A Guide to FRB/US: A Macroeconomic Model of the United States

A Guide to FRB/US AMacroeconomicModeloftheUnitedStates MacroeconomicandQuantitativeStudies (cid:3) DivisionofResearch andStatistics Federal ReserveBoard Washington,D.C.20551 version1.0,October1996 Abstract: FRB/US is a large-scale quarterly econometric model of the U.S. economy, developed to replace the MPS model. Most behavioral equations are based on specifications of optimizing behavior containing explicit expectations of firms, households, and financial markets. Although expectations are explicit, the empirical fits of the structural descriptions of macroeconomic behavior are comparable to those ofreduced-form timeseriesmodels. Inmostinstances, tests donotreject overidentifying restrictions of rational expectations or the hypothesis of serially independent residuals. As modeled, private sector expectations ofpolicyconstitute amajortransmission channel ofmonetarypolicy. Keywords: Macroeconomic models,privatesectorlearning, rationalexpectations, vectorautoregressions. (cid:3) Edited by F. Brayton andP. Tinsley. Authorsof sectionsare: F. Brayton(section 1); P. Tinsley(sections2 and 4); A. Bomfim, D. Reifschneider,and P.vonzur Muehlen(section3); B. TetlowandJ. Williams(section5). Other members of the Macroeconomicand Quantitative Studies section who contributed to developmentof FRB/US and provided comments on this summary are M. French, E. Mauskopf, and S. Schuh. In addition, D. Battenberg, T. Grunwald, and T. Horvath provided significant technical assistance. Views presented are those of the editors and authorsanddonotnecessarilyrepresentthoseoftheFederalReserveBoard.

Contents 1 Introduction and Overview 1 1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Characteristics ofequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Simulationcapabilitiesandproperties . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Organizationofremainingsections . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Types of Economic Behavior 7 2.1 Arbitrageequilibria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Rationalbehaviorunderfrictions. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Heterogeneoushouseholdsandfirms. . . . . . . . . . . . . . . . . . . . . . . . . 14 3 A Bird's Eye View of FRB/US 15 3.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Financialmarkets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Foreigntradeandgovernmentsectors . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Expectations 26 4.1 Thescopeofsectoral information. . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Long-runexpectations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 ThehistoricalVAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 Full-System Properties 34 5.1 Systemresponsestotransitoryshocks . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Systemresponsestopermanentshocks . . . . . . . . . . . . . . . . . . . . . . . . 40 A Appendix: Testing the Theory 43 B References 45

1 1 Introduction and Overview FRB/US is a new quarterly econometric model of the U.S. economy developed to replace the MPS model. The new model has three distinctive features. First, the expectations of private sectors are explicit, and these expectations, especially market perceptions of policy, constitute a major transmission channel of monetary policy. Second, information assumed to be accessed by private sectors can vary in scope and timing, and can include “perfect foresight” or learning from past observations. Third, although expectations are explicit, the empirical fits of the structural descriptions of macroeconomic behavior are comparable to those of reduced-form time series models. In contrast to the inability of MPS to disentangle forecast and response lags, the structure of FRB/US parses dynamics into contributions of expectations and adjustment frictions. This decomposition enables the new model to provide sharper interpretations of macroeconomic developments and to examine the sensitivity of forecasts and policy scenarios to a range of assumptions about how sectoral expectations are formed. However, the more explicit theoretical structure has been achieved while also incorporating newer statistical techniques aimed at improvingthegoodnessoffit andreliabilityofempiricalestimates. Despite changes in structure and updatedempirical estimates,FRB/US retains several notable characteristics of MPS. One important similarity is the blend of long-run neoclassical conditions for equilibrium with short-run sticky-price disequilibria where monetary policy actions have significantshort-runeffects onthelevelofreal activity. FRB/US is a large-scale model, containing some 300 equations and identities. However, the number of stochastic “core” equations or estimated descriptions of the economic behavior of firms, households, and investors is much smaller, around 50 equations. In the current edition of FRB/US (version1.0),abouthalf ofthesebehavioral equationsare basedonformal specifications of optimizing behavior containing explicit estimates of the forward expectations of firms and households. Most of the discussion in this guide will focus on this subset of core equations because the format ofthese structural equations is new. Althoughthe explicitexpectationsformat may be extended to additional core equations in future model editions, most key macroeconomic relationships in version 1.0 of FRB/US now include explicit expectations. These include the estimatedstructural equations for aggregateconsumption,twocomponentsof consumerdurables, residential construction, investment in producers' durable equipment, inventory investment, labor hours, measures of aggregate price and labor cost, three long-term interest rates, and the value of corporateequity.

2 1.1 Objectives In the 25 years since the initial version of the MPS model was completed and brought into operational use at the Board, the practice of macroeconomics has evolved substantially. Notable developmentsinclude: greater emphasisonmodelingexpectations,especiallytheassumptionthat expectations are rational or consistent with modeled outcomes; more extensive use of dynamic optimization theory to characterize responses of households and firms to shocks; expansion of the types of models in general use to include atheoretic vector autoregressions (VARs) and theoretically-based, general equilibrium models of business cycles; and development of new statisticaltechniquesforestimationoflong-runrelationshipsamongtrendingseriesandfortesting facets ofequationperformance. Although the design of FRB/US draws on many of these developments,other adaptations and new concepts were required in the construction of the model in order to meet a list of goals. The mainobjectivesthatguidedthedevelopmentofFRB/US are: 1. Use: (a) The model should be suitable for both forecasting and policy simulations. (b) It should also be able to run simulations of policy and other scenarios under a variety of assumptionsabouthowhouseholds,firmsandfinancialmarketsformexpectations,including theextentofavailableinformation. 2. Conceptual design: (a) Expectations should be explicit. (b) Structural equations for households,firms,andfinancialmarketsshouldbebasedoneconomictheoriesofoptimizing behavior. 3. Statistical implementation: (a) Estimation of equations in the model should be based on modern timeseries techniques. (b) Equations shouldhave satisfactory statistical properties, includinggoodnessofhistoricalfit. 4. Simulationproperties: Forshocksthatarenotunusualinhistoricalperspective,themodel's simulated responses should be close to those obtained from atheoretic models that do not impose strong priors, such as VARs. The model should also be able to match established rulesofthumbregardingeconomicrelationshipsunderappropriatecircumstances. Some of these objectives are complementary: Explicit representation of expectations, 2a, assists in identifying the contributions of alternative expectations, 1b. However, there are well-known conflicts between the objective for more theoretical rigor, 2b, and the objectives relatedtoforecasting,1a,andimprovedstatisticalproperties,3b. Indeed,majorinnovationsinthe designofFRB/US have beenmotivatedtoimprovethetradeoffbetween theoretical and empirical properties.

3 1.2 Implementation KeyequationsinFRB/US arebasedonfourfundamentalbuildingblocks: (cid:15) arbitrageequilibria (cid:15) equilibriumplanning (cid:15) dynamicadjustments (cid:15) forecasting(expectationsformation) The way these elements are combined in key equations varies. Equations for financial variables are based on arbitrage equilibria. For example, arbitrage equates the rate of return on a bond to a weighted average of expected future values of a short-term interest rate plus a term premium. Becausetransactionscostsarerelativelysmallinfinancialmarkets,arbitrageisassumed within each quarter. In contrast, nonfinancial variables are costly to adjust and, thus, move only gradually to eliminate any disequilibria. Equations for nonfinancial variables are based on equilibrium plans—values that would be desired in the absence of adjustment frictions—and dynamic adjustments. The latter employ a general model of dynamic frictions to estimate the optimalrateat whichdeviationsfrom equilibriaare eliminated. Theapproachtooptimaldynamic adjustments used in FRB/US is a generalization of a costly adjustment approach commonly used inappliedmacroeconomics. The mainprice equation in FRB/US is convenient for illustratingthe structure of nonfinancial equations. The equilibrium condition is derived from profit maximizationand makes the planned price level a markup over unit factor costs. A timeseries (cointegration) regression is used to estimate the long-run weights on unit labor and energy costs. Once the equilibrium relationship is specified, a generalized frictions model is estimated for price dynamics, including the degree of cyclical variation of the markup. There are several ways to represent price adjustments, as discussed in section 2, but for now it is sufficient to indicate that they depend on expectational terms (future values of the equilibrium price level) and inertial terms (lags of the actual price level). Equations for financial and nonfinancial variables alike contain expectations of the private sectors. As indicated above, expectations enter financial equations through the definition of arbitrage equilibria. For nonfinancial equations, expectations may enter two ways. First, components of the equilibrium plans of some equations are present values. For example, the desiredlevelofconsumptionisderivedfrom thelifecyclemodeland,thus,dependsonthecurrent value of tangible assets and current and expected future values of household income. Second, the dynamic adjustment component of each nonfinancial equation depends on expectations because

4 households and firms aim, not at the current equilibrium value, but at the trajectory that the equilibriumisexpectedtofollow. The last building block, forecasting, describes how sectoral expectations are formed. Here, it is useful to distinguish between the assumption made about expectations when FRB/US was estimated,andthevariousoptionsforspecifyingexpectationswhenthemodelissimulated. For estimation purposes, sectoral expectations were derived from forecasts of small VARs. AlthoughthestructureoftheVARsvariesacross equations,theVARshaveasimilarorganization. Eachcontainsacommonsetofvariables—consumptionpriceinflation,outputgap,andthefederal funds rate—along with one or more sector-specific variables. Because each VAR contains an equationfor the federal funds rate, this form of expectationsincorporates an average sampleview ofhowmonetarypolicywasconductedhistorically. For simulation purposes, several options for expectations are currently available. One is the approach of VAR expectations just described. Under a second option, expectations are equal to forecasts from the FRB/US model—an option labeled full-model expectations. Both types of expectations can be viewed as reflecting rational behavior, but under different assumptions about thescope ofinformationavailabletoindividuals. Anotherdimensionofexpectations along which FRB/US has flexible capabilities is the speed with which individuals learn about changes in the economic environment. The specific application developed so far pertains to how quickly the private sector catches on to changes in the long-run inflation objective of monetary policy—an issuecloselyrelatedtothetopicofpolicycredibility. 1.3 Characteristics of equations The economic behavior described by FRB/US can be summarized most easily by focusing on threemainsectors: (cid:15) Households choose equilibrium aggregate consumption based on the lifecycle model, an approach motivated by utility maximization, but are assumed to be quite risk averse and, thus, to discount the future heavily in computing expected income. The dynamic equation for aggregate consumption contains sluggish adjustment of actual consumption toward its equilibrium as well as modest effects of liquidity constraints. Investment in consumer durablesandresidentialconstructionvarieswithaggregateconsumptionaswellaswithreal interestrates andrelativeprices. (cid:15) Firms chooseinvestment,inventories,laborhours, andprices basedonprofit maximization under imperfect competition. Firms also are involved, along with households, in the short-run determination of wages. Adjustment dynamics are estimated to be most rapid

5 for inventories and labor hours and slowest for wages and investment in producers' durable equipment. The speed of adjustment of the aggregate price is intermediate. In addition to its sensitivitytothecost ofcapital, investmentinproducers' durableequipmentis modestly sensitivetocashflow. (cid:15) Financial markets set bondrates, stockprices, andtheexchange rate bystandardarbitrage conditions. Thus, bond yields depend on values of short-term interest rates expected to prevail over the maturity of the bond, and the stock market valuation depends on expected dividends. Termpremiumsinthebondequationsvarycountercyclically;theriskpremiumin the equity market is modeled as a constant. All asset price equations have significant serial correlation in their residuals, which is interpreted as an additional time-varying component oftermorriskpremiums. FRB/US also contains “traditional” equations—without explicit expectations—for imports, exports, nonresidential construction, employment, labor force participation, and the relative price ofconsumption. With regard to interactions among sectors, an important set of linkages describes the transmissionchannels of monetarypolicy. As inMPS, keytransmissionchannels operate through medium- and long-term interest rates directly in equations for investment in producers' durable equipment, residential construction, and consumer durables, and indirectly through effects of the valueofthestockmarketonaggregateconsumptionandeffectsoftheexchangerateonexportsand imports. Asindicatedabove,assetpricesandbondratesaredirectlylinkedtoexpectationsoffuture federal funds rates which, in turn, are perceived to reflect policy responses to macroeconomic indicators. Inadditiontoforwardfundsrates,thetransmissionofmonetarypolicythroughsectoral expectations is extended in FRB/US to private sector forecasts of future equilibrium values. For example,undereitherVARorfull-modelexpectations,anincreaseinthecurrentfundsratelowers expectedfutureoutputandincome,therebyrestrainingcurrent investmentandconsumption.1 1.4 Simulation capabilities and properties FRB/US currently can be simulated with either VAR or full-model expectations, alternatives that vary the scope of information available to the private sector. Although a common view is that economic models have “stark” properties under the assumption of full-model expectations, because of the extensive amount of information provided individuals under this assumption, this 1NotethattheadditionalpolicytransmissionchannelsidentifiedinFRB/USdonotnecessarilyimplylargerpolicy “multipliers”butmayreflectonlytheexplicitdecompositionofestimatedresponsesamonglagsduetofrictionsand lagsstemmingfromtheformationofexpectations,includinglearning.

6 characterization is not generally correct for FRB/US. The model contains significant adjustment frictionsthatslowresponsesofnonfinancialvariables,eventoanticipatedevents. Also,properties of FRB/US under full-model expectations can be similar to those under VAR expectations, if the shockorchangebeingsimulatedisnotunusualinanhistoricalcontext. Oneexampleisatransitory changeinthefederal fundsrate. UndereitherVARorfull-modelexpectations,outputmovesfora periodoftimeintheoppositedirectionoftheinterestratechange,asdoesinflation,andlong-term interest rates change by a fraction of the movement in the funds rate. In this instance, the VAR contains the essential macroeconomic responses so VAR expectations are similar to full-model expectations. In contrast, unusual shifts can yield quite different outcomes under VAR and full-model expectations—an example is a future change in fiscal policy that is perfectly anticipated under full-modelexpectationsbutrecognizedonlyasitoccursunderVARexpectations. Inthisinstance, macroeconomicvariablesmoveinadvanceofthefiscal changeunderfull-modelexpectations,but onlyafterthepolicychangeunderVARexpectations. AseconddimensionofFRB/USsimulationsthatpertainstoexpectationsisthespeedatwhich theprivatesectorlearns about changes inpolicyobjectives,suchas ashiftinmonetarypolicythat seeks to reduce the rate of inflation. The private sector's perception of the inflation objective—as distinct from the actual policy objective—is included explicitly in the structure of FRB/US. Thus, consequences of a disinflationary policy can be compared under different assumptions about the credibility of the shift. Under full credibility, perceptions of the inflation objective respond immediately and the output cost of reducing inflation is relatively small. Alternatively, ifperceptionsoftheinflationtargetadjustmoreslowly—andonlyafterpolicyactionstoachievea lowerrate ofinflationare instigated—theoutputcostofreducinginflationishigher. 1.5 Organization of remaining sections Four sections follow. Section 2 presents the specifications of economic behavior in FRB/US, using a bond rate equation to illustrate the structure of the typical financial equation and the aggregate price equation to portray the specification of adjustment dynamics in nonfinancial equations. Section3reviewshowtheseapproacheshavebeenappliedtocorebehavioralequations. Expectations are explicit in the structure of FRB/US and important to its properties. Section 4 discusses options for expectations that are currently used in simulations of FRB/US; develops the concept of long-run expectations, which are termed expectation endpoints; and presents the core equations of the vector autoregression used for VAR-based expectations. Finally, section 5 presents simulations of FRB/US to illustrate its responses to key types of shocks under different assumptionsabouthowexpectationsare formed.

7 2 Types of Economic Behavior ThissectionintroducesthetypesofequationsusedtodescribeeconomicbehaviorinFRB/US. Discussion of each equation type includes a brief summary of assumptions about economic behavior that motivate the equation format; significant differences, if any, with specifications in alternative macro models; and an example to illustrate how the equation is referenced in later sections. There are only four standard types of equations, each describing a different activity by individuals: (cid:15) Arbitrage equilibria. This category includes equations for bond yields and the price of equity, basedonstandardformulationsofefficientmarketpricing. Undertheassumptionthatarbitrage profitsarezero,expectedreturnsonassetstradedinfinancialmarketsarerelatedtoanexpected baselinerateofreturnandtermorriskpremiums. (cid:15) Equilibrium planning. Variables not determined in financial markets are controlled by individualsinaparticularsector,suchasconsumptionbyhouseholdsorfixedcapitalinvestment byfirms. Eachsectorselectsequilibriumvaluesforitsownactivitiesthatwouldbeundertaken in theabsence offrictions. These equilibriumsettingsare functions ofvariables not controlled byindividualswithinthesector. Examplesofpredeterminedexplanatoryvariablesareexpected income for household consumption and sales for business capital investment. Equilibrium equationsareessentiallysteady-staterelationshipsandcomprisethesetofequationsmostlikely tobesubjecttofunctionalandcoefficient restrictionsfromeconomictheory.2 (cid:15) Dynamic adjustments. Unlike financial market behavior, where asset prices are assumed to reflectequilibriumvaluations,activitiesinremainingsectorsaresubjecttoavarietyofdynamic frictions. A shift in an equilibrium setting for a variable controlled by a sector will initiate a response to reach the new desired value. However, when responses are constrained, the optimal approach to the revised equilibrium may be spread over many quarters. The format of dynamic adjustment equations in FRB/US is dissimilar to that in other macroeconomic models of rational behavior in that the extent of dynamic frictions is not imposed by priors but determinedby statisticaltesting; consequently,the empirical goodness of fit is comparable tothoseofatheoretictimeseries models. (cid:15) Forecasting. Structural equations in FRB/US require forecasts of explanatory variables. In the caseof arbitrageequilibria,principaldeterminantsoffinancial market yields,suchas bond rates, are multiperiod forecasts of the federal funds rate. For variables subject to dynamic 2The explicitness of equilibrium equations in FRB/US is useful in empirical checks of theoretical long-run restrictions,usingtestsfortrendingvariablesdevelopedinrecentyearsbytimeseriesanalysts,vid.EngleandGranger (1987).

8 frictions, the formulation of optimal adjustment plans requires multiperiod forecasts of the relevantequilibriumvalues. Asindicatedlater,rationalexpectationsareassumedinestimation where sector expectations are generated by a VAR model of the economy. There are two advantagesinassumingrationalexpectations. First,theunobservedexpectationsthatcondition actions of households and firms are replaced by forecasts from an explicit forecast model. Second, the use of explicit expectations permits identification of frictions that impede sectoral dynamicadjustments.3 The remainder of this section provides brief discussions of the arbitrage equilibrium equation for the 10-year bond rate, the equilibrium and dynamic adjustment equations for the aggregate price equation, and examples of additional constraints on behavior. Because sectoral expectations requireaforecastmodeloftheeconomy,discussionofthistopicispostponeduntilanoverviewof thefullmodelispresented. 2.1 Arbitrage equilibria. The simplest form of economic behavior that is assumed to reflect rational expectations is pricing of financial assets in auction markets. The assumption that auction market forecasts are rational requires that the model's multiperiod forecasts of fundamental variables, such as a representative short-term interest rate, should explain significant movements in the auction prices oflong-maturityassets. Assumingbondvaluationsreflect rationalforecasts ofthefuturepathofthefederal fundsrate, the yield to maturityon a 10-year Treasury bond, r 1 0 , is a weighted movingaverage of the funds rates expectedoverthenext40quarters, r 1 0 t = (cid:22) 1 0 + w 1 r t + w 2 r e +t 1 + w 3 r e +t 2 + : : : + w 4 0 r e +t 3 9 ; (1) = (cid:22) 1 0 + 1 : 0 leads 4 0 ( r e +t i ) : The first term on the right hand side of the equal sign in equation 1 is the term premium for the 10-year bond, (cid:22) 1 0 . The remaining explanatory variables are forecasts of the federal funds rate in future periods. Current market forecasts based on information available in period t are denoted by the superscript “e”. In this instance, no superscript is attached to the funds rate of the current periodbecausethequarterlybondrateisassumedtoincorporatecurrent-quarterinformation;thus, 3Ofcourse,theassumptionthatexpectationsofsectorsarerationalmaynotbeconsistentwithobservedbehavior. Rational expectations (RE) impose very tight restrictions on the way that sectoral expectations influence sector responses. In contrast to rejections of RE restrictions in many empirical macroeconomic studies, RE restrictions aregenerallyacceptedinFRB/US.

9 r t istherealizedfundsrateinthecurrent period,notaforecast. Theweights, w i ,onexpectedfundsratesoverthe40-quartermaturityofthe10-yearbondsum to unity. This is indicated in the second line of equation 1 where the relevant weight sum (1.0 in this equation) multiplies a condensed notation for the weighted average of 40-quarter funds rate forecasts,leads 4 0 (.). Foradiscountbond,therelationshipinequation1wouldbeasimplemoving average with each funds rate forecast receiving the same weight of 1/40. However, in the current example of a 10-year Treasury coupon bond, the quarterly weights decline geometrically at a 2% rateconsistentwithapplyinganannualdiscountrateof8%(approximatelythesamplemeanofthe 10-yearbondrate) tofuturecouponpayments. Table1: 10-YearGovernmentBond RateEquation (r10) equilibrium relationship: r 1 0 =.46+1.0leads 4 0 ( r e )-.79leads 4 0 ( ~x e )+ .85lags 1 ( ~(cid:22) 1 0 ). R 2 .98 SEE.32 properties: meanresponselagtosurprise=0quarters. span: 63q1-94q4 definitions: r 1 0 -ten-yeargovernmentbondrate. r -federal fundsrate. ~x -aggregateoutputgap. ~(cid:22) 1 0 -termpremiumresidual. Theestimatedequationforthe10-yearbondrateinFRB/USisdisplayedintable1.4 Thejoint assumptions of rational expectations and the absence of frictions on portfolio adjustments leave only the term premium to explain predictable movements in the bond equation residual. Some of the predictable variation in the term premium is explained by a 40-quarter lead of expected deviations of aggregate output from trend output, ~x e +t i , where the negative coefficient sum, -.79, indicates countercyclical movements in the term premium. Predictable variation in the term premium residual, after accounting for the effect of expected output deviations, is approximated by a one-quarter autoregressive lag, lags 1 ( ~(cid:22) 1 0 ), where the notation for lags parallels that used for leads. Asindicatedintable1,themeanlagresponsetounanticipatedshocksiszerobecausethebond rate is assumed to incorporate current-quarter news. Additional reported statistics include the 4The construction of expectations used in estimating model equations, such as the funds rate forecasts for the 10-yearbondrateequation,isdiscussedinsection4.

10 proportionofsamplevariationinthe10-yearbondratethatisexplainedbytheestimatedequation, R 2 =.98,andthestandarddeviationofthequarterlyequationresidual,SEE=.32. 2.2 Rational behavior under frictions. In contrast to financial markets, where quarterly outcomes are assumed to reflect equilibrium valuations, behavior in most other sectors of FRB/US is constrained by costs of adjustment. This section describes the basic features of behavior in FRB/US when actions planned by sectors are subject to dynamic frictions. For concreteness, the aggregate price equation of FRB/US will illustratethebasicfeatures ofrationalbehaviorunderfrictions. As indicated earlier, optimal behavior under frictions is described as a two-stage decision process. In the first stage, a sector decides on the equilibrium setting of a variable under its control. The equilibrium setting is the value that would be selected in the absence of frictions and is derived from standard conditions for profit or utilitymaximization. In the case of the price equation, profit maximization under imperfect competition requires the optimal (log) producer price to be proportional to the (log) marginal cost of production. Because long-run production is Cobb-Douglas in FRB/US, the marginal cost of production can be expressed as a weighted average of unit labor and energy costs. As shown in the first equation of table 2, the long-run elasticitiesoftheaggregatepriceequilibriumwithrespect tounitcostsoflaborandenergyare.98 and.02,respectively. Toallowforpossiblecyclicalvariationsinperceiveddemandelasticitiesand marginalcostsofproduction,theequilibriumconditionalsospecifiesthattheoptimalpricemargin may vary with the aggregate unemployment rate. The negative coefficient of the unemployment rate, u ,indicatesprocyclicalvariationintheequilibriumprice,withadeclineintheunemployment rateofonepercentagepointraisingthelevelofthedesiredpricemarkupby0.3percent. In the second decision stage, after selecting an equilibrium setting, the sector formulates an optimal approach to the equilibrium price. A standard model of optimal price adjustment under frictionsisthatbyRotemberg(1987),wherecostsassociatedwithchangingpricesareproportional tothesquareddeviationofthecurrentpricefromlastperiod'sprice, ( p t (cid:0) p (cid:0)t 1 ) 2 . Moregenerally, this quadratic penalty on changing the level of sector activity is the standard approximation of frictions used in all areas of applied macroeconomics, in part because it provides a rationale for thefamiliarpartialadjustmentmodelinwhichafixedfractionofthedistancetotheequilibriumis eliminatedineachperiod.5 The basic paradigm in FRB/US of adjustment dynamics employs a generalization of this standard adjustment cost specification, which includes not only penalties for changing the level 5A recent historical review of dynamic friction specifications in empirical macroeconomics may be found in BraytonandTinsley(1995).

11 Table 2: AggregatePriceEquation (p) equilibrium relationship: p (cid:3) = : 9 8 ( w (cid:0) (cid:26) ) + : 0 2 p e (cid:0) : 0 0 3 u : remarks: (cid:15) equilibriumconditionincludesalsoeffects offarm andimportprices. dynamic adjustment: (cid:1) p t = (cid:0) : 1 0 ( p (cid:0)t 1 (cid:0) p (cid:3) (cid:0)t 1 ) +.57lags 2 ( (cid:1) p (cid:0)t i )+.43leads 1 ( (cid:1) p (cid:3) e +t i ). R 2 .88 SEE.0025 properties: meanresponselagtosurprise=3.3quarters. span: 63q1-94q4 remarks: (cid:15) dynamicequationincludesanaccelerated responsetoenergypriceinflation. definitions: p -logpriceoffinal salesplusimports lessgov'tlaborandindirect businesstaxes. w -logcompensationperhour(ECI). (cid:26) -logtrendlaborproductivity. p e -logcrudeenergyprice. u -demographically-weightedunemploymentrate.

12 of an action but also for changing its growth rate or for altering a moving average of recent actions.6 There are three reasons to consider a more general description of dynamic frictions: First, in contrast to equilibrium relationships, economic theory offers relatively little guidance on thenatureofdynamicfrictions. Second,thestandardpriorthat firms andhouseholdssmoothonly the levels of macroeconomic aggregates is arbitrary and generally strongly rejected by postwar data. Third, the extendedmodel of frictions developedfor FRB/US convenientlyturns out tobe a restrictedversionofthefamiliarvectorautoregression(VAR)developedbySims(1980),enabling FRB/US to take advantage of the data-oriented techniques of modern time series analysis while estimatingstructuraldescriptionsofrationalbehavior. Under the generalized adjustment cost specification, the basic equation in FRB/US describing rational adjustment under frictions contains three sets of regressors: a single regressor consisting of the distance to the equilibrium that remains at the start of the current quarter, p (cid:0)t 1 (cid:0) p (cid:3) (cid:0)t 1 ; a second set of regressors consisting of lags of the dependent variable; and a third set of regressors containingexpectedfuturechanges inthedesiredequilibriumprice. Forconvenientreference, the dynamicadjustmentequationreportedintable2fortheaggregatepricelevelisreproducedhereas equation2: (cid:1) p t = (cid:0) : 1 0 ( p (cid:0)t 1 (cid:0) p (cid:3) (cid:0)t 1 ) + : 5 7 lags 2 ( (cid:1) p (cid:0)t i ) + : 4 3 leads 1 ( (cid:1) p (cid:3) e +t i ) : (2) The first term after the equal sign indicates that 10 % of the distance to the equilibrium price level is eliminated in each quarter. Additional inertia is indicated in the next set of regressors which includes two lags of the inflation rate, with a weight sum of .57. The coefficients of these lagged terms would be zero if frictions were only associated with costs of changing the level of the price. Thus, an implication of the joint presence of the initial level adjustment term and the lagged inflation terms is that both the level and the inflation rate of the aggregate price index are “sticky.” The third set of regressors are forecasts of changes in the equilibrium price over the planning horizon of firms. As indicated by the subscript notation for forecasted leads, leads 1 ( : ) , theplanninghorizonisinfinitebecausefirms are assumedtobeinfinitely-livedentities. In fact, the effective planning horizon of firms is considerably shortened by the rapid rate of decay in weights assigned to distant forecasts of equilibrium price changes. Under rational planning,there are twosources of influence on therelevance of future events. One is thestandard rate of return required by investorssuch as the discount factor applied to future earnings to derive the current market value of equity. In the case of business firms, a fixed quarterly discount factor of .98 is assumed, which is consistent with a postwar annual real rate of return to equity of about 6Issues in specifying and estimating more general descriptions of adjustment frictions are discussed in Tinsley (1993)andKozickiandTinsley(1995).

13 Figure1: LeadandLagResponseWeightsoftheAggregatePrice Equation sthgiew 21.0 01.0 80.0 60.0 40.0 20.0 0.0 Quarters 21.0 01.0 80.0 60.0 40.0 20.0 0.0 -11 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 8 % . The second and more important source of rapid decay in weights assigned to future periods is the influence of frictions in constraining sectoral adjustments. Intuitively, if frictions are large then the planning horizon must be lengthy because the equilibrium will not be reached quickly. Conversely,iffrictionsare small,therequiredplanninghorizonwillbeshort. Lead and lag response weights of the estimated price equation are reproduced by the downward-sloping solid lines in figure 1. These weights are obtained by a rearrangement of the dynamic adjustment equation so the price in the current quarter, p t , is expressed as an infinite distributedlead of future price equilibria, p (cid:3) +t i , and an infinite distributedlag of past price equilibria, p (cid:3) (cid:0)t i .7 Because the aggregate price eventually converges to the equilibrium price, the combined sum of lead and lag weights is unity. Note that the lead and lag response weights resembleatentwiththecenterpoleanchoredinthecurrentquarter,designatedbytheorigin. Both theleadandlagresponseweightsareapproximatelyzerowithintwelvequarters,movingforwards orbackwardsintimefromthecurrentquarter. Theleadresponseweightsareslightlymoreconcave downward due to the additional tilt introduced by the discount factor. As reported in table 2, the mean response lag to an unanticipated shock (surprise) is 3.3 quarters; this estimate is the mean lag defined by the left half of the weights in figure 1 that extend over past quarters. By contrast, theresponselagtoafutureeventthathasbeenperfectlyanticipated(perfect foresight)isthemean lagassociatedwiththefulldistributionofleadandlagweightsandis approximatelyzero. 7SeederivationsofresponseweightsandadditionaldiscussioninvonzurMuehlen(1996).

14 The alternative dotted line in figure 1 indicates the surprisingly modest effect on the response weights of the aggregate price if the quarterly discount factor of the business sector were to be reduced from .98to .94,consistent withan increase of the annual discount rate from 8 % to 24 % .8 Giventhesizeabletriplingofthediscountrate,thisexercisedemonstratesthatfrictionsonbehavior are generally the dominant influence in determining the effective planning horizons of rational firms andhouseholds. Anotherwaytohighlightdifferencesininterpretingthedynamicresponsesofarationalpricing model under frictions is to contrast the FRB/US equation in table 2 with the dynamic structure of a standard reduced-form Phillips curve. If the distinction between actual and equilibrium prices is suppressed except for the contribution of the unemployment rate to the equilibrium, then the FRB/USaggregatepriceequationcanbeviewedasatwo-sidedPhillipscurvewithcurrentinflation depending on both past inflation and expected future inflation as well as on a cyclical indicator. If it were further assumed that expected future inflation depended only on lagged inflation, then the format of the price equation would resemble that of a simple Phillips curve in which inflation is a function of inflation lags and a cyclical variable. Although the inflation lags appearing in a typical Phillips curve are frequently identified only with inflation expectations, the discussion of rational planning under frictions indicates that reduced-form inflation lags combine the effects of both frictional inertia and expectations. An advantage of the rational adjustment format used in FRB/US is that it provides an empirically tested separation of response lags attributable to frictions on price adjustment from lags associated with forecasting. As implemented in FRB/US, the forecasting lags are conditioned not only on lagged inflation but also on the business sector's perceptionsofresponselagsinothersectors,includingthoseofthecentral bank. 2.3 Heterogeneous households and firms. Althoughequations based on the assumptionof identical individualsoften provide reasonable estimates of macroeconomic aggregates, in some cases there is no recourse but to acknowledge measurable differences among individuals. This section discusses how the standard equation formatsare modifiedtoaccommodateheterogeneityamongfirms andhouseholds. Itisoftenpossibletoaccountforheterogeneityifthereexistsanobservableproxyforacategory ofconstrainedbehavior. Forexample,inthecaseofimperfectcapitalmarkets,investmentspending of some firms may be limited to that which can be financed by retained earnings. Similarly, consumptionplansofsomeindividualsmaybeconstrainedbycurrent income,ratherthanwealth, due to an inabilityto borrow on expected future labor income or an absence of tangible assets for 8As noted in section 3, an annual 25 % discountrate is used in the householdsector to discountexpectationsof uncertainfutureincome.

15 collateralizedborrowing. Inbothexamples,thenetresponsecanbemodeledasaweightedaverage oftheproxyforliquidity-constrainedbehavior(eitherretainedearningsorcurrentincome,inthese examples) and the behavior that would be predicted by the standard equation format for rational adjustmentunderdynamicfrictions. Another way to accommodate heterogeneity is to disaggregate explanatory variables. For example, unlike firms, households have finite lifetimes. Under the assumption of lifecycle averaging of planned consumption, an individual's propensity to spend in each period out of the sum of tangible and intangible wealth is approximately the inverse of the expected remaining lifespan of the individual. Consequently,evenfor a stable age distributionof households,cyclical orsecularchangesinthedistributionsoftypesofwealthoveragecohortswillinducechangesinthe aggregatepropensitytospendfromaggregatehouseholdwealth. Forexample,theexpectedpresent value of labor income is more important for young households, who have lower propensities to spend; by contrast, the expected present value of distributions from pension funds and social security is more important for retirees who, in the absence of bequest motives, have higher propensitiestospend. Effectsofheterogeneityinhouseholdspendingresponsesareapproximated by including the composition of wealth, in addition to total wealth, in equilibrium equations for householdconsumption. 3 A Bird's Eye View of FRB/US This section presents the estimated structure of all equations that have been reformulated using specifications required for either of the two primary forms of rational behavior in FRB/US—arbitrage equilibria and dynamic adjustment subject to frictions. As described later in section 4, these equations were estimated under the assumption of rational expectations, implemented by representing sectoral expectations with VAR forecasts. For each equation, the restrictions imposed by rational expectations were tested, as was the hypothesis that the equation residuals were serially independent. Results of these tests, reported in Appendix A, generally supportthenullhypothesesofrational expectationsandseriallyindependentresiduals. Beforesurveyingindividualestimatedequations,abriefpreviewofthreecategoriesofdynamic adjustment equations may be useful. First, as noted in section 2, the generic format for dynamic adjustment under frictions expresses the first difference of a variable as a function of the gap between the variable and its equilibrium value, lags of first differences of the variable, and leads of expected first differences of its equilibrium. Three dynamic equations that use this standard templateare thequarterlyadjustmentsininventories,price,andwage. A second category of dynamic adjustment formats is due to heterogeneous behavior, when some households and firms follow the standard adjustment model and others behave according

16 to another criterion. Three equations exhibit heterogeneous behavior: aggregate consumption includes effects of liquidity-constrained households; investment in producers' durable equipment incorporates effects of liquidity-constrained firms; and labor hours include some workers whose hours are costless to adjust. In each case, extra regressors are added to represent the alternative behavior, and coefficient restrictions are imposed so that the proportions of each type of behavior appearingintheaggregateadjustmentequationsumtoone. A third dynamic adjustment category is also represented in investment equations. Although investment decisions aim at achieving a desired capital stock, the estimated equations do not include capital stocks among the explanatory variables. This omission is a consequence of a current lack of data on stocks; nevertheless, even if stock data were available, doubts about its quality might argue against its use. Investment equations have been modified to approximately captureeffectsthatarisewhentheunderlyingtargetisastock. Themodifications,whichappearin equationsforinvestmentinmotorvehicles,otherdurables,residentialconstruction,andproducers' durable equipment, involve the inclusion of extra growth rate variables which operate like the outputacceleratorfoundinmanytraditionalinvestmentequations. 3.1 Households Mosthouseholdsbaseconsumptiondecisionsonexpectedlifetimeassets,which equal current property wealth plus the present value of expected after-tax labor and transfer income. Risk aversion causes future income flows to be discounted quite heavily with the consequence that households do not act to fully offset government fiscal actions, as suggested by the Ricardian equivalencehypothesis. Anothercharacteristic ofthe householdsectoris that movementsinlabor force participation are driven by changes in social norms in the long run, represented by time trends, and by the availability of jobs in the short run. The real, after-tax wage has no effect on laborsupply. Thus,therearenoeffectsoffiscalpolicyonthelaborforce,norareconsumptionand laborsupplydecisionsintertwined. Aggregateconsumption(table3). AsinMPS,aggregateconsumptionisdefinedastheservice flow derived from the stock of consumer durables plus spending on nondurables and services. In the steady state, aggregate consumptiondepends on total wealth—defined as the after-tax present value of the sum of labor, property, and transfer income—and its composition. Expected future income flows are discounted at a 25 percent annual rate in computing present values due to the aversion of households to the uncertainty of future income. The composition of wealth matters because the lifecycle hypothesis suggests that, absent strong bequest motives, the propensity to spend out of expected lifetime resources at the individual level increases with age. Types of incomeandassetsarenotevenlydistributedoveragegroups;thus,consumptionaggregatedacross

17 individuals varies with the composition of total wealth. Based on a linear approximation of the logarithmic equilibrium condition, marginal propensities to consume out of categories of income and tangible wealth are .51 for labor income, 1.05 for transfer income, .39 for property income, .030 for corporate equities, and .075 for other net tangible assets.9 In addition to total wealth and its composition, desired aggregate consumption also depends positively on the output gap, approximating the effect of countercyclical variation in the perceived riskiness of future income flows. Table3: AggregateConsumption Equation ( c ) equilibrium relationship: c (cid:3) = 1 : 0 v + : 6 2 s tr a n s (cid:0) : 1 5 s p r o p + : 5 2 s s to c k + 1 : 2 8 s o + : 0 1 3 ~x : dynamic adjustment: (cid:1) c t = (cid:0) : 1 2 ( c (cid:0)t 1 (cid:0) c (cid:3) (cid:0)t 1 ) +.17lags 1 ( (cid:1) c (cid:0)t i ) +.75leads 1 ( (cid:1) c (cid:3) e +t i ) +.09 (cid:1) y t : span: 63q1-95q4 R 2 : .54 SEE: .0032 MRL a : 7.9quarters definitions: c -logconsumption(includingserviceflowofstockofdurables). Y -income(labor+transfer+property). y -log Y . V -wealth=leads 1 ( Y e ) . v -log V . s tr a n s -transferwealth/ V . s p r o p -propertywealth/ V . s s to c k -valueofcorp. equity/ V . s o -othernetfinancial andtangibleassets/ V . ~x -aggregateoutputgap. a Meanresponselagtoasurprise. The dynamic consumption equation is a weighted average of the behavior of lifecycle and liquidity-constrainedhouseholds. Theshareofincomeassociatedwiththelattergroupisabout10 percent,basedontheestimatedcoefficient oncontemporaneousincomegrowth. Thisdirect effect ofincomegrowthisinadditiontothecontributionofincomegrowthintheVARforecastingmodel for expectations of target consumption that was used in estimation of the dynamic consumption equation. Lifecycle consumers adjust spending sluggishly, with a mean response lag to shocks (surprise)ofabouttwoyears. 9The equilibrium equation contains both the present value of property income and a Flow of Funds estimate of property wealth because the latter controls for differences between the market valuation of property assets and the valuationofpropertyincomeusingafixeddiscountrate. Foramoredetaileddescriptionofhouseholdconsumption andinvestment,seeReifschneider(1996).

18 Table4: HouseholdInvestment Equations ( c d v , c d o ,and i h ) equilibrium relationships: c c i (cid:3) d (cid:3) d (cid:3) h v o = = = 1 1 1 : : 0 : 0 0 c c c (cid:3) (cid:3) (cid:3) (cid:0) (cid:0) (cid:0) : : 4 : 5 1 3 6 6 r ( ( h p p d d (cid:0) v o (cid:0) (cid:0) : 0 0 p p 3 ) c ) c 4 t (cid:0) (cid:0) 7 + : 4 : 0 : 1 2 0 ( r 0 p d 3 g o t a + 8 s 8 (cid:0) : : 0 p 0 c 4 ) t (cid:0) 8 2 : : 0 3 r d v : dynamic adjustment: (cid:1) c d v ;t = (cid:0) : 3 0 ( c d v (cid:0) ;t 1 (cid:0) c (cid:3) d v (cid:0) ;t 1 ) -.28lags 1 ( (cid:1) c d v (cid:0) ;t i ) +3.22leads 1 ( (cid:1) c (cid:3) d e v + ;t i ) + 7.46lags 4 ( (cid:1) c (cid:3) d v (cid:0) ;t i ) . span: 63q1-94q4 R 2 : .43 SEE: .0054 MRL a : 3.3quarters (cid:1) c d o ;t = (cid:0) : 1 0 ( c d o (cid:0) ;t 1 (cid:0) c (cid:3) d o (cid:0) ;t 1 ) +.17lags 1 ( (cid:1) c d o (cid:0) ;t i ) +2.15leads 1 ( (cid:1) c (cid:3) d e o + ;t i ) + 1.12lags 4 ( (cid:1) c (cid:3) d o (cid:0) ;t i ) . span: 64q1-95q4 R 2 : .34 SEE: .0016 MRL a : 7.3quarters (cid:1) i h ;t = (cid:0) : 0 9 ( i h (cid:0) ;t 1 (cid:0) i (cid:3) h (cid:0) ;t 1 ) +.38lags 1 ( (cid:1) i h (cid:0) ;t i ) +6.10leads 1 ( (cid:1) i (cid:3) h e + ;t i ) + 4.15lags 4 ( (cid:1) i (cid:3) h (cid:0) ;t i ) . span: 63q1-95q4 R 2 : .60 SEE: .034 MRL a : 5.9quarters remark: (cid:15) dynamicequationfor i h alsoincludesvariablesfordeposit disintermediationin1966-7andforcredit controlsin1980. definitions: c d v -logconsumerexpendituresonmotorvehicles(constantdollars). c d o -logconsumerexpendituresonotherdurables (constantdollars). i h -loginvestmentinresidentialconstruction(constantdollars). c (cid:3) -logaggregateconsumptiontarget (seetable3). p d v and p d o -logprices ofmotorvehiclesandotherdurables. p c -logpriceofaggregateconsumption. p g a s -logretail gasolineprice,adjustedforvehiclefuel efficiency. r d v , r d o and r h -costs ofcapital,motorvehicles,otherdurables,andhousing. t 8 2 , t 4 7 and t 8 8 -quarterlytimetrends starting82q1,47q1and88q1. a Meanresponselagtoasurprise. Household investment (table 4). The equilibrium or long-run ratios of household investment in motor vehicles, other durables, and housing to aggregate consumptionare functions of relative prices, user costs of capital, and time trends (for the equilibrium ratios of other durables and

19 housing). User costs incorporate effects of depreciation and the real rate of interest. The latter is defined as a nominal interest rate—the auto loan rate in each of the durables equations and the fixed-rate mortgage yield in the housing equation—less an expected consumer inflation measure ofcorrespondingmaturity,adjustedforthemarginal taxrate. Asnotedearlier,thedynamicadjustmentequationforeachcomponentofhouseholdinvestment includes a modification of the standard rational adjustment format to account for the requirement that investment decisions are aimed at achieving a desired stock of capital, in addition to optimal rates of investment. This modification introduces accelerator effects to the investment equations by adding lags and additional leads in the growth rates of desired investment as regressors. As indicated by the large coefficient sums on the growth rates of equilibrium investment in the dynamic adjustment equations of table 4, accelerator effects are most pronounced for two categoriesofhouseholdinvestment,motorvehiclesandhousing. 3.2 Firms Firms produce with a long-run Cobb-Douglas technology, setting equilibrium factor input quantities (labor, capital, and energy) and the price of output to maximizeprofits. The production function and first-order conditions for profit maximization define the equilibrium equations for investmentin producers' durable equipment,laborhours,and theprice ofoutput. Energy demand isdeterminedasanexogenousratiotooutputratherthanbyafirst-ordercondition. Thefirmsector also includes equations for inventories and hourly labor compensation. The structure of the latter iscloselyrelatedtothatofthepriceequation. Investment in producers' durable equipment (table 5). The desired capital stock varies proportionally with output and, based on the Cobb-Douglas production structure, inversely with the user cost of capital. In the latter, financing costs are measured with weights of 0.8 on the cost ofdebtand0.2onthecostofequity;theweightswerechosentomaximizethefitoftheequipment equation. Theratiooftargetinvestmenttotargetcapitalequalsthesumofthedepreciationrateand thegrowthrate ofoutput. In theequilibriumequationfor investment,whichis writteninlogs,the log of the sum of depreciation and output growth has been linearized so that output growth—the accelerator—appears separately.

20 Table5: Business Investment Equations ( i p d and k i ) equilibrium relationships: i k (cid:3) p d (cid:3) i = = 1 1 : 0 : 0 x x b : b (cid:0) 1 : 0 r p d + 1 : 0 z p d + 1 9 : 5 (cid:1) x b : dynamic adjustment: (cid:1) i p d ;t = (cid:0) : 0 7 ( i p d (cid:0) ;t 2 (cid:0) i (cid:3) p d (cid:0) ;t 2 ) +.26lags 2 ( (cid:1) i p d (cid:0) ;t i ) +.47leads 1 ( (cid:1) i (cid:3) p e d + ;t (cid:0)i 1 ) + .22lags 2 ( (cid:1) c f (cid:0)t i ) . span: 64q1-94q4 R 2 : .40 SEE: .0022 MRL a : 7.0quarters (cid:1) k i;t = (cid:0) : 2 3 ( k (cid:0) i;t 1 (cid:0) k (cid:3) (cid:0) i;t 1 ) +.47lags 3 ( (cid:1) k (cid:0) i;t i ) + .53leads 1 ( (cid:1) k (cid:3) e + i;t i ) . span: 62q3-94q4 R 2 : .42 SEE: .0065 MRL a : 1.3quarters remarks: (cid:15) dynamicequationfor i p d is aweightedaverageofadjustment model(.78)andcashflowmodel(.22). (cid:15) adjustmentmodelcomponentfor i p d includes1-quarterdeliverylag. definitions: i p d -loginvestmentinproducers' durableequipment(constantdollars). k i -logstockofmanufacturingandtradeinventories(constantdollars). x b -logoutput,businesssector(constantdollars). r p d -logusercostofcapital,producerdurables. z p d -log(depreciationrate+meanof (cid:1) x b ). c f -logcorporate cashflow(constantdollars). a Meanresponselagtoasurprise. Thedynamicinvestmentequationaugmentsthegeneralizedadjustmentmodel,whichdoesnot fully capture cyclical investment fluctuations, with cash flow. The added variable is motivated by recent empirical literature suggesting that some firms are constrained in their access to capital markets. The resulting equation places 20 percent weight on the growth of cash flow and 80 percent weight on the standard adjustment specification. Considering only the adjustment part of the equation, investment is found to be fairly sluggish, with a mean response lag of almost 2 years. The response includes a one-quarter delivery lag which was found to improve the fit of the equation; the delivery lag causes the timing of the level correction term and the expectations variablestobeshiftedbackonequartermorethaninthestandardadjustmentspecification. Inventory investment (table 5). The model's main inventory equation determines the stock ofmanufacturingand tradeinventories,where theinventorytarget stock, k (cid:3) ,is proportionaltothe

21 outputofthebusinesssector, x b . Relativetotheadjustmentsofmostnonfinancialaggregates,firms adjustinventoryholdingstorevisedtargetlevelsrapidlywithameanlagresponsetounanticipated shocksof1.3quarters. Laborinput(table6). InMPS,theequilibriumlevelofaggregatelaborhourswasbasedonthe sumoflaborrequirementsacrossthevintagesofcapitalneededtoproduceagivenlevelofoutput. Thiscondition,derivedfromanassumedputty-claycharacteristicofcapital,wasapproximatedby making target hours a function of output, total factor productivity (specified as a time trend), and contributionsofaveragecapitalandenergyintensitiesofexistingcapacitytothelabor-outputratio. Coefficientsbasedoncapitalandenergyintensitieswereimposed,basedonaveragefactorshares, without causing too much deterioration in the equation's goodness of fit. However, with revised NIPA data, imposing the factor-intensity effects in FRB/US led to a substantial deterioration of fit. Thus, the equilibrium condition has been simplified to make target hours a function of output and a pair of time trends capturing the well-known slowdown of productivity growth in the early 1970s.10 Laborhoursaremodeledasheterogeneouswithrespecttoadjustmentcosts. Aboutone-thirdof hoursisestimatedtorespondimmediatelytochangesintargethours,withtheremainderfollowing the generalized adjustment cost model. Aggregatedacross bothtypes of labor, the mean response lagtoasurpriseisveryshort,less thanaquarter. Prices and Wages (table 6). The steady-state structures of the price and wage equations are derived from the equilibrium Cobb-Douglas assumption that the share of income received by capital is constant in the steady state. If the share of income received by energy producers is stable, then constancy of the capital share implies constancy of the labor share. Price and wage targetsalsovaryprocyclically. Thetwoequilibriumconditionsjointlydeterminetherealwageand the NAIRU. The latter is constant when defined by a demographically-weighted unemployment rateandabitless than6percent,currently,intermsofthecivilianunemploymentrate. The generalized frictions model yields level error correction terms in the dynamic adjustment equations for both price and wage. The level correction term in the price equation is a standard feature of level price markup equations and captures variations in price margins. By contrast, a level correction term in wage equations is less common, but consistent with models of wage bargaining. The dynamics of wages and prices are strongly interrelated because the target wage is largely determinedbythe priceleveland,reciprocally,thetarget price isa functionprimarilyof thewage level. Withregardtoadjustmentspeeds,wages areestimatedtobemoresluggishthanprices. The mean response lag in the wage equation (8.7 quarters) is more than twice the mean response lag 10A simulation option is available to make labor productivity respond gradually to movements in relative factor prices,similartothecorrespondingMPSequation.

22 Table6: Aggregate LaborHours,Wages,and Prices ( h , w ,and p ) equilibrium relationship: h w p (cid:3) (cid:3) (cid:3) = = = 1 1 : 9 : 0 x : 0 (cid:26) 8 ( w g + (cid:0) (cid:0) 1 : 0 0 6 : 0 2 p (cid:26) ) + 4 9 t (cid:0) g : 0 7 2 + : p 0 e : 2 (cid:0) 0 p 0 e : 4 (cid:0) 0 2 0 t : 3 7 3 : 0 1 u : u : remark: (cid:15) equilibriumconditionfor p alsoincludeseffects offarm andimportprices. dynamic adjustment: (cid:1) h t = (cid:0) : 1 5 ( h (cid:0)t 1 (cid:0) h (cid:3) (cid:0)t 1 ) +.38lags 1 ( (cid:1) h (cid:0)t i ) +.41leads 1 ( (cid:1) h (cid:3) e +t i ) . +.31 (cid:1) h (cid:3) t -.12lags 1 ( (cid:1) h (cid:3) (cid:0)t i ) . span: 63q1-94q4 R 2 : .76 SEE: .0046 MRL a : 0.7quarters (cid:1) w t = (cid:0) : 0 3 ( w (cid:0)t 1 (cid:0) w (cid:3) (cid:0)t 1 ) +.71lags 3 ( (cid:1) w (cid:0)t i ) +.29leads 1 ( (cid:1) w (cid:3) e +t i ) . span: 63q1-94q4 R 2 : .82 SEE: .0028 MRL a : 8.7quarters (cid:1) p t = (cid:0) : 1 0 ( p (cid:0)t 1 (cid:0) p (cid:3) (cid:0)t 1 ) +.57lags 2 ( (cid:1) p (cid:0)t i ) +.43leads 1 ( (cid:1) p (cid:3) e +t i ) . span: 63q1-94q4 R 2 : .88 SEE: .0025 MRL a : 3.3quarters remarks: (cid:15) dynamicequationfor h isaweightedaverageofstandardadjustment model(.69)andimmediateresponsemodel(.31) . (cid:15) dynamicequationfor w alsoincludesvariablesforwage andpricecontrols, employersocialinsurancecontributions,andtheminimumwage. (cid:15) dynamicequationfor p alsoincludesanaccelerated responsetoenergypriceinflation. definitions: h -loghours,nonfarm businesssector(employeesandself-employed). w -logcompensationperhour(ECI). p -logpriceoffinal salesplusimports lessgov'tlaborandindirect businesstaxes. x g -logoutput,nonfarmbusinesssectorplusoilimports lesshousingproduct (constantdollars). t 4 7 and t 7 3 -quarterlytimetrendsstarting47q1and73q1. (cid:26) -logtrendlaborproductivity. p g -logpriceof x g less indirectbusinesstaxes. p e -logcrudeenergyprice. u -demographically-weightedunemploymentrate. a Meanresponselagtoasurprise.

23 (3.3 quarters) in the price equation. Wages are also less responsive to forward expectations, with a coefficient sum of .29ontheexpected rate ofgrowthof theequilibriumwage comparedwithan expectations coefficient sum of .43 in the price equation. The dynamic wage and price equations contain inflation-neutrality restrictions to insure that the equilibrium real wage and NAIRU are independentoftherate ofinflation.11 Other equations. Nonresidential construction is assumed to move with business output and a time trend. In the current version, no effects of interest rates or tax policy are included. Employmentisdeterminedbyastandarderror-correctionequationinwhichtheemploymenttarget dependsonaggregatehoursandtrends intheworkweek. 3.3 Financial markets Equationsforthreelong-terminterestratesandthestockmarketcomprisethecoreofthefinancial marketsectorofFRB/US.Unlikenonfinancialbehavior,wherefrictionsmakeittoocostlytomove immediatelytoequilibriumvalues,asset prices areassumedtobeinequilibriumcontinuously. Long-term bond rates (table 7) are determined according to the expectations theory of the term structure. The theory posits that, up to a term premium, the yield on a long-term bond is given by the expected future path of short-term interests rates. Abstracting from term premia, the yield on a 5-year government bond is a weighted average of expected federal funds rates over the next 20 quarters.12 The yields on the other two long-term bonds—the 10-year government bond andtheMoodyAAAcorporatebond—aremodeledinananalogousway. Term premia for all three bond equations are estimated to vary negatively with a weighted average of the output gap expected over the maturity of the bond. One can interpret this negative relationship to mean that investors require larger risk premiums when they expect a deterioration in the average performance of the economy over their investment horizon. Consistent with this notion, the sensitivity of the term premium to expected economic conditions increases with the maturityofthebond. 11Theserestrictionssetthesumofthecoefficientsontheleadandlaginflationtermstoone.Analogousrestrictions werenotimposedonotheradjustmentequations,becausetheyexplainthebehaviorofrealvariableswhosesteady-state growthratesarelikelytohaverelativelysmallvariations. 12Theweightsattachedto eachfuturevaluedeclineabout2 percentperquarter. Theimpliedaveragedurationof the5-yeargovernmentbondrateisapproximately4years.The10-yeargovernmentbondandthecorporatebondhave durationsof7and12years,respectively.

24 Table7: Financial Sector Equations ( r 5 , r 1 0 , r c b ,and v s ) 5-yeargov't bondrate a : r 5 ;t = : 3 4 + 1 : 0 leads 2 0 ( r e +t i ) (cid:0) : 6 2 leads 2 0 ( ~x e +t i ) + : 8 3 lag 1 ( ~(cid:22) 5 (cid:0) ;t i ) span: 63q1-94q4 R 2 : .97 SEE: .47 MRL b : 0quarters 10-yeargov't bondrate a : r 1 0 ;t = : 4 6 + 1 : 0 leads 4 0 ( r e +t i ) (cid:0) : 7 9 leads 4 0 ( ~x e +t i ) + : 8 5 lag 1 ( ~(cid:22) 1 0 (cid:0) ;t i ) span: 63q1-94q4 R 2 : .99 SEE: .32 MRL b : 0quarters corporate bondrate a : r c b ;t = 1 : 2 1 + 1 : 0 leads 1 2 0 ( r e +t i ) (cid:0) 1 : 2 1 leads 1 2 0 ( ~x e +t i ) + : 8 7 lag 1 ( ~(cid:22) 3 0 ;t ) span: 63q1-94q4 R 2 : .99 SEE: .27 MRL b : 0quarters stockmarket wealth: v s ;t (cid:0) p g ;t = 4 : 7 + d t + 5 0 leads 1 ( (cid:1) d e +t i ) (cid:0) 5 0 ( ( r c b ;t = 4 0 0 ) (cid:0) leads 1 2 0 ( (cid:1) p e c + ;t i ) ) span: 65q1-95q4 R 2 : .97 SEE: .20 MRL b : 0quarters definitions: r -federal fundsrate. ~x -outputgap. ~(cid:22) 5 , ~(cid:22) 1 0 ,and ~(cid:22) 3 0 -termpremiumresidualsfor r 5 , r 1 0 ,and r c b . v s -logstockmarketwealth(current dollars,flowoffundsaccounts). d -lognationalincomedividends(constantdollars,deflated by p g ). p g - logprice,businesssectoroutput. (cid:1) p c c -inflationrate, householdconsumptionprice. a c Forthethreebondequations,thereportedSEEandR 2 arecomputedafter adjustmentforfirst-orderserialcorrelationoftheterm-premiumresiduals. b Meanresponselagtoasurprise. c Priceindexesdividedby100beforetakinglogarithms. Stock market wealth (table 7). Similar to the log linearized model in Campbell and Shiller (1989), the real value of the stock market is determined by expectations of the future flow of real dividend payments. Future expected dividends are discounted by the expected opportunity yield oncorporate bonds,measuredbythecurrent corporatebondrateless expectedconsumerinflation rates over a 30-year horizon. Both the stream of expected real dividends and the real corporate bond rate are multiplied by a normalization factor due to linearization about the sample average of the real return on equity. As noted in the fourth equation of table 7, on a quarterly basis, the

25 normalizationfactor is approximately 1/.02 = 50. Additional compensationfor equity ownership, apart from the cyclical risk premium already embedded in the corporate bond rate, is captured by freely estimating the intercept of the equity equation. However, the substantial residual serial correlationofthisequationsuggeststheadditionalriskpremiumrequiredbyhouseholdsforequity holdingis notconstantovertime. Mortgage and car loan rates. The equilibrium relationships for the mortgage rate and the loan rate on new cars are based on the 10-year and 5-year government bond rates, respectively. In the short run, movements in the mortgage and new car loan rates reflect partial adjustments to long-runequilibria;themortgageratealsovaries countercyclically. 3.4 Foreign trade and government sectors In the current model version, these sectors are constructed using unrestricted error correction regressionsandnot therestrictedrationaladjustmentspecificationsdescribedinsection2. Abrief descriptionfollowsofthetheoretical motivationsofthemainequationsinthesesectors. The equations describing the long-run determinants of real exports and imports are standard. Real exports depend on foreign GDP and the real exchange rate. The latter is defined by open-interest parity arbitrage with an expected long-term real rate of interest and a country risk premium,whichisa functionofU.S.netforeignindebtedness. Real nonoilimportsare afunction of domestic GDP and the relative price of imports. Each trade equation contains a time trend and is formulated as an error correction; long-run income elasticities are constrained to unity and long-runpriceelasticitiestominusunity. The government sector of FRB/US is disaggregated into two tiers: federal and state & local. Most variables in this sector are either exogenous or defined through identities. However, tax paymentsareendogenous,asaretransferstopersons,whichareestimatedtohavecomponentsthat varycountercyclically,andnetinterestpayments,whicharefunctionsofstocksofdebtoutstanding andinterestrates.

26 4 Expectations ExpectationsoffutureeventsareimportantdeterminantsofprivatesectorbehaviorinFRB/US, especiallyovermultiperiodhorizonsastheinitialrestraininginfluencesoffrictionsdissipate. This sectiondiscussestheformulationofexplicitexpectationsinFRB/USandoptionsthatmaybeused inhypotheticalscenarios as wellas thoseusedinestimation. Inestimatingthestructuraldescriptionsofbehaviordescribedinsection3,explicitexpectations ofthedesiredequilibriaoffirms andhouseholdswererequiredtoidentifythefrictionsthatinhibit dynamic adjustments. Although explicit expectations need not be rational, the assumption that sectors formulate rational expectations using a condensed model of the economy was imposed in estimation and not empirically rejected in most instances. Rational expectations (RE) denote forecasts of variablesthat are consistentwithexpected outcomesof themodeledmechanisms that generatethesevariables.13 Althoughtheassumptionofrationalexpectationshasprovedtobeausefulorganizingprinciple inestimatingthemodel—aswastheassumptionthatfinancialauctionmarketsareefficient—there are obviously instances when such an assumption is unlikely to provide a good prediction of short-run behavior, as in the aftermath of an unusual event which is not well understood by individuals in any sector, such as the prospect of war or the collapse of a market. Also, prior reasoning cannot indicate the level of detailed information about the general economy that is neededtomakerationaldecisionsinaparticularsector. Therefore,severaloptionsexist(orarebeingdeveloped)forFRB/USthatvarytwodimensions in testing or imposing rational expectations in estimation and simulation. One dimension is the scope of rational expectations, where sector forecasts may be consistent for summary aggregates but not necessarily for all forecast components. The other is the speed of rational expectations formationwhere,inthemosttypicalcase,privatesectorperceptionsmaybeconsistentwithpolicy goalsinthelongrunbutnotnecessarilyovershortforecast horizons. Discussion in this section is organized as follows: The first subsection outlines the main options that specify the scope of information shared by sectors. The second subsection discusses specificationsoflong-runexpectationsinFRB/US,includingprivatesectorperceptionsoflong-run policygoals. Finally,athirdsubsectiondescribestheVARmodelusedintheestimationofFRB/US togenerate sectorexpectations. 13Note that rational behavior by individuals, in the sense of optimal dynamic planning, need not imply rational expectations.AsdiscussedbySargent(1993),bothrationalplanningandconsistentforecastsarerequiredforrational expectations. ThefirstconditionofrationalbehaviorisalwaysimposedinFRB/USstructuralequations,whereasthe second conditionof consistentperceptionsby firms and householdsis a simulation option. Other optionsthat limit theinformationorcomputingabilityofindividualsareexamplesofboundedorlimitedrationality. Notallboundson rationalityadmitrationalexpectations,eveninthelongrun.

27 4.1 The scope of sectoral information. The two main options for the scope of information shared in common by firms, households, andinvestorsare: Full-model expectations. Under full-model expectations, all sectors use the same forecast model of the economy. The common forecast model is a closed version of FRB/US, meaning that that thebehavioral responses ofall sectors—includingpolicyresponses—arespecified.14 The forecast of a variable appearing anywhere in the model will be equal to the forecast generated by the full FRB/US model. The full-model expectations option is the conventional assumption regarding sharedinformationinREmodelsandprovidesausefulbenchmarkofpolicyeffectsundercomplete informationandperfect foresightofselectedfutureevents. Because FRB/US is nonlinear, full-model expectations are obtained by iterative, numerical simulations under the assumption that future shocks are either known (perfect foresight) or equal to zero.15 In discussing the method of solving for full-model expectations, the 10-year bond rate equationisagainusedasanexample. Thebondraterequires40-quarterforecastsofthefundsrate andtheoutputgap(trenddeviation). r 1 0 = : 4 6 + 1 : 0 leads 4 0 ( r e ) (cid:0) : 7 9 leads 4 0 ( ~x e ) + : 8 5 lag 1 ( ~(cid:22) 1 0 ) : (3) Thefull-modelexpectationssolutionproceedsbyiterativerevisionsofexpectations.16 Assume that initial“guesses” byinvestors of 40-quarter forecasts of the funds rates and the output gapare available for each quarter of the forecast horizon. These will define investorforecasts of the bond rate in each forecast period. However, after also solving the full model, where each sector uses its own “guesses” to initialize forecasts of explanatory variables, the forecasts of the funds rate and output generated by the full model will not generally match the initial investor guesses used toconstructthebondrateforecasts. Similarly,theinitialbondrateguessesusedinotherequations of the model, such as the guesses by the business sector to forecast the cost of capital for fixed investment, will generally not match the bond rate forecasts produced by equation 3. So, in the 14Obviously,individualsinprivatesectorscannotformulaterationalforwardplanswithoutadescriptionofexpected policies. For purposes of discussion in this section, monetary policy is summarized by an equation describing the responsesofthefederalfundsratetorecentmovementsininflationandthetrenddeviationinoutput. Otherwaysto characterizemonetarypolicyarediscussedinsection5. 15The practice of setting future residuals to zero (thus ignoring nonzero expectations of nonlinear functions of residuals) has become a convention in macroeconomic constructions of full-model expectations, which are often termedmodel-consistentexpectations. ThebiasofexpectationsgenerallyissmallbecauseFRB/USisapproximately linearforsmallshocksand,indeed,formanypurposesmaybeviewedasalarge,restrictedVAR. 16Thefollowingisasimplified descriptionoftheFair-TaylormethodofsolvingRE models, whichis aniterative Jacobimethodofsolvingsystemsofdynamicequations.

28 next iteration, each sector uses the multiperiodforecast of the model to replace its initial guesses. Thisiterativecycleofupdatingsectoralexpectationsbymodelforecastsolutionsfromtheprevious iteration and then solving the full model for a new multiperiod forecast sequence continues until sectoralexpectationsexactlymatchthefull-modelforecast. SinceFRB/USisapproximatelylinear for small shocks,this iterativeprocess willconverge to a uniqueforecast, if initialguesses are not toodistantfromthesolution. This cursory description ignores an important issue involving the forecast horizon. Note that the bond rate in the last period of the forecast horizon requires 40 more quarters of funds rates and output gap forecasts. But these variables, in turn, require accompanying bond rate forecasts. Thissecondsequenceofmoredistantbondratesrequires40additionalquartersoffundsratesand output gap forecasts, and so on. This infinite recursive sequence is approximated by solving the model for a large number of periods, say 25 years (100 quarters). In practical terms, the impact of distant variables on present actions dissipates rather quickly for most variables. For example, as noted in section 2, the lead weights in the aggregate price equation are approximately zero at the end of a three-year forecast horizon even though the theoretical planning horizon of firms is infinite. Although the effective planning horizon for most variables does not exceed three or four years,notableexceptionsarelong-maturityinstrumentssuchasbondsandequity. VAR expectations. Under this option, a small vector autoregression (VAR) model of summary macroeconomicaggregatesreplacesthefullFRB/USmodelasthedescriptionoftheeconomythat conditionssectoral forecasts. The sameVAR modelis used byall sectors,althoughdisaggregated information may be added to provide additional local information within a sector. The VAR model most commonly used for expectations in FRB/US is the historical VAR. The historical VAR provides an average-history summary of the dynamic behavior of the economy and was usedtogeneratesectoral expectationsin estimationofFRB/US arbitrageanddynamicadjustment equations.17 Using the same description of policy, general multiplier properties are similar for the historical VAR and the full FRB/US model. This correspondence suggests that the many restrictions in FRB/US required for structural interpretations are not noticeably inconsistent with historicaldata,reinforcingtheresultsofdirectstatisticaltestsofRErestrictionsshowninAppendix A. Given the well-behaved and generally white noise residuals of equations in FRB/US under historical VAR expectations,the historicalVAR is the standardexpectations optionfor short-term forecast analysis. Ifahypotheticalmonetarypolicyissimulatedthatdeviatesmarkedlyfromthepolicyresponses captured by the historical VAR, it is unlikely that forecasts of summary aggregates from the 17Conditionsfor maximumlikelihoodpropertiesof estimated friction parametersbased on VAR expectationsare discussedinBraytonandTinsley(1995)andKozickiandTinsley(1995).

29 historical VAR model and from FRB/US using historical VAR expectations would agree. In this instance,testsofsimulatedbehaviorfromthefullmodelwouldbelikelytorejectahypothesisthat thehistoricalVARexpectationswererational.18 Given the linear structure of VAR models, VAR expectations can be directly obtained by analytical solutions rather than numerical iterations. Thus, the calculation of sectoral VAR expectations, even for infinite forecast horizons, is significantly less computationally demanding thanfull-modelexpectations. Thematrixmanipulations19 requiredforVARexpectationstypically require only a few seconds on a Sparc10 computer, whereas a standard forecast simulation using full-model expectations can take 25-30 minutes. In the example of the bond rate, equation 3, the funds rate andoutputgapare explicitvariablesinthehistorical VAR.Theresultant fundsrate and output gap forecasts will be the same for all sectors since all sectors use the same VAR model to generateexpectations. 4.2 Long-run expectations. As noted in previous sections, long-horizon forecasts of the federal funds rate and inflation rate are important determinants of the equity price and bond rates in the current period. Thus, long-horizon expectations constitute a significant policy transmission channel in FRB/US. Short-runpolicyactions,suchasfundratealterations,canhaveverydifferenteffectsdependingon howthefundsratemovementsinfluence privatesectorperceptionsoflong-runpolicyobjectives. Long-horizon expectations, succinctly called expectation endpoints in FRB/US, are generally definedbythesteady-statevaluesofequilibriumplanningequations. However,theendpointsofthe nominalfundsrateandconsumerinflationrateareexplicitlydefinedinFRB/US,inparttoindicate theroleofpolicyindeterminingthesevariablesbutalsotobettercapturehistoricalshiftsinprivate sector perceptions of the endpoints of these variables. These shifting endpoint expectations apply tobothfull-modelexpectationsandVARexpectations. A default assumptionin manyRE modelsis symmetricpolicyinformation,where the plans of policymakersareknownbyallsectorsoftheeconomy. Bycontrast,asymmetricpolicyinformation is also an option in FRB/US, where private sectors either do not know policy objectives or are sceptical of policy announcements. Explicit endpoint expectations allow the often fuzzy topic 18AnotheroptionunderdevelopmentforVARexpectationsisavirtualVAR,whichisaminiaturizationofanyclosed version of FRB/US. If a hypothetical alteration in the full model changes dynamic relationships among summary aggregates included in the VAR, the reduced-form implications of these changes are captured by the virtual VAR. Virtualor“mapped”VARsareiterativelyestimatedfromtheoutcomesoffullmodelsimulationsusingvirtualVAR expectations.Thisexpectationsoptionisverycomputer-intensiveandthesubjectofongoingwork. 19ThesearesimplemanipulationsofamatrixcontainingtheVARcoefficients.TheVARmodelistransformedinto a first-order autoregressiveformat. A two-periodforecast requires the square of the matrix; a three-periodforecast requiresthecubeofthematrix,andsoon.

30 of policy “credibility” to be precisely framed in FRB/US. If endpoint expectations are based on symmetric policyinformation, then the inflation endpoint of privatesector forecasts will be equal to the long-run inflation goal of a fully credible policy. If policy information is asymmetric then the long-run inflation goal of policy is not known or believed and must be inferred by the private sectorfromobservableindicators,suchas pastrates ofinflation. Endpoints are defined, usually implicitly, in conventional macroeconomic models as either constants (if variables are detrended) or moving averages (if variables contain a random walk component). The differences in summary statistics and one-period prediction errors between equations estimated with these two endpoint alternatives are often small or insignificant, but profounddifferencesappearinlong-horizonforecasts,asshownbelow. Athirdcategoryofshifting endpoints, developed for FRB/US, draws on private sector perceptions of long-run expectations. Theremainderof thissubsectionindicates typicaleffects of alternativeendpointsonlong-horizon forecasts and reviews the historical measurements used in FRB/US to represent shifting endpoint perceptionsoffirms, households,andinvestors. Constant and moving-average endpoints. To illustrate long-horizon forecast effects of the two endpointsusedinconventionalforecastmodels,alternativeforecastsofthefundsratearedisplayed inthetwopanels offigure 2. Forecasts inthetoppanel are generatedbya four-lagautoregression inthelevelofthefundsrate. Thismodelisappropriateifthevariableiswithoutatrend(stationary). Notethatbothforecastsshowninthetoppanel,onestartingfromthehighlevelofinterestratesin 1980:Q1 and the other starting from the relatively low level in 1986:Q4, converge to and remain at a common forecast. This constant endpoint is approximately the mean of the funds rate in the sampleusedtoestimatetheautoregressioninthelevelofthefunds rate. Forecasts in the second panel of figure 2 are generated by a four-lag autoregression in the first-difference of the funds rate. This model is often selected if the variable contains a random walk component (making it nonstationary).20 In this panel, each forecast rapidly converges to and remains at a constant that is near the level of the funds rate at the start of the forecast. This is because, in a first-difference autoregression, the endpoint moves over time and is defined by a weightedmovingaverageoffundsratesintheperiodsimmediatelypriortothestartoftheforecast. A characteristic of the forecasts in both panels of figure 2 is that endpoints are reached rather quickly. Typically,inlinearforecastingmodelsofthefundsrate,thefundsrateendpointisreached by the fifth or sixth year of the forecast horizon.21 Consequently, in the case of long-maturity 20Manyempiricalstudiesinmacroeconomicsandmacrofinanceassumepostwarnominalinterestratesandinflation ratescontainrandomwalkcomponents. 21Obviously,closureisfasterforforecaststhatstartneartheendpointandslowerforinitialforecaststhatarefarther away,suchastheforecastinthetoppaneloffigure2thatbeginsin1980:Q1.

31 Figure2: Autoregressivefundsrateforecasts withalternativeendpoints 22 20 CONSTANT ENDPOINTS 18 16 14 12 10 8 6 4 2 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 FFR 22 20 MOVING AVERAGE ENDPOINTS 18 16 14 12 10 8 6 4 2 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 FFR bonds such as the 10-year bond equation discussed in section 2, the selection of the expected funds rate endpoint will determine much of the variation in the predicted bond rate. As can be inferred from the behavior of the funds rate predictions in figure 2, bond rate predictions from theautoregressioninthelevelofthefundsrateunderstatehistoricalbondmovementsbecausethe long-horizon forecasts of the funds rate converge to the fixed endpoint. On the other hand, bond ratepredictionsfromtheautoregressioninthefirst-differenceofthefundsrate overstatehistorical bond rate variation because the moving-average endpoint is too sensitive to recent levels of the fundsrate. A shifting endpoint for the funds rate. The combination of the rational expectations assumption that bond rates are an average of expected funds rates over the appropriate maturity horizon, as discussed in section 2, and the fact that forward rates in the second five years of a 10-year bond appeartobedominatedbythefundsrateendpointsuggeststhatanaverageofdistantforwardrates intheobservedtermstructurewillprovideadirectestimateofinvestors' time-varyingperceptions of the funds rate endpoint. The historical time series in FRB/US of the perceived funds rate endpoint is based on forward rates in the 10-30 year segment of the term structure. For future

32 reference, thisexpectationsendpointforthefundsrateisdenotedbythesubscriptconvention, r 1 . The empirical fit of bond rate equations using this endpoint, r 1 , is substantially better than those using fixed or moving-average endpoints.22 However, use of this shifting endpoint measurement merely postpones the task of selecting a model that can predict investorperceptions ofthenominalinterestrateendpoint. Ashiftingendpointforinflation. BytheFisheridentity,theexpectedendpointofthenominalfunds rate is a weighted average of the expected real rate endpoint (determined by the marginal product of capital) and the expected inflation endpoint.23 Thus, a source of sizeable movements in the nominal interest rate endpoint is the shifting of investor perceptions of the endpoint of expected inflation. Although survey estimates of individual perceptions of the inflation endpoint such as the Philadelphia survey of 10-year inflation expectations are available in recent years, none are available prior to the major shift in policy in late 1979. In order to estimate a longer historical series and to provide a behavioral description of investors' evolving perceptions of endpoints, KozickiandTinsley(1996)developaninvestorlearningmodelwhereindividualssequentiallytest for statistically significant shifts in the endpoint of expected inflation, given a null hypothesis of an unchanged endpoint. Learning is nonlinear with faster responses to signals of a large change in the inflation endpoint than to signals of small changes. However, movements in the aggregate perception of the inflation endpoint are smoothed because the rate of learning varies among individuals in the economy. The inflation endpoint constructed by this learning model is quite similar to the (discontinued) Hoey survey estimates of inflation expected in the second five years of a 10-year horizon and accounts for most of the sample variabilityof the shiftingnominal interest rate endpoint. The Kozicki-Tinsleyseries of the inflation rate endpoint is spliced withthe Philadelphia estimate of expected 10-year inflation to provide the FRB/US historical estimate of the inflation rate endpoint, (cid:25) 1 , perceived by the private sector; note that this need not be equal to thelong-runpolicytarget forinflation. 4.3 The historical VAR. Under VAR expectations, all sectors share a condensed description of the aggregate economy representedbyathree-variablevectorautoregressioninaggregateoutput,inflation,andthefederal funds rate (where the latteris selected as a summaryindicatorof monetarypolicy). The historical 22Explicitcontrastsofbondratepredictionsfromfundsrateforecastmodelswithalternativeendpointsarepresented inKozickiandTinsley(1996). 23Undertheassumptionthatinvestorsarbitrageafter-taxrealrates,adjustedfordifferentialriskpremia,theweights arefunctionsofthetaxrateoninvestorearnings.

33 VARisanaverage-historyestimateofreducedformrelationshipsamongthesethreevariablesover the32year samplebeginningin1963. Estimated equations of the historical VAR are listed in table 8 for the three summary aggregates: thefederalfundsrate, r ;consumptioninflation, (cid:25) ;andthetrenddeviationinaggregate output(the“outputgap”), ~x . Table8: The HistoricalVAR (cid:1) r = .03lag 1 ( (cid:25) (cid:0) (cid:25) 1 )+.12lag 1 ( ~x (cid:0) 0 )-.05lag 1 ( r (cid:0) r 1 ) SEE1.14 +.33lag 3 ( (cid:1) (cid:25) ) +.22lag 3 ( (cid:1) ~x )-.27lag 3 ( (cid:1) r ). R 2 .30 (cid:1) (cid:25) =-.17lag 1 ( (cid:25) (cid:0) (cid:25) 1 )+.13lag 1 ( ~x (cid:0) 0 )-.01lag 1 ( r (cid:0) r 1 ) SEE1.13 -.27lag 3 ( (cid:1) (cid:25) ) -.17lag 3 ( (cid:1) ~x )+.02lag 3 ( (cid:1) r ). R 2 .26 (cid:1) ~x =-.02lag 1 ( (cid:25) (cid:0) (cid:25) 1 )-.04lag 1 ( ~x (cid:0) 0 )-.21lag 1 ( r (cid:0) r 1 ) SEE1.12 +.09lag 3 ( (cid:1) (cid:25) ) +.19lag 3 ( (cid:1) ~x )+.08lag 3 ( (cid:1) r ). R 2 .33 remarks: (cid:15) span1963q1-1994q4. definitions: r -federal fundsrate. (cid:25) -inflationrate ofpersonalconsumptiondeflator(chainweights). ~x -trenddeviationofoutput. Theequationsintable8differfromthoseinconventionalVARsduetothepresenceofexplicit endpoints for each of the variables in the historical VAR. As discussed above, each endpoint represents private sector perceptions of the long run outcome for that variable. The format of theequationsinthehistoricalVARenforces thisviewwhere,intuitively,alltermsontheleft-hand andright-handsidesoftheVARequationsarezerointhelongrun. Thus,inthelongrun,thefunds rate will reach thefunds rate endpoint, r 1 ; the inflationrate willattainthe inflationrate endpoint, (cid:25) 1 ; andtheoutputgapwillconvergetoitsendpoint,whichiszero.24 In contrast to analysis of individual equations, the average speed of adjustment of variables toward long run values in a fully interdependent system, such as the historical VAR, is the same for all variables in the system. This is evident in table 8, since each endpoint deviationappears in 24Furtherdiscussionofendpoint-deviationformulationsofVARsmaybefoundinBraytonandTinsley(1995)and KozickiandTinsley(1996).

34 allequations;thus,ifoneendpointdeviationisnonzero,thenallendpointdeviationsgenerallywill benonzero. UnlikethestructuralequationsinFRB/US,theequationsinVARmodelsarereducedformsso direct behavioral interpretations are ordinarily not possible. However, under the assumption that thefundsrateisthevariablemostresponsiveto“news”,thefundsrateequationmaybeinterpreted as an average-history representation of policy responses to shocks and observed movements of inflationand output. Under this interpretation,the first equationin table8 indicates that historical policygenerallyincreasedthefundsrate ifeitherinflationoroutputwereabovetheirendpointsor thefundsratewas belowitsendpoint.25 5 Full-System Properties This section provides an overview of the system properties of FRB/US and demonstrates through a few examples how the model can be used to analyze a rich set of forecast and policy questions,includingcaseswherethepublichasimperfectknowledgeofpolicyobjectivesorwhere monetaryorfiscalpolicylackcredibility. Asdiscussedinprevioussections,thedynamicbehavior of sectors depends significantly on expectations of households, firms, and financial markets, including anticipations of policy. Assumptions about how expectations are formed, such as the scope of informationor the speed of learning, can be tailored in FRB/US to address a wide range ofissues. Unless otherwise indicated, simulations in this section assume that monetary policy responds to economic conditions according to the VAR equation for the federal funds rate in table 8 that reflectstheaverage-samplebehaviorofhistoricalpolicy. Infiguresbelow,thissimulatedmonetary policy is referenced as the average historical policy. To highlight the role of private sector expectations in the transmission of policy effects, simulation results are shown for two cases of expectations formation. Under VAR expectations, firms and households use the same estimated VAR togenerate forecasts of thefuture that was usedinestimatingtheFRB/US equations. Under full-model expectations, expected values of future variables equal the values forecast by the full FRB/US model. 5.1 System responses to transitory shocks To provide a brief introduction of the system properties under different assumptions regarding expectation formation, the initial simulations indicate responses of the economy to transitory 25Notethisinterpretationisbasedonsymmetricpolicyinformationundertheimplicitassumptionthattheendpoints ofoperationalpolicyinthefundsrateequationarethesameasthoseperceivedbyprivatesectorsintheremainderof themodel.

35 shocksinthefederalfundsrateandinaggregatedemand(governmentspending). Beyondshowing representative dynamic responses of key macroeconomic variables in FRB/US, these exercises demonstrateunderwhatconditionsVARexpectationsdifferfromfull-modelexpectationsandhow such differences affect macroeconomic outcomes. The general rule that can be extracted from these(andsimilar)simulationsoftransitoryshocks is: RuleofThumb 1 For transitory shocks, the dynamic response of most main macroeconomic aggregates differs between VAR expectations and full-model expectations to the extent the shocks appliedtothesystemdeviatefromaveragehistoricalexperience. VAR expectations are based on the historical behavior of macroeconomic aggregates. For shocks that are not unusual in an historical perspective, the summary macroeconomic responses provided by the VAR contain most of the information needed to predict the responses of the full FRB/US model. In contrast, when the simulation experiment strays from the typical pattern of shocksintheeconomy,differences betweenVARandfull-modelexpectationsemerge. Figure 3 shows the responses of inflation, output, the federal funds rate, and the 10-year government bond rate to a one-quarter, 100-basis point positive shock to the VAR equation for the federal funds rate.26 In this and most of the following figures, the economy's response under full-modelexpectationsisshownasthesolidline,whilethedashedlinerepresentstheresultsunder VARexpectations. Resultsinallinstancesaredisplayedasdeviationsfromabaselineforecast. As thefigureshows,thepresenceoflaggedendogenousvariablesinthefundsrateequationamplifies the initial impulse, and the funds rate stays above baseline for about two years. The higher level of the funds rate brings about a decline in aggregate demand and downward pressure on prices. Although not shown in the figure, eventually all variables return to their baseline values as the effects oftheshockwearoff. 26Inordertosimplifythedesignofthetransitoryshocks,eachsimulationofthistypeisbasedontheassumption thatthelong-runinflationobjectiveofmonetarypolicyisunchanged,asareprivateperceptionsofthepolicyobjective.

36 Figure 3 One-quarter, 100-basis-point shock to the federal funds rate Average Historical Policy (deviations from baseline, per cent) consumption price inflation output gap 0.10 0.1 0.05 0.0 0.00 -0.1 -0.05 -0.2 -0.10 -0.15 -0.3 -0.20 -0.4 -0.25 -0.5 -0.30 -0.6 -0.35 -0.40 -0.7 0 2 4 6 8 years 0 2 4 6 8 years nominal federal funds rate 10-year government bond rate 1.2 0.30 1.0 0.25 0.8 0.20 0.6 0.15 0.4 0.10 0.2 0.05 0.0 -0.2 0.00 -0.4 -0.05 0 2 4 6 8 years 0 2 4 6 8 years full-model expectations (solid); VAR expectations (dashed) The responses to the interest rate shock under full-model and VAR expectations are quite similar, except for the response of inflation which is somewhat damped under full-model expectations. Therelativelyminordifferencesbetweenthedynamicresponsesunderthetwotypes of expectations can be traced to two sources. First, the dynamics of the VAR model, while close, arenotidenticaltothedynamicsofthefullFRB/USsysteminthisinstance,introducingsomebias inVARexpectations. Tosomedegree,agentsarenotusingtherightmodeltoformexpectationsof futureevents. Second,aone-perioddifferenceintimingofsomeresponsesisduetotheassumption underVARexpectationsthatanticipationsgenerallyareformedatthebeginningofthequarterand donotdependoncontemporaneousinformation,whilefull-modelexpectationsimplicitlytakeinto considerationobservationsinthecurrent period.

37 Figure 4 Four-quarter shock to government spending equal to 1% of GDP Average Historical Policy (deviations from baseline, per cent) consumption price inflation output gap 0.15 1.5 0.10 0.05 1.0 0.00 0.5 -0.05 -0.10 0.0 -0.15 -0.20 -0.5 -0.25 -0.30 -1.0 0 2 4 6 8 years 0 2 4 6 8 years nominal federal funds rate 10-year government bond rate 1.0 0.25 0.8 0.20 0.6 0.15 0.4 0.10 0.2 0.05 0.0 -0.2 0.00 -0.4 -0.05 0 2 4 6 8 years 0 2 4 6 8 years Full-model expectations (solid); VAR expectations (dashed) Figure 4 shows the response to an increase in government purchases, equal to one percent of GDP, lasting one year. In this experiment, an additional source of expectations bias is introduced into VAR expectations: The hypothesized duration of the demand shock does not correspond to theaverage historical serial correlationof output. Historically,deviationsof outputfrom potential tend to amplify themselvesinitiallyand then to die out gradually. Under VAR expectations,firms and households do not “see through” the four-quarter design of the shock and expect output (and inflation)tofollowtypical historicalpatterns. The perceptionofahighleveloffutureactivityand inflation drives up prices today. Once the shock ends, expectations of future activityand inflation are reviseddownward,dampeningwageandpriceinflation. Bycontrast,underfigure 4'sexample offull-modelexpectations,firmsandhouseholdshaveperfectforesightaboutthisshockandknow the spending impulse will only last one year. This leads to a smaller rise in the bond rate and no discernable rise in inflation. In fact, for nearly all of the first eight years, inflation is below baseline. Thefallininflationisduetoeffectsoffuturelowactivityandinflationandalsotoeffects of the appreciation of the exchange rate caused by the rise in bond yields. Agents make ex post expectational errors using VAR expectations; but, if the true duration of the shock is not known

38 beforehand,theresponsesunderVARexpectationsmaybeconsideredreasonable. The government purchases scenario shown in figure 4 demonstrates how deviations of hypothetical shocks from historical behavior introduce a wedge between VAR expectations and full-model expectations. Figure 5 illustrates another way differences can arise between the two types of expectations—a shift in the actual responsiveness of monetary policy to output and inflation deviations that is not reflected in VAR expectations, at least over the simulationinterval. The policy used in this simulation is one estimated over the shorter sample period from 1979 to 1995 (termed the post-1970's policy) and is more aggressive in combatting output and inflation deviationsthanis theaverage historicalpolicy. Figure 5 One-quarter, 100-basis-point shock to the federal funds rate Post-1970s Policy (deviations from baseline, per cent) consumption price inflation output gap 0.05 0.10 0.05 0.00 0.00 -0.05 -0.05 -0.10 -0.10 -0.15 -0.15 -0.20 -0.25 -0.20 -0.30 -0.25 -0.35 0 2 4 6 8 years 0 2 4 6 8 years nominal federal funds rate 10-year government bond rate 1.0 0.25 0.8 0.20 0.6 0.15 0.4 0.10 0.2 0.05 0.0 0.00 -0.2 -0.05 0 2 4 6 8 years 0 2 4 6 8 years full-model expectations (solid); VAR expectations (dashed) The simulations reported in figure 5 repeat the interest rate increase shown in figure 3, but with actual monetary policy determined by the average post-1970s policy. In this example, the revised policy is captured by full-model expectations but not by VAR expectations. Differences in responses under VAR expectations from those under full-model expectations are larger than in figure 3, because an additional source of expectations bias has been introduced due to the

39 misperception of monetary policy under VAR expectations, which continue to be based on the average historical characterization of monetary policy. In particular, under VAR expectations the federal funds rate is anticipated to persist at an elevated level far longer than the actual policy entails. This leads to an overly pessimistic view of future output and prices, and an exaggerated responseofoutputandinflationtotheshock. Figure 6 Anticipated future four-quarter shock to government spending Average Historical Policy (deviations from baseline, per cent) consumption price inflation output gap 0.15 1.5 0.10 0.05 1.0 0.00 0.5 -0.05 -0.10 0.0 -0.15 -0.20 -0.5 -0.25 -0.30 -1.0 0 2 4 6 8 years 0 2 4 6 8 years nominal federal funds rate 10-year government bond rate 1.0 0.25 0.8 0.20 0.6 0.15 0.4 0.10 0.2 0.05 0.0 -0.2 0.00 -0.4 -0.05 0 2 4 6 8 years 0 2 4 6 8 years Full-model expectations (solid); VAR expectations (dashed) Finally, implications of perfect foresight are indicated by contrasting the effects of a future shockthatisforeseeninonecaseandunexpectedintheother. Figure6showsthesamefour-quarter governmentpurchases shockas infigure 4,withthe exceptionthat itoccurs oneyear inthefuture (atthedatedesignatedbytheverticalline). Forfull-modelexpectations,theshockisassumedtobe foreseeninadvancewhereasunderVARexpectationsitisunexpectedandthereisnoreactionuntil the shock occurs. In both cases, monetary policy follows the average historical policy. For VAR expectations, the responses are identical to those in figure 4 aside from a one-year delay. Under full-model expectations, inflation and output rise with the announcement of the future spending increase. The initial rise in inflation is due to labor and product markets that are foreseen to

40 be tighter in the future. Once the fiscal expansion is in full swing, however, inflation begins to fall because of the anticipated weakening of output after the end of the temporary increase in government spending. Also, long bond rates rise upon the announcement of the policy change underfull-modelexpectations,duetotheanticipatedriseinthefederal fundsrate. 5.2 System responses to permanent shocks Thus far, simulations have illustrated the effects of expectation formation on system dynamics for temporary disturbances. Discussionnow turns to analysis of a permanent change in monetary policy, where the experiment is a policy that aims to reduce the inflation rate permanently by one percentage point within ten years. Any number of funds rate paths can achieve this objective; in the simulationsthat follow,the funds rates set bypolicyare consistent withthe planned reduction ininflationbutotherwiserespondtomovementsinobservedoutputandinflationusingtheaverage post-1970's responses discussed above. We consider two cases of policy credibility. In the first case of “perfect credibility,” the private sector fully believes that the announced disinflationary policy will occur as planned. In the second case of “learning,” the private sector only slowly adjusts its views about the probability that the full disinflationary program will be carried out. In the latter case, the rate of adjustment in the inflation endpoint is 5% per quarter, so that long-run inflationexpectationswillhavefallenbyone-halfofonepercentagepointafter31/2years.27 These simulationsprovidethebasisforasecondruleofthumb: RuleofThumb 2 The cost of disinflation, in terms of lost output and employment, is decreasing inthedegreeof credibilityofthepolicy. Figure 7 shows the consequences of a credible policy of disinflation. As with the transitory monetary policy shift of figure 3, there is little difference between the outcomes under VAR and full-modelexpectations. Withthechange inthepolicytakenas known,the informationcontained in the VAR is sufficient to understand the responses of the full FRB/US model. Sacrifice ratios (cumulativeannualincreaseintheunemploymentratedividedbythepercentagepointdecreasein the inflation rate) are also similar. For VAR expectations, the sacrifice ratio is 1.3; for full-model expectationsit is1.7.28 27Thisrateof inflationendpointlearningis consistentwith thefallin long-runexpectationsmeasuredby surveys duringthedisinflationofthe1980's. 28Sacrificeratiosarecomputedattheendofthetenthyearofthesimulations.

41 Figure 7 Permanent disinflation of one percentage point Post-1970s Policy (deviations from baseline, per cent) consumption price inflation output gap 0.2 0.1 0.0 0.0 -0.1 -0.2 -0.2 -0.4 -0.3 -0.6 -0.4 -0.8 -0.5 -1.0 -0.6 -1.2 -0.7 -1.4 -0.8 0 2 4 6 8 years 0 2 4 6 8 years nominal federal funds rate 10-year government bond rate 0.4 0.2 0.2 0.0 0.0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1.0 -1.0 -1.2 -1.2 -1.4 -1.4 0 2 4 6 8 years 0 2 4 6 8 years Full-model expectations (solid); VAR expectations (dashed) Instantaneous recognition of change in target inflation The assumption of perfect credibility of the disinflationary policy is removed in figure 8.29 Inflation declines more gradually in this case, as the dampening effects of the credible policy on expected inflation are attenuated in the case of gradual learning. Also, the rapid decline in bond rates in the case of perfect credibility is absent under imperfect credibility. Bond traders, like all agents in the economy, only gradually adjust their views about the long-run objectives of policy. Thehigherrealinterestratesgeneratedbythisdisinflationarypolicyleadtolossesofoutputthatare significantly greater than those under perfect credibility. In terms of the sacrifice ratio, the effect of imperfect credibilityis toincrease the cost of disinflationfrom 1.3to 2.6for VARexpectations andfrom1.7to2.3forfull-modelexpectations. The credibility of monetary policy is not the only aspect of policy that affects the output and employment cost of disinflationin FRB/US. Also important is the speed at which policyattempts toreduce inflation,withthecost beinghigherthefasteris thedesiredreduction 29For easeof comparison,thefigurealso repeatsthesimulatedresponsesfoundunderfullcredibility. Unlikethe otherfigures,thefederalfundsrateisplottedinrealtermsinfigure8.

42 Figure 8 Disinflation, with and without learning Post-1970s Policy (deviations from baseline, per cent) consumption price inflation output gap 0.2 0.2 0.0 0.0 -0.2 -0.2 -0.4 -0.6 -0.4 -0.8 -0.6 -1.0 -0.8 -1.2 -1.4 -1.0 0 2 4 6 8 years 0 2 4 6 8 years real federal funds rate 10-year government bond rate 1.2 0.2 1.0 0.0 0.8 -0.2 0.6 -0.4 0.4 -0.6 0.2 -0.8 0.0 -1.0 -0.2 -1.2 -0.4 -1.4 0 2 4 6 8 years 0 2 4 6 8 years Full-model expectations, instantaneous recognition (thick solid) Full-model expectations, learning, 5% rate (dashed) VAR expectations, instantaneous recognition (dotted) VAR expectations, learning, 5% rate (thin solid) ininflation. Inamodelwhereinflationdependsonlyonpastobservations,thecostofapermanent disinflation is invariant to the speed at which the disinflation occurs. In a model where inflation depends on the expected future values, this invariance disappears. A credible policy to reduce inflation that affects variables in the future will also affect the present. Thus, an effective way of reducing current inflation at little cost in terms of lost output is to “announce” a restrictive future policy that reduces expectations of inflation. The private sector is not being “fooled” because policy must generate a reduction in output below potential at some point to be consistent with the reduction in inflationary expectations. The key is that much of the reduction in inflation is accomplishedthrough lower inflation expectations,as opposedto operatingonly through reduced aggregate demand. The effect of the speed of disinflation on the sacrifice ratio is illustrated by comparing outcomes for the two “average” descriptions of historical policy. Under VAR expectations with full endpoint credibility, the sacrifice ratio falls from 1.3 to 1.1 if the more gradualaveragehistoricalpolicyissubstitutedfortheaveragepost-1970'spolicyusedinfigures7 and8.

43 Even though FRB/US is an empirically estimated model with well-behaved statistical properties, the simulations reported in this section demonstrate that the structural design of FRB/US is suitable for analyses aimed at a broad range of macroeconomic policy questions. The model has the flexibility to examine policy issues under different assumptions about policy credibilityandtheextentofeconomicinformationuponwhichexpectationsare based. A Appendix: Testing the Theory The goodness of fit of the main structural equations in the model is summarized by the proportions of explained variation, R 2 , and the standard deviations of equation residuals, S E E , reported in tables 1 through 7 in the main text. This appendix presents two additional empirical tests directed at assessing the adequacy of the theoretical specifications of dynamic adjustments andtheassumptionofrationalexpectations. The first is a test for serial independence of the residuals to determine if the generalized adjustment cost specifications are able to describe dynamic behavior adequately or if significant correlations in the data remain unexplained. The test in the first column of numbers in table A1 indicatesthesignificanceofautocorrelationsofanequationresidualwithanyofitsfirsttwelvelags. The entries in this column are rejection probabilities (p-values) of the null hypothesis of serially independentresiduals. A p-valueof .05(.01)or less indicatesrejectionof theserial independence hypothesis with at least a 95 % (99 % ) level of confidence. The entries in this column suggest that eight(ten)ofthetwelveequationsexaminedhavewhitenoiseresiduals. The second test examines coefficient restrictions imposed by the VAR-based implementation of rational expectations (RE). Using the example of the aggregate price equation, the FRB/US price equation presumes that firms use the VAR to generate predictions of the equilibrium price. These predictions are then weighted by the lead response weights of the price equation (shown earlier in figure 1) to determine the estimated price change in the current period. The potential information in the VAR consists of lagged values of all variables included in the VAR model. Although this information is organized by the VAR to produce minimum mean square errors in predicting the equilibrium price, it may be that firms prefer to organize this information in some otherway,suchasrule-of-thumbextrapolations. Thetestinthesecondcolumnofnumbersintable A1augmentstheFRB/US dynamicadjustmentequationwithlaggedvaluesofthevariables inthe sectoral VARas additionalregressors. IftheadditionalVARregressorsare statisticallysignificant then the p-values in the second column will be low, indicating that firms are not using rational expectations(at least as defined byVARforecasts) intheir dynamicadjustmentequations. Again, ap-valueof.01orlessindicatesrejectionofRErestrictionswithatleasta99 % levelofconfidence. Theentriesinthesecondcolumnsuggestthatrationalexpectationsrestrictionsarenotrejectedfor

44 tenofthetwelveequationsexamined. Table A1: Tests forSeriallyIndependent Residuals and RERestrictions a seriallyindependent RE equation residuals restrictions aggregateconsumption .24 .30 consumerdurables,motorvehicles .28 .12 otherconsumerdurables .28 .19 residentialinvestment .01 .01 producers' durableequipment .67 .72 inventoryinvestment .24 < .01 laborhours .02 .09 aggregateprice .20 .71 wage .04 .27 5-year Treasurybondrate .01 .78 10-yearTreasurybondrate .07 .68 corporatebondrate .37 .91 atable entries are rejection probabilities (p-values); a low p-value indicates rejection of a null hypothesis(whitenoiseresidualsorRErestrictions).

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