Exchange Rates and Fundamentals: A Generalization

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 948 September 2008 Exchange Rates and Fundamentals: A Generalization James M. Nason and John H. Rogers NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from Social Science Research Network electronic library at http://www.ssrn.com/.

Exchange Rates and Fundamentals: A Generalization James M. Nason and John H. Rogers Abstract: Exchange rates have raised the ire of economists for more than 20 years. The problem is that few, if any, exchange rate models are known to systematically beat a naive random walk in out of sample forecasts. Engel and West (2005) show that these failures can be explained by the standard-present value model (PVM) because it predicts random walk exchange rate dynamics if the discount factor approaches one and fundamentals have a unit root. This paper generalizes the Engel and West (EW) hypothesis to the larger class of open economy dynamic stochastic general equilibrium (DSGE) models. The EW hypothesis is shown to hold for a canonical open economy DSGE model. We show that all the predictions of the standard-PVM carry over to the DSGE-PVM. The DSGE-PVM also yields an unobserved components (UC) models that we estimate using Bayesian methods and a quarterly Canadian--U.S. sample. Bayesian model evaluation reveals that the data support a UC model that calibrates the discount factor to one implying the Canadian dollar--U.S. dollar exchange rate is a random walk dominated by permanent cross-country monetary and productivity shocks. Keywords: Exchange Rates; present-value model fundamentals; random walk; DSGE model; unobserved components model; Bayesian model comparison. JEL Codes: E31, E37, and F41 * Author notes: Nason is Research Economist and Policy Adviser in the Research Department of the Federal Reserve Bank of Atlanta. He can be reached at Jim.Nason@atl.frb.org. Rogers is Deputy Associate Director in the Division of International Finance of the Federal Reserve Board, and can be reached at John.Rogers@frb.gov. We wish to thank Toni Braun, Fabio Canova, Menzie Chinn, Frank Diebold, John Geweke, Fumio Hayashi, Sharon Kozicki, Adrian Pagan, Juan F. Rubio-Ramírez, Tom Sargent, Pedro Silos, Ellis Tallman, Noah Williams, Farshid Vahid, Kenji Wada, Ken West, Tao Zha, the Federal Reserve Bank of Atlanta macro lunch study group, and seminar participants at the 2006 NBER Summer Institute Working Group on Forecasting and Empirical Methods in Macroeconomics and Finance, the 2007 Norges Bank Workshop on, "Prediction and Monetary Policy in the Presence of Model Uncertainty", the 2007 Bank of Canada-European Central Bank Exchange Rate Modeling Workshop, the 2007 Reserve Bank of Australia Research Workshop, "Monetary Policy in Open Economies", Ohio State, Houston, Tokyo, and Washington University for useful comments. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

Exchange Rates and Fundamentals: A Generalization James M. Nason John H. Rogers† Research Department International Finance Division Federal Reserve Bank of Atlanta Board of Governors of the 1000 Peachtree St., N.E. Federal Reserve System Atlanta, GA 30309 Washington, D.C. 20551 email: jim.nason@atl.frb.org email: john.h.rogers@frb.gov May21,2008 Abstract Exchangerateshaveraisedtheireofeconomistsformorethan20years. Theproblemisthatfew,ifany, exchangeratemodelsareknowntosystematicallybeatanaiverandomwalkinoutofsampleforecasts. Engel and West (2005) show that these failures can be explained by the standard-present value model (PVM) because it predicts random walk exchange rate dynamics if the discount factor approaches one and fundamentals have a unit root. This paper generalizes the Engel and West (EW) hypothesis to the largerclassofopeneconomydynamicstochasticgeneralequilibrium(DSGE)models. TheEWhypothesis is shown to hold for a canonical open economy DSGE model. We show that all the predictions of the standard-PVM carry over to the DSGE-PVM. The DSGE-PVM also yields an unobserved components (UC) modelsthatweestimateusingBayesianmethodsandaquarterlyCanadian–U.S.sample. Bayesianmodel evaluationrevealsthatthedatasupportaUCmodelthatcalibratesthediscountfactortooneimplying theCanadiandollar–U.S.dollarexchangerateisarandomwalkdominatedbypermanentcross-country monetaryandproductivityshocks. JELClassificationNumber: E31,E37,andF41. Key Words: Exchange rates; present-value model and fundamentals; random walk; DSGE model; unobservedcomponentsmodel; Bayesianmodelcomparison. †We wish to thank Toni Braun, Fabio Canova, Menzie Chinn, Frank Diebold, John Geweke, Fumio Hayashi, Sharon Kozicki, AdrianPagan,JuanF.Rubio-Ramírez,TomSargent,PedroSilos,EllisTallman,NoahWilliams,FarshidVahid,KenjiWada,Ken West, Tao Zha, the Federal Reserve Bank of Atlanta macro lunch study group, and seminar participants at the 2006 NBER Summer Institute Working Group on Forecasting and Empirical Methods in Macroeconomics and Finance, the 2007 Norges BankWorkshopon,“PredictionandMonetaryPolicyinthePresenceofModelUncertainty”,the2007BankofCanada-European Central Bank Exchange Rate Modeling Workshop, the 2007 Reserve Bank of Australia Research Workshop, “Monetary Policy inOpenEconomies”, OhioState, Houston, Tokyo, andWashingtonUniversityforusefulcomments. Theviewsinthispaper representthoseoftheauthorsandarenotthoseofeithertheFederalReserveBankofAtlanta,theBoardofGovernorsofthe FederalReserveSystem,oranyofitsstaff. Errorsinthispaperaretheresponsibilityoftheauthors.

1. Introduction The search for satisfactory exchange rate models continues to be elusive. This paper studies a workhorse theory of currency market equilibrium determination, the present-value model (PVM) of exchangerates,inthespiritofEngelandWest(2005). StartingwiththePVMandusinguncontroversial assumptions about fundamentals and the discount factor, Engel and West (EW) hypothesize that the PVM generates an approximate random walk in exchange rates if the PVM discount factor approaches one and fundamentals are I(1). An important implication of the EW hypothesis is that fundamentals havenopowertoforecastfutureexchangerates,evenwiththePVMdictatingequilibriuminthecurrency market. EWsupporttheirhypothesiswithakeytheoremandempiricalandsimulationevidence. This paper complements Engel and West (2005) by generalizing their main hypothesis in two ways. First,theEWhypothesisisgeneralizedusingacanonicaltwo-countrymonetarydynamicstochastic general equilibrium (DSGE) model. Its linearized uncovered interest parity (UIP) and money demand equations yield the DSGE-PVM that coincides with the standard PVM of the exchange rate. Second, we show the standard- and DSGE-PVMs make equivalent predictions for exchange rates. The predictions are summarized in five propositions: (1) the exchange rate and fundamental cointegrate [Campbell and Shiller (1987)], (2) the PVM yields an error correction model (ECM) for currency returns in which the lagged cointegrating relation is the only regressor, (3) the PVM predicts a limiting economy (i.e., the PVM discount factor approaches one from below) in which the exchange rate is a martingale, (4) given fundamental growth depends only on the lagged cointegrating relation, the exchange rate and fundamentalhave acommontrend-common cycledecomposition[Vahid andEngle(1993)], and(5) the EW hypothesis is also satisfied when the exchange rate and fundamental share a common feature and the PVM discount factor approaches one. A corollary to (5) is that the exchange rate is unpredictable whenthePVMdiscountfactorgoestoone. We report evidence from vector autoregression (VARs) about the propositions using quarterly floating rate Canadian–, Japanese–, and U.K.–U.S. samples. The VAR evidence rejects cointegration and revealssubstantialserialcorrelationfortheexchangerateandthefundamental. Thereisalsoevidence 1

thatacommonfeatureexistsbetweentheCanadiandollar–,Yen–,andPound–U.S.dollarexchangerates andtherelevantfundamentals. Nonetheless,theVARapproachisunabletoaddresstheEWhypothesis questionofwhetherthePVMdiscountfactorapproachesone.1 TheDSGE-PVMpossessesadeepstructuretiedtotheprimitivesoftheunderlyingopeneconomy unlikethestandard-PVM.RatherthanrelyontheentiresetofDSGEoptimalityandequilibriumcondition, we give empirical content to the DSGE-PVM by placing restrictions on its fundamentals (cross-country money and consumption). We restrict these fundamentals with permanent-transitory decompositions. ThisdecompositionallowsustocasttheDSGE-PVMasatri-variateunobservedcomponents(UC)model in the exchange rate and observed fundamentals. The UC model also incorporates DSGE-PVM crossequation restrictions conditional on whether the discount factor is calibrated or estimated. Three UC models calibrate the discount factor to one, which disconnects the exchange rate from the transitory component(s) of fundamentals. Transitory fundamentals restrict the exchange rate in three other UC modelsinwhichtheDSGE-PVMdiscountfactorisestimated. We estimate six UC models on a Canadian-U.S. sample running from 1976Q1 to 2004Q4. The UC models yield state space systems for the DSGE-PVM, which allows us to recruit the Kalman filter to evaluate likelihoods. We compute likelihoods of the UC models using the Metropolis-Hastings (MH) simulator described by Rabanal and Rubio-Ramírez (2005) to draw Markov chain Monte Carlo (MCMC) replications from posteriors. We conduct model comparisons using marginal posterior likelihoods of thesixUCmodelstofindwhichisfavoredbythedata. WefindthatthedatafavorstheUCmodelthat calibratesthediscountfactortooneandinwhichcyclicalfluctuationsaredrivenonlywiththetransitory shocktocross-countryconsumption. FavorednextistheUCmodelwiththesametransitoryshockand inwhichtheestimatedposteriormeanoftheDSGE-PVMdiscountfactoris0.9962. Theposteriorofthis UC model reveals that permanent shocks to fundamentals dominate exchange rate fluctuations. Thus, 1Actualdatamostoftenrejectsthestandard-PVM.TypicalaretestsMeese(1986)reportedthatemployedthefirsttenyears ofthefloatingrateregime. Hefindsthatexchangeratesareinfectedwithpersistentdeviationsfromfundamentals,whichreject thestandard-PVManditscross-equationrestrictions. However,Meeseisunabletouncoverthesourceoftherejections. Instead ofacondemnationofthestandard-PVM,weviewresultssuchasMeese’sasachallengetoupdateanddeepenitsanalysis. 2

thedatapreferUCmodelsthatareconsistentwiththeEWhypothesis. Moreover, wefindthatthedata failtosupportUCmodelsthattietheexchangeratetothetransitorymonetaryshock. Rogoff(2007)also notes that exchange rates appear disconnected from ‘mean reverting monetary fundamentals’. These results stand in contrast to those of open economy DSGE models which assign key roles to nominal rigidities,UIPshockpersistence,andmonetarydisturbances.2 The next section constructs the standard- and DSGE-PVMs of the exchange rate. Section 3 presents five propositions that generalize the EW hypothesis. Our Bayesian econometric strategy is discussedinsection4. Section5reportsestimatesofsixUCmodels. Weconcludeinsection6. 2. Two Present-Value Models of Exchange Rates ThissectionfleshesoutthestandardPVM,inwhichtheequilibriumexchangerateisdetermined bymeldingaliquidity-moneydemandfunction,UIPcondition,purchasingpowerparity(PPP),andflexible prices. Thisisaworkhorseexchangeratemodelusedby,amongothers,Dornbusch(1976),Bilson(1978), Frankel(1979),Meese(1986),Mark(1995),andEngelandWest(2005). ThissectionalsodevelopsaPVM oftheexchangeratederivedfromacanonicaloptimizingtwo-countrymonetaryDSGEmodel. Weshow thattheEWhypothesisgeneralizestothiswiderclassofmodels. 2a. The Standard Present-Value Model of Exchange Rates Thestandard-PVMoftheexchangeratestartswiththeliquidity-moneydemandfunction (1) m − p = ψy − φr , 0 < ψ, φ, h,t h,t h,t h,t wherem p , y ,andr denotethehomecountry’snaturallogarithmofmoneystock,pricelevel, h,t h,t h,t h,t output, and the level of the nominal interest rate. The parameter ψ measures the income elasticity of money demand. Since the nominal interest rate is in its level, φ is the interest rate semi-elasticity of moneydemand. Definecross-countrydifferentialsm =m −m , p =p −p , y =y −y , t h,t f,t t h,t f,t t h,t f,t 2TheopeneconomyVARliteratureoffersmixedevidenceontheimportanceofvariousshocksfortheexchangerate. Early papersincludingEichenbaumandEvans(1995),Rogers(2000)andKimandRoubini(2000)foundsomesignificanceforidentified monetaryshocks. Recentcontributions,however,suggestthatmonetarypolicyshockshaveonlyaminorimpactonexchange ratefluctuations,consistentwithRogoff’sview,forexample,seeFaustandRogers,(2003)andSchollandUhlig(2005). 3

and r =r − r ,wheref denotestheforeigncountry. AssumingPPPholds,e =p ,wheree isthe t h,t f,t t t t logofthe(nominal)exchangerateinwhichtheU.Sdollaristhehomecountry’scurrency. UnderUIP,thelawofmotionoftheexchangerateisapproximately (2) E t e t+1 − e t = r t . Substituteforr inthelawofmotionoftheexchangerate(2)withthemoneydemandfunction(1)and t (cid:0) (cid:1) imposePPPtoproducetheEulerequatione t −ωE t e t+1 =(1−ω) m t −ψy t ,wherethestandard-PVM φ discountfactorisω≡ andm −ψy isthestandard-PVMfundamental,whichnetscross-country 1+φ t t money with its income demand. Iterate on the Euler equation through date T and recognize that the transversalityconditionlim T→∞ωT+1E t e t+T =0toobtainthestandardPVMrelation ∞ (cid:88) (cid:110) (cid:111) (3) e t = (1 − ω) ωjE t m t+j − ψy t+j . j=0 ThestandardPVM(3)setsthelogexchangerateequaltotheannuityvalueofthefundamentalm −ψy t t atthestandard-PVMdiscountfactorω.3 2b. The DSGE Model The optimizing monetary DSGE model consists of the preferences of domestic and foreign economies and their resource constraints. For the home (h) and foreign (f) countries, the former objectstaketheform  (1−ϕ) (cid:18) (cid:19)(1−ν) C ν M i,t  (cid:32) M (cid:33) i,t P i,t (4) U C , i,t = , 0 < ν <1, 0 < ϕ, i,t P 1−ϕ i,t whereC andM representtheithcountry’sconsumptionandtheithcountry’sholdingsofitsmoney i,t i,t stock. Theresourceconstraintofthehomecountryis (5) B h h ,t + s t B h f ,t + P h,t C h,t + M h,t = (1+r h,t−1 )B h h ,t−1 + s t (1+r f,t−1 )B h f ,t−1 + M h,t−1 + P h,t Y h,t , where B i i ,t , B i (cid:96) ,t , r i,t−1 , r (cid:96),t−1 , Y i,t , and s t denote the ith country’s nominal holding of its own bonds at the end of date t, the ith country’s nominal holding of the (cid:96)th country’s bonds at the end of date t, 3Thepresent-valuerelation(3)yieldstheweakpredictionthateGranger-causesz. EngelandWest(2005)andRossi(2007) reportthatthispredictionisoftennotrejectedinG–7data. 4

the return on the ith country’s bond, the return on the (cid:96)th country’s bond, the output level of the ith country, and the level of the exchange rate. The two-country DSGE model is closed with Bh + B f h,t h,t + Bh + B f = 0. This condition forces the world stock of nominal debt to be in zero net supply, f,t f,t period-by-period,alongtheequilibriumpath. In section 2, analysis of the standard-PVM relies on I(1) fundamentals. Likewise, we assume thattheprocessesforlabor-augmentingtotalfactorproductivity(TFP),A ,andM satisfy i,t i,t Assumption1: ln[A ]andln[M ]∼I(1), i = h, f. i,t i,t Assumption2: Cross-countryTFPandmoneystockdifferentialsareI(1)anddonotcointegrate. Assumptions1and2imposestochastictrendsonthetwo-countryDSGEmodel. 2c. Optimizing UIP and Money Demand Thehomecountrymaximizesitsexpecteddiscountedlifetimeutilityoveruncertainstreamsof consumptionandrealbalances, E t  (cid:88) ∞ (1+ρ)−jU (cid:32) C h,t+j , M h,t+j (cid:33)  , 0 < ρ,  j=0 P h,t+j  subject to (5). The first-order necessary conditions of economy i yield optimality conditions that describeUIPandmoneydemand. Theutility-basedUIPconditionofthehomecountryis (cid:40) (cid:41) (cid:40) (cid:41) U U (1 + r ) (6) E C,h,t+1 (1 + r ) = E C,h,t+1 f,t , t h,t t P h,t+1 P f,t+1 s t where U is the marginal utility of consumption of the home country at date t. Given the utility C,h,t specification(4),theexactmoneydemandfunctionofcountryiis M (cid:18)1 − ν (cid:19)1 + r (7) i,t = C i,t , i = h, f. i,t P ν r i,t i,t Theconsumptionelasticityofmoneydemandisunity,whiletheinterestelasticityofmoneydemandis anonlinearfunctionofthesteadystatebondreturn. TheUIPcondition(6)andmoneydemandequation(7)canbestochasticallydetrendedandthen linearizedtoproduceanequilibriumDSGE-lawofmotionfortheexchangerate. Beginbycombiningthe 5

utilityfunction(4)andtheUIPcondition(6)toobtain (cid:40) (cid:41) (cid:40) (cid:41) U U (1 + r ) E h,t+1 (1 + r ) = E h,t+1 f,t , t h,t t P h,t+1 C h,t+1 P f,t+1 C h,t+1 s t where U is the utility level of country i at date t. Prior to stochastically detrending the previous i,t expression, define U(cid:99)i,t = U i,t /A i,t , P(cid:98)i,t = P i,t A i,t /M i,t , C(cid:98)i,t = C i,t /A i,t , γ A,i,t = A i,t /A i,t−1 , γ M,i,t = M i,t /M i,t−1 , s (cid:98)t =s t A t /M t , A t =A h,t /A f,t ,andM t =M h,t /M f,t . NotethatC(cid:98)i,t isthetransitorycomponent of consumption of the ith economy, γ (γ ) is the TFP (money) growth rate of country i, and the A,i,t M,i,t cross-countryTFP(moneystock)differentialA (M )areI(1). Applyingthedefinitions,thestochastically t t detrendedUIPconditionbecomes     E  U(cid:99)h,t+1 γ A 1− ,h ϕ ,t+1  (1 + r ) = E  U(cid:99)h,t+1 γ A,f,t+1 (1 + r f,t ) . t γ M,h,t+1 P(cid:98)h,t+1 C(cid:98)h,t+1  h,t t γ A ϕ ,h,t+1 γ M,f,t+1 P(cid:98)f,t+1 C(cid:98)h,t+1  s (cid:98)t AloglinearapproximationofthestochasticallydetrendedUIPconditionyields r∗ (cid:8) (cid:9) (8) E t e (cid:101)t+1 − e (cid:101)t = 1+r∗ r (cid:101)t + E t γ (cid:101)A,t+1 − γ (cid:101)M,t+1 , where,forexample,e =ln[s ]−ln[s∗]andr∗(=r∗ =r∗)denotesthesteadystateworldrealrate. (cid:101)t (cid:98)t h f 2d. A DSGE-PVM of the Exchange Rate We use the linear approximate law of motion of the exchange rate (8), and a stochastically detrended version of the money demand equation (7) to produce the DSGE-PVM. When linearized, the 1 unit consumption elasticity-money demand equation (7) produces −p = c − r . Impose PPP (cid:101)t (cid:101)t 1+r∗(cid:101)t onthe stochasticallydetrendedversion ofthemoney demandequationand combineit withthelaw of motion(8)ofthetransitorycomponentoftheexchangeratetofind (cid:20) 1 − 1+ 1 r∗ E t L−1 (cid:21) e (cid:101)t = 1+ 1 r∗ E t (cid:8) γ (cid:101)M,t+1 − γ (cid:101)A,t+1 (cid:9) − 1+ r∗ r∗ c (cid:101)t . Solving this stochastic difference equation forward gives a present value relation for the transitory componentoftheexchangerate ∞ ∞ (cid:88) (cid:110) (cid:111) (cid:88) (9) e (cid:101)t = κjE t γ (cid:101)M,t+j − γ (cid:101)A,t+j − (1−κ) κjE t c (cid:101)t+j , j=1 j=0 6

1 wheretherelevanttranversalityconditionsareinvokedandtheDSGE-PVMdiscountfactorκ ≡ . 1+r∗ NotethattheDSGE-PVMandpermanentincomehypothesisdiscountfactorsareequivalent. The DSGE-PVM relation (9) is the equilibrium law of motion of the cyclical component of the exchange rate. Transitory movements in the exchange rate are equated with the future discounted expected path of cross-country money and TFP growth and the (negative of the) annuity-value of the transitory component of cross-country consumption. The DSGE model identifies the exchange rate’s unobservedtime-varyingriskpremiumwiththeexpectedpathofcross-countryTFPgrowthandtransitoryconsumption,whichsuggestadditionalsourcesofexchangeratefluctuations. The DSGE model produces a present value relation that resembles the standard-PVM (3). The DSGE-PVMfollowsfromunwindingthestochasticdetrendingofthepresentvalue(9) ∞ (cid:88) (cid:110) (cid:111) (10) e t = (1−κ) κjE t m t+j − c t+j . j=0 Thus,thestandard-PVM(3)andDSGE-PVM(10)areidenticaluptodifferencesintheirdiscountfactors and real fundamentals. The standard-PVM discount factor ω is tied to the interest rate semi-elasticity of money demand, φ, while the DSGE-PVM sets κ to the inverse of the gross steady state real world interest rate, 1+r∗. For the standard-PVM (DSGE-PVM), the real fundamental is cross-country output y (consumptionc ). Table1summarizesthenotableelementsofthestandard-andDSGE-PVMs. t t 3. Generalizing the Engel–West Hypothesis ThissectionpresentsfivepropositionsthatgeneralizetheEWhypothesis. Thisallowsabroader empiricalanalysisoftheEWhypothesis,anddoessousingstandardtimeseriestools. Thepropositions applytothestandard-PVMand theDSGE-PVMbecausetheirpresentvaluerelationscoincide. Thus,we generalizetheEWhypothesistothelargeclassoftwo-countrymonetaryDSGEmodels. Wecollapsethedifferencesinthediscountfactorandrealfundamentalofthestandard-PVM(3) and DSGE-PVM (10) to stress their mutual predictions in this section. These differences are put aside by defining a PVM discount factor B equal to either ω or κ, while the fundamental z is equivalent to t eitherm −ψy orm −c . Withtheseassumptions,thefocusisonthePVM t t t t 7

∞ (cid:88) (11) e t = (1−B) BjE t z t+j , j=0 whichsubsumesthestandard-andDSGE-PVMs. ThePVM(11)providesseveralpredictionsgiven Assumption3: z ∼I(1). t Assumption4: (1−L)z t hasaWoldrepresentation,(1−L)z t = ∆ z∗ + ζ(L)υ t ,whereLz t =z t−1 .4 EngelandWest(2005)employAssumption3,buttheydonotrequirerestrictionsasstrongasAssumption4. However,Assumption4isstandardforlinearrationalexpectationmodels;seeHansen,Roberds, and Sargent (1991). Assumption 4 is also an implication of a linear approximate solution of the open economyDSGEmodel,whileAssumption3isconsistentwithAssumptions1and2. 3a. Cointegration Restrictions The first prediction is that e and z share a common trend. This follows from subtracting the latter t t from both sides of the equality of the present-value relation (11) and combining terms to produce the exchangerate-fundamentalcointegratingrelation ∞ (cid:88) (12) e t − z t = BjE t∆ z t+j , ∆ ≡ 1−L. j=1 Equation (12) reflects the forces – expected discounted value of fundamental growth – that push the exchangeratetowardlong-runPPP.Theexplanationis Proposition 1: If z satisfies Assumptions 3 and 4, X = β(cid:48)q forms a cointegrating relation with t t t cointegratingvectorβ(cid:48) =[1 −1],whereq ≡[e z ](cid:48). t t t ThepropositionisavariationofresultsfoundinCampbellandShiller(1987). WeinterpretthecointegrationrelationX asthe‘adjusted’exchangeratebecausemovementsinfundamentalsareeliminatedfrom t it. Accordingtothecointegrationpresentvaluerelation(12),the‘adjusted’exchangerateisstationary andforward-lookinginfundamentalgrowth. Moreover,thecointegrationrelationX isaninfinite-order t moving average, MA(∞) equal to B(L)ζB(L)υ t , where B(L) = (cid:80)∞ j=0 BjLj and ζB(L) = (cid:80)∞ j=0 (Bζ)jLj−1 4Therestrictionsonthemovingaverageare∆z∗islinearlydeterministic,ζ0=1,ζ(L)isaninfiniteorderlagpolynominal withrootsoutsidetheunitcircle,theζisaresquaresummable,andυtismeanzero,homoskedastic,linearlyindependentgiven historyandisseriallyuncorrelatedwithitselfandthepastof∆zt. 8

under Assumptions 3 and 4 (i.e., z is I(1) and its growth rate has a Wold representation). Thus, the t ‘adjusted’exchangerateisa“cyclegenerator”–asdefinedbyEngleandIssler(1995)–becauseshocks toseriallycorrelatedfundamentalgrowthcreatepersistentPPPdeviations. Thestandard-andDSGE-PVMrequireAssumptions3and4tosatisfyProposition1. Ratherthan theseassumptions,wecanconstructacointegrationrelationfromtheDSGEmodelusingAssumptions 1and2becauseX isimpliedbythebalancedgrowthrestriction,e ≡ln[s ]=e +m −a ,wherem = t t t (cid:101)t t t t ln[M ] and a = ln[A ]. In this case, PPP deviations arise from the DSGE-PVM because of restrictions t t t thepresent-valuerelation(9)placesonthetransitorycomponentoftheexchangerate,e . (cid:101)t 3b. Equilibrium Currency Return Dynamics The second PVM prediction is that currency returns depend only on the lagged ‘adjusted’ exchange rate and fundamental forecast innovation. We show this by first rewriting the PVM of (11) as e t − (1−B)z t = (1−B) (cid:80)∞ j=1 BjE t z t+j . Differencing this equation produces, ∆ e t −(1−B) ∆ z t = ∞ (cid:88) (cid:104) (cid:105) (1−B) Bj E t z t+j −E t−1 z t+j−1 . Next,addandsubtractE t−1 z t+j insidethebrackets,andsubstitute j=1 withthecointegration-present-valuerelation(12)toobtain ∞ (13) ∆ e t − 1− B B X t−1 = (1−B) (cid:88) Bj(cid:2) E t − E t−1 (cid:3) z t+j . j=0 In equilibrium, currency return are generated by the lagged cointegration relation, X , and the ext−1 pectedannuityvalueoftheforecastinnovationsofthefundamental. Thelaggedcointegrationrelation is the error correction mechanism of (13) that reflects the only force that restores currency returns to equilibriumandPPPinresponsetotheshockinnovationu . Theseideasaresummarizedby ∆e,t Proposition 2: Under Proposition 1, the PVM predicts that the equilibrium currency return is an error correctionmechanisminwhichthelagged‘adjusted’exchangerate(orcointegrationrelation)istheonly factorthatdrivestheexchangeratetoPPPinresponsetofundamentalshockinnovations. Equation (13) is an ECM that regresses currency returns only on the lagged ‘adjusted’ exchange rate. 1−B Theregressionis ∆ e t = ϑX t−1 +u ∆e,t withfactorloadingϑ= B andcurrencyreturnforecasterror u ∆e,t =(1−B) (cid:80)∞ j=0 Bj (cid:2) E t − E t−1 (cid:3) z t+j .5 5Theerroru ∆e,t isalsojustifiediftheeconometrician’sinformationsetisstrictlywithinthatofcurrencytraders. 9

3c. A Limiting Model of Exchange Rate Determination Proposition 2 relies on B < 1 to define short- to medium-run currency return dynamics. This raisesthequestionoftheimpactofrelaxingthisbound. Proposition 3: The exchange rate approaches a martingale (in the strict sense) as B (cid:45)→ 1, according tothepresent-valuerelation(13)assumingProposition1. Proposition 3 relies on B (cid:45)→1 to produce the martingale E t e t+1 = e t and random walk behavior in the exchangerate.6 Thisbehaviorsuggestsanequilibriumpathfore t+1 inwhichitsbestforecastise t ,given relevantinformation,becausethesourceofserialcorrelation,X disappearsasB (cid:45)→1.7 t 3d. PVM Exchange Rate Dynamics Redux Engel and West (2005) show that the PVM of the exchange rate yields an approximate random walk as B approaches one. This section affirms the EW hypothesis, but unlike Proposition 3 does not rely on Proposition 2. Rather than follow the EW proof exactly, we invoke Assumptions 3 and 4, the present-value relation (3), the Weiner-Kolmogorov prediction formula, and the conjecture e = az to t t findthatcurrencyreturnsareunpredictable. The EW hypothesis is plimB (cid:45)→1[ ∆ e t −aζ(1)υ t ] = 0. Its hypothesis test begins by noting ∞ (cid:88) e t = z t−1 + BjE t∆ z t+j ,whichisobtainedfromthepresent-valuerelation(3). Usethisequationto j=0 construct ∆ e t −E t−1∆ e t =ζ(B)υ t ,givenAssumptions3and4andtheWeiner-Kolmogorovprediction formula. ThePVMof(11)alsosetscurrencyreturnsequaltotheannuityvalueoffundamentalgrowth, ∞ (cid:88) ∆ e t =(1−B) Bj E t∆ z t+j . Thelasttwoequationsyield j=0 ∞ (cid:88) (14) ∆ e t = ζ(B)υ t + (1−B) BjE t−1∆ z t+j . j=0 BylettingB (cid:45)→1,therandomwalkhypothesisofEWisverifiedindependentoftheECMofProposition 2(andcointegrationpredictionofProposition1).8 6Maheswaran and Sims (1993) show that the martingale restriction has little empirical content for tests of asset pricing modelswhendataissampledatdiscretemomentsintime. 7Hansen,Roberds,andSargent(1991)studylinearrationalexpectationsmodelsthatanticipateProposition3. 8ThisanalysismatchesequationsA.3−A.11andthesurroundingdiscussionofEngelandWest(2005). 10

The ECM (13) and Proposition 2 maps into the EW currency return generating equation (14). First, apply the change of index j = i − 1 to the present value of (14) to obtain the present-value cointegrationrelation(12)laggedonce. FortheECM(13),itspresentvalue(1−B) (cid:80)∞ j=0 Bj (cid:2) E t −E t−1 (cid:3) z t+j equals ζ(B)υ subsequent to evoking Assumptions 3 and 4 and the Weiner-Kolmogorov prediction t formula. Thus, when the PVM discount factor B is arbitrarily close to one, the EW hypothesis predicts e = ζ(1)υ which is consistent with currency returns following an ECM with no own lags or lags ∆ t t of fundamental growth. Since the standard- and DSGE-PVMs produce the ECM, the EW hypothesis is generalizedtothelargerclassoftwo-countrymonetaryDSGEmodels. 3e. A Common Trend-Common Cycle Model of Exchange Rates and Fundamentals Proposition 2 predicts an ECM for currency returns that is consistent with the EW currency return generating equation (14). These results rely, at most, on assumptions 3 and 4 under which fundamentals are I(1) and have a Wold representation in growth rates. However, empirical work on exchangeratesoftenemploymultivariatetimeseriesmodels(i.e.,VARs)insteadofthedeepernotionof aWoldrepresentation. This section studies the impact on the bivariate exchange rate-fundamental process, q = t (cid:2) (cid:3)(cid:48) e z ofendowinganECMonfundamentalgrowth. Inthiscase, q formsaVECM(0) t t ∆ t     (15) ∆ q t =    ϑ    X t−1 +    u ∆e,t    ,     η u ∆z,t where η is the factor loading on X t−1 for ∆ z t and u ∆z,t is its forecast innovation. Pre-multiplying the (cid:48) (cid:104) ϑ(cid:105) VECM(0)byβ = 1 − createsthecommonfeature η (cid:48) (cid:48)(cid:2) (cid:3)(cid:48) (16) β q = β u u . ∆ t ∆e,t ∆z,t (cid:48) Thevectorβ satisfiestheEngleandKozicki(1993)notionofacommonfeaturebecauseitcreatesalinear combinationof e and z thatisunpredictableconditionalontheirhistory. Giventhiscommonfeature ∆ t ∆ t restrictionandthecointegrationrelationofProposition1,VahidandEngle(1993)provideamethodto constructaStockandWatson(1988)multivariateBeveridgeandNelson(1981)commontrend-common cycledecomposition. Wesummarizetheseresultswith 11

Proposition4: AssumefundamentalgrowthistheECMprocess ∆ z t =ηX t−1 +u ∆z,t ,wheretheforecast (cid:48) innovationu isGaussian. WhenProposition2holds, q hasacommonfeature,β q ,inthesenseof ∆z,t t ∆ t (cid:48) (cid:104) ϑ(cid:105) EngleandKozicki(1993),whereβ = 1 − . Thecointegratingandcommonfeaturevectorsβandβ η restrictthetrend-cycledecompositionofq ,asdescribedbyVahidandEngle(1993). t ThecommonfeatureofProposition4endowsq =[e z ](cid:48)withacommontrendandacommon t t t cycleBeveridge-Nelson-Stock-Watson(BNSW)decomposition. VahidandEngle(1993)provideanexample in which the cointegration and common feature vectors restrict the trend of q to I − β(β(cid:48)β)−1β(cid:48), t 2 which gives trend and cycle components −Bη β (cid:48) q and 1−B β(cid:48)q , respectively.9 The 1−B(1+η) t 1−B(1+η) t BNSWdecompositionimposesacommoncycleone andz intheshort-,medium-,andlong-run,which t t restrictstheexchangeratetobe unpredictableatallforecasthorizons. Thispredictionisatodds with theempiricalevidenceofMark(1995). Thecommonfeaturerelation(16)alsoprovidesanotherapproachtoverifytheEWhypothesis, plimB (cid:45)→1[ ∆ e t −aζ(1)υ t ]=0. Proposition5: LettheexchangerateandfundamentalhavetheVECM(0)(15). Then,theEWhypothesis (cid:48) ϑ requirescurrencyreturnsandfundamentalgrowthtoshareacommonfeaturedefinedbyβ =[1 − ] η andthatϑ (cid:45)→0orB (cid:45)→1. Proposition 5 differs from other approaches to the EW hypothesis. First, the common feature relation (16) imposes cross-equation restrictions on q because its cycle generator, the lagged cointegrating ∆ t (cid:48) relationX t−1 ,isannihilatedbyβ . HavingeliminatedX t−1 ,theEWhypothesisdecouplestheexchange rate from fundamental growth and its forecast innovation u (= ζ(1)υ ). Finally, observe that when ∆z,t t (cid:48) ϑ (cid:45)→0(orB (cid:45)→1),β (cid:45)→[1 0]. Thisleavesonlytheforecastinnovationu togeneratemovements ∆e,t in e . Thus,theEWhypothesisisaffirmedbyProposition5.10 ∆ t A corollary of Proposition 5 is that changes in fundamentals do not Granger cause currency 9VahidandEngleshowan–dimensionVAR(1)withdcointegratingrelationshasn−dcommonfeaturerelations. 10Proposition5canalsobecastasanimplicationoftheBNSWrepresentationof∆qt. Inthiscase, β (cid:48) removesthevector MA(∞)inu ∆e,tandu ∆x,tfromtheBNSWrepresentationof∆qt. Onlyalinearcombinationofpureforecastinnovations,u ∆e,t andu ∆x,t,arelefttodrive∆qt. Letϑ (cid:45)→0toobtaintherandomwalkexchangeratewithinnovationu ∆e,t =ζ(1)υt. 12

returnsasB (cid:45)→1. OnlyifB∈(0, 1),domovementsinfundamentalshavepredictivepowerforcurrency returns according to the PVM. However, currency returns Granger cause growth in the fundamental as longasitispredictedbyitsownlaggedforecastinnovations. Theequilibriumcurrencyreturngenerating equation(13)andProposition2showsthatthisholdsevenifB (cid:45)→1. 3f. Reduced Form Evidence The propositions suggest testable restrictions on exchange rates and fundamentals. Table 3 describes details of the tests and summarizes results. Fisrt, if the lag length of the levels VAR of the exchangerateandfundamentalexceedsone,theVECM(15)isrejected. Second,cointegrationtestsare sufficient to examine Proposition 1. Finally, common feature tests are used, following Vahid and Engel (1993)andEngelandIssler(1995),thatyieldinformationaboutProposition4. WeestimateVARsofforeigncurrency-U.S.dollarexchangeratesandfundamentalsusingCanadian, Japanese, U.K., andU.S.dataona1976Q1–2004Q4sample.11 VARlaglengthsarechosenusing likelihood ratio (LR) statistics, given a VAR(8), ..., VAR(1).12 As described in Table 3, the Canadian–, Japanese–, and U.K.–U.S. samples yield a VAR(8), VAR(5), and VAR(4), respectively.13 Thus, the Canadian, Japanese, U.K., and U.S. data reject the VECM (15) because q has more serial correlation than ∆ t explainedbythelaggedcointegrationrelationX . t−1 Table3alsopresentsJohansen(1991,1994)traceandλ−maxteststatisticsthatfailtoconfirm thecointegrationpredictionofProposition1fortheCanadian–,Japanese–,andU.K.–U.S.samples. This finding is consistent with Engel and West (2005), who argue there is little evidence that exchange rates andfundamentalscointegrate. Finally, the common feature test is described in Table 3. This uses squared canonical corre- 11Fundamentalsequalcross-countrymoneyminuscross-countryoutput,whichimpliesanincomeelasticityofmoneydemand, ψ,calibratedtoone. ThiscalibrationisconsistentwithestimatesreportedbyMarkandSul(2003). Themoneystocks(outputs) aremeasuredincurrent(constant)localcurrencyunitsandpercapitaterms. 12TheVARsincludeaconstantandlineartimetrend. TheLRstatisticsemploytheSims(1980)correctionandhavestandard asymptoticdistributionaccordingtoresultsinSims,Stock,andWatson(1990). 13TheCanadian-U.S.andJapanese-U.S.VARsareselectedwhenthep−valueoftheLRtestisfivepercentorless. Sincethe U.K.-U.S.VARoffersambiguousresults,wesettleonaVAR(4). 13

lations of currency returns and fundamental growth. The common feature null is that the smallest correlationequalszero. Weuseaχ2 statisticofVahidandEngle(1993)andaF−statisticdevelopedby Rao(1973)totestthisnull. Thetestsrejectthenullforthelargestcanonicalcorrelation,butnotforthe smalleroneinthethreesamples. ThisisevidencethatcurrencyreturnsandfundamentalshareacommonfeatureintheCanadian–,Japanese–,andU.K.–U.S.samples. Givenacommonfeature,theexchange rateapproximatesarandomwalkwhenB (cid:45)→1. Thenextsectionexplorestheempiricalcontentofthis hypothesisintheCanadian–U.S.data. 4. Econometric Models and Methods Propositions 1–5 broaden our understanding of the EW hypothesis, which is also generalized to hold for the DSGE-PVM. Although the previous section discusses VAR methods that yield evidence about the joint behavior of the exchange rate and standard-PVM fundamentals, this approach is not informativeaboutestimatesofthePVMdiscountfactor. This section presents methods to estimate a PVM discount factor and test the EW hypothesis. Instead of relying on VARs, we employ unobserved components (UC) models to estimate the DSGE- PVM and test the EW hypothesis using Bayesian methods. A brief example motivates our approach. Consider the PVM (11) where the fundamental z has the permanent-transitory decomposition z = t t τ t + z (cid:101)t , τ t+1 = τ t + ε τ,t+1 , (1− (cid:80)p i= z 1 A z(cid:101),i Li)z (cid:101)t = ε z(cid:101),t , E t ε τ,t+1 = E t ε z(cid:101),t+1 = 0, E t ε τ 2 ,t+1 = σ τ 2, E t ε z 2 (cid:101),t+1 = σ z(cid:101) 2, and E t ε τ,t+i ε z(cid:101),t+j = 0 for all i and j.14 Combining the PVM (11) and the permanent-transitory decomposition of z gives an equilibrium permanent-transitory decomposition of the exchange rate, t e =τ +(1−B)ι (cid:2) I − BA (cid:3)−1 z ,whereι isa1×p rowvectorwithafirstelementofoneandzeros t t z(cid:101) z(cid:101) (cid:101)t z(cid:101) z elsewhereandA isthecompanionmatrixoftheARofz . Theexchangeratetrendisidentifiedwiththe z(cid:101) (cid:101)t randomwalkofz underitspermanent-transitorydecomposition. Transitoryexchangeratefluctuations t are driven by the fundamental cyclical component, z , which is the common dynamic factor of the (cid:101)t exchangerateandobservedfundamental. Thepermanent-transitorydecompositionoftheexchangerate is useful for the EW hypothesis because it becomes possible to estimate B, along with the coefficients 14WethankFarshidVahidforsuggestingthisexample. 14

of the permanent-transitory decomposition of z . Note also that as B approaches one, the permanent t componentτ comestodominateexchangeratefluctuationsaspredictedbytheEWhypothesis. t We use Bayesian methods to estimate multivariate UC models of the DSGE-PVM. The models representdifferentcombinationsofrestrictionsimposedbytheDSGE-PVMontheexchangerate,crosscountry money, and cross-country consumption. For example, κ is estimated for three UC models, which ties the exchange rate to the transitory component(s) of fundamentals. The exchange rate is disconnected from transitory shocks in remaining three UC models because κ is calibrated to one. We casttheUCmodelsinstatespaceformtoevaluatenumericallythelikelihoods. Weuseasampleofthe Canadiandollar–U.S.dollar(CDN$/US$)exchangerateandtheCanadian–U.S.moneyandconsumption differentials from 1976Q1–2004Q4. The random walk MH simulator is used to generate MCMC draws from the UC model posterior distributions conditional on this sample. We compute model moments, such as parameter means, unconditional variance ratios, permanent-transitory decompositions, and forecast error variance decompositions (FEVDs), from the posterior distributions. Model comparisons are based on marginal likelihoods, which we construct by integrating the likelihood function of each modelacrossitsparameterspacewheretheweightingfunctionisthemodelprior. 4a. State Space Systems of the UC Models The state space systems of the six UC models begin with the balanced growth restriction the DSGE model imposes on the exchange rate. This restriction is equivalent to the permanent-transitory decompositione =m −a +e . TheDSGE-PVM(9)placescross-equationrestrictionsonthestationary t t t (cid:101)t componentoftheexchangerate,e . (cid:101)t Cross-equation restrictions are conditioned on the permanent and transitory components of cross-country money and cross-country consumption. The permanent components of money and consumptionareµ t+1 =µ∗+µ t +ε µ,t+1 , ε µ,t+1 ∼N(0,σ ε 2 µ ),anda t+1 =a∗+a t +ε a,t+1 ,ε a,t+1 ∼N(0,σ ε 2 a ), respectively. Notethatµ∗ anda∗ arethedeterministictrendgrowthratesofcross-countrymoneyand TFP. We assume m (cid:102)t is a MA(k m(cid:102) ), m (cid:102)t = (cid:80)k j= m(cid:102) 0 α j ε m(cid:102),t−j , where α 0 ≡ 1 and ε m(cid:102),t ∼ N(0,σ ε 2 m(cid:102) ). For c (cid:101)t , we employ a AR(k c(cid:101) ), c (cid:101)t = (cid:80)k j= c(cid:101) 1 θ j c (cid:101)t−j +ε c(cid:101),t , where ε c(cid:101),t ∼ N(0,σ ε 2 c(cid:101) ). Put these elements together to form 15

thebalancedgrowthversionoftheDSGE-PVM ∞ (cid:88) (cid:110) (cid:111) (17) e t = µ t − a t + (1−κ) κjE t m (cid:102)t+j − c (cid:101)t+j , j=0 whichsatisfiestheDSGEbalancedgrowthpathrestrictions. ThebalancedgrowthDSGE-PVM(17)implies the cointegrating relation of Proposition 1. Thus, the exchange rate responds only to trends in crosscountrymoney,µ ,andTFP,a ,inthelong-run. Serialcorrelationintheexchangerateisproducedbythe t t transitory components of cross-country money and consumption, m and c . Also, if a common cycle (cid:102)t (cid:101)t generates these transitory components, the exchange also shares the restriction. Thus, the permanent andtransitorycomponentsofcross-countrymoneyandconsumptiondriveexchangeratefluctuations, whichgiverisetocross-equationrestrictionsintheUCmodels. TheUCmodelsareclassifiedaccordingtowhethertherearetwocyclesoracommoncycleand whether κ is calibrated to one or estimated. Three UC models follow from solving the DSGE-PVM (17) givenm ∼MA(k )andc ∼AR(k )oracommoncycleisimposedusingeithertheMA(k )orAR(k ). (cid:102)t m(cid:102) (cid:101)t c(cid:101) m(cid:102) c(cid:101) ThethreeUCmodelsareestimatedwhenκ iscalibratedtoone. WealsousetheUCmodelstoestimate κ. ThesixUCmodelshaveincommonthecross-countrymoneytrend,µ ,andTFPtrend,a . t t Arichsetofcross-equationrestrictionsarisesinthe2-trend,2-cycleUCmodelwithκ ∈(0, 1). Inpart,itsstatespacesystemconsistsoftheobservationequations     e 1 −1 δ δ ... δ δ δ ... δ   t     m(cid:102),0 m(cid:102),1 m(cid:102),km(cid:102) c(cid:101),0 c(cid:101),1 c(cid:101),kc(cid:101) −1       (18)    m t    =    1 0 1 α 1 ... α km(cid:102) 0 0 ... 0    S m(cid:102),c,t ,         c 0 1 0 0 ... 0 1 0 ... 0 t (cid:104) (cid:105)(cid:48) whereS m(cid:102),c,t = µ t a t ε m(cid:102),t ε m(cid:102),t−1 ... ε m(cid:102),t−km(cid:102) c (cid:101)t c (cid:101)t−1 ... c (cid:101)t−kc(cid:101) +1 ,factorloadingsonε m(cid:102),t anditslagsare (cid:88) km(cid:102) (19) δ = (1 − κ) κj−iα , i = 0, ..., k , m(cid:102),i j m(cid:102) j=i factorloadingsonc (cid:101)t ,...,c (cid:101)t−kc(cid:101) areelementsoftherowvector (cid:20) (cid:21)−1 (20) δ c(cid:101) = −s c(cid:101) (1 − κ) I kc(cid:101) − κ Θ , s c(cid:101) = [1 0 1×kc(cid:101) −1 ], 16

and isthecompanionmatrixoftheAR(k )ofc . Thesystemoffirst-orderstateequationsis Θ c(cid:101) (cid:101)t      1 0 ... 0 0 ... 0    (21) S m(cid:102),c,t+1 =                      a µ 0 0 . . . . . ∗ ∗                      +                        0 0 0 . . . . . . 1 0 0 . . . . . . I . . . k . . . m(cid:102) . . . 0 0 0 . . . . . . θ 0 0 . . . 1 I k . . . c(cid:101) . . . − . . . 1 0 (k θ c(cid:101) − 0 0 . . . k 1 c(cid:101) )×1                        S m(cid:102),c,t +                      ε 0 ε ε ε m(cid:102) a µ k c(cid:101), m , , (cid:102) , t t t t + + + × + 1 1 1 1 1                      , .     0 (kc(cid:101) −1)×1 withcovariancematrix Ωm,c(cid:101) =ε m(cid:102),c,t ε m (cid:48) (cid:102),c,t ,whereε m(cid:102),c,t =[ε µ,t+1 ε a,t+1 ε m(cid:102),t+1 0 km(cid:102) ×1 ε c(cid:101),t+1 0 (kc(cid:101) −1)×1 ](cid:48). We also study UC models that impose one common transitory factor on m and c . When the t t common component is m , the response of c to m is denoted π . This implies c = π m and (cid:102)t t (cid:102)t m,c(cid:101) (cid:101)t m,c(cid:101) (cid:102)t givesrisetothe2-trend,moneycycleUCmodel. Identifyingthecommontransitorycomponentwithc (cid:101)t definesm =π c whichrestrictsthe2-trend,consumptioncycleUCmodel. Theappendixdescribes (cid:102)t m,c(cid:101) (cid:101)t thestatespacesystemsofthe2-trend,moneycycleand2-trend,consumptioncycleUCmodels. The three remaining UC models set κ = 1. The restriction on the state space of the 2-trend, 2-cycleUCmodelisthattheexchangerateisdecoupledfromtransitorycross-countrymoneyandconsumption shocks. Similar restrictions arise in the observer equation of the 2-trend, money cycle and 2-trend,consumptioncycleUCmodels. Thus,weareabletocompareDSGE-PVMsinwhichκisestimated tothoseinwhichκ iscalibratedtoone. ThisprovidesanempiricalappraisaloftheEWhypothesis. 4b. The UC Model and Its Likelihood Function We label the 2-trend, 2-cycle UC model with κ ∈ (0, 1) UC . Likewise, UC and UC 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ denotethe2-trend,moneycycleand2-trend,consumptioncycle,κ ∈(0, 1)UCmodels. Thestatespace systemofUC is(18)and(21),whiletheappendixpresentsthesesystemsfor UC andUC . 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ (cid:2) (cid:3)(cid:48) These state space systems represent the dynamics of Y = e m c restricted by the DSGE-PVM t t t t and permanent-transitory specifications of m t and c t . We calibrate κ = 1 in UC 2,2,κ=1 , UC 2,m(cid:102),κ=1 , and UC . The state space systems are mapped into the Kalman filter to evaluate likelihood functions 2,c(cid:101),κ=1 17

asproposedbyHarvey(1989)andHamilton(1994).15 DenotethelikelihoodL (cid:0) Y | , UC (cid:1) ,where t Γ2,i,κ 2,i,κ i=2, m, c,κ iseithercalibratedtooneorestimated,and istheparametervectorofUC . (cid:102) (cid:101) Γ2,i,κ 2,i,κ (cid:2) Thelargestparametervectoris . Itcontains11+k +k elements, = κ α ... α Γ2,2,κ m(cid:102) c(cid:101) Γ2,2,κ 1 km(cid:102) θ ... θ µ∗ a∗ σ σ σ σ (cid:37) π π π (cid:3)(cid:48) . We add the parameters (cid:37) , π , π , and 1 kc(cid:101) µ a m(cid:102) c(cid:101) a,c(cid:101) e,0 e,t e,a a,c(cid:101) e,0 e,t π to tobetterfitUC tothedata. Forexample,theCanadian-U.S.TFPdifferentialexhibitsmore e,a Γ2,2,κ 2,2,κ variationthanc ifthecorrelationcoefficientofinnovationstoa andc ,E{ε ε }=(cid:37) ,isnegative.16 t t (cid:101)t a,t c(cid:101),t a,c(cid:101) Theremainingthreeparametersallowforanunrestrictedexchangerateintercept,π ,alinearexchange e,0 ratetrend,π ,andafactorloadingontheCanadian-U.S.TFPdifferential,π ,ratherthansetthe(1, 2) e,t e,a element in the matrix of the observation system (18) to negative one.17 We estimate π to ask if the e,a datasupportsthecointegration-balancedgrowthpathrestrictionimposedontheDSGE-PVM(17). The parameter vectors of the other five UC models are smaller. The UC model drops 2,m(cid:102),κ two plus k parameters from = (cid:2) κ α ... α µ∗ a∗ σ σ σ π π π π (cid:3)(cid:48) , c(cid:101) Γ2,m(cid:102),κ 1 km(cid:102) µ a m(cid:102) e,0 e,t e,a c,m(cid:102) while adding the factor loading on m for c , π . The factor loading π enters the parameter vec- (cid:102)t t c,m(cid:102) m,c(cid:101) tor of UC , while α ... α and σ are dropped from = (cid:2) κ θ ... θ µ∗ a∗ σ σ 2,m(cid:102),κ 1 km(cid:102) m(cid:102) Γ2,c(cid:101),κ 1 kc(cid:101) µ a (cid:3)(cid:48) σ c(cid:101) (cid:37) a,c(cid:101) π e,0 π e,t π e,a π m,c(cid:101) . The parameter vectors of the UC models UC 2,2,κ=1 , UC 2,m(cid:102),κ=1 , and UC areidenticalto , ,and exceptthatκ =1. 2,c(cid:101),κ=1 Γ2,2,κ Γ2,m(cid:102),κ Γ2,c(cid:101),κ 4c. The Data The sample runs from 1976Q1 to 2004Q4, T = 116. We have observations on the Canadian dollar–U.S.dollarexchangerate(averageofperiod). TheCanadianmonetaryaggregateisM1incurrent Canadian dollars, while for the U.S. it is the Board of Governors monetary base (adjusted for changes in reserve requirements) in current U.S. dollars. Consumption is the sum of non-durable and services expendituresinconstantlocalcurrencyunits.18 Theaggregatequantitydataisconvertedtopercapita 15ArelatedexampleisHarvey, Trimbur, andvanDijk(2007)whouseBayesianmethodstoestimatepermanent-transitory decompositionsofaggregatetimeseries,butwithoutrationalexpectationscross-equationrestrictions. 16Morley,Nelson,andZivot(2003)showthatthisrestrictionappliedtoanunivariateUCmodelresolvesitsdifferenceswith theBeverageandNelson(1981)decomposition. 17Thefactorloadingonthepermanentcomponentofmt remains(normalizedto)one. 18Theappendixdiscussesthedataandexplains,forexample,thatCanadianconsumptionincludessemi-durableexpenditures. 18

units. Thedataisloggedandmultipliedby100,butisneitherdemeanednordetrended. 4d. Estimation Methods ThelikelihoodfunctionsoftheUCmodelsdonothaveanalyticsolutions. Weapproximatethe likelihoodsL(Y t | Γ2,i,κ=1 , UC 2,i,κ=1 )andL(Y t | Γ2,i,κ , UC 2,i,κ )withposteriordistributionsof Γ2,i,κ=1 and , generated by the MCMC replications of the random walk MH simulator. Our estimates of Γ2,i,κ Γ2,i,κ=1 and and marginal likelihoods build on the Bayesian estimation tools of Fernández-Villaverde and Γ2,i,κ Rubio-Ramírez (2004), Rabanal and Rubio-Ramírez (2005), Geweke (1999, 2005), An and Schorfheide (2007),andGelman,Carlin,Stern,andRubin(2004). TheMHsimulatorcreates1.5millionMCMCdraws from the posterior. The initial 750,000 draws are treated as a burn-in sample and therefore discarded. Webaseourestimatesontheremaining750,000drawsfromtheposteriorsofthe UC 2,2,κ=1 , UC 2,m(cid:102),κ=1 , UC , UC , UC ,andUC models.19 2,c(cid:101),κ=1 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ 4e. Priors Thesecondcolumnoftable4(5)listthepriorsof Γ2,i,κ=1 ( Γ2,i,κ ),i=2, m (cid:102) , c (cid:101) . Underanormal prior, the first element is the degenerate mean and second its standard deviation. The inverse-gamma priorsareparameterizedbyitsdegreesoffreedom,thefirstelement,anditsmean,thesecondelement. Theleftandrightendpointsofauniformpriorisdenotedbyitsfirstandsecondelements. WechoosedegeneratepriorsforthelaglengthsoftheMA(k )ofm andAR(k )ofc thatset m(cid:102) (cid:102)t c(cid:101) (cid:101)t k =k =2. Normal priors for the MA (α and α ) and AR (θ and θ ) coefficients allow for disparate m(cid:102) c(cid:101) 1 2 1 2 transitory behavior in m and c . The prior means of α , α , θ , and θ guarantee that the relevant (cid:102)t (cid:101)t 1 2 1 2 eigenvalues are strictly less than one. The eigenvalues of the MA(2) (AR(2)) of m (c ) are 0.60±0.20i (cid:102)t (cid:101)t (0.95and-0.10). ThestandarddeviationofthenormalpriorsoftheMAandARcoefficientsprovidefor 19Theposteriordistributionsarebasedonacceptanceratesofbetween25and36percent. Besidesthe750,000MCMCdraws usedtocomputethemomentsreportedbelow,fourmoresequencesof750,000MCMCsaregeneratedfromdisparatestarting valuestoassessacrosschainandwithinchainconvergence. WecomputetheR(cid:98) statisticofGelman,Carlin,Stern,andRubin (2004)toevaluateacrosschainacrossandtheseparatedpartialmeanstestofGeweke(2005)convergence,whichisdistributed asymptoticallyχ2. Acrossthe77parametersofthesixUCmodels,thetwolargestR(cid:98)sare1.20and1.04,whileGelman,etal suggestaR(cid:98) ofabout1.10. Onfivesubsamples, theGewekeseparatedpartialmeanstesthasnop−valuesmallerthan0.21 acrossthesixUCmodelsandfiveMCMCsimulationsequences. 19

awidesetofrealizationsforα ,α , θ ,andθ . However,whenadrawgeneratesaneigenvaluegreater 1 2 1 2 than one (in absolute value) for either the MA or AR coefficients, the draw is discarded. Nonetheless, the MA and AR priors admit transitory cycles in cross-country money and consumption that allow for poweratthebusinesscyclefrequencies,ifthedatawants. We opt for priors of µ∗ and a∗ that rely on the Canadian–U.S. money stock and consumption differentials samples. Since µ∗ and a∗ represent deterministic trend growth, we ground the priors on normaldistributions. Thepriorstandarddeviationsofµ∗ anda∗ matchsamplemoments. Priors on the standard deviations of the shock innovations reflect standard practice for estimating DSGE models with Bayesian methods. For example, Adolfson, Laséen, Lindé, and Villani (2007) employinverse-gammapriorsforthestandarddeviationsoftheshockinnovationsoftheirstickyprice openeconomyDSGEmodel. However,thereisalackofgoodinformationaboutσ , σ ,σ ,andσ . This µ a m(cid:102) c(cid:101) explainswhyweimposeapriorwithtwodegreesoffreedom,whichforcesthesestandarddeviationsto bepositive. Ontheotherhand,weattachanormallydistributedpriortothecorrelationofinnovations to a and c , (cid:37) . Its mean is negative to capture our prior that the TFP differential, a , is smoother t (cid:101)t a,c(cid:101) t thancross-countryconsumption,c . Sincewehavenoinformationabouttheextentofthesmoothness, t themeanis−0.5withastandarddeviationof0.2thatplacesdrawsclosetonegativeoneorzerointhe 95percentcoverageintervaloftheprior. Drawsgreaterthanoneorlessthannegativeoneareignored. Thecorrelationofinnovationstoµ andm isfixedatzero,reflectingourassumptionthatthesources t (cid:102)t andcausesofpermanentandtransitorymonetaryshocksareorthogonal. Theexchangerateinterceptandlineartimetrendpriorsaresetaccordingtoalinearregression oftheexchangerateontheseobjects. Thismotivatesourchoiceofnormallydistributedpriorsfor π e,0 andπ andoftheirdegeneratemeansandstandarddeviations. e,t The remaining factor loadings have priors that reflect a dearth of information on our part. The uniform priors of π , π , and π are wide and include zero. If, for example, π is small it e,a c,m(cid:102) c,m(cid:102) e,a indicates the inadequacy of the balanced growth restriction and the impact of permanent fluctuations inCanadian–U.S.TFPdifferentialsontheexchangerate. Thesameholdsfortheresponseofc (m )to t t 20

transitorymovementsintheCanadian–U.S.moneystock(consumption)differential. The UC models have only one ‘economic’ parameter, the DSGE-PVM discount factor κ ≡ 2,i,κ 1 ,incommon. WeadopttheEngelandWest(2005)priorforκ. Theyarguethatitisnecessaryfor 1+r∗ κ ∈[0.9, 0.999]togenerateanapproximaterandomwalkexchangeratefromthestandard-PVM.Hence, ourprioronκ isconstructedtoprovideinformationabouttheEWhypothesisfromtheposteriorsofthe UC models. We impose an inverse-gamma prior on the DSGE-PVM discount factor κ, which follows 2,i,κ DelNegroandSchorfheide(2006). Thedegeneratepriormeansofκ =0.988andexp([a∗ =0.158]/400) implyanannualaveragerealworldinterestrateofaboutfivepercent. Althoughafivepercentrealworld interest rate is large for the floating rate period, the standard deviation of 0.038 guarantees draws for κ thatcoverawideinterval. However,MCMCdrawsfromtherandomwalkMHsimulatoroftheUC 2,i,κ modelsobeytheEWpriorbecauseweignoredrawsforwhichκ ∉ [0.9, 0.999]. 5. Results Thissectionpresentstheresultsofimplementingourempiricalstrategy. Tables4and5provide the posterior means and standard deviations of Γ2,i,κ=1 and Γ2,i,κ vectors, i = 2, m (cid:102) , c (cid:101) , for the six UC models. WeincludemarginallikelihoodsofthesixUCmodels,usingmethodsdescribedbyFernández- VillaverdeandRubio-Ramírez(2004),RabanalandRubio-Ramírez(2005),andGeweke(1999),toconduct inferenceacrossthesemodels. Wepresentdensitiesofthepriorandposteriorsofκfor UC , UC , 2,2,κ 2,m(cid:102),κ andUC infigure1. Tables6,7,8,and9reportfactorloadingsontheexchangerateofthetransitory 2,c(cid:101),κ componentsofmoneyandconsumptiondifferentials, unconditionalvarianceratiosofthepresentdiscountedvalueoftheshockinnovationstotheexchangerate,FEVDsofthetrend-cycledecompositionof theexchangeratewithrespecttotheseshocks,andsummarystatisticsoftrend-cycledecompositions, respectively. Figures2,3,and4plotthetrend-cycledecompositionoftheCDN$/US$exchangerate. 5a. Parameter Estimates Tables 4 and 5 list the posterior means and standard deviations of the parameters of the six UCmodels. Estimatesof UC 2,2,κ=1 , UC 2,m(cid:102),κ=1 , UC 2,c(cid:101),κ=1 appearintable4. Thesethreemodelsexhibit persistence in the transitory components of the money and consumption differentials, m and c . For (cid:102)t (cid:101)t 21

example,the UC (UC )modelyieldsAR(MA)estimatesthatimplythehalflifeofashockto 2,m(cid:102),κ=1 2,c(cid:101),κ=1 c (cid:101)t (m (cid:102)t )is17(7)years.20 However,onlyc (cid:101)t ispersistentinthe UC 2,2,κ=1 model. Thehalflifeofashockto m islessthantwoquarters,whileforc itisbetweennineandtenyears. Notealsothatthepriorsand (cid:102)t (cid:101)t posterior means of the MA coefficients, α and α , only differ for the UC model. Although the 1 2 2,m(cid:102),κ=1 posteriormeansoftheARcoefficientshavemovedawayfromthepriormeans,aonestandarddeviation oftheposteriorofθ 2 coverszeroforthe UC 2,2,κ=1 and UC 2,c(cid:101),κ=1 models. The posterior means of µ∗ and a∗ show that Canada experiences slower (faster) trend money (TFP) growththan the U.S. overthe sample. Trend U.S. moneygrowth is onaverage about 0.05 percent higher annually according to the posteriors of the UC 2,2,κ=1 , UC 2,m(cid:102),κ=1 , and UC 2,c(cid:101),κ=1 models. Across thesemodels,a∗ ≈0.16indicatesCanadiandeterministictrendTFPgrowthdominatesitsU.S.counterpartbyabout0.06percentatanannualrate. The UC 2,2,κ=1 , UC 2,m(cid:102),κ=1 , and UC 2,c(cid:101),κ=1 models show differences across estimates of the posteriormeansoftheshockinnovationstandarddeviations. Onlytheestimatedimpulsestructureofthe UC model is dominated by movements in the permanent innovations of the money shock, σ . 2,m(cid:102),κ=1 µ The converse is that this model yields the smallest posterior means of the standard deviation of the TFP differential shock and m shock innovations, σ and σ . The UC model yields the largest (cid:102)t a m(cid:102) 2,c(cid:101),κ=1 estimates of σ and σ , but these posterior means are about the same magnitude. Note also that the a c(cid:101) correlationoftheinnovationstotheTFPshockandc shockisestimatedtobe(cid:37) =−0.88and−0.95 (cid:101)t a,c(cid:101) by the UC 2,2,κ=1 and UC 2,c(cid:101),κ=1 models, respectively. Thus, these models are consistent with the TFP trendbeingmorevolatilethanobservedCanadian–U.S.consumption. Estimates of the exchange rate intercept and linear time trend indicate that the UC 2,m(cid:102),κ=1 model provides the largest value for the US$ in steady state and the largest deterministic growth rate for the CDN$/US$ exchange rate. The posterior means of π and π imply that the steady state e,0 e,t CDN$/US$exchangerateis1.23withadeterministicannualgrowthrateofabout0.8percent. Forthe UC 2,2,κ=1 (UC 2,c(cid:101),κ=1 )model,theanalogousvaluesare1.10(1.03)and0.3(0.2)percentperannum. Thus, 20Thehalflifeequalslog[0.5]/log[q],whereqisthelargestmodulusofthecompanionmatrixoftheARorMAcoefficients. 22

the UC model places more emphasis on deterministic elements to fit the data compared to the 2,m(cid:102),κ=1 othertwoUCmodelsthatcalibrateκ toone. Theremainingcoefficientsarethefactorloadingsπ , π ,andπ . Posteriormeanestimates e,a e,m(cid:102) e,c(cid:101) of π −2.68 and −9.02 reveal that there are statistically and economically large deviations from the e,a balancedgrowthpathbytheUC 2,2,κ=1 andUC 2,m(cid:102),κ=1 models. TheUC 2,c(cid:101),κ=1 modelisclosertosatisfying the balanced growth hypothesis that π =−1. This UC model has a posterior mean of −0.72 for π e,a e,c(cid:101) whose two standard deviation interval contains the balanced growth restriction. The response of the Canadian–U.S.moneystockdifferentialtoc isalsoclosetonegativeonefortheUC modelbecause (cid:101)t 2,c(cid:101),κ=1 theposteriormeanofπ =−0.90withastandarddeviationof0.21. TheUC modelrevealsthat m,c(cid:101) 2,m(cid:102),κ=1 aonepercentriseinc resultsina4.4percentriseintheCanadian–U.S.consumptiondifferential. (cid:101)t ThekeyeconomicparameteroftheDSGE-PVMisitsdiscountfactorκ. Table5liststheposterior means and standard deviations of the and , and vectors that include estimates of κ. Γ2,2,κ Γ2,m(cid:102),κ Γ2,c(cid:101),κ Asidefromtheinclusionoftheprior,posteriormean,andstandarddeviationκ atthetopoftable5,the posteriormeansandstandarddeviationsoftheremainingcoefficientsresemblethosereportedintable 4. The only notable exceptions are that the posterior means of σ , σ , π , and π are smaller for a c(cid:101) e,a e,c(cid:101) theUC modelcomparedtoitscousinwiththecalibrationκ =1. Theresultisthatσ isthelargest 2,c(cid:101),κ µ innovation shock standard deviation of the UC model. Also, this UC model and the data produce 2,c(cid:101),κ an estimate of π whose one standard deviation coverage interval contains negative one. Thus, the e,a UC model is closer to the balanced growth hypothesis and relies to a greater extent on permanent 2,c(cid:101),κ shockstotheCanadian–U.S.moneystockdifferential. Theposteriormeansofκrangefrom0.966fortheUC model,to0.974fortheUC model, 2,2,κ 2,m(cid:102),κ tothelargestestimateof0.9962fortheUC model. Theseestimatesareconsistentwithannualworld 2,c(cid:101),κ realinterestratesof15.1,11.4,and1.7percentgiventheposteriorsoftheUC , UC andUC 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ models,respectively. AlthoughtheUC , UC modelshaveposteriorsthatsuggestunreasonably 2,2,κ 2,m(cid:102),κ large world real interest rates, these UC models yield 95 percent coverage intervals whose upper end is0.999. TheUC modelproduces aposterior ofκ witha 95percent coverageinterval whoselower 2,c(cid:101),κ 23

end equals 0.987. This value of κ is greater than the posterior means of κ for the UC and UC 2,2,κ 2,m(cid:102),κ models. Thus, the UC model generates a posterior distribution of κ that is to the right of those 2,c(cid:101),κ producedbytheUC andUC models. 2,2,κ 2,m(cid:102),κ Figure1reinforcestheviewthattheUC modelposteriorsyieldestimatesofκthataretothe 2,c(cid:101),κ rightofthoseoftheUC andUC models. Posteriordensitiesofκ appearinfigure1fortheseUC 2,2,κ 2,m(cid:102),κ models,alongwiththedensityoftheinverse-gammapriorrestrictedtotheEWpriorofκ ∈[0.9, 0.999]. The solid (black) line is the κ prior density. It is close to the posterior density of κ derived from the UC model, which is the dashed (blue) line. The UC model generates a posterior density of κ, 2,2,κ 2,m(cid:102),κ thedot-dash(green)plot,thatmovesoffthepriorbyplacinglessweightonκslessthan0.97andmore weightaboveit. Thedot-dot(red)plotisthedensityofκ fromtheUC modelposterior. Thisdensity 2,c(cid:101),κ isdeflatedbytenpercenttoeasecomparisontotheotherdensities. Astrikingfeatureoffigure1isthat theUC modelposteriorpushesκ offofitspriorbecauseitsmasslaysbetween0.98and0.999. 2,c(cid:101),κ Table 6 contains the posterior means of the exchange rate factor loadings with respect to m (cid:102)t and c , the δ s and δ s.21 A striking aspect of the estimates of δ , δ , and δ is that the (cid:101)t m(cid:102),i c(cid:101),i m(cid:102),0 m(cid:102),1 m(cid:102),2 response of the CDN$/US$ exchange rate to innovations in m is economically small for either the (cid:102)t UC orUC models. Thelargeposteriorstandarderrorsonthesefactorloadingalsoindicatethe 2,2,κ 2,m(cid:102),κ imprecision the Canadian–U.S. data give to these estimates. The data yield a more precise estimate of δ fortheUC model. Theposteriormeanofthisfactorloadingshowsthattheexchangeratefalls c(cid:101),0 2,2,κ by0.6percentgivenaonepercentincreaseinc . Theseestimatesdropto−0.33fortheUC model. (cid:101)t 2,c(cid:101),κ Also,theassociated95percentcoverageintervalcontainszero. Insummary,theUC , UC ,and 2,2,κ 2,m(cid:102),κ UC modelshaveposteriorsinwhichthereiseitheranegligibleexchangerateresponsetom shocks 2,c(cid:101),κ (cid:102)t oraneconomicallylargenegativereactionbytheCDN$/US$exchangeratetoc fluctuations. However, (cid:101)t thelatterexchangerateresponseissometimesestimatedimprecisely. 5b. Unconditional Variance Ratios and FEVDs of the Exchange Rate Tables7and8presentunconditionalvarianceratiosandFEVDscomputedusingtheposteriors 21FortheUC andUC models,therelevantfactorloadingsaremultipliedby1−π or1−π . 2,m(cid:102),κ 2,c(cid:101),κ e,m(cid:102) e,c(cid:101) 24

of the UC , UC , and UC models. We calculate the variances of the present discounted val- 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ ues (PDVs) of the money, TFP, and consumption shock innovations using the DSGE-PVM version of the equilibrium currency return generating equation (14) and UC model restrictions when κ is estimated. The variance ratios are these values divided by the sample variance of the CDN$/US$ exchange rate (= 2.04). According to the unconditional variance ratios, only permanent shocks to the Canadian–US moneydifferential,ε ,andtheTFPdifferential,ε ,explainvariationintheCDN$/US$exchangerate. µ,t a,t The variances of the PDVs of shock innovations to m and c are small and lack precision. Note that (cid:102)t (cid:101)t exceptfortheUC model,thevarianceofthePDVofε islargerthanthatofε . 2,m(cid:102),κ a,t µ,t WereportFEVDsintable8withimplicationssimilartotheunconditionalvarianceratios.22 The toppaneloffigure8showsthattheposterioroftheUC modelyieldsaFEVDinwhichtheTFPshock 2,2,κ ε makes a large and increasing contribution to exchange rate fluctuations at longer horizons. The a,t money shock ε remains economically important for exchange rate movements out to a three to five µ,t year horizon, but shocks to m and c are unimportant at any horizon. Much the same is true for the (cid:102)t (cid:101)t FEVDsfoundusingtheUC modelposterior. However,therelativesharesoftheε andε shocks 2,m(cid:102),κ µ,t a,t areunchangedatatwo-thirds/one-thirdsplitfromtheonequartertotenyearhorizons. TheposterioroftheUC modelimbuesasluggishdynamictotheexchangerateFEVDsfound 2,c(cid:101),κ in the bottom panel of table 8. The ε and ε shocks are responsible for about 60 and 40 percent, µ,t a,t respectively,offluctuationsintheexchangerateatshorthorizons. Ata10yearhorizon,thecontribution of ε (ε ) only falls (rises) to 55 (45) percent. Thus, only the posterior of the UC model predicts µ,t a,t 2,c(cid:101),κ thatpermanentshockstomoneydominateexchangeratemovementsatlongerhorizons. 5c. Trend-Cycle Decompositions Trend-cycle decompositions of the CDN$/US$ exchange rate and Canadian–U.S. money and consumption differentials are plotted in figures 2 and 3 with summary statistics given in table 9. We run UC , UC , and UC model posteriors through the Kalman smoother to create trend-cycle 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ decompositionsandsummarystatistics. Figures2and3andtable9containmomentsthatareaverages 22TheFEVDsarecomputedusingtheVECMimpliedbyequation(13). TheVECMisplacedinstatespaceformasoutlinedby Heqc,Palm,andUrbain(2000)anditeratedtocreatetheFEVDsoftable8. 25

over750,000drawsfromUCmodelposteriors. Trendexchangerategrowthislabeled eτ intable9. ∆ The top window of figure 2 contains plots of the exchange rate and smoothed trends taken fromtheposteriorsoftheUC andUC models.23 Thesolid(black)lineise ,thelogoftheactual 2,2,κ 2,c(cid:101),κ t exchangerate. ThesmoothedtrendsoftheUC andUC modelsarethedashed(blue)anddotted 2,2,κ 2,c(cid:101),κ (red) plots, respectively. Note that these UC models generate smoothed exchange rate trends that are more volatile than the actual exchange rate. The top row of table 9 indicate that the posteriors of the UC and UC models generate standard deviations of eτ equal to 2.66 and 2.44, respectively. 2,2,κ 2,c(cid:101),κ ∆ Thestandarddeviationof eτ equals2.04. ∆ Thesmoothedexchangeratecyclesappearinthebottomwindowoffigure2. Thedotted(blue) line is the smoothed exchange rate cycle, e , based on the posterior of the UC model, while the (cid:101)t 2,2,κ dotted (red) line is associated with the UC model. Although the former e exhibits more variability 2,c(cid:101),κ (cid:101)t thanthelatter(thestandarddeviationsare3.68and2.44),thesee sarepersistentwithAR1correlation (cid:101)t statisticsof0.97and0.98. NotethatonlytheposterioroftheUC modelyieldsa(closeto)non-zero 2,m(cid:102),κ correlationfor eτ ande,accordingtotable9. ∆ (cid:101) Figure 3 depicts smoothed permanent-transitory decompositions of the Canadian-U.S. money and consumption differentials. The actual differentials and smoothed trends appear in the top row of windows, while smoothed cycles are found in the bottom row of windows. The money (consumption) differentials are the right (left) side windows. The posterior of the UC model produces a money 2,2,κ trend,µ thatalmostperfectlymimicstheactualCanadian-U.S.moneydifferentials,asshowninthetop t leftwindowoffigure3. Theresultisthatsmoothedm ismuchlessvolatile,withastandarddeviation (cid:102)t of 0.68 compared to a standard deviation of 1.62 for µ . The bottom left window of figure 3 shows a t saw-toothed pattern in m , conditional on the posterior of the UC model. This explains the AR1 (cid:102)t 2,2,κ correlationstatisticof−0.68form (middleofthesecondcolumnoftable9). (cid:102)t 23Wedonotpresentthetrend-cycledecompositionsbasedontheposterioroftheUC modelbecauseitslogmarginal 2,m(cid:102),κ likelihoodisfarbelowthoseoftheotherUCmodels. Table9includesstandarddeviationsof∆eτ ande(cid:101)fromtheposteriorof theUC 2,m(cid:102),κ modelthatarelargerbyafactorof30orcomparedtothesestatisticsfromtheUC2,2,κ andUC 2,c(cid:101),κ models. This signalsthelackofacceptanceoftheUC modelbyourCanadian–U.S.sample. 2,m(cid:102),κ 26

Table9revealsthattheposteriorofUC modelproducesasmoothedmoneytrend,µ ,that 2,c(cid:101),κ t isaboutasvolatileasintheUC model. Therelevantstandarddeviationsare1.62and1.71(second 2,2,κ andfourthcolumnsoftable9). However,thesmoothed µ and eτ haveapositivecorrelationof0.62 ∆ ∆ onlyintheposterioroftheUC model(bottomhalfofthefourthcolumnoftable9). 2,c(cid:101),κ TheposteriorsoftheUC andUC modelsyieldqualitativelysimilarplotsforthesmoothed 2,2,κ 2,c(cid:101),κ TFP trend differential, a . These plots appear in the top right window of figure 3 as dashed (blue) and t dotted (red) lines for the UC and UC models, respectively, where observed cross country con- 2,2,κ 2,c(cid:101),κ sumptionisthesolid(black)line. ThesmoothedTFPgrowthdifferential, a,is50percentmorevolatile ∆ fortheUC modelthanitisfortheUC model. TheposteriorsofthesetwoUCmodelsalsopro- 2,c(cid:101),κ 2,2,κ ducecorrelationsof−0.71and−0.85between eτ and a. Thus,arisingU.S.TFPisassociatedwithan ∆ ∆ appreciationoftheU.S.dollar. Thelate1970sisonesuchperiodbecauseCanadaexperiencedagreater relative productivity slowdown. By the 1980s, Canadian TFP is growing more rapidly than in the U.S., which continues into the early 1990s. Subsequently, U.S. TFP recovers relative to Canadian TFP. At the endofthesample,theCanadian–U.S.TFPdifferentialisexpandingoncemore. Thebottomrightwindowoffigure3presentsthesmoothedc oftheUC andUC models. (cid:101)t 2,2,κ 2,c(cid:101),κ The former cycle is the dashed (blue) line and the latter is the dotted (red) plot. These cycles are persistent,withAR1correlationstatisticsof0.97and0.98,buttheUC modelgeneratesathirdless 2,2,κ volatilityinsmoothedc thanfoundfortheUC model. (cid:101)t 2,c(cid:101),κ Thesmoothedc haspeaksandtroughsthatcoincidewithseveralU.S.-Canadianbusinesscycle (cid:101)t dates. Forexample,troughsintheposteriormeanofc appearin1981and1990,whichalsorepresent (cid:101)t recessionsdatesintheU.S.andCanada. Sincetheendofthe1990–1991recession,theriseinc points (cid:101)t to a persistent, but transitory, rise in U.S. consumption relative to Canada. Nonetheless, c has been (cid:101)t fallingrapidlysinceapeakinlate2001,whichcorrespondstotheendofthelastU.S.recession. The bottom row of table 9 shows that c and e are perfectly negatively correlated in the pos- (cid:101)t (cid:101)t teriors of the UC and UC models. The negative correlation of the transitory component of the 2,2,κ 2,c(cid:101),κ exchange rate with c helps to interpret exchange rate fluctuations. Peaks in the transitory component (cid:101)t 27

of the exchange rate occur either at or shortly after the end of recessions. For example, the smoothed exchangeratecycleshaveatendencytopeakandtrougharounddatesusuallyassociatedwithU.S.and Canadian business cycle dates (i.e., the late 1970s, early 1990s, and 2001). A specific case is the peak in e during the 1990–1991 recession in the U.S., which is a moment at which the Canadian dollar ap- (cid:101)t proachedparagainsttheU.S.dollar. Anexceptionistheendofthe2001recessionwhentheCanadian dollarreachedalowofnearly0.62totheU.S.dollar. 5d. Comparing the UC models Thebottomrowoftable4reportsthelogmarginallikelihoods,lnL(cid:98),oftheUC 2,2,κ=1 , UC 2,m(cid:102),κ=1 , andUC models. ThesemarginallikelihoodsshowthatourCanadian-U.S.samplegivesmostsup- 2,c(cid:101),κ=1 port to the UC 2,c(cid:101),κ=1 model. The difference between this model and the UC 2,2,κ=1 model is about 29, so that the Bayes factor prefers the UC model with only transitory consumption. For the data to give more credence to the latter model, its prior probability must be raised by the prior probability of the UC model multiplied by 4.7×1012[= exp(29.18)]. Since the magnitude of this factor is large, it 2,c(cid:101),κ=1 seemsunreasonabletoincludethetransitorymoneyshockintheUCmodelwhenκ iscalibratedtoone. Thelastrowoftable5containsthelogmarginallikelihoodsoftheUC , UC ,andUC 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ models. The ranking of these models matches that of the UC models with the κ = 1 calibration. The UC modeldominatestheUC and UC models. Akeyreasonisthattheposteriorsofthese 2,m(cid:102),κ 2,2,κ 2,m(cid:102),κ modelsyieldeconomicallyimplausibleestimatesoftheDGSE-PVMdiscountfactorκ. This raises the question of whether it is difficult to choose between the UC and UC 2,c(cid:101),κ=1 2,c(cid:101),κ models. OursamplefavorstheUC andUC modelscomparedtotheotherfour. TheUC 2,c(cid:101),κ=1 2,c(cid:101),κ 2,c(cid:101),κ=1 model has the largest marginal likelihood, which suggests that the data support it over the UC 2,c(cid:101),κ model. This choice relies on the belief that scaling up the prior probability of the UC model by 2,c(cid:101),κ 167.3 =exp(5.12)istoolargetobejustified. If,ontheotherhand,thisfactorisregardedasinconclusive in rejecting the UC model, it could be argued that our Canadian–U.S. sample cannot pick between 2,c(cid:101),κ theUC andUC models.24 Nonetheless,theseUCmodelssupportfortheEWhypothesis. 2,c(cid:101),κ=1 2,c(cid:101),κ 24Jeffreys(1998,p. 432)contendsthatBayesfactorsdifferingby3.16isevidenceaboutthetwomodelsjustbetween‘not worthmorethanabaremention’andsubstantiallyinfavorofthemodelwiththelargermarginallikelihood. 28

5e. Exchange Rate Dynamics as κ (cid:45)→ 1 Engel and West (2005) argue that the exchange rate will approximate a random walk when the discountfactorisclosetooneandfundamentalshaveaunitroot. Propositions3and5alsopredictthat e willcollapsetorandomwalkasκ (cid:45)→1. (cid:101)t We extract evidence about the EW hypothesis from the UC and UC model posteriors. 2,2,κ 2,c(cid:101),κ ThefocusisontheseUCmodelsratherthantheUC modelbecauseitattributesallexchangerate 2,c(cid:101),κ=1 movementstopermanentshocks. Weconductthiscomparisonwithκsatthe16thand84thpercentiles, along with the largest κs, from the and vectors. For the UC (UC ) model, the 16th Γ2,2,κ Γ2,c(cid:101),κ 2,2,κ 2,c(cid:101),κ percentile,84thpercentile,andlargestκsare0.9425,0.9883,and0.9990(0.9943,0.9987,and0.9990), respectively. Fixingκ atthesevalues,wesimulatetheUC andUC modelsextracting2000draws 2,2,κ 2,c(cid:101),κ fromtheposteriors,discardthefirst1000,runtheKalmansmootherontheremaining1000,andaverage theensembletogenerateexchangeratecyclesthatrespecttherationalexpectationshypothesis. Figure 4 plots the smoothed exchange rate cycles. The top (bottom) window contains the e (cid:101)t createdfromtheposterioroftheUC (UC )model. Thedot-dash(blue),dotted(green),anddotted 2,2,κ 2,c(cid:101),κ (red) lines are conditional on the 16th percentile, 84th percentile, and largest κs, respectively. Across thetopandbottomwindows,thevolatilityofe iscompressedasκ approaches0.999. Thisisreflected (cid:101)t inthestandarddeviationsofe thatequal4.44,2.84,and0.57movingfromthesmallesttolargestκ for (cid:101)t the UC model. The equivalent standard deviations are 3.74, 1.64, and 1.30 for the UC model. 2,2,κ 2,c(cid:101),κ AlthoughtheUC modelgeneratesexchangeratecyclesthataresmootherthanatitsposteriormean 2,2,κ only for the largest κs, this UC model is able to produce smoother exchange rate cycles at the 84th percentileandlargestκs. Thus,pushingκ increasesthesmoothnessoftheexchangeratecycle. Thisis evidencethatlendscredencetotheEWhypothesis. 6. Conclusion Economists have little to say about the impact of policy on currency markets without an equilibriumtheoryofexchangeratedeterminationthatisempiricallyrelevant. AccordingtoEngelandWest (2005),thenearrandomwalkbehaviorofexchangeratesexplainsthefailureofequilibriummodelstofit 29

thedataortofindanymodelthatsystematicallybeatsitatout-of-sampleforecasting. Theyconjecture thatthestandard-presentvaluemodel(PVM)ofexchangeratesyieldstherandomwalkpredictionwhen fundamentalsarepersistentandthediscountfactorisclosetoone. ThispapergeneralizestheEngelandWest(EW)hypothesisbyconstructingaPVMfromatwocountrymonetarydynamicstochasticgeneralequilibrium(DSGE)model. Thestandard-andDSGE-PVMs yieldidenticalpredictionsfortheexchangerate. Thesepredictionsaresummarizedbyfivepropositions. Thus,wegeneralizetheEWhypothesistothelargerclassofopeneconomyDSGEmodels. Our empirical results support the view that the Canadian-U.S. data prefer a random walk exchange rate and a DSGE-PVM with a discount factor calibrated to one. At the same time we obtain evidence on the nature of the shocks driving exchange rates. Bayesian estimates of the DSGE-PVM indicate that the Canadian dollar–U.S. dollar exchange rate is dominated by permanent shocks, whether the discount factor is estimated or calibrated to one, which supports the EW hypothesis. Our evidence is also consistent with the recent VAR literature suggesting that monetary policy shocks have only a minorimpactonexchangeratefluctuations. Monetarypolicyshocksarealsofoundtobeunimportant forexchangeratemovementsbyLubikandSchorfheide(2006)withinthecontextofanestimatedopen economyDSGEmodel. WhetherthisresultholdsacrossawidersetofopeneconomyDSGEmodelsisa worthygoaloffutureresearch. References Adolfson, M., S. Laséen, J. Lindé, and M. Villani. 2007. Bayesian Estimation of an Open Economy DSGE ModelwithIncompletePass-Through.JournalofInternationalEconomics 72,481–511. An,S.,F.Schorfheide.2007.BayesiananalysisofDSGEmodels.EconometricReviews 26,113–172. Beveridge,S.,C.R.Nelson.1981.ANewApproachtoDecompositionofEconomicTimeSeriesintoPermanent and Transitory Components with Particular Attention to Measurement of the Business Cycle. JournalofMonetaryEconomics 7,151–174. Bilson, J.F.O. 1978. Rational Expectations and the Exchange Rate. in Frenkel, J.A., H.G. Johnson, The EconomicsofExchangeRates: SelectedStudies,Addison-Wesley,Reading,MA. Campbell, J.Y., R. Shiller. 1987. Cointegration and Tests of Present Value Models. Journal of Political Economy 93,1062–1088. DelNegro,M.,F.Schorfheide.2006.FormingPriorsforDSGEModels(andHowitAffectstheAssessment ofNominalRigidites).WorkingPaper2006–16,FederalReserveBankofAtlanta. 30

Dornbusch,R.1976.ExpectationsandExchangeRateDynamics.JournalofPoliticalEconomy 84,1161– 1176. Eichenbaum, M., C. Evans, 1995. Some Empirical Evidence on the Effects of Monetary Policy Shocks on ExchangeRates.QuarterlyJournalofEconomics 110,975–1010. Engel,C.,K.D.West,2005.ExchangeRatesandFundamentals.JournalofPoliticalEconomy 113,485–517. Engle, R.F., J.V. Issler, 1995. Estimating Common Sectoral Cycles. Journal of Monetary Economics 35, 83–113. Engle,R.F.,S.Kozicki,1993.TestingforCommonFeatures.JournalofBusinessandEconomicsStatistics 11,369–395. Faust, J., J. Rogers, 2003. Monetary Policy’s Role in Exchange Rate Behavior. Journal of Monetary Economics 50,1403–1424. Fernández-Villaverde, J., J.F. Rubio-Ramírez. 2004. Comparing Dynamic Equilibrium Models to Data: A BayesianApproach.JournalofEconometrics 123,153–187. Frankel, J.A., 1979. On the Mark: A Theory of Floating Exchange Rates Basedon Real Interest Rate Differentials.AmericanEconomicsReview 69,610–622. Gelman,A.,J.B.Carlin,H.S.Stern,D.B.Rubin,2004,BayesianDataAnalysis,SecondEdition,ChapmanandHall,NewYork,NY. Geweke, J., 1999. Using Simulation Methods for Bayesian Econometric Models: Inference, Development andCommunication.EconometricReviews 18,1–126. Geweke, J., 2005. Contemporary Bayesian Econometrics and Statistics, J. Wiley and Sons, Inc., Hoboken,NJ. Hamilton,J.D.,1994,TimeSeriesAnalysis,PrincetonUniversityPress,Princeton,NJ. Hansen, L.P., W. Roberds, T.J. Sargent. 1991. Time Series Implications of Present Value Budget Balance andofMartingaleModelsandConsumptionandTaxes.inHansen,L.P.,T.J.Sargent(eds.),Rational ExpectationsEconometrics,WestviewPress,Boulder,CO. Harvey, A.C., Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge UniversityPress,Cambridge,England. Harvey,A.C.,T.M.Trimbur,H.K.vanDijk.2007.TrendsandCyclesinEconomicTimeSeries: ABayesian Approach.JournalofEconometrics 140,618–649. Hecq,A.,F.C.Palm,J-P.Urbain.2000.Permanent-TransitoryDecompositioninVARModelswithCointegrationandCommonCycles.OxfordBulletinofEconomicsandStatistics 62,511–532. Jeffreys,H.,1998.TheoryofProbability,OxfordUniversityPress,Oxford,UK. Johansen,S.1991.EstimationandHypothesisTestingofCointegrationVectorsinGaussianVectorAutoregressiveModels.Econometrica 59,1551–1580. Johansen, S. 1994. The Role of the Constant and Linear Terms in Cointegration Analysis and NonstationaryVariables.EconometricReviews 13,205–229. Kim,S.,N.Roubini.2000.ExchangeRateAnomaliesintheIndustrialCountries: ASolutionwithaStructuralVARApproach.JournalofMonetaryEconomics 45,561–586. Lubik,T.,F.Schorfheide.2006.ABayesianLookatNewOpenEconomyMacroeconomics.inGertler,M., K.Rogoff(eds.),NBERMacroeconomicsAnnual2005.MITPress,Cambrigde,MA. 31

MacKinnon,J.G.,A.A.Haug,L.Michelis.1999.NumericalDistributionsofLikelihoodRatioTestsofCointegration.JournalofAppliedEconometrics 14,563–577. Maheswaran,S.,C.A.Sims.1993.EmpiricalImplicationsofArbitrage-FreeAssetMarkets.inPhillips,P.C.B. (ed.),Models,MethodsandApplicationsofEconometrics,BlackwellPublishing,Oxford,UK. Mark, N. 1995. Exchange Rates and Fundamentals: Evidence on Long-Horizon Predictability. American EconomicsReview 85,201–218. Mark, N., D. Sul, 2003. Cointegration Vector Estimation by Panel DOLS and Long-Run Money Demand. OxfordBulletinofEconomicsandStatistics 65,665–680. Meese,R.A.,1986.TestingforBubblesinExchangeMarkets: ACaseofSparklingRates.JournalofPolitical Economy 94,345–373. Morley,J.C.,C.R.Nelson,E.Zivot.2003.WhyAreBeveridge-NelsonandUnobservedComponentDecompositionsofGDPSoDifferent? ReviewofEconomicsandStatistics 85,235–234. Osterwald-Lenum,M.1992.QuantilesoftheAsymptoticDistributionoftheMaximumLikelihoodCointegrationRankTestStatistics,OxfordBulletinofEconomicsandStatistics 54,461–472. Rabanal, P., J.F. Rubio-Ramírez. 2005. Comparing New Keynesian Models of the Business Cycle: A BayesianApproach.JournalofMonetaryEconomics 52,1151–1166. Rao,C.S.1973.LinearStatisticalInference,Wiley,Ltd.,NewYork,NY. Rogers, J.H., 2000. Monetary Shocks and Real Exchange Rates. Journal of International Economics 49, 269–288. Rogoff, K. 2007. Comments on: Engel, Mark, and West’s Exchange Rate Models Are Not as Bad as You Think.inAcemoglu,D.,K.Rogoff,M.Woodford(eds.),NBERMacroeconomicsAnnual2007,vol. 22,MITPress,Cambridge,MA. Rossi, B. 2007. Comments on: Engel, Mark, and West’s Exchange Rate Models Are Not as Bad as You Think.inAcemoglu,D.,K.Rogoff,M.Woodford(eds.),NBERMacroeconomicsAnnual2007,vol. 22,MITPress,Cambridge,MA. Scholl,A.,Uhlig,H.,2005.NewEvidenceonthePuzzles.ResultsfromAgnosticIdentificationonMonetary PolicyandExchangeRates.SFB649DiscussionPaperNo.2005–037. Sims,C.A.1980.MacroeconomicsandReality.Econometrica 48,1–48. Sims,C.A.,Stock,J.H.,M.W.Watson.1990.InferenceinLinearTimeSeriesModelswithSomeUnitRoots. Econometrica 58,113–144. Stock, J.H., M.W. Watson. 1988. Testing for Common Trends. Journal of the American Statistical Association 83,1097–1107. Vahid, F., R.F. Engle. 1993. Common Trends and Common Cycles. Journal of Applied Econometrics 8, 341–360. 32

Table 1: Summary of Standard PVM and DSGE-PVM Standard-PVM ECM(0): (13) e − 1−ω X = (1−ω) (cid:80)∞ ωj[E −E ]z . ∆ t ω t−1 j=0 t t−1 t+j EW Equation: (14) e = ζ (ω)υ + (1−ω) (cid:80)∞ ωjE z . ∆ t t j=0 t−1∆ t+j φ Parameters: ω ≡ = Discount Factor, 1+φ φ = Money Demand Interest Rate Semi-Elasticity, ψ = Money Demand Income Elasticity. Fundamentals: X = e −z , z = m −ψy , t t t t t t m = Cross Country Money, t y = Cross-Country Output. t DSGE-PVM (1−κ) ECM(0): (13) e − X ∆ t κ DSGE,t−1 (cid:110) (cid:111) = (1−κ) (cid:80)∞ κj [E −E ] m − c . j=0 t t−1 t+j t+j (cid:110) (cid:111) EW Equation: (14) e = (cid:80)∞ κj [E −E ] m − c ∆ t j=0 t t−1 ∆ t+j ∆ t+j (cid:110) (cid:111) + (1−κ) (cid:80)∞ κjE m − c . j=0 t ∆ t+j ∆ t+j 1 Parameters: κ ≡ = Discount Factor, 1+r ∗ r∗ = Steady State Real World Interest Rate. Fundamentals: X = e −m +c , DSGE,t t t t m = Cross-Country Money, t c = Cross-Country Consumption. t 33

Table 2: Summary of Propositions for Standard- and DSGE-PVMs Proposition 1: PVM Predicts Exchange Rate and Fundamentals Cointegrate; Campbell and Shiller (1987). Proposition 2: Currency Returns Are an ECM(0). Proposition 3: Exchange Rate Approximates a Martingale as B (cid:45)→ 1. Proposition 4: VECM(0) Imply Common Trend and Common Cycle for Exchange Rate and Fundamental. Proposition 5: EW’s (2005) Hypothesis Needs Currency Returns and Fundamental Growth Share a Co-Feature and B (cid:45)→ 1. 34

Table 3: Tests of Propositions 1, 3, and 5 Sample: 1976Q1 – 2004Q4 Canada Japan U.K. & U.S. & U.S. & U.S. Proposition 3: VECM(0) Levels VAR Lag Length 8 5 4 LR statistic p−value (0.02) (0.01) (0.09) Proposition 1: Common Trend Cointegration Tests Model Case 2∗ Case 1 Case 1 λ−Max statistic 4.86 0.20 2.27 17.28 4.64 12.32 Trace statistic 4.86 0.20 2.27 12.42 4.43 10.04 Proposition 5: Common Cycle Sq. Canonical Correlations 0.30 0.44 0.19 0.09 0.08 0.07 χ2−statistic p−value (0.01) (0.00) (0.00) (0.69) (0.21) (0.12) F−statistic p−value (0.00) (0.00) (0.00) (0.61) (0.19) (0.11) Theleveloffundamentalsequalscross-countrymoneynettedwithcross-countryoutputcalibratedtoa unitaryincomeelasticityofmoneydemand. Themoneystocks(outputs)aremeasuredincurrent(constant)localcurrencyunitsandpercapitaterms. Aconstantandlineartimetrendareincludedinthelevel VARs. The LR statistics employ the Sims (1980) correction and have standard asymptotic distribution according to results in Sims, Stock, and Watson (1990). The case 2∗ and case 1 model definitions are based on Osterwald-Lenum (1992). MacKinnon, Haug, and Michelis (1999) provide five percent critical values of 8.19 (8.19) and 18.11 (15.02) for the case 2∗ model λ−max (trace) tests and 3.84 (3.84) and 15.49(14.26)forthecase1model. Thecommonfeaturetestscomputethecanonicalcorrelationsof e ∆ t and m − y . The common feature null is all or a subset of the canonical correlations are zero. See ∆ t ∆ t EngleandIssler(1995)andVahidandEngle(1993)fordetails. 35

Table 4: UC Model Posterior Means, κ = 1 Parameter Priors UC UC UC 2,2,κ=1 2,m(cid:102),κ=1 2,c(cid:101),κ=1 α Normal −1.1906 −0.8691 − 1 [−1.2, 0.10] (0.0613) (0.0470) α Normal 0.4133 0.9501 − 2 [0.40, 0.17] (0.1092) (0.0528) θ Normal 0.9407 − 0.9830 1 [0.85, 0.10] (0.0491) (0.0296) θ Normal 0.0403 − 0.0069 2 [0.10, 0.15] (0.0488) (0.0291) µ∗ Normal −0.1260 −0.1258 −0.1258 [−0.126, 0.015] (0.0150) (0.0150) (0.0120) a∗ Normal 0.1615 0.1645 0.1571 [0.158, 0.025] (0.0229) (0.0213) (0.0199) σ Inv-Gamma 1.8838 2.4629 1.6784 µ [2.0, 1.5] (0.1436) (0.1507) (0.1264) σ Inv-Gamma 1.0471 0.3971 1.9461 a [2.0, 0.4] (0.2206) (0.0345) (0.3831) σ Inv-Gamma 0.6002 0.4728 − m(cid:102) [2.0, 0.6] (0.0899) (0.1168) σ Inv-Gamma 1.2874 − 2.0135 c(cid:101) [2.0, 0.7] (0.2354) (0.3823) (cid:37) Normal −0.8758 − −0.9475 a,c(cid:101) [−0.5, 0.2] (0.0462) (0.0234) π Normal 80.1528 138.8984 62.9442 e,0 [100.0, 15.0] (6.8283) (5.7281) (2.9991) π Normal 0.7038 1.9366 0.3831 e,t [1.0, 0.5] (0.1710) (0.1106) (0.1306) π Uniform −2.6822 −9.0223 −0.7208 e,a [−10.0, 0.0] (0.6316) (0.5377) (0.1886) π Uniform − 4.3973 − c,m(cid:102) [−2.0, 7.5] (1.1057) π Uniform − − −0.8985 m,c(cid:101) [−7.5, 2.0] (0.2099) lnL(cid:98) −53.95 −226.76 −24.76 36

Table 5: UC Model Posterior Means, κ ∈ [0.9, 0.999] Parameter Priors UC UC UC 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ κ Inv-Gamma 0.9658 0.9738 0.9962 [0.988, 0.038] (0.0219) (0.0196) (0.0046) α Normal −1.1892 −0.8828 − 1 [−1.20, 0.10] (0.0673) (0.0369) α Normal 0.4131 0.8465 − 2 [0.40, 0.17] (0.1179) (0.0223) θ Normal 0.9396 − 0.9799 1 [0.85, 0.10] (0.0431) (0.0315) θ Normal 0.0421 − 0.0018 2 [0.10, 0.15] (0.0422) (0.0314) µ∗ Normal −0.1256 −0.1260 −0.1260 [−0.126, 0.015] (0.0148) (0.0149) (0.0147) a∗ Normal 0.1621 0.1630 0.1578 [0.158, 0.025] (0.0227) (0.0205) (0.0243) σ Inv-Gamma 1.8914 2.4241 1.7188 µ [2.0, 1.5] (0.1484) (0.1663) (0.1112) σ Inv-Gamma 1.0842 0.4043 1.5738 a [2.0, 0.4] (0.2477) (0.0378) (0.2727) σ Inv-Gamma 0.6068 0.5752 − m(cid:102) [2.0, 0.6] (0.0865) (0.0975) σ Inv-Gamma 1.3828 − 1.6742 c(cid:101) [2.0, 0.7] (0.2346) (0.2772) (cid:37) Normal −0.8990 − −0.9256 a,c(cid:101) [−0.5, 0.2] (0.0409) (0.0291) π Normal 85.2271 135.7241 67.8714 e,0 [100.0, 15.0] (6.7620) (6.3363) (3.7506) π Normal 0.7955 1.9076 0.4526 e,t [1.0, 0.5] (0.1774) (0.1211) (0.1112) π Uniform −3.2825 −8.8023 −1.2605 e,a [−10.0, 0.0] (0.6426) (0.5900) (0.3087) π Uniform − 4.4186 − c,m(cid:102) [−2.0, 7.5] (0.7952) π Uniform − − −1.0380 m,c(cid:101) [−7.5, 2.0] (0.2179) lnL(cid:98) −53.94 −253.03 −29.88 37

Table 6: UC Model Posterior Means, κ ∈ [0.9, 0.999], Factor Loadings on Money and Consumption Cycles Parameter UC UC UC 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ (1−π )δ 0.0086 −0.0841 − c,m(cid:102) m(cid:102),0 (0.0096) (0.0676) (1−π )δ −0.0269 0.0064 − c,m(cid:102) m(cid:102),1 (0.0188) (0.0082) (1−π )δ 0.0143 −0.0762 − c,m(cid:102) m(cid:102),2 (0.0108) (0.0624) (1−π ) δ −0.0040 −0.1542 − c,m(cid:102) Σi m(cid:102),i (0.0202) (0.1237) (1−π )δ −0.6044 − −0.3252 m,c(cid:101) c(cid:101),0 (0.2042) (0.2052) (1−π )δ −0.0238 − −0.0004 m,c(cid:101) c(cid:101),1 (0.0261) (0.0111) (1−π ) δ −0.6283 − −0.3256 m,c(cid:101) Σi c(cid:101),i (0.2117) (0.2053) †Thefactorloadingsπ andπ equalzerofortheUC model. c,m(cid:102) m,c(cid:101) 2,2,κ Table 7: UC Model Posterior Means, κ ∈ [0.9, 0.999], Variance(PDV–ε) / Variance( e) ∆ Parameter UC UC UC 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ Var(PDV −ε )/Var( e) 0.92 1.48 0.71 µ ∆ (0.15) (0.22) (0.09) Var(PDV −ε )/Var( e) 3.04 0.36 0.96 a ∆ (0.73) (0.06) (0.53) Var(PDV −ε )/Var( e) 0.00 0.00 − m(cid:102) ∆ (0.00) (0.00) Var(PDV −ε )/Var( e) 0.22 − 0.12 c(cid:101) ∆ (0.16) (0.22) 38

Table 8: UC-Models, κ ∈ [0.9, 0.999], Exchange Rate FEVDs† UC Model 2,2,κ Forecast Horizon ε ε ε µ A c(cid:101) 1 0.23 0.77 0.00 4 0.23 0.77 0.00 12 0.21 0.79 0.00 20 0.19 0.80 0.01 40 0.15 0.82 0.03 UC Model 2,m(cid:102),κ Forecast Horizon ε ε ε µ A c(cid:101) 1 0.32 0.68 − 4 0.32 0.68 − 12 0.32 0.68 − 20 0.32 0.68 − 40 0.32 0.68 − UC Model 2,c(cid:101),κ Forecast Horizon ε ε ε µ A c(cid:101) 1 0.59 0.40 0.01 4 0.58 0.41 0.01 12 0.57 0.42 0.01 20 0.57 0.43 0.01 40 0.55 0.45 0.01 †The summary statistics are the means of the ensemble of FEVDs with respect to permanent and transitory Canadian-U.S. money differential and Canadian-U.S. consumption differential shocks generated fromtheUC ,UC ,andUC modelposteriordistributions. 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ 39

Table 9: UC-Models, κ ∈ [0.9, 0.999], Summary of the Trend-Cycle Decomposition† Parameter UC UC UC 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ STD( eτ) 2.66 104.28 2.44 ∆ STD(e) 3.68 106.84 2.55 (cid:101) AR1(e) 0.97 0.55 0.98 (cid:101) Corr( eτ, e) −0.06 −0.37 −0.02 ∆ (cid:101) STD( µ) 1.62 4.54 1.71 ∆ STD(m) 0.68 3.03 − (cid:102) AR1(m) −0.68 0.13 − (cid:102) Corr( µ, m) 0.26 −0.24 − ∆ (cid:102) STD( a) 0.99 12.18 1.56 ∆ STD(c) 5.73 − 7.73 (cid:101) AR1(c) 0.97 − 0.98 (cid:101) Corr( a, c) −0.16 − −0.14 ∆ (cid:101) Corr( eτ, µ) 0.01 −0.40 0.62 ∆ ∆ Corr( eτ, a) −0.85 −0.93 −0.71 ∆ ∆ Corr( µ, a) 0.52 0.55 0.10 ∆ ∆ Corr(e, m) 0.04 −0.73 − (cid:101) (cid:102) Corr(e, c) −1.00 − −1.00 (cid:101) (cid:101) †The summary statistics are the means of the ensemble of CDN$/US$ exchange rate, Canadian-U.S. money differential, and Canadian-U.S. consumption differential trends and cycles generated from the UC ,UC ,andUC modelposteriordistributions. 2,2,κ 2,m(cid:102),κ 2,c(cid:101),κ 40

Figure 1: Prior and Posterior PDFs of DSGE-PVM Discount Factor 50 Prior of DSGE-PVM Discount Factor Posterior 2-Trend, 2-Cycle Model Posterior 2-Trend, M-Cycle Model 45 Posterior 2-Trend, C-Cycle Model (PDF deflated by 0.1) 40 35 30 25 20 15 10 5 0 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 .0

Figure 2: CDN$/US$ Exchange Rate Trend and Cycle, 1976Q1 - 2004Q4 60 ln[CDN$/US$ Ex Rate] 50 2-Trend, 2-Cycle Trend 2-Trend, C-Cycle Trend 40 30 20 10 0 -10 1975 1979 1983 1987 1991 1995 1999 2003 2006 8 6 Ex Rate Cycle from 2-Trend, 2-Cycle Model 4 2 0 -2 -4 Ex Rate Cycle from 2-Trend, C-Cycle Model -6 -8 1975 1979 1983 1987 1991 1995 1999 2003 2006

Figure 3: CDN-US Money, Consumption Trends and Cycles, 1976Q1 - 2004Q4 -20 40 35 -30 30 -40 25 -50 20 15 -60 10 ln[CDN-US Money Stock] ln[CDN-US Consumption] -70 2-Trend, 2-Cycle Model 5 2-Trend, 2-Cycle Model 2-Trend, C-Cycle Model 2-Trend, C-Cycle Model -80 0 1975 1979 1983 1987 1991 1995 1999 2003 2006 1975 1979 1983 1987 1991 1995 1999 2003 2006 2 20 1.5 Money Cycle from 15 Consumption Cycle from 2-Trend, 2-Cycle Model 2-Trend, C-Cycle Model 1 10 0.5 5 0 0 -0.5 -5 -1 -10 Consumption Cycle from 2-Trend, 2-Cycle Model -1.5 -15 -2 -20 1975 1979 1983 1987 1991 1995 1999 2003 2006 1975 1979 1983 1987 1991 1995 1999 2003 2006

Figure 4: CDN$/US$ Ex Rate Cycles at Different DSGE-PVM Discount Factors 11 Discount Factor, 16th Percentile 2-Trend, 2-Cycle UC Model = 0.943 8 5 Discount Factor, 84th Percentile = 0.988 2 -1 -4 Discount Factor = 0.999 -7 -10 1975 1979 1983 1987 1991 1995 1999 2003 2006 11 Discount Factor, 16th Percentile 2-Trend, C-Cycle UC Model 8 = 0.994 5 Discount Factor, 84th Percentile = 0.9987 2 -1 -4 Discount Factor = 0.999 -7 -10 1975 1979 1983 1987 1991 1995 1999 2003 2006