the esti-mated ordem-s of integration of the relevant ser-ies. Simple t-tests can he used to test forlong-run PPP by dividing one minus the estimat-ed order of integration of the real exchangei-ate by its standard error. Doing this, we seethat the null hypothesis that b< 1, where b is

rate, is rejected for only two of the six pairs:United States/Japan and Britain/Japan. These arethe significantly negative t-statistics in tahle 1.Thus long-run PPP is not rejected as a nullhypothesis in four of six cases. If we reversethe null, however, we can reject the null thatb 1 for onl one pair: Britain/Germany.12 Thisis the significantly positive t-statistic in table 1.The results for United States/Germany are hot’-derline with a t-statistic of 1.45, but this is notsignificant at the usual 5 percent level of sig-nificance in a one-tailed t-test, where the criticalvalue is 1.658.

Overall, the om’ders of integration of real ex-change rates are often close enough to one thatneither null hypothesis is rejected. This explainssome discrepancies hetween past tests of long-run PPP. Coughlin and Koedijk (1990) usedDickey-Fuller’ unit-m’oot tests on real exchangerates and could not i-eject the null that long-runPH’ fails to hold. Cheung and Lai (1993b), onthe other hand, tested tile miull that long-runPPP holds and did not find many rejections oflong-run PPP. ‘I’he results from the long-memorymodel reconcile these findings.

The estimates from the long-memory modelsdo more than give unit root tests, however, by

method of estimating cointegrating relationships the order of integration of the real exchange

‘2Orders of integration greater than one for data in logs im-ply that the growth rates of the series have long memory.

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providing an estimate of the order of integra-tion of the real exchange rate on a continuousscale. Whichever null is used, one result isclear: Even if long-run PPP holds, it is very slowin developing. Assuming that the order of in-tegration of the real exchange rate is 0.9, 73percent of a shock is still present after 12months; 68 percent after 24 months; 65 percentafter 36 months; and 63 percent after 48 months.As a practical matter, it seems fair to concludethat PPP does not hold within a time horizonthat is economically relevant. Uncovering thistype of information is the chief advantage of es-timating the order of integration on a continu-ous scale. With other unit-root testing methods,we are fotced to view a series as being either1W) or 1(1). 5uch a polar characterization maynot provide practical information about the per-sistence of the shocks.

Table I also provides information about thepersistence of shocks to the nominal exchangerate relative to shocks to the ratio of the pricelevels. In five of six cases, point estimates sug-gest that the nomninal exchange rate has a lowerorder of integration than the price ratio.” ForBritain/Japan the point estimate of the order ofintegration is 1.15 for the nominal exchangem’ate vs. 1.12 for the price ratio, hut this differ-ence does not appear to he statistically signifi-cant.14 Thus the conjecture that nominalexchange rites in the post-Bretton Woods erahave shown excess persistence appears to befalse. In general, greater persistence in thenominal rate would he needed to offset the per-sistence in the price ratios for the real exchangerate to he rendered 1(0). This is because no 1W)lineam’ combination can exist, for example, be-tween a series that is 1(8) and one that is 1(2).The series that is 1(2) does not have enough ofa trend with which to offset the relativelystrong trend in the variable that is 1(8).

Another finding from table 1 is that inflationdifferentials are fractionally cointegrated insome cases. For example, the estimates indicatethat (p,..~—p,,,)is 1(1.62) and (p,~—p~~)is 1(1.60),but the difference (p,,~—p,,,)is only 1(1.12). Thismeans that the inflation differentials betweenthe United States and Britain and between the

United States and Japan appear much more per-sistent than the inflation differential betweenBm’itain and Japan. In other words, inflationrates in Britain and Japan come closer to shar-ing a common trend with each other than withinflation in tile United States.

Tables 2 through 4 report the fractionalARMA parameter estimates fully only for thebilateral relationships for the United States forthe sake of brevity. The key result in these ta-bles is that in fractional ARM±kmodels the frac-tional differencing parameter can captum’e thelong-run behavior of the data, freeing ARparameters to match the short-run dynamics. If,on the other hand, an ARMA model instead of afractional ARMA model were fit to the data, theautoregressive polynomial would he forced tohave a near-unit root.

In table 2 several AR parameters are negative,and the largest equals 0.53 in the fractionalARMA model of the ratio of the price levels be-tween the tnited States and Britain. In table 3both estimated AR parameters for the nominalexchange rate between the United States andGermany are negative, implying that all positivedependence in the exchange rate beyond thefirst lag is due to the large positive value of thefractional-diffem’encing parameter. The largestroot in an AR polynomial in table 3 is 0.37which is far from the unit circle, in the real cx-chamige rate between the United States and Ger-mammy. Estimates of the model of the realexchange rate between the United States andJapan, found in table 4, also show two miegativeAR coefficients. In fact all of the AR polynomi-als in table 4 have roots with real parts that arevery far from the unit circle. ‘They are — .05,— .03, and .08, respectively, for the real ex-change rate, the nominal exchange rate and theprice ratio. With the imiclusion of the fractionaldifferencing parameter, the AR parameter-s cantake values which allow the fractional ARMAmodel to capture both long-i-un dependence andshort-run dynamics in the data. The shaded in-sert and figures I thm’ough 3 provide a visualcheck of the match between the covariancestructure of the data and that implied by the es-timated fractional ARMA mnodel.

‘3An order of integration above one for variables in logsmeans that the growth rates display long memory. In thecase of the price ratio, the corresponding growth rate isthe inflation differential across the two countries, For thenominal exchange rate, it is the rate of exchange rate ap-preciation.

14A formal test would require loint estimates of the two frac-tional ARMA models, however.

44

a 0-I

46

Figure 1Spectrum of Differenced Real Exchange Rate: U.S./Germany

ARFIMA (2,d,1) Spectrum Smoothed Periodogram

2.20 350

Figure 2

Spectrum of Differenced Nominal Exchange Rate: U.SiGermany

ARFIMA (2,d,1) Spectrum Smoothed Periodogram40

3.5

3.0

25

2.0

1.5

1.0

05

0.0

3.0

2.5

2.0

1.5

1.0

-0.5

.0

0 1 2 3Frequency

0 1 2 3

Frequency

Figure 3

47

Spectrum of Differenced Price Level Ratio: U.SiGermany

ARFIMA (2,d,1) Spectrum0.12

0.10

0.08

0.06

0.04

0.02

0.00

Smoothed Pellodogram0.08

0.06

~i04

0.02

.00

CONCLUSIONS

This article illustrates the key role played bythe null hypothesis in testing for unit roots inreal exchange rates. If b is the order of integra-tion of the real exchange rate, then the nullthat b>i is difficult to reject, in which case onewould presume that lomig-run purchasing-powerparity does not hold. When the null is thath< I, we also find few rejections, so long-runPPP apparently holds. When this type of am-biguity appears, it is helpful to estimate the ord-er of integration on a continuous scale. Thefractional ARMA models presented here do thisand the standard errors on h for the six realexchange mates studied show that even if b< 1,it is not far enough away from one to make astrong case that purchasing-power parity is em-pirically relevant.

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0 1 2 3Frequency

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The Fox and Taqqu EstimatorDahlhaus (1988) discusses why the Fox and

Taqqu (1986) frequency domain estimator is anapproximate maximum-likelihood estimator,sharing the same optimality properties as exactmaximum-likelihood estimation. ‘l’he Fox andTaqqu estimator is derived from the followimigminimnizatiomi problem:

flAk)(12) rain ~2~[’tfbf(k6)) + o2f(A~6)

where lUrk) is the vector of periodogram or-

density function implied by the parameterizedmodel. Fot- the fractional ARMA model in equa-tion (8), the spectral density equals

(13) fiX) 8(e)] (1 —eH” (1—c~h”,

where A and B are polynomials defined in equa-tions (9) and (10). The objective function isminimized with respect to 6 and ~2, An intuitivedescription of the objective function is that onewants to choose parameters that will make thespectral density function implied by the model

dinates of the data and 02f(Ak) is the spectral look like the periodogram of the data.