NBER WORKING PAPER SERIESARE EXCHANGE RATESEXCESSIVELY VARIABLEJeffrey - PDF document

1 NBER WORKING PAPER SERIESARE EXCHANGE RA
NBER WORKING PAPER SERIESARE EXCHANGE RATESEXCESSIVELY VARIABLE?Jeffrey A. Franke]Richard MeeseWorking Paper No. 2249NATIONAL BUREAU OF ECONOMICRESEARCH1050 Massachusetts AvenueCambridge, MA 02138May 1987Support from the Sloan Foundation isgratefully acknowledged. The researchreported here is part of the NBER'sresearch program in InternationalStudies. Any Opinionsexpressed are those of the authors andnot thoseof the National Bureau ofEconomic Research. NBER Working Paper #2249May 1987Are Exchange RatesExcessively Variable?ABSTRACT"Unnecessary variation" isdefined as variationnot attributable tovariation in fundamentalsIn the absence of agood model of macroeconomicfundamentals, the question "areexchange rates excessivelyvariable?," cannotbe answered bycomparing the variance of theactual exchange rate to thevariance of a set of fundamentals.This paper notes thefailure of regressionequations to explain exchangerate movements evenusing contemporaneousmacroeconomic variablesIt notes as well thestatistical rejections of theunbiasedness of the forwardexchange rate as a predictor ofthe spot rate.Itthen argues that, giventhese results, there isnot much to be learned fromthevariance-bounds tests and bubblestests.The paper also discussesrecent results on variation inthe exchange riskpremiums arising from variationin conditionalvariances, both as a source ofthe bias in the forwardrate tests and as a source ofvariation in the spot rate.It finishes with a discussionof

2 whether speculators'expectations arestab
whether speculators'expectations arestabilizing or destabilizing,as measured by survey data.The paper concludesthat it is possible thatexchange rates have beenexcessively variable --as,for example, when thereare speculative bubbles -- butthat if policy-makerstry systematically to exploittheir credibility in orderto stabilize exchangerates, they may see theircurrent credibility vanish.Jeffrey A. Franke].Richard MeeseDepartment of EconomicsSchool of BusinessAdministrationUniversity of CaliforniaUniversity of CaliforniaBerkeley, CA 94720Berkeley, CA 94720(415) 642-8084(415) 642-1435 Are Rachange Rates RacessivelyVariable?JeffreyA. FrankelRichard MeeseApril 1987This paper was written for theNBER Macroeconomics AnnualConference,Cambridge, Ma., March 13—14, 1987. Anabridged version will appear inthe Macroeconoajcs Annual, editedby Stanley Fischer. The authors wouldlike to thank Joe Mattey forresearch assistance, the Institute ofisiness and Economic Research fortyping, tharles Engel for dataassistance, Andrew Rose forcomments, and Takeo Hoshi for a carefulreview of the manuscript.OU U IN EI. The Meaning of "ExcessiveVariability"1. Introduction2. Factors Determining "ExcessiveVariability"(a) disturbances unrelated to fundamentals?(b) degree of Substitutability?(c) are expectations stabilizingor destabilizing?II. Random Walk Results1. Nonstatlonarity of Nominal andReal Exchange Rates(a) nominal(b) real2. Regressions of Exchange RateDetermination3. Interpreting Tes

3 ts of Bias inthe Forward Discount(a) the
ts of Bias inthe Forward Discount(a) the stationarityargument for changes rather than levels(b) real returns rather than nominalreturns(c) peso problem(d) risk premium4. Variance—Bounds and &ibbleg Ièsts(a) bubbles tests(b) variance—bounds testsIII. The Exchange RiskPremium1. Implications of Portfolio—optimizatjowith ConstantVariance2. Implications of Time—VaryingCovariancegIV. Survey Data and HeterogenousExpectations1. Measuring Stabilizing andDestabilizing Expectations2. Conclusion I. The Pan1ng of 'ExcessjveVariabi1ityThe proponents of floatingexchange rates before 1973 didnotpromise that exchange rates wouldnecessarily be stable under such asystem, but only that they would beas stable as the underlyingmacro-economic fundamentais.lNevertheless, the widespreadfeeling is thatexchange rates have turned outto be more volatile thannecessary. Manypractitioners believe thatexchange rates are driven byPsychologicalfactors and other irrelevantmarket dynamics, rather thanby ecomomicfundamentals. Support seemsto have grown in the l980sfor "target—zone" proposals, or some othersort of government action tostabilizeexchange ratesEconomists have understood forsome time that under conditionsofhigh international capitalmobility, currency values willmove sharplyand unexpectedly inresponse to new information. Evenso, actual move-ments of exchange rates have beenpuzzling in two major respects.First, the proportion ofexchange rate changes that weare able to pre-dict seems

4 to be not justlow, but zero. According t
to be not justlow, but zero. According torational expecta-tions theory we should be ableto use our models to predictthat propor-tion of exchange ratechanges that is correctly predictedby exchangemarket participants. Yetneither models based oneconomic fundamentals,nor simple time series models,nor the forecasts of marketparticipantsas reflected in the forward discountor in survey data, seem abletopredict better than the laggedspot rate. Second, the proportionofexchange rate movements that can beexplained even after the fact,usingcontemporaneous macroeconomicvariables, is disturbingly low.—1— —2—1. IntroductionWhen we return to the basics, to ask what is actually known aboutthe crucial building blocks of exchange markets, we find one questionthat has been answered and three that have not. The one conclusionabout which there is no longer serious disagreement is that monetarydisturbances have real effects. The 1980s have witnessed very largeswings in real money supplies, real interest rates, and——moredemonstrably——real exchange rates, convincing most observers that thereis not an exogenous instantly—equilibrating optimally—functioning realeconomy that can be divorced from monetary factors.Itfollows thateven if financial markets do operate efficiently——a proposition that weneither automatically presuppose nor rule out in this paper——one cannotmake an argument against government intervention in foreign exchangemarkets on the grounds of Arrow—Debreu optimality a

5 lone.There are three questions that have
lone.There are three questions that have yet to be satisfactorilyanswered, and that are examined in this study. Question 1: Howresponsive are investors' demands for domestic and foreign assets toexpected rates of return, that is, what is the degree of substitu-tability? Question 2: How do investors form expectations? Inparticular, how much weight do they give to the contemporaneous spotrate and how much to other factors? Question 3: How does the actualprocess governing the spot exchange rate correspond tothe processembodied in investors' expectations, that is, are expectationsrational? As we will see, these questions together contain theessential elements necessary to evaluate claims of excessive exchangerate variability. —3—We will be trying to shedlight on these questions bydrawing onseveral areas of the existingempirical literature on thespot andforward exchange markets, as wellas on some new empirical results.Empirical topics to be covered, ifonly briefly, are non—stationarity ofthe nominal and real exchangerates, regression tests of exchange ratedetermination, forward market efficiency,variance—bounds tests andbubbles tests, portfolio_optjmizatjoand the exchange riskpremium, andexpectations survey data.However, we begin by considering themore general motivation foranswering the three questions stated above: howknowing the answers tothem might help answer whetherexchange rate fluctuations have beenunnecessarily large.2. Factors In lterainIng'Rxcesslye

6 Variability'In seeking to get a handle o
Variability'In seeking to get a handle on thequestion of alleged excessivevariability, we specify as general a model ofthe spot exchange rate aspossible.(1)s =S(Z,ii*, ASu).We represent fundamental determinantssuch as asset supplies by t, theinterest differential by i—i*, investors'expected future change inthe exchange rate by ASeand any short—term random factorsby u. Thelast will be foundnecessary if we are to confront regressionresults,which under the best of circumstanceshave relatively low R2s. It willalso be found necessary if weare to consider the issue of allegedshort—term volatility unrelated to fundamentals. —4—The equation is so general that it could be interpreted as theold balance—of—payments flow approach to exchange rate determination,where9.represents factors affecting the current account and the otherthree variables are determinants of the capital account •Weshallfollow the stock approach here however, in which the focus is on stocksof assets rather than flows.We can impose additional structure on equation (1) by defining9.to be specifically the log of the supply of domestic assets minus thelog of the supply of foreign assets, defining s to be the log of thespot price of foreign exchange, imposing homogeneity, and assuming alsothat the two components of expected returns enter with coefficients ofequal magnitude:(2)s =9.—L(i_i*_Me;u)In equation (2),9.is the relative supply of domestic assets and Lis the relative demand for domestic assets,

7 which depends positively onrpl*_ASe ,the
which depends positively onrpl*_ASe ,therisk premium or expected excess rate of return ondomestic assets. In a portfolio—balance approach, for example, we couldassume that the share of the portfolio allocated to foreign assets,x, is linearly related to the risk premium on foreignassets:4(3)x =A—B(rp)Then (2) would hold, withlog(x(.)) —log(l—x(.)),anddL/d(rp) =(!+1JB —5—We can now use equation (2) to consider thequestion of exchangerate variability. it seems likely thatregardless whether the funda—nientals term2.is defined to include onlymoney supplies or alsosupplies of bonds and other assets, one cannot in factexplain observedvariability in s by variability in9..This is the Implication ofboth volatility tests and regressions of thespot rate against funda-mentals such as asset supplies .5The same conclusion seems to holdaswell if the fundamentals term9.isdefined to include the currentaccount6We are thus led to consider the other twoterms in equation (2),which are determinants of asset demands ratherthan asset supplies:and u. The expectations formationprocess is key to thequestion of variability, whether as a source of fluctuationsor as"stabilizing speculation," moderating the effect ofdisturbances thatoriginate in the other terms •Toallow for both the bandwagon effectsthat are so often cited by participants in theexchange market,7 and thestabilizing expectations that are the norm in models basedon economicfundamentals, we can specify expected depreciation

8 to be thesum of anextrapolative or "ban
to be thesum of anextrapolative or "bandwagon" expectations term anda regressiveexpectations term:(5)As =(s—sri)+Q(s—se).ODnsider a disturbance in u, orany of the other terms, thatincreases s. lb the extent that�g 0, "speculators," or investors,will expect future increases ins, will shift their asset demand intoforeign assets, and will thereby exaggerate the increase Ins. To —6—�the extent that 0 0, they will expect a return back to the equilib—rium valueand�will thereby dampen the fluctuation. If g 0,expectations are said to be elastic or destabilizing; if g are said to be inelastic or stabilizing.It is important to note that equation (5) could be fullyconsistent with rational expectations in a variety of models. Forexample, in the case g =0,regressive expectations can be rational inthe sticky—price monetary ("overshooting") model of Dornbusch (1976),where the rational value of 0 depends on the speed of adjustment ofthe price level, or in the portfolio—balance model of Kouri (1977) andRodriguez (1980), where it depends on the speed of adjustment of thestock of foreign claims.8 In the case 0 =0on the other hand,extrapolative or "bandwagon" expectations could be rational in models ofspeculative bubbles such as Blanchard and Watson (1982) or Dornbusch(1982) .Finally,in the case g =00, static expectations(ASe =0)could be rational if the true exchange rate process is arandom walk, a result consistent with recent empirical findings.9An

9 alternative interpretation of equation
alternative interpretation of equation (5) is a speculativebubble: speculators think that there is a probability0 that thebubble will burst within the coining period and the spot rate will returnall the way to equilibrium s, and a probability (1—8) that it willcontinue on a bubble path (in which case it will increase at a rateequal to g/(1—e) times the rate at which it has already beenincreasing) .oAgain,such a bubble could be rational, if the actualspot rate turns out to follow the same process. —7--Friedman (1953) argued persuasively thatspeculators who had adestabilizing effect would be "buying high andselling low," and thuswould lose money and be driven out of the market.In modern terms, heargued that destabilizing speculation would be inconsistentwithrational expectations. &t the modern realizationthat one can haverational stochastic speculative bubbles, in whicheach speculator standsto lose money if he doesn't go along with theothers, has all butdestroyed the classic Friedman argument.A linearized form of the equation ofspot rate determination (2)is now(6)s =—— 1—g(s_s—O(st—s))+whereis the degree of substitutability dL/d(rp) (as, forexample,in equation (4)).Volatility will be high, in the sense that the variability ofs will be high with9.and i—it given, if the variability of u ishigh, if g is high, and if O is low. Indeed ifwe were interestedin the one—period effect ofu alone, on the theory that this is thesource of short—term uncertainty, then the

10 conditional variance ofwould be given b
conditional variance ofwould be given by2(7)l+(y)var(u)Equation (7) illustrates in a simple way a conflict that exists in dis-cussions of excessive exchange rate volatility. Someeconomists, suchas Tobin (1978), argue that exchange rates are too variable becausefinancial markets are "excessively efficient," that capital sl0shes back —8—and forth among countries in response to trivial disturbances, and thata tax on foreign exchange transactions would reduce volatility. Thisview says that volatility is high because 3,the degree of substituta-bility, is high. t there is another view, associated with McKinnon(1976), that exchange rates are too variable because of a "deficiency ofstabilizing speculation,' in other words, becauseis too low. Theapparent paradox can be resolved by noting that the variance is posi-tively related to(the Ibbin case) if8 (&#x g, ; nd0;1 (g—8))because in that case the expectations to which investors react aredestabilizing. The variance is negatively related to(the McKinnon&#x g, ; nd0;case) if B g, because in that case expectations are stabilizing. Toanalyze the possible sources of exchange rate volatility, we need toconsider both the degree of substitutability and whether expectationsare stabilizing.In this paper we will be seeking enlightenment on the empiricalmagnitude of these parameters. In doing so we are leaving out much thatis central to the macroeconomics of exchange rates •Evenif the expres-sion in (7) is small, one

11 could still make many arguments for or a
could still make many arguments for or againstrestricting the movements of the exchange rate that result from changesin macroeconomic factors such as2. or i—i' (assuming it could bedone)For example, many economists believe that the strong real appre-ciation of the dollar in the early l980s can be attributed to a shift inthe monetary—fiscal policy mix that raised the interest differentialj_j* and made U.S. assets more attractive to world investors. Somebelieve further that it would not have been desirable to force down the —9—value of the dollar, for example byforeign exchange intervention or theimposition of controls to shut off the capitalinflow (even assuming itpossible), in the absence of reductions in the fiscalbudget deficit.The argument is that U.S. Interestrates would have risen even more thanthey in fact did, crowding Out domestic investment.''To evaluate sucharguments, one should probably specify an objec-tive function, including such variablesas output, inflation, tradebalance and investment, andtry to judge whether letting the marketdetermine the exchange rate is likely toresult in a higher value of theobjective function than proposed plans to stabilizethe exchange rate.Such questions are beyond thescope of this paper.'2Our interest here is only In thequestion whether foreignexchange markets can fairly be said to beworking well. If allegationsare found justified that speculative bubbles,a failure of market eff I—ciency, or random fluctuations,

12 are raisingexchange rate variabilityneed
are raisingexchange rate variabilityneedlessly, then it could be said that the marketsare not working well.The possibility might in that case exist ofobtaining lower exchangerate variability without cost. There isa wealth of empirical resultsthat can be brought to bear. —10—It. RandomWalkIsu1tsAvariety of different econometric approaches seem to end up atthe same conclusion, that the exchange rate follows a random walk. Inthis part of the paper we discuss the apparent Inability to forecastfuture changes In the exchange rate using either(i) the past time series of the process itself (section 11.1),(ii) macroeconomic fundamentals (section 11.2), or(iii) the forward exchange market (section 11.3).We then discuss what else, if anything, can be learned from thecurrently popular variance—bounds and bubbles tests.1. NonstationarIty of Noainal and al Fxchange RatesIt is now widely recognized that the linear time seriesrepresentation of the natural logarithm of either spot or forwardexchange rates is best described by a random walk process13 Formalstatistical tests for the presence of a unit root in the autoregressiverepresentation of the logarithms of spot and forward exchange rates werefirst conducted by Meese and Singleton (1982) .Theseunit root tests,pioneered by Fuller (1976) and his students, are known to have low poweragainst borderline stationary alternatives. 1-wever, we find thesuperior out—of—sample forecasting performance of the random walk model,over

13 time series models where the unit root I
time series models where the unit root Is not imposed, to be power-ful evidence in favor of the unit root null. Finally, more recentstatistical tests of the unit root hypothesis that are robust toconditionally heteroskedastic disturbances (Phillips (1985)) also —11—supportthe unit root hypothesis. This is an importantmethodologicaladvance, since it is also widely recognized that exchange rate vari-ability tends to be episodic; see Qimby and Obstfeld (1984) for tests ofconditional heteroskedasticity in nominal exchange rates.'4Nonstationarity in the nominal exchange rate does not createproblems for standard theories of exchange rate determination. In themonetary models, if the money supply is nonstationary in levels, or evenin changes, then the exchange rate will be nonstationary in levelsorchanges. We have only to be careful how we specify our econometrictests of nominal exchange rates, preferring first differences overlevels in general. Nonstationarity in the real exchange rate isconsidered by some to be a more serious matter however. If the realexchange rate follows a random walk, then there is no tendency to returnto purchasing power parity, and seemingly no limit on how far out ofline one country's prices can get from another's.'5Not long ago, purchasing power parity was widely accepted. Itwas argued on a priori grounds that the law of one price should beenforced at least for traded goods once transport costs were accountedfor: If automobiles were sel

14 ling at a lower price in Germany than in
ling at a lower price in Germany than inthe United States, International arbitrageurs should buy them in Germanyand sell them in the United States, raising the price in one country orlowering it In the other until equality was restored.The empirical evidence against PPP in level form is overwhelm-ing. The enormous real appreciation of the dollar in the early l980sconvinced the remaining doubters, but abundant statistical evidence wasthere all along. For example, Krugman (1978, p. 406) computed for the — I )_L.floatingrate period July 1973 —December1976 standard deviations ofthe (logarithmic) real exchange rate equal to 6.0 percent for thepound/dollar rate and 8.4 percent for the mark/dollar rate. He alsocomputed serial correlation coefficients for PPP deviations of .897 and.854, respectively, on a monthly basis, equal to .271 and .150 on anannual basis •Theserial correlation coefficient is of interest becauseit is equal to one minus the speed of adjustment to PPP.Table 1 shows updated annual statistics on the real exchange ratebetween the United States and Great Britain. During the floating rateperiod 1973—84, there is a significant time trend, and a standarddeviation of I5.4 percent. The serial correlation in the deviationsfrom PPP is estimated at .720, with a standard error of .248. (Theequation estimated is (er—) = AR(er—)+c,wheret+1t+1ttt+1er is the real exchange rate and er is the long—run equilibriumlevel, alternatively estimated as the sample mean

15 or a time trend, andAR is the autoregres
or a time trend, andAR is the autoregressive coefficient.) This means that the estimatedspeed of adjustment to PPP is .280 per year and that one can easilyreject the hypothesis of instantaneous adjustment.From the ashes of PPP, a phoenix has risen. In response tofindings such as those reported here, some authors have swung from oneextreme, the proposit1on that the tendency of the real exchange rate toreturn to a constant is complete and instantaneous, to the oppositeextreme that there is no such tendency at all. &t there is even lessof an a priori case why PPP should hold in rate—of—change form than inthe level form. Table 1Purchasing Power Parity between the United States and the UnitedKingdom1869—198241973—19841945—19741945—19841869—1984Mean absolute deviation.121.075.106.093Standard deviation.154.092.146.122Time trend—.ooi*.006*—.0004.009( .0003)C .002)( .0022)( .013)Autoregressionof deviations from mean.720*.706*.829*.860*( .248)( .132)C .090)( .048)of deviations from trend•734*.710*.750*.846*( .277)C .133)C .106)( .050)Note: Standard errors are reported in parentheses.*Signiflcant at the 95 percent level. —13—Several authors16 have claimed that the random walk model of thereal exchange rate has a basis in efficient markets theory. Theirargument is apparently that if the expected inflation rate is lower inthe United States than In Germany, allowing for expected exchange ratechanges, then an arbitrageur can contract to buy automobiles In theUnited Stat

16 es and ship them to Germany and expect t
es and ship them to Germany and expect to sell them there ata higher price; such profitable arbitrage would then eliminate theopportunity to begin with, enforcing PPP in expected rate—of—changeform. If this arbitrage is intended to be different from the oldarbitrage in level form which has been empirically rejected, and theseauthors clearly intend it to be different, then it Is a remarkablestrategy to recommend to international traders. Measures of expectedreal depreciation of the dollar as of 1985 showed a short—term depreci-ation rate of about 1.0 percent per annum. As of 1985, the level of fWprices was said to be almost twice as high in the United States as inGermany, as a result of the five—year appreciation of the dollar againstthe mark. Yet the apparent strategy tells people to buy 1Ws in theUnited States and ship them to Germany because in the three months Ittakes to complete the shipment their relative prices will have increased0.25 percent in expected value! The near—50—percent loss would seem tooutweigh the 0.25—percent gain.Even though ex ante relative PPP has little basis in theory, itdoes appear to have some empirical support. Typically, the estimatedspeeds of adjustment during the floating rate period, .27 or .28 on anannual basis in Table 1 (1973—84), while not so low as to be Implausibleas point estimates, are nevertheless so low that one statistically —14—cannot reject the hypothesis that they are zero. In otherwords, onecannot reject the

17 hypothesis that theautoregressive coeff
hypothesis that theautoregressive coefficient is 1.0.A 95—percent confidence interval on theautoregressivecoefficient covers the range 0.17 to 1.27 (in theno—trend case). Ifthe null hypothesis is an autoregressive coefficientof 1 .0, one cannotlegitimately use the standard t—test derived from aregression wherethe right—hand variable is the level of the realexchange rate, becauseunder the null hypothesis its variance is infinite.There are a numberof ways of dealing with this nonstationarityproblem. Here we simplyapply the corrected Dickey—Fuller (1979) cumulativeprobability distri-bution for the t—test appropriate for thisproblem. The t—ratio totest an autoregressive coefficient of 1.0 is 1.13, which fallsfar shortof the Dickey—Fuller 95—percent significancelevel, 3.00.This failure to reject a random walk in the realexchange rate isthe same result found by Roll (1979), Frenkel (1981,p. 699), Darby(1981), Adler and Lehman (1983), Mishkin (1984,pp. 1351—53) and Pigottand Sweeney (1985) .Hakkio(1984) provides evidence of a unit root inthe real exchange rate using the Dickey—Fuller (1979)statisticalprocedures. Most of these studies used monthly data rather thanyearly,and the statistical procedures employed weregenerally not powerfulenough to reject the random walk.'7A more promising alternative is to choose a longer timesample.Table 1 also reports statistics for the entirepostwar period 1945—84.PPP held slightly better during the etton Woodsyears than it dida

18 fter 1973, as measured either by the mea
fter 1973, as measured either by the mean absolute deviation andstandard deviation of the real exchange rate, or by the ability to —15—reject the hypothesis of zero autocorrelation. t,despitethe longertime sample, one is still unable to reject the random walk. The 95—percent confidence interval runs from .65 to 1.01, and the t—ratio of1 .9 falls short of the Dlckey—Fuller 95—percent significance level of2 .93.The last column of Table 1 presents an entire 116 years of U.S.—U.K. data. With this long a time sample, the standard error is reducedconsiderably. The rejection of no serial correlation in the realexchange rate Is even stronger than in the shorter time samples. Moreimportant, one is finally able to detect a significant tendency for thereal exchange rate to regress to PPP, at a rate of 14 percent a year.The confidence interval for AR runs from .77 to .95, safely less thanunity, and the t—ratio of 2.92 exceeds the Dickey—Fuller significancelevel of 2.89.If the speed of adjustment to PPP is indeed on the order of 20percent a year, and the standard deviation of the real exchange rate ison the order of .15, then the standard deviation of new shocks is on theorder of /1 —.802)(.152) =10percent. With such a large error termin the regression equation, it is not surprising that most econonietri—clans have been unable statistically to reject zero adjustment using thedata from a mere 14 years of post—1973 data. The tests simply haveinsufficient power. This con

19 clusion is supported by Monte Carlo test
clusion is supported by Monte Carlo testsas performed by Hakkio (1986). He hypothesizes an ARIMA (1,1,2) processfor the real exchange rate (with the first—order autoregressive coef-ficient equal to .9) and manufactures 100 months of data. He finds thatthe standard tests are not powerful enough to reject the randomwalk.'8 —16--The problem distinguishing betweena stationary and a random walkprocess can be considered by calculating anapproximate value of samplesize N that would give rise toa 95 percent confidence interval forAR (the first order autoregressivecoefficient) that excludes the valueone. With a little hand—waving and theuse of the conventional formulafor the variance of the estimate ofAR these values of N would beroughly 156, 196, and 759 for actual values ofAR equal to .95, .96,and .99 respectively. N=156 isroughly the number of months in themodern floating rate period. In terms ofthe half—life of deviationsfrom PPp it would take about 13,16, and 68 observations (months) for 50percent of the deviations from PPP to disappear, whenAR for the realexchange rate was .95, .96 and .99respectively.Thus in our view the evidence fora unit root in real exchangerates is much less convincing than the evidencefor a unit root innominal exchange rates, suggesting that PPPis still a reasonable anchorfor long—run exchange rateexpectations.The implications of thenonstationarity of the logarithms ofnominal exchange rates and the nearnonstationarity of the real excha

20 ngerate for tests of spot rate determina
ngerate for tests of spot rate determination,forward rate bias, andvariance bounds will be discussed at theappropriate places in the nextthree sub—sections respectively.2. gress1onsof &changete1ter1natjonRegressions of equations of exchange rate determinationwere thefirst sort of tests to become popular inthe iuld—l970s. The flexible—price monetary model,'9 for example, was represented by theequation —17—(8)s =—+X(i_i*) + uwhere s is the log of the spot exchange rate (domestic currency!foreign), mt is the log of the domestic money supply relative to theforeign, " is the log of domestic income relative to foreign,(i_i*) is the interest differential, and u is the regressionerror. The model is derived from the assumption of instantaneousadjustment and perfect substitutability in the goods market (implyingpurchasing power parity) as well as in the bond market (implyinguncovered interest parity) 2O Under the assumptions, (i_i*)t could aseasily be replaced by the forward discount fdt, or by investors'expected rate of depreciation ts.(9)s =m—+X(s) +Intuitively, an increase in the relative supply of the domesticcurrency m will lower its value, or raise the price of foreigncurrency s. Anything that raises the relative demand for domesticcurrency, like an Increase in relative income y or a decreaseinexpected future capital losseswill have the opposite effect.Other authors argued that important elements were missing fromthe equation. As we saw in the last section, d

21 eviations from purchasingpower parity ar
eviations from purchasingpower parity are in fact very large. If they were purely random, theycould just be subsumed in the regression error u (as could randomshifts in money demand) .&itwe also saw that they are in fact highlyautocorrelated. If the deviations are thought to have anautocorrelation coefficient of 1, i.e., If the real exchange rate is —18—thought to follow a random walk, we have the version ofthe monetarymodel used by feese (1986).Theequation could simply be estimated onfirst differences.the other hand, if deviations from PPP ariseprimarily from price level stickiness and thus arethought to be dampedover time, e.g., to follow an AR(1), and ifexpectations correctlyreflect this tendency to return to long—runequilibrium, then a morecomplete model is needed. The real interest differential,which isequal to expected real depreciation, will be proportionateto thecurrent deviation from equilibrium.2' In the sticky—pricemonetaryniodel,22 we can simply add the real interestdifferentiale**e(i—u)— (i), toequation (8): When the interest differentialrises without a rise in expected inflation (lTe)itattracts anincipient capital inflow that causes thecurrency to appreciate. Thecoefficient is 1/9, where 0 is the expected rate ofadjustment ofthe spot rate to equilibrium.Another alternative to the simple monetary model is theportfolio—balance model23 which relaxed the assumption of uncoveredinterest parity, and as a consequence introduced the stocksof bondsin

22 to the model. Some synthesis versions re
to the model. Some synthesis versions requiredonly adding a variablefor the cumulation of government deficits and currentaccount deficitsto the earlier equations 24These models have all been grouped under the name "asset marketapproach" because they all assume that exchange rates are determined infinancial markets in which investors are able to shift theirassetholdings instantaneously. it is important to note that the modelsalready build in a high degree of exchange rate volatility, even without —19—any special factors such as irrational expectations, speculative bub-bles, or an error term. In the flexible—price monetary model, forexample, a one percent change in the money supply will have a more—than—proportionate effect on the contemporaneous exchange rate, if it leadsinvestors to expect more money growth and currency depreciation in thefuture. (This has been called the magnification effect.)In the sticky—price overshooting model of Dornbusch, even aonetime change in the money supply can have a more—than—proportionateeffect, because it transitorily lowers the interest rate and as a resultdrives the value of the currency below the new long—run equilibriumlevel.25 Sometimes, especially in policy circles, the overshootingmodel has been mistakenly invoked to support the idea that irrationalityor speculative bubbles increase exchange rate variability. &t mostreaders of the Dornbusch paper have realized that its beauty liesprecisely in the fact that overshoot

23 ing occurs even when investorsbehave wel
ing occurs even when investorsbehave well in the sense that their speculation equates the forwarddiscount to the rationally expected rate of depreciation. Indeed, whenexpectations are rational in the Dornbusch model, the conditionalvariance of the spot rate is given by2(10)(1 +-—-) cwhereisthe variance of changes in the money supply.26 There is asense in which this much volatility, if not necessarily optimal for theallocation of resources (a question on which we have demurred), is anatural and inevitable consequence of money supply changes in a sticky—price world. —20—The econometric evidence from regression tests can only beinterpreted as saying that either expected depreciation is notadequately captured by the forward discount (or Interest differential),or else there is some other substantial error term u in an equationlike (8) that will enter the variance of s in addition to thefundamentals variables. e can always postulate the existence ofvariables that must have been incorrectly omitted, as the real interestdifferential would be if the sticky—price monetary model is correct butequation (8) is tested without it. &t it is fair to say that everyequation that has been proposed, or that is likely to be proposed in thefuture, has a substantial error term left over. Much- has been made(appropriately) of the models' inability to predict out—of—sample. <many of the regression estimates have shown very poor fits, not tomention unsensible coefficients, within

24 the sample period as well. Evenfor the
the sample period as well. Evenfor the sample period during which the sticky—price monetary model fitthe mark/dollar data remarkably well, the R2 was only .80.27Subsequent sample periods usually showed less sensible coefficients andworse fits for all varieties of models (except when a lagged endogenousvariable Is used. In such cases a more informative goodness of fitstatistic would be one that measured the contribution of the explanatoryvariables after accounting for the past history of the spot rate.)28Unsensible coefficients are often attributable to endogeneity ofrlghthand side variables. For example, negative coefficients on themoney supplies can be attributed to central bank reaction to theexchange rate when setting monetary policy. Income, interest rates andother variables are also almost certainly endogenous 29 Unsensible —21—coefficients would in turn explain the inability to predict evendirections of movement out—of—sample. Such econometric problems haveencouraged many to go on to other testing procedures, such as thosediscussed in later sections. &t it is Important to note at this stagethat the endogeneity problems alone cannot explain the poor fits •Tosee this, one need not rely on instrumental variables estimates, whichare only as good as the instruments used. One can impose a unit coef-ficient on the money supply and reasonable values on the other coeffi-cients; the fits are still poor.3° In the limit, if the error termin the regression were

25 indeed always close to zero, one should
indeed always close to zero, one should get aperfect fit regardless of whether the rIghthand—side variables aredetermined in other equations. This is true even If sophisticatedtheories of the expectations term are built from rational expectations,speculative bubbles, etc. Assuming expected depreciation is measurableby the forward discount, then some function of the forward discount andother fundamentals should give a good fit, unless there are largeomitted factors.Why emphasize so much the poor fits? The first reason is italready gives us our first conclusion: no set of macroeconomic vari-ables that has been proposed is capable of explaining a very highpercentage of the variation in the exchange rate. One can always postu-late, In the manner of "real business cycle theory" some unobservableportfolio shifts or productivity shocks that must be determining theexchange rate. &it if the shocks cannot be measured or even describedmeaningfully, then they probably belong in the error term u. Ourconclusion that the magnitude of u Is large Is evidence, for example, —22—undermining any defense of exchange rate variability made on the groundsthat it is appropriate given changes in monetary policy. If allexchange rate changes were in truth explainable by changes in money sup-plies, either contemporaneous or anticipated, we would have much betterresults In our regressions of the monetary equation (1) than we do.The second reason why we flag here the poor fits and simu

26 ltaneityproblems is that some of the alt
ltaneityproblems is that some of the alternative tests that econometricianshaveturned to, though seemingly more sophisticated than these regressions,are very sensitive to the assumed behavior of the error term.These arethe variance—bounds and bubbles tests, which are discussed in section11.4 below.Faced with poor econometric results for our models based onmacroeconomic fundamentals, the proper response is to test components ofthe models in isolation. (It is not to test the models jointly withother assumptions!) Tests of unbiasedness in the forward market are onesuch approach, as almost all of the models include rational expectationsas a key element, or at least as a special case. They arealso thoughtto shed light on the question whether the forward discount canlegitimately be used to measure expected depreciation. We now turntothese tests3.InterpretIng 1sts of Biasinthe Forward DiscountThe literature testing the unbiasedness of the forward discountis by now truly voluminous. Typically, the ex post error made bytheforward discount in predicting the change in the spot rate is regressedagainst information available at the beginningof the period, such as —23—the lagged prediction error.3' It often turns out that a statisticallysignificant portion of the prediction errors can be explained using theavailable information, which constitutes a rejection of the null hypo-thesis of unbiasedness.The most common test in this literature takes the forward dis-count itse

27 lf to be the information set on which ex
lf to be the information set on which expectations areconditioned.32 The regression equation is(11)As÷1 =a+bfd +Underthe null hypothesis that the forward discount is an unbiasedpredictor of actual depreciation, the coefficient b should be one.33It is important to consider tests of the bias in the forward ratein difference rather than levels form. Earlier versions of the testregressed the level of the realized future spot rate against the levelof the forward rate.34 The argument in favor of the difference versionsof the forward rate bias test can be made as follows •Sincethe loga-rithms of the levels of spot and forward rates contain unit roots, eachof these series are highly autocorrelated. In addition, the samplecontemporaneous cross correlation of the log levels of spot and forwardrates is essentially one, over any subset of the modern floating rateperiod. This common stochastic trend in the log levels of spot andforward rates ensures that they are highly cross—correlated, even whenthe two series are sampled at slightly different points in time.Finally, the sample variance estimates of the log levels of spot andforward rates are essentially the same over any subset of the recentfloating rate period. These "empirical regularities" guarantee that the —24—slope coefficient from a regression ofonwill be nearunity, as this coefficient is equal to the sample correlation coeffi-cient when the variables have the same sample variance. In modern timeseries parlan

28 ce, the log levels of spot and forward r
ce, the log levels of spot and forward rates are cointe—grated with a cointegrating constant near unity. The set of empiricalregularities that suggests the levels regression will have a coefficientnear one has no implications for either of the difference versions ofthe forward rate bias test.35The null hypothesis in equation (11) is usually rejected. Thecoefficient is significantly less than one; the implication is that onecould expect to make money by betting against the forward discountwhenever it is nonzero. Often the estimated coefficient is close tozero or even negative, which would say that the forward discount doesnot even get the direction of movement of the exchange rate right.Bilson (1981) interprets this finding as "excessive speculation:"investors would do better if they would routinely reduce toward zero themagnitude of their expectations of exchange rate changes.Most economists have not followed Bilson in the large step fromthe statistical finding of bias to the conclusion that the rationalexpectations hypothesis should be rejected. The economist's usual apriori argument——that any incipient opportunity for earning excessiveprofits would quickly attract investors who would eliminate it——isconsidered sufficiently strong that other explanations for the findingof bias are sought.Easily the most common explanation given is exchange risk. Risk—averseinvestors will demand some extra expected return for taking an —25—open position in a currency that

29 they perceive asriskier.36 Whether orno
they perceive asriskier.36 Whether ornot the optimal statistical predictor equals the expectation that inves-tors have in mind (rational expectations), if the investors' expectationis not in turn equal to the forward rate (because of a risk premiumseparating them), then the forward raje will be biased. This explana-tion is discussed at some length in Part III.Here we discuss two other explanations that are sometimes givenfor the finding of bias in the forward rate. One is easily covered.Under the joint null hypothesis of rational expectations and riskneutrality, it is expected real profits that should be zero; but thecondition usually tested is the absence of expected nominal profits.The reason for the distinction is not that goods price indices aredifferent in different countries. The same price index, determined bywhatever consumption basket is relevant for the investor in question,should be applied to both currencies. The problem is rather that theprice index goes in the denominator of the expression for the expectedreal value of an asset, and by Jensen's Inequality it does not drop outof the expected difference in returns on countries' assets. (This isthe resolution of the famous "Siegel's Paradox;" see Frankel and Razin(1980), Fngel (1984) or the references cited in these papers for a morecomplete explanation.) Doing the tests in real terms rather thannominal does not seem to make much difference however. This is what onewould expect from the fact t

30 hat the short—term variability of goodsp
hat the short—term variability of goodsprices is much smaller than the variability of exchange rate; presumablyit is also true that uncertainty regarding goods prices is much smallerthan uncertainty regarding exchange rates. Indeed, in the special case —26--where goods prices are nonstochastic when expressed in the currency ofthe producer country and the spot rate is distributed log—normally, theusual method of running the regression on logs of the nominal spot andforward rate is exactly correct .Amore serious obstacle to interpreting findings of forward ratebias as evidence against the joint hypothesis of rational expectationsand risk neutrality is the "peso problem." As is widely known by now,the peso problem arises when there is the possibility of a large depre-ciation in the currency contingent on an exogenous event that may nothave occurred in the sample period. In the context of the surprisinglysustained period of dollar appreciation in the early 1980s, with theforward market all the while forecasting a depreciation, it has beensuggested that either the collapse of a rational speculative bubble or asudden shift in the fiscal and monetary policy mix could be such anexogenous event. Unfortunately, the term "peso problem" is sometimesused indiscriniinantly to explain away any rejections of unbiasedness,leaving one to wonder why the test is run in the first place. It isimportant to remind ourselves of the familiar fact that standardstatistical significa

31 nce tests take into account the possibil
nce tests take into account the possibility of anevent by chance failing to occur in the sample. (This assumes that thesample period was dictated by exogenous considerations such as dataavailability, as Is the case in most of the tests.) One cannot say, forexample, that "the forward market repeatedly mis—forecast the apprecia-tion of the dollar in 1981—84 because it could not know that the WhiteHouse or Qrngress would repeatedly fail to correct the structural budgetdeficit." If investors repeatedly mis—forecast fiscal policy in the —27—same direction, that itself Is a violation of the rational expectationshypothesis.The correct definition of the peso problem is that, because ofthe possibility of a discretely—large change in the exchange rate, ausually—respectable number of observations might not in fact be largeenough to give an approximately normal distribution to the coefficientestimate, with the result that the usual significance levels applied tothe t—statistic may be inappropriate.38 When one suspects that such afailure of normality may be a problem, one response is to use tests thatdo not require that distributional assumption. Nonparametric tests ofthe dollar in the 1981—1985 period show that statistical rejections ofunbiasedriess need not necessarily depend on normality: the dollar re-peatedly moved upward in value while the forward discount was predictingthe reverse (Frankel (l985b), Evans (1986)). It is true, however, thatnonparanietric tests fr

32 equently depend on a random sampling ass
equently depend on a random sampling assumption.If we leave behind Jensen's Inequality and the peso problem, theexchange risk premium remains the major explanation——short of arejection of rational expectations——for the findings of bias In theforward rate. We will consider exchange risk, and the information to begained from the theory of portfolio optimization, in Part III.4. Variance &unds andB.ibbles1kstsVariance bounds tests have been found intuitively appealing fortwo reasons. First, they have the appearance of more generality thanregression tests. Second, they appear to hook up neatly with thepopular feeling——which is the main motivation of the present study——thatmarkets have been in some sense too volatile. —28—It has been pointed out repeatedly that the variance—bounds andbubbles tests require the assumption that the economic fundamentals havebeen correctly identified. Hamilton and Whiteman (1986) criticize thebubble tests on the grounds that "one can always relax restrictions onthe dynamics of the fundamental driving variables so as to interpretwhat appears to be a speculative bubble as instead having arisen fromrational agents responding solely to economic fundamentals not observedby the econornetrician." Similarly, Meese (1986) and Flood, HodrickandKaplan (1986, p. 32) argue that the tests are actually tests of thejoint hypothesis of (1) a correct model, (ii) no regime changes,and(iii) no bubbles.These criticisms have also been levelled at the va

33 riance—boundstests applied to the stock
riance—boundstests applied to the stock market by Shiller (1981). For example, Marshand Merton (1986) argue that their assumption about the dynamic processgoverning the payout of dividends will result in a violation ofShiller's sample variance bounds even when the simple stock pricepresent value is in fact true. &it it has not entirely sunk in,for thecase of the foreign exchange market, how damaging is the dependenceofthe tests on having correctly specified the macroeconomic fundamentals.(The same could be said for specifying the money demand function cor-rectly in tests of hyperinflation.) In the case of the stock market, atleast modelling the price as the present discounted value of expectedfuture dividends is fairly airtight, subject only to the possibleproblem of a risk premium.39We now spell Out briefly the steps in deriving a bubbles test,starting from a model such as equation (9), and the perilsthat lie —29—therein. If agents are assumed to have rational expectations, Asecan be replaced by E(st÷i —s)in the equation:(9')s =—+ X(Es+i—s)+ uEquation (9') could be estimated by McCallum's (1976) method of replac-ing Est+i by the expost realization st+1 plus a random predictionerror et and then using an instrumental variables (IV) techniquesuch as Generalized Method of Moments or two—step two—stage leastsquares. Alternatively, if the regression errorwere thought to besmall relative to the expectational errorit would be preferableto solve forbefore estimat

34 ing.(9'')s1 =St—m+ -y—- u—Equation(9') o
ing.(9'')s1 =St—m+ -y—- u—Equation(9') or (9'') will hold——under the joint hypothesis of rationalexpectations and the rest of the model——regardless whether there is aspeculative bubble term or not.To test the special case of no bubble, we estimate the model adifferent way. We solve for s as a function of expectations:1A1(12)St =m—r '+jr(Es+i) +UtWe then note thatE ————--- +—E+—E1tst+l —1+Am+i 1+A t't+11+A 't5t+2'1+A t''t+l' 'substituteinto (12), and continue to substitute recursively. The well—known result is that the (no—bubble) solution for today's exchange ratecan be written as the present discounted sum of the entire expected —30--future path of monetary conditions:T*xI(13)s =(Tj-)(-)E+T —t+T+ t+TFor example, if far—sighted agents expect an increase inthe moneysupply to take place four years in the future, itwill have an effect onthe exchange rate today. The reason is that they expectthe currency todepreciate (whether in terms of goods or foreign currency)in fouryears, and thus expect that agents inthree years will seek to move outof domestic currency in anticipation of capital losses, causing adepreciation in that period. Agents in two yearswill in turn seek tomove out of domestic currency, and so on. The depreciationis passedall the way back to the present.Note that setting the price of foreign exchange to the presentdiscounted sum of expected future monetary conditions (wherethediscount factor is iI(1+x)) is analogous to themodel In the stockmarket

35 that sets the price of equity to the pr
that sets the price of equity to the presentdiscounted value ofexpected future dividends (where the discount factoris one over oneplus the real interest rate) .Themajor difference is that we are muchless confident about having the right fundamentals inthe foreignexchange market. In addition, estimation of equation(13) requires thatthe disturbance term u be uncorrelated with the appropriatelydatedfundamentals (or else an IV procedure must beutilized40)Equation (13) gives only the particularfundamentals solution,which sets the coefficient on the speculative bubble termto zero. Theintent of the bubbles tests is to test the equation againstthealternative more general solution —31—(14)s =s+where a is any stochastic process satisfying E,a+ =a.Theextra term can arise from self—fulfilling expectations: if everyoneexpects the dollar to appreciate, even if for a reason unrelated tofundamentals ("sunspots"), they will buy dollars and drive up the price,so that the expectation turns out to have been rational. In a singledeterministic bubble of the sort Flood and Garber (1980) test for,ais a constant •Witthere are other possibilities. In thestochastic bubble model of Rianchard and Watson (1982) at has aprobability of collapsing to zero each period.The next step in the bubbles tests of West (1984), Meese (1986)and Casella (1985) is a non—trivial assumption in any context: somestable dynamic process must be assumed for the fundamentals variablesand y, such as a vector

36 autoregression. Then the Hansen—Sargent(
autoregression. Then the Hansen—Sargent(1980) prediction formula can be applied to (13) so that the expectedfuture values of m andare substituted out. This results in amultiple equation system with nonlinear cross equation constraints thatwe shall refer to as (13').The trick behind the bubbles test Is the recognition that underthe null hypothesis of "no bubble term" the estimator of the parametersof equations (13') will be more efficient than the estimator of theparameters of equation (9') .Underthe alternative hypothesis thatthere is a bubble term as in equation (14), the estimator of theparameters of equation (9') will still be consistent, whereas theestimator of the parameters of equations (13') will be inconsistent. —32—Thus a Hausman (1978) specification test can be used to choose betweenthe two possibilities.At least four propositions are being maintained when estimatingthe system (13'); (a) the macroeconomic model such as equation (8) iscorrect, (b) the interest differential or forward discountis anunbiased predictor in the sense of equalling the realization within thesample period, up to a random prediction error (this requiresrationalexpectations, no peso problem or regime changes, and norisk premium),41(c) there are no bubbles, and (d) the dynamic model assumedfor theexplanatory variables is correct. (The last assumption canbe checkedindependently using standard procedures.) Assumptions (a)and (b) arealso maintained when estimating (9') .Thusthe

37 bubbles test procedureonly makes sense
bubbles test procedureonly makes sense if diagnostic checks of the estimatedfit of (9') donot indicate misspecification. Testing propositions (c) and (d)whilemaintaining (a) and (b) has the obvious difficulty thatif the nullhypothesis Is rejected one does not know why. &it Inthe presentcontext, it seems particularly tenuous, since propositions(a) and (b)can be tested individually, and few people interpretthe evidence assupporting them.In addition, the small sample properties of the bubble specifi-cation test have been questioned (in the context of the presentvaluerelation for stock prices) by Mattey and Meese (1987) .Intheir sitnu—lation experiments the nonparametric tests for bubbles turned out tobemuch more reliable tests for the presence of bubbles than the specifi—cation test. Evans (1986) has employed nonparametric testsof bubbles —33.—on the dollar—pound exchange rate over the recent floating rate period,and reports finding a bubble in the dollar—pound rate.We now consider the weaknesses of variance bound tests •Werepeat equation (1), or its incarnation as the monetary model equation(9), ass =+—s)whereis the sensitivity of the current spot rate to the expectedchange in the spot rate (the same as A, the semi—elasticity of moneydemand, in the monetary model), anddenotes all the fundamentals.Now the results from Meese and Singleton (1983) allow us to deduce(15)var(s) ()in the absence of exchange market bubbles. }recan include thestructural disturbance

38 u in (1) or the variance bound can be t
u in (1) or the variance bound can be takenafter projecting s andon an information set that excludesu. The variance bound in (15) can be written in termsof conditionalvariances or, if equation (9') holds in first differences with u asthe structural disturbance, then a bound analogous to (15) holds for thefirst difference of s andThus nonstationarity of theexchange rate or fundamentals will not undermine the following discus-sion. The relation (15) makes It clear that It is meaningless to com-pare the variability of s with an individual component ofunless SL contains a single variable, or we know all the values of thestructural parameters on the variables inand the covariancesbetween all the fundamentals. Actual variance bounds tests of (1) are —34—generally uninteresting because they test whether thevariance of alinear combination of the variables in 5L is an upper bound onthevariance of s, and the tests are conditioned on knowing the correctvariables and the correct values of the structural coefficients.Whileit is true that the Generalized Method of Moments (GMM) methodologyofHansen (1982) can be used to construct a statistical test of (15)thatincorporates the sampling variability of theestimated parameters, thishas not been done in the exchange rate context •Webelieve that such anexercise is futile since It is already known that asset marketmodels ofexchange rate determination fit poorly.A more obvious problem with variance bounds tests can be seenfrom

39 the application of variance bounds proc
the application of variance bounds procedures to testsof forwardrate bias. Recall that the unbiasedness equation(16)s+i =a+ b(fd) +with b =1implies(17)�var(As+i) var(fd)The variance bounds test has no power to detect thealternativecov(fdi c) =cov{fd,(s—f)]� 0, since (17) would hold afortiori .Themost common empirical finding in regression testsof (16)is that cov(As÷i fd) (However, the variance of the lefthand sideof (17) is typically so muchlarger than the variance of the righthandside that a test of (17) failsto uncover a significant negative covarianceof the forward discountwith the forecast errorForexample, take the published resultsin Huang (1984) •Hisregressiontests of (16), reported in his Table 1(p. 157), indicate two rejectionsof b =1when —35—rejection of b =1�when b 0, out of a total of nine currencies.In his following Table 4 (p. 160), none of the variance bounds testsreject (17) for the same currencies and sample periods. It is true thatall of Huang's point estimates of the bound�var(ts+i) var(e1)are violated, but none of the violations is statistically significant.These "small sample" results illustrate the large—sample theoreticalresults of Fraukel and Stock (1987) who show that the most powerfulconditional volatility test is equivalent to the analogous regressiontest in terms of asymptotic power. See also Froot (1987)Finally, we conclude this section by noting that it is the lowervariance bound on the forward

40 rate (or forward discount) that isviolat
rate (or forward discount) that isviolated in exchange markets, not the upper bound. Properties ofconditional variances allow us to deduce var(f)var(s) whereStisthe predictor of s based on the limited informatin set of pastspot rates. The forward rate is the predictor of the spot based on themarket's larger information set. (The regression analogy is that R2cannot decrease when you add explanatory variables.) Since the varianceof the prediction error from forecasting s on just its own past valueis less than the variance of the prediction error from forecasting thespot rate withthe market's forecast, the lower variance bound onthe forward rate is violated. (See for example the root mean squareerror statistics in Meese and Rogoff (1983a) for the forward rate versusthe random walk model as predictors of future spot rates and the otherresults discussed in section 11.3.) —36—III. The change Risk Pre1uaWeare interested in the size and variability of therisk premiumfor two reasons •Firstif the size and variability are thought to besmall, as argued in Frankel (1986a), then it is difficult toattributethe results of regression tests of forward rate unbiasedness (describedin section 11.3), or the results of variance bounds tests (describedinsection 11.4), to the risk premium. This would leave onlythe explana-tion that expectations cannot be assumed rational (in the senseoflending themselves to representation by the ex post sampledistribution)Even if expectations are

41 thought to be rational, thereIs asecond
thought to be rational, thereIs asecond motivation for looking at the variability of therisk premium.Since the risk premium, rp in equation (2), togetherwith thesubstitutability parameter ,canbe a key determinant of the exchangerate, estimating the variability of therisk premium will help usanalyze the sources of variability in the spot rateHere we willbe particularly interested in the effects on swhen there is anexogenous change in asset supplies L,expectations As, or thesubstitutability parameter .Untilrelatively recently, empirical work on the risk premiumwaslimited almost entirely to the estimates of bias inthe forward market'sprediction of future spot rates discussedin section 11.3. The problemwas that rational expectations had to beassumed a priori in order tointerpret the systematic component ofthe prediction errors as equal tothe risk premium. For those who were willing tomake that assumption, —37—the conclusion was that the risk premium is large and variable. Forexample, the finding of zero coefficients in the regression of exchangerate changes against the forward discount implied that the rationallyexpected rate of depreciation was zero (random walk), and 100 percent ofthe forward discount was made up by the risk premium, rather than byexpected depreciation. Since the dollar's forward discount against themark or yen has moved over a range of roughly 2 percent to 4 percent inrecent years, this would imply that the risk premium was substantial in

42 both magnitude and variability.It has be
both magnitude and variability.It has been argued that if the systematic component of theprediction errors is indeed properly interpreted as the risk premium,then it ought to be related statistically to those variables on whichtheory tells us that the risk premium depends. We now turn to thetheoretical determinants of the risk premium and the correspondingeconometric tests.1. Implications of Portfolio—Optimization with Qnstant VarianceIf investors maximize single period utility that is a function ofmean and variance of end of period wealth, asset demands can be writtenas a linear function of expected relative rates of returns:(3')x =A—Brpwhere=pc2and A is the minimum variance portfolio. Theparameter p is the coefficient of relative risk aversion and c isthe variance (covariance matrix in general) of exchange returns.Several authors43 have inverted equation (3') without imposing the —38—theoretical restrictions of mean—variance analysis, andhave attemptedto explain the ex post risk pretniuni (forecast errors) byvariables towhich portfolio balance theory says that the risk premiumshould berelated. This line of research has uniformly found norelationbetween rp and x. The difficulty n predicting ex postrisk premiumsby asset shares can be appreciated by inspecting figures1—5, where thetwo variables are plotted for five currencies versusthe dollar overApril 1973 to December 1984. The mean driftsin the Japanese, German,French and British shares are not accompani

43 ed by meandrifts in thecorresponding exc
ed by meandrifts in thecorresponding excess return variables.Using the constraints implied by mean—varianceanalysis, andreasonable coefficient estimates for the parameters in (3t)Frankel(1986a) has argued that the exchange rate risk premium(and also itsvariability) must be very small. The argument can besummarized asfollows: the unconditional monthly variance of therelative return ondollars over the period August 1973 —August1980 is roughly .001.Ifwe take .001 as an upper bound on theconditional variance of relativedollar returns, and a coefficient on risk aversion equalto two, thenthe term tpc]is .002. An increase in the supply of foreign assetsequal to 1% of the portfolio would only requirean increase in the riskpremium of 2.4 basis points on an annualbasis!44 The argunient does,however, assume that the conditional varianceof returns is constant; wetake up this subject in the next subsection.Hansen and Idr1ck (1983), and Wdrick and Srivastava(1984,1986), among others, have attempted to conductinference regarding themagnitude and variability of the risk premium using amore general —39--intertemporal utility valuation model of the risk premium. In thissetting a linear equation relating asset supiles to the risk premiumwould only obtain if investors' preferences were logarithmic or assetreturns are interternporally independent. We would not a priori expectto be able to explain the risk premium by relative asset shares alone,so these models offer an alterna

44 tive theory of rp.The intertemporal gene
tive theory of rp.The intertemporal general equilibrium model of Lucas (1982), asamended by Fbdrick and Srivastava (1984), can be used to derive arelation between the forward rate and the expected future spot rate.While this model relies on some very restrictive assumptions, Itprovides considerable insight into the nature of the exchange rate riskpremium. In this type of individual utility maximization model theequilibrium price of an asset Is found by equating the foregone marginalutility from purchasing an asset to the conditional expectation of thepresent discounted value of the marginal utility of return from holdingthe asset. In order to price a nominal forward exchange contract, moneyis introduced into the asset pricing model by the restriction thatagents purchase a country's idiosyncratic endowment with that country'smoney. Arbitrage ensures that the next period forward price of foreignexchange be equal to the expected present value of a known return attime t of investing in a nominally risk free bond with payoff inperiod t+l, i, multiplied by the spot rate that will prevail in thenext period,The asset pricing model provides the interpretationof the discount factor as the interteinporal marginal rate of substitu-tion (Richard and Sundaresan (1981)) between period t and t+ldollars, +1' where, for convenience, a U.S. perspective has been -40-adopted. Thus the forward rate can be written as:(18)=Usingthe definition of conditional covariance, (18) may

45 berewritten as:-(19)=E{st+ijI(t)]+ cov[Q
berewritten as:-(19)=E{st+ijI(t)]+ cov[Q+ii, s11I]where cov[ .,.1I] denotes covariance conditional on the informationset I, and the derivation of (19) makes use of the first ordercondition for utility maximization E[ QI=1.The second termt t+1 t ton the righthand side of (19) has the Interpretation of a risk premium.However, it is useful to Introduce a benchmark return, i1, definedexplicitly in Hansen and Hodrick (1983), so that (19) can be rewrittenas a conditional capital asset pricing equation (CAPM):(20)E(f —s+1I1)=tE(1+i—iIIt)whereis the conditional covariance of the risk premium andbenchmark return divided by the conditional variance of the benchmarkreturn.Implications of this model have been tested by Hansen and Hodrick(1983), Hodrick and Srivastava (1984, 1986) and cumby (1986), amongothers. Ekapirical work is typically conducted assuming that conditionalsecond moments and/ordo not vary across time. While statisticaltests of the "consumption beta" model usually indicate a rejection ofthe model, qualitative features of the data are explained by this para-digm; see the discussion in Oimby (1986) .Wenow turn our attention to —41—the implications of time variation in return second moments on variabil-ity of the risk premium and in turn on the variability of the spot rate.2. Implications of lime—Varying turn (bvariance8Anumber of authors have in effect argued that the assumption ofa constant covariance matrix of exchange returns should berelaxed.45Pagan

46 (1986) argues, in a context where the c
(1986) argues, in a context where the conditional variance changesover time, that there may be some points when it exceeds the samplevariance (.01 on an annual basis), and that the risk premium at such apoint will exceed the upper bound claimed in Frankel (1986a) .Witif weallow the conditional variance to vary over time, then one can stillapply the upper bound to the average conditional variance and thereforeto the average risk premium. If the conditional variance is 10 timeslarger than .01 one period in ten (for example, when the precedingsquared realization was particularly large), then it is true that a onepercent change in the portfolio in that period will change the riskpremium by as much as 0.2 percent per annum, and that the magnitude ofthe risk premium could be as large as 20 percent annum (if close to 100percent of the portfolio is in one asset or the other) .Witin theother nine periods out of ten, these magnitudes would have to be zerofor the variance to average out to .01.When we allow for return variances to vary over time, variationin the risk premium derives from this additional source and can thusexhibit considerable volatility. This point is made by Giovannini andJorion (1986) .Ifwe are interested in the question of how big aneffect foreign exchange intervention has on average, then the —42—observation that the conditional variance and the risk premium may attimes be higher and at times lower maynotbe very relevant. &it forother questions

47 , such as explaining the variability of
, such as explaining the variability of the exchangerate, the observation that the risk premium changes overtime is quiterelevant.Recent work by Oiiaby and Obstfeld (1984), Hsieh (1984), Domowitzand Hakkio (1985), and Giovannini and Jorion (1987a), rejects the hypo-thesis that the conditional variance of exchange returns is constantover time. Supporting evidence is provided by Implicitvariancesextracted from options data in studies by Lyons (1986) and HsiehandManas—Anton (1986): these estimated variances, which are to be thoughtof as characterizing Investor's conditional beliefs, clearly vary overtime.Giovannini and Jorion (1987a) specify the conditional variance asa function of the levels of domestic and foreigninterest rates. Theiraim is to argue that their estimate of variation in theconditionalvariance corresponds to large variation in the risk premium,In contrastto Frankel (1986a) .&itthey appear to have fallen into a (remarkablycommon) pitfall in their calculations: their estimates imply atruevariance of the monthly risk premium equal to 1.1 x1O, not 1.1(Giovannini and Jorlon, 1987b)Perhaps the most popular approach to modelingthe conditionalvariance of returns Is to employ variants of Engle's (1982)autore-gressive conditional heteroskedasticity (ARCH) process •Inthe contextof the single period mean—variance model, Engeland Rodriguez (1987)show how to extend the econometric procedure ofFrankel (1982) to —4')—account for time variation in return seco

48 nd moments. However, the basicmessage fo
nd moments. However, the basicmessage for the relation between asset shares and the risk premiumisunaltered when the Engel—Rodriguez procedure is employed.Table 2 provides evidence of the variability in exchange returnsecond moments; the maximum, minimum, and average values of the samplestandard deviations and correlations are tabulated for a sample of 100months. The autocorrelatlon function of the annualized standard devi-ation of exchange return calculated from the $/DM rate is displayed inTable 3. The pattern of autocorrelations in Table 3 is typical of all21 elements in the second moment matrices displayed in Table 2. Areasonable characterization of this process is simple exponentialsmoothing or an ARIMA (0,1,1):(21)—=+÷ 016t—1with=Theestimate ofvaries by currency second moment, but=—.9isrobust to the series under analysis. The stochastic model (21) impliesthat all shocksto the standard error of exchange return secondmoments are permanent. The large negative coefficient onimpliesthat after one period, the level of the standard deviation of return isincreased by 10% of the initial shock forever.We are now ready to consider the implications of time variationin return second moments for the larger question of exchange ratedetermination. We can infer the effects of changes in exchange ratereturn variance on the demand for asset shares by looking at ourequation for the optimally—diversified portfolio: Table 2Exchange Rate Return Second MomentsMonthly Echan

49 ge Returns Calculated from Daily Observa
ge Returns Calculated from Daily Observationsover the period December 1977 —April1986(Monthly return standard deviations run down the diagonal,and monthly correlations are displayed off the diagonal.)The following entries are maximum values over the sample of 100 months:Australian Dollar36.24British Pound.9445.23Canadian Dollar.81.8514.80German Mark.93.96.8249.87Japanese Yen.93.93.77.9623 .78Swiss Franc.93.94.83.99.9542.58The following entries are minimum values over the sample of 100 months:Australian Dollar1.70British Pound—.462.42Canadian Dollar—.46—.63.76German Mark—.63—.42—.722.72Japanese Yen—.34—.65—.72—.212.74Swiss Franc—.47—.37—.65—.20—.043 .02The following entries are average values over the sample of 100 months:AustralianDollar6.69British Pound.3410.91Canadian Dollar.19.254.03German Mark.37.60.2611.06JapaneseYen.42.43.22.629 .89Swiss Franc.39.58.27.81.6313.07Notes: A daily rate of return is defined as (s(t) —s(t—1))/s(t—1)for tinthe same month. All monthly return standard deviations are expressed asannual percentages and are calculated as if daily returns within a monthconstitute a random sample. Table 3Autocorrelation Functions for the Level and Differenceof Return VolatilityLevel of Standard Deviation of $/DM Annual Return:,/ciMeanof Ii= 11.06Sample Variance of I?i= 50.05Sample Size =100AutocorrelatlonAutocorrelation1.147.142.168.073.29*9,Ø54.0910.115.0711.026.0412—.03Difference of $/DM Annual Return Standard DeviationMean=.02Sample Variance=8

50 6.94Sample Size =99of—f,/?Autocorrelatio
6.94Sample Size =99of—f,/?AutocorrelationAutocorrelation1_.52*7.112—.068—.043.199—.044—.1110.085.0111—.026—.0812—.03*denotes significance at a 5% level —44—(3')x =A-(pt)'rptWe will try several alternative cases regarding the assumed permanenceof a change in the variance. Using equations (2) and (3') we can calcu-late first the effect on the spot rate of a once and for all change inthe variance of exchange returnsholding the interest differential*47i —iconstant:ttdsrp(22)___--= +1J__÷'t)d*ex(1—x)2tttThisanalysis can be justified by assuming that the composition ofmonetary and nonmonetary assets is varied in whatever way is necessaryto hold the interest differential constant. Since the change inispermanent we know that the effect on tomorrow's spot rate will be thesame as the effect on today's spot rate. Thus the risk premium—i—(sefl—se))is held fixed in this experiment. The analysisis in the same spirit as our earlier attempts to quantify loosely theeffects of changes in the disturbance terni u in (2) and in expec-tations when macroeconomic fundamentals are held constant.The sign of the effect, equation (22), of the return variance onthe spot rate depends on the sign of the initial risk premium. If theforeign asset initially pays a positive risk premium over the domesticasset (because the supply that must be held exceeds the demand consti-tuted by the minimum—variance portfolio A; rpt as we have defined itis negative), then the permanent increase in uncert

51 ainty reduces thedemand for foreign asse
ainty reduces thedemand for foreign assets and thus reduces their priceThe effect —45--on st is zero if the initial risk premium is zero. &it the effect canbe very large in magnitude if the initial risk premium is non—zero, forexample if the initial risk premium is on the order of .03 (as it mightbe if the entire 3 percent discount at which the dollar sold against themark or yen in the early l980s is attributed to a risk premium ratherthan to expected depreciation). For our benchmark parameter values(xt =1/2=.01on an annual basis, and p =2),we cancalculate the linearized effect on the spot rate s of a change incLConsider a permanent increase in the annual variance cfrom .01to .02. (One standard deviation of the monthly standard deviation forthe $/DM exchange return is estimated at roughly 9.3% per year over theDecember 1977 to April 1986 period in the bottom panel of Table 3; thesquare is .0087.) Such a shock will have a possible linearized effectOflSt ofroughly (—4)[ .02__joi =—600%,alarge number. Of2( .01)course, permanent changes in 2t cannot occur very often.The autocorrelations in Table 3 are also consistent with thehypothesis thatis white noise around its mean. Under such ahypothesis no shocks are permanent. A purely transitory disturbance towill have an effect which is very much smaller than that calculatedabove: calculations based on (22) are mitigated by the presence of asecond term that arises because the spot rate is expected to go back toits p

52 revious level.48The results of this subs
revious level.48The results of this subsection suggest that additional researchbe directed at the spot rate—conditional volatility nexus. In appendix1 we consider the model (21) foras well as an ARCH process, in —46—which the initial shock to the variance dies outgradually over time.The algebra is considerably more complicated then for thetransitorydisturbance, because there is a third effect, via the rationalexpecta-tion of an effect on the spot rate next period, when the innovationtothe variance will have only partially died out. The effecton theexchange rate lies between the effects of a permanent and transitorychange in(Poterba and Summers (1986) calculate the effect ofvariability in the variance of stock market prices on the level of stockmarket prices •)49IV.Survey Itaand Heterogeneous pectat1onsOf the factors suggested as determining "excessivevariability"in section I of this paper, we have considered the role of fundamentalsversus the disturbance term, and we have considered risk and the degreeof substitutability. We have still to consider the role ofexpectationsper se. The idea of destabilizing speculation——that investors, respon-ding to non—zero expectations of exchange rate changes, work to raisethe variability of the exchange rate——is what is often meantby descrip-tions of the market as excessively variable. The variance—bounds testsand bubbles tests at first sounded like a promisingway to shed light onquestions of destabilizing speculation an

53 d bandwagons. If we were con-fident abou
d bandwagons. If we were con-fident about having specified the fundamentals correctly inequation (2),then we might look to the variance—bounds and bubbles tests tosee ifthe expectations term ASe is formed in a destabilizingway. More —47—simply, we could compare the variance when ASe in equation (9) isconstrained to zero with the unconstrained variance: this is the testfor "destabilizing speculation" performed by Kohlhagen (1979) andEichengreen (1981). &it, as we argued in section II, we are not at allconfident about having specified the fundamentals correctly, which meansthat there is no new information to be gained from these tests.At the end of Part I we suggested that the best way to get at thequestion of whether speculation is destabilizing or not is to considerwhether expected future depreciation responds positively or negativelyto a current change in the exchange rate. If a current depreciation,originating in fundamentals or in anything else, generates anticipationsof further depreciation, speculators will sell the currency and therebyexaggerate the depreciation. If it generates anticipations of futureappreciation, back in the direction of some long—run equilibrium,speculators will buy the currency and thereby dampen the depreciation.In this part of the paper we consider this question of how expectationsare formed.A way of defining stabilizing expectations is that the expectedfuture spot rategives a weight less than one to the contem-poraneous spot

54 rate, s, that it Is a convex combination
rate, s, that it Is a convex combination of thecontemporaneous rate and other factors. We have the case of regressiveexpectations when the "other factor" is the equilibrium rate :=(l—8)s+ o(s) .Or,in terms of expected depreciation,(23)As+i =s) —48—Stabilizing expectations are the case 0 expectations the case 0 expectations, 0 =0.1. Measuring Stabilizing and Istabilizing pectations1.zo alternative ways of measuring expected exchange rate changesare common in the literature. The first is the forward discount. Thesecond is ex post changes in the sample period, allowing only for apurely random error term. The first is valid only if there is no time—varying risk premium, and the second only under the rational expecta-tions assumption (including the absence of regime changes, pesoproblems, etc.).When we use the rational expectations approach of substitutingt+1 for ts1 in the above equation (23) and interpreting theregression error as a random expectational error, a test of 0 =0is atest of a random walk in the true spot process, as in section II. Table4a, which is drawn from Frankel and Froot (1985), looks for a tendencyof the dollar to regress toward equilibrium, where equilibrium isalternatively measured in rows 1—4 by a constant and in rows 5—8 by thevaluethat would give purchasing power parity (with 1973—1980 asthe base period) .Weare unable to reject a random walk in several(limited) post—1976 sample periods. We argued in section tI.1 that thetheoretical ca

55 se for a reversion to equilibrium rather
se for a reversion to equilibrium rather than a randomwalk is much stronger in the case of the real exchange rate than it isfor the nominal exchange rate. Rows 9—12 of Table 4a report tests ofthe tendency to regress towardwhere the change in the exchanget —49—rate is adjusted for expected inflation rates.5° If we wanted to applythe same rational expectations assumption to the price levels as to thenominal exchange rate, we would be back in section 11.1's tests ofchanges in the real exchange rate. The results on our limi.ted post—1976sample periods are the same as in most tests of ex ante purchasing powerparity on the floating rate period: we can find no statisticallysignificant tendency to regress toward an equilibrium.These findings are exactly what we would expect from our earlierconclusion that the speed of return to PPP may in fact be in the rangeof 15—25 percent a year, but that the floating rate period does notoffer enough data to reject zero statistically. If we are sufficientlyconfident about both the gradual tendency to return to equilibrium inthe longer run——based, for example, on the 115 years of U.S.—U.K. data——and the rational expectations assumption, we can infer that speculatorsmust have stabilizing expectations. &t it would be better to have moretangible evidence, since the 115 year autoregressions cover a number ofdifferent economic policy regimes.Table 4b uses the forward discount to measure expected depreci-ation in the same regression

56 equations. Such regressions will only be
equations. Such regressions will only bevalid in the unlikely case that the risk premium is uncorrelated withthe rlghthand—side variable (the current spot rate relative to itsequilibrium). Not surprisingly, the results are poor. Even when thecorrection for expected inflation is applied, which makes the equationessentially a regression of the short—term real interest differentialagainst the real exchange rate, the coefficient is statisticallyinsignificant. Others have gotten better results on larger data sets. Table 4a Estimates of Regressive Expectations — 8(s — Expected Depreciation Measured by s Measured by Forecast Horizon Estimation Technique 8 Standard Error t—test for 8 = 0 D d f . R2 Ex post changes t9t+l Constant 1. 2. 6 'no. I 6 'no. II OLS OLS —.08 +.02 .08 .14 —1.01 +0.17 3.05 2.83 29 24 .59 .27 3. 3 'no. OLS —.04 .05 —0.80 2.28 59 .15 4. 3 'no. SUR +.03 .02 +1.35 59 PPP 5. 6 no. I OLS —.10 .09 —1.20 3.08 29 .59 6. 6 no. II OLS +.04 .16 +0.26 2.79 24 .27 7. 3 'no. OLS —.01 .06 —0.16 2.21 59 .14 8. 3 'no. SUR +.10 .04 +2.54** 59 Ex post changesa (adjusted for expected inflation) e *e — — ' PPP 9. 10. 11. 6 mo. I 6 'no. It 3 'no. OLS OLS OLS —.02 —.01 —.00 .06 .09 .04 —0.44 —0.16 —0.08 2.92 1.94 2.42 22 18 46 .00 .00 .00 12. 3 'no. SUR —.01 .01 46 aSeparate dummy variables were reported for each currency. All estimates of constant terina and dummy variables are omitted to save space, but can be found in Frankel and Froot (1986) Tables 6a, 6b, 7a,

57 7b, 9a, 9b. * Significant at the 95 per
7b, 9a, 9b. * Significant at the 95 percent confidence level. **Significant at the 99 percent confidence level. Standard errors reported for Seemingly Unrelated Regressions are asymptotic standard errors. Dates of sample observations are reported In Frankel and Froot (1985), Table Al. Table 4b Estimates of Regressive Expectations — e(s — Expected Depreciation Measured by s Measured by Forecast Horizon Estimation Technique I I 0 Standard Error t—test for 0 0 D.W. d.f. R2 Forward Discount fdt Constant 1. 2. 6 6 mo • mo. I II OLS OLS — .6 —.01 .09 .02 —1 .20 —0.31 3 .08 1.24 29 24 .59 .85 3. 3 no. OLS + .00 .01 +0 .49 1 .26 59 .79 4. 3 mo. SUR +.01 .01 +1.09 59 PPP 5. 6 mo • I OLS + .02 .02 +0 .99 1 .67 29 .79 6. 6 no. It OLS —.02 .03 —0.66 1.17 24 .86 7. 3 mo. OLS +.00 .01 +0.72 1.43 59 .79 8. 3 mo. SUR +.00 .01. +0.30 59 Forward Discounta (adjusted for expected inflation) e *e fd - ( - x PPP 9. 10. 11. 6 6 3 mo. too. mo. I II OLS OLS OLS —.02 —.05 —.01 .02 .02 .01 —1.15 0.80 1.01 0.81 22 18 46 .01 .25 .08 12. 3 ma. SUR —.01 .00 46 (For Notes, See Table a) Table 4c Estimates of Regressive Expectations 8(s — Expected Measured Depreciation by s Measured by Forecast Horizon Estimation Technique Standard Error t—test D d f R2 for e 0 Economist Constant 1. 6 inn. I OLS +.li .03 +3.77** 1.53 29 .89 Survey Data s 2. 6 mo. II OLS +.14 .03 +5•49** 2.71 24 .95 3. 3 mo. OLS +.06 .12 +4.82** 1.60 59 .85 4. 3 inn. SKUR +.05 .01 +4.13** 59 PPP 5. 6 mo. I OLS .1.1 .03

58 3•53** 2.55 29 .84 6. 6 inn. II OLS .16
3•53** 2.55 29 .84 6. 6 inn. II OLS .16 .03 4.92** 2.56 24 .94 7. 3 mo. OLS .05 .01 3.83** 2.12 59 .84 8. 3 ian. SUR .06 .02 3.56** 59 Economist8 PPP 9. 6 mo. I OLS .06 .02 2.50** 1.30 22 .91 Survey Data (adjusted for 10. 6 mo. II OLS .04 .03 1.46 1.54 18 .06 expected inflation) 11. 3 mao. OLS .03 .01. 2.54** 1.65 46 .10 S e ** — 12. 3 mo. SUR .00 .00 0.15 46 (For Notes, See Table a) TABLE 4dREGRESSIVE EXPECTATIONSIndependent variable: s(t) —s(t)s measured by PPPSUR Regressions(1) of Survey Expected Depreciation:E(s(t+1)) —s(t)=a+ Os(t) —s(t))CoefficientData SetDatest:0 =0DW(2)DFR2MMS 1 Week10/84—2/86—0.0283—3.53 **2.102190.58(0 .0080)MMS2 Week1/83—10/84—0.0299—3.78 **2.151790.61(0 .0079)MMS1Month10/84—2/86—0.0782—5.84 **1.401510.79(0 .0 134)MMS3Month1/83—10/84—0.0207—1.411.551790.18(0 .0146)Economist 3 Month6/81—12/850.02231.78 *1.661840.26(0 .0126)Amex 6 Month1/76—8/850.03151.561.22450.21(0.0202)Economist 6 Month6/81—12/850.06003q77 **1.321840.61(0.0159)Amex 12 Month1/76—8/850.12364.48 **0.60450.69(0.0276)Economist 12 Month 6/8 1—12/850.17508 .10 **1.251840 .88(0.0216)(1) Amex 6 and 12 Month regressions use OLS due to the small number of degreesof freedom(2)The DW statistic is the average of the equation by equation OLS Durbin—Watson statistics for each data set.*representssignificance at the 10 percent level.**representssignificance at the 1 percent level.R2 corresponds to an F test on all nonintercept parameters.The above results are reported In Fran

59 kel and Froot (1985)Constant terms for e
kel and Froot (1985)Constant terms for each currency were included in the regressions, but notreported above. —50—Hooper (1985), Hutchison and Throop (1985), Golub et al (1985), Sachs(1985) and Feldstein (1986) report regressions with the two variablesreversed, and find that the long—term real interest differential, inparticular, Is a significant factor explaining the real exchange rate.&it when Meese and Rogoff (1986) examine real versions of the stickyprice monetary model of section 11.2, they find that real interestdifferentials cannot explain movements in real exchange rates signi-ficantly better than a random walk model. (The advantage of testing thesticky price monetary model In real terms is that one abstracts frominstability in money demand parameters. Given the assumptions employedin their paper, monetary shocks have proportionate effects on the realexchange rate and real interest differential, and hence net out of theanalysis.) In any case, the entire approach of using the interestdifferential or forward discount to measure expected depreciation willbe viewed as suspect by those concerned by the risk premium.What is sorely needed is an alternative to measuring expecteddepreciation either by ex post exchange rate changes or by the forwarddiscount, one that does not require pre—judging either the unbiasednessof expectations or the existence of the risk premium. A good candidatefor such a measure is offered by surveys of the exchange rate expecta-t

60 ions of market participants. One such su
ions of market participants. One such survey has been conducted everysix weeks since 1981 by the Fconomist—affiliated Financial Report. Thedata are discussed and analyzed at length in Frankel and Froot (1985,87)and Froot and Frankel (1986).Table 4c reports regressions of regressive expectations withexpected depreciation measured by the Economist survey data, for the —51--samesamples of observations that were used in the preceding regressionsof ex post changes and forward discounts.5' Almost all the results showa highly significant expectation of regression toward equilibrium, at arate of about 10 percent in the case of expectations at a six—monthhorizon, which is (1 —•92)=20percent per year. (The expected speedof adjustment per year is somewhat lower in the case of the three—monthhorizon.) This expected speed of adjustment to PPP is in the range ofthe actual speeds of adjustment estimated in Table 1, suggesting thatthe low power in the regressions of ex post changes in Table 4a mightalone have been responsible for the failure to reject a random walk.52Other tests reported in Frankel and Froot (1987), Dominguez(1986), and Froot and Frankel (1986), show that the prediction errormade by the survey numbers is not in fact random. The tests constitutea rejection of rational expectations (jointly with the hypothesis of noregime changes or other peso problem) that is free from any concernsabout the risk premium. Generally, the true spot process behaves morelike a

61 random walk than the survey respondents
random walk than the survey respondents realize. In terms of thelanguage attributed to Rhlson (1981) in section 11.3 above, there isexcessive speculation: investors would do better to reduce theirexpectations of exchange rate changes toward zero. In terms of thespecific regressive expectations model estimated in Table 4c, surveyrespondents overestimate the speed of return to equilibrium.One might think that such a failure of market efficiency would beevidence of the sort we are looking for, that "exchange markets are notworking properly." it a tendency for speculators to expect theexchange rate to regress toward the equilibrium at a faster rate than is —52—correct is stabilizing. An increase in the value of the currency, duein the context of equation (2) to an increase in the interest differ-ential i_i* or the error term u for example, will be damped becauseof the effect on expectations. We saw in equation (10) that the varia-bility of the exchange rate in the DQrnbusch overshooting model isinversely related to the value ofe.Ahigh 0 means that, for anygiven positive interest differential, it takes less of an appreciationto generate the necessary expectations of future depreciation.53One cannot work with the survey data on expectations withoutpondering the issue of heterogeneous expectations. Almost all of theexchange rate literature, theoretical as well as empirical, presupposesthat market participants all share the same expectation. &it the truthis that

62 people disagree. Disagreement can expla
people disagree. Disagreement can explain the very high volumeof trading in the spot and forward exchange markets. The Economist'sFinancial Report shows quite a range of variation in their surveyresponses; the high—low spread for the six—month expectations averages15.2 percent. (The regressions reported in the tables here are based onthe median response.)The possibility of heterogeneous expectations introduces anotherpossible source of variability into the exchange rate: the market inthe aggregate may shift over time the weights it assigns to differentforecasting mechanisms, for example the weight assigned to regressiveversus bandwagon expectations in equation (5). The market may increasethe weight it gives to one of these formulations if it has recently beenforecasting better than the other. This could happen if portfolio mana-gers update in a Bayesian way the weights they place on the forecasts of —53—different models. Alternatively, it could happen when those investorswho bet correctly gain wealth and receive more weight in the market inthe next period. As the weight placed by the market on different expec-tations shifts, the aggregate demand for foreign currency and thereforethe exchange rate will change over time. Even if no single forecasterholds destabilizing bandwagon expectations, any factor pushing up thevalue of the currency, such as an increase in (i_i*) or Ut willproduce a drawn—out appreciation as the weight placed on the optimisticforecast

63 s gradually increases. Although none of
s gradually increases. Although none of the actors in such amodel is satisfying the rational expectations assumption in the sense ofknowing the complete process that is driving the exchange rate, neitheris any of the actors behaving foolishly. Putting more weight on band-wagon expectations than on regressive expectations would have given theright answer in the case of the dollar from 1981 to February 1985, forexample, but would have lost the investor a lot of money thereafter. Insuch a changing world it is difficult to see what variables it would be"rational" for the investors to grant more weight.We can offer some evidence for the idea that forecasters don'tconcur on a single stabilizing sort of expectations model as nicely asthe estimates of regressive expectations in Table 4c would suggest.Money Market Services, Inc., has conducted since 1983 a weekly survey ofcurrency traders as to their forecasts at shorter—term horizons than theEconomist survey. Estimates of regressive expectations on these twosets of survey data, together with a third conducted by the AmericanExpress Bank Review irregularly between 1976 and 1985, are reported friTable 4d. The nine data sets are ordered by forecast horizon. The —54—results are striking. The longer—term forecasts are strongly regressive:the &onotnist 12—month forecasts show that a 10 percent appreciationtoday generates the expectation of a 1 .75 percent depreciation over thesubsequent year. The shorter—term forecasts

64 show precisely the reversehowever: a 10
show precisely the reversehowever: a 10 percent appreciation today generates the expectation of0.78 percent further appreciation over the next month. This suggeststhe possibility that the forecasters who subscribe to bandwagon expecta-tions ("chartists," or technical analysts, who use time series analysisto extrapolate past trends) tend to be traders with a shorter—term out-look, while those who subscribe to regressive expectations ("fundatnen—talists," who forecast a return to macroeconomic equilibrium) tend to beeconomists with a longer—term outlook. A small change in the weightthat the market gives to two such different forecasts could have a bigeffect on the exchange rate, especially if asset demands are as sensi-tive to expected rates of return as was suggested by some of the substi-tutability arguments in section iii.2• OnclusionThe question, "are exchange rates excessively variable?", cannotbe answered by comparing the variance of the actual exchange rate to thevariance of a set of macroeconomic fundamentals. Overshooting theories,which are based on fundamentals and are consistent with market eff 1—ciency, allow exchange rate variability to exceed monetary variabilityby an indefinitely high multiple.More damaging is the finding of regression tests that measurablefundamentals do not explain the exchange rate well, even contempo— —55—raneousLy. The proposition that the exchange rate follows a randomwalk, in the sense that first differences cannot be

65 explained by any-thing, may be unnecess
explained by any-thing, may be unnecessarily nihilistic. There does appear to be sometendency for the exchange rate to return to equilibrium at longerhorizons •Eventhe dollar came home to earth in 1985—86. &it the errorterm is very large in the shorter—term (e.g. monthly) data that isavailable for most econometric studies.It is tempting to argue that there must exist fundamentals ofwhich market investors are aware but the econometrician is not •Suchanargument might be supported by any evidence that the market could pre-dict future exchange rates better than the models; but there is no suchevidence. Expectations measured by the forward exchange market (or bysurvey data) contain no useful information for predicting exchange ratechanges. The very bad performance of the forward discount, either as apredictor of future changes in the forward exchange rate or as a contem-poraneous determinant of the level of the exchange rate in equationssuch as (8), could in theory be attributed to an exchange risk premium.The framework of optimal portfolio diversification and conventionalestimates of the degree of risk—aversion seem to imply that the riskpremium does not vary enough to acquit the forward exchange market •Awelcome recent line of research takes the source of variation to bechanges in the conditional variance rather than changes in asset sup-plies. &t this literature has yet to demonstrate that such variationin the conditional variance can explain the bias in t

66 he forward rate.We have seen that small
he forward rate.We have seen that small changes in the variance, unless transi-tory, could produce large changes in the demand for assets and therefore —56—in the exchange rate as opposed to the risk premium.. &itunless"changes in the variation in the exchange rate" is considered anenlightening explanation of variation in the exchange rate, we are stillmostly in the dark.Oneneednot explain all the fluctuations in the exchange rate toevaluate the scope for government policy.55 Policy—makers could affectthe foreign exchange market through three different channels. First,macroeconomic policy, for example interest rates and the mixbetweenmonetary and fiscal policy, has large effects (the poor regressionreults not withstanding) .Wehave not explored the resulting policy-making tradeoff between the exchange rate and other macroeconomicobjectives in this paper. They are thoroughly discussed elsewhere, andthe regression tests in any case suggest that the major determinants ofthe exchange rate, at least on a shorter—term basis, lie elsewhere.Second, Tobin (1978) and Dornbusch (1986) have argued that a taxon International borrowing or on other foreign exchange transactionswould reduce the extent to which investors could react to small changesin the attractiveness of different countries' assets, and would therebyreduce exchange rate volatility. As we noted in section 1.2, thisargument requires that expectations be destabilizing. If expectationsare instead stabilizing

67 , then a decrease in the degree of subst
, then a decrease in the degree of substituta-bility would increase exchange rate volatility rather than thereverse. On the other hand a Ibbin tax on foreign exchange transactionsshould work to discourage short—term trading more than long—termtrading, since the given tax is a higher percentage of the return forshorter term transactions. If our survey data findings are correct and —57—short—term expectations (less than 3 months) are destabilizing but long—term expectations (greater than 3 months) stabilizing, then a Ibbin taxmight reduce volatility after all.Third, others argue that central banks should intervene inforeign exchange markets to dampen fluctuations. Foreign exchangeintervention of course is generally thought to affect the exchange rateto the extent it changes the relevant macroeconomic fundamentals,particularly nonsterilized intervention that allows the change inreserves to change the money supply. &t effects via current macro-economic fundamentals should be subsumed in the first category above.If foreign exchange intervention is to have an independent effect,particularly if sterilized intervention is to have a substantial effect,it will be via investor expectations of future exchange rate changes.The strongest case for steps toward reform of the floating rate systemwould be if one could demonstrate that expectations are destabilizing,producing bandwagons in the exchange rate, and that a change ingovernment policy might alter these expectatio

68 ns even without alteringasset supplies,
ns even without alteringasset supplies, for example, by bursting a speculative bubble. Theannouncement on September 22, 1985, that the G—5 had decided at thePlaza Hotel to work to bring the dollar down caused an instant 5 percentdepreciation of the dollar. While the fall in demand for dollars couldbe explained as a rational re—evaluation of the future expansionarinessof U.S. monetary policy, it might also be explained as the bursting of abubble. .ir theories of rational speculative bubbles have virtuallynothing to say about what causes the price to jump from one bubble pathto another •Witthis is precisely the sort of effect for which many —58—proponents of a more activist policy are looking. Proponents of atarget zone argue that the stabilizing effect would be even greater ifthe government announced a change in policy regime, rather than a one-time initiative of the sort that took place at the Plaza.The key question, then, seems to be the behavior of investorexpectations. In particular, much hinges on whether expectations whenleft to themselves are destabilizing. The question whether the truespot process matches up with the expected one, i.e., whether expecta-tions are rational, Is not as directly relevant. The evidence appearsto be that expectations are stabilizing, at least at horizons greaterthan three months. The survey data at a six month horizon reported inTable 4c, for example, show that a 10 percent appreciation todaygenerates an expected future d

69 epreciation of about 1 percent, or 2perc
epreciation of about 1 percent, or 2percent at a per annum rate. If speculators are investing on the basisof these expectations, then they are acting to stabilize the exchangerate. A Tobin tax to discourage speculation might then raisevariability.Survey data at short horizons show quite different resultshowever •Itseems likely that expectations are in fact heterogeneous.Oneconsequenceis that "the" expectation can't be rational if investorsdo not agree on asingleexpectation. A second implication follows fromthe high degree of substitutability (for an average value of thevariance) that we found in section III: small changes in the weightsthat the market assigns to competing exchange rate forecasts willproduce large changes in portfolio preferences and thus large changes inthe exchange rate. This source of exchange rate variability could be —59—classed as a speculative bubble in the sense that it arises from self—confirming changes in expectations rather than from fundamentals, thoughit is not the rational speculative bubble that has been extensivelystudied recently.As Krugman (1985) has argued, .when the market has temporarily"lost its moorings,' It is possible that a more activist policy canrestore the anchor to expectations. Investors might be persuaded toexpect more of a tendency to return to equilibrium. Wit central bankgovernors and finance ministers of major countries will only be able toaffect expectations If they have credibility. They did not havec

70 redibility in 1973. In this sense the br
redibility in 1973. In this sense the breakdown of the fixed exchangerate system was inevitable. They have more credibility today; this muchis clear from the market's sensitivity to every utterance of theTreasury Secretary and the thairman of the Federal Reserve, and theirJapanese and German counterparts. Whether this credibility would stillbe there if policy—makers tried to exploit it more systematically with areform of the world monetary system is another question, especially ifone allows for the usual politicization of any process of choosingtargets for an economic price that affects people's livelihoods. —60—Appendix 1Consider an ARIMA (0,1,1) model for2—1+ 6t —c1cS_iwith0 (( 1 •Thelinearized effect on s of a shocktocan be obtained from the following expression.ds11rp dcds ds+=(i— +i- )[— r --d61)]ttt 1ttttterp dst+i_____= [__— +,wheredt =(i+ pc/(------ + ii_)), a numberslightly less than one. Now assuming the initial position representedan equilibrium we can take 'xand rp to be constant so thateee_____________t+11t+1t+2ds+i—(1+1rpdsds-j-——)[ — (1—c)—____ ____ ____d6t—x1t+i—d6eeerpdsdsrpdstt+2_______________t+2tt+3=(1—cs)+d6•Likewise,d6—(1—n)+d5tttCombining these results we obtaindstrp={i+ (l-a) + (1a)4 +rp_____=+-1)]Using our benchmark values for x, rp,and= (1.005)iIf we can also assume that a =.9as in equation (21) for,thends63.3. Therefore, the linearized effect on the spot rate of a .01change inis an aproximately 63% appreciation of the less risky

71 tcurrency. —61—For the case wherefollows
tcurrency. —61—For the case wherefollows an AR(1) process,eeds+irpds+2=ci+withdo=c+do].Therefore,tt—1ttdsrprp1=—j_{i +++ ...]c—— (i)•If=.9,thendst—30%for a .01 change inassuming our benchmark parametervalues. 'Friedman (1950)—62—Footnote82See, for example, Williamson (1985)3Ainong manypossiblecitations are Isard (1977), Krugman (1978),Frenkel (1981), &igel and Frankel (1984), and Hardouvelis (1987)Variation in the real exchange rate is documented in section 11.1.41n Section 111.1 below, we will see that this linear form is thecorrect one for an asset demand function under the assumption of mean—variance optimization by investors.5For example, Meese and Rogoff (1983a,b)the framework of equationsinsert a role for the (cumulated) currentdemand of residents of country i to beaggregating: (3') x =wjAi—B(rp)world wealth held by residents of countrycumulated claims on foreigners.(2) and (4), we can easilyaccount by defining the assetx =A1—B(rp),andwhere Wjisthe share ofi, which includes their7See, for example, Dooley and Shafer (1983)8Frankel and Froot (1986) show the equivalence. Note that in themonetary model, dL/d(rp) =9Forexample, Meese and Rogoff (l983a,b).'OThe probability of collapse mayriseas s gets farther fromequilibrium. If not, the ratio g/(1—e) would have to be greater than —63—one, unless the expected rate of depreciation (equal to the interestdifferential under uncovered interest parity) is to be declining overtime as the bubble proceeds.11For exa

72 mple, (buncil of Economic Advisers (1984
mple, (buncil of Economic Advisers (1984)'2&iison Miller and Williamson (1986) is an attempt to show howa target zone would actually work in a macroeconomic model and McKibbinand Sachs (1986) does the same for a variety of regimes. Also beyondthe scope of this paper is the question whether it is even a feasibleoption for government intervention in foreign exchange markets tostabilize the exchange rate without changing monetary and fiscal policy,as are related questions of international policy coordination. Howeverthe magnitude of our substitutability parameteris relevant to thequestion of the effectiveness of sterilized intervention, as elaboratedin Frankel (1986a)'3Poole (1967), Giddy and Dufey (1975), Mussa (1979) and eseand Rogoff (1983a,b), among others.'4We consider time—varying variances more in section 111.2.'5The following discussion draws on Frankel (l986b)16Roll (1979), Adler and Lehman (1983), Pippenger (1986)17cumby and Obstfeld (1984, p. 146) used a Q—statistic to testfor higher order serial correlation in monthly real exchange ratechanges and found none. However, they also found that expected —64—inflation differentials are unrelated to expected exchange rate changes,rejecting the random walk characterization of the real exchange rate.Huizinga (1986) is also able to reject the random walk.'8llakkio's unrestricted process allows for a unit root.Similarly the last line in Table 1 allows for the possibility of atrend. Fromamacroeconomic viewpoi

73 nt, rather than a statistical one,one mi
nt, rather than a statistical one,one might consider the important question to be whether most changes inthe real exchange rate die out over time, rather than whether there aresometimes also permanent changes in the real exchange rate, for example,when the price of oil goes up.19Frenkel (1976), Mussa (1976) and Bilson (1978).20ThiS is the case where B in equation (3) is infinite.21This proportionality is tested directly in Part IV below.22Dornbusch (1976), Frankel (1979)23Branson (1977), Kouri (1976), Girton and Henderson (1977).24Frankel (1984), Hooper and rton (1982)25Overshooting can occur also in the portfolio—balance model,where it can be viewed as the consequence of a finite rate of adjustmentin the stock of claims on foreigners, just as in the monetary modelovershooting can be viewed as the consequence of a finite rate ofadjustment in the general price level. —65—261In Dornbusch,-representsthe amount of overshooting. Forelaboration, see Frankel (1983, P. 42).27Frankel (1979) .Thesample went up to February 1978.28For example, Meese and Rogoff (1983a), Frankel (1984), andackus( 1984) .Sornanath(1986) reports updated fit statistics for theabove—mentioned models both insainpie (1975—1982) and out of sample(1983) .Schinasiand Swamy (1986) use a method that allows coefficientsto change over time. The two more recent papers claim relatively goodresults with a lagged endogenous variable.29The rate—of—return term (j_i*) in equation (8), seems forsome reason

74 to generate more econometrician anxiety
to generate more econometrician anxiety over endogeneityproblems when it is expressed as expected depreciation,inequation (9), even though it is the same variable when covered interestparity holds.30Meese and Rogoff (l983b) try a grid of parameter values. (kit—of—sample performance, while better than a random walk at horizonsexceeding 18 months, is never good.3100oley and Shafer (1983) and Hansen and Hodrick (1980) are twoof the tests that take the lagged prediction errors to be the availableinformation.32Studies regressing against the forward discount Include Tryon(1979), Levich (1980), Bilson (1981), Longworth (1981), Longworth,Boothe and clinton (1983), Fama (1984) and Huang (1984) .Qimbyand —66—Obstfeld (1984) and Obstfeld (1986) regressed against the Eurocurrencyinterest differential and again found that for most exchange rates thecoefficient was significantly less than 1.0 and even less than zero.(One would expect the same finding as in the forward discount tests,given that the interest differential is equal to the forward discount bycovered interest parity.) These findings are also consistent with thoseof Meese and Rogoff (1983a) that the random walk predicts, not onlybetter than other models, but better than the forward market as well.33Equlvalently, in a regression of the prediction error—fdagainst fdt, the coefficient under the null hypothesisshould be zero.34E.g. Frenkel (1978) and Frankel (1980)35See Meese (1987) for a more rigorous argument.

75 exception is the unlikely case where, ev
exception is the unlikely case where, even though investorsare risk—averse, exchange rates are like the outcome of a bet on a foot-ball game in that they are completely uncorrelated with other rates ofreturn (on all "outside" assets), so that exchange risk is completelydiversifiable.37This statement assumes a Cobb—Douglas price index with weighta on foreign goods. The constant term a in equation (11) can then beinterpreted as the (conditional) variance of the exchange rate times(a —1/2),which is a quite small number.38Frankel (1980), Krasker (1980), or Rogoff (1979) —67—39Recall that the tests of forward rate unbiasedness in thepreceding section are also joint tests of rational expectations and norisk premium.401f u is known to be correlated with the monetary fundamentalsbut an appropriate instrumental variable is available, then equation(13) can still be estimated by the appropriate techniques, the same asthe standard regression equation (8). Casella (1985), for example,allows for endogeneity of the money supply in her bubbles test of theGerman hyperinflation.41one could presumably relax the assumption of no risk premium byspecifying a portfolio—balance model, in place of (8), with bond sup-plies appearing on the righthand—side. Assumption (b) would then be theless—stringent condition that expected depreciationis an unbiasedpredictor in the sample (requiring rational expectations and no pesoproblem or regime changes). Woo (1984) has attempted this sort o

76 fbubbles test.42Fama (1984), Hodrick and
fbubbles test.42Fama (1984), Hodrick and Srivastava (1986) and Sweeney (1986)provide evidence of b Notethatcov[(f —se),(s+1 —= cov[(f—(s+1—s)——= _var(f—s)+cov{(f—se),(s+j —se)].The sum of the last twoterms is less than zero whenever b 435ee Frankel (1982), Rogoff (1984), Dooley and Isard (1982)among others. —68--441f investors are assumed to calculate their portfolios once ayear, the numbers are slightly different. The annual variance inreturns is observed to be about .01 •Thenan increase in the supply offoreign assets equal to 1 percent of the portfolio of 2.0 basis points.45Frankel (1982, P. 260) describes this assumption as one madefor convenience, to focus on variation in asset supplies and the riskpremium, with variation in the variances and covariances considered apriority for future research.46This model marginally outperforms (i) an autoregressive modelwith lag length estimated, (ii) the sample mean or median, and (iii) arandom walk model, on the basis of out—of—sample fit.47For simplicity we are leaving out the effect of a change in thereturn variance on the minimum variance portfolio A via the convexityterm.48Suppose=+whereis now a purely transitorydisturbance toThe effect ofon the exchange rate will beconsiderably smaller than that implied by (22). Besides the directeffect onfrom (22) we must recognize that the spot rate in thesubsequent period will return to its previous level, so that the riskpremium will rise by the full amount of the increa

77 se in s• Takingaccount of this second of
se in s• Takingaccount of this second offsetting term we get:dsrpdst11.t1t=+) [—i - -) --1tttttdctt —69—Note that the effect on expectations is much more important than theportfolio valuation effect, due to the high degree of substitutability.Again, if the initial risk premium is close to zero, the effect on achange in the return variance is close to zero. &it if the initial riskpremium is .03 and we consider a transitory change infrom .01 to.02, the change in the spot rate will be roughly —(.03/.01)( .01) =—.03,or a 3 percent appreciation of the less risky currency.49Note that the mean—variance model (3') used to derive (22) isless applicable whenvaries over time.50The expected inflation rates used here are three—year forecastsby DRI.51lndeed, the samples were dictated by the availability of thesurvey data.is noteworthy that the survey data regressions are valideven if the survey responses measure true expectations with error, solong as the measurement error is random. The standard error of theregression (and statistically significant coefficient estimates) implythat the magnitude of the measurement error is small, relative, forexample, to the magnitude of the expectatiorial errors introduced when expost exchange rate changes are used as the lefthand—side variable.5311weask what happens when the true speed of regression to PPPis held constant but investors have a higher expected speed of regres-sion 0, it turns out that the effect is still to reduce vari

78 ability.The effect on the conditional va
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