Exchange Rate Volatility And Deviations From Unbiasedness In A Cash-in .
Journalof InternationalEconomics36 (1994) 29952. North-HollandExchange rate volatility and deviationsfrom unbiasedness in a cash-in-advancemodelGeert Bekaert*GraduateSchool oJ’Business,StanfordReceived July 1991, revised versionUniuersify,receivedStanford,MarchCA 94305-501.5,USA1993This paper examines time-series properties of exchange rate changes, the forward premium andthe forward bias in the context of a variant of Svensson’s cash-in-advancemodel. The model issolved and simulatedusing realistic forcing processes whose law of motion is estimated fromU.S.-Japandata and then approximatedby a Markov chain. Althoughmethod of momentsestimationshows that the over-identifyingrestrictionsimplied by the model are not rejected, itfails dramaticallyin producinga suflicientlyvariablerisk premiumon forwardmarketspeculation.This result is robust to various perturbationsto the model’s parameters,forcingprocesses and preference structure.The model also fails to match exchange rate and forwardpremium volatility simultaneously.1. IntroductionThe floatingexchangerate period has generatednumerouspuzzlingregularities.One of the stylized facts is the high variabilityof exchange ratemovementswhich has raised concern about ‘excessive’ volatilityof foreignexchangerates. Anotherwell-knownpuzzle is the clear rejectionof the‘unbiasednesshypothesis’,which postulatesthat the forwardrate is anunbiased predictor of the future spot rate. Attempts to model the deviationfrom unbiasednessas a time-varyingrisk premium in the context of simplegeneral equilibriummodels have not met with much empirical success. Thevariabilityof the risk premium predicted by the models is typically orders ofmagnitudesmaller than what is observed in the data.Most of these studies are based upon the two-country,cash-in-advanceCorrespondence10: Geert Bekaert, GraduateSchool of Business, Stanford University, Stanford,CA 94305-5015.*I wish to thank Tim Bollerslev, Mark Watsonand three anonymousreferees for theircommentsand suggestionsand Craig Burnsidefor helpful discussionson the estimationprocedure.I am especially indebted to Robert Hodrick for his many suggestions and continuousencouragementwhile I wrote this paper. Partial financial support from the NBER’s Committeeon DissertationSupportAwards and a Sloan DissertationFellowshipis gratefullyacknowledged. All remaining errors are my own.0022-1996/94/ 07.00#; 1996ElsevierSSDI 0022-1996(93)01283-KScience B.V. All rights reserved
30G. Bekaert,Exchangerate uolatility(CIA) model of Lucas (1982).’ The goal of this paper is to judge the‘excessiveness’ of both exchange rate volatility and deviations from unbiasedness from the perspective of Svensson’s (1985a) two-countryCIA model. TheSvensson model differs from the Lucas model primarilyin the timing ofinformationarrival. In the Lucas model, all uncertaintyis resolved at themoment when consumerschoose the money holdings with which they willbuy consumptiongoods and, given positive nominal interest rates, the CIAconstraintsare always binding.In the Svenssonmodel non-bindingCIAconstraintswith positive interest rates are possible as consumersdecide ontheir cash holdings before the state is revealed. As a consequence,velocitydirectly enters the exchangerate and risk premiumexpressionsand ispotentiallyan importantfactor in their determinationwhich the Lucas modellacks. The particulartiming of the Svensson model also induces a wedgebetween the marginal utility of wealth and the marginal utility of consumption except when the CIA constraintsare slack. As a consequence,realinterest rates depend on monetarypolicy, which is not true in the Lucasmodel. Moreover,as agents have to trade currenciesbefore the state isrevealed, the Svensson model also induces a forward-lookingspot exchangerate. This implies that, even with binding CIA constraints,the exchange rateand risk premium differ across the models.This paper differs from related studies in other aspects as well. First, ratherthan imposing the real world exchange rate process as Macklem (1991) andBackus et al. (1992) do, I solve for exchange rate changes and its moments asa function of the exogenous processes (money and endowmentshocks) andmodel parameters.A focus on several exchange rate moments, jointly withthe issue of risk premium variability,reveals several trade-offs in the model’sability to match different aspects of the data.Second, the law of motion for the forcing processes is estimated from avector autoregression(VAR) on U.S.-Japanesedata on money and consumption. This law of motion is approximatedby a first-orderMarkov chainusing a discretizationtechnique developed by Tauchen and Hussey (1991). Inthe data, there is correlationbetween monetaryand real shocks, and Engel(1992), in the context of the Lucas model, stresses the importanceof these comovementsin the determinationof the risk premium.The Markov chainreplicates the actual correlationstructure of the data.Third, rather than simulatingthe model at pre-specifiedpreference parameters, I estimatethe preferenceparameterswith a variantof Hansen’s(1982) GeneralMethodof Moments(GMM).The estimationtechniqueminimizes a weighted sum of the deviationsbetween the sample momentsand numericallyobtainedmodel momentsof exchange rate changes, the‘See the studies of Canova and Marrinan(1991), Macklem (1991), Backus et al. (1992) andEngel (1992). Hakkioand Sibert (1991) performsimulationsof an overlapping-generationsmodel and also find the variability of the model’s risk premium to be very small.
G. Bekaert,Exchangerate volatilily31forward premium and the forward bias. * Mark (1985) and Hodrick (1989)performEuler equationtests of a representativeagent economy,usingforward market returns. The point estimates of the coefficient of relative riskaversionare large and the standarderrors encompassa wide range ofpossible values. The curvature parametersobtained here are more reasonableand more precisely estimated. This suggests that the exchange rate momentsemployed here may be more informativeabout preferences. Although I fail toreject the over-identifyingrestrictions,the implied momentsreveal somedimensionsalong which the model fails. In particular,the implied riskpremium is far from variable enough.The paper is organized as follows. Section 2 contains a brief discussion ofa growth version of the Svensson model and a descriptionof a solutiontechniquethat allows the use of realistic forcing processes and does notimpose bindingCIA constraints.Solutionsfor endogenousvariablesforseveral preference specificationsare presented.In section 3 the estimationmethodologyand results are discussed. Theempirical results are shown to be robust to a number of perturbationsto theforcing processes and parameters.One robust finding is the failure of theinternationalSvensson model to deliver variable velocity. This generalizes thesimulationresults of Hodrick et al. (1991), who examine the closed economySvensson (1985b) model. They find the CIA constraintto be always bindingfor a wide range of parametervalues when the forcing processesarecalibrated to correspondwith U.S. data on consumptionand money growth.The results are further examined in section 4. The failure to match riskpremiumvariabilityis . The latter are bounds on intertemporalmarginalrates of substitution (IMRS) that can be derived from asset market data. Backus et al. (1992)note that habit-formingutility increases the variabilityof the risk premium. Ishow that this is also true in the Svensson economy but that there is a tradeoff betweenmatchingexchangerate and risk premiumvolatility.Habitpersistence also leads to many states with a large precautionarydemand formoney and hence non-bindingCIA constraints.Finally, I address one of theobvious limitationsthat the Svensson model shares with most models in thisliterature: the implict assumptionof purchasingpower parity (PPP). As PPPis grossly violated in the data, it is importantto establish whether the lack ofa channelto generatePPP deviationsmight partiallyexplain the poorperformanceof this class of models with respect to the forward market risk‘Bansal (1989) estimates the parametersof a transactioncost model with standard GMM, buthe focuses on the terms of trade and investigatesthe forward market risk premium implied bythe model solely in terms of Hansen-Jagannathanbounds (see below). Canova and Marrinan(1991) used a Simulated Moments Estimatorto estimate the parametersof a Lucas-typemodelusing moments of the forward bias.
32G. Bekaert,premium.The concludingresearch possibilities.Exchangesectionsumsrate uolatilit)upanddiscussessomefurther2. Svensson’s two-country model2.1. First-order conditions und solution algorithmsIn Svensson’s model an infinitely lived representativeconsumer in each oftwo countries maximizes a time-separableutility function. The representativeagent enters each period with predeterminedholdings of home and foreignmoney and of the assets that are traded in the economy. He then learns thecurrent state and purchases home and foreign goods with home and foreigncurrency, respectively. At the end of the period, there is an asset market inwhich currencies and assets are traded and at which time monetary transfersare received. Marketsare perfectly competitiveand agents have rationalexpectations.Let xf (yf) denote the home (foreign) country’sstochastic,non-storableendowmentof goods and Ms and Nf the home and foreign money supply attime t. Since growth rates will follow a Markov chain, I use mc, to denotethe vector that contains the (gross) rates of the variables described above, i.e.mc, [gx,,gy,,w,, w:]‘, where gi, i,/i, I (i x, y) and w, Mf, ,/Ms, w: The state vectorfor this economyis then given by 0, N; ,lN;.CxS,yS, W, N;, 4 1’.Agents can purchase/sellclaims to all of the endowmentprocesses and tothe monetarytranfers. There is one perfectly divisible share of each asset.Asset prices are given by Q, [Qf,Q:,Qf”,Q;“]’ and asset holdingsaresummarizedby c(, [c , cc:, c?’t ,a;“]’ for the home consumer (@ for the foreignconsumer). The dividends are nominal and expressed in the currency of thehome country, i.e. D, [P:x;, S,P:‘ys,(w, - l)M;, S,(u:l)N;]’ with S, indicating the level of the spot exchange rate between the two currencies(i.e.home currency per unit of the foreign currency) and (Pf, Pj’) denotingtherespective goods prices in home and foreign currency units.The home consumer’sdecision problemcan now be describedby thefollowing equations,representingrespectivelyhis preferences, the CIA constraints and the budget constraint:M , &W’ , %; IQ, 5 4(Q, S,( N f - P:y:‘). D,) (M:’ -P:x:‘)(3)
G. Bekaert,Exchangeratevolatility33By adding the value of the cash goods consumption on both sides, theright-hand side of the modified eq. (3) defines nominal wealth. It consists ofthe proceeds of the sale of the asset holdings, of dividends and of moneyholdings. Note that all constraints are expressed in units of the homecurrency. I only investigate the standard perfectly pooled equilibrium asdefined in Lucas ( 1982).3Because of the law of one price, the exchange rate equates the value of aforeign currency unit in today’s asset market with the value of a homecurrency unit in today’s asset market. Since a currency unit acquired intoday’s asset market can only be used for consumption next period, the valueof a home (foreign) currency unit today equals the expected marginal utilityof the home (foreign) good per unit of the home (foreign) currency. Hence,the exchange rate is a forward-looking asset price in the Svensson model:IU2(XS 1,YS 1)Efs, [p: 1(4)7ul(xs l?Ys l)Ef[p: 11where the subscripts on U denote partial derivatives.Note that the expected marginal utility of the home good per unit of thehome currency also equals the Lagrange multiplier on the budget constraintII, as this represents the marginal utility of wealth in home currency terms.The nominal (home currency) intertemporal marginal rate of substitution(IMRS), denoted by n,,,, is the ratio of the discounted value of a unit of thehome currency tomorrow (/?A, ,) and the value of a home currency unittoday (A,). Therefore, it is given byU1(XS 29YS d1p: 2[19(5) l(x l Y l)[p: 11To solve the model without assuming the CIA constraints to be binding, Iadapt a technique from Giovannini and Labadie (1991). The crucial step inthe solution algorithm is to solve for the inverse of velocity for home andforeign money, denoted respectively by K(8,) and K*(O,). Manipulation ofthe first-order conditions yields3A formal statement of the optimization problem and the first-order conditions is given in anunpublished appendix, which is available upon request.
G. Bekaert, Exchange rate volatilitys-G 1 ul(xs l Ys l)(J-4(6)An analogousexpression can be derived for K*(O,). It can be shown that,under suitable conditions,the mapping defined in (6) is a contraction.4Bydefinition,P:xfK(O,) Msso that K(8,) [and analogouslyK*(O,)] and thestate vector determine prices, which in turn can be combined with the statevector to yield solutions for the exchange rate and the IMRS accordingtoeqs. (4) and (5).The forward rate, F,, and the forward premium, FP, (F, - S,)/S,, can bededuced from covered interest rate parity. Nominalinterest rates are foundby pricing a nominal bond. The price of such a bond, yielding one unit ofthe home (foreign) currencynext period equals the conditionalexpectedvalueof the home (foreign)nominalIMRS.The erestrates,it(S, ,/S,)& ,.follows:F,S(l i:) l i,,f(7)l i: {E,[n, iy]}-‘.The variablesof interestare now completelycharacterized.Currencydepreciation,DS, i (S, i -S,)/S,, can be found from eq. (4); the normalizedforward bias, FL?, i (S,, i - F,)/S, DS,. 1 - FP,, from eqs. (4) and (7). Thepredictablecomponentin the forward bias, E,[FB, i] E,[DS, i] - FP,,equals zero when the unbiasednesshypothesisholds and is usually termedthe ‘risk premium in the forward market’. I will denote it by RP,.’2.2. Endogenous variables for homothetic and addilog utilityTwo widely used preferencespecificationsare4The proof is similar to the proof in Giovanniniand Labadie (1991) and is omitted.sThe nominal risk premium as defined here might not be equal zero, even if consumersarerisk neutral, because of the stochastic inflation effect. Macklem (1991) finds that the stochasticinflation effect in the Lucas model is only relatively importantat low levels of risk aversion andin general remains small, Engel (1992) discusses the problemsassociatedwith appropriatelydefining the risk premium in the forward market in a multigood economy.
G. Bekaert,qx,,y,) E.whangerate uolatility35(xpd-“,‘-”’l-aI-au(x,,y,) x’- l-x1-y(8)l-y’The first utility function is homotheticand strictly concave in its argumentsfor 6 in (0, 1) and r strictlypositive.The intratemporalelasticityofsubstitutionbetween x and y is 1. The intertemporalelasticity of substitutionwith respect to the composite good, x v ’ m6), is (l/a). Addilog preferences areseparable in the two goods. When 1 and y go to 1 they reduce to logarithmicpreferences.Strict positivityof cx and y ensures strict concavity,and theintertemporalelasticity of substitutionis l/x for the home good and l/r forthe foreign good.’Expressionsfor the exchange rate and the nominalIMRS under thesepreferencesare summarizedin table 1. For comparison,I includetheexpressionsfor the Lucas model. With homotheticpreferences and bindingCIA constraints,the exchange rate in the Svensson model reduces to theexchange rate in the Lucas model. Moreover, the models reproduce the verysimplest version of a monetary exchange rate model.Both for addilog and homotheticutility, the inverse of velocity of home orforeign money constitutesan additionalsource of variation which the Lucasmodel lacks. Moreover, the particulartiming of the Svensson model impliesthat expected marginal utilities determine the exchange rate and the nominalIMRS. Since the risk premiumdepends both on expected exchange ratechanges and the home and foreign IMRS, it differs from the risk premium inthe Lucas model even when the CIA constraintsbind.Note that DS, 1 and rr, 1 have been written in terms of the sub-statevector mc,. As this sub-state vector contains stationarygrowth rates, DS, i,n, 1 and the other endogenousvariables derived from them will be stationarytoo. The law of motionfor mc, then completelydefines the stochasticstructure of the model and allows the computationof the (joint) populationmoments of the stationary endogenousseries.2.3.The law of motion ,for the ,forcing processesTo implement the solution procedure, I determine the law of motion of thestate variable mc, and then convert it into a discrete Markov chain. Tomeasurehome and foreign transactionsmonies on a quarterlybasis, Iaverage end-of-monthU.S. and Japanese stocks obtained from International‘Both homotheticutility and addilog utility when z is restricted to equal 7 are special cases ofthe general multi-goodutility function defined in Eichenbaumand Hansen (1990).
G. Bekaert, Exchange rate oolatility36TableEndogenousLucasvariablesIin the Lucas and .-DssE, ,C(gxp;2gYP:2)‘-aKT 21E,C(gxf;,gyp:,)‘-*K, ,l1 1 , w’ ’4 I W&t cw;‘:I PK: ,lE, ,[( xP; syls )‘-“K1 23)(g&,gyk1)‘-” ,I pE,*,I(gy 2gYP;2)‘-“K, 21E,C(gxp;,gyp:,)‘-“K, ,lWI 1Svensson-addilogOS; , I w’ E, ,IdJK: ,lE,kx:;PK Jgy:;:& I Wg&:K: 11 E, ,I.P: ; K J gx:;P’ns -, B Et ,Lv:ZK 21gx:;;E,l&:,P K, ,l w,,I’Notes: The L-superscriptdenotes the Lucas model, the S-superscripttheSvensson model and SB the Svensson model with binding CIA constraints.K,(Kt)representK(t?,)(K(fl:)). Their values in terms of the exogenousprocesses are completely determined by eq. (6) in the text.Financial Statistics (IFS) of the IMF (the sum of the money and quasimoney series). As an empirical proxy to the endowment series, I use data onconsumption of non-durables and services from the OECD QuarterlyNational Accounts. All the series are expressed per capita and are deseasonalized. More details can be found in a data appendix.’The joint distribution of the exogenous variables is assumed to beappropriately described by a finite-order vector autoregression (VAR) withGaussian errors. Table 2 reports the VAR estimation results. To conformwith the Svensson timing, the money growth series were lagged one period,‘The use of quarterly data stems from the fact that decent empirical measures for the forcingprocesses of the theoretical models are only available at the quarterly level. In Bekaert (1992), Iexplore the effects of temporal aggregationin a dynamic economy similar to the one analyzedhere.
G. Bekaert,ExchangeTableEstimated2VAR and its Markov counterpartfor the SvenssonSample period: 1975:2-1989:4.PanelOrderOrderOrderLikelihoodand 2(0.156)0.5140.210(0.053)0.20 IJPMRZ 0.1310.210(0.252)0.215JPCR’ 1.052(0.343)1.050USCR’ O.283-0.013PanelQ2(4)0.019(0.098)0.0173.1 I2(0.539)4.423(0.110)0.191(0.100)0. 60(0.1 .046(0.148)0.048- 0.028(0.280) 0.0280.144(0.154)0.144-0.207(0.137)- PCmeansandUSC-0.018-0.0170.003270.003 I3JPMBJJPCC: Unconditional-tests0.006(0. I �� O.427- 39.94- 39.09- 38.7414.23 (0.581)34.33 (0.005)USMUSMSchwarzcriterion-40.52-40.25- 40.48I23I vs. 22 vs. 3ConstantmodelA: Test of VAR lengthAkaikecriterionPane1 B: Estimates37rate volatilitycorrelationJPMmatrixof theMeansJPC-0.081 0.077-0.016-0.015-0.088- 0.086-0.011-0.01residuals1.016I.016I1.004I .0040.007000.006930.01 I0.01 I1.0221.0220.008500.008491.009I .009Nores: In Panel A, the likelihood ratio statistics incorporatethe degrees of freedom correctionrecommendedby Sims (1980). Marginal levels of significance are given in parentheses,In PanelB, M stands for the (M -2) money measure (per capita) in gross growth rates. C for growthrates of real consumptionper capita. US for United States and JP for Japan. In Panel B, thefirst line for each variablereportsthe OLS parameterestimateswith heteroskedasticityconsistentstandarderrors on the second line. The third line contains the induced parameterscomputedfrom the approximatingMarkov chain (see text). RZ is the adjusted R’ (coefficient ofdetermination).The last columnreportsthe tionsof the squared residuals. Under the null of conditionalhomoskedasticitythestatistics should have a X’(4)-distribution.The last column reports the Bera-Jarque(1982) testfor normalitywhich has a X’(2)-distributionunder the null. P-valuesare reportedbetweenparentheses.In Panel C, the first line refers to the original VAR estimation.the second line tothe induced VAR from the Markov chain. The diagonal elements are standard deviations of thecorrespondingresiduals. The last column reports the unconditionalmeans implied by the VARand the approximatingMarkov chain.
38G. Bekaert,Exchangerate volatilityso that w, Mf, 1/Mf and u N;, i/N; enter jointly with gx, x,/x,1 andgy, y,/y,i. I assess the order of the VAR with likelihoodratio tests andSchwartz and Akaike criteria, which are reported in Panel A of table 2. TheAkaike and Schwarz criteria both select the first-order VAR and a likelihoodratio test also does not reject the restrictionsof the first-orderVAR vs. asecond-orderVAR. I therefore choose to work with a first-order VAR. Theparameterestimates are given in Panel B. While there are few significantcross-countrylinkages, the predictabilityof the U.S. series and Japanesemoney is strong. Japanese consumption,on the other hand, is not predictableby any of the VAR variables. Tests for conditionalhomoskedasticityandnormality support the assumptionof homoskedasticnormal errors.The next step is to approximatethis continuousstate space economy withan auxiliary,discrete economy.Tauchenand Hussey (1991) describeaprocedure for approximatingintegral operators which can be used to converta continuousdistributioninto a discrete Markov chain. The multivariatenormal distribution,implicit in the above VAR estimation,can be rewrittenas the product of univariatenormal densities with an appropriatechange ofvariables. The univariatedensities are then approximatedwith a Gaussianquadraturerule. The state space therefore expands exponentially.I choose 3states of nature for each variable which results in a total of 34 81 states.Although the approximationgets better with liner state spaces, it is alreadyquite accurate for very coarse state spaces. One way to judge the accuracy ofthe approximationis to comparethe originalVAR estimateswith theautoregressiveparametersand the residualcovariancesinducedby theMarkovchain. As Panel B of table 2 indicates,all inducedparameterestimates are within at most half a standard error of the original estimates.Panel C likewise shows that the Markov chain replicates the correlationstructure of the actual shocks in the economy.The discretizationprocedureyields a vector of 81 state values for theforcing variables,a vector of stationaryprobabilitiesand a transitionprobabilitymatrix, which are sufficient to solve the models, to evaluate theconditionalexpectationsin the expressions for the forward premium and riskpremium, and to compute moments for the endogenousvariables of interest.3. Estimationand empirical results3.1. EconometricmethodologyDenote the utility parametersby di and the parameters governing the lawof motion of the forcing processes, i.e. the VAR parameters,by I . Toestimate the utility parameters,I minimize a quadratic form in the deviationsof the sample momentsof interestfrom the correspondingnumericallyobtainedmodel moments. The model moments depend both on the struc-
G. Bekaert, Exchange rate volatility39tural parametersand on the VAR parameters.Joint estimationof (4rr *)with Hansen’s (1982) GMM requires solving the model by discretizingthestate space for each evaluationof the objective functionand is computationally too burdensome.However, a consistentestimate 2,T of &2 can beobtainedby OrdinaryLeast Squares, as was done in the previous section.The sampling error in that estimationmust be taken into account when 4r isestimatedholding the VAR parametersfixed at ,,r.To see how this isdone, let g,, be the difference between the sample moments from the dataand the model moments, let g,, be the sample means of the orthogonalityconditionscorrespondingto the VAR, and let g, [g’rT,g;r]‘.Then theestimator for b1 satisfies l,T argming,,( ,, ,,,)‘W,,,,g,.( ,, ,,,),(9)where the minimizationis over r E P, a compact set, and where W,, 11 isa positive definite weighting matrix. Burnside (1991) shows that the optimalmatrix ofchoice for W,, 11 depends on S, the asymptotic variance-covariancethe orthogonalityconditionsg, evaluated at the true parametervalues andon the derivativesof g, with respect to both 4r and 42. A consistentestimate of S can be constructedas in Newey and West (1987) and consistentestimates of the derivativesof the orthogonalityconditionsare found bytaking numericaland/or analyticalderivativesof the sample orthogonalityconditionsat the parameter estimates. These are the channels through whichof 4r. Using this optimalthe sampling error in ,,, influences the estimationweightingmatrix, the standardHansen (1982) test of the over-identifyingrestrictionsremains valid. A more formal and detailed descriptionof thissequential GMM technique can be found in Burnside (1990, 1991).The parameterswere estimated by iterating on the weighting matrix untilconvergence.Convergenceis defined as maxijI W,(i, j) - W,(i, j)l 10m6 withW,(i,j),W,,(i,j) the elements of the new, respectivelyold, weighting matrix.Four sets of different starting values led to the same parameter estimates.3.2. Estimation resultsTable 3 containsresults for two estimationexercises on the addilogpreference specification.The model moments used in the first experimentarethe means, variances and first autocovariancesof exchange rate changes andof the forward premium together with the covariancebetween the forwardpremium at time t and exchange rate changes at time t 1. This providesseven orthogonalityconditionsto estimate the two preference parameters(c1,y). The second experimentuses the means, variancesand first autocovariancesof exchange rate changes and of the forward bias. The samplemoments are computed from quarterly /yen rates. The curvature parameters
G. Bekaert, Exchange rate uolarility40Table 3EstimationSamplez . .AL IAL II4.398(1.329)7.985(2.051)results for addilog utility.period: .928(0.112)8.1 I2(0.088)6.287(0.012)7.303(0.007)Notes: The tirst set of momentsused in estimationare themeans,variancesandfirstautocovariancesof currencydepreciationand the forwardpremiumand the covariancebetween currency depreciationand the lagged forward premium(AL I row). The second set of moments include mean, varianceand first autocovariancefrom boththe forwardbias andexchange rate changes (AL II row). Parameterestimates of theutility parametersof the addilog utility specification(a, ) areobtained according to the estimation procedure described in text.In computingan estimate for S, the variancecovariancematrixof the orthogonalitycondtions at the optimum, four NeweyyWest(1987) lags were used. The J, statistic has live degrees of freedomfor the first set of moments and four for the last set of moments,with asymptoticp-values given in parentheses.The &I)teststatistic tests the equality of s( and 1’.of the utility function are quite precisely estimated, and are in the ‘admissiblerange’ proposedby Mehra and Prescott (1985).8 A Wald test rejects thehypothesis that the estimated values for a and y are equal at the 5 percentlevel. The over-identifyingrestrictionsare not rejected at the 5 percent levelfor either estimationexercise, but they would be at the 10 percent level forthe second set of moments. Given the small sample size, the failure to rejectthe over-identifyingrestrictionsmay reflect the low power of the test, ratherthan substantialevidence in favor of the models.The implied moments at the parameterestimates are comparedwith thesample moments in table 4 (columns 1 through 3). Two features of the datastand out. First, exchange rate changes and the forward bias are far morevariable and less persistent than the forward premium. To facilitate interpretation, note that the forward premium can be written as the difference of thetwo predictablecomponentsin exchange rate changes and the forward bias,i.e. FP, E, [OS, r] - E, [FB, r]. Hence, this feature of the data is indicativeof high variability of the forecast error associated with exchange rate changesand highly autocorrelatedpredictablecomponentsin exchange rate changesand/or the forward bias [see also Macklem (1991)]. The model’s first-ordercorrelationcoefficient for all three series is always within two standard errorsof the data moment, but the model tends to under-predictexchange rate and‘A value of 10 is the upperaversion.limit of their ‘admissiblerange’ for the coefticientof relativerisk
821(3.989)26.033(2.639)0.164(0.122)mom.AL IAL II1.0011.0000.0370.567*0.553*0.0115.8414.127**- 20.0020.102- .83521.283*0.000*0.000*- 18.72619.6954.75822.759*0.105**0.466SENSand .825- eriod:0.567*8.261- 1.0160.000*18.7370.542**0.146**20.48 1-0.474*IISampleSENSanalysis.1.000
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ANATOMI & HISTOLOGI JARINGAN PERIODONTAL Oleh: drg Ali Taqwim . terbentuk dari tulang haversi (haversian bone) dan lamela tulang kompak (compacted bone lamellae). drg Ali Taqwim/ KG UNSOED 29 Lamina dura Alveolar bone proper GAMBARAN HISTOLOGIS GAMBARAN RADIOGRAFIS It appears more radiodense than surrounding supporting bone in X-rays called lamina dura . drg Ali Taqwim/ KG UNSOED 30 1. Cells .
