1 Effective exchange rate indexes were developed by the Interna- tional Monetary Fund. The seminal work was by Hirsch and Higgins (1970). 2 For all indexes discussed in this article, percentage changes are calculated on a logarithmic basis. Thus the percentage change in an index that in- creases from 100.0 to 111.2 is the natural logarithm of the ratio of 111.2 to 100 or 10.6 percent. 3 The issues involved in con- structing effective exchange rate indexes have been dis- cussed by many authors, in- cluding Rhomberg (1976), Rosensweig (1987), and Turner and Van ‘t dack. (1993). FEDERAL RESERVE BANK OF ST. L OUIS 3 JULY/AUGUST 1996 Board, the U.S. dollar fell in value by 62 percent between March 1985 and Decem- ber 1995. 2 In contrast, the index produced by the Federal Reserve Bank of Dallas shows the dollar rising in value by 60 per- cent during the same period. Even when the indexes show the dollar moving in the same direction, they gener- ally do not agree on the overall magnitude of that change. Why don’t these indexes provide a consistent view of changes in the value of the dollar? This article answers this question by examining the way in which exchange rate indexes are con- structed. We begin by exploring the basic issues of constructing effective exchange rates using the six indexes shown in Figure 1 for illustration. After discussing the dif- ferences in constructing these indexes, we examine some factors that might account for the contrasting views of the dollar by focusing on two specific indexes—the Fed- eral Reserve Board and the Federal Reserve Bank of Dallas indexes. CONSTRUCTING EFFECTIVE EXCHANGE RATE INDEXES The construction of effective exchange rate indexes requires a number of deci- sions. 3 Because many of the decisions have more than one defensible alternative, it is not surprising that a number of effective exchange rate indexes are used. Six deci- sions are examined: (1) which formula is used to calculate the average, (2) which foreign currencies are used in the calcula- tion, (3) which measure of economic activ- ity is used as the basis for weighing the im- portance of individual currencies, (4) how to calculate the weights for individual cur- rencies, (5) the base period for calculating the weights, and (6) the base period for calculating exchange rate changes. These decisions are illustrated with specific refer- ences to how six well-known effective ex- change rate indexes are constructed. These indexes are identified by their producers— Cletus C. Coughlin is associate director of research and Patricia S. Pollard is an economist at the Federal Reserve Bank of St. Louis. Jerram Betts provided research assistance. A Question of Measurement: Is the Dollar Rising or Falling? Cletus C. Coughlin and Patricia S. Pollard I n March 1985 one U.S. dollar could buy 258 Japanese yen and 0.21 Mexican pesos. In December 1995 the same dol- lar could buy only 102 yen, but could now buy 7.7 Mexican pesos. Though the change in the value of the dollar against each of these currencies was exceptionally large, the behavior of the dollar—rising against one currency and falling against another—was not uncommon. Over the past 10 years the dollar has appreciated against many currencies and depreciated against others. How then can one deter- mine what has happened to the overall value of the dollar? Is the dollar stronger or weaker than it was 10 years ago? To begin answering this question, economists construct effective exchange rate indexes. Effective exchange rates, commonly termed trade-weighted exchange rates, mea- sure the average foreign exchange value of a country’s currency relative to a group of other currencies. 1 Unfortunately, looking at effective exchange rate indexes may not provide a consistent answer to the preced- ing questions. The effective exchange value of the dollar as measured by six commonly used indexes is shown in Figure 1. Accord- ing to four of these indexes, the dollar has fallen in value since March 1985, whereas two other indexes show a rise in the value of the dollar since March 1985. For exam- ple, according to the effective exchange rate index produced by the Federal Reserve
Transcript
1 Effective exchange rate indexeswere developed by the Interna-tional Monetary Fund. Theseminal work was by Hirschand Higgins (1970).
2 For all indexes discussed in thisarticle, percentage changes arecalculated on a logarithmicbasis. Thus the percentagechange in an index that in-creases from 100.0 to 111.2is the natural logarithm of theratio of 111.2 to 100 or 10.6percent.
3 The issues involved in con-structing effective exchangerate indexes have been dis-cussed by many authors, in-cluding Rhomberg (1976),Rosensweig (1987), andTurner and Van ‘t dack.(1993).
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JULY/AUGUST 1996
Board, the U.S. dollar fell in value by 62percent between March 1985 and Decem-ber 1995.2 In contrast, the index producedby the Federal Reserve Bank of Dallasshows the dollar rising in value by 60 per-cent during the same period.
Even when the indexes show the dollarmoving in the same direction, they gener-ally do not agree on the overall magnitudeof that change. Why don’t these indexesprovide a consistent view of changes in thevalue of the dollar? This article answersthis question by examining the way inwhich exchange rate indexes are con-structed. We begin by exploring the basicissues of constructing effective exchangerates using the six indexes shown in Figure1 for illustration. After discussing the dif-ferences in constructing these indexes, weexamine some factors that might accountfor the contrasting views of the dollar byfocusing on two specific indexes—the Fed-eral Reserve Board and the Federal ReserveBank of Dallas indexes.
CONSTRUCTING EFFECTIVEEXCHANGE RATE INDEXES
The construction of effective exchangerate indexes requires a number of deci-sions.3 Because many of the decisions havemore than one defensible alternative, it isnot surprising that a number of effectiveexchange rate indexes are used. Six deci-sions are examined: (1) which formula isused to calculate the average, (2) whichforeign currencies are used in the calcula-tion, (3) which measure of economic activ-ity is used as the basis for weighing the im-portance of individual currencies, (4) howto calculate the weights for individual cur-rencies, (5) the base period for calculatingthe weights, and (6) the base period forcalculating exchange rate changes. Thesedecisions are illustrated with specific refer-ences to how six well-known effective ex-change rate indexes are constructed. Theseindexes are identified by their producers—
Cletus C. Coughlin is associate director of research and Patricia S. Pollard is an economist at the Federal Reserve Bank of St. Louis. JerramBetts provided research assistance.
A Question ofMeasurement:Is the DollarRising orFalling?Cletus C. Coughlin and Patricia S. Pollard
In March 1985 one U.S. dollar could buy258 Japanese yen and 0.21 Mexicanpesos. In December 1995 the same dol-
lar could buy only 102 yen, but could nowbuy 7.7 Mexican pesos. Though thechange in the value of the dollar againsteach of these currencies was exceptionallylarge, the behavior of the dollar—risingagainst one currency and falling againstanother—was not uncommon. Over thepast 10 years the dollar has appreciatedagainst many currencies and depreciatedagainst others. How then can one deter-mine what has happened to the overallvalue of the dollar? Is the dollar strongeror weaker than it was 10 years ago? Tobegin answering this question, economistsconstruct effective exchange rate indexes.
Effective exchange rates, commonlytermed
trade-weighted exchange rates, mea-sure the average foreign exchange value ofa country’s currency relative to a group ofother currencies.1 Unfortunately, looking ateffective exchange rate indexes may notprovide a consistent answer to the preced-ing questions. The effective exchange valueof the dollar as measured by six commonlyused indexes is shown in Figure 1. Accord-ing to four of these indexes, the dollar hasfallen in value since March 1985, whereastwo other indexes show a rise in the valueof the dollar since March 1985. For exam-ple, according to the effective exchangerate index produced by the Federal Reserve
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Federal Reserve Board, J.P. Morgan (broadand narrow), International Monetary Fund(IMF), Federal Reserve Bank of Dallas, andFederal Reserve Bank of Atlanta. Themovement of these indexes over time ispresented in Figure 1, and a summary oftheir construction characteristics is pro-vided in Table 1. In sorting through thevarious choices in constructing an index, itmay be helpful to keep in mind a generalprinciple: The use of the index shouldguide its construction.4
Which Formula?Suppose the world has three curren-
cies—the dollar, Currency x and Currency y.Further suppose that in the first year one
dollar could buy 25 units of Currency x. Inthe second year one dollar could buy 50units of Currency x, and in the third year adollar could buy 100 units of Currency x.With respect to Currency y, one dollarcould buy 40 units in the first year, 20units in the second year, and 10 units inthe third year. The dollar rose in valueagainst Currency x—over time one dollarcould buy more and more units of this cur-rency. In contrast, the dollar fell in valueagainst Currency y—over time one dollarcould buy fewer and fewer units of thiscurrency. Note that compared with the firstyear, one dollar could buy twice as manyunits of Currency x and half as many unitsof Currency y in the second year, and fourtimes as many units of Currency x and one-quarter as many units of Currency y in thethird year.
What happened to the overall valueof the dollar? There are two methods ofcalculating an average value for the dol-lar: an arithmetic mean or a geometricmean. Each method compares the effec-tive value of the dollar with its value in agiven period, for example, relative to thefirst year. An arithmetic mean computes asimple average. In Year 1 the effective ex-change rate using the arithmetic mean is
,
where ex,1 is the Currency x/dollar ex-change rate in Year 1, and ey,1 is the Cur-rency y/dollar exchange rate in Year 1. InYear 2 the effective exchange rate usingthe arithmetic mean is
,
where ex,2 and ey,2 are the Currency x/dollarexchange rate and the Currency y/dollar ex-change rate, respectively, in Year 2. Similarly,in Year 3 the effective exchange rate usingthe arithmetic mean is
e
e
e
e
x
x
y
y
,
,
,
,
+
= +
=
e
e
e
e
x
x
y
y
,
,
,
,
+
= +
=
4 Following this general principlewill not necessarily mean thatthe constructed exchange ratemeasure will generate superiorresults when used in a specificcase. See Belongia (1986) foran empirical demonstration sup-porting such a conclusion in thecontext of U.S. agricultural ex-ports. See Deephouse (1985)and Hooper and Morton(1978) for an elaboration ofthe uses of effective exchangerate indexes.
where ex,3 and ey,3 are the Currency x/dollarexchange rate and the Currency y/dollar ex-change rate, respectively, in Year 3. The re-sulting number in each year is generallymultiplied by 100 to create an easily usableindex. Thus the effective exchange rate in-dex for the three years is 100, 125, and212.5.
The geometric mean in Year 1, againusing the first year as the base year, is
.
In the Year 2 the geometric mean is
.
In Year 3 the geometric mean is
.e
e
e
e
x
x
y
y
,
,
,
,
×
= ×
=
e
e
e
e
x
x
y
y
,
,
,
,×
= ×
=
e
e
e
e
x
x
y
y
,
,
,
,
×
= ×
=
e
e
e
e
x
x
y
y
,
,
,
,
+
= +
=
Multiplying the resulting number in eachyear by 100 produces the following index forthe three years: 100, 100, 100.
Using the arithmetic mean, the effectivevalue of the dollar rose over the three-yearperiod, whereas using the geometric mean,the effective value of the dollar was un-changed. The result based on the geometricmean seems more reasonable, given that therise in the value of the dollar against Cur-rency x is offset by the fall in the value ofthe dollar against Currency y. The arithmeticmean created an upward bias.5 The Board ofGovernors of the Federal Reserve System,when it switched from using an arithmeticmean to a geometric mean to construct itseffective exchange rate index for the dollar,noted that “as currencies diverged from eachother over time, changes in currencies thatrose against the dollar had a reduced impacton the index while changes in currenciesthat fell against the dollar had an increasedimpact on the index. As a result, arithmeticaveraging imparted a systematic upward biasto the measurement of changes in the dol-lar’s average exchange value.”6
Because of the bias inherent in an indexbased on arithmetic averaging, all the effec-tive exchange rate indexes shown in Figure1 use a geometric averaging technique. Ofthe six decisions involved in constructing aneffective exchange rate index, this choice of
5 It is not mandatory that the di-rection of the bias be upward.If Year 3 had been used as thebase year, the index using thearithmetic average would be212.5, 125, 100 and theindex using geometric averag-ing would be 100, 100, 100.In this example, arithmetic av-eraging would have created adownward bias.
6 See Board of Governors(1978), p. 700.
Table 1
Construction Features of Effective Exchange Rates for the Dollar
Number of Trade-WeightProducer Years Covered Countries Period Weighting Scheme
Federal Reserve Board 1967–present 10 1972–1976 MultilateralJ.P. Morgan (narrow) 1970–1986 15 1980 Double (manufactures)
Bank of Dallas averageFederal Reserve 1973–present 18 1984 Bilateral
Bank of Atlanta
1
2
1
2
100
25
10
402 1253
1
3
1
x
x
y
y
e
e
e
e
,
,
,
,. ,+ = + =
x
x
y
y
e
e
e
e
,
,
,
,.1
1
1
1
1
21
225
25
40
401+ = + =
x
x
y
y
e
e
e
e
,
,
,
,.2
1
2
1
1
21
250
25
20
401+ = + =
x
x
y
y
e
e
e
e
,
,
,
,. .3
1
3
1
1
21
2100
25
10
401+ = + =
a geometric average is the only one onwhich there is consensus.
The generic formula, using geometricaveraging, for the value of the effective ex-change rate index at time t is
(1) Indext = 100n
Πi=1
1 2wit
,
where Π is the product over the n for-eign currencies in the index, eit is thenumber of units of Currency i per dollarat time t; eib is the number of units ofCurrency i per dollar in the base period;and wit is the weight assigned to Cur-rency i at time t.
In the above example, each currencywas given equal weight in each period, wit = 1/2 and the base period was Year 1. In actually constructing an exchange rateindex, developers must make numerousdecisions involving the currencies in-cluded, the weights for the currencies, andthe base periods. An elaboration of the keydecisions is provided below.
Which Currencies?Ideally, an effective exchange rate for
the dollar should include all currenciesfor which the dollar is exchanged. Suchan ideal, however, is tempered by the re-ality that the construction of the index re-quires timely, reliable data. As a result,most indexes are limited to the currenciesof the principal industrial economies.Table 1 shows that most indexes use dataon the dollar relative to the currencies ofbetween 10 and 20 countries. The majorexceptions are the broad index producedby J.P. Morgan that uses the currencies of44 countries relative to the dollar and theindex produced by the Federal ReserveBank of Dallas that currently uses thecurrencies of 128 countries.
The index produced by the Federal Re-serve Board uses data on the dollar relativeto the currencies of the other nine mem-bers of the Group of Ten—Belgium,Canada, France, Germany, Italy, Japan,Netherlands, Sweden, United Kingdom—
plus Switzerland. These countries were se-lected for several reasons.7 First, eachcountry has a well-developed foreign ex-change market with exchange rates that de-pend primarily on the supply and demanddecisions of private individuals and firms.Second, these countries are involved in themajority of U.S. trade and capital flows.Third, many of the countries excludedfrom the index either attempt to keep theircurrencies pegged to an included currencyor use one of the included currencies fortheir international transactions.
The countries whose currencies are in-cluded in the index produced by the Fed-eral Reserve Board are located in Europe,except for Canada and Japan. Clearly, thisindex includes the major traded currenciesand consequently allows an assessment ofchanges in the value of the U.S. dollar rela-tive to the other major currencies. Theother five indexes discussed here use the10 currencies in the Board’s index, but theyadd other currencies as well.8 For example,the narrow index produced by J.P. Morganadds currencies from seven Europeancountries—Austria, Denmark, Finland,Greece, Norway, Portugal and Spain—plusAustralia. The currencies of Finland,Greece, and Portugal did not appear in theindex until 1987. The IMF index adds thecurrencies of Ireland and New Zealand tothe J.P. Morgan narrow index. The IMFindex therefore contains the currencies ofall the major industrialized countries.9 TheAtlanta index adds the currencies of Tai-wan, Hong Kong, South Korea, Singapore,and China, as well as those of Australia,Spain, and Saudi Arabia, to the Board’sindex. The addition of the currencies of thefirst five countries is justified by the shift-ing pattern of U.S. trade toward developingcountries in Asia.10 In addition to a narrowindex for the United States, J.P. Morganproduces a broad index that uses the cur-rencies of most member countries of theOrganization for Economic Cooperationand Development plus numerous develop-ing countries.11 The ultimate in inclusive-ness is the index produced by the FederalReserve Bank of Dallas, which currently includes 128 currencies.12
eit
}eib
7 See Hooper and Morton(1978).
8 Whether indexes with a broadrange of currencies are superiorto those using a small range ofcurrencies is an empirical ques-tion. See Batten and Belongia(1987) for an empirical studyof U.S. trade flows indicatingthat measures based on morecurrencies performed no betterthan the measures based onfewer currencies.
9 J.P. Morgan and the IMF pro-duce effective exchange rateindexes for each of the curren-cies included in the U.S. dollarindexes.
10 For more on the choice of cur-rencies in the Atlanta index, seeRosensweig (1986a and b).
11 The 26 countries included inJ.P. Morgan’s broad, but not itsnarrow, index are Ireland, NewZealand, Turkey, Argentina,Brazil, Chile, Colombia,Ecuador, Mexico, Peru,Venezuela, Hong Kong, Indone-sia, South Korea, Malaysia,Philippines, Singapore, Taiwan,Thailand, India, Kuwait, Mo-rocco, Nigeria, Pakistan, SaudiArabia, and South Africa.
12 Cox (1986) stressed that theindex contained all U.S. tradingpartners; however, the indexcontains few currencies fromEastern European countries andcountries that were formerlypart of the Soviet Union.
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Which Measure of Economic Activity?
Deciding how many countries to in-clude in the index also requires decisionsconcerning how much importance shouldbe attached to the currency from a particu-lar country. In other words, the relativeimportance of a currency is determined byits weight in the average. Before determin-ing the weight of a particular currency, re-searchers must decide which measure ofeconomic activity is used in the calcula-tion of the weights.
Because effective exchange rate in-dexes are most often constructed to mea-sure changes in a country’s internationalcompetitiveness, generally some measureof international trade is used to calculatethe weights. For this reason, effective ex-change rates are frequently termed trade-weighted exchange rates. Internationaltrade, however, is not the only measure ofinternational economic activity that couldbe used. The exchange value of the dollaris determined by supply and demandforces involving the international ex-change of goods, services, and assets. Indi-viduals, firms, and governments demand(buy) dollars in foreign exchange marketsto purchase goods, services, or assets de-nominated in U.S. dollars. Likewise, indi-viduals, firms, and governments supply(sell) dollars in foreign exchange marketsto purchase goods, services, or assets de-nominated in foreign currencies. For ex-ample, a U.S. auto dealer wanting to im-port BMWs must first obtain Germanmarks and so supplies dollars and de-mands marks. Any country wanting to im-port petroleum must pay in U.S. dollarsand so must first exchange its own cur-rency for dollars, supplying its currencyand demanding dollars. A Japanese in-vestor who wants to buy U.S. Treasury se-curities must first obtain U.S. dollars andso supplies yen and demands dollars.
Though trade flows are used to calcu-late the weights given to each currency inan effective exchange rate index, based oninternational financial movements, onecould use international capital flows to de-
termine the weights. Both the absolute lev-els and the rapid growth rates of interna-tional capital flows suggest that capitalflows might currently be a more importantdeterminant of exchange rates than tradeflows. Thus using capital flows, the curren-cies of countries with larger investmentand portfolio flows are more important inthe determination of the value of the dollarthan are the currencies of countries withsmaller investment and portfolio activity.Even though such a calculation is reason-able on theoretical grounds, no major pro-ducer of effective exchange rates uses capi-tal flows to construct its measures.13
A key reason trade is used for weight-ing purposes is that, although trade data aresubject to errors, they are much easier toobtain on a timely basis than capital flows.Different indexes, however, use differentmeasures of international trade. Generallyspeaking, most indexes are constructedusing total merchandise trade and do notinclude services, which have tended to in-crease rapidly in recent years. The indexesproduced by J.P. Morgan and the IMF, how-ever, use only trade in manufactures.
Which Weighting Method?Another issue in weighting the impor-
tance of a specific currency involves theselection of a weighting scheme. If the ef-fective exchange rate index is to reflectchanges in a country’s international com-petitiveness, then ideally the weightsshould be chosen to reflect the responsive-ness of a country’s trade flows to changesin exchange rates. A theoretically basedindex was previously produced by theIMF: the Multilateral Exchange RateModel (MERM) index. In the U.S. dollarMERM index, for example, the weightgiven to each currency was chosen so thatany combination of changes in the curren-cies against the dollar leading to a one per-cent change in the index would have thesame effect on the U.S. trade balance (overa 2-3 year period) as a one percent changein the dollar against each currency in theindex. Estimation of the weights requiredthe use of an econometric model incorpo-
13 See Ott (1987) for a more ex-tensive discussion and illustra-tion of a capital-weighted ex-change rate.
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rating information on price elasticities, ex-change rate effects on domestic prices, andthe policy response of the economy. Con-cerns about the unreliability of the modeldetermining the weights led to the aban-donment of the MERM and similarly con-structed indexes.14
Three other methods of weighting re-main in use: bilateral, multilateral, and dou-ble weights.15 With bilateral weighting, eachcountry is weighted by the proportion of itsshare of the total trade flows to and fromthe United States of the countries used toconstruct the index. Thus the weight forCountry i is simply the sum of U.S. exportsto and imports from Country i divided bythe sum of U.S. exports to and imports fromall countries included in the index. Assum-ing that n countries are used to constructthe index, the weight for Country i is:
(2) wi = ,
where USXi is the exports from the UnitedStates to Country i and USMi is the importsof the United States from Country i.16
With multilateral weighting, eachcountry is weighted by the proportion ofits share of total trade flows throughoutthe world. Thus the weight for each Coun-try i is the sum of Country i’s worldwideexports and imports divided by the sum ofthe worldwide exports and imports of allthe countries included in the index. Onceagain, assuming that n countries are usedto construct the index, the weight forCountry i is:
(3) wi = ,
where WXi is the worldwide exports ofCountry i and WMi is the worldwide im-ports of Country i.
Neither alternative is obviously supe-rior. The multilateral weighting approachattempts to capture the competition be-
tween two countries in countries outside oftheir domestic markets. For example, achange in the Japanese yen-U.S. dollar ex-change rate can affect relative prices ofJapanese goods, American goods, andgoods from other countries besides Japanand the United States, such as Canada. The multilateral approach used in the con-struction of the index produced by the Fed-eral Reserve Board seems more suitable foraccounting for these third-country effects.On the other hand, it is possible that themultilateral weighting approach gives toomuch weight to nations that trade more ex-tensively with each other than with theUnited States. For example, EuropeanCommunity countries that trade exten-sively with each other are likely to receivehigher-than-warranted weights in the construction of an index for the UnitedStates. A possible result in the case of an effective exchange rate for the United Stateswould be that Canada, the largest U.S. trad-ing partner, would be weighted less thanwarranted. In this case, a bilateral weight-ing approach that is used in the indexesproduced by the Federal Reserve Bank ofDallas and the Federal Reserve Bank of At-lanta might be more appropriate.
The double weighting method, whichis used in the indexes produced by theIMF and J.P. Morgan, attempts to combinethe advantages of both the bilateral andmultilateral weighting schemes: recogni-tion of competition in third markets andthe strength of links between particulartrading partners. In addition, the doubleweighting method recognizes the com-petitive position of domestic producers of import substitutes and therefore re-quires information on production for local consumption as well as on tradeflows.17 In the dollar index, the weights reflect both the competition U.S. ex-portersface from other countries’ exporters andfrom the local countries’ producers.
Which Base Period for Weights?The fifth major issue in the construc-
tion of an effective exchange rate is thechoice of a base period for the trade flows
WXi + WMi}}
^n
i=1
(WXi + WMi)
USXi + USMi}}
^n
i=1
(USXi + USMi)
14 Turner and Van ‘t dack (1993)provide a good overview of theconstruction and problems as-sociated with the MERM index.
15 Bilateral weights were used inthe original work on effectiveexchange rates, see Hirsch andHiggins (1970).
16 To simplify the discussion wehave omitted all references totime. Obviously, the tradeflows cover a particular periodand the weight for a countrypertains to a particular period.As indicated by equation 1 anddiscussed in the next section,the weight for a country maychange over time.
17 See Hargreaves (1993) for de-tails on how the J. P. Morganindex is constructed. Turner andVan ‘t dack (1993) provide ageneral analysis of the doubleweighting method.
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on which the weights are based. The indexmay use fixed weights, weights that are updated periodically, or weights that areupdated annually. For example, the FederalReserve Board’s index uses fixed weightsthat have remained unchanged; the J.P.Morgan indexes use different weights forthe period from 1970 to 1986 and the pe-riod from 1987 to the present; and the in-dex produced by the Federal Reserve Bankof Dallas uses a three-year moving averageto continually update its weights.18 If fixedweights are used, then researchers must de-cide which year or years should be used.For example, the Federal Reserve Bank ofAtlanta index uses 1984 trade figures, theFederal Reserve Board index uses tradedata from 1972 to 1976, and the IMF indexuses trade data from 1989 to 1991.
The existence of various base periodssuggests that there is no obviously superiorbase period. Fixing the base period for thetrade weights means that the index doesnot incorporate the effect of changing trade patterns. Thus a shifting pattern oftrade raises the possibility that a fixed-weight index becomes a less reliable ex-change rate measure over time. On theother hand, a potential problem stemmingfrom updating the weights annually is thatthe effects of exchange rate changes maybe confounded with changes caused byshifting weights in the index. It is possible,because of shifts in trade shares, that an ef-fective exchange rate may change even if no individual exchange ratechanges.
Table 2 illustrates this point. The upper half of the table shows the results of calculating a hypothetical trade-weighted exchange rate index for the U.S.dollar assuming fixed weights for each currency based on trade shares at somepoint. The weight for Country 1 is 0.7,whereas the weight for Country 2 is 0.3.The lower half of the table shows the re-sults of calculating a hypothetical trade-weighted exchange rate index for the U.S.dollar assuming that the weights given toeach currency are updated annually. In the example, the weight for Country 1 declines from 0.7 in Year 1 to 0.3 in Year 7,
whereas the weight for Country 2 increasesfrom 0.3 in Year 1 to 0.7 in Year 7.
Between Year 5 and Year 6, the valueof the dollar was unchanged against bothcurrencies as 61 units of Country 1’s cur-rency and 17 units of Country 2’s currencycould be traded for one U.S. dollar in eachyear. The index calculated using fixedweights shows no change in the effectiveexchange value of the dollar. For example,assuming that the effective exchange ratein Year 1 equals 100, then the rate in bothYear 5 and Year 6 is 144.4. When weightsare updated often, however, the effectiveexchange value of the dollar does change.For example, assuming that the effectiveexchange rate in Year 1 equals 100, thenthe rate in Year 5 is 93.3 and the rate inYear 6 is 78.4.
Thus changes in an index withweights that are updated annually alwaysleave doubt as to whether changes in theindex reflect exchange rate changes orshifting trade weights. On the other hand, if trade patterns shift, then the useof fixed weights may cause the index toproduce misleading signals. This is highly likely over long periods. A com-promise is to change the weights period-ically; however, it is not obvious how fre-quently weights should be changed.
Which Base Period for Exchange Rates?
The effective exchange rate indexshown in Equation 1 calculates changes inthe exchange rate of the domestic currency(for our purposes the U.S. dollar) relative toeach foreign currency from a base exchangerate. The Federal Reserve Board uses theMarch 1973 exchange rates as the baserates.19 The Federal Reserve Bank of Atlantauses 1980. The Federal Reserve Bank ofDallas uses the exchange rate averages forfirst quarter 1985 as the base. The IMF andJ.P. Morgan use the exchange rate averagesfor 1990 as the base. As Equation 1 indicates,the index in the base period equals 100.
The creation of effective exchange rateindexes differs from that of most price in-
18 For example, trade data for1992–94 is used for calculat-ing the index in 1995.
19 This period reflects the start ofthe flexible exchange rate era.
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dexes in the use of two base periods. Forexample, in the consumer price index thebase period for prices is exactly the sameas the base period for quantities. In effec-tive exchange rate indexes the base periodsfor weights and for exchange rates are gen-erally different. The Atlanta index, for ex-ample, uses 1984 as the base period for thetrade data used to construct the weightsbut uses first quarter 1985 as the base pe-riod for exchange rates.
The choice of the base exchange rateperiod is irrelevant to the picture of thedollar’s strength or weakness as measuredby indexes with fixed trade weights. Whenthe weights are updated annually, however,the calculated percentage changes in thevalue of the dollar become sensitive to thebase period for the exchange rates.20 The
example in Table 2 can be used to illus-trate this problem. Two versions of thefixed trade weights and annually updatedtrade weights indexes are calculated. Oneversion uses the exchange rates in Year 1as the base rates. The other version uses
the exchange rates in Year 7 as the baserates. When the trade weights are fixed,changing the base year does not affect thepercentage change in the exchange rateindex. As shown in the last two columnsof the top panel of Table 2, the percentagechange in the effective exchange rate be-tween any two years is the same regardlessof whether Year 1 or Year 7 is used as thebase year. As shown in the top panel ofFigure 2 under either base year for the ex-change rate index, the index indicates anappreciation of the dollar through Year 5, a
20 This issue is explored exten-sively in Coughlin, Pollard andBetts (1996).
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Exchange Rate Indexes: Alternative Updating Procedures for Weights*
Fixed Trade WeightsExchange Percent Change
Rates Weights Index in IndexYear e1 e2 w1 w2 Year 1 5 100 Year 7 5 100 Year 1 5 100 Year 7 5 100
* Note that e = foreign currency per dollar. Percentage changes are calculated on a logarithmic basis from the preceding year to the current year.
Table 2
constant value of the dollar from Year 5 toYear 6, and a slight appreciation of thedollar in Year 7.
The effective exchange value of thedollar, however, is affected by the choiceof the base period for the exchange rate when the trade weights are updated annu-ally. As shown in the bottom halves ofTable 2 and Figure 2, if exchange rates inYear 1 are used as a base, the effective ex-change value of the dollar appreciates untilYear 3 and depreciates thereafter. If ex-change rates in Year 7 are used as the base,the effective exchange value of the dollarrises through Year 6 and falls in Year 7.Note that whereas the value of the dollaris constant between Year 5 and Year 6using fixed trade weights, when the tradeweights are continuously updated, the ef-fective exchange rate index indicates either a depreciation or an appreciation ofthe dollar, depending on the base periodfor the index.
WHAT ACCOUNTS FOR DIFFERENCES IN THE EXCHANGE RATE INDEXES?
Because exchange rates indexes areconstructed differently, it is not surprisingthat the picture they give of the value ofthe dollar may differ. The previous sectionexplained the choices creators of effectiveexchange rate indexes face in designing anindex. This section concentrates on twopopular indexes––the Federal ReserveBoard (Board) index and the Federal Re-serve Bank of Dallas (Dallas) index––to illustrate which factors are the most impor-tant in accounting for differences in the be-havior of the two indexes. As Figure 1shows, these two indexes were qualita-tively similar between January 1976 andMarch 1985 but differed sharply betweenMarch 1985 and December 1995. Accord-ing to Table 3, during the early period theBoard index showed a 43 percent apprecia-tion of the U.S. dollar, whereas the Dallasindex showed a substantially larger appre-ciation of the dollar, 77 percent. During thelater period the Board index showed a 62
percent depreciation of the dollar. In sharpcontrast, the Dallas index showed a 60 per-cent appreciation of the dollar. Over thesample period 1976–95 there was little correlation between the two indexes, asshown by the correlation coefficient of 20.27 in Table 4. In the early period theindexes were highly positively correlated(0.91), but exhibited a negative correlation(20.50) in the later period.
The construction of the Board andDallas indexes differs in three aspects: themethod used to calculate the trade weights, the base period for the tradeweights, and the choice of currencies ineach index.21 The Board index uses multi-lateral trade shares, whereas the Dallasindex uses bilateral trade shares. The
21 The Board and Dallas indexesalso differ in their choice ofbase period used for their ex-change rates. To eliminate anyproblems caused by this differ-ence, we recalculated theBoard index using the March1985 exchange rates as thebase rates.
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Figure 2
Exchange Rate Indexes:Fixed WeightsUsing Different Base Years for the Exchange Rates
160
140
120
100
80
601 2 3 4 5 6 7
Year
Inde
x
Year 1=100 Year 7=100
Exchange Rate IndexesAnnually Updated WeightsUsing Different Base Years for the Exchange Rates
1 2 3 4 5 6 7Year
120
110
100
90
80
70
60
50
Year 1=100 Year 7=100
Inde
x
weight assigned to each currency in theBoard index is fixed, whereas the weightsin the Dallas index are updated annually.Specifically, the weights used in the Boardindex were determined by the averagetrade share of each country whose cur-rency is included in the index for the pe-riod 1972–76. In contrast, in the Dallasindex, the weights used in a given year arebased on the average trade shares over theprior three-year period. Last, the curren-cies of 10 countries are used in the Boardindex, whereas the currencies of 128countries are used in the Dallas index.
This section examines the importanceof each of these three aspects in account-ing for the differences between the two in-dexes. It does so by creating five variationson the Board index—BilBoard, MupBoard,BupBoard, CmBoard, and CmupBoard—
shown in Figure 3. Each variation modi-fies the construction of the Board index sothat it is more closely in accord with theDallas index. These new indexes are usedto determine what causes the differencesbetween the Board and the Dallas indexes.
Table 5 presents an overview of thesefive indexes, comparing them with theBoard and the Dallas indexes. The BilBoardindex is constructed using the same 10 cur-rencies as in the Board index and the fixedweights based on 1972–76 trade shares ofeach country. However, whereas the Boardindex uses the world trade of each countryto determine the weight given to its cur-rency in the index, the BilBoard index usesonly the bilateral trade flows of the 10countries with the United States. Contrast-ing this index with the Board and Dallas in-dexes allows us to determine the impor-
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Table 4
Correlations Among Trade-Weighted Exchange Rate Indexes
Correlation with the Board IndexPeriod Dallas BilBoard MupBoard* BupBoard CmBoard CmupBoard
* The data period for the MupBoard index ends in December 1994.
tance of the multilateral/bilateral tradeshare choice in explaining the differencesbetween the latter two indexes.
The MupBoard index differs from the Board index solely in the type of thebase period for the weights given to eachcurrency. Trade weights in the MupBoardindex are updated annually, using a three-year moving average as in the Dallasindex. The MupBoard index can be con-trasted with the Board and Dallas indexesto determine the importance of the updat-ing of weights in accounting for the differ-ences between the latter two indexes.
The remaining difference between theBoard and Dallas indexes is the choice ofcurrencies used in each index. We createdthree variations on the Board index to ex-amine the importance of currency choice.First we created BupBoard, an index thatwas identical to the Dallas index exceptthat only the ten currencies used in theBoard index were included in its calcula-tion. Thus any differences in the behaviorof the BupBoard and Dallas indexes couldbe attributed to the difference in currencychoice between the Board and Dallas in-dexes. To further explore the importance of currency choice, we added the curren-cies of China and Mexico to a bilateral–trade share version of the Board index.Mexico was chosen because it has consis-tently been the most important U.S. tradingpartner excluded from the Board index.China is currently the next most importanttrading partner missing from the Board in-dex. Its relative importance, as shown inTable 6, has grown substantially over thelast 20 years. In 1976 the Chinese yuan re-ceived a weight of only 0.4 percent in theDallas index, but its weight rose to 3.9 per-cent by 1995. Using the Chinese yuan andMexican peso, we created two more in-dexes. In the CmBoard index, the weightsgiven to each of the 12 currencies are de-termined by each country’s share of tradewith the United States. This index there-fore differs from the Board index in twoways: it includes China and Mexico anduses bilateral trade shares. The CmupBoardindex is constructed in the same manner asthe CmBoard index except that the weightsassigned to each currency are updated an-
nually using a three-year moving average.The CmupBoard index therefore is identi-cal to the Dallas index except that it in-cludes only 12 currencies, not 128.
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Table 5
Overview of Variations on the Board and Dallas Indexes*
Trade Base PeriodIndex Shares for Weights Currencies
* Country whose currency is included in the Board index.† Not in the top 10 in this year.
Bilateral vs. Multilateral TradeShares—BilBoard
As shown in Table 7, the weights as-signed to each currency in the Board andthe BilBoard indexes vary substantially.For example, the weight given to theCanadian dollar is more than 30 percent-age points higher in the BilBoard indexthan in the Board index. The reason forthis difference is that although Canada isthe most important U.S. trading partner, itis less important in worldwide trade. Japanalso holds a higher share of U.S. trade thanworldwide trade, but the other eight coun-tries rank higher in worldwide trade ratherthan in trade with the United States. As aresult, the weight given to the Japanese
yen is more than seven percentage pointshigher in the BilBoard index than in theBoard index, whereas the other eightcountries receive less weight in the Bil-Board index than in the Board index.
These weight changes produce somenoteworthy differences in the two indexesthat are shown in the top panel of Figure3. Table 3 reveals that between January1976 and March 1985, the dollar appreci-ated 43 percent according to the Boardindex and 30 percent according to the Bil-Board index. Accounting for this differ-ence is relatively straightforward. The U.S.dollar rose by less against the Canadiandollar during the 1976–85 period than itdid against some currencies that receivedhigher weights than the Canadian dollar inthe Board index (for example, the Frenchfranc and the British pound). With respectto the Japanese yen, the U.S. dollar fellduring the 1976–85 period. Furthermore,since March 1985, the dollar has changedlittle relative to the Canadian dollar, fallingonly 1 percent. The dollar has fallen farmore against the remaining nine curren-cies since 1985. As a result, the BilBoardindex shows a less pronounced change inthe dollar over the sample period thandoes the Board index.
The direction of the movement in theBilBoard index, however, closely matchesthat of the Board index as shown by thehigh degree of correlation between the twoin Table 4. The correlation was 0.98 overthe entire period. Meanwhile, the correla-tion between the BilBoard index and theDallas index, even though high during1976–85, is negative during 1985–95 andnegative over the entire sample period1976–95. In sum, the differences betweenthe Board and the Dallas indexes cannotbe primarily attributed to a difference inthe method used to calculate the weightsof each currency.
Base Period for Trade Weights—MupBoard
The multilateral trade shares of thecountries used in the MupBoard index for1976, 1985, and 1994 are shown in Table 7.
These trade shares did not change substan-tially over time. As a result, the MupBoardindex closely mimics the Board index, asshown in the middle panel of Figure 3. Bothindexes show the same percentage apprecia-tion of the dollar between January 1976 andMarch 1985 and nearly the same deprecia-tion from March 1985 through 1994.22 Like-wise, the two indexes were nearly perfectlycorrelated. Thus one can conclude that thefrequency of updating weights is not the dri-ving force for differences in the Board andDallas indexes.
Currency Choice—BupBoard, CmBoard and CmupBoard
The top panel of Figure 3 shows thatthe BupBoard index closely mimics the be-havior of the BilBoard index, particularlyin the 1976–85 period when the weightsfor the two indexes, listed in Table 7, aresimilar. In the 1985–95 period, as Japan’sshare of U.S. trade rises, the BupBoardindex shows a slightly larger deprecia-tion of the dollar than the BilBoard index.This result follows from the fact that during this period the U.S. dollar fell bymore against the yen than against any of
the other currencies included in the index.
The behavior of the BupBoard indexresembles that of the Board index. For ex-ample, Table 3 shows a 17 percent depreci-ation of the dollar using the BupBoardindex from January 1976 to December1995, whereas the Board index shows a 19 percent depreciation of the dollar. Dur-ing this period the Dallas index shows thedollar appreciating by 137 percent. Theseresults are reinforced by the correlation coefficients shown in Table 4. The Bup-Board index is highly correlated with theBoard index in the 1976–95 period (0.97)but negatively correlated with the Dallasindex (20.45). Changing the manner andfrequency with which the weights are cal-culated to accord with the Dallas index didnot create an index that resembled the Dal-las index. Thus the primary cause of thedifferences between the two indexes mustbe the selection of countries in each index.
The CmBoard index allows us to fur-ther explore the importance of countrychoice. In the CmBoard index, the weightsgiven to each currency are determined bythat country’s share of trade with the
22 Worldwide trade data for someof the countries used in theindex were not available for1994; therefore, the Mup-Board index ends in 1994.
* Note that weights in the Board index are based on multilateral trade shares during 1972–76. Weights in the BilBoard and CmBoard indexes are based on bilateral trade shares during 1972-76. Weights in theMupBoard, BupBoard, and CmupBoard indexes are based on three-year moving average bilateral trade shares, updated annually. Thus, the weights in the three columns: 1976, 1985, and 1995 (1994 forMupBoard), are based on trade shares during 1973–75, 1982–84, and 1992–94, (1991–93 for MupBoard), respectively.
United States.23 This index therefore differsfrom the Board index in two ways: its in-clusion of China and Mexico and the useof bilateral trade shares. The behavior ofthe CmBoard index, shown in the bottompanel of Figure 3, is similar to the Boardindex over the January 1976–March 1985period. As shown in Table 3, the CmBoardindex appreciated 46 percent, whereas theBoard index appreciated 43 percent. Agreater difference between the CmBoardand the Board indexes occurs over the pe-riod from March 1985 to December 1995.The CmBoard index shows an 18 percenttrade-weighted depreciation of the dollarduring this period, while the Board indexshows a 62 percent depreciation. The Cm-Board index, however, does not show anappreciation of the dollar as the Dallasindex does during this period. That thechanges embedded in the CmBoard indexcause it to become more similar to theDallas index and less similar to the Boardindex is reinforced by the correlation coef-ficients in Table 4. For the entire period,the correlation of the CmBoard index withthe Board index is much lower than theBilboard, MupBoard, and BupBoard in-dexes, whereas its correlation with the Dallas index is positive rather thannegative.
The CmupBoard index, which also in-cludes China and Mexico, still does notshow the magnitude of the appreciation ofthe dollar in the bottom panel of Figure 3that the Dallas index indicates in the Janu-ary 1976–March 1985 period. In contrast,however, to all of the previously con-structed indexes, it does show an apprecia-tion of the dollar during the March1985–December 1995 period, althoughthis appreciation is less than that indicatedby the Dallas index. For the entire period,the CmupBoard index shows little correla-tion with the Board index but is highlycorrelated with the Dallas index.
The CmBoard and the CmupBoard in-dexes illustrate two key points. The first isthat the Dallas index differs from theBoard index primarily because the Dallasindex includes currencies whose behavior,particularly during the March 1985–
December 1995 period, was in sharp con-trast to the behavior of the currencies in-cluded in the Board index. Specifically, theDallas index includes currencies againstwhich the dollar appreciated substantiallyduring this period. Between March 1985and December 1995, the dollar rose by362 percent against the Mexican peso. Incontrast, the dollar fell against all of thecurrencies included in the Board indexduring this period.
The second point is that in an index inwhich there are sharp differences in thebehavior of the currencies (such as theDallas index), the weights assigned to eachcurrency matter. In the Board index thebehavior of the currencies was relativelysimilar: The dollar rose against all 10 cur-rencies with the exception of the Japaneseyen during the early period and fellagainst all 10 currencies during the laterperiod. Given such similarities in the be-havior of the currencies, the manner inwhich the weights were calculated—bilat-eral or multilateral trade shares—and thefrequency of updating of the weights hadlittle effect on the behavior of the indexes.However, when the behaviors of the cur-rencies in the index differ greatly, as evi-denced by the enormous appreciation of the dollar against the Mexican peso during the same period in which the dollar was depreciating against the curren-cies of the major industrialized countries,the method of calculating the weights assigned to each currency increases in im-portance.
This latter point is illustrated by thedifferences in the CmBoard and the Cmup-Board index. The dollar appreciatedagainst the Chinese yuan by 107 percentbetween March 1985 and December 1995.This appreciation, however, has little effecton the trade-weighted value of the dollarwhen the weight assigned to the yuan isbased on China’s share of U.S. trade overthe 1972–76 period (as in the CmBoardindex). With annual updates of theweights, as in the CmupBoard index, thegrowth in China’s share of U.S. tradeplaces increased importance on the appre-ciation of the dollar against the yuan.
23 We were unable to constructan index using multilateraltrade shares that includedChina and Mexico becauseworld trade data for China before 1982 are unavailable.
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Likewise, the appreciation of the dollaragainst the peso is given greater weight inthe index with annual updates. If theweights used in the CmBoard index hadbeen based on the 1992–94 trade shares,the index would have shown a sharper ap-preciation of the dollar than that evi-denced by the CmupBoard index.
The difference between the Board andthe Dallas indexes does not simply resultfrom the fact that the Dallas index in-cludes more countries than the Boardindex. Two factors make the countrychoice important: (1) the Board index ex-cludes (the Dallas index includes) coun-tries that account for a significant share ofU.S. total merchandise trade; and (2) thebehavior of the excluded currenciesagainst the dollar has been substantiallydifferent since 1985 from that of the cur-rencies included in the Board’s index. Theimportance of the first factor has increasedover time. In 1976, as shown in Table 6,seven of the 10 currencies that constitutethe Board index were among the 10 mostheavily weighted currencies in the Dallasindex. By 1995, only five of the countriesincluded in the Board index also were inthe top 10 of the Dallas index.
Our analysis indirectly identifies animportant consideration in using trade-weighted exchange rate indexes as a mea-sure of international competitiveness.Generally speaking, changes in real (thatis, nominal exchange rates adjusted for in-flations difference), rather than nominalexchange rates, are commonly used for as-sessing changes in international competi-tiveness. Since the inflation experience ofthe countries whose currencies are in theBoard index has been roughly similar overtime, the nominal Board index mimics itsreal counterpart. The Dallas index, how-ever, includes countries that have experi-enced periods of hyperinflation. As a resultof this hyperinflation, the currencies ofthese countries depreciated sharply againstthe dollar during these periods, driving theappreciation of this index between 1985and 1995. After adjusting for the inflationdifferences, the real Dallas index declinesbetween 1985 and 1995.
CONCLUSIONOur examination of effective exchange
rates reveals the many decisions underly-ing their construction. These decisions canproduce substantially different views ofchanges in the average foreign exchangevalue of a currency. The actual effect ofthese decisions was investigated by com-paring the Board index with the Dallasindex.
The difference between the Boardindex and the Dallas index is driven pri-marily by the choice of currencies. Thisdoes not mean, however, that issues suchas the determination of trade shares andthe frequency with which weights are up-dated are unimportant. What makes theselatter factors unimportant in the Boardindex is the similarity in the behavior ofthe currencies that make up the index.This also illustrates why all of the trade-weighted exchange rate indexes covered inthis article show an appreciation of thedollar between 1976 and 1985. Duringthis period, and particularly after 1980,the dollar was appreciating against mostother currencies. Since 1985, the behaviorof the dollar has been markedly differentagainst the currencies of the industrializedcountries from its behavior against thecurrencies of the developing countries.Thus even though we have not provided adefinitive answer to the question posed inthe title of this article, the reasons for themeasurement differences have been illumi-nated.
REFERENCESBatten, Dallas S., and Michael T. Belongia. “Do the New Exchange Rate
Indexes Offer Better Answers to Old Questions?” this Review (May1987), pp. 5–17.
Belongia, Michael T. “Estimating Exchange Rate Effects on Exports: A Cautionary Note,” this Review (January 1986), pp. 5–16.
Board of Governors. “Index of the Weighted-Average Exchange Value ofthe U.S. Dollar: Revision,” Federal Reserve Bulletin (August 1978), p. 700.
Coughlin, Cletus C.; Patricia S. Pollard; and Jerram C. Betts. “To Chain or Not to Chain Trade-Weighted Exchange Rate Indexes,” Federal Reserve Bank of St. Louis Working Paper No. 96-010A(1996).
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Cox, W. Michael. “A New Alternative Trade-Weighted Dollar ExchangeRate Index,” Federal Reserve Bank of Dallas Economic Review(September 1986), pp. 20–8.
Deephouse, David L. “Using a Trade-Weighted Currency Index,” Federal Reserve Bank of Atlanta Economic Review (June/July 1985), pp.36–41.
Hargreaves, Derek. “Effective Exchange Rates: OECD Currencies,” Mor-gan Guaranty Trust Company Economic Research Note (December 30,1993).
Hirsch, Fred, and Ilse Higgins. “An Indicator of Effective ExchangeRates,” IMF Staff Papers (November 1970), pp. 453–87.
Hooper, Peter, and John Morton. “Summary Measures of the Dollar’sForeign Exchange Value,” Federal Reserve Bulletin (October 1978),pp. 783–9.
Ott, Mack. “The Dollar’s Effective Exchange Rate: Assessing the Impactof Alternative Weighing Schemes,” this Review (February 1987), pp. 5–14.
Rhomberg, Rudolf R. “Indices of Effective Exchange Rates,” IMF StaffPapers (March 1976), pp. 88–112.
Rosensweig, Jeffrey A. “Constructing and Using Exchange Rate In-dexes,” Federal Reserve Bank of Atlanta Economic Review (Summer1987), pp. 4–16.
______. “A New Dollar Index: Capturing a More Global Perspective,”Federal Reserve Bank of Atlanta Economic Review (June/July1986a), pp. 12–22.
______. “The Atlanta Fed Dollar Index and its Component Sub-Indexes,” Federal Reserve Bank of Atlanta Working Paper No. 86-7(August 1986b).
Turner, Philip and Jozef Van ‘t dack. “Measuring International Price andCost Competitiveness,” BIS Economic Papers (November 1993).
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