"Forecasting the Forecasts of Others:" Expectational Heterogeneity and Aggregate Dynamics

“Forecasting the Forecasts of Others:” Expectational Heterogeneity and Aggregate Dynamics AntulioN.Bomfim (cid:3) DivisionofResearchandStatistics,MailStop61 BoardofGovernorsoftheFederalReserveSystem Washington, DC,20551 abomfim@frb.gov LatestDraft: October17,1996 Abstract I construct a dynamic generalequilibrium modelwhere agents differ in the way they form expectations. Sophisticatedagentsformmodel-consistentexpectations. Rule-of-thumbagents' expectations arebasedonanintuitiveforecastingrule. Allagentssolvestandarddynamicoptimizationproblemsand facestrategiccomplementarityinproduction.ExtendingtheworkofHaltiwangerandWaldman(1989), I showthatevenaminorityofrule-of-thumbforecasterscan havea significanteffectontheaggregate properties of the economy. For instance, as agents try to forecast each others' behavior they effectively strengthenthe internalpropagationmechanism of the economy. I solve the modelby assuming a hierarchicalinformationstructure similar to the one in Townsend's(1983)modelof informationally dispersed markets. The quantitative results are obtained by calibrating the model and running a batteryofsensitivitytestsonkeyparameters. Theanalysishighlightstheroleofstrategiccomplementarity in theheterogeneousexpectationsliteratureand precisely quantifiesmanyqualitativeclaims aboutthe aggregateimplicationsofexpectationalheterogeneity. JELClassification: E32,D82,C61 Keywords: business cycles,expectations, strategic complementarity, propagation, boundedrationality (cid:3) IreceivedvaluablecommentsfromSatyajitChatterjee,RussellCooper,FrankDiebold,JohnHaltiwanger,RobertKing,Michael Waldman,andananonymousreferee. SteveSumnerprovidedexcellentresearchassistance. Thispaperisasubstantialrevisionof mypreviouspaper“Forecast-HeterogeneityintheBusinessCycle: SmallDeviationsfromRationality,LargeDynamicEffects.” I amsolelyresponsibleforanyremainingerrors.TheopinionsexpressedherearenotnecessarilysharedbytheBoardofGovernors oftheFederalReserveSystemoranyothermembersofitsstaff.

1 Introduction Theassumptionthatagentsformrational(model-consistent) expectations isastandardfixtureofmostwork inmodernmacroeconomics. Though widespread, such reliance onthe rational expectations hypothesis has not gone unquestioned. For example, Frydman and Phelps (1983), Board (1994), and Arthur (1994) argue that human rationality isbounded and thus the rational expectations assumption imposes extreme informational and computational requirements on agents (see also Sargent, 1993). In addition, other researchers base their reservations on concerns about the observable characteristics of rational expectations equilibria. For instance, De Long, Schleifer, Summers, and Waldman (1990) construct a model in which behavior based onirrational noise trading helps explain anumber ofobserved phenomena infinancial markets, such as the excess volatility of asset prices and the equity premium puzzle. Also in the finance literature, Roll (1996) mentions incomplete (bounded) rationality as a possible explanation for the observation of large tradevolumesindebtmarkets. Whereasthesereservations haveledsomeeconomiststodiscardtherational expectations hypothesis altogether, others have sought to reconsider it in the context of environments with less demanding informational assumptions and more plausible observable implications. Lucas (1975) and Townsend (1983) were among the first to take up this line of inquiry. Townsend, in particular, analyzed a model where agents form heterogeneous expectations because their forecasts are conditioned on different subsets of the relevant data. He showed that, as agents attempt to “forecast the forecast of others,” the economyconverges toarational expectations equilibrium. More recently, the work of Haltiwanger and Waldman (1985, 1989) brought a new perspective to the analysisofforecastheterogeneity. Ratherthanworkingwithdynamicclassicalmodelswithnoexternalities (theLucas-Townsend approach), Haltiwanger andWaldmananalyzed environments thatallowedforstrategiccomplementarity.1 Inaddition, insteadoffocusing onagentswithdifferentaccesstothedata,theybuilt models where forecast heterogeneity arises because some agents may use more sophisticated forecasting methods than others. Accordingly, while some may form model-consistent expectations, others rely less on the structure of the model and form expectations based on a simple rule-of-thumb. Unlike the Lucas- Townsend tradition, the class of simple, static models analyzed by Haltiwanger and Waldman gave rise to environmentswhereforecastheterogeneitydidcastsomedoubtontheaggregateimplicationsoftherational expectations hypothesis. They showed that, with strategic complementarity, less sophisticated forecasters may have asizable effect on the evolution ofaggregate output, effectively driving the economy awayfrom itspurerationalexpectations equilibrium. 1CooperandJohn(1988)definestrategiccomplementarityanddiscussitsimplicationsformacroeconomics. Forthepurpose of my paper, strategic complementarity involves a situation where an individual's output decision is increasing on the level of aggregateoutput. 1

InthispaperIintroduceandsolveamodelthatbringstogetherimportantissuesstemmingfromboththe Lucas-Townsend and Haltiwanger-Waldman approaches to analyzing heterogeneous-expectations models. Capturing thekey insights ofHaltiwanger andWaldman, myanalysis allows forstrategic complementarity andheterogeneousforecastingrules. However,tobringthediscussionmoreintothecurrentstageofmacroeconomic thought—which emphasizes dynamic rather than static frameworks—I extend the Haltiwanger- Waldmananalysistoaricherdynamicgeneralequilibrium model,whichIsolvewithamethodology thatis verycloseinspirittoTownsend's. Inchoosingaspecificdynamicmodelingframework,Ioptedfortheclassofmodelsintherealbusiness cycle tradition. The advantages of this choice are two-fold. First, the virtues and limitations of the RBC framework are well understood by the profession. Thus, the results I obtain under forecast heterogeneity and strategic complementarity can be directly and quantitatively compared to those generated by standard RBCmodels with homogeneous, rational expectations. Second, by introducing forecast heterogeneity into theRBCframework, Iamabletoaddress arecurring themeintheliterature: theweakinternal propagation mechanism thatunderlies manyequilibrium business cyclemodels(KingandPlosser,1988). The paper's main results and methodology can be summarized as follows. With a sufficiently strong degreeofstrategiccomplementarity, Ishowthatevenifonlyasmallsubsetofagentsforecasts accordingto asimplebutreasonableruleofthumb,aggregateoutputexhibitsmorepersistencethanwarrantedbyeither(i) thedegreeofserialcorrelationintheproductivityshockprocess,or(ii)theshareofrule-of-thumbforecasters in the total population. More to the point, because agents engage in forecasting each others' forecasts, rule-of-thumb forecasters have a quantitatively important impact on the serial correlation properties of the business cycle. The results are obtained by calibrating all standard RBC parameters and then running a battery of sensitivity tests on the forecast-heterogeneity and strategic-complementarity parameters. The sensitivity tests highlight the role of strategic complementarity in the forecast-heterogeneity debate and precisely quantify thequalitative claimsmadebyHaltiwangerandWaldmanandothers. 2 The Model Apart from the issues of heterogeneity, the model described here is very close to that of Baxter and King (1991). 2

2.1 Strategic Complementarityand the Production Function Individual output is a function not only of inputs and a productivity shock, but also of an index of per capitaaggregateoutput. Thisindexisageometricaverageofthepercapitaoutputdecisionsoftwotypesof agents:2 Y t = Y (cid:18) s S ;t Y ( R (cid:0) 1 ;t (cid:18) s ) (1) where (cid:18) s is the proportion oftotal population represented by sophisticated forecasters (type S agents), and Y S ;t is the per capita output decision of these agents. The R subscript denotes variables pertaining to the rule-of-thumb forecasters. Anagentoftype i facestheproduction function, y i;t = e x p ( A t ) F ( k (cid:0) i;t 1 ; n i;t ) Y (cid:30) t i = R ; S (2) where y i;t denotes individual output, and k (cid:0) i;t 1 and n i;t correspond to capital and labor inputs. The shock A t isassumedtocapturestochastic shiftsintotalfactorproductivity. Itfollowsafirst-orderautoregression: A t = (cid:26) A (cid:0)t 1 + a t (3) where j(cid:26) j (cid:20) 1 ,and f a t g isazero-mean, normallydistributed whitenoiseprocess withvariance (cid:27) 2 a . F ( :) isaCobb-Douglas production function: F ( k (cid:0) i;t 1 ; n i;t ) (cid:17) k (cid:18) k (cid:0) i;t 1 n (cid:18) n i;t whichishomogeneous ofdegree1 ( (cid:18) k + (cid:18) n = 1 ) .3 The (cid:30) parameter inequation (2)embodies thecomplementarityassumption;itdeterminestheextenttowhichindividualoutput, y i;t ,dependsonaggregateoutput, Y t . ( 0 (cid:20) (cid:30) < 1 ) 2Lucas(1972,1973) usesthisindexinsteadoftheconventionaldefinitionofaggregateoutput.This,asnotedinBlanchardand Fischer(1989),amountsto“definingaggregateoutputastheproductofindividualoutputs,ratherthantheirsum”(p. 358). (See alsoSargent1987,p.442).Theuseofsuchindexsimplifiestremendouslythealgebra,bothhereandinthereferencedworks. 3Including labor-augmenting technical change intheproduction function toallowfor economicgrowthwouldhaverequired “detrending”themodelbeforeproceedingtothesolutionalgorithm;theresultswouldhavebeenunchanged. AsinKing,Plosser, andRebelo(1988),detrendingwouldinvolvenothingmorethandividingallgrowingvariablesbythelaboraugmentingfactor.For thesakeofbrevity,wehavechosentostartwithastationarymodelfromthebeginning. 3

2.2 Evolutionofthe CapitalStock Outputnotconsumedconstitutes grossinvestment, i i;t . With k i;t representing thecapitalstockattheendof period t andassumingthatthisstockdepreciates attherate (cid:14) , 0 (cid:20) (cid:14) < 1 , k i;t = ( 1 (cid:0) (cid:14) ) k (cid:0) i;t 1 + i i;t (4) 2.3 Preferences Preferences are homogeneous throughout theeconomy. Themomentary utility function ofarepresentative agentis: u ( c i;t ; l i;t ) = l o g ( c i;t ) + (cid:18) l l o g ( l i;t ) (5) where c i;t denotes consumption, and l i;t isleisure—expressed asaproportion oftheunittimeendowment. 3 Individual Behavior Theeconomy ispopulated by infinitely-lived, forward-looking agents whodiscount thefuture attherate (cid:12) and maximize expected utility overaninfinite horizon. AsinTownsend (1983), wecan think of individual behavior as the outcome of two separate problems: dynamic optimization and inference.4 The solution to the first problem yields the perfect-foresight equilibrium laws of motion of all choice variables, which expresstheoptimalvaluesofthesevariablesasafunctionofpast,current,andfuturestatesoftheeconomy. Bysubsequently solving theirinference problems, theagents converttheseequilibrium lawsofmotioninto decision rulesthatexpressallchoicevariables asfunctions onlyoftheobservedstatesoftheeconomy. My assumption of heterogeneous expectations amounts to saying that agents rely on the same mechanismtosolvetheirdynamicoptimizationproblem,butnottheirinferenceproblem. InthenextsectionIwill discusshowdifferentagentstackletheirinferenceproblem,butfirstIwillfocusontheaspectsofindividual behavior thatarecommontobothtypesofagents. 4Invokingtheseparationorcertainty-equivalenceprincipleiscommon-placeinstandardrealbusinesscyclemodelswithhomogeneous,rationalexpectations(see,e.g.,Kingetal.,1988). 4

3.1 The DynamicOptimizationProblem Groupingallagentsaccordingtothewaytheyformexpectations,Iassumethateachindividualtakesasgiven boththegroup-andeconomy-widelevelsofallrelevantvariables.5 Inadditiontoequations(1)through(5), thesolutiontotheagents' dynamicoptimization problemmustsatisfytheusualtimeandgoodsconstraints, l i;t + n i;t (cid:20) 1 (6) c i;t + i i;t (cid:20) y i;t (7) and a symmetry condition that says that individuals who rely on the same forecasting mechanism must behave identically. This last condition implies that the equilibrium quantities for a given type i agent are equaltotherespectivepercapitaquantities forallagentsofthattype( y S ;t = Y S ;t , n R ;t = N R ;t ,etc). When appliedtotheequilibrium levelsofcapital, labor, andoutput, thesymmetrycondition implies Y i;t = (cid:7) i ( A t ; K (cid:0) i;t 1 ; N i;t ; Y j t ) (cid:17) [e x p ( A t ) F ( K (cid:0) i;t 1 ; N i;t ) (cid:17) ] i Y s j j t (8) where,for i = S ; R , j 6= i , (cid:17) i (cid:17) 1 = [1 (cid:0) (cid:30) ( 1 (cid:0) (cid:18) j ) ] and s j (cid:17) (cid:30) (cid:18) j (cid:17) i Euler Equations. Given equations (1) through (7), the agent's dynamic optimization problem reduces to solving max 1 X t= 0 (cid:12) t n u ( c i;t ; 1 (cid:0) n i;t ) + (cid:21) t [e x p ( A t ) F ( :) Y (cid:30) t (cid:0) c i;t (cid:0) k i;t + ( 1 (cid:0) (cid:14) ) k (cid:0) i;t 1 ] o (9) subjectto k (cid:0) i; 1 and f A t ; Y t g 1 t= 0 given;and l i m !t 1 (cid:12) t (cid:21) t k i;t = 0 .6 Thederivation ofthesystem ofEulerequations thatcorresponds to(9)isstraightforward. Afterimposingthesymmetryconditions onthissystem weobtain: 7 u C ( C i;t ; 1 (cid:0) N i;t ) (cid:0) (cid:3) i;t = 0 (10) 5Notethatgiventhatthesolutiontothedynamicoptimizationproblemrequiresnoforecasting,bothagentsbehaveidentically insofarasthisproblemisconcerned.Forcompleteness,Iwillretain,however,the i indexthroughoutthissubsection. 6Thelastequationisthetransversalitycondition.Itsfinitehorizonanalogsaysthatindividualswouldplacenovalueinholding capital after the end of the last period of the planning horizon. (cid:21) t is the discounted Lagrange multiplier relevant for time t ( (cid:21) (cid:3) t (cid:17) (cid:12) t (cid:21) t istheundiscountedone) 7 u i ( :) [F i ( :) ] correspondstothefirstderivativeoftheutility[production]functionwithrespectto i . Note,e.g.,thattheprivate marginalproductoflaborcanbewrittenas: e x p ( A t ) F N ( k (cid:0) i;t 1 ; n i;t ) Y (cid:30) t = ( 1 (cid:0) (cid:18) k ) y i;t = n i;t 5

u L ( C i;t ; 1 (cid:0) N i;t ) (cid:0) ( 1 (cid:0) (cid:18) k ) (cid:3) i;t (cid:7) i ( A t ; K (cid:0) i;t 1 ; N i;t ; Y j t ) = N i;t = 0 (11) (cid:12) (cid:3) i;t+ 1 [(cid:18) k (cid:7) i ( A t+ 1 ; K i;t ; N i;t+ 1 ; Y j t+ 1 ) = K i;t + ( 1 (cid:0) (cid:14) ) ] (cid:0) (cid:3) i;t = 0 (12) (cid:7) i ( A t ; K (cid:0) i;t 1 ; N i;t ; Y j t ) (cid:0) C i;t (cid:0) K i;t + ( 1 (cid:0) (cid:14) ) K (cid:0) i;t 1 = 0 (13) whicharethefamiliaroptimalityconditionsalsofoundinotherequilibriumbusinesscyclemodels(see,e.g., BaxterandKing,1991). Equilibrium Laws of Motion. The perfect-foresight equilibrium paths of consumption, investment, and laboreffortaregivenbythesolutiontothesystemformedbyequations(10)through(13). Itiswellknown, however, that there is no closed-form solution to this system so I will focus instead on an approximate solution, obtainable by log-linearizing the system around its steady state.8 The resulting (approximate) equilibrium lawsofmotiontaketheform x i;t = (cid:5) i;k ^K (cid:0) i;t 1 + (cid:5) i;(cid:21) ~(cid:21) i;t + (cid:5) i;e e i;t ; i ; j = S ; R , j 6= i (14) where x i;t (cid:17) [N ^ i;t ; ^K i;t ; ^C i;t 0 ] , e i;t (cid:17) [A t ; ^Y j ;t 0 ] ,and ~(cid:21) i;t (cid:17) 1 X h = 0 (cid:22) (cid:0) i h ( F i;1 e i;t+ h + 1 + F i;2 e i;t+ h ) (15) A“caret”overasymboldenotesthatthevariableisexpressedinpercentagedeviationsfromthesteadystate (e.g., ^Y i;t (cid:17) l o g ( Y i;t = (cid:22)Y i ) ). Thematrices (cid:5) i and F i ,aswellasthe (cid:22) i parameter, are allfunctions ofvarious steady-state properties ofthemodel, such astheeconomy's capital-output ratio andthesteady-state labor's shareoftotalincome.9 Equation(14)corresponds tothesolutiontotheagents' dynamicoptimizationproblem,whichabstracts from stochastic considerations. To fully characterize individual behavior, I still need to explicitly address theissuesofuncertainty andexpectational formation. 3.2 The Inference Problem By solving their inference problem, agents convert the perfect-foresight equilibrium laws of motion just derivedintothedecisionrulesthatmakeuptheirbehavior. Inthissubsection Idescribethoseaspectsofthe information structure thatarecommontobothsophisticated andrule-of-thumb forecasters. 8Thelog-linearapproximationmethodusedhereisdescribedindetailinKingetal.(1990). 9Thederivationof(14)followsKingetal.(1990)veryclosely. 6

Foreachperiod t ,Iassumethatallagentsfollowatwo-stagedecisionprocess.10 Atthebeginningofthe period, thefirststagetakesplace: agentsmaketheirlaborsupplyandcapitalaccumulation decisions before being able to observe the current value of the productivity shifter, A t , or the current output decision of the otheragentsintheeconomy. Thefactor-allocation decision rulestaketheform: z i;t = (cid:25) i;k ^K (cid:0) i;t 1 + (cid:25) i;(cid:21) E ( i) h ~(cid:21) i;t j (cid:10) (cid:0)t 1 i + (cid:25) i;e E ( i) [e i;t j (cid:10) (cid:0)t 1 ] (16) where z i;t (cid:17) [N ^ i;t ; ^K i;t 0 ] , and the (cid:25) i parameters correspond to the appropriate elements of the (cid:5) i matrices fromequation (14). (cid:10) (cid:0)t 1 istheinformation setavailable atthebeginning ofperiod t ;itcontains thewhole historyoftheeconomyuptoperiod t (cid:0) 1 . E ( i) [:j(cid:10) (cid:0)t 1 ] denotestheexpectationofatype i agentconditioned on (cid:10) (cid:0)t 1 . Once the factor-allocation decisions are made, production takes place and the agents move to the second and last stage of their decision making process. Assuming that both sophisticated and rule-of-thumb forecasters observe each others' output as soon as production takes place, each agent can use its knowledge of the production function to deduce the current value of the productivity shifter ( A t ). Therefore, the consumption decision isbasedonalargerinformation set, (cid:10) 0 ;t (cid:17) f (cid:10) (cid:0)t 1 ; A t ; Y S ;t ; Y R ;t g . Equation (16) makes explicit two points advanced earlier in this paper. First, agents of different types areinformationally linked; togenerate theirowndecision rulestheymustforecast thebehavior oftheother agents in the economy—recall that ^Y j ;t is an element of e i;t . Second, equation (16) highlights the channel through which agents' expectations affect their behavior. Accordingly, the different expectational rules embeddedin E ( S ) and E ( R ) canleadtopotentially different responses tothesamefundamental shocks. 4 Expectational Heterogeneity As stated before, type S agents are sophisticated forecasters; their forecasts are fully consistent with the rational expectations hypothesis. Accordingly, they have full knowledge of the structure of the model, including the expectational behavior oftype R agents. Foranygiven variable t , their expectations can be formallydefinedas E ( S ) [ t j(cid:10) (cid:0)t 1 ] = E [ t j(cid:10) (cid:0)t 1 ] (17) where E [ t j(cid:10) (cid:0)t 1 ] isthemathematicalexpectationof t conditionedonthetruestructureoftheentiremodel (Muth,1961). 10KydlandandPrescott(1982)assumeasimilarinformationstructure. 7

4.1 Rule-of-Thumb Forecasting Type R agents arecalledrule-of-thumb forecasters because theirexpectations arebasedonasimpleexpectational rule. In allowing for the existence of these agents, I am motivated by a number of considerations. First,manyresearchers,especiallythoseinthenoisetradingliterature, haveimplicitlyassumedframeworks where agents are endowed with different expectational formation capabilities. Forinstance, DeLong et al. (1990) assume that some agents' misperceptions lead them to form incorrect expectations about the price of risky assets, while others form model-consistent expectations. Second, a series of recent papers in the macroeconomics literature has argued that even agents who are equally endowed with respect to their expectation formation capabilities could optimally adopt different expectational behaviors. According to this view, if the implementation of rational expectations is costly, otherwise identical agents who face different constraints and opportunities maychoose different levels ofsophistication whengenerating their forecasts. For instance, Evans and Ramey (1992) and Sethi and Franke (1995) developed theoretical models that include explicit costs of forming rational expectations and show that equilibria with a mix of rational and rule-of-thumb expectations arepossible.11 Though partly motivated by both the noise-trading and costly-computation literatures, my analysis of the business-cycle implications of forecast heterogeneity is also driven by a third question: whether or not thegeneral topicofbounded rationality mattersforquantitative equilibrium business cycleanalysis. Along these lines, one can think of this paper as an attempt to quantitatively assess the robustness of the RBC framework to a partial relaxation of the rational expectations hypothesis. Thus, while implicitly taking as giventheexistenceofagentswho,inastatisticalsense,makeless-than-efficient forecasts, mygoalistouse thetoolsoftheRBCliteraturetoassesstheirpotentialimpactonaggregatedynamicsandtodeterminewhat featuresoftherealworld,ifany,aremissedbyexclusively considering modelswithhomogeneous, rational expectations.12 4.2 AForecastingRuleforType R Agents Three criteria guided the specification of anillustrative forecasting rule fortype R agents. First, tocapture the concerns of the costly-implementation literature, the rule must be simple to implement. Second, the forecastingruleshouldgenerate“reasonable”forecasts,i.e.,itshouldbeconsistentwithsimple,well-known characteristics of the economy. Finally, the expectational scheme assumed for type R agents must not be at odds with their ability to solve their dynamic optimization problem. In other words, when solving their 11For tractability, however, Evans and Ramey and Sethi and Franke had toassume ahigher level of abstraction than the one assumedhere,makingtheirmodelsnotassuitableforexercisesinquantitativeanalysis. 12KydlandandPrescott(1996)andKing(1995)discusstheuseofRBCmodelsinquantitativeanalysis. 8

dynamic optimization and inference problems, type R agents must rely on a single, consistent pool of information aboutthebehavior oftheeconomy. We saw in equation (16) that a type i agent must forecast current and future movements in total factor productivity ( A t ) and the per capita output of the other agents in the economy ( ^Y j ;t ). For illustrative purposes, supposethattype R agentsassumeanautoregressive forecasting modelforthesevariables: E ( R ) [A t+ h j(cid:10) (cid:0)t 1 ] = (cid:11) h + 1 A (cid:0)t 1 (18) E ( R ) ^ [Y S ;t+ h j(cid:10) (cid:0)t 1 ] = (cid:11) h + 1 ^Y S (cid:0) ;t 1 (19) Toseehowtheabove forecasting structure fareswiththethree criteria listed above, note thefollowing. First, especially for (cid:11) close or equal to (cid:26) , the forecasting model in equation (18) is not only “simple, reasonable, and consistent with their ability to solve their dynamic optimization problem,” but also, for (cid:11) = (cid:26) , perfectly rational. Second, turning to equation (19), it is obvious that it corresponds to a less-thanperfectapproximation tothetrueprocessgoverningtheevolution of ^Y S ;t . However,despiteitssimplicity, it capturesawell-knownfeatureoftraditionalRBCmodels: thefactthatoutputpersistenceistightlylinkedto the degree of serial correlation inthe productivity shock series (King and Plosser, 1988). Thus, rather than taking the time and resources to compute a fully model-consistent forecast for ^Y S ;t , a practice that would considerably complicatethesolution tothemodel,type R agentsusethesimplerulegivenby(19). Finally, note that the ability of type R agents to solve their dynamic optimization problem with the same of level of sophistication as type S agents is not inconsistent with the relatively unsophisticated methods they use in solving their inference problem. The solution to the dynamic optimization problem requires structural informationonlyaboutone'sownconstraintsandopportunities;byassumption,bothtypesofagentsusethis information. However, to solve their inference problem in a way consistent with the rational expectations hypothesis,individualsalsoneedcompletestructuralinformationontheconstraintsandopportunitiesfacing theotheragentsintheeconomy;byassumption, type S agentshavethisinformation, type R agentsdonot. Thus,agentssolvetheirdynamicoptimization andinference problems inawaythatisfullyconsistent with theinformation assumedtobeintheirinformation sets. 4.3 Hierarchical ForecastingStructure In several aspects, the expectational assumptions made so far are similar toTownsend's (1983) description of a hierarchical informational structure. Sophisticated forecasters are placed higher in the hierarchy; in addition tothe information that enables them to solve their dynamic optimization problem, they also know 9

the precise nature of the dynamic optimization and inference problems being solved by the rule-of-thumb agents. Thus, their forecasts incorporate structural information about the whole economy. In contrast, the structural information embodied in the forecasting behavior of the rule-of-thumb forecasters is selfcontained; they use information about their own constraints and opportunity sets, but not those of type S agents.13 5 Decision Rules under Heterogeneous Expectations Giventhe hierarchical information structure, the model can besolved sequentially in twosteps. Starting at thebottomoftheforecasthierarchy,Ifirstderivethedecisionrulesoftype R agentsandthenusetheresults tocomputethemuchmorecomplicated decision rulesofthesophisticated forecasters.14 5.1 Rule-of-Thumb Agents Thefactor-allocation decision rulesoftype R agents, canbewrittenas z R ;t = (cid:25) R ;k ^K R (cid:0) ;t 1 + (cid:25) R ;A A (cid:0)t 1 + (cid:25) R ;Y ^Y S (cid:0) ;t 1 (20) whichisobtainedbyreplacingtheexpectationsformulae—equations(18)and(19)—intotheperfect-foresight equilibrium law of motion of z R ;t —equation (16).15 Given the factor-allocation decisions, production and consumption takeplace. 13IntheillustrativecaseanalyzedbyTownsend,allagentsusethesameinformationaboutthestructureoftheeconomy.However, theyhavedifferentaccesstothedata;certainagentsonlyhadobservationsontheirownlocalmarkets,whileothersobservedalarger dataset.ThedifferencebetweenTownsend'sassumptionsandmineisrelativelystraightforward.WhileTownsendfocusedonagents whoseforecastswereconditionedondifferentsubsetsofthedata,Iexamineagentswhoseexpectationsaremoreorlessstructurally baseddependingonwhetherornottheyformsophisticatedforecasts. 14Notethatthissequentialschemeisvalidforanyhierarchicalforecastingstructure,regardlessofthespecificforecastingrules usedbytype R agents. Alsonotethatahierarchicalforecastingstructureeliminatesconcernsaboutthe“infiniteregressproblem” (Townsend,1983). 15Thecoefficientsofequation(20)aredefinedasfollows: (cid:25) (cid:25) R R ;A ;Y (cid:17) (cid:17) (cid:25) (cid:25) R R ;x ;x = = ( ( 1 1 (cid:0) (cid:0) (cid:22) (cid:22) (cid:0) b (cid:0) b 1 1 (cid:11) (cid:11) ) ) ( ( F F (1 ) R ;1 (2 ) R ;1 (cid:11) (cid:11) + + F F (1 ) R ;2 (2 ) R ;2 ) ) (cid:11) (cid:11) + + (cid:25) (cid:25) (1 ) R ;e (2 ) R ;e (cid:11) (cid:11) where F m( ) R ;1 denotesthe m th elementof F R ;1 ,andallotherparameterscomedirectlyfrom(16). 10

5.2 Sophisticated Agents Likethe rule-of-thumb forecasters, type S agents make their labor, investment, and consumption decisions intwostatesandsubjecttothesameinformationsets, (cid:10) (cid:0)t 1 and (cid:10) 0 ;t . Thus,theirfactor-allocation decisions are made before they can observe either the output decision of the rule-of-thumb agents or the current state of productivity. However, to form expectations about Y R ;t , type S agents look not only at their own dynamicoptimizationproblem,butalsoattheforecastinganddecisionrulesadoptedbytype R agents. This information issubsumed inequations (14), (15), (18)through (20), and(3), whichcanbegrouped together toformatwo-sidedmatrixdifference equation, H (cid:0) 1 X t+ 1 + H 0 X t + H 1 X (cid:0)t 1 = (cid:15) t (21) where the H i are square coefficient matrices, X t (cid:17) h x 0 S ;t ~(cid:21) S ;t x 0 R ;t ^Y S ;t ^Y R ;t A t i 0 and (cid:15) t is a vectorofzeroseverywhere,exceptfortherowcorresponding to A t ,whichcontains a t . The first-stage inference problem of type S agents entails solving (21) for X t and computing expectations subject to (cid:10) (cid:0)t 1 . To solve this problem I use the generalized saddle-path algorithm described in AndersonandMoore(1985). Thealgorithm mapsequation(21)intoitsstableVARrepresentation:16 X t = (cid:0) X (cid:0)t 1 + S (cid:15) t (23) where (cid:0) and S arefunctions ofthe H i matrices. Given(23),the j -stepaheadforecastof Y R ;t ,madeattime t basedontime t (cid:0) 1 information, is E ( S ) ^ [Y R ;t+ j j(cid:10) (cid:0)t 1 ] = { (cid:3) (cid:0) j + 1 X (cid:0)t 1 (24) where { (cid:3) is the vector that selects the row of (cid:0) j + 1 X (cid:0)t 1 that corresponds to ^Y R ;t . By plugging the above expression—along withthecorresponding prediction formula for theproductivity shock ( (cid:26) j + 1 A (cid:0)t 1 )—into (16), I obtain the labor supply and capital accumulation decision rules of type S agents. Given these decisions, production takes place; ^Y S ;t , ^Y R ;t , and A t become observable; and the consumption decision is implemented. 16ThefirststepintheAnderson-Moorealgorithmistofindatransformationof(21)suchthatacompanionmatrixrepresentation exists: V t+ 1 = G V t (22) with V t definedas [X 0 (cid:0)t 1 ; X 0 0 ] t Thestabilityconditionsofthetransformedsystemarecombinedwiththeoriginal“untransformed” systemtogenerateequation(23). 11

5.3 Aggregation The previous subsections described how different types of agents go about solving their respective utility maximizationproblems. Ourultimateinterest,however,liesontheanalysisofthedynamicsoftheeconomy as a whole. As it turns out, the evolution of aggregate variables can be easily derived from the individual decision rules computed above—see equation (1). Moreover, itcanalsobeshownthat movementsinthese variables obeyastate-space formthathasthefollowinggeneralrepresentation: P t = A P (cid:0)t 1 + B v t (25) Z t = Q P t (26) where P t and Z t arevectorsofindividualandaggregatevariables,respectively, and A , B ,and Q areappropriatelydefinedmatrices.17 6 Quantitative Business-Cycle Analysis Themainquestionaskedinthispaperiswhethertheintroductionofboundedrationalityaffectsthedynamic properties of a real-business-cycle economy. In particular, given the forecasting rule of type R agents, my goalistogaugehowthe (cid:18) R and (cid:11) parameters affectthecyclical properties of Z t . Toanswerthisquestion I runwhatKydlandandPrescott(1996)callacomputationalexperiment—seealsoKing(1995). Theprevious sections implemented the initial steps in the experiment: (i) posing the question that the experiment will address; (ii) constructing the theoretical model where the analysis will be carried out, and (iii) solving the model to compute its equilibrium properties. The final steps involve calibrating the model to allow for meaningful quantitative analysis andrunningtheexperiment itself. 6.1 Model Calibration With the exception of the forecast-heterogeneity and complementarity parameters, [(cid:18) R ; (cid:11) ; (cid:30) ] , all model parametersarecalibratedasinKingetal. (1988). Thefirstpaneloftable1showsthisbasicparameterization. Theparameters showninthesecondpanelarediscussed below. Strategic complementarity parameter. The calibration of the strategic complementarity parameter ( (cid:30) ) is 17Equation(25)containstheexpressionsdescribingthedecisionrulesofbothtypesofagents. Forinstance,therowsof A and B thatcorrespond to z R ;t aresetaccordingtoequation(20). Equation(26)mapsindividualvariablesintoaggregateonesusing expressionssimilartoequation(1),expressedinpercentagedeviationsfromsteady-statevalues. 12

guided by the empirical work of Baxter and King (1991), Caballero and Lyons (1992), and Cooper and Haltiwanger(1993). Baxter and King (1991) estimated (cid:30) using aggregate data by running instrumental-variable regressions ofoutputgrowthoninputgrowth. Inprinciple, theirestimationapproachisverystraightforward: choosean appropriateinstrumentsetandruntheusualfirst-andsecond-stageregressions. Inpractice,however,Baxter and King expressed concern about the lack of precision of their estimation results: of the three instrument setstheyexperimented with,neithergenerated afirst-stage R 2 higherthan0.08,andtheresulting estimated values of (cid:30) ranged from 0.1 to 0.45. For their final simulation results, however, Baxter and King set (cid:30) at 0.23, about the mid-point of their range of estimates and in line with the early estimates obtained by Caballero and Lyons (1989). However, based on the more recent work of Caballero and Lyons (1992) and Cooper and Haltiwanger (1993), there is some reason to believe that this number might be higher, perhaps evenabovetheupperendoftheBaxter-Kingestimates. Caballero and Lyons (1992) found evidence of a strong reduced-form relationship between disaggregated(two-digit)productivityandaggregateactivity. Theyofferedtwopossibleexplanationsforthisfinding. First,theypositedthattheestimatedexternaleffectsaresimplytheresultoftrueexternalities,anassumption thatisconsistentwiththemodeldiscussedbyBaxterandKingandinthispaper(equation(2)). Basedonthe true-externalities specification, Caballero andLyonsestimated (cid:30) tobeinthe0.32-to-0.49 range, depending on the set of instruments and on how energy-price effects are modeled. Their second explanation for the estimated external effects allows for the possibility that unobservable variations in effort, and not just true externalities, might also be behind the measured “external effects.” If confirmed, this second explanation would implythat the estimates obtained under the true-externalities specification arepotentially biased upwards. Heretheirresultsaremixed: thoughtheydidfindsomeevidenceforunobserved effortvariation, the coefficientononeoftheireffortproxies,whilestatisticallysignificant,actuallycameinwiththewrongsign acrossalloftheirestimated equations. Usingindustry andaggregate manufacturing data, Cooperand Haltiwanger (1993) reported evenlarger values for their own estimates of (cid:30) . Moreover, their estimates remained large even after accounting for potential biases associated with measurement error. In fact, the Cooper-Haltiwanger estimates for certain manufacturingsectorsweresolargethattheywouldviolatethesaddle-pathstabilityconditionsofthemodel presented in this paper. Therefore, rather than using any of their estimates, I take the Cooper-Haltiwanger resultsasevidencefavoring thehigherrangeofestimatesobtained byCaballeroandLyons(1992). Anobviousconclusionofthisbriefoverviewoftheempiricalstrategiccomplementarityliteratureisthat it is hard to pin down with confidence what the true value of (cid:30) actually is. For the purposes of this paper, rather than defending any particular parameterization of (cid:30) , Irun myexperiments over the full of estimates 13

reported above. Thus, my range of values will include Baxter and King's (1991) Caballero and Lyons's (1992) lower bonds—0.10 and 0.32, respectively—as well as their estimated upper bounds, 0.45 and 0.49. Mygoalistotracetheconsequences ofthesedifferent estimated values of (cid:30) fortheaggregate implications ofexpectational heterogeneity. Serial correlation and volatility of technology shocks. Two parameters that do not affect the steady-state properties of the model, but play a crucial role in aggregate fluctuations, are the innovation variance and the autoregressive coefficient of A ^ t , ( (cid:27) 2 a and (cid:26) ). To calibrate (cid:27) 2 a , I simply set it to a value that makes the model'soutputvariance equaltoitsempiricalcounterpart—this valueisshowninthesecond paneloftable 1.18 Nevertheless, itisimportant tonotethat, aslong asmyfocus restsonthepropagation mechanism, my resultsareinvariant totheparticular parameterization of (cid:27) 2 a . The parameterization of (cid:26) is designed to highlight a recurring weakness of the standard RBC model withhomogeneous expectations: thelackofaquantitatively importantinternalpropagation mechanism. As is well known, in order to be able to mimic the degree of serial correlation in the data, most RBC models require near-unit root processes for A t , effectively implying that persistence is exogenously imposed on the system, rather than explained by it. To isolate the role of expectational heterogeneity in the persistent generation process, I start by setting (cid:26) at 0.5, about half the usual parameterization adopted in other RBC models—later inthispaperIwillexperiment ofothervaluesof (cid:26) .19 Expectational parameters. Ifthereweresomeprecisionconcerns surrounding theavailable estimatesofthe strategic complementarity parameter, wearehard pressed tofind anyestimates, howeverimprecise, forthe expectationalparametersofthemodel( (cid:18) R and (cid:11) ).20 Thus,ratherthantryingtodefendanyparticularvalues for these parameters, I will treat (cid:18) R and (cid:11) as semi-free parameters and experiment with a wide range of values foreach ofthem.21 Therefore, the focus ofmyanalysis isnot todetermine justhow muchandwhat type of rule-of-thumb forecasting is out there. My emphasis is on assessing the quantitative implications of different degrees of rule-of-thumb forecasting. Accordingly, I view my results as a mapping from the magnitudes of the (cid:30) , (cid:18) R and (cid:11) parameters to the properties of the artificial time series generated by the 18ItiscustomarytoassessthevalidityofRBCmodelsaccordingtoitsabilitytocapturetheobservedamplitudeofaggregate fluctuations(King,1995KydlandandPrescott,1991).Obviously,thisisnottheapproachtakeninthispapersince,byconstruction, Iobtainaratioofmodelandempiricaloutputvariancesthatisequaltoone. 19Prescott(1986)andPlosser(1989)havearguedthatthenear-unitrootassumptiononthetechnologicalshockprocessisjustifiedbyestimatedautocorrelations of Solowresiduals. However, theassumption that Solowresidualsactuallycapture shiftsin technologyisnotuniversallyaccepted(see,e.g.,Hall,1987). 20Anaturalstrategytoparameterize (cid:11) istosetitequalto (cid:26) ,anapproachItakeinseveralofthecasesanalyzedbelow. 21Kydland and Prescott (1982) adopted a similar approach to deal withthe difficulty in obtaining estimates of the time nonseparabilityparameteroftheutilityfunction(King,1995). 14

model. Therestofthissectionprovides asensitivity analysis thatreflectsthismapping.22 6.2 Expectational Heterogeneityand the PropagationMechanism In analyzing the aggregate implications of heterogeneous expectations, I primarily focus on how the existence oflesssophisticated forecasters affects thepropagation mechanism oftheeconomy. Towardstheend ofthissectionIdiscusstheimplications offorecastheterogeneity forotherselected momentsofthedata. Model without external returns. I start by examining an economy without strategic complementarity ( 0 (cid:30) = ). Figure 1 shows the autocorrelation function of aggregate output under alternative values of the expectational parameters. In particular, the figure summarizes the results of a computational experiment that explores different degrees oftype R agents' misperception about thepersistence oftechnological shocks— for (cid:26) = 0 :5 , (cid:11) is allowed to vary from 0.01 to 0.95—and for different shares of type R agents in the total population— (cid:18) R variesfrom0to0.9. Twomainresultsareevidentinfigure1. First,exceptforthecasewherethebeliefsoftype R agentsare wildly at odds with reality ( (cid:11) = :0 1 ), their impact on the autocorrelation function of output is very small, even if we allow these agents to make up the vast majority of the population ( (cid:18) R = 0.9). Second, though quantitatively small, the particular expectational model used by type R agents has a noteworthy property: theseagents' misperceptionsaboutthepersistenceofthetechnologicalshockarereflectedintheactualserial correlation pattern of output. Whenever they expect the shocks to be more [less] persistent than warranted bythedatagenerating process, aggregateoutput endsupslightlymore[less]persistent thanotherwise. Onthewhole,however,theinclusion ofunsophisticated forecasters intheRBCmodelwithoutcomplementarityproducedquantitativelyinsignificantresults. With (cid:30) settozero,smalldeviationsfromtherational expectations assumption produce onlysmalldeviations fromthestandard RBCresults, shownasthedotted linesinfigure1. Moreover,forthecasewheretype R agentsknowthecorrectvalueof (cid:26) — (cid:11) = (cid:26) ,notshown infigure1—thebehavior ofaggregateoutputisvirtuallyunaffected bythepresence oftype R agents. ExpectationalHeterogeneityunderStrategicComplementarity. Figure2summarizestheresultsofacomputationalexperimentidenticaltotheonedescribedabove,exceptthatnowIsetthestrategiccomplementarity parameter at 0.49, the upper end of the range of values discussed in the previous section. The plots in this figurestandinstarkcontrasttotheonesinfigure1. Accordingtofigure2,evenifonlyaminorityofagents forecasts according to the rule of thumb, the serial correlation properties of aggregate output may be affected in a quantitatively important way. For instance, as shown in the lower-left panel of figure 2, if the 22Iamgratefultoananonymousrefereeforsuggestingthiscourseofinquiry. 15

rule-of-thumbagentsover-estimatethepersistenceofthetechnologicalshockbyonlytwodecimalpoints( (cid:11) =0.70, (cid:26) =0.50),aggregateoutputbecomesmorepersistentthaninthecaseofpurelyrationalexpectations. Moreover, the impact of forecast-heterogeneity remains sizable even when the unsophisticated forecasters represent aslittle as30percent ofthetotalpopulation. Thus,unlike thecaseofnostrategic complementarity, whenexternal returns arehighevensmalldeviations from therational expectations hypothesis canlead tosizable deviations fromstandard RBCresults. Somemightquestionthedifferentparameterizationsof (cid:11) infigure2. Inparticular,giventhesimpledata generating process for A t , one might want to see the effects of eliminating type R agents' misperception about (cid:26) . Theresultsofthisexperimentareshowninfigure3,whereIallowtherule-of-thumb forecastersto correctlyinferthedegreeofserialcorrelationinthetechnologyshock( (cid:11) = (cid:26) ). Asshown,evenwithoutany misperceptions aboutthenatureofthetechnological shockprocess,type R agentsstillhaveasignificanteffectontheserialcorrelationofaggregateoutput;deviationsofoutputfromitstrendbecomemorepersistent thaninthecasewithnorule-of-thumb forecasting (shownasthedottedlineinfigure3). Figures 2and 3 highlight an important feature of the model. Notethat, for given (cid:11) and (cid:30) , the effect of rule-of-thumbforecastingonthepersistenceofaggregateoutputdoesnotgenerallymonotonicallyincreases with (cid:18) R . Forinstance, asshowninthetwolowerpanels offigure2,outputactually becomes lesspersistent as the share of rule-of-thumb forecasters in the population increases from 0.60 to 0.90. This finding has important implications for the study of the macroeconomic effects of expectational heterogeneity. What it says is that persistence is not simply being exogenously generated as a result of the introduction of type R agents. Undoubtedly, even in isolation, these agents do affect the serial correlation properties of output (see figure 1); however, in addition to this exogenous factor, there is also an endogenous component of thetypeofexpectations-induced propagation mechanismfeaturedinthispaper. Aswesawbefore,strategic complementaritystrengthenstheinformationallinksbetweenthetwotypesofagents,whichinturnleadsthe sophisticatedforecasterstorespondtotheperceivedbehavioroftherule-of-thumbforecasters. Accordingly, output persistence is affected not just by the ad hoc introduction of type R agents, but by the interactions among agents operating under different expectational assumptions. Now, as we allow the share of ruleof-thumb forecasters to increase further, we dampen the extent of interactions between the two types of agents—obviously, therelessofthesophisticated forecasterstointeractwith. Thisexplainswhytherelative effectofrule-of-thumb forecasting actually decreases athighervaluesof (cid:18) R . SofarIhavereported ontheeffects ofrule-of-thumb forecasting under whatmightbecalled twopolar assumptionsaboutthemagnitudeofthestrategiccomplementarityparameter: thezero-lowerboundfeatured inmostRBCmodelsand0.49,atthehighendoftheestimatesdiscussedintheprevioussection. Aquestion of interest is what happens at intermediate values of (cid:30) . Accordingly, figure 4 shows the autocorrelation 16

function of aggregate output under four different parameterizations of strategic complementarity. These alternativevaluesof (cid:30) ,[0.10,0.32,0.45,0.49],correspond totherangeofestimatesobtainedbyBaxterand King (1991) and Caballero and Lyons (1992), but do not include the higher estimates reported by Cooper and Haltiwanger (1993). As shown in this figure, the impact of the type of expectational heterogeneity examined in this paper is still quite sizable for (cid:30) = 0 :4 5 , but the results are clearly not as dramatic for the twolower values ofthe external returns parameter (cid:30) = 0 :1 0 and (cid:30) = :3 2 . Figure 4 highlights the fact that more precise estimates of the actual degree of strategic complementarity are crucial for a more definitive assessment oftheaggregate(quantitative) implications offorecastheterogeneity. 6.3 Model Evaluation ItiscustomaryintheRBCliteraturetousethedatatocalibrateallparametersofthemodelandthencompare itstimeseries properties withselected momentsofthedata. Thisapproach cannot befullyimplemented in the model presented here because of the two semi-free parameters discussed above ( (cid:18) R and (cid:11) ) and the uncertainty surrounding the magnitude of the strategic complementarity parameter ( (cid:30) ).23 Nevertheless, it would be useful to verify whether some plausible parameterization of the heterogeneous-expectation RBC modelwithstrategic complementarity canmakeitroughly consistent withthedata. Table2-A,extractedfromKingetal. (1988),summarizestheselectedmomentsoftheU.S.datathatthe modelwilltrytomatch. Thecorresponding modelmomentsareshownintable2-B.Theresultsreportedin thistableareobtainedbyassumingarelativelyhighdegreeofcomplementarity( (cid:30) =0.49),whilepotentially allowing for only a limited role for bounded rationality ( (cid:18) R = 0.30). To highlight the internal propagation mechanism coming from heterogeneous expectations under strategic complementarity, I arbitrarily set the persistenceparameter( (cid:26) )at0.70,lowerthanwhatastandardRBCmodelwouldrequiretocapturetheserial correlation observed in the data. Assuming no misperceptions from the part of type R agents ( (cid:11) = (cid:26) ), the modelreplicateswelltheserialcorrelationofthedata,especiallyforconsumptionandoutput. Theobserved relative volatilities of output, consumption, and investment are also largely captured by the model, though consumption andinvestment areabittoovolatileandhoursdonotvaryasmuchasinthedata.24 To compare the internal propagation of the model with the standard RBC framework, table 2-C shows thesameselected momentsshownintable2-B,butnowthecomplementarity andexpectational parameters areallsettozero. Asexpected,withoutstrongserialcorrelationintheshocks,thestandardRBCmodelfails 23Thoughonecouldarguethat (cid:11) couldbesetto (cid:26) ,wearestillleftwithveryimprecisemeasuresof (cid:30) andnomeasuresatallfor (cid:18) R . 24Again, these results are only suggestive since the semi-free parameters can not be calibrated with an acceptable degree of confidence. 17

tocapture theserialcorrelation thatcharacterizes observed business cycles. 7 Interactions Between Sophisticated and Rule-of-Thumb Agents In a statistical (mean-squared-error) sense, the forecasts formed by rule-of-thumb forecasters are less efficient than the model-consistent expectations of the sophisticated forecasters. When motivating this difference, I appealed to, among other things, the issues raised in the computation literature, whereby some agents might face atrade-off between forecast efficiency and computational costs. The goal of this section is to informally check whether this reliance on statistically suboptimal forecasts translates into significant behavioral differences between sophisticated and rule-of-thumb forecasters. If these differences are large, either the (unspecified) computational costs are large or the rule-of-thumb forecasters are inherently irrational. Ontheotherhand, ifthebehavior oftype R agents resembles thatoftype S agents, thenevensmall computational costs could implicitly justify the existence of “rational” rule-of-thumb forecasting. In other words, small differences would suggest that the rule-of-thumb forecasters behave like near-rational agents (AkerlofandYellen,1985a,1985b). Theanalysisperformedinthissectionissuggestive, however,andonly looksattheresultingbehaviorofeachtypeofagent;neitherthecostsofbecomingasophisticatedforecaster northepotential utilitylossfromforecasting withrules-of-thumb aremodeledexplicitly. The left-hand-side panel of figure 5 plots the simulated output paths for representative sophisticated and rule-of-thumb agents.25 As shown in this panel, despite significant differences in the way they form expectations, sophisticated andrule-of-thumb forecasts behave inanalmostidentical manner. Inparticular, as sophisticated agents anticipate and react to the imperfections embedded in the forecasting schemes of type R agents, they effectively end up mimicking their actions. In the presence of complementarities, it paystoproducemore[less]wheneveraggregateoutputishigher[lower],eveniftherise[decline] inoutput is largely due to the suboptimal forecasts of type R agents (and not warranted by true fundamentals). The left-hand-side panel of figure 5 shows that even in their investment decisions, which correspond to much morevolatileseries, theactions ofsophisticated andrule-of-thumb agentsareremarkably similar. Takentogether, theplots showninfigure5suggest thatthepotential utility lossesfrom beingarule-ofthumb forecaster are likely small, implying that this course of action might well constitute a near-rational strategy. Moreover, this figure reinforces a notion introduced earlier in this paper. The aggregate effects of rule-of-thumb forecasters cannot be solely explained by simply looking at their actions in isolation; a muchmoreinteresting andimportantfactorliesintheendogenous responsethattheiractionselicitfromthe sophisticated forecasters. As noted in the previous section, it was primarily this endogenous response that 25Theparametersettingsarethesameusedintable2-B( (cid:11) = (cid:26) = 0 :7 0 , (cid:30) = 0 :5 0 , (cid:18) R = 0 :3 0 ). 18

accounted forthestrongerpersistence foundintheheterogeneous expectations model. 8 Summary and Concluding Remarks Thepaperintroducedexpectational heterogeneity inadynamicgeneralequilibrium modelofstrategiccomplementarity. I found strong quantitative effects at the aggregate level from allowing even a minority of agents to form expectations according to a sensible rule of thumb. Namely, when combined with strategic complementarity, forecastheterogeneity canstrengthen theinternalpropagationmechanismofthemodel.26 As sophisticated agents try to forecast the forecasts (and actions) of others, they effectively end up reinforcing the perceptions of less sophisticated forecasters, even if these are not entirely consistent with the structure ofthe economy. Bydefinition, strategic complementarity raises the individual reward forproducing morewhenever aggregate output ishigher. Intuitively, the representative sophisticated agent ultimately cares about aggregate state ofthe economy; this agent will produce more whenever it foresees gains in aggregateoutput,regardlessofwhetherornotthesegainsarecomingfromtheunsophisticated forecastsofthe rule-of-thumb agents. The above findings are quantitatively relevant only if the degree of strategic complementarity is sufficientlyhigh(intheupperhalfoftherangeofestimatesobtained intheempiricalstrategic complementarity literature).27 Nevertheless, the sensitivity of the results to the strategic complementarity parameterization should not be a basis for concluding that expectational heterogeneity does not matter for macroeconomic analysis. First, there isstill muchuncertainty surrounding theempirical measures ofthedegree ofcomplementarity: forinstance, manyofthe(sectoral)estimatesobtainedbyCooperandHaltiwanger(1993)would place (cid:30) well above the range of values analyzed in this paper. Second, this paper can be interpreted as a relatively conservative approach to the issue of bounded rationality: after all, the rule-of-thumb forecasters depicted herein are still highly sophisticated individuals. In particular, they are smart enough to solve their dynamic optimization problem optimally and sufficiently well informed to know the exact nature of the stochastic process governing the evolution of the technology shocks and observe the contemporaneous actions of all the agents in the economy.28 Given how much the type R agents are allowed to know, one mightevenbesurprisedastotheextenttowhichsuchasmalllimitationintheirbehavior matteredasmuch 26ThisconfirmsthequalitativefindingsofHaltiwangerandWaldman(1989)andOhandWaldman(1994). 27Forlowdegreesofstrategiccomplementarity,forecastheterogeneitymatteredonlywhenthebeliefsofrule-of-thumbforecastersarewildlyatoddswiththedatageneratingprocessoftheexogenousproductivityshock,acasemosteconomistswouldfindless interesting. 28Krusell and Smith (1996) analyze an artificial economy where agents are allowed to adopt simple savings rules of thumb insteadofbasingtheirsavingsbehavioronthesolutionofastandarddynamicoptimizationproblems.Theyshowthattheaggregate timeseriespropertiesoftherule-of-thumbeconomyaresubstantiallydifferentfromthoseofatraditionalartificialeconomy. 19

asitdidforanydegreeofstrategic complementarity. Inadditiontoanalyzingtheeffectsofheterogeneous expectations intheRBCframework,themodeldeveloped inthis paper wasdesigned toencompass aspects oftwodifferent approaches toanalyzing macroeconomic models of expectational heterogeneity. The Lucas-Townsend approach is centered on dynamic, classical modelsofincomplete information: agents haveonlyalimitedaccess tothedataandformrational expectations accordingly (Lucas, 1975; Townsend, 1983). Theresult is a rational expectations equilibrium with heterogeneous expectations. The Haltiwanger and Waldman (1989) approach was developed in the context of simple, static models of strategic complementarity: although potentially sharing an equal access to the data, agents differ in the extent to which their expectations are based on the structure of the model (model-consistent vs. rule-of-thumb). The result is a heterogeneous-expectations economy whose equilibrium characteristics are, at least in a qualitative sense, fundamentally different from a pure rational expectationseconomy. BybringingtheissuesraisedbyHaltiwangerandWaldmantobearontoamethodologyandmodelingenvironment thatarecloser inspirittotheworkofLucasandTownsend,Ifound thatthe differences in the results arrived by the two lines of inquiry are more a matter of degree than of substance. Tobeprecise,whetherornottheproperties oftheheterogeneous-expectations economyareconsistent with those of a pure rational expectations economy is largely a function of the degree of strategic complementarity displayed by the economy. With strong enough complementarity, the findings of Haltiwanger and Waldman prevail; without it, they lack quantitative relevancy. In the end, the debate is likely to be settled empirically as we develop a better understanding of the nature and extent of strategic complementarity in themacroeconomy. Fornow,theavailable rangeofestimates ofthestrategiccomplementarity parameteris stilltoowidetoallowustoquantify withprecision theaggregate effectsofexpectational heterogeneity. 20

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Table1—ParameterValuesandDefinitions Parameter Definition A.StandardRBCParameters (cid:18) n = 0 :5 8 a long-run shareoflaborincome (cid:14) = 0 :0 2 5 quarterly rateofdepreciation N (cid:22) = 0 :2 0 steady-state hours(proportion oftimespentworking) (cid:12) = 0 :9 8 8 utilitydiscount rate (cid:26) AR(1)coefficient oftechnology shock (cid:27) 2 a b variance oftechnology innovation c ( a t ) B.Complementarity andExpectational Parameters (cid:30) strategic complementarity parameter (cid:11) d perceived valueof (cid:26) (type R agents) (cid:18) R e proportion oftype R (rule-of-thumb) agents intotalpopulation e aSource:King,Plosser,andRebelo(1988),unlessotherwisenoted. bSetaccordingtotheexperimentbeingrun.Seetextandfigures. cCalibratedsothatthemodelmimicstheobservedvarianceofoutput dCalibratedaccordingtotherangeofvaluesestimatedbyBaxterandKing(1991),CaballeroandLyons(1992), andCooperandHaltiwanger(1993). (seetext,tables,andfigures) eSemi-freeparameter. Asensitivityanalysiswasrunoverawiderangeofparametervalues. (seetext,tables, andfigures) 24

Table2—ComparingSelectedMoments a Series StdDev Rat. SD auto(1) auto(2) auto(3) A.U.S.PostwarQuarterlyData b Output 5.62 1.00 .96 .91 .85 Consumption 3.86 0.69 .98 .95 .93 Investment 7.61 1.35 .93 .78 .62 Hours 2.97 0.52 .94 .85 .74 B.ModelwithS.C.andHeterogeneous Expectations c Output 5.62 1.00 .97 .91 .86 Consumption 4.73 0.84 .96 .95 .95 Investment 10.96 1.95 .85 .68 .54 Hours 1.89 0.34 .81 .59 .42 PrdvtyShock 0.53 0.09 .70 .49 .34 C.StandardRBCModel d Output 5.62 1.00 .85 .63 .47 Consumption 4.46 0.79 .16 .21 .25 Investment 16.24 2.89 .67 .44 .29 Hours 3.19 0.57 .66 .43 .26 PrdvtyShock 3.85 0.69 .70 .49 .34 aThefirstcolumnofnumbersshowsthestandarddeviationofeachseries;thesecondcolumnshowsratiosof standarddeviationsofeachserieswithoutput. Columns3through4showfirst,second,andthirdautocorrelation coefficients. bSource:King,Plosser,andRebelo(1988). c (cid:18) R = 0 :3 0 , (cid:30) = 0 :5 0 , (cid:11) = (cid:26) = 0 :7 0 .AllotherparameterscalibratedasshowninTable1,panelA. d (cid:18) R = 0 :0 0 , (cid:30) = 0 :0 0 , (cid:26) = 0 :7 0 . AllotherparameterscalibratedasshowninTable1,panelA. 25

Figure1 ModelwithoutStrategicComplementarity( (cid:30) = 0 ) -Effectofalternativeexpectationalassumptionsontheautocorrelationofaggregateoutput* ( (cid:26) = 5: 0 ) a =.01 a =.30 1.0 1.0 q R = 0 q R = 0 0.8 q q R R = = . . 3 6 0.8 q q R R = = . . 3 6 q R = .9 q R = .9 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 1 2 3 1 2 3 lags lags a =.70 a =.95 1.0 1.0 q R = 0 q R = 0 0.8 q q R R = = . . 3 6 0.8 q q R R = = . . 3 6 q R = .9 q R = .9 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 1 2 3 1 2 3 lags lags *Fordifferentvaluesof (cid:11) and (cid:18) R ,thefiguresshowtheautocorrelationfunctionofaggregateoutput.Eachexpectational assumptionisdefinedbya[ (cid:11) , (cid:18) R ]pair. (cid:11) istheperceivedvalueof (cid:26) (theAR(1)coefficientofthetechnologyshock). (cid:18) R isthe proportionofrule-of-thumbforecastersinthepopulation.Allotherparametersaresetasintable1. 26

Figure2 ModelwithStrategicComplementarity( (cid:30) = 0 4: 9 ) -Effectofalternativeexpectationalassumptionsontheautocorrelationofaggregateoutput* ( (cid:26) = 0 5: 0 ) a =.01 a =.30 1.0 1.0 q R = 0 q R = 0 q R = .3 q R = .3 0.8 q q R R = = . . 6 9 0.8 q q R R = = . . 6 9 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 1 2 3 1 2 3 lags lags a =.70 a =.95 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 q R = 0 q R = 0 q R = .3 q R = .3 q R = .6 q R = .6 0.2 q R = .9 0.2 q R = .9 0.0 0.0 1 2 3 1 2 3 lags lags *Fordifferentvaluesof (cid:11) and (cid:18) R ,thefiguresshowtheautocorrelationfunctionofaggregateoutput.Eachexpectational assumptionisdefinedbya[ (cid:11) , (cid:18) R ]pair. (cid:11) istheperceivedvalueof (cid:26) (theAR(1)coefficientofthetechnologyshock). (cid:18) R isthe proportionofrule-of-thumbforecastersinthepopulation.Allotherparametersaresetasintable1. 27

Figure3 ModelwithStrategicComplementarity,butnomisperceptions aboutthepersistenceofthetechnologyshock* a =r =0.50,f =0.49 1.0 0.8 0.6 0.4 q R =0 q R =.3 q R =.6 0.2 q R =.9 0.0 1 2 3 lags *Fordifferentvaluesof (cid:18) R ,thefigureshowstheautocorrelationfunctionofaggregateoutput. Parametersnotmentionedherearesetaccordingtotable1,whichalsodefinesallparameters. 28

Figure4 TheaggregateeffectsofExpectationalHeterogeneity* ( (cid:11) = 0 7: 0 , (cid:26) = 0 5: 0 ) f=.10 f=.32 1.0 1.0 q R = 0 q R = 0 0.8 q q R R = = . . 3 6 0.8 q q R R = = . . 3 6 q R = .9 q R = .9 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 1 2 3 1 2 3 lags lags f=0.45 f=0.49 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 q R = 0 q R = 0 q R = .3 q R = .3 q R = .6 q R = .6 0.2 q R = .9 0.2 q R = .9 0.0 0.0 1 2 3 1 2 3 lags lags *Thefigureshowstheautocorrelationfunctionofaggregateoutput.Parametersnotmentionedherearesetaccording totable1,whichalsodefinesallparameters. 29

Figure5 ComparingthebehaviorofSophisticatedandRule-of-Thumbagents Output Investment 6 30 Sophisticated Agents Sophisticated Agents 4 Rule-of-Thumb Agents Rule-of-Thumb Agents 20 2 10 0 -2 0 -4 -10 -6 -20 -8 0 20 40 60 80 100 0 20 40 60 80 100 quarters quarters *Thechartsshowtheoutputandinvestmentdecisionsofrepresentativesophisticatedandrule-of-thumbagents. Theparameterizationisthesamedescribedfortable2b.( (cid:11) = (cid:26) = 0 :7 0 , (cid:30) = 0 :4 9 , (cid:18) R = 0 :3 0 ) 30