LEARNING WITH EXPERT ADVICE

LEARNING WITH EXPERT ADVICE

Krisztina MolnárNorwegian School of Economics andBusiness Administration

AbstractSurveys of inflation forecasts show that expectations combine forward-looking and backward-looking elements. This contradicts conventional wisdom: In the presence of rational agents,adaptive agents would be driven out of the market. In our paper, we rationalize this finding in anequilibrium framework. Our model has two types of agents, one having rational expectationsand the other using adaptive learning. The proportion of these agents in the population evolvesaccording to their past forecasting performance. We show that even an underparameterizedlearning algorithm survives competition with rational expectations. (JEL: C62, D83, D84)

1. Introduction

The importance of forward-looking behavior in economic decision-making haslong been recognized in economics. However, the modeling of expectationsremains a matter of controversy. Rational expectations is criticized for placingunreasonable computational and informational demands on economic agents;also, a vast empirical literature rejects rational expectations.1 However, once wedepart from fully rational expectations, there are many ways to do so. Thus,agents should be allowed to abandon their ad hoc expectation rule if they coulddo better otherwise. Indeed, the early literature motivated rational expectationsby saying that if agents did not behave rationally they would disappear from themarket.

In this paper we address the question: What happens if agents follow least-squares learning but have access to the forecast of an expert who can actually dobetter? Least-squares learning departs from rationality in a way that still attributes

Acknowledgments: I am grateful to my supervisor Albert Marcet for his careful reading andextremely helpful comments and suggestions. I would also like to thank Massimo Guidolin, OmarLicandro, Gábor Lugosi, Ramon Marimon, Karl Schlag, and participants at the second PhD Con-ference in Research in Economics: Aims and Methodologies, Pavia 2004, and the EEA Meeting,Vienna, 2006. All remaining errors are mine.E-mail address: [email protected]. See, for example, Lovell (1986), Baghestani (1992), and Ball and Croushore (2003).

Journal of the European Economic Association April–May 2007 5(2–3):420–432© 2007 by the European Economic Association

Molnár Learning with Expert Advice 421

a lot of rationality to agents;2 still, the choice of a learning algorithm is necessarilyarbitrary and subject to the caveat already raised: Agents would abandon theirlearning rule when they have access to a better forecasting algorithm. We assumethat the forecasts of an extremely clever expert are also available and that thisexpert has fully rational expectations.

Weights on learners and experts evolve dynamically. The key assumptionis that we allow private agents to choose between the learning and the ratio-nal predictor, assuming private agents do not ex ante differentiate between thembut choose depending solely on past predictor performance. Whoever made thebetter forecasts in the past will have a higher weight in the population. Animportant feature of this weighting is that heterogeneity can be an equilibriumoutcome.

The main result is that, when private agents must find the good forecasterfrom past data, learners can survive. Even if learners know much less aboutthe economy than rationals, they can learn to make a good forecast and witha positive probability even provide a better forecast than rationals. This resultrationalizes empirical work on survey expectations suggesting that expectationsactually combine backward- and forward-looking elements.3

A second result is that the presence of rational experts can introduce stabilityto the economy. There are parameter regions for which learning would explodewithout rational expectations agents, but with the help of rational experts learningcan converge.

In designing the weighting algorithm we build on a recent approach inlearning theory known as prediction with expert advice.4 This literature con-siders problems of sequential decision making under uncertainty without makingassumptions about the data-generating process. Instead, the decision maker com-petes with a wide set of forecasting experts, and the goal is to design decisionalgorithms that are almost as good as the best expert in a wide benchmark class.5

We propose a simple dynamic predictor selector in this spirit that has the further

2. They “behave as working economists or econometricians” (Sargent 1993, p. 22). For a surveyof adaptive learning and boundedly rational modeling see Marimon (1996).3. See Roberts (1998), Baak (1999), and Chavas (2000).4. This idea of forecasting with the use of expert advice is applied in several branches of economictheory (for a summary see Cesa-Bianchi and Lugosi 2006). In game theory, the concept of correlatedequilibria takes experts to be pure strategies; in finance, portfolio choice models regard experts asdifferent portfolio strategies.5. In other words, the goal of the predictor is not to minimize his loss function but to minimize hisrelative loss (or regret) compared to the best “expert” in a fixed, possibly infinite, set of experts. Themain focus is to provide general upper bounds on the cumulative regret. In the terminology of thisliterature, “experts” in our paper are the two forecasting algorithms (forecasting with least-squaresregression and rational expectations). To avoid confusion we will consider only the rational agent tobe an expert. (In the terminology of this literature, “predictors” in our paper are simulatable experts:functions that use data accessible to the forecaster himself; thus the forecaster can simulate theexperts’ future reactions.)

422 Journal of the European Economic Association

advantage of being analytically tractable for least-squares learning; it thereforegives a clear intuition about convergence results.

In addition to the weighting algorithm, our paper differs in two main respectsfrom earlier literature with endogenous weighting. First, least-squares learnersexhibit more rationality than adaptive agents coexisting with rationals in otherpapers (naive agents in Brock and Hommes [1997] and Sethi and Franke [1995]and simple adaptive agents in Branch [2002]).6 Second—and in contrast to Brockand Hommes, Sethi and Franke, and Branch—our agents do not have to pay forthe forecasts of the rational agent. In Sethi and Franke (1995), costly optimizationof the rational agent is needed to guarantee survival of naive agents; with costlessrational forecasts, naive agents are driven out of the economy. In Brock andHommes (1997), coexistence may prevail under zero optimization cost whenagents do not fully optimize.7 In contrast, in our paper the main idea is that, whenit takes time to find out who is the good forecaster, learners can learn to producegood forecasts and survive; this holds true even if private agents pay attention toany infinitesimal difference in predictor performance.

The paper is organized as follows. Section 2 presents the model. Section 3establishes properties of the equilibrium and conditions of convergence for thebenchmark economy with two types of predictors: fully rational and a learningalgorithm that is able to learn the rational expectations equilibrium. Section 4extends the results to the case when the learning algorithm is underparameterized.Section 5 concludes.

2. The Model

It is interesting to model the game between learners and rationals in a self-referential model, where expectations influence the outcome. A self-referentialsetup makes the convergence of learning nontrivial, because exploding expecta-tions can be self-confirming as expectations make the outcome also exploding.There are several models that would suit our analysis. In order to keep our focus onthe expectations side, we choose a simple underlying model, the cobweb model.

Consider an economy populated by a set of firms [0,1] who maximizeexpected profit. Firms consider themselves to be atomistic, and so assume notto influence the aggregate price level. Firm i produces a nonstorable good and

6. Adam (2005) and Branch and Evans (2006) are examples of learning with endogenous weighting.In Adam agents choose between two learning algorithms. Branch and Evans build a heterogeneouslearning model where learners are constrained to underparametrize.7. In the terminology of this literature, agents choose between predictors with a finite intensity ofchoice.


faces a production lag; thus the firm makes its supply decision with a lag. That is,

S(Eit−1[pt ]) = arg max

qit

Eit−1π

it

= arg maxqit

Eit−1[pt ]qi

t − c(qit

) = (c′)−1(Eit−1[pt ]

),

where p denotes the price level, q the quantity of the nonstorable good, π profit,S(·) the supply function, and c(q) a cost function increasing in q. We assumea quadratic cost function c(q) = q2/2b, b > 0, which implies that the supplydecision of firm i is a simple linear function of the firm’s expectation about theprice level next period: Si

t = bEit−1[pt ], and the profit of firm i is

πit = ptE

it−1[pt ] −

(Ei

t−1[pt ])2

2b. (1)

Firms face a stochastic demand on a competitive market:8

D(pt , µt ) = Aµt − Bpt , A, B ∈ R, µ ∼ AR(1).

Market equilibrium is given by equating demand and supply:

pt = λ

∫ 1

0Ei

t−1pt di + mt, (2)

where λ = −b/B. For convenience we have redefined the stochastic processmt = (A/B)µt , mt = �mt−1 + εt , ε ∼ i.i.d.N(0, σ 2

ε ).9

Firms choose a predictor from a set of two types of predictors, the least-squares (LS) predictor and the rational expectations (RE) predictor. Marketexpectations formed at t −1 depend on the fraction of firms using the LS predictorand the RE forecast at time t − 1:

∫ 1

0Ej,t−1pt dj = ωt−1E

LSt−1pt + (1 − ωt−1)E

REt−1pt , (3)

where ωt−1 ∈ [0, 1] is the population weight of the LS predictor. Notice that thepopulation weights depend on time; we allow firms to switch predictors in eachperiod.

8. We assume B can be any real number in order to keep the model more general. In a cobwebmodel it would be reasonable to assume that demand depends negatively on the price level (B < 0).9. There are several economic models that have a price dynamics similar to (2). In a simple versionof the Cagan (1956) model of inflation, pt corresponds to the log price level, which depends onexpectations about the next period’s price level and an exogenous mt , the log money supply. Thebasic model of asset pricing under risk neutrality takes the same form, with pt interpreted as the priceof stock, and mt as its dividend; λ = 1/(1 + r) is the one-period discount factor, and r is the rate ofreturn on the riskless asset.


The idea of the algorithm governing the evolution of population weightscomes from the “prediction with expert advice” literature,10 and we also build onthe empirical research of Branch (2004). The prediction with expert advice liter-ature considers problems of sequential decision-making when nothing is knownabout the underlying environment, and it shows how cumulative losses can beupper bound if the forecasts of a set of experts is available. The goal is to designdecision algorithms that weight the expert forecasts in such a way that the resultis almost as good as the best expert. We borrow the main idea from this litera-ture and assume that the aggregate forecast (3) is a weighted average of the LSand RE forecast and the weights evolve dynamically according to past predictorsuccess. Similar dynamic expectation formation was found by Branch (2004) inthe Michigan survey of inflationary expectations. He found evidence that agentsswitch predictor use as the relative mean-squared errors change: agents’ predictorchoices respond negatively to increases in relative mean square error.

We assume that both predictors are free of charge but that firms don’t knowwhich one is better.11 Firms must find out from past data whether the LS or theRE “expert” is the better; therefore we assume that population weights dependon the past performance of the two type of predictors. This has an evolutionaryinterpretation: If a predictor is successful then an increasing number of firms willswitch to use it, and eventually the type with more successful forecasts will bethe more dominant type.12 Because firms are profit maximizers, we assume theychoose between LS and RE depending on the how much profit these predictorsgenerated.

The weight of LS evolves in a recursive fashion:

ωt = ωt−1 + 1

t

[F

(πLS

t − πREt

) − ωt−1], (4)

whereELS0 p1,ERE

0 p1, andω0 are given, and whereF : R → [0, 1],F(x) ≤ F(y)

for x ≤ y, and F(x) = 1 − F(−x).Weight in period t equals the weight in the previous period, as adjusted by a

measure of the forecasting performance of LS in t −1. Adjusting the weights onlypartially is suitable in a stochastic environment, where even if one predictor werecloser to the true outcome today, with a positive probability it will be worse thenext period. The forecasting performance is measured by the function F , which

10. For an excellent summary see Cesa-Bianchi and Lugosi (2006).11. We believe that assuming zero cost for the rational predictor is realistic. For example, the centralbank inflation forecasts typically are both publicly available and the closest to rationality.12. Observe that because the price dynamics (2) is linear in expectations, an alternative interpre-tation of the model could be a representative agent one: There is a representative firm who assumesitself to be atomistic and uses a weighted average of the two predictor forecasts. Weights correspondto how much this representative firm “believes” each predictor.


compares the relative profit of LS and RE. The time-t weight on LS is adjustedtoward the value of F taken at time t − 1.13

Now let us discuss what properties F(·) must fulfill. Because ω denotesthe percentage of firms using the LS predictor, weights are naturally within theinterval [0,1]. Because F(·) is monotone, a better LS forecast implies a biggerF , so the weight on LS is adjusted toward a higher value than the weight on RE.A key feature of F that the “expert” literature imposes is symmetry around 0;formally, F(x) = 1 − F(−x). This condition means that firms do not ex antedifferentiate between the two predictors: A similar performance of LS or RE isvalued the same way. Finally, for some theorems we will also assume continuityof F , which together with the symmetry condition implies that F(0) = 0.5.

Alternatively, the weighting algorithm can be written as

ωt =∑t

k=1 F(πLS

k − πREk

)t

.

The weight on LS indicates how the success of LS over RE was valued on average.One example of F is an indicator function that takes the value 1 whenever

LS has a larger profit than RE and 0 otherwise. Here F at time t simply indicateswhether LS was better than RE at time t , and ωt measures how many times LSforecasted better than RE up to time t . In the limit, ω has an intuitive interpretation:ω converges to the probability that LS has smaller forecast error than RE.

When F is the indicator function, any infinitesimal difference between profitsis rewarded. Also, any small difference is rewarded in the same way as biggerdifferences. Whenever LS is better its weight is adjusted toward 1, and wheneverRE is better its weight is adjusted toward 1. By choosing another functional formfor F we can also give a measure to how the representative agent evaluates therelative forecasting success of LS. Let us consider an example. For σ1 < σ2, F1is the cumulative distribution function (c.d.f) of N(0, σ 2

1 ) and F2 is the c.d.f. ofN(0, σ 2

2 ). Then, whenever LS makes a better profit, F1 gives a higher value thanF2 and then F1 adjusts the weight of LS to a higher value than F2.14 In otherwords: F1, the distribution with a smaller variance, places more value on smallprofit differences than F2. When σ goes to zero F(·) converges to an indicatorfunction,15 taking the value 1 whenever LS is better, 0.5 when LS and RE made

13. Instead of relative profit we could also choose the most recent relative forecast error to measurepredictor success. Similarly to Brock and Hommes (1997), this would not alter our main results.14. A similar example could be F(x) = (1/π) arctan(αx) + 1/2, α > 0. Then the larger α is, themore a good profit performance is rewarded in F .15. The normal density function in the limit of zero variance is the Dirac delta function; thecumulative distribution function of it is the unit step function. The value at 0 is often set to 0.5 byconvention. Considering the function F(x) = (1/π) arctan(αx) + 1/2, α > 0, with α → ∞, it iseasy to show that for x > 0 the limit is 1, for x < 0 the limit is 0 and for x = 0 the limit is 0.5 (withthe convention limα→∞(0 α) = 0).


equal forecasts, and 0 whenever RE is better. When the LS and RE predictor yieldsthe same profit, the value of F1 and F2 is 0.5 and the weight on both predictors isadjusted toward 0.5. Observe that if F = 0.5 (i.e., if πLS = πRE) infinitely manytimes, then the weights converge to 0.5.

The dynamic predictor selection algorithm developed in this paper has theadvantage that the joint dynamics of the learning algorithm and the weight can beanalytically examined with stochastic approximation, which makes the intuitionclearer. The first papers in dynamic predictor selection used either a multinomiallogit16 (see, e.g., Brock and Hommes 1997) or an algorithm that closely resem-bles the replicator dynamic in evolutionary game theory (see Sethi and Franke1995); none of these algorithms allow using stochastic approximation in a learningenvironment.17

To sum up, the underlying model consists of

pt = λ[ωt−1E

LSt−1pt + (1 − ωt−1)E

REt−1pt

] + mt, (5)

mt = �mt−1 + εt , (6)

and the evolution of the population weights (4). Equations (5), (6), and (4),together with the assumptions about the expectations of LS and RE, completelydetermine the price level.

3. Benchmark: Clever Learner and Rational Expert

In this section, we examine the benchmark economy with a set of predictors: (1)a learner who is able to learn the rational expectations equilibrium and (2) a fullyrational expert.

Least-squares learner. We assume that the clever learner (henceforth LSβ )is learning in the form of the rational expectations equilibrium. Under ratio-nal expectations the minimum state variable (MSV) solution of (5)–(6) ispt = �/(1 − λ)mt−1 + εt , Et−1pt = �/(1 − λ)mt−1, therefore a learningalgorithm has a chance to learn the MSV rational expectations equilibrium only

16. In a random utility model, and under certain assumptions on shocks, the limiting probabilitythat a given individual chooses an alternative is given exactly by multinomial logit.17. In these papers, the competing predictors were rational and naive agents in a non stochasticenvironment; hence, the model reduces to a system of deterministic difference equations. In oursetup the model reduces to a system of stochastic difference equations, and stochastic approximationcan be used to obtain an analytical solution. Learning algorithms with dynamic predictor selectionhave been examined by Branch and Evans (2006), who used multinomial logit law of motion; thecompeting forecasting algorithms were learners with underparameterized regressions. They use theunconditional expected relative profit to evaluate predictor selection. This choice allows them toconsider the fixed point of a map rather than a solution to a difference equation as in Brock andHommes (1997) or in our paper.


if it conditions its expectations on m. The clever learner correctly hypothesizesthat the last period’s m is a leading variable of the price level today and runs theOLS regression pi = βimi−1 + νi . At t the clever learner observes {p1, . . . , pt },{m1, . . . , mt } and uses the m process up to mt−1 to estimate βt . The forecast isthen generated using mt as follows:18

EtpLSβ

t+1 = βtmt , βt =∑t

i=1 pimi−1∑ti=1 m2

i−1

. (7)

The recursive formulation of the regression coefficient is

βt = βt−1 + 1

t − 1

1

Rt−1mt−1(pt − mt−1βt−1), (8a)

Rt = Rt−1 + 1

t

(m2

t−1 − Rt−1), (8b)

where Rt is the moment matrix.19

Rational expert. Let’s assume the rational predictor is fully rational. It is ratio-nal not in the sense of the traditional rational expectations equilibrium, for thiswould provide bad forecasts: First, because the economy might converge to a dif-ferent equilibrium; second, because even if the economy converges to the rationalexpectations equilibrium (REE) these forecasts would be wrong during the tran-sition. The RE predictor is rational within the learning equilibrium: It knows thestructure of the economy and forms expectations by conditioning on the forecastsof learners. The rational expert is fully rational: At time t she knows ELS

t pt+1,ωt , the process for m, and the price dynamics (5), so she can calculate the trueunconditional expectation of pt+1 as20

EREt pt+1 = Etpt+1 = λωtβt + �

1 − λ(1 − ωt)mt . (9)

Our benchmark economy can be viewed as an economy where firms haveno sophisticated model at hand to form their expectations but instead observe theforecast of two institutions and decide which one to use. One institution is runningOLS regressions; let’s say this is a forecasting agency. The other institution has

18. The equilibrium remains unchanged if a constant is also included in the regression.19. Rt = (

∑ti=1 mim

′i )/t .

20. The fully rational expert forecasts the true conditional expectation of p, so one can simplyfind its forecast by using ERE

t−1pt = Et−1pt in (5) and taking expectations Et−1 of both sides:Et−1pt = Et−1{λ[ωt−1E

LSt−1pt + (1 − ωt−1)Et−1pt ] + mt }. This yields Et−1pt = (λωt−1E

LSt−1 +

�mt−1)/(1 −λ(1 −ωt−1)). Alternatively, one can use the method of guess and verify with the guessERE

t pt+1 = cmt , or the guess of Nunes (2005), EREt pt+1 = c1E

LSt pt+1 + c2mt .


perfect knowledge about the economy and can calculate the true unconditionalexpectation; let’s say this is the central bank with a good research department. Thecentral bank does not have full credibility. It must gain credibility by publishingits forecasts and making good forecasts. In this interpretation the weight on therational agent will be higher when firms give greater credibility to its forecasts.Furthermore, this central bank can investigate the credibility of its forecasts (byobserving the population weights) and then use this knowledge to improve itsfuture forecasts.

Equilibrium. Next we examine the equilibrium and establish convergenceresults. For comparability we first describe convergence results when the economyis populated only with learners and then examine how the results are modified inthe presence of rational expectations agents.

The model populated only with learners is examined by several papers, for aproof of convergence we refer to these.21 It can be shown that βt converges to therational expectations solution βREE = �/(1 − λ) if λ < 1 and the equilibrium isthe MSV rational expectations equilibrium22

pt = �

1 − λmt−1 + εt . (10)

The values of β, R, and ω evolve over time, and their joint dynamical systemis a stochastic difference equation23 that can be solved via the theory of stochasticapproximation.24

Proposition 1. Let the economy (4)–(6) be populated with learners with theperceived law of motion (7) and fully rational agents (9), and let F be continuous.Then, if λ < 1 or λ > 2, the vector [βt , ωt ] converges to [βREE, 1

2 ], whereβREE = �/(1−λ) is the β corresponding to the rational expectations equilibrium.

The main result is that learners survive even in the presence of a costlessfully rational expert. In equilibrium half of the population are learners and theother half are rationals. The intuition behind this is simple: If agents do not knowwhich predictor is the RE expert then they must evaluate performance of the LSβ

and RE predictor from past data; this takes time, during which LSβ can learn

21. See, for example, Marcet and Sargent (1989).22. Strictly speaking, in Marcet and Sargent (1989) pt depends on expectations formed at time tabout pt+1, whereas in our case it depends on expectations formed at t − 1 about pt . In their modelβ converges to �/(1 − λ�) given λ� < 1.23. The stochastic recursive algorithm form consists of (8) governing the dynamics of β and Ras well as the dynamics of the predictor weights (4), where pt is substituted out by (5) and whereprofits of LS and RE are given by (1) using the expectations of the predictors (7), and (9).24. For the proof we refer to the working paper version available online at http://www.nhh.no/sam/cv/molnar-krisztina.html.


the rational expectations equilibrium. As a result,the forecast of LSβ and REwill eventually be negligibly close to each other and close to the REE. Becausethe shocks are symmetric about the mean, the learner and the expert have thesame probability of producing the better forecast. Thus shocks guarantee that insome periods learners in other periods rationals get closer to the actual outcome.In the limit, both LSβ and RE generate the same profit and the population weightconverges to F(0) = 0.5.

A second result is that the presence of rational expectations predictors intro-duces stability to the economy. Expectations of learners without rationals wouldexplode whenever λ > 1. When we introduce rational experts to the economyfor λ > 2, an interesting interaction between LSβ and RE prevails. To illustratethis, observe that for mk > 0, βk > 0 the LSβ forecast βkmk is positive, whereasthe RE forecast (9) can be negative. In particular, when ω ≈ 0.5 and λ > 2 thenumerator in (9) is negative, so for a positive LSβ forecast the RE forecast is neg-ative. In other words, the rational expert dampens the explosive nature of the LSβ

forecast by giving a forecast with the opposite sign. The rational agent can do sowhen both λ and the weight on the rational predictor are sufficiently high—whenthe RE expectations influence the outcome to a sufficiently high extent.

The equilibrium of this economy is the rational expectations equilibrium(10). When learners are able to learn the rational expectations equilibrium, thepresence of rationals alters stability conditions of the learning algorithm but doesnot alter the equilibrium itself.

4. Extension: Less Clever Learners and Rational Experts

We have shown that when it takes time for agents to find out which predictoris better, learners in the meantime are able to learn the rational expectationsequilibrium and can thereby survive competition with a costless rational agent.In this section, we briefly discuss the case whether less clever learners could alsosurvive this competition. For details we refer to Molnar (in preparation).

Let’s assume that the less clever learners are not conditioning their forecaston m and run an OLS regression of p on a constant pt = αt + νt . At time t

this predictor (henceforth LSα) observes {p1, . . . , pt } and runs a regression on aconstant; in other words, LSα takes the average of the available p data:

ELSαt pt+1 = αt , αt =

∑ti=1 pi

t. (11)

In a recursive formulation:

αt = αt−1 + 1

t(pt − αt−1). (12)


This predictor has a worse information set than the RE predictor, and, moreover,cannot converge even to the rational expectations equilibrium because it is runningan underparameterized regression (not conditioning on m).25 Still, predictor LSα

can do quite well: It can learn the true unconditional expectation of the price level.The fully rational predictor conditions its forecast on the LSα prediction and

forecasts the true unconditional expectation26

EREt−1pt = Et−1pt = λωt−1

1 − λ(1 − ωt−1)αt−1 + �

1 − λ(1 − ωt−1)mt−1. (13)

In the limit α converges to zero and ω converges to a positive number. This isa surprising result: Even an underparameterized learning algorithm can survivecompetition with a fully rational agent.27 Of course, LSα is a worse predictorthan a rational agent, but he is still quite clever in the sense that he eventuallylearns the true unconditional expectation of p under rational expectations. In thelimit the price level varies about its mean, so the forecasts of the learner will bebetter than the rational forecast with positive probability. The equilibrium weightcannot be solved analytically unless � = 0. In this case the weights of LSα willbe equal to the weights of the rational expert, ω = 0.5. This is intuitive: whenm is a random noise, conditioning on m does not help forecasting p; the rationalforecast (13) is 0, just like the learning forecast.

The equilibrium is different from the REE and depends on the equilibriumweight

pt = �

1 − λ(1 − ωt−1)mt−1 + εt .

Observe that the equilibrium depends intrinsically on the type of agents in theeconomy. It is interesting to note that the equilibrium can get close to the MSVrational expectation solution when the equilibrium ω is close to zero. This happenswhen the persistence of m is very high; then the LSα forecast will perform so badlythat its weight will be close to zero and experts will dominate the equilibrium.28

A result similar to the benchmark economy is that the presence of fullyrational agents can introduce stability into the economy. If expectations have ahigh influence on the price level (λ > 1) then explosive expectations can be selfinforcing and learning would explode without RE. With RE experts, learning canconverge whenever RE has enough influence in the economy to counterbalancethe otherwise explosive forecasts of learners.

25. In the terminology of Evans and Honkapohja (2001), LSα can achieve a “restricted perceptions”equilibrium. In a restricted perceptions equilibrium expectations are optimal within a restricted classof misspecified beliefs.26. The solution proceeds similarly as in the previous section; see footnote 20.27. This result holds true also for σ → ∞; in other words, if firms pay attention to any smalldifference in the forecasting performance of the two types of predictors.28. The negative relationship between the equilibrium ω and � can be shown numerically.


5. Conclusion

This paper has shown that the coexistence of learners and rationals can be ratio-nalized in an equilibrium framework. If agents forecast with a learning algorithmand have access to forecasts of a rational agent, then they will not rush to aban-don their ad hoc learning rule even if rational forecasts are costless. Surprisingly,learning survives forecasting competition with a rational agent even if it is under-parameterized. When it takes time to find out who is the good forecaster, learnershave time to learn to produce good forecasts and in a stochastic environment willeventually produce a better forecast then the fully rational agent with positiveprobability.

These results coincide with recent empirical research, which find that sur-vey inflation expectations are well approximated as being a weighted averageof forward-looking and backward-looking expectations. We believe that ourresults strengthen the case for modeling expectations as a mixture of adaptiveand forward-looking expectations—especially because it is well documented thatmodeling expectations this way improves the empirical performance of standardmodels.

A further result is that the equilibrium depend intrinsically on the type ofagents in the economy. This might have important policy implications. Learningand rational expectations call for different policies, so we consider it importantfor future research to examine policy implications of modeling expectations as amixture of them.

The type of agents in the economy also affects stability conditions. Makingfully rational forecasts accessible to learners can introduce stability to the econ-omy. When welfare loss is associated with being out of the steady state, as in theNew Keynesian framework, a diverging economy clearly leads to huge welfarelosses.29 A practical example could be a central bank that provides fully ratio-nal forecasts to private agents in order to stabilize expectations, where privateexpectations are formed by learning and the weights in aggregate expectationscorrespond to how much agents believe central bank forecasts. However, ourresults show that rational forecasts can stabilize the economy only if the ratio-nal forecasts already carry significant weight in aggregate expectations—that is,when the central bank forecasts are already seen as highly credible.

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