James B. Bullard

James B. Bullard is an economist at the Federal Reserve Bankof St. Louis. David H. Kelly provided research assistance.

N THE THREE DECADES since the publica-tion of the seminal work on rational expecta-tions, a steely consensus has been forged in theeconomics profession regarding acceptablemodeling procedures.’ Simply stated, the con-sensus is that economic actors do not persist inmaking foolish mistakes in forecasting overtime. People are presumed to be able to bothdetect past patterns in their prediction errorsand base their behavior on the “best possible”forecast of future economic conditions. In theclassic phrase of Robert Lucas, the predictionsof individuals must be “. . - free of systematicand easily correctable biases.” The currentwide acceptability of this notion is testament tothe success of the rational expectationsrevolution.

Unfortunately, the consensus that in equilibri-um systematic forecast errors should be elimi-nated has been insufficient to end the debate inmacroeconomics over expectational assumptions.In particular, some current research examinesthe idea that how systematic forecast errors are

eliminated may have important implications formacroeconomic policy. Researchers who focuson this question are studying what is called“learning,” because any method of expectationsformation is known as a learning mechanism.’

This paper provides a synopsis of some of therecent research on learning. Three importantpoints are emphasized within the context of thesurvey. The most salient point is the close rela-tionship between learning issues and macroeco-nomic policy. In fact, the topic attracts attentionprecisely because of its perceived policy implica-tions. The second point is more subtle: learningis implicitly an integral part of rational expecta-tions models, and current research only makesthis fact explicit. There is little prospect thatone can avoid the study of learning by assumingrational expectations. The third point is that in-cluding learning in macroeconomic models isunlikely to either confirm or overturn completelythe results from rational expectations macro-models. Instead, the concept of rational expecta-tions equilibrium seems to provide the appropri-

for different readers. This terminology developed becauserational expectations is justified by the notion that peopleeliminate systematic forecast errors over time, and thedynamic elimination of errors is a definition of learning.

50 II1I

Learning, Rational Expecta-tions and Policy: A Summaryof Recent Research

IIIIIIIIIIIIIIa

1The seminal work is Muth (1960, 1961).2Lucas (1977), p. 224.

‘The phrase “learning mechanism” is used in this paper ina broad way, even though it conjures up different images

FEDERAL RESERVE BANK OF St LOUIS

I 51

ate benchmark for the study of learning in that,when systems converge under learning, theytypically converge to stationary rational expecta-tions equilibria.

The next section provides a non-technical in-troduction to learning and rational expectations.The subsequent section looks at the effects ofintroducing learning through a simple exampleattributable to Albert Marcet and ThomasSargent. Some interpretation and discussionof the example is offered in the third section,along with a review of other attempts to intro-duce learning into macroeconomic models. Thefinal section provides summary comments.

THE LEARNING IMPLICIT IN RA-TIONAL EXPECTATIONS

Why Study Learning?

Since decisions made today affecting produc-tive behavior are presumed to be based, in part,on individual assessments of the future, macro-economic theories and models generally haveprovided a role for expectations. Around 1970,however, researchers began to realize that thepolicy implications of their models were oftenquite sensitive to the choice of expectationalassumptions.4 This failure of robustness hasbeen increasingly apparent in the last 20 yearsand has drawn increased attention to the prob-lem of how expectations are formed. This lineof research constitutes what has been called therational expectations revolution.

A capsule characterization of rational expecta-tions contains the following themes: (1) In equi-librium (a steady state in dynamic terms), ex-pectations are “correct” in the sense that in-dividuals make no systematic forecast errors;(2) Individuals use all available information (asdefined by the researcher) in forming forecasts;(3) Expectations vary with changes in govern-ment policy; and (4) Individuals know “the

4Sargent (1987) has documented some of this earlyresearch in rational expectations.

‘A more formal approach to defining rational expectationsis pursued in the next section.

6For a discussion of policy along the transition path, seeTaylor (1975).

‘This is the same as saying that learning provides a stabili-ty theory for rational expectations equilibria. In the bestcase, given initial conditions and some parameter values,statements claiming that the dynamic evolution of theeconomy always leads to a particular stationary equilibrium

model” and thus can predict as well as theeconomist manipulating the model.’ These arethe tenets of the theory and are often espousedby its advocates.

The first tenet, the heuristic notion that in-dividuals eliminate systematic forecast errors, isthe one most responsible for the rise of the ra-tional expectations hypothesis. In a deterministicenvironment, this idea implies that, once learn-ing is complete, people have perfect foresight.In a stochastic environment, it means that theremaining forecast errors are white noise.

Since the consensus is that the elimination ofsystematic forecast errors is a sensible postu-late, all reasonable long-run equilibria must berational expectations equilibria. Macroeconomistsgenerally are interested in the effects of changesin policy parameters at these steady-state equi-librium points.’ Given such equilibria, there is atleast one reason why the explicit specification oflearning could matter: in a model with multiplerational expectations equilibria, the learningmechanism may serve to select the actual out-come.7 The next section contains an example oflearning as a selection mechanism.

Representing Learning Via Econ”ometric Techniques

Macroeconomists generally have avoided speci-fying explicitly the optimal learning mechanismunderlying rational expectations for a numberof reasons. First and foremost, they consideredrational expectations a shortcut in expectationsmodeling that made explicit specification un-necessary.8 Further, ascertaining the full im-plications of the rational expectations hypothesisturned out to be a difficult problem; presumably,these implications must be understood beforethe issue of learning can be investigated.

There was, however, at least one additionalreason why learning was essentially ignored—explicit specifications of how agents form expec-

can be made. This is called global stability. Alternatively,one can define local stability, where a particular stationaryequilibrium is stable only if the initial conditions are nearthat equilibrium.

‘Lucas’ comment, “ - - take the rational expectationsequilibrium - - - as the model to be tested and view [learn-ing] as - - . an adjunct to the theory that serves to lend itplausibility,” hints at this pragmatism. See Lucas (1987a),p.231.

IIIIIIIIIIIIIIIIIa JANUARY/FEBRUARy 1991

tations have often been attacked. In particular,one key criticism of the adaptive expectationsformulation was that it allowed systematic er-rors to persist over time.’

The notion of adaptive expectations is essen-tially that people predict the future value of avariable via a geometric distributed lag (or Koycklag) of its past values. One of Muth’s results wasthat adaptive expectations is an optimal predic-tion method if the variable being forecast followsa random walk.”’ Muth’s result makes it clearthat, from the beginning, rational expectationstheorists had econometric techniques in mindwhen thinking about ways to model optimalforecasting.

The idea of looking to econometrics to modellearning obviously can be extended, since theKoyck lag is only one of many available econ-ometric techniques. In principle, it should bepossible to take advantage of the developmentsin econometric theory to shed light on the prob-lem of how expectations are formed. Further-more, if econometric methods are to be appliedto solve the inference problem faced by individ-uals, a good place to start would be some vari-ant of least squares regression. This is the logicbehind the recent work on least squares learn-ing, an example of which is presented below.

‘l’he application of econometric theory to theexpectations problem was vigorously pursuedrecently because the mathematical knowledge toanalyze the problem became available.h1 Thistechnology was developed in the engineeringliterature and has been extended to economicsby Albert Marcet and ‘thomas Sargent.’2 Econo-mists can now determine, therefore, whether itmakes any difference for policy implications ifthe learning mechanism behind rational expecta-tions is explicitly specified. Viewed this way, thework on learning can be considered part of acontinuing effort to understand the microfoun-dations of expectations formation.

‘the impetus for a detailed analysis of learninghas come from both advances in research tech-nology and one pressing problem: many rationalexpectations models are characterized by multi-ple equilibria. Moreover, these different equilib-ria can have different policy implications, as thenext section illustrates.”

LEARNING AND THE UNPLEAS-ANT MONETARIST ARITHMETIC

This section examines a simple expository ex-ample of Marcet and Sargent;14 it is a simplifiedversion of the “unpleasant monetarist arithmetic”of Sargent and Neil Wallace.” The example ismeant only to illustrate the types of issues thatcan arise; it is not intended as a definitive state-ment of the effects of including learning in eco-nomic models. As the next section will discuss,these effects are still uncertain.

The model’s most important feature is thatthere are two steady states (high inflation andlow inflation) with differing policy implications.At the low-inflation steady state, a permanentincrease in the government deficit implies a per-manent increase in the stationary inflation rate.However, increasing the government deficitleads to lower steady-state inflation if the econ-omy is in the high-inflation stationary equilibri-um. Because the policy implications of the modelhinge on the choice, it is desirable to name oneof the stationary states as the more likely out-come. The policy conclusions turn out to be dif-ferent under learning as opposed to perfectforesight, even though systematic forecast errorsare eventually eliminated under learning andthe only possible limit points of the system arethe two perfect foresight stationary states. Thisoccurs because the stability properties of thetwo steady states are altered under learning asopposed to perfect foresight.

In Marcet and Sargent’s model, fiat currencyis the only money. ‘the model consists of two

‘See Cagan (1956) and Nerlove (1958) for expositions ofthe adaptive expectations hypothesis.

“See Muth (1960).‘‘Formally, the problem is one of stability in a recursive

stochastic environment. This means that the economistwants to take uncertainty, time and feedback (beliefs affectoutcomes and outcomes affect beliefs) into account in thesame model to find out the conditions under which con-vergence will occur.

‘25ee Marcet and Sargent (1988, 1989a.b,c).

“The selection of an actual outcome among these potentialoutcomes is the subject of a spirited debate in the

macroeconomics literature which is not considered indetail here. A number of authors have suggested methodsof choosing a unique equilibrium through deeper theoriz-ing, that is, by applying certain restrictions to the model toeliminate unsavory equilibria. See, for instance, McCallum(1983), Taylor (1977) and Evans (1985, 1986). Some recentevaluations of these selection criteria suggest they areless than satisfactory. See, for instance, Pesaran (1988)and Boyd and Dotsey (1990).

“See Marcet and Sargent (1989b).“See Sargent and Wallace (1981, 1987).

52 II1IIIIIII1IIIIIIIaFEDERAL RESERVE BANK OF St LOWS

I 53

equations relating the price level to the stock ofcurrency. Let p be the price level and c, be thestock of currency per capita at time t. The modelis given by

(1) p, = AF,p~, + ye,

(2) c, = c,_, + Op,

(3) F,p,÷, fJ,p,

where (1 < A < 1; y,d > 0; p,,c, > 0; and c, > 0is given. The term ~,p,÷, is the forecast at time

of the price at time t+ 1, where /3, is the ex-pected (gross) inflation rate at time r. Rearrang-ing equation 1 with c, on the left-hand sideyields a linear money demand equation. Equa-tion 2 can be interpreted as a government bud-get constraint, where the government chooses c,to maintain a fixed real deficit 6. The model hassome microfoundations, since it can be derivedin an overlapping generations framework withutility-maximizing individuals. The model gener-ates time paths for c, and p, given an additionalassumption on how expectations are formed.The remainder of the section will develop andcompare the results under two alternative as-sumptions: rational expectations (or perfectforesight) and learning as described by a leastsquares autoregression.

Perfect Foresight DynamicsFirst, close the model by assuming individuals

have rational expectations, which in this case isequivalent to perfect foresight because the modelis deterministic. Then

(4) ~ =

Equations 1-4 can he rearranged to yield:”

(5) P~÷,= [A~ + I dyA~’] - A-’p~ -

A rational expectations equilibrium is a se-quence (/~jt, that satisfies (5), and it determinesthe equilibrium sequences for e, and p. The dif-ference equation 5 has two roots, /37 and /3,where 1 < /3,~< (3 < A~’;these roots are thestationary states of the model. The real govern-ment deficit 6 is a shift parameter in equation 5.

The differing policy implications at the twosteady states are illustrated in figure 1, which isborrowed from Marcet and Sargent’s paper. A

“See the appendix for the detailed derivation of this andsubsequent statements in this section.

“The appendix also describes how to determine stabilityproperties in figures 1 and 2. Only initial values betweenzero and A~are feasible.

pt

permanent increase in the government deficit 6shifts the entire curve down in the figure. Atthe low-inflation steady state, f3 this parameterchange raises the stationary inflation rate. Butat the high-inflation steady state, /3, a perma-nent increase in 6 lowers the stationary inflationrate. In the first case, the comparative staticsare “classical,” while in the second they are“perverse.” Any policy advice about inflationbased on this model differs depending on whichasymptotic outcome is considered more likely tobe observed.

The high-inflation stationary state, /3, is theattractor for all initial values /3, between /3’ andA”.” The low-inflation stationary state is at-tainable only if /3, = /37 If the initial conditionsare between zero and /37 no equilibrium se-quence exists. Altogether, there are many possi-ble equilibrium sequences in the model, andthey can be indexed by /3,. Of these, all but oneconverge to the high-inflation stationary state/7’, ~yhere the comparative statics are perverse.

In related work, Sargent and Wallace haveused similar arguments to claim that monetaristmodels can yield “unpleasant” results. Under ra-tional expectations, the “bad” stationary statewith the perverse comparative statics is the

Figure 1Perfect Foresight

IIIIIIIIIIIIII1IIa JANUARY/FEBRUARY 1991

eventual outcome for virtually all initial condi-tions, provided an equilibrium exists.

Perfect foresight is a strong assumption, but itcan be made palatable by arguing that peopleeliminate systematic errors in their forecastsover time. Therefore, perfect foresight may pro-vide a good approximation once the steady statehas been attained and learning is complete.

Marcet and Sargent have developed techniquesto analyze this argument in detail. Operationallyspeaking, people can be viewed as using sometype of statistical technique to infer a futureprice from available data. One widely knowntechnique is ordinary least squares (OLS). Thenext portion of the paper analyzes the modelusing the assumption of least squares learning.

Least Squares Learning DynamicsTo develop results analogous to the perfect

foresight case under least squares learning,Marcet and Sargent replace equation 4 with

(4,) /3, = ~ [~~8~-] -

People form their forecasts of future inflationby calculating /3,, which is found via a firstorder autoregression using available data throughtime t— 1. The difference equation that describesthe evolution of /3, is in this case given by:

(5’)~=fl,,+ ~

~:,~where

(1-Afl,,) G(fl,,)-fi,,I

(5”) ~ = 1 -

1 — 1J3,_, —

While equation 5’ is quite complicated, it can beinterpreted without too much difficulty. Near astationary equilibrium, the term (1 — A/3,~jI(1—

A/3,,,,) is close to unity, while the term multiply-ing the brackets is always between zero andone. Therefore, near a steady state, equation 5’states that /3, is approximately a convex com-bination of /3,, and G(/3,,). That is, near a sta-tionary equilibrium, the projected gross inflationrate is a weighted average of last period’s pro-jected gross inflation rate and a certain function

54

Figure 2Least Squares Learning

G(/3,,.,). Alternatively, one canperiod’s prediction “updated”amount.

The possible steady states under learning, /37and j3, are the same steady states possible underperfect foresight. This captures the notion thatsystematic forecast errors are eliminated overtime and suggests that stationary rational ex-pectations equilibria provide a good benchmarkfor the study of learning. Marcet and Sargentclaim, in fact, that when least squares learningmechanisms converge, they always converge torational expectations equilibria.’~

Now that a method of expectations formationhas been added to the model, however, someaspects have changed. Marcet and Sargent havecompleted an analysis of the complicated dif-ference equation given by (5’). They show thatequation 5’ describing the dynamics under leastsquares learning is closely related to the follow-ing simpler equation:

(6) /3, = G(/3,,j -

For the purposes of this exposition, it will suf-fice to analyze (6).” The graph that Marcet andSargent use to describe the dynamics of (6) isgiven in figure 2.

view (5’) as lastby a certain small

“See Marcet and Sargent (1989a,b,c).“Readers interested in the actual dynamics under least

squares learning are referred to Marcet and Sargent(198gb).

IIIIII

(1 “vi) A”

IIIIIIIIIIIIaFEDERAL RESERVE BANK OF St LOUlS

I 55

The approximate dynamics under least squareslearning as described by (6) indicate that thelow-inflation steady state /37 is the attractor forall initial conditions /3,, between zero and /3;.The high-inflation stationary state is attainedonly if /3, = /3;. If /3, is between /3; and A’’, noequilibrium exists under least squares learning.The policy advice emanating from this analysisis therefore approximately the opposite fromthat offered in the perfect foresight case. Underleast squares learning, the low-inflation sta-tionary equilibrium with the classical com-parative statics is the asymptotic outcome forvirtually all initial conditions, provided that anequilibrium exists.

It is important to emphasize at this point thatno generic result links learning with “good”equilibria and rational expectations with “bad”equilibria. In some other model, rational expec-tations may be associated with good outcomesand learning with bad outcomes. Nevertheless,some may take comfort in the fact that, at leastfor the present case, the comparative statics areonce again “classical” under a plausible assump-tion about how expectations are formed.

Experimental EvidenceSo far, the example has illustrated that, while

the potential stationary equilibrium inflationvalues are exactly the same under perfect fore-sight and least squares learning, the stabilityproperties are approximately reversed, leadingto opposite policy implications. Nothing hasbeen said about whether perfect foresight orleast squares learning is the better descriptionof human behavior in this context. Recently,however, Ramon Marimon and Shyam Sunderhave gathered some experimental evidence thatbears on this issue.”

Marimon and Sunder summarize the resultsfrom a set of controlled experiments that usestudents as subjects. Their model is an overlap-ping generations version of the one used byMarcet and Sargent. The students were orientedto the context of the model and asked to fore-cast inflation, with monetary rewards for moreaccurate predictions. The authors were especial-ly interested in characterizing the actual out-comes for inflation in the model where the deci-sions are made by humans.

The results of these experiments indicate thatactual inflation tends to cluster around the low-

inflation stationary equilibrium. Marimon andSunder never observed a tendency of the infla-tion rate to converge to the high inflation sta-tionary equilibrium in any of their experiments.Based on the forecasts made by their subjects,the authors conclude that least squares learningprovides a good approximation to observedbehavior. While the work has some caveats andis open to interpretation, it at least providespreliminary evidence on the viability of assum-ing least squares learning.

LEARNING IN OTHERMACROECONOMIC CONTEXTS

The notion of a rational expectations revolu-tion stresses the much greater emphasis thatmacroeconomists have placed on the plans ofindividuals in macroeconomic models since1970. One way to think about current researchin the area is to categorize models according totheir treatment of expectations formation. Whilemost authors want to analyze models where ex-pectations are “rational,” several different ap-proaches have been taken. These different ap-proaches within rational expectations macroeco-nomics reflect unresolved issues in the theoryof expectations formation. Various views havebeen espoused about the specific informationthat individuals take into account when they areplanning for the future; such issues becomesalient partially because rational expectationstheory does not specify a method of learning.Thus, at least three different categories ofmacromodels that attempt to analyze individualswith rational expectations can be identified.

Forecast Functions That Use 0,1EvHistorical Data

The first group contains models in which in-dividuals forecast using only the history of theeconomy as a guide. In the previous example,for instance, learning was modeled as being bas-ed on the historical price sequence that peopleobserved. The approach could be extendedrelatively easily to include historical time serieson other variables in the individuals’ forecastfunctions. While authors assuming rational ex-pectations normally do not make statementsabout the learning mechanism implicitly under-lying their models, it seems that an appropriate

IIIIIIIIIIIIIIII

“See Marimon and Sunder (1990).

a JANUARY/FEBRUARY 1991

use of available history is what many have inmind.

Considering the use of historical data only,one might be tempted to conclude (say, fromlooking at the results of the model from the lastsection) that the only possible long-run outcomesare stationary states. In fact, most discussion ofrational expectations applies to dynamic steadystates. This notion of long-run equilibrium as astationary state, which dominates most macro-economic thought, was challenged recently byan argument that the dynamic equilibrium of aneconomy might be periodic or chaotic, evenunder rational expectations and perfect competi-tion.2’ Therefore, the artificial economy may notconverge to any of the multiple rational expec-tations steady states, instead remaining in a per-manent periodic equilibrium. Cyclical equilibriaare important because, presumably, policy im-plications are altered if long-run equilibria areperiodic.”

Forecast Functions That Includethe Beliefs of Others

A second group of macromodels containsforecast functions where, in addition to histori~cal time series, the beliefs of others also play arole.2a Consider, for instance, the view of Cassand Shell: “In seeking to optimize his own ac-tions, an economic actor must attempt to pre-dict the moves of all other economic actors.”4

Such consideration adds a new element to theinference process.’5 Of course, the notion of in-teraction among individuals (especially individualfirms) has a long history in economics; it is elimi-nated in the Arrow-Debreu competitive equilib-rium framework by the assumption of a largepopulation.

Still, a participant contemplating a forecastmay well be concerned with the aggregate ex-pectations of the remaining players, or merelywith the beliefs of one other player, such as thegovernment in a monetary policy game.” The

game—theoretic macromodels that take this lat-ter view are structurally very different fromthe traditional models, even though they areboth based on rational expectations. Most im~portantly, they are different in terms of policyimplications: these models often yield “. . - anequilibrium that is extremely sensitive to thepublic’s beliefs about the monetary authority’spreferences.” This conclusion does not hold ina model in which individual forecasts are afunction of historical data only, such as themodel of section two, because it excludes thepreferences of the government and the public’sbeliefs about them.

Forecast Functions That IncludeFrivolous Variables

When individuals forecast based only on rele~vant past history, they are sometimes said to bebasing expectations on “fundamentals.” Whendeciding what is relevant, however, some peoplemay rationally take into account what seems toan observer to be irrelevant information in theform of a frivolous variable (typically called thesunspot variable). This variable acquires impor-tance in determining the actual outcomes of theeconomy only because people think it is impor-tant. The frivolous variable serves only to signalchanges in expectations. Once some individualstake the frivolous variable into account in form-ing expectations, it becomes rational for allothers to do so, since the variable actually doesinfluence outcomes.”

Predictions in models with this type of indi-vidual forecast function are based not only onrelevant and objectively irrelevant historical in-formation, but also on the expectations of others,since the beliefs of others are taken into accountwhen the frivolous variable is assessed. Theliterature on sunspot equilibria provides a thirdbut distinctly different strain of rational expec-tations macroeconomics, with distinctly differentpolicy implications.” In particular, “. - . a con-

“See, for instance, Grandmont (1985). Chaos means thatthe equilibrium sequence is aperiodic but bounded anddisplays sensitive dependence on initial conditions. For ageneral discussion of chaos, see Butler (1990).

“In the overlapping generations model, for example, the ex-istence of periodic equilibria is disturbing because it im-plies that welfare varies from generation to generation.

“John Maynard Keynes, for instance, discussed this type offorecast function in some detail. See Keynes (1936),p. 156.

2Cass and Shell (1989).

“But not necessarily a strategic element. See Rogoff (1989).26This theme is outlined in detail in the volume edited by

Frydman and Phelps (1983).

“Rogoff (1989).“See, for instance, Azariadis (1981).“See Cass and Shell (1989).

IIII

56 IIIIIIIIIIIIII

rrnrnai nrsr~vrRMJ~(nr ~t flth~ a

I 57

sideration of the complete set of possible equi-libria, [including the sunspot equilibria], associ-ated with a given policy regime may alter one’sevaluation of the relative desirability of alterna-tive policies, relative to the conclusion that onemight reach if one considered only a singlepossible equilibrium ...“

The existence of sunspot equilibria raises thequestion of whether models with learning havedynamics converging to them. One author,Michael Woodford, has shown that exactly thissort of dynamics is possible.” Also, in general,the extent of the sunspot phenomenon is wide-ranging since there are no limits to the numberof possible frivolous variables.

These three approaches to macroeconomicsdiffer according to their alternative assumptionsabout what it means to assume rational expecta-tions. Since the theory provides no method ofexpectations formation (that is, no learning pro-cess), researchers are free to provide their own:perhaps individuals base their expectations onsunspots, or the expectations of others, or astraightforward application of classical or Baye-sian inference to historical time series. As partof the legacy of the rational expectations revolu-tion, all three approaches place heavy emphasison the role of individuals’ views of the future ininfluencing current macroeconomic equilibria.

Unfortunately, there is little prospect thateconometric analysis wili decide which versionof rational expectations is correct. Because thetheory is not well-defined, the empirical testsare unconvincing.” While economists want toassume that expectations are rational, the im-plications of this consensus for modeling andfor policy are in doubt.

SUMMARY: THE DIFFICULTY OF

DEFINING OPTIMAL BELIEFSSeveral lines of research, each in its own way,

are attempting to extend the rational expecta-tions hypothesis to include learning. The firstand most obvious is a direct attempt to findmild assumptions showing that most reasonablemethods of expectations formation converge toparticular types of rational expectations equilib-ria. If this can be done, the concept of rationalexpectations equilibrium can be said to provide

“Woodford (1988).

“See Woodford (1990)

a good approximation to the concept of long-runcompetitive equilibrium. This literature, how-ever, generally ignores the problem of the ex-pectations of others and of frivolous variables.

Even simple learning mechanisms often havenot yielded the outcomes that intuition suggestsmight have occurred. The notion that a reason-able method of expectations formation must giverise to a dynamic system that always convergesto a certain a priori plausible rational expecta-tions equilibrium has been gradually eroded.The general conclusion so far seems to be thatexplicitly introducing learning into macroeco-nomic models is unlikely to provide a widely ap-plicable selection criterion for rational expecta-tions equilibria. That is, in a rational expectationsmodel with multiple equilibria, introducing alearning mechanism does not appear to reduce theset of potential outcomes in any meaningful way.

One reason for this disappointing result isthat it is difficult to define optimal learning. Notonly is the class of plausible mechanisms quitelarge, it is also hard to limit the learning techni-ques under study to one that can be justified bysome optimality argument. One of the biggestproblems is that the usual statistical techniquesare, strictly speaking, inapplicable to the pro-blem of individual inference in the context ofmacroeconomic models.” The source of difficul-ty is that, in models with expectations, there isan aspect of simultaneity in the sense that be-liefs affect outcomes and outcomes affect beliefs.In order to apply standard inference techniques,people must be unaware of the effects of beliefson outcomes. Making this assumption is unsatis-factory, however, because it means that individ-uals ignore relevant and potentially useful in-formation when forming their forecasts.

Nevertheless, work continues on ways to ex-plicitly model learning; Marcet and Sargent pro-vide one example. Some attempt is made tochoose the learning mechanism via an optimiza-tion criterion, and the asymptotic properties ofthe implied systems are then analyzed. Thisresearch agenda is difficult and relies to a largeextent on mathematical machinery only recentlydeveloped to study such systems.

The policy implications of including learninghave been emphasized in this summary of therecent research. In models with multiple ra-

“Webb (1988).“See Marcet and Sargent (1989a,b,c).

IIIIIIIIIIII

IIIIIa .lANllaRV/FFRRtIARV leqi

58

tional expectations equilibria, the consensus opi- - Models of Business Cycles, Yrjo Jahnsson Lec-nion that people eliminate systematic forecast tures, (Basil-Blackwell, 19~b).

errors is generally not enough to determine the Marcet, Albert, and Thomas J. Sargent. “The Fate of Sys-actual outcome. Moreover, because different ra- tems with Adaptive Expectations,” American EconomicReview (1988), pp. 168-71.tional expectations equilibria have different im-

_______ - ‘Convergence of Least-Squares Learning Mechan-plications for policy, merely stating that the isms in Self Referential, Linear Stochastic Models:’ Journalecoribmy will converge to one of the possible of Economic Theory (1989a), pp. 337-68.

_______ - “Least-Squares Learning and the Dynamics ofstationary equilibria, without saying which one,is insufficient for useful policy advice. Work on Hyperinflation,” in William A. Barnett, John Geweke, andhow expectations are formed has only just be- Karl Shell, eds., Economic Complexity: Chaos, Sunspots,gun; one hopes that it will lead economists Bubbles, and Nonlinearity (Cambridge: Cambridge Universi- Ity Press, 1989b), pp 119-40.eventually to understand how equilibrium isachieved and what their policy advice is likely . - “Convergence of Least-Squares Learning in En-vironments with Hidden State Variables and Private Infor-to produce. mation:’ Journal of Political Economy (1989c), pp. 1308-22.

Marimon, Ramon, and Shyam Sunder. “Indeterminacy ofEquilibria in a Hyperinflationary World: ExperimentalEvidence[ Working paper, University of Minnesota and

REFERENCES Universitat Autonoma de Barcelona, and Carnegie-MellonUniversity (January 1990).

Azariadis, Costas. “Self-Fulfilling Prophecies:’ Journal of McCallum, Bennett I “On Non-Uniqueness in Rational Ex-Economic Theory (1981), pp. 380-96. pectations Models: An Attempt at Perspective:’ Journal of

Monetary Economics (1983), pp. 139-6&Boyd, John H., Ill, and Michael Dotsey. “Interest Rate Rules

and Nominal Determinacy’ Working paper No. 901, Muth, John F. “Optimal Properties of Exponentially WeightedUniversity of Rochester and the Federal Reserve Bank of Forecasts:’ Journal of the American Statistical AssociationRichmond (1990). (1960), pp. 299-306.

Butler, Alison. “A Methodological Approach to Chaos: Are_______ - “Rational Expectations and the Theory of Price

Economists Missing the Point?” this Review (March/April Movements:’ Econometrica (1961), pp. 315-35.1990), pp. 36-48.

Cagan, Phillip. “The Monetary Dynamics of Hyperinflation[ Nerlove, Marc. “Adaptive Expectations and Cobwebin Milton Friedman, ed., Studies in the Quantity Theory of Phenomena:’ Quarterly Journal of Economics (1958),Money (University of Chicago Press, 1956), pp. 25-120. pp. 227.40.

Cass, David, and Karl Shell. “Sunspot Equilibrium in an Pesaran, M. Hashem. The Limits to Rational ExpectationsOverlapping-Generations Economy with an Idealized Con- (Basil-Blackwell, 1988).tingent Commodities-Market,” in William A. Barnett, JohnGeweke, and Karl Shell, eds., Economic Complexi- Rogoff, Kenneth. “Reputation, Coordination, and Monetaryty: Chaos, Sunspots, Bubbles, and Nonlinearity (Cam- Policy:’ in Robert J. Barro, ed., Modern Business Cyclebridge: Cambridge University Press, 1989), pp. 3-20. Theory (Harvard University Press, 1989), pp. 236-64,

Evans, George. “Expectational Stability and the Multiple Sargent, Thomas J. Macroeconomic Theory (AcademicEquilibria Problem in Linear Rational Expectations Press, 1987).Models:’ Quarterly Journal of Economics (1985),pp. 1217-33. Sargent, Thomas J. and Neil Wallace. “Some Unpleasant

Monetarist Arithmetic:’ Federal Reserve Bank of Mm-Evans, George. “Selection Criteria for Models with Non-

_____ Ineapolis Quarterly Review (Fall, 1981), pp. 1-17.uniqueness:’ Journal ot Monetary Economics (1986),pp. 147-57. _______ - “Inflation and the Government Budget Constraint:’

Frydman, Roman, and Edmunds Phelps. Individual in Assaf Razin and Efraim Sadka, eds., Economic Policy inForecasting and Aggregate Outcomes: Rational Expecta- Theory and Practice (MacMillan, 1987), pp. 170-200.tions Examined, (Cambridge University Press, 1983). Taylor, John B. “Monetary Policy during a Transition to Ra-

Grandmont, Jean-Michel. “On Endogenous Competitive Busi- tional Expectations:’ Journal of Political Economy (1975),ness Cycles:’ Econometrica (1985), pp. 995-1045. pp. 1009-21.

Harvey, A. C. The Econometric Analysis of Time Series, - “Conditions for Unique Solutions in Stochastic(Philip Allan, 1981). Macroeconomic Models with Rational Expectations:’

Keynes, John Maynard. The General Theory of Employment, Eoonometrica (1977), pp. 1377-85.Interest and Money (MacMillan, 1936). Webb, Roy H. “The Irrelevance of Tests for Bias in Series of

Lucas, Robert E., Jr. “Understanding Business Cycles:’ in Macroeconomic Forecasts:’ Federal Reserve Bank of Rich-Karl Brunner and Allan H. Meltzer, eds., Stabilization of the mond Economic Review (November/December 1988),Domestic and International Economy, Carnegie-Rochester pp. 3-9.Conference Series on Public Policy (Amsterdam: North-Holland, 1977), pp. 215-39. Woodford, Michael. “Monetary Policy and Price LevelIndeterminacy in a Cash-in-Advance Economy:’ Working

_______ - “Adaptive Behavior and Economic Theory:’ in paper, University of Chicago (December 1988).Robin M. Hogarth and Melvin W. Reder, eds., RationalChoice: The Contrast Between Economics and Psychology . “Learning to Believe in Sunspots:’ Econometrica(University of Chicago Press, 1987a), pp. 217-42. (1990), pp. 277-308.

tnn~,,rc,cnu ~ nr Cr , C~,,~ a

I__________________________1 AppendixDetails of the Marcet and Sargent Model

I This appendix provides the mathematical (1 + A-~ ydA~’)±v~i i-~~~yoA-uj2_~details for Marcet and Sargent’s model outlined 2

in section two. For convenience, the model is

I reproduced here; it consists of the three The level of the real deficit 6 > 0 must beequations: chosen such that these roots are real. This

(1) p, = AF,p,÷ + yc, requires

I (2) c, = c,_, + (A.1O) 6 < Ay (A’ + I — (4A~)l~z]= 0,,,,,.(3) F,p,÷ =

with Q<A<i; y,d > 0; p,,c~> 0; and c0 > ~ The roots also satisfygiven. First close the model under perfectforesight: (All) 1 < 13; p -c A.

I ~ = - These facts provide the basis for the qualitative

P graph of figure 1.

I Substituting (4) in (1) and rearranging shows To analyze the dynamics of the model usingthat J3~< A’ is required to be compatible with the graph, consider some initial condition J3~on> 0. Substitute (3) in (1) to obtain the horizontal axis. Find the value /3, by tracing

I (Al) p = Afip + ~,, up from J3~to the plotted function. The value ofnow serves as the input for the next period.(AZ) p,, = AJ3,p,_ + yc,.. To transfer the 13 value to the horizontal axis,

I Rearranging (AZ) trace horizontally from the plotted function tothe 45 degree line, and from the 45 degree line

(A.3) c, = y[p,, — A/3t,p,_,] down to the horizontal axis. Now repeat theprocedure as though J3~is the initial condition.

I Substituting (2) into (Al) gives Next, solve the model under least squares(A.4) p = Afl,p, + y[c,_, + dpi. learning. Use

I Now substitute (A.3) into (A.4): = -t ~

(AS) p, = Afl,p + y[y(p~., — Afl,,p,) + Op,]In order to get to equation Sin the text, (4’)

(AG) Ap. = 1 — fl~,+ A — yd must be written in recursive form. To do this,

I or, iterating forward and rearranging, define temporarily two vectors

(5) f3’~, = (I + A’ — y61’) — A’fi,’ (A.12) P,_, = [p0

,-.., p,_,]’

as in the text. A rational expectations equilibri-um is defined as a sequence [13,)r, that solves (A.13) P,1 = [p,,-.. p,_21’-(5). Proceed to analyze (5) as follows:I (A.7) dp,÷, A-’p~2>0 Then

(A.14) /3, = [P~2P,_2]’ P~_2P,_,

I (AS) d213,÷, = —2A’fl[’< 0. andd/3~ (A.15) /3, = [P;,P,_3] ‘

I The roots of (5) are found by setting /3÷,= 13, where nowand applying the quadratic formula: (A.lG) ~,_, = [p ,---~

I (AS) ~ = (A.17) P,~,= [p0 p,-3].

• JANUARY/FEBRUARY 1991

60 IThe additional information in /3, is the observa- Substituting (A.23) into (A.ZZ) gives the equationtion p,,. The relationship between /3, and /3,, is in the text:given by a well-known recursive least squaresformulal which can be applied as follows: (5) ~ p~2 [ (I—A/3,2) G(/3,.,)—/3,_, -

(Ala) /3, = fl,, + [P~,P,-t’p,3(p,, — ~:~, ~ (l—Afl,,)

A set of positive sequences 1/3,, c,, pj: Iwhere satisfying (l)-(3) and (4’) is an equilibrium under

least squares learning. To approximate the(A.lS)f, = I + ~ ~~

3Pt~

3~’ Pt-2 dynamics under least squares learning, consider

is a scalar. Since the simpler but closely related differenceequation:

(A.Z0) [P3P,3~’ = [:~,P5k] -, ‘ (8) /3, = G(fl,,) = I - fl,i yd - Ithe scalar f, can be written as

(A.211f, = [t? ~~1]-, [j1 ~, + - The derivatives are given by ISubstituting into (Ala) yields (A.24) = Ayd(l — A(3,, — ydY2> 0 p(A.22) fi,=fl,,+ ‘-~ [p,, — p,, /3,_I.

~:,P~, (A.Z5) ~ = ZA2yd(l — Afl,, — yd)’ > 0

To obtain equation (5’), use (1), (2) and (3) tofind provided /3 < A’ (1 — yd). This provides the in-

formation for the second qualitative graph,(A.23) p’, = G(fl,,) I—Afl,2 ~ given by figure 2.

I —A/3,_,

IIIIIII

1See Harvey (1981), p. 54.

FEDERAL RESERVE BANK OF St LOUIS