Moving Endpoints and the Internal Consistency of Agents' Ex Ante Forecasts

MOVINGENDPOINTSANDTHE INTERNALCONSISTENCY OFAGENTS' EX ANTEFORECASTS SharonKozickiandP.A.Tinsley (cid:3) Version: November1996 Abstract: Forecasts by rational agents contain embedded initial and terminal boundary conditions. Standard timeseriesmodelsgeneratetwotypesoflong-run“endpoints”—fixedendpointsandmovingaverageendpoints. Neither can explain the shifting endpoints implied by postwar movements in the cross-section of forward rate forecasts in the term structure or by post-1979 changes in survey estimates of expected inflation. Multiperiod forecastsbyabroaderclassof“movingendpoint”timeseriesmodelsprovidesubstantiallyimprovedtrackingof the historical term structure and generally support theinternal consistency ofthe ex ante long-run expectations ofbondtradersandsurveyrespondents. Keywords: Boundaryvalues, expectedinflation, termstructure. (cid:3) Authors' addresses are: Federal Reserve Bank of Kansas City, 925 Grand Boulevard, Kansas City MO 64198, skozicki@frbkc.org;andFederalReserveBoard,Washington,D.C.20551,ptinsley@frb.gov.Theviewsexpressedherein aresolelythoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheFederalReserveBankofKansasCityorthe BoardofGovernorsoftheFederalReserveSystem. ThispaperisforthcominginComputationalEconomics.

1 1. Introduction One of the most neglected topics of empirical modeling in macroeconomics and finance is the specificationofterminalboundaryvalues. Inmodelsofrationalbehavior,agentforecastsaresolutions of first-order conditions for intertemporal optimization, subject to split boundary conditions at the beginningandtheendoftheforecasthorizon. Thispaperdemonstratesthatconventionalspecifications of long-run boundary values or endpoints are inconsistent with the endpoints implied by ex ante multiperiod forecasts of economic agents, such as bond traders and households. An alternative formulation of moving endpoints is proposed to account for the shifting perceptions by agents of predictedsteady-stateoutcomes. Agent forecasts of interest rates and inflation rates are selected in this paper to illustrate the significant consequences of endpoint specifications, even on medium-term forecasts. In the case of interestrates,thetermstructureisacross-sectionofmarketforecasts ofmultiperiodoutcomes. Under the expectations model of the term structure, the yield on each bond maturity is linked to an implicit endpointof forward interest rates. It is a well-knownproblem infinance that conventionaltimeseries models of interest rates are generally not capable of reproducing the term structure at a giveninstant. Likewise, empirical models that fit the term structure in one period do not track shifts in the term structure in subsequent periods. Empirical examples in this paper indicate that much of the problem appears to be associated with the long-run endpoints that are implicitly imposed by conventional time series models of the representative short-term interest rate. Neither mean-reverting nor unit-root time series models generate cross-sections of bond rate predictions that are internally consistent with market-generatedterm structures. The Fisher equation is another area where an internal consistency is assumed among agent forecasts, in this instance among ex ante interest rates and expected inflation. Empirical examples of inflation models will illustrate that time series models with conventional endpoint formulations do notsupportacorrespondencebetweenlong-runinflationforecastsandlong-maturitybondrates. Also, the long-run inflation forecasts of these models are inconsistent with available survey evidence on long-runinflationexpectations. Themovingendpointformulationoutlinedinthispaperappears toresolvetheempiricalproblems notedabove,providingbothimprovedcross-sectionpredictionsofthetermstructureasitevolvesover time and long-run inflation forecasts that mimic survey estimates of expected long-run inflation. In contrasttothemovingaveragecharacterizationofthepermanent effect ofshocksinunit-rootmodels, movingendpoint time series models are an alternativeway to specify persistent change in time series where the expected “permanent” trend component or steady-state outcome of a variable is constant over any given forecast horizon but may shift, either rapidly or infrequently, over time. As noted later, there are a number of ways to measure moving endpoints, ranging from statistical estimation

2 of changepoints to extraction of time-varying endpoints implicitly embedded in agent forecasts. The empirical analysis in this paper features examples of implicit endpoint estimates in order to illustrate theinternalconsistencyofagents' forecastsofbondratesandexpectedinflation,onceaccountistaken ofshiftsinagents' perceptionsoflong-runoutcomesorendpoints. Alternative endpoint concepts are derived and compared in the remaining sections of the paper. Implicationsof conventionalcharacterizations of long-run behavior inforward-looking models of the term structure are discussed in section 2. Bond rate forecasts from an autoregressive model of the one-month Treasury rate are compared for constant and moving-average endpoints. Both endpoint models are shown to provide poor descriptions of the historical evolution of the term structure. Alternative moving endpoint models of interest rates and inflation are developed in section 3. The movingendpointmodelsprovideimprovedpredictionsofacross-sectionofbondratesinthehistorical term structure and also illustrate the ex ante consistency of the model forecasts of inflation with observedbondrates andwithsurveyestimatesofexpectedlong-runinflation. Section4concludes. 2. AutoregressiveModels withConventional Endpoints This section illustrates the strong effects on long-horizon forecasts of the implicit terminal endpoints defined by conventional time series models. The first subsection explores the fixed endpoints associatedwith mean-revertingautoregression(AR) models of timeseries, and the secondsubsection indicates the weighted moving average property of endpoints defined by unit-root AR models. Both models are used to generate predictions of a cross-section of bond rates in the term structure. The linearizedexpectationsmodelofthetermstructureisaparticularlyusefulempiricalexamplebecause, under the assumption of constant term premia, long-horizon forecasts of the short-term interest rate are directly translated to predicted movements in bond rates. The cross-section of poor bond rate fits generated by both the mean-reverting and unit-root AR models of the short-term interest rate are interpretedinthecontextoftheforecast boundaryvaluesorendpointsimpliedbytheseARmodels. 2.1AnExpectationsModeloftheTermStructurewithConstantEndpoints Forcoupon-bearingbonds,thelinearized expectationsmodeloftheterm structureis E (cid:0)t 1 f r n ;t g (cid:24) = D (cid:0) n 1 (cid:0) n X i= 1 0 (cid:12) i E (cid:0)t 1 f r t+ i g + b n ; ( 1 ) where E (cid:0)t 1 f : g denotes expectations based on information available at the end of t (cid:0) 1 ; r n ;t denotes the nominal yield to maturity of the n -period bond; r t is the one-monthbond rate; D n is the duration associated with an n -period coupon bond;1 and b n is the constant term premium associated with the 1 D n (cid:17) ( 1 (cid:0) (cid:12) n ) = ( 1 (cid:0) (cid:12) ) ,where (cid:12) isthediscountfactor,seediscussioninShiller,Campbell,andSchoenholtz(1983)

3 n -period bond. The discount factor, (cid:12) , is also maturity-specific, where ( 1 (cid:0) (cid:12) ) = (cid:12) equals the coupon rateofthe n -periodbond.2 As noted by Campbell (1986) and Shea (1992), equation 1 is consistent with a variety of asset pricing theories, ranging from simple no-arbitrage conditions to solutions of optimal intertemporal portfolio conditions. An enormous empirical literature tests for the validity of various implications of equation1, with mostlynegativeresults.3 By contrast, we will assume the expectations hypothesis is true (indeed, apart from more elaborate specifications of term premia, we are unable to imagine a useful alternative interpretation of the term structure), and explore the performances of bond rate predictionsbyalternativeautoregressionmodelsoftheshort-terminterestrate, r t . Most models of the term structure in the finance literature, such as Cox, Ingersoll, and Ross (1985), are based on the assumption that the short rate, r t , is a mean-reverting stochastic process.4 A discrete-time, mean-reverting ( p + 1 ) -order autoregression in the short rate, r t , can be represented as (cid:1) r t = a 0 + (cid:13) r (cid:0)t 1 + A ( L ) (cid:1) r (cid:0)t 1 + e t ; ( 2 ) where e t isawhitenoiseinnovation, (cid:13) isthecoefficientofthelaggedleveloftheshortrate,5 and A ( L ) denotesa ( p (cid:0) 1 ) -orderpolynomialinthelagoperator, L , where A ( L ) = a 1 + a 2 L + : : : + a p L p (cid:0) 1 . A tractable description of agent expectations generated by the autoregression in equation 2 is providedbythefirst-ordercompanionsystem, E (cid:0)t 1 f z t g = H z (cid:0)t 1 + h ; ( 3 ) wherethe ( p + 1 ) -vector, z (cid:0)t 1 ,is aconvenientsummaryoftheagents' laggedinformationset, [ r (cid:0)t 1 ; r (cid:0)t 2 ; : : : ; r (cid:0)t p (cid:0) 1 ] 0 z (cid:0)t 1 (cid:17) ; h is a ( p + 1 ) -vector containing the intercept of the autoregression, [ a 0 ; 0 ; : : : ; 0 ] 0 h = ; and H denotes the ( p + 1 ) (cid:2) ( p + 1 ) companion matrix of the agents' AR forecast andShiller(1990). 2Thesampleaverageoftheparbondrateisusedasanestimateoftheaveragecouponrate;seediscussionandreferences inShiller(1990). 3Representative studies of the expectations hypothesis of the term structure include Sargent (1979), Shiller(1979), Shiller, Campbell, and Shoenholtz (1983), Fama (1984), Mankiw and Miron (1986), Campbell (1986), Fama and Bliss (1987),Hardouvelis(1988),andRudebusch(1995).Rejectionsareofcompositehypothesesthatincludespecificstochastic processesforforcingterms. 4Although the short rate model is typically ergodicfor the mean, specifications in finance often link the conditional variancetorecentlevelsofinterestratessotheinterestratemodelsaregenerallynotcovariance-stationaryprocesses. 5Theshortrateautoregressionwillbestationaryandexhibitmean-reversionif 1j + (cid:13) j < 1 .

4 model, H (cid:17) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 + (cid:13) 1 0 + a 1 a 2 (cid:0) 0 1 a 1 : : : : : : : : : a p (cid:0) 0 0 a p (cid:0) 1 (cid:0) a 0 0 p . . . . . . . . . . . . . . . 0 0 : : : 1 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : The first-order structure of the forecast model in equation 3 facilitates a compact formulation of multiperiodforecasts oftheshort rate. E (cid:0)t 1 f r t+ k g = = (cid:19) (cid:19) 0 1 0 1 H H k k + + 1 1 z z (cid:0)t (cid:0)t 1 1 + + (cid:19) (cid:19) 0 1 0 1 ( ( I I p p + + 1 1 + (cid:0) H H + k + H 1 ) ( 2 I + p + : 1 : (cid:0) : + H H ) (cid:0) k 1 ) h h ; ; ( 4 ) where (cid:19) 1 isa selector ( p + 1 ) -vectorthat containsaoneinthefirstelement andzeroes elsewhere. Empirical data on par bond yields are drawn from McCulloch and Kwon (1993). This dataset provides monthlyobservations on yields for bonds of various maturities from one month to 40 years, but is incomplete due to some missingobservations. The estimatedmodels in this paper use monthly yield data from 1960m1-1991m2. Time series sequences of cross-section predictions of the term structurearepresentedlaterfor3-and12-monthand5-and10-yearmaturities. Theone-monthrateis usedastheshortrate, r t . The first column in table I, labeled Constant, indicates the coefficient estimates and summary statistics of the autoregression description of the one-month rate, r t , provided by equation 2, using twelve autoregressive lags in the first-differences of the short rate, p = 1 2 ; results in this paper are not sensitiveto variations in the choice of p . The point estimate of the coefficient of the lagged level, (cid:13) = (cid:0) : 0 2 7 ,isconsistentwithmean-reversionoftheshortrateprocess. Two examples of multiperiod forecasts of the short rate provided by the mean-reverting autoregression in equation 2 are shown in the top panel of figure 1. The first multiperiod forecast sequence beginsin1970m12,wheninterest rates were near thesamplemean,andthe secondforecast sequencebeginsin1980m12,aboutayearafterthedramaticriseinratesduetothelate1979alteration in monetary policy. Despite the rather different policy contexts, both multiperiod forecasts converge tothesamelong-runforecastorendpointbecausethelimitinglong-runforecast ofthemean-reverting AR model is independent of the initial information set. As represented by equation 4, the limiting

5 forecast orendpointoftheshortrate isaconstant, lim k ! 1 E (cid:0)t 1 f r t+ k g = = = (cid:19) (cid:19) (cid:22)r 0 1 0 1 1 ( (cid:22)z I 1 ; p + ; 1 (cid:0) H ) (cid:0) 1 h ; ( 5 ) andequaltothesamplemeaninlengthysamples. The speed of convergence of forward short rate forecasts to the short rate endpoint, (cid:22)r 1 , can be gaugedbyrewritingequation2intheendpointdeviationformat ~r t = ( 1 + (cid:13) ) ~r (cid:0)t 1 + A ( L ) (cid:1) ~r (cid:0)t 1 + e t ; ( 6 ) where ~r t denotes the deviation of the short rate from the endpoint, ~r t = r t (cid:0) (cid:22)r 1 . According to the parameter estimates in the first column of table I, the mean lag of an endpoint deviation is about 40 months.6 Thus, by the fifth or sixth year of the forecast horizon, forward short rate predictions are largelydeterminedbytheshortrateendpoint, (cid:22)r 1 ,as showninthetoppanel offigure1. To illustrate the effects of the constant short rate endpoint on bond rate predictions, we substitute multiperiodshortratepredictionsfromequation 4 intotheexpectationsmodelofthetermstructurein equation 1 . This yields the following closed form solution of the term structure for a mean-reverting shortrate,7 E (cid:0)t 1 f r n ;t g = = = = D D 0 (cid:19) 1 + 0 (cid:19) 1 (cid:0) n (cid:0) n ( (cid:22)z (cid:0) n 1 1 ( (cid:12) X i= 0 (cid:0) n 1 0 1 (cid:19) [ 1 X i= 0 I (cid:0) p + 1 0 (cid:0) 1 D (cid:19) ( n 1 1 + b n i (cid:12) H I + E p (cid:0)t i [ H (cid:0) ) + 1 D f 1 i+ 1 h (cid:0) (cid:0) 1 n r g ) t i + 1 (cid:0) z t 1 + b n ( (cid:12) H ) 0 (cid:19) ( I p + 1 + + n ) 1 b ; n ( I p + ( I p + (cid:0) ( (cid:12) 1 1 H (cid:0) (cid:0) ) H (cid:12) n ) H ( i+ ) I p 1 (cid:0) + ) ( I 1 H (cid:0) 1 p + [ z (cid:12) 1 (cid:0)t H (cid:0) 1 ) H (cid:0) (cid:0) 1 ) ( H (cid:0) I 1 h p + [ z ] 1 (cid:0)t ] (cid:0) 1 + (cid:0) H b n ) (cid:22)z ; (cid:0) 1 1 ] h : ] ; ( 7 ) 6Fortheautoregressionequation 2 ,themeanlag(inmonths) = (cid:0) ( 1 + (cid:13) (cid:0) A ( 1 ) ) = (cid:13) . 7Estimates ofthetermpremium, b n , arebasedonsample averagesof theexcessreturntoan n -periodbondoverthe one-monthbondyield.

6 This compact formulation of the predicted term structure is applied in each month of the sample, 1960m1-1991m2, to generate a sequence of monthly forecasts for each of four yields in the term structure: 3- and 12-month and 5- and 10-year maturities. The influence of the constant endpoint on bondratepredictionsbythemean-revertingARmodelisshowninfigure2. Thepanelsinfigure2plot concatenated time series of monthly bond rate predictions, along with plots of historical bond rates. Thus, for any month, the four panels show a cross-section of term structure predictions for the four bond rate maturities. Although the dampening influence of the constant endpoint, (cid:22)r 1 , is noticeable even for the 12-month bond rate, the excessively smooth predictions of the constant-endpoint AR model are quite evident for the longer maturities. Predictions of long-horizon yields are based on long-horizon forecasts of short rates, in addition to shorter horizon forecasts. Because long-horizon predictions of short rates converge to the constant endpoint, predictions of long-maturityyields place alargerweightontheconstantendpointthanpredictionsofshort-maturityyieldsand,thus,variations inpredictionsare moremutedforlongermaturities. 2.2BondRatePredictionsunderMovingAverageEndpoints In contrast to the mean-reverting model of a representative short-term interest rate that is generally assumedinfinance,recentstudiesofthetermstructureinmacrofinance,suchasCampbellandShiller (1987), Choi and Wohar (1991), Hall, Anderson, and Granger (1992), Mougoue (1992), and Shea (1992),arepredicatedontheassumptionthatarepresentativenominalinterestratecontainsaunitroot. Indeed, because the format of equation 2 is thesame as that required for an augmented Dickey-Fuller (ADF) test for a unit root, the first column of table I indicates that the magnitude of the t -statistic associated with (cid:13) is below the critical value ( 2 : 5 6 for a p -value of 1 0 % ) that would be required to reject thehypothesisthat theshort rate, r t ,containsaunitroot. The second column of table I, under the heading Moving Average, contains the estimated parameters ofthedifferencedshortratemodel (cid:1) r t = a 0 + A ( L ) (cid:1) r (cid:0)t 1 + e t ; ( 8 ) where the coefficient of the lagged level of the short rate, (cid:13) , is restricted to zero. As indicated by the rootmeansquarederror(RMSE)showninthesecondcolumnoftableI,thedeteriorationinempirical fit duetothiszerorestrictionis negligible. Inthecaseoftheunit-rootARmodelinequation 8 ,itisconvenienttoslightlyalterthecompanion form oftheARforecast modeltothedifferencedformat E (cid:0)t 1 f (cid:1) z t g = H (cid:1) (cid:1) z (cid:0)t 1 + h ; ( 9 )

7 wherealldefinitionsremainthesameexceptforthe ( p + 1 ) (cid:2) ( p + 1 ) companionmatrix,whichisnow definedby H (cid:1) (cid:17) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 a 1 1 a 0 2 : : : : : : a (cid:0) p 0 1 a 0 p 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 0 : : : : : : 1 0 0 1 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : Usingthisforecast model,multiperiodforecasts oftheshortrateare nowgenerated by E (cid:0)t 1 f r t+ k g = = = = (cid:0) r t 0 (cid:19) z 1 0 (cid:19) z 1 0 (cid:19) z 1 + 1 (cid:0)t (cid:0)t (cid:0)t ( k + 1 1 1 + k X i= + + + 1 E 0 k X i= 0 k X i= 0 [ I p + 0 ) (cid:19) 1 (cid:0)t 0 (cid:19) 1 0 (cid:19) 1 1 [ I f 1 E H (cid:0) p + (cid:1) (cid:0)t i+ (cid:1) H 1 r t+ f (cid:1) 1 1 [ (cid:1) k + (cid:1) (cid:0) H g i z z 1 ] (cid:1) ; g t i + (cid:0)t 1 [ I p + (cid:0) 1 ] ; (cid:0) 1 h [ (cid:0) ; I p H + 1 (cid:1) (cid:0) ] (cid:0) H 1 H (cid:1) (cid:1) (cid:0) ] [ (cid:1) 1 h z ] (cid:0)t + 1 ( (cid:0) k + [ I p 1 + ) 1 0 (cid:19) 1 (cid:0) [ I H p + (cid:1) 1 ] (cid:0) (cid:0) 1 H h ] (cid:1) ] (cid:0) 1 h ; ( 1 0 ) where,asbefore, h isa ( p + 1 ) -vectorcontainingtheinterceptoftheautoregression, h = [ a 0 ; 0 ; : : : ; 0 ] 0 . Examples of multiperiod forecasts of the short rate provided by the unit-root autoregression in equation 8 are shown in the bottom panel of figure 1. Once again, the first multiperiod forecast sequence beginsin1970m12,andthesecondforecast sequence beginsin1980m12. Incontrast tothe forecastssubjecttotheconstantendpointinthetoppanel,eachforecastnowremainsinaneighborhood of the short rate in the months just prior to the start of each forecast. This is because the effective endpointoftheunit-rootARmodelisaweightedmovingaverage ofrecent shortrates. Partly to develop a well-defined endpoint for the unit-root autoregression model and partly to condense notation, we will assume h is zero in the remainder of the discussion.8 Thus, dropping 8The analoguesof subsequent formulaefor the case of nonzerodrift, h , are available from the authors. A non-zero interceptinaunit-rootARimpliesthatexpectedfutureshortrateswilleventuallyincrease( a 0 > 0 )ordecrease( a 0 < 0 ) withoutlimit. Infact,asnotedinthesecondcolumnoftableI,theestimatedinterceptisinsignificantlydifferentfromzero, andsimilarbondratepredictionsareobtainedusingthepointestimateoftheconstantinsteadofzero.

8 the intercept and taking the limit of the last line in equation 1 0 indicates that the endpoint for the unit-rootautoregressionis a ( p + 1 ) -orderweightedmovingaverageofrecent shortrates, E (cid:0)t 1 f r m 1 a g . lim k ! 1 E (cid:0)t 1 f r t+ k g = = (cid:19) E 0 1 (cid:0) z t (cid:0)t 1 1 f + m r 1 (cid:19) a 0 1 g [ I : p + 1 (cid:0) H (cid:1) ] (cid:0) 1 H (cid:1) (cid:1) z (cid:0)t 1 ; ( 1 1 ) It is noteworthy that the moving average endpoint, E (cid:0)t 1 f r m 1 a g , of a unit-root autoregression is precisely the “permanent” component of a unit-root stochastic process suggested by Beveridge and Nelson(1981). Theseauthors startwithamovingaverageformulationofa unit-rootprocess,suchas (cid:1) r t = = 1 X i= (cid:30) 0 ( L (cid:30) ) (cid:0) e i t e ; t i ( 1 2 ) where the (cid:30) i arethemovingaverageweights. TheanalogueoftheBeveridge-Nelsondefinitionofthe permanentcomponentofaninterestratethat containsaunitrootis E (cid:0)t 1 f r b t n g (cid:17) r (cid:0)t 1 + ( 1 X i= 1 (cid:30) i ) e (cid:0)t 1 + ( 1 X i= 2 (cid:30) i ) e (cid:0)t 2 + : : : ( 1 3 ) To demonstrate equivalence of this “permanent” component to the moving average endpoint, we convert to vector notation, letting (cid:15) denote the ( p + 1 ) -vector, (cid:15) t = [ e t ; 0 ; : : : ; 0 ] 0 . Thus, using the unit-root autoregressive model of equation 8, the short rate can be expressed in the moving average format (cid:1) r t = = = (cid:19) (cid:19) (cid:30) 0 ( I 1 0 [ (cid:15) 1 ( L p t ) + + e 1 t (cid:0) H : (cid:1) H (cid:15) (cid:1) (cid:0)t L 1 ) + (cid:0) 1 H (cid:15) ; t 2 (cid:1) (cid:15) (cid:0)t 2 + : : : ] ; ( 1 4 ) Substituting from equation 1 4 into equation 1 3 , demonstrates that the Beveridge and Nelson (1981) definitionofthe“permanent”componentoftheshortrateisequivalenttothemovingaverageendpoint E (cid:0)t 1 f r b t n g = (cid:19) 0 1 z (cid:0)t 1 + ( 1 X i= 1 (cid:19) 0 1 H i (cid:1) ) (cid:15) (cid:0)t 1 + ( 1 X i= 2 (cid:19) 0 1 H i (cid:1) ) (cid:15) (cid:0)t 2 + : : : ; ( 1 5 )

9 = = = (cid:19) (cid:19) E 0 1 0 1 (cid:0) z t (cid:0) z t (cid:0)t 1 1 1 f + + r m 1 (cid:19) (cid:19) a 0 1 0 1 g ( ( ; I I p p + + 1 1 (cid:0) (cid:0) H H (cid:1) (cid:1) ) ) (cid:0) (cid:0) 1 1 [ H H (cid:1) (cid:1) (cid:1) (cid:15) (cid:0)t z 1 (cid:0)t + 1 ; H 2 (cid:1) (cid:15) (cid:0)t 2 + : : : ; Returningtotheissueofthecross-sectionofbondratepredictionsprovidedbythemovingaverage endpoint, we substitute multiperiod short rate predictions from the unit-root AR forecast model, equation 1 0 , into the expectations model of the term structure in equation 1 . This gives the closed form expressionofthetermstructureforaunit-rootAR modeloftheshortrate as E (cid:0)t 1 f r n ;t g = = = = D D 0 (cid:19) 1 (cid:0) 0 (cid:19) 1 (cid:0) n 1 (cid:0) i 1 (cid:0) (cid:12) E t n X i= 0 (cid:0) n 1 0 (cid:0) i 1 (cid:19) [ (cid:12) [ z n 1 X i= 0 (cid:0) [ z + ( I t p 1 + 1 0 (cid:0) 1 D (cid:19) ( I p + 1 n 1 m a (cid:0) E f z g + 1 t 1 f 1 (cid:0)t (cid:0) (cid:0) b r 1 n t+ + H ( (cid:12) + g + i ( I p + (cid:0) 1 ) (cid:1) H ) (cid:1) (cid:0) D n b ; n (cid:0) 1 H (cid:1) n ) ( I 0 1 (cid:19) ( 1 H (cid:1) p + I p i+ (cid:1) (cid:0) z t (cid:0) 1 + 1 1 ) ( ] 1 (cid:12) (cid:0) I p + H ( (cid:12) + b (cid:1) H 1 n ) (cid:1) (cid:0) (cid:0) ) 1 n H ) (cid:1) ( I p + ) ( I p (cid:0) 1 + 1 (cid:0) 1 H (cid:0) (cid:1) H (cid:1) (cid:1) (cid:12) H (cid:0) z t 1 (cid:0) 1 ) H (cid:0) ) (cid:1) ] ] (cid:1) 1 + (cid:1) [ z b z (cid:0)t ; n (cid:0)t 1 (cid:0) 1 ; E (cid:0)t 1 f z m 1 a g ( ] 1 : 6 ) The concatenated time series of the four bond rates predicted by the moving average endpoints model are displayed in the four panels of figure 3. Although the 3-month predictions of the moving averageendpointmodelinfigure3resemblethosegeneratedbytheconstantendpointmodelinfigure 2, the long-maturity predictions differ markedly. The similarity of near-term predictions, such as the 3-month rates, is foreshadowed in table I where the summary statistics such as R 2 and RMSE are similar for both the mean-reverting and unit-root autoregressions, suggesting little noticeable differences inshort-horizonpredictionsbythetwoAR models. In contrast to the excessively damped bond rate predictions in figure 2, the problem in figure 3 is thatthecross-sectionsofpredictedbondratesgeneratedbytheunit-rootautoregressionareexcessively sensitive to recent bond rate history. Especially in the case of the longer maturities, the bond rate predictions in figure 3 mirror rather closely the recent levels of the short rate. Of course, the reason for the higher sensitivity of the bond rate predictions in figure 3 to variations in the short rate is that theunitroot intheAR forecast model ofequation 9 inducesunit-rootbehaviorinthemovingaverage endpointaswell as intheshortrate.

10 3. AutoregressiveModels withMovingEndpoints Although it is evident from figure 2 that the interest rate endpoint is not fixed, it is equally apparent from figure 3 that the unit-root description of nonstationarity in the interest rate endpoint is also deficient. Unit-root stochastic processes are only one subset of the general class of nonstationary time series, and it is well-known that standard tests for unit roots have low power against other descriptions of nonstationarity,such as the episodic shifts analyzed in Perron (1989). There are ways other than unit roots to characterize nonstationary movements. These include the shifting-regimes modelsofHamilton(1989),anddetectionofstructuralshiftsorchangepoints,suchasHinkley(1970) andAndrews(1993).9 Rather than explore alternative statistical descriptions of nonstationary endpoints, this section demonstratesthatestimatesofshiftingperceptionsbyagentsofendpointscanoftenbeextractedfrom observable information on the ex ante forecasts of agents. Two examples of ex ante estimates of moving endpoints are discussed in this section. The first subsection indicates that implicit forward rates in the term structure provide an effective measure of the interest rate endpoints embedded in market multiperiod forecasts. The second subsection relates the moving endpoint of nominal interest rates to the implied endpoint of expected inflation. Time series forecasts of these moving endpoint models are compared with time series of a cross-section of historical bond rates in the term structure andwithtimeseries ofsurveyestimatesofexpectedlong-runinflation. 3.1AMovingEndpointModelof theShort-TermInterest Rate The autoregressive model for the short rate is now extended to include the effects of an explicit movingendpointfor theshortrate, r ( 1 (cid:0)t 1 ) . E E (cid:0)t (cid:0)t f 1 f 1 (cid:1) r r t t ( ) 1 g g = = a r + 0 (cid:0)t ( 1 1 (cid:13) ) : ( r (cid:0)t 1 (cid:0) r ( 1 (cid:0)t 1 ) ) + A ( L ) (cid:1) r (cid:0)t 1 ; ( 1 7 ) Two equations now characterize the short rate process. Iterations of the first equation describe the evolutionofshortrateforecasts overtheforecast horizonthatbeginsinperiod t . Thesecondequation indicatesthat theendpointforecast is fixedoverthehorizonofforecasts originatinginagivenperiod, t . It is important to observe that 1 7 is not a closed system because the second equation does not indicatehowtheconditionalexpectationoftheendpointmaybealteredovertimeatthestartoffuture forecast horizons. Indeed, as indicated later, the short rate endpoint perceived by agents has shifted 9AnexampleofchangepointlearningthatinducesshiftsinagentperceptionsofendpointsisdevelopedinKozickiand Tinsley(1996).

11 substantiallyoverthehistoricalsample.10 Agents' exanteforecastsofthenominalrateendpointarereadilyavailablefromtheobservedterm structure of nominal rates. One such measure is the average of the expected short rates from t + n to t + n 0 ,for n 0 > n , ^r ( 1 t) = D n 0 ( r n 0 ;t (cid:0) b D n n 0 ) 0 (cid:0) (cid:0) D D n n ( r n ;t (cid:0) b n ) ; ( 1 8 ) where, as before, D n denotes the duration of an n -period coupon bond, D n (cid:17) ( 1 (cid:0) (cid:12) n ) = ( 1 (cid:0) (cid:12) ) .11 Estimatesoftheshortrateendpointarebasedontheaverageofexpectedshortratesbetweenthe5-year and10-yearmaturities. Characteristics of the estimated moving endpoint autoregression in equation 1 7 are displayed in the third column of table I, under the heading Moving Endpoints. The estimate of the coefficient of the lagged level of the short rate, (cid:13) = (cid:0) 0 : 0 7 1 , is consistent with endpoint-reversion or long-horizon conditional forecasts that converge to the expected endpoint, which may vary with the date of the conditioninginformationset. Themeanlagofadjustmenttothemovingendpointisabout10months, considerablyfasterthanthemeanlagof40monthsestimatedearlierforadjustmenttoafixedendpoint. If we have correctly identified the moving endpoint, then the estimated intercept, a 0 , should be zero, as it isinthethirdcolumnoftableI.Finally,notethatthestandardsummarystatistics,suchas the R 2 and RMSE, are similar across all three autoregressions in table I, indicating that the one-step-ahead forecast properties of the fitted equations are relativelyinsensitiveto assumptionsabout long-horizon endpoints. The forecasting system for the moving endpoint system in 1 7 is similar to that shown earlier in equation 3 for the constant endpoint companion matrix system. The only change is that the 1 ) ( p + -vector, h ,is nowtime-subscriptedtoincorporatethelatestestimateofthemovingendpoint. E (cid:0)t 1 f h t g = = [ h a 0 (cid:0)t ( t (cid:0) 1 ) (cid:13) : ^r ( 1 (cid:0)t 1 ) ; 0 ; : : : ; 0 ] 0 ; ( 1 9 ) Thus, using the revised definition of the forcing term, h t , of the moving endpoint autoregression, 10Temporalshiftsintheperceivedendpointofthenominalinterestrateshouldnotbesurprising,givenwell-publicized alterationsintheoperatingproceduresandtargetsofmonetarypolicyinpostwardecades,suchasdiscussedbyHuizinga andMiskin(1986). 11Notethat ( D n 0 r n 0 ;t (cid:0) D n r n ;t ) = ( D n 0 (cid:0) D n ) providesabiasedestimateoftheendpointoftheshortrateprocess,unless D n b n (cid:0) D n 0 b n 0 = 0 .Theexpressionfor ^r ( 1 t) inequation 1 8 isunbiased,onaverage,underthehypothesisofconstantterm premia.

12 multiperiodforecasts oftheshort ratearenowprovidedby E (cid:0)t 1 f r t+ k g = = (cid:19) (cid:19) 0 1 0 1 ( z I ( 1 p + (cid:0)t 1 1 ) (cid:0) + H (cid:19) 0 1 ) H (cid:0) 1 k h + ( t 1 (cid:0)t [ z 1 ) (cid:0)t + 1 (cid:0) (cid:19) 0 H 1 ( z 1 k (cid:0)t + 1 1 [ z ) ] ; (cid:0)t 1 (cid:0) ( I p + 1 (cid:0) H ) (cid:0) 1 h ( t (cid:0)t 1 ) ] ; ( 2 0 ) wheretheeffectivemovingendpointwillbeequivalenttothemeasuredshortrateendpoint, ^r ( 1 (cid:0)t 1 ) (cid:19) 0 1 z ( 1 (cid:0)t 1 ) = iftheestimatedintercept iszero, a 0 = 0 . Finally, the closed form solutions for term structure predictions under the moving endpoint autoregressionareobtainedbysubstitutingthebondratepredictionsfromequation 2 0 intoequation 1 , giving E (cid:0)t 1 r n ;t = = D (cid:19) 0 1 (cid:0) n z 1 (cid:19) (cid:0)t ( 1 (cid:0) n 0 ( 1 X i= 1 ) + 1 0 (cid:12) b n i E + (cid:0)t D 1 f (cid:0) n r 1 t+ 0 (cid:19) 1 g i ( I ) p + + 1 b (cid:0) n ; ( (cid:12) H ) n ) ( I p + 1 (cid:0) (cid:12) H ) (cid:0) 1 H [ z (cid:0)t 1 (cid:0) z ( 1 (cid:0)t 1 ) ] : ( 2 1 ) The panels of figure 4 display the concatenated time series of the cross-section of four bond rates predictedbythemovingendpointautoregression. Unlikethetermstructurepredictionsoftheconstant and moving average endpoint models in figures 2 and 3, the bond rate predictions of the moving endpoints model closely track the historical bond rates at each of the selected four maturities in the termstructure. Sinceeachmaturityrateinthepredictedtermstructurecross-sectionisgeneratedbythe same autoregression model, the moving endpoints model generally supports the internal consistency ofagent exanteforecasts containedinthehistoricaltermstructure. In principle, the moving endpoint model should be expected to dominate the other two endpoint specifications because it is less restrictive in use of available information. The structural bond yield model with the moving short rate endpoint, r ( 1 (cid:0)t 1 ) , can be interpreted as a two-factor model of the term structure of interest rates where one factor is identified as the endpoint ^r ( 1 (cid:0)t 1 ) and the second factor is identified as the deviation of the short rate from that endpoint. By contrast, the constant and moving average endpoint autoregressions are one-factor models where the short rate is identified as the single factor. Because the performance of two-factor models should be expected to dominate that of one-factor models, yield predictions generated by the model using the movingshort rate endpoint, ^r ( 1 (cid:0)t 1 ) , should generally outperform yield predictions based on constant or moving average endpoint characterizations.12 12Subsequentworkwillindicatethatstandardliteraturesuggestionstoincreasethedimensionoftheinformationsetof agents,suchastheadditionofaspreadbetweenalongandshortratetotheshortrateequation,areunlikelytoapproximate

13 3.2AlternativeForecastModelsof ExpectedInflation Anotherrelationshipthatisbasedonanassumedinternalconsistencyintheexanteforecastsofagents is the standard Fisherian decomposition of a nominal interest rate into the expected real rate and expected inflation. In the current context, the after-tax nominal rate endpoint can be partitioned into anexpectedreal rateendpointandanexpectedinflationendpoint. ( 1 (cid:0) (cid:28) ) r ( 1 t) = (cid:26) ( 1 t) + (cid:25) ( 1 t) ; ( 2 2 ) where (cid:28) denotesthemarginaltaxrateonbondearnings,and (cid:26) ( 1 t) isthet-periodestimateoftheafter-tax real rate endpoint. Thus, equation 2 2 suggests that a transformation of the nominal rate endpoint, such as that used in the preceding section, might provide an internally consistent estimate of agents' evolvingperceptionofexpectedlong-runinflation, (cid:25) ( 1 t) . The procedure used here to extract the implicit ex ante inflation endpoint from the nominal rate endpoint rests on two additional approximations. First is the assumption that the after-tax real rate endpoint is a constant. This is tantamount to the assumption that the after-tax real interest rate is mean-reverting. Mean-reverting behavior, or even constancy, of the real rate is not an uncommon specification infinance, Fama (1975); theoretical reasons and empirical evidence for this assumption, absent marked shifts in fiscal policy not captured by changes in the tax rate, (cid:28) , are discussed in Kozicki and Tinsley(1996). Consequently,we will assume the constant after-tax real rate endpoint is amaintainedhypothesis. A more nonstandardassumptionin analyses of real rate arbitrage is theuse of a nonzero marginal taxrate. However,studiesbyMcCulloch(1975)andCrowderandHoffman(1996)suggestthatthetax rateonTreasuryearningsfallsintheinterval, : 2 0 (cid:20) (cid:28) (cid:20) : 3 0 . Usingflowoffundsestimatesofsectoral holdings of Treasury securities, we estimate that the effective tax rate on earnings of US Treasuries, averagedoverthe1960-90sample,isaboutthirtypercent, (cid:28) = 0 : 3 0 . Analogous to the alternative models estimated for the monthly interest rate, three endpoint autoregression models are estimated for monthly inflation, (cid:25) t . Monthly inflation of the deflator for personal consumption expenditures is expressed at annual rates. The formats of the regressions are identical to those used for the monthly interest rate, except for a smaller number of autoregressive lags, p = 6 ,andasamplespanof1960m1-1990m12. The estimated parameters and summary statistics of the constant endpoint, moving average endpoint,andmovingendpointARmodelsofinflationarelisted,respectively,inthethreecolumnsof tableII.Aswiththeinterestrateexamples,thereislittletochooseamongtheequationsonthebasisof thebehaviorofatwo-factormovingendpointmodel.

14 conventionalsummarystatisticssuchas the R 2 and RMSE. Adjustmentof inflationpredictions tothe inflationendpointsaresluggishforboththemodelswithconstantandmovingendpoints,withamean lagofabout30monthsfortheformerand38monthsforthelatter. Althoughtheestimatedinterceptis notsignificantlydifferentfromzerointhemovingendpointautoregression,thepointestimatesuggests anafter-taxannualrealrateendpointfor1-monthUSTreasuriesofabout80basispoints, ^(cid:26) 1 (cid:17) (cid:0) ^a 0 = ^(cid:13) . In contrast to the term structure, available measurements of agent multiperiod predictions of inflation are more limited. Several surveys of expected inflation in the United States are available but most focus on short-term expectations of inflation. For the purpose of drawing comparisons among the three inflation forecast models, this is a serious drawback since we know from the results for multiperiodinterestrateforecasts that short-horizonforecasts willbesimilarforallthreemodels. Two notable exceptions are a survey of market participants conducted in the 1980s by Richard Hoey, an economist at Drexel Burnham Lambert, which asked for 10-year inflation forecasts and a survey of household estimates of inflation in the next 5-10 years, conducted by the Michigan Survey Research Center. Although the Hoey survey has been discontinued, a contiguous quarterly series of expected 10-year inflation is assembled for the span, 1980q4-1990q4. Similarly, although there are manymissingobservationsfortheMichigansurvey,aquarterlyseries oflong-termexpectedinflation for the span, 1982q3-1990q4. is assembled by using the “last known estimate” to fill in missing observations. Predictionsof10-yearinflationratesaregeneratedbytheequationsusedtogenerate10-yearbond rates, with adjustments for undiscounted averages. In other words, similar to alterations required for zero-coupon bonds, the closed-form solutions for the coupon-bond term structure in equations 7 , 1 0 , and 1 6 are adjusted for 10-year averaging by setting the discount factor to unity, (cid:12) = 1 : 0 , and the duration to 120 months, D n = 1 2 0 . The three endpoint autogression forecast models are used to generate concatenated time series of monthly predictions of 10-year inflation rates. These are averaged to quarterly observations, to match the frequency of the assembled survey estimates of expectedlong-terminflation,andplottedinthethreepanelsoffigure 5. In somewhat of a replay of the properties of the bond rate predictions, the first panel of figure 5 indicates that the 10-year inflation rates predicted by the constant endpoint model are too smooth relativetothesurveyestimates. Likewise,the10-year inflationrates predictedbythemovingaverage autoregression,showninthesecondpaneloffigure5,morecloselytrackthepathofhistoricalinflation thanthesurveyestimates. By contrast, the third panel of figure 5 indicates that the predictions of 10-year inflation rates generatedbythemovingendpointmodelofinflationmovequitecloselywiththeavailableobservations onsurveyestimatesofexpectedinflation,slightlyfavoringtheMichigansurveyestimateovertheHoey surveyestimateinthe last few years of the sample. Since none of theinformationineither of the two surveyswas used toestimatethe movingendpointmodel of inflation, this is a very remarkable result.

15 Also,thefactthatthisrelativelyclosecorrespondenceisgeneratedbyamovingendpointsmodelbased on a linear transformation of the endpoint in the term structure suggests that the ex ante forecasts of inflationthat are plausiblyembeddedin bondrates are internallyconsistent withthe ex ante forecasts oflong-terminflationmeasuredbythesurveys. 4. ConcludingComments Results in this paper suggest that more flexible specifications of boundary values or transversality conditionsarerequired,alongwithestimationofEulerequationdescriptionsofdynamicadjustments, if empirical models in macroeconomics and finance are to be successful in rational interpretations of agent behavior. In particular, the two alternatives of constant or moving average long-run boundary values that are implied by standard time series models are shown to be inconsistent with the implicit endpointsofagents' exanteforecasts. The examplesusedinthis papertoillustratethestrongeffects ofterminal boundaryconditionson multiperiod forecasts are simple autoregression implementations of expectations models of the term structure and expected inflation. These examples are useful because measurements of agent forecasts of forward interest rates and expected inflation are available for comparison with model predictions. Also, assuming auction markets are efficient at least at monthly frequencies, measured outcomes are unlikelytobesignificantlydistortedbytransactioncostsorotherinhibitingfrictionsonagentbehavior. Neither mean-reverting autoregressions, with constant endpoints, nor unit-root autoregressions, with moving average endpoints, are able to adequately track the historical movements of a cross-sectionofbondratesinthetermstructureorvariationsinsurveyestimatesoflong-terminflation. Incontrast,thepredictionsbyinterestrateandinflationautoregressionswithmovingendpointsareable toreproducehistoricalshiftsinboththetermstructureandsurveymeasuresofexpectedinflation. The implicit endpoint estimates used in this paper, inferred from the far end of the term structure, vary considerablyovertimebutnotinawaythatis consistentwithunitrootspecifications. Implicit endpoints, extracted from observable agent forecasts, are useful in testing the internal consistency of market ex ante forecasts, such as the demonstration that the same moving endpoint model of the short rate can capture time variations in the cross-section of bond rates in the term structure, and that a Fisherian decomposition of bond rates into an after-tax real rate and expected inflationappearstobebroadlyconsistentwithsurveyestimatesoflong-terminflation. Thealternative estimates of expected inflation by conventional time series models would deliver very different estimatesofexanterealinterestratestomonetarypolicyadvisors,withpotentiallyerroneousestimates ofagents' perceptionsofcurrent andexpectedpolicy. Moving endpoints are consistent with learning by agents of unpredictable changes in steady-state or long-run outcomes, such as alterations in government long-run policies or permanent changes in the aspirations or behaviors of other market participants. Unlike the gradual outcomes of Kalman

16 filtering modeling, expanding-sample regressions, or other forms of incremental learning by linear projections,theimplicitendpointsusedhereseemtoindicatethatendpointlearningmaybenonlinear, with intervals of small changes followed abruptly by large shifts in beliefs. Developing endogenous modelsofagents' perceptionsoflong-runendpointsisachallengingtaskforfuturework.13 References Andrews,D.,“TestsforParameterInstabilityandStructuralChangewithUnknownChangePoint,” Econometrica,61,4,July1993,821-56. Beveridge, S. and C. Nelson, “A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the 'Business Cycle',” JournalofMonetaryEconomics,7,2,March1981,151-74. Campbell, J., “A Defense of Traditional Hypotheses about the Term Structure of Interest Rates,” JournalofFinance,41,1986,183-93. Campbell,J.andR.Shiller,“CointegrationandTestsofPresentValueModels,”JournalofPolitical Economy,95,5,1987,1062-88. Choi,S.,andM.Wohar,“NewEvidenceConcerningtheExpectationsTheoryfortheShortEndof theMaturitySpectrum,”TheJournalof FinancialResearch,14,1991,83-92. Cox,J.,J.Ingersoll,andS.Ross,“ATheoryoftheTermStructureofInterestRates,”Econometrica, 53,2,March1985,385-407. Crowder, W. and D. Hoffman, “The Long Run Relationship Between Nominal Interest Rates and Inflation: The Fisher Equation Revisited,” Journal of Money, Credit, and Banking, 28, 1,February 1996,102-18. Fama,E.,“Short-TermInterestRatesasPredictorsofInflation,”AmericanEconomicReview,65,3, June1975,269-82. Fama, E., “The Information in the Term Structure,” Journal of Financial Economics, 13, 1984, 509-28. Fama, E. and R. Bliss, “The Information in Long-Maturity Forward Rates,” The American EconomicReview,77,1987,680-92. Hall, A., H. Anderson, and C. Granger, “A Cointegration Analysis of Treasury Bill Yields,” The Reviewof EconomicsandStatistics,74,1,February1992,116-26. Hamilton, J., “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle,”Econometrica,57,1989,357-84. Hardouvelis,G.,“ThePredictivePoweroftheTermStructureDuringRecent MonetaryRegimes,” JournalofFinance,43,1988,339-56. Hinkley,D.,“InferenceAbouttheChange-PointinaSequenceofRandomVariables,”Biometrika, 57,1,April1970,1-17. 13Anexampleofchangepointlearningbyagentsofperceivedshiftsintheinflationtargetofmonetarypolicyisdiscussed inKozickiandTinsley(1996).

17 Huizinga, J. and F. Mishkin, “Monetary Policy Regime Shifts and the Unusual Behavior of Real Interest Rates,” inK.Brunner andA.Meltzer(eds.),Carnegie-RochesterConference Series on PublicPolicy,24,1986,231-74. Kozicki, S. and P. Tinsley, “Moving Endpoints in the Term Structure of Interest Rates,” FRBKC/FRB staffworkingpaper, May1996. Mankiw, G., and J. Miron, “The Changing Behavior of the Term Structure of Interest Rates,” QuarterlyJournalofEconomics,101,1986,211-28. McCulloch,H.,“TheTax-AdjustedYieldCurve,” The Journalof Finance,30,1975,811-30. McCulloch, H. and H. Kwon, “U.S. Term Structure Data, 1947-1991,” Ohio State University WorkingPaper93-6,March1993. Mougoue,M.,“TheTermStructureofInterestRatesasaCointegratedSystem: EmpiricalEvidence from theEurocurrencyMarket,”TheJournalofFinancialResearch,15,3,1992,285-96. Perron, P., “The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis,” Econometrica, 57,1989,1361-401. Rudebusch, G., “Federal Reserve Interest Rate Targeting, Rational Expectations, and the Term Structure,” Journalof MonetaryEconomics,35,1995,245-74. Sargent, T., “A Note on Maximum Likelihood Estimation of the Rational Expectations Model of theTerm Structure,”JournalofMonetaryEconomics,5,1979,133-43. Shea, G., “Benchmarking the Expectations Hypothesis of the Interest-Rate Term Structure: An Analysis of Cointegration Vectors,” Journal of Business and Economic Statistics, 10, 3, July 1992,347-66. Shiller, R., “The Volatility of Long-Term Interest Rates and Expectations Models of the Term Structure,” Journalof PoliticalEconomy,67,6,December 1979,190-219. Shiller, R., “The Term Structure of Interest Rates,” in B. Friedman and F. Hahn (eds.) Handbook of MonetaryEconomics,vol. I,1990,627-722. Shiller, R., J. Campbell, and K. Schoenholtz, “Forward Rates and Future Policy: Interpreting the Term StructureofInterest Rates,” BrookingsPapers,1983,173-223.

18 TableI:Autoregressive Models oftheShort-Term Interest Rate, (cid:1) r t = a 0 + (cid:13) r (cid:0)t 1 + A ( L ) (cid:1) r (cid:0)t 1 (cid:0) (cid:13) 1 ^r ( 1 (cid:0)t 1 ) + e t : r t EndpointCharacterization parametersa Constant Moving Moving Average Endpointb a 0 0.173 0.006 -0.000 (0.089) (0.036) (0.033) (cid:13) -0.027 -0.071 (0.013) (0.033) A ( 1 ) -0.097 -0.249 0.203 (0.232) (0.221) (0.304) (cid:13) 1 -0.071 (0.033) R 2 0.069 0.059 0.071 RMSE 0.690 0.693 0.689 aAll regressions estimated using monthly par yields from McCulloch and Kwon (1993) over the sample, 1960m1-1991m2. Standarderrorsinparentheses. bMovingendpoint, ^r ( 1 (cid:0)t 1 ) , impliedbytermstructureforwardrates. Themovingendpointmodelimposestherestriction, (cid:13) 1 = (cid:13) .

19 TableII:AutoregressiveModels ofMonthlyInflation, (cid:1) (cid:25) t = a 0 + (cid:13) (cid:25) (cid:0)t 1 + A ( L ) (cid:1) (cid:25) (cid:0)t 1 (cid:0) (cid:13) 1 ( 1 (cid:0) (cid:28) ) ^r ( 1 (cid:0)t 1 ) + e (cid:25) : t t EndpointCharacterization parametersa Constant Moving Moving Average Endpointb a 0 0.543 0.038 0.069 (0.235) (0.111) (0.112) (cid:13) -0.108 -0.088 (0.044) (0.052) A ( 1 ) -2.37 -2.68 -2.40 (0.297) (0.269) (0.318) (cid:13) 1 -0.088 (0.052) R 2 0.368 0.357 0.362 RMSE 2.12 2.14 2.13 aAll regressions estimated using monthly inflation (at annualized rates) of the seasonally adjusted deflator for personal consumption expenditures over the sample, 1960m1-1990m12. Standard errors in parentheses. bUptoanunknownconstant, ( 1 (cid:0) (cid:28) ) ^r ( 1 (cid:0)t 1 ) isanexanteestimateof themovingendpointofinflation,where (cid:28) denotesthemarginaltaxrate and ^r ( 1 (cid:0)t 1 ) istheinterestratemovingendpointimpliedbytermstructure forward rates. The moving endpoint model imposes the restriction, (cid:13) 1 = (cid:13) .

20 FFiigguurree 11:: MMuullttiippeerriioodd pprreeddiiccttiioonnss ooff 11--mmoonntthh bboonndd rraattee bbyy AARR mmooddeellss 16 constant endpoints historical 14 predicted predicted 12 10 8 6 4 2 16 moving average endpoints 14 12 10 8 6 4 2 1960 1965 1970 1975 1980 1985 1990 FFFFiiiigggguuuurrrreeee 2222:::: BBBBoooonnnndddd rrrraaaatttteeee pppprrrreeeeddddiiiiccccttttiiiioooonnnnssss ((((ccccoooonnnnssssttttaaaannnntttt eeeennnnddddppppooooiiiinnnntttt AAAARRRR)))) 3-month maturity 15 historical predicted 10 5 12-month maturity 15 10 5 5-year maturity 15 10 5 10-year maturity 15 10 5 1960 1965 1970 1975 1980 1985 1990

21 FFFFiiiigggguuuurrrreeee 3333:::: BBBBoooonnnndddd rrrraaaatttteeee pppprrrreeeeddddiiiiccccttttiiiioooonnnnssss ((((mmmmoooovvvviiiinnnngggg aaaavvvveeeerrrraaaaggggeeee eeeennnnddddppppooooiiiinnnntttt AAAARRRR)))) 3-month maturity 15 historical predicted 10 5 12-month maturity 15 10 5 5-year maturity 15 10 5 10-year maturity 15 10 5 1960 1965 1970 1975 1980 1985 1990 FFFFiiiigggguuuurrrreeee 4444:::: BBBBoooonnnndddd rrrraaaatttteeee pppprrrreeeeddddiiiiccccttttiiiioooonnnnssss ((((mmmmoooovvvviiiinnnngggg eeeennnnddddppppooooiiiinnnnttttssss AAAARRRR)))) 3-month maturity 15 historical predicted 10 5 12-month maturity 15 10 5 5-year maturity 15 10 5 10-year maturity 15 10 5 1960 1965 1970 1975 1980 1985 1990

22 FFFiiiggguuurrreee 555::: AAAlllttteeerrrnnnaaatttiiivvveee AAARRR ppprrreeedddiiiccctttiiiooonnnsss ooofff 111000---yyyeeeaaarrr iiinnnffflllaaatttiiiooonnn rrraaattteeesss 15 constant endpoint AR model historical inflation predicted inflation 10 Michigan survey Hoey survey 5 0 15 moving average endpoint AR model 10 5 0 15 moving endpoints AR model 10 5 0 1960 1965 1970 1975 1980 1985 1990