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The Purchasing Power Parity Debate

Alan M. Taylor and Mark P. Taylor

Our willingness to pay a certain price for foreign money must ultimately and essentiallybe due to the fact that this money possesses a purchasing power as against commoditiesand services in that country. On the other hand, when we offer so and so much of ourown money, we are actually offering a purchasing power as against commodities andservices in our own country. Our valuation of a foreign currency in terms of our own,therefore, mainly depends on the relative purchasing power of the two currencies in theirrespective countries.

Gustav Cassel, economist (1922, pp. 138–39)The fundamental things applyAs time goes by.

Herman Hupfeld, songwriter (1931; from the film Casablanca, 1942)

P urchasing power parity (PPP) is a disarmingly simple theory that holds thatthe nominal exchange rate between two currencies should be equal to theratio of aggregate price levels between the two countries, so that a unit of

currency of one country will have the same purchasing power in a foreign country.The PPP theory has a long history in economics, dating back several centuries, butthe specific terminology of purchasing power parity was introduced in the yearsafter World War I during the international policy debate concerning the appro-priate level for nominal exchange rates among the major industrialized countriesafter the large-scale inflations during and after the war (Cassel, 1918). Since then,the idea of PPP has become embedded in how many international economists thinkabout the world. For example, Dornbusch and Krugman (1976) noted: “Under theskin of any international economist lies a deep-seated belief in some variant of thePPP theory of the exchange rate.” Rogoff (1996) expressed much the same

y Alan M. Taylor is Professor of Economics and Chancellor’s Fellow, University of Californiaat Davis, Davis, California, and Research Associate, National Bureau of Economic Research,Cambridge, Massachusetts. Mark P. Taylor is Professor of Macroeconomics, University ofWarwick, United Kingdom. Both authors are Research Fellows, Centre for Economic PolicyResearch, London, United Kingdom. Their e-mail addresses are �[email protected]� and�[email protected]�, respectively.

Journal of Economic Perspectives—Volume 18, Number 4—Fall 2004—Pages 135–158

sentiment: “While few empirically literate economists take PPP seriously as ashort-term proposition, most instinctively believe in some variant of purchasingpower parity as an anchor for long-run real exchange rates.”

The question of how exchange rates adjust is central to exchange rate policy,since countries with fixed exchange rates need to know what the equilibriumexchange rate is likely to be and countries with variable exchange rates would liketo know what level and variation in real and nominal exchange rates they shouldexpect. In broader terms, the question of whether exchange rates adjust toward alevel established by purchasing power parity helps to determine the extent to whichthe international macroeconomic system is self-equilibrating.

Should PPP Hold? Does PPP Hold?

The general idea behind purchasing power parity is that a unit of currencyshould be able to buy the same basket of goods in one country as the equivalentamount of foreign currency, at the going exchange rate, can buy in a foreigncountry, so that there is parity in the purchasing power of the unit of currencyacross the two economies. One very simple way of gauging whether there may bediscrepancies from PPP is to compare the prices of similar or identical goods fromthe basket in the two countries. For example, the Economist newspaper publishes theprices of McDonald’s Big Mac hamburgers around the world and compares themin a common currency, the U.S. dollar, at the market exchange rate as a simplemeasure of whether a currency is overvalued or undervalued relative to the dollarat the current exchange rate (on the supposition that the currency would be valuedjust right if the dollar price of the burger were the same as in the United States).In January 2004, the cheapest burger was in China, at $1.23, compared with anaverage American price of $2.80. According to the Big Mac index, therefore, thisimplied that China’s yuan was 56 percent undervalued. The average price of a BigMac in the euro area countries was $3.48, suggesting that the euro was 24 percentovervalued against the dollar. In contrast, the Japanese yen was 12 percent under-valued on the Big Mac PPP standard. The Big Mac index has proved so popular thatthe Economist has also started to publish prices around the world of another globallyinvariable standard of value: a tall latte cup of coffee from Starbucks.

While the Big Mac and tall latte indices are an immediately engaging and funway to think about exchange rates, it is easy to come up with good reasons why theprices of coffee and burgers might differ internationally, most of which are relatedto the fact that many of the inputs into a tall latte or a Big Mac cannot be tradedinternationally or not easily at least: each good contains a high service component—thewages of the person serving the food and drink—and a high property rentalcomponent—the cost of providing you with somewhere to sit and sip your coffee ormunch your two beef patties on a sesame seed bun with secret-recipe sauce. Neitherthe service-sector labor nor the property (nor the trademark sauce) is easilyarbitraged internationally, and advocates of PPP have generally based their viewlargely on arguments relating to international goods arbitrage. Thus, while these

136 Journal of Economic Perspectives

indices may give a lighthearted and suggestive idea of the relative value of curren-cies, they should be treated with caution.

The idea that purchasing power parity may hold because of internationalgoods arbitrage is related to the so-called Law of One Price, which holds that theprice of an internationally traded good should be the same anywhere in the worldonce that price is expressed in a common currency, since people could make ariskless profit by shipping the goods from locations where the price is low tolocations where the price is high (for example, by arbitraging). If the same goodsenter each country’s market basket used to construct the aggregate price level—and with the same weight—then the Law of One Price implies that a PPP exchangerate should hold between the countries concerned.

Possible objections to this line of reasoning are immediate. For example, thepresence of transactions costs—perhaps arising from transport costs, taxes, tariffsand duties and nontariff barriers—would induce a violation of the Law of OnePrice. Engel and Rogers (1996), for example, looked at the price differentialsbetween similar goods in cities across the United States and Canada and reportedevidence broadly in support of this hypothesis: they found that the volatility of theprice differential tended to be larger the greater the distance between the citiesconcerned, and it increased substantially when prices in cities in different countrieswere compared (the so-called “border effect”).

Moreover, not all goods are traded between all countries, and the weightattached to similar goods in aggregate price indices will differ across countries. Inaddition, different countries tend to produce goods that are differentiated ratherthan perfectly substitutable. Some of these problems could be addressed, at least inprinciple, with better data. Also, since PPP is based on traded goods, it might bemore usefully tested with producer price indices that tend to contain the prices ofmore manufactured tradables, rather than consumer price indices, which tend toreflect the prices of relatively more nontradables, such as many services. A recenttheoretical and empirical literature, discussed below, has attempted to allow forshort-run deviations from PPP arising from sources such as these, while retainingPPP in some form as a long-run average or equilibrium.

These objections notwithstanding, however, it is often asserted that the PPPtheory of exchange rates will hold at least approximately because of the possibilityof international goods arbitrage. There are two senses in which the PPP hypothesismight hold. Absolute purchasing power parity holds when the purchasing power ofa unit of currency is exactly equal in the domestic economy and in a foreigneconomy, once it is converted into foreign currency at the market exchange rate.However, it is often difficult to determine whether literally the same basket of goodsis available in two different countries. Thus, it is common to test relative PPP, whichholds that the percentage change in the exchange rate over a given period justoffsets the difference in inflation rates in the countries concerned over the sameperiod. If absolute PPP holds, then relative PPP must also hold; however, if relativePPP holds, then absolute PPP does not necessarily hold, since it is possible thatcommon changes in nominal exchange rates are happening at different levels of

Alan M. Taylor and Mark P. Taylor 137

purchasing power for the two currencies (perhaps because of transactions costs, forexample).

To get a feel for whether PPP in either its relative or its absolute versions is amoderately good approximation to the real world, start with Figure 1. The toppanel plots data on the U.S. and UK consumer price indices (CPIs) over the period1820–2001. Both are expressed in U.S. dollar terms, which means that the UK CPIwas multiplied by the number of U.S. dollars exchanging for one UK pound at thatpoint in time. The bottom panel shows the comparison using producer priceindices, using data for a slightly longer period 1791–2001.1 We have (arbitrarily)normalized each of the series to be equal to zero in 1900.

At least three points are worth raising from a consideration of these graphs.First, absolute PPP did not hold perfectly and continuously: the correlation betweenthe two lines is less than perfect in both cases. In other words, there are substantialshort-run deviations from PPP. Second, the national price levels of the two coun-tries, expressed in a common currency, did tend to move together over these longperiods. Third, the correlation between the two national price levels is muchgreater with producer prices than with consumer prices.

Figure 2 shows several graphs using data for a large number of countries overthe period 1970–1998.2 Consider first the two graphs in the top row of Figure 2. Foreach country we calculated the one-year inflation rate in each of the 29 years andsubtracted the one-year U.S. inflation rate in the same years to obtain a measure ofrelative inflation (using consumer price indices for the figures on the left andproducer price indices for the figures on the right). We then calculated thepercentage change in the dollar exchange rate for each year and, finally, we plottedrelative annual inflation against exchange rate depreciation for each of the 29 yearsfor each of the countries. If relative PPP held perfectly, then each of the scatterpoints would lie on a 45o ray through the origin. In the second row, we have carriedout a similar exercise, except we have taken averages: we have plotted 29-yearannualized average relative inflation against the average annual depreciation of thecurrency against the U.S. dollar over the whole period, so that there is just onescatter point for each country.

Figure 2 confirms some of the lessons of Figure 1. For small differences inannual inflation between the United States and the country concerned, the corre-lation between relative inflation and depreciation in each of the years seems low.Thus, relative PPP certainly does not appear to hold perfectly and continuously inthe short run, although it appears to hold more closely for countries experiencingrelatively high inflation.

When we take 29-year averages, however, the scatter plots tend to collapse ontothe 45o ray. (This effect is only slightly less marked if we take averages over shorter

1 See Lothian and Taylor (1996, 2004) for a guide to the sources for these data series.2 Data are from the IMF’s International Financial Statistics database over the period 1970–1998 (data werenot available after 1998 for some countries). The sample included data on consumer price indices for20 industrialized countries and 26 developing countries, while that based on producer price indicesincludes 14 industrialized countries and twelve developing countries. The data set is identical to thatused in Coakley, Flood, Fuertes and Taylor (2004), which contains more precise details.


periods of ten years or so.) Thus, relative PPP seems to hold in a long-run sense. Withthe longer-run averages, the degree of correlation between relative inflation andexchange rate depreciation again seems higher when using PPIs than when using CPIs.

So the conclusions emerging from our informal eyeballing of Figures 1 and 2seem to be the following. Neither absolute nor relative PPP appear to hold closelyin the short run, although both appear to hold reasonably well as a long-runaverage and when there are large movements in relative prices, and both appear tohold better between producer price indices than between consumer price indices.

Figure 1Dollar-Sterling PPP Over Two Centuries

�0.5

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PPI U.S.PPI UK

1831 1851 1871 1891

(b) U.S. and UK PPIs in dollar terms

1911 1931 1951 1971 19911791

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CPI U.S.CPI UK

1860 1880 1900

(a) U.S. and UK CPIs in dollar terms

1920 1940 1960 1980 20001820

Notes: This figure shows U.S. and UK consumer and producer price indices expressed in U.S. dollarterms over roughly the last two centuries using a log scale with a base of 1900 � 0.

The Purchasing Power Parity Debate 139

In other words, as far as exchange rates are concerned, the fundamental things—relative price levels—apply increasingly as time goes by. These conclusions are infact broadly in line with the current consensus view on PPP. So where does all thecontroversy arise?

The PPP Debate: A Tour of the Past Three Decades

The Rise and Fall of Continuous PPPUnder the Bretton Woods agreement that was signed after World War II, the

U.S. dollar was tied to the price of gold, and then all other currencies were tied, or

Figure 2PPP at Various Time Horizons

�200

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Annual Consumer Price Inflation Relative to the U.S.versus Dollar Exchange Rate Depreciation,

1970–1998

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Annual Producer Price Inflation Relative to the U.S.versus Dollar Exchange Rate Depreciation,

1970–1998

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Consumer Price Inflation Relative to the U.S. versusDollar Exchange Rate Depreciation,

29-Year Average, 1970–1998

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Producer Price Inflation Relative to the U.S. versusDollar Exchange Rate Depreciation,

29-Year Average, 1970–1998

Notes: This figure shows countries’ cumulative inflation rate differentials against the United States inpercent (vertical axis) plotted against their cumulative depreciation rates against the U.S. dollar inpercent (horizontal axis). The charts on the left show CPI inflation, those on the right PPI inflation.The charts in the top row show annual rates, those in the bottom row 29-year average rates from1970 to 1998.


http://pubs.aeaweb.org/action/showImage?doi=10.1257/0895330042632744&iName=master.img-003.png&w=144&h=110

“pegged,” to the U.S. dollar. However, in 1971, President Nixon ended the con-vertibility of the U.S. dollar to gold and devalued the dollar relative to gold. Afterthe failure of attempts to restore a version of the Bretton Woods agreement, themajor currencies of the world began floating against each other in March 1973.

At this time, the dominant approach to determination of exchange rates wascalled the “monetary approach.” This approach assumed that the purchasing powerparity exchange rate held continuously (Frenkel, 1976; Taylor, 1995; Frankel andRose, 1995). Advocates of this approach argued that since the exchange rate is therelative price of two monies, that relative price should be determined by the relativebalance of supply and demand in the respective money markets in an asset marketequilibrium. Exactly how percentage changes in relative money supplies translated,other things equal, into exactly matching exchange rate movements was notimmediately obvious, however, unless one resorted to the earlier argument basedon goods arbitrage: that is, changes in the relative money supply affect relativeprices, including relative traded goods prices, which then leads to internationalgoods arbitrage.

A wave of empirical studies in the late 1970s tested whether continuouspurchasing power parity did indeed hold, as well as other implications of themonetary approach to the exchange rate, and the initial results were encouraging(Frenkel and Johnson, 1978). With the benefit of hindsight, it seems that theseearly encouraging results arose in part because of the relative stability of the dollarduring the first two or three years or so of the float (after an initial period ofturbulence) and in part because of the lack of a long enough run of data with whichto test the theory properly. Toward the end of the 1970s, however, the U.S. dollardid become much more volatile and more data became available to the econome-tricians, who subsequently showed that both continuous PPP and the simplemonetary approach to the exchange rate were easily rejected. One did not have tobe an econometrician, however, to witness the “collapse of purchasing powerparity” (Frenkel, 1981): one could simply examine the behavior of the real ex-change rate.

The real exchange rate is the nominal exchange rate (domestic price offoreign currency) multiplied by the ratio of national price levels (domestic pricelevel divided by foreign price level); since the real exchange rate measures thepurchasing power of a unit of foreign currency in the foreign economy relative tothe purchasing power of an equivalent unit of domestic currency in the domesticeconomy, PPP would, in theory, imply a real, relative-price-level-adjusted exchangerate of one (although the nominal rate—the rate that gets reported in the Wall StreetJournal or the Financial Times—could, of course, differ from one even if PPP held).In practice, if we are working with aggregate real exchange rates and henceaggregate price indices with arbitrary base periods, it may be difficult to pin downexactly when PPP held true, in order to normalize the measured real exchange rateto unity. What is clear, however, is that there will be some level of the measured realexchange rate that is consistent with PPP and—most importantly—that variation inthe real exchange rate must indicate deviations from PPP (since otherwise it wouldbe constant at the level consistent with PPP). The real trade-weighted value of the


U.S. dollar didn’t change too much from 1973 to 1976, thus lending a degree ofplausibility to the continuous PPP argument. But the real value of the dollardropped sharply starting in 1977, and from then on it became increasingly clearthat continuous PPP could not hold as nominal exchange rates were patently farmore volatile than relative national price levels.

Formal Tests of (the Failure of) Long-Run PPP: Random Walks and Unit RootsOne reaction to the failure of purchasing power parity in the short run was a

theory of exchange rate overshooting, in which PPP is retained as a long-runequilibrium while allowing for significant short-run deviations due to sticky prices(Dornbusch, 1976). However, the search for empirical evidence of long-run PPP alsomet with disappointment.

Formal tests for evidence of PPP as a long-run phenomenon have often beenbased on an empirical examination of the real exchange rate. If the real exchangerate is to settle down at any level whatsoever, including a level consistent with PPP,it must display reversion toward its own mean. Hence, mean reversion is only anecessary condition for long-run PPP: to ensure long-run absolute PPP, we shouldhave to know that the mean toward which it is reverting is in fact the PPP realexchange rate. Still, since much of this research has failed to reject the hypothesisthat even this necessary condition does not hold, this has not in general been anissue.

Early empirical studies, such as those by Roll (1979) or Adler and Lehmann(1983), tested the null hypothesis that the real exchange rate does not mean revertbut instead follows a random walk, the archetypal non–mean reverting time seriesprocess where changes in each period are purely random and independent. Someauthors even argued that the random walk property was an implication of theefficiency of international markets, in the sense of prices and exchange ratesreflecting all available information and all arbitrage opportunities being quicklyexploited. Under this “efficient markets PPP” view, Roll (1979), for example,argued that the change in the real exchange rate, since it is effectively a measure ofthe one-period real return from arbitraging goods between countries, should havean expected value of zero if markets are efficient. However, this early strand of theempirical literature suffered from logical and econometric weaknesses. The theo-retical underpinnings of “efficient markets PPP” failed to adjust the expectedreturn for the real cost of financing goods arbitrage (Taylor and Sarno, 2004).Once this arbitrage adjustment is made, efficiency requires that the expected realexchange rate change be equal to the expected real interest rate differential, andlong-run PPP will be implied if the latter differential is stationary.3 On the empirical

3 See Obstfeld and Taylor (2004) for evidence on the stationarity of real interest rate differentials. Underreasonable assumptions, the real interest rate differential will be negatively correlated with the level ofthe real exchange rate if a loss in international competitiveness (a real appreciation) has a netdeflationary impact on the economy, reducing inflation and, other things equal, raising the real interestrate. This in turn implies mean reversion of the real exchange rate, since it implies that changes in thereal exchange rate are negatively correlated with the level of the real exchange rate.


side, the evidence in favor of a random walk was at best mixed, and resultsdepended on the criteria employed (Cumby and Obstfeld, 1984). Sharper econo-metric tools were still being fashioned, but—as we discuss below—they also sufferedfrom low power just like these early tests, so for many years researchers wereunlikely to reject the null hypothesis of a random walk even if it were false.

In the late 1980s, a more sophisticated econometric literature on long-run PPPdeveloped, at the core of which was the concept of a “unit root process.” If a timeseries is a realization of a unit root process, then while changes in the variable maybe to some extent predictable, the variable may still never settle down at any oneparticular level, even in the very long run. For example, suppose we estimated thefollowing regression equation for the real exchange rate qt over time, where �t is arandom error and � and � are unknown parameters:

qt � � � �qt�1 � �t .

If � � 1, we say that the process generating the real exchange rate contains a unitroot. In that case, changes in the real exchange rate would be predictable—theywould be equal on average to the estimated value of �. The level of the realexchange rate would, however, not be predictable, even in the long run: since thechange each period would be equal to a constant plus an unpredictable randomelement, the long-run level will be equal to the sum of the constant changes eachperiod plus the sum of a large number of random elements. As these randomshocks get cumulated, there is no way of telling in advance what they will add up to.(In fact, the real exchange rate would be following a random walk with drift, whichis an example of a so-called unit root process.) Thus, testing the null hypothesis that� � 1 is a test for whether the path of the real exchange rate over time does notreturn to any average level and thus that long-run PPP did not hold.

The flurry of empirical studies employing these types of tests on real exchangerate data among major industrialized countries that emerged toward the end of the1980s were unanimous in their failure to reject the unit root hypothesis for majorreal exchange rates (for example, Taylor, 1988; Mark, 1990), although—as we shallsee—this result was probably due to the low power of the tests.

In any case, at the time, this finding created great uncertainty about how tomodel exchange rates. At a theoretical level, there was still a consensus belief inlong-run PPP coupled with overshooting exchange rate models; now the data wereraising the possibility that even long-run PPP was a chimera. Some economistsposited theoretical models to explain why the real exchange rate could in fact benon–mean reverting (as in Stockman, 1987). Others questioned the empiricalmethodology.

The Power ProblemFrankel (1986, 1990) noted that while a researcher may not be able to reject

the null hypothesis of a random walk real exchange rate at a given significancelevel, it does not mean that the researcher must then accept that hypothesis.Furthermore, Frankel pointed out that the statistical tests typically employed to


examine the long-run stability of the real exchange, at that time based on datacovering just 15 years or so since 1973, may have low power to reject the nullhypothesis of a unit root even if it is indeed false. This criticism would apply bothto the early random walk studies and to the subsequent literature testing moreformally for unit roots. It was further taken up and examined by other authors(Froot and Rogoff, 1995; Lothian and Taylor, 1996, 1997). The argument is thateven if the real exchange rate tends to revert toward its mean over long periods oftime, examination of one real exchange rate over a relatively short period may notyield enough information to detect this mean reversion. Using simulations in whichthe real exchange rate is assumed to mean revert by about 11 percent per year, theprobability of rejecting at the 5 percent level the null hypothesis of a random walkreal exchange rate, when in fact the real rate is actually mean reverting, is extremelylow—somewhere between about 5 and 8 percent—when using 15 years of data(Lothian and Taylor, 1997; Sarno and Taylor, 2002a). With the benefit of theadditional 10 to 15 years or so of data that are now available, the power of the testincreases by only a couple of percentage points, and even with a century of data,there would be less than an even chance of correctly rejecting the unit roothypothesis.

Moreover, increasing the sample size by increasing the frequency ofobservation—moving from, say, quarterly to monthly data—won’t increase thepower because increasing the amount of detail concerning short-run movementscan only give you more information about short-run as opposed to long-runbehavior (Shiller and Perron, 1985).

More Statistical Power: More Years, More CountriesIf you want to get more information about the long-run behavior of a partic-

ular real exchange rate, one approach is to use more years of data. However, longperiods of data usually span different exchange rate regimes, prompting questionsabout how to interpret the findings. Also, over long periods of time, real factorsmay generate structural breaks or shifts in the equilibrium real exchange rate.Once these issues are recognized, there are ways to look for possible effects ofdifferent regimes and structural shifts.

In one early study in this spirit, using annual data from 1869 to 1984 for thedollar-sterling real exchange rate, Frankel (1986) estimates a first-order autoregres-sive process for the real exchange rate q of the form

�qt � q� � ��qt�1 � q� � �t ,

where q is the assumed constant equilibrium level of q, �t is a random disturbance,and � is the autocorrelation coefficient—an unknown parameter governing thespeed of mean reversion. Notice that a proportion of � times the random shock attime t � 1, �t�1 , will still be part of the real exchange rate deviation at time t.Hence, we can say that shocks die out—or the real exchange rate reverts toward itsmean of q—at the rate of (1 � �) per period. (If the real exchange rate followeda random walk, then � � 1 and shocks would never die out.) Frankel’s point


estimate of � is 0.86, and he is able to reject the hypothesis of a random walk at the5 percent level.

Similar results to Frankel’s were obtained by Edison (1987), based on ananalysis of data over the period 1890–1978, and by Glen (1992), using a datasample spanning the period 1900–1987. Lothian and Taylor (1996) use two cen-turies of data on dollar-sterling and franc-sterling real exchange rates, reject therandom walk hypothesis and find point estimates of � of 0.89 for dollar-sterling andof 0.76 for franc-sterling. Moreover, they are unable to detect any significantevidence of a structural break between the pre– and post–Bretton Woods period.Taylor (2002) extends the long-run analysis to a set of 20 countries over the1870–1996 period and also finds support for PPP and coefficients that are stable inthe long run. Studies such as these provide the formal counterpart to the informalevidence of long-run relative PPP like eyeballing Figure 1.

Another approach to providing a convincing test of real exchange rate stabil-ity, while limiting the timeframe to the post–Bretton Woods period, is to use morecountries. By increasing the amount of information employed in the tests acrossexchange rates, the power of the test should be increased. In an early study of thistype, Abuaf and Jorion (1990) examine a system of 10 first-order autoregressiveregressions for real dollar exchange rates over the period 1973–1987, where theautocorrelation coefficient is constrained to be the same in every case. Their resultsindicate a marginal rejection of the null hypothesis of joint non–mean reversion atconventional significance levels, which they interpret this as evidence in favor oflong-run PPP. An academic cottage industry sprang up in the 1990s to apply unitroot tests to real exchange rate data on panels of countries for the post–BrettonWoods period. A number of these studies claimed to provide evidence supportinglong-run PPP. Taylor and Sarno (1998), however, issued an important warning ininterpreting these findings. The tests typically applied in these panel-data studiestest the null hypothesis that none of the real exchange rates under consideration aremean reverting. If this null hypothesis is rejected, then the most that can beinferred is that at least one of the rates is mean reverting. However, researcherstended to draw a much stronger inference that all of the real exchange rates weremean reverting—and this broader inference is not valid. Some researchers havesought to remedy this shortcoming by designing alternative tests; for example,Taylor and Sarno (1998) suggest testing the hypothesis that at least one of realexchange rates is non–mean reverting, rejection of which would indeed imply thatthey are all mean reverting. However, such alternative tests are generally lesspowerful, so that their application has not led to clear-cut conclusions (Taylor andSarno, 1998; Sarno and Taylor, 1998).

PPP PuzzlesThe research on the evolution of exchange rates from the 1970s to the turn of

the century has generally been interpreted as supporting the hypothesis thatexchange rates adjust to the PPP level in the long run. But the evidence is weak. Forexample, rejecting at standard levels of statistical significance the null hypothesisthat a unit root exists certainly doesn’t prove that a long-run PPP exchange rate


exists, either. The long-span studies raise the issue of possible regime shifts andwhether the recent evidence may be swamped by history. The panel-data studiesraise the issue of whether the hypothesis of non–mean reversion is being rejectedbecause of just a few mean-reverting real exchange rates within the panel. Ifexchange rates do tend to converge to PPP, economists have—at least so far—hada hard time presenting strong evidence to support the claim. Following Taylor, Peeland Sarno (2001), we see the puzzling lack of strong evidence for long-runPPP—especially for the post–Bretton Woods period—as the first PPP puzzle.

A second related puzzle also exists. In the mid-1980s, Huizinga (1987) andothers began to notice that even the studies that were interpreted as supporting thethesis that PPP holds in the long run also suggested that the speed at which realexchange rates adjust to the PPP exchange rate was extremely slow. A few yearslater, Rogoff (1996, p. 647) presented the puzzle this way: “The purchasing powerparity puzzle then is this: How can one reconcile the enormous short-term volatility ofreal exchange rates with the extremely slow rate at which shocks appear to damp out?”

As we noted above, this speed is related to the estimated coefficient in theautoregressive process described earlier: a proportion of � of any shock will stillremain after one period, �2 of it remains after two periods, and in general, �n of theshock will remain after n periods. One way to get a feel for how fast the realexchange rate mean reverts is by asking how long it would take for the effect of ashock to die out by 50 percent—in other words, we can compute the half-life ofshocks to the real exchange rate.

Based on a reading of the panel unit root and long-span investigations oflong-run PPP, Rogoff (1996) notes a high degree of consensus concerning theestimated half-lives of adjustment: they mostly tend to fall into the range of threeto five years. Moreover, most such estimates were based on ordinary least squaresmethods, which may be biased because ordinary least squares will tend to push theestimated autocorrelation coefficient away from one to avoid nonstationarity; usingdifferent estimation methods to correct for bias, but still in a linear setting, someauthors have argued that half-lives are even longer (for example, Murray andPapell, 2004; Chen and Engel, 2004). Now, although real shocks to tastes andtechnology might plausibly account for some of the observed high volatility in realexchange rates (Stockman, 1988), Rogoff argues that most of the slow speed ofadjustment must be due to the persistence in nominal variables such as nominalwages and prices. But nominal variables would be expected to adjust much fasterthan a half-life of three to five years for exchange rates would suggest.

The apparently very slow speed of adjustment of real exchange rates—from 0to about 10 percent or so per annum—has been the source of considerabletheoretical and empirical research in recent years.

Nonlinearity?

One approach to resolving the PPP puzzles lies in allowing for nonlineardynamics in real exchange rate adjustment. In a linear framework, the adjustment


speed of PPP deviations from parity is assumed to be uniform at all times and, inparticular, for all sizes of deviation, and implicitly the econometric problem isreduced to the estimation of a single parameter—the half-life. While this frame-work is very convenient, there are good reasons for suspecting that the speed ofconvergence toward the PPP exchange rate should be greater as the deviation fromPPP rises in absolute value. Indeed, some 90 years ago, Heckscher (1916) suggestedthat adjustment may be nonlinear because of transactions costs in internationalarbitrage. For example, if two goods differ in price (expressed in a commoncurrency) in different countries because PPP does not hold, it won’t be wortharbitraging and therefore correcting the price difference unless the anticipatedprofit exceeds the cost of shipping goods between the two locations. This insightbegan to be expressed more formally in the theoretical literature starting in the late1980s (for example, Benninga and Protopapadakis, 1988; Williams and Wright,1991; Dumas, 1992). The qualitative effect of such frictions is similar in all of theproposed models: the lack of arbitrage arising from transactions costs such asshipping costs creates a “band of inaction” within which price dynamics in the twolocations are essentially disconnected. Such transactions costs might take the formof the stylized “iceberg” shipping costs (“iceberg” because some of the goodseffectively disappear when they are shipped and the transaction cost may also beproportional to the distance shipped), fixed costs of trading operations or ofshipments or time lags for the delivery of goods from one location to another.

In empirical work on mean reversion in the real exchange rate, nonlinearitycan be examined through the estimation of models that allow the autoregressiveparameter to vary. For example, transactions costs of arbitrage may lead to changesin the real exchange rate being purely random until a threshold equal to thetransactions cost is breached, when arbitrage takes place and the real exchange ratereverts back toward the band through the influence of goods arbitrage (althoughthe return is not instantaneous because of shipping time, increasing marginal costsor other frictions). This kind of model is known as “threshold autoregressive.”

The model applies straightforwardly to individual commodities. Focusing ongold as foreign exchange, Canjels, Prakash-Canjels and Taylor (forthcoming) stud-ied the classical gold standard using such a framework applied to daily data from1879 to 1913; they found dollar-sterling exchange-rate adjustment consistent witha threshold autoregressive model. Examining subindices of the consumer priceindex, Obstfeld and Taylor (1997) modeled price adjustment in various interna-tional cities in the post-1973 period and also found significant nonlinearities. Theimplied transaction cost bands and adjustment speeds were also found to be of areasonable size (consistent with direct shipping cost measures) and to vary system-atically with impediments such as distance, tariffs, quotas and exchange-rate vola-tility. Sarno, Taylor and Chowdhury (2004) employ this approach with disaggre-gated data across a broad range of goods in the G-7 countries (Canada, France,Germany, Italy, Japan, the United Kingdom and the United States). Zussman(2003) uses a threshold autoregressive model with the postwar Penn World Tabledata to show nonlinear adjustment speeds for a very wide sample of countries.

Using a threshold autoregressive model for real exchange rates as a whole,


however, could pose some conceptual difficulties. Transactions costs are likely todiffer across goods, and so the speed at which price differentials are arbitraged maydiffer across goods (Cheung, Chinn and Fujii, 2001). Now, the aggregate realexchange rate is usually constructed as the nominal exchange rate multiplied bythe ratio of national aggregate price level indices and so, instead of a singlethreshold barrier, a range of thresholds will be relevant, corresponding to thevarious transactions costs of the various goods whose prices are included in theindices. Some of these thresholds might be quite small (for example, because thegoods are easy to ship) while others will be larger. As the real exchange rate movesfurther and further away from the level consistent with PPP, more and more of thetransactions thresholds would be breached and so the effect of arbitrage would beincreasingly felt. How might we address this type of aggregation problem? One wayis to employ a well-developed class of econometric models that embody a kind ofsmooth but nonlinear adjustment such that the speed of adjustment increases asthe real exchange rate moves further away from the level consistent with PPP.4

Using a smooth version of a threshold autoregressive model, and data on real dollarexchange rates among the G-5 countries (France, Germany, Japan, the UnitedKingdom and the United States), Taylor, Peel and Sarno (2001) reject the hypoth-esis of a unit root in favor of the alternative hypothesis of nonlinearly mean-reverting real exchange rates—and using data just for the post–Bretton Woodsperiod, thus solving the first PPP puzzle. They also find that for modest realexchange shocks in the 1 to 5 percent range, the half-life of decay is under threeyears, while for larger shocks the half-life of adjustment is estimated muchsmaller—thus going some way toward solving the second PPP puzzle.

While transactions costs models have most often been advanced as possiblesources of nonlinear adjustment, other less formal arguments for the presence ofnonlinearities have also been advanced. Kilian and Taylor (2003), for example,suggest that nonlinearity may arise from the heterogeneity of opinion in theforeign exchange market concerning the equilibrium level of the nominal ex-change rate: as the nominal rate takes on more extreme values, a greater degree ofconsensus develops concerning the appropriate direction of exchange rate moves,and traders act accordingly. Taylor (2004) argues that exchange rate nonlinearitymay also arise from the intervention operations of central banks: intervention ismore likely to occur and to be effective when the nominal—and hence thereal—exchange rate has been driven a long distance away from its PPP or funda-mental equilibrium.

In sum, the nonlinear approach to real exchange rate modeling offers someresolution of the PPP puzzles. Moreover, simulations show that if the true data are

4 The smoothly adjusting extension of the threshold autoregressive or TAR model is the aptly namedsmooth-transition autoregressive, or STAR, model (Granger and Terasvirta, 1993) and the exponentialSTAR or ESTAR has proved very successful in the application to real exchange rates (Michael, Nobayand Peel, 1997; Taylor and Peel, 2000). The ESTAR model can be thought of as a TAR with an infinitenumber of regimes and a continuously varying and bounded adjustment speed; Sarno and Taylor (2002)offer a textbook treatment. Alternative treatments of the goods-aggregation problem are currently beingexplored, but not without controversy (Imbs, Mumtaz, Ravn and Rey, 2002; Chen and Engel, 2004).


generated by a nonlinear process, but then a linear unit root or other linearautoregressive model is estimated, problems can easily arise. Standard unit roottests, already weak in power, are further enfeebled in this setting and half-lives canbe dramatically exaggerated (Granger and Terasvirta, 1993; Taylor, 2001; Taylor,Peel and Sarno, 2001).

The transactions-cost approach to real exchange rates is also generating influ-ential intellectual spillovers into current research flourishing at the nexus of thetwo fields of international trade and international macroeconomics. The study ofinternational trade has currently been enlivened by a new focus on the role ofphysical and other barriers, such as distance, remoteness, borders or variouspolicies (Anderson and van Wincoop, 2004). International macroeconomists arealso now making explicit the role of these trading frictions in new models wheresuch refinements may yet help us solve some of the major puzzles in the field(Obstfeld and Rogoff, 2001; Betts and Kehoe, 2001; Bergin and Glick, 2003;Ghironi and Melitz, 2003).

Remaining Puzzles: Short-Run Disturbances, Long-Run Equilibrium

Empirical work that focuses on the path of real exchange rates must grapplewith three key factors: the reversion speed; the volatility of the disturbance term;and the long-run, or equilibrium, level of the real exchange rate. Most of ourdiscussion to this point, in keeping with the focus of the literature, has pertainedto the reversion speed, which is a medium-run phenomenon. But in the future, weexpect to see more attention given to the disturbances, which are a short-runphenomenon measured over months and also to the very long run question of whatis the equilibrium real exchange rate. Thus, questions about the real exchange rateare likely to shift—from not so much “how fast is it reverting?” to “how did it deviatein the first place?” and “what is it reverting to?”5

Exchange Rate DisturbancesEven if current work can establish that exchange rates do revert to the PPP rate

over the medium term at a more reasonable speed, the volatilities present in thedata in the short run, at least under floating rate regimes, still cause considerablemystification. Over short periods, nominal exchange rates move substantially andprices do not, so real and nominal exchange rate volatilities in the short term arecorrelated almost one for one, and the Law of One Price for traded goods is oftenviolated (Flood and Rose, 1995). This pattern holds across different monetaryregime types over a wide swathe of historical experience (Taylor, 2002). Despite

5 Of course, with reference to the PPP puzzle, the literature on real exchange rates has generally beenfocused on monthly or lower frequencies, since this is where price index data are available. There areplenty of nominal exchange rate puzzles at even higher frequency—weekly, daily, even intraday—butthese are more properly in the domain of finance than international macroeconomics and are outsidethe scope of this survey.


efforts to explain such phenomena as a result of taste or technology shocks inflexible price models, it is implausible that the patterns could be traced wholly toreal factors. History shows that volatilities also differ systematically across monetaryregimes.6 Instead, it is likely that some combination of monetary policy and pricestickiness plays a role in the short-run volatility of exchange rates, mechanisms thatcould be amplified by other types of frictions.

Various new strands in the literature seek to address these issues. One commontheme recognizes the important role of frictions in trade, not just for generatingno-arbitrage bands for traded goods, but for delineating traded from nontradedvarieties. Economies may be more closed than we once thought, since large sectorslike retailing, wholesaling and distribution are nontraded—even if the price ofimports as measured in official data typically includes some element of thesedomestic costs (Obstfeld, 2001; Obstfeld and Rogoff, 2001; Burstein, Neves andRebelo, 2003).

In an economy with price stickiness and a nontraded sector, small monetaryshocks can generate high levels of exchange rate volatility. For example, Burstein,Eichenbaum and Rebelo (2003) model a price sticky nontraded sector with a shareof the economy well above 0.5, which yields a large devaluation response of anemerging market country even for a small monetary shock. When the nontradedshare of the economy rises, the economy is less “open” and the Law of One Priceassumption applies to a smaller fraction of goods (here, imported varieties),implying a larger role for exchange rate overshooting and other sources of ex-change rate volatility (Obstfeld and Rogoff, 2000; Hau, 2000, 2002). Considerablefuture research remains to be done in this area to establish a general frameworkthat will apply to a wide range of cases.

Long-Run Movements of the Real Exchange RateThe PPP exchange rate theory is built on the concept that the exchange rate

is based on actual buying power over a basket of goods, and so changes in thenominal exchange rate should reflect changes in the price of goods—with the realexchange rate staying fixed. But a nation’s equilibrium real exchange rate may notremain fixed forever.

One of the textbook explanations for changes in the level of the real exchangerate focuses on the net international asset position. Consider a small, open econ-omy in long-run equilibrium. Now impose a shock in the form of an increase inexternal debt. The country must run a trade surplus in the future to service theinterest payments due. To encourage foreign consumers to import more, and toencourage its own consumers to import less, the country’s competitiveness mustimprove in equilibrium. With the nominal exchange rate defined as the domesticprice of foreign currency, this means that the equilibrium level of the real exchange

6 For example, it is hard to imagine that, say, the real shocks to the Argentine economy were small inthe 1960s, suddenly became several times larger during the 1980s hyperinflations, then shrank again inthe 1990s currency board epoch, only to mysteriously reappear in late 2001. But we do know that to afirst approximation Argentina’s turbulent monetary history matches the observed volatilities very closely.


rate must rise, making the country’s exports cheaper to foreigners and its foreignimports more expensive to domestic residents. This example supposes differenti-ated goods at home and abroad, but the logic holds in a wide range of models (likethe portfolio balance model of Taylor, 1995; Sarno and Taylor, 2002b).

Lane and Milesi-Ferreti (2002) present some empirical confirmation of thisargument. They find that countries with larger positive net asset positions havemore negative trade balances and stronger real exchange rates, controlling forother factors and allowing for real return differentials across countries. This resulthas implications for the study of real exchange rate dynamics. If the equilibriumPPP exchange rate changes as a result of changes in net wealth, and if these shiftsare not controlled for in an autoregression, then the exchange rate will appear todeviate from what is falsely assumed to be a fixed PPP rate for too much or for toolong.

The other textbook story for trending real exchange rates is built aroundnontraded goods. Standard arbitrage arguments may lead PPP to hold for tradedgoods, but these arguments fail for nontraded goods, so that we must eitherabandon PPP theory, or else modify it. The Harrod-Balassa-Samuelson model ofequilibrium real exchange rates is attracting renewed interest as a desirable mod-ification after languishing for some years in relative obscurity (Harrod, 1933;Balassa, 1964; Samuelson, 1964).

In this model, rich countries supposedly grow rich by advancing productivity intraded “modern” sectors (say, manufacturing). Meantime, all nontraded “tradi-tional” sectors, in rich and poor countries alike, remain in technological stasis (say,haircuts). Suppose the Law of One Price holds among traded goods and we live ina world where labor is mobile intersectorally, but not internationally. As produc-tivity in the modern sector rises, wage levels rise, so prices of nontraded goods willhave to rise (as there has been no rise in productivity in that sector). If we measurethe overall price index as a weighted average of traded and nontraded goods prices,relatively rich countries will tend to have “overvalued” currencies. The example offast-growing Japan springs to mind, where the trend real exchange rate has steadilyappreciated by about 1.5 percent per year since 1880; the opposite trend hassometimes been observed in slow growth eras, for example, in Argentina (Taylor,2002; Froot and Rogoff, 1995; Rogoff, 1996).

Early studies of the Harrod-Balassa-Samuelson effect such as Officer (1982)found little support in the data from the 1950s to the early 1970s. But newerresearch dealing with later periods has often found support for this hypothesis, andit is now textbook material (for example, Micossi and Milesi-Ferreti, 1994; DeGregorio, Giovannini and Wolf, 1994; Chinn, 2000).

Why have more recent studies provided stronger evidence of the Harrod-Balassa-Samuelson effect? At some level, the reasons reflect developments in thePPP literature more broadly: more data, of longer span, for a wider sample ofcountries, coupled with more powerful univariate and panel econometric tech-niques, has allowed researchers to take a once-fuzzy relationship in the data andmake it tighter.

In addition, recent work suggests that the magnitude of the Harrod-Balassa-


Samuelson effect has been variable over time—certainly in the postwar period andperhaps going back several centuries (Bergin, Glick and Taylor, 2004). Considerthe relationships in Figure 3. The horizontal axis shows the per capita income levelof countries as a ratio of the U.S. per capita income level, expressed in log terms.The vertical axis shows the common currency Penn World Table Consumer PriceIndex price level of other countries as a ratio of the U.S. CPI price level—that is, thereal exchange rate level—again expressed as a log. (Note that most countries haveincomes and price levels lower than the United States, so the ratios are less thanone, and the logs are negative.) The Harrod-Balassa-Samuelson hypothesis suggeststhat as per capita income rises, driven by productivity growth in tradables, thenprice levels should also rise: there should be a positive correlation in the scatterplot.The graphs show that the cross-country relationship between income per capita andthe price level has been intensifying since 1950; once close to zero, and statisticallyinsignificant, the elasticity is now over one half. The null hypothesis of a zero slopecan be rejected beginning in the early 1960s when Balassa and Samuelson wrotetheir seminal papers (albeit with no knowledge of these hypothesis tests).

It is not clear why the Harrod-Balassa-Samuelson effect has altered over time.One possible explanation is that the nontraded share has increased over time, butthis effect does not seem to have enough magnitude to match the changes that haveoccurred, nor to match the timing of the changes (remember that global trade in1950, after world wars and depression, was a lower share of output than in 1913 orin 2000). Perhaps the productivity advances of traded and nontraded goods havediffered at various times? This may better help us explain the data, but over longtime frames begs the question of which goods are traded and why. Bergin, Glickand Taylor (2004) advance the hypothesis that trade costs determine tradabilitypatterns, which allows a variety of possible productivity shocks to give rise eventuallyto an endogenous Harrod-Balassa-Samuelson effect.

Figure 3Harrod-Balassa-Samuelson Effects Emerge: Log Price Level versus Log Per CapitaIncome

�3

�2

�1

0

1

�4 �3 �2 �1 0 1�5

ln(p

/pU

S)

ln(y/yUS)

(a) 1995 data (N � 142)

�2

�1

0

1

�3 �2 �1 0�4

ln(p

/pU

S)

ln(y/yUS)

(b) 1950 data (N � 53)

�1

0

1

�2 �1 0 1�3

ln(p

/pU

S)

ln(y/yUS)

(c) 1913 data (N � 24)

Notes: This figure shows countries’ log price level (vertical axis) against log real income per capita for1995, 1950 and 1913, with the United States used as the base country.





But the key point here is that if the equilibrium exchange rate is movinggradually over time, and our statistical analysis presupposes that the PPP exchangerate is fixed over time, then estimates of the speed of reversion to the PPP will bebiased. An allowance for such long-run trends can make a material difference inresolving the puzzles about whether and how fast the exchange rate moves to itsPPP level. For example, Taylor (2002) finds relatively low half-lives in a 20-countrypanel when an allowance is made for long-run trends in the equilibrium exchangerate. Allowing for nonlinear time trends, Lothian and Taylor (2000) suggest thatthe half-life of deviations from PPP for the U.S.-UK exchange rate may be as low as21

2years. Lothian and Taylor (2004) show that the Harrod-Balassa-Samuelson effect

may account for about a third of the variation in this real exchange rate.The equilibrium real exchange rate could shift for a number of reasons over

the very long run: wealth effects, productivity effects and other forces could all beimportant. Models that allow for a time-varying equilibrium real exchange rate, andpermit an exploration of its causes and consequences, are likely to be a busy areafor future research. Coordinated progress on these fronts will not only refine ourresolution of the PPP puzzles, they will also help us address related puzzles in themacroeconomic literature. For example, Engel (1999, 2000) decomposed thevariance of the real exchange rate into external traded goods prices and internaltraded-versus-nontraded goods prices. The component related to external tradedgoods can be viewed as related to the Law of One Price in the basic version of thePPP theory where the real exchange rate is assumed to be fixed, while the compo-nent related to traded and nontraded goods can be related to the Harrod-Balassa-Samuelson effect or any approach where the real exchange rate has a trend. Inlooking at post–Bretton Woods samples of data, Engel found that both of thesecomponents have persistence, but the Law of One Price component seems toexperience larger shocks. The two insights may also be unified, as in recent modelsof endogenous tradability (Betts and Kehoe 2001; Bergin and Glick, 2003). Theresearcher can use nonlinear models with trade costs to understand the volatilityrelated to traded goods—as Parsley and Wei (2003) do with the Engel puzzle—andthen use some version of the Harrod-Balassa-Samuelson effect to model andestimate the slower and often obscured drift in the prices of nontraded goods.

Conclusion

Since the early 1970s, the purchasing power parity theory of exchange rates hasbeen the subject of an ongoing and lively debate. For much of that period,theoretical work suggested that exchange rates should be linked to relative changesin price levels with deviations that might be only minimal or momentary, whileempirical work could find only the flimsiest evidence in support of purchasingpower parity, and even these weak findings implied an extremely slow rate ofreversion to PPP of, at best, three to five years.

After a struggle to find common ground, the gap between theory and empiricsis being closed from both directions. After early disappointments with dynamic


general equilibrium models, recent applications with nominal price rigidities showhow monetary shocks may have large and long-lasting effects on the real exchangerate (Bergin and Feenstra, 2001). When such insights are combined with theoret-ical work on transactions costs and nonlinearity, we can now better understand thevolatility and persistence of the real exchange rate. Going further, the presence ofnontraded goods (a manifestation of extreme transaction costs) enriches ourmodels further. A renewed attention to the Harrod-Balassa-Samuelson effect andwealth effects leads to a modified view of PPP where the equilibrium real exchangerate itself may move over time.

The idea that transaction costs matter for PPP is an old one. For Hume(1742–1752 [1987]), goods arbitrage caused countervailing flows of specie, with theanalogy that “all water, wherever it communicates, remains always at a level,” exceptthat if markets are separated “by any material of physical impediment. . .there may,in such a case, be a very great inequality of money.” Heckscher (1916) developedthe idea further, introducing the concept of “commodity points.” Keynes (1923,pp. 89–90, 91–92) highlighted transaction costs as a key substantive issue for thePPP theory:

At first sight this theory appears to be one of great practical utility. . . . Inpractical applications of the doctrine there are, however, two further difficul-ties, which we have allowed so far to escape our attention,—both of themarising out of the words allowance being made for transport charges and imports andexport taxes. The first difficulty is how to make allowance for such charges andtaxes. The second difficulty is how to treat purchasing power of goods andservice which do not enter into international trade at all. . . . For, if we restrictourselves to articles entering into international trade and make exact allow-ance for transport and tariff costs, we should find that the theory is always inaccordance with the facts. . . . In fact, the theory, stated thus, is a truism, andas nearly as possible jejune.

As these venerable ideas start to be incorporated into formal theory andempirics—in particular, via nonlinear adjustment—results are suggesting morestrongly that exchange rates do revert to a certain level determined by the pricelevel in the long run and that the half-life of this reversion is short enough, atperhaps one to three years for moderately sized shocks of more than 1 or 2 percent,to seem theoretically plausible.

Several new directions could now be taken. A very general theoretical modelcould be developed to incorporate all of the above refinements simultaneously andits predictions studied. Empiricists could attempt to include both nonlinearitiesand Harrod-Balassa-Samuelson effects to get even tighter estimates of convergencespeeds. Introducing trade costs and real shocks into a Clarida and Galı (1994)decomposition might advance the potential for reconciliation even further.

In sum, however, our interpretation of the consensus view of the PPP debate—that short-run PPP does not hold, that long-run PPP may hold in the sense thatthere is significant mean reversion of the real exchange rate, although there may


be factors impinging on the equilibrium real exchange rate through time—ishighly reminiscent of the consensus view that held sway in the period before the1970s. In that sense, this paper may be taken as evidence of mean reversion ineconomic thought.

y For their helpful comments we thank, without implicating, Menzie Chinn, Richard Clarida,Bradford DeLong, Charles Engel, Jeffrey Frankel, James Hines, James Lothian, BennettMcCallum, Michael Melvin, Peter Neary, Maurice Obstfeld, Lawrence Officer, David Papell,David Peel, Phillip Lane, Kenneth Rogoff, Andrew Rose, Lucio Sarno, Timothy Taylor, Ericvan Wincoop and Michael Waldman.

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