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We propose behavioral learning equilibria as a plausible explanation of coordination of individual expectations and aggregate phenomena such as excess volatility in stock prices and high persistence in inflation. Boundedly rational agents use a simple univariate linear forecasting rule and correctly forecast the unconditional sample mean and first-order sample autocorrelation. In the long run, agents learn the best univariate linear forecasting rule, without fully recognizing the structure of the economy. The simplicity of behavioral learning equilibria makes coordination of individual expectations on such an aggregate outcome more likely. In a first application, an asset pricing model with AR(1) dividends, a unique behavioral learning equilibrium exists characterized by high persistence and excess volatility, and it is stable under learning. In a second application, the New Keynesian Phillips curve, multiple equilibria co-exist, learning exhibits path dependence and inflation may switch between low and high persistence regimes.

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... The agents in the economy form subjective expectations using a linear forecasting model, which are sometimes called "perceived laws of motion" or "PLM". Hommes and Zhu (2014), working in a similar framework, assign to agents forecasting models of the AR(1) form ...

... Examples of hidden shocks depend on the precise model environment and include drifts in a central bank's inflation target, aggregate mark-up shocks, or asset float. Hommes and Zhu (2014) define a behavioral learning equilibrium as a stochastic process for y t satisfying (1), given that expectations are formed from (3), and with b equal to the first-order autocorrelation coefficient of y t . ...

... A fundamentals RPE is a fixed point,b, of (8) and, it should be noted, is equivalent to the behavioral learning equilibrium in Hommes and Zhu (2014). ...

This paper shows that belief-driven economic fluctuations are a general feature of many determinate macroeconomic models. Model misspecification can break the link between indeterminacy and sunspots by establishing the existence of "statistical sunspots" in models that have a unique rational expectations equilibrium. Building on the insights of Marcet and Sargent (1989) and Sargent (1991), with some state variables 'hidden' to a set of agents the state vector expands to include agents' expectations and, in a restricted perceptions equilibrium, agents form beliefs by projecting the state vector onto their set of observables. This set of observables can include serially correlated non-fundamental factors (e.g. sunspots, judgment, expectations shocks, etc.). Agents attribute, in a self-fulfilling way, some of the serial correlation observed in data to extrinsic noise, i.e. statistical sunspots. This leads to sunspot equilib-ria in models with a unique rational expectations equilibrium. Unlike rational sunspots, these equilibria are stable under learning. Applications are developed in the context of a New Keynesian, an asset-pricing, and a pure monetary model.

... In the simplest model applying this idea, agents run an univariate AR(1) regression to generate out-of-sample forecasts of the state of the economy. Hommes and Zhu (2014) provide the first-order SCEE with an intuitive behavioral interpretation and refer to them as a Behavioral Learn-ing Equilibrium (BLE). Although it is possible for some agents to use more sophisticated models, one may argue that these practices are neither straightforward nor widespread. ...

... A simple, parsimonious BLE seems a more plausible outcome of the coordination process of individual expectations in large complex socio-economic systems (Grandmont, 1998). Hommes and Zhu (2014) formalize the concept of BLE in the simplest class of models one can think of: a one-dimensional linear stochastic model driven by an exogenous linear stochastic AR(1) process. Agents do not recognize, however, that the economy is driven by an exogenous AR(1) process y t , but simply forecast the state of the economy x t using an univariate AR (1) Numerous empirical studies show that overly parsimonious models with little parameter uncertainty can provide better forecasts than models consistent with the actual datagenerating complex process (e.g. ...

... In Hommes and Zhu (2014), F is a one-dimensional linear function. In this paper F F F may be a general n-dimensional linear vector function. ...

We generalize the concept of behavioral learning equilibrium (BLE) to a general high dimensional linear system and apply it to the standard New Keynesian model. Boundedly rational agents learn to use a simple AR(1) forecasting rule for each variable with parameters consistent with the observed sample mean and autocor-relation of past data. Agents do not fully recognize the more complex structure of the economy, but learn to use an optimal simple AR(1) rule. We find that BLE exists, under general stationarity conditions, typically with near unit root autocor-relation parameters. BLE thus exhibits a novel feature, persistence amplification: the persistence in inflation and output gap is much higher than the persistence in exogenous fundamental driving factors. In a boundedly rational world, coordination of individual expectations on an aggregate outcome described by our simple, parsimonious BLE seems more likely. We also consider monetary policy under BLE for different Taylor interest rate rules and study whether inflation and/or ouput gap targeting can stabilize coordination on near unit root BLE.

... We discuss the concept of behavioral rationality through three case studies; more examples may be found in Hommes (2013). The first is in macroeconomics, a New Keynesian Philips curve (NKPC) with a representative agent learning the simplest, but misspecified, univariate AR(1) rule to forecast future inflation in an economy that is too complex to fully understand (Hommes and Zhu, 2013). The parameters of the AR(1) rule are pinned down by simple and intuitive consistency requirements: the mean and the first-order autocorrelation of the AR(1) forecasting rule coincide with the realizations. ...

... The inflation persistence, as measured by the first-order autocorrelation β * is typically much larger than the RE persistence (which equals ρ, the autocorrelation of the exogenous driving force). Moreover, Hommes and Zhu (2013) show that for the New Keynesian Philips curve (1) multiple SCEE may coexist. Coexistence of multiple SCEE is illustrated in Figure 1. ...

... Hommes and Zhu (2013) derive the recursive constant gain version of SAC-learning. Under SAClearning, the "gain", i.e., the change in the updating of the parameter values α t and β t converges to 0 because of the 1/t terms averaging over the entire past. ...

Rational expectations assumes perfect, model consistency between beliefs and market realizations. Here we discuss behaviorally rational expectations, characterized by an observable, parsimonious and intuitive form of consistency between beliefs and realizations. We discuss three case-studies. Firstly, a New Keynesian macro model with a representative agent learning an optimal, but misspecified, AR(1) rule to forecast inflation consistent with observed sample mean and first-order autocorrelations. Secondly, an asset pricing model with heterogeneous expectations and agents switching between a mean-reverting fundamental rule and a trend-following rule, based upon their past performance.The third example concerns learning-to-forecast laboratory experiments, where under positive feedback individuals coordinate expectations on non-rational, almost self-fulfilling equilibria with persistent price fluctuations very different from rational equilibria.

... (1996), Barberis et al. (1998), Lansing (2006Lansing ( , 2010, Adam et al. (2008), Branch and Evans (2010), Fuster et al. (2012), and Hommes and Zhu (2014), among others. ...

... Motivated by the survey evidence described in Section 2, we consider a forecast rule that is based on an exponentially-weighted moving-average of past observed values of the relevant forecast variable. Such a forecast requires only a minimal amount of 16 For other applications of the SAC learning algorithm, see Lansing (2009Lansing ( , 2010 and Hommes and Zhu (2014). 17 Otrok et al. (2002) employ a similar procedure which they describe (p. ...

We investigate the behavior of the equilibrium price-rent ratio for housing in a standard asset pricing model. We allow for time-varying risk aversion (via external habit formation) and time-varying persistence and volatility in the stochastic process for rent growth, consistent with U.S. data for the period 1960 to 2011. Under fully-rational expectations, the model significantly underpredicts the volatility of the U.S. price-rent ratio for reasonable levels of risk aversion. We demonstrate that the model can approximately match the volatility of the price-rent ratio in the data if near-rational agents continually update their estimates for the mean, persistence and volatility of fundamental rent growth using only recent data (i.e., the past 4 years), or if agents employ a simple moving-average forecast rule for the price-rent ratio that places a large weight on the most recent observation. These two versions of the model can be distinguished by their predictions for the correlation between expected future returns on housing and the price-rent ratio. Only the moving-average model predicts a positive correlation such that agents tend to expect higher future returns when house prices are high relative to fundamentals - a feature that is consistent with survey evidence on the expectations of real-world housing investors.

... (1996), Barberis et al. (1998), Lansing (2006Lansing ( , 2010, Adam et al. (2008), Branch and Evans (2010), Fuster et al. (2012), and Hommes and Zhu (2014), among others. ...

... Motivated by the survey evidence described in Section 2, we consider a forecast rule that is based on an exponentially-weighted moving-average of past observed values of the relevant forecast variable. Such a forecast requires only a minimal amount of 16 For other applications of the SAC learning algorithm, see Lansing (2009Lansing ( , 2010 and Hommes and Zhu (2014). 17 Otrok et al. (2002) employ a similar procedure which they describe (p. ...

We investigate the behavior of the equilibrium price–rent ratio for housing in a standard asset pricing model and compare the model predictions to survey evidence on the return expectations of real-world housing investors. We allow for time-varying risk aversion (via external habit formation) and time-varying persistence and volatility in the stochastic process for rent growth, consistent with the U.S. data for the period 1960 to 2013. Under fully-rational expectations, the model significantly underpredicts the volatility of the U.S. price–rent ratio for reasonable levels of risk aversion. We demonstrate that the model can approximately match the volatility of the price–rent ratio in the data if near-rational agents continually update their estimates for the mean, persistence and volatility of fundamental rent growth using only recent data (i.e., the past 4years), or if agents employ a simple moving-average forecast rule for the price–rent ratio that places a large weight on the most recent observation. These two versions of the model can be distinguished by their predictions for the correlation between expected future returns on housing and the price–rent ratio. Only the moving-average model predicts a positive correlation such that agents tend to expect high future returns when prices are high relative to fundamentals—a feature that is consistent with a wide variety of survey evidence from real estate and stock markets.

... (1996), Barberis et al. (1998), Lansing (2006Lansing ( , 2010, Adam et al. (2008), Branch and Evans (2010), Fuster et al. (2012), and Hommes and Zhu (2014), among others. ...

... Motivated by the survey evidence described in Section 2, we consider a forecast rule that is based on an exponentially-weighted moving-average of past observed values of the relevant forecast variable. Such a forecast requires only a minimal amount of 16 For other applications of the SAC learning algorithm, see Lansing (2009Lansing ( , 2010 and Hommes and Zhu (2014). 17 Otrok et al. (2002) employ a similar procedure which they describe (p. ...

We investigate the behavior of the equilibrium price-rent ratio for housing in a standard asset pricing model and compare the model predictions to survey evidence on the return expectations of real-world housing investors. We allow for time-varying risk aversion (via external habit formation) and time-varying persistence and volatility in the stochastic process for rent growth, consistent with the U.S. data for the period 1960 to 2013. Under fully-rational expectations, the model significantly underpredicts the volatility of the U.S. price-rent ratio for reasonable levels of risk aversion. We demonstrate that the model can approximately match the volatility of the price-rent ratio in the data if near-rational agents continually update their estimates for the mean, persistence and volatility of fundamental rent growth using only recent data (i.e., the past 4. years), or if agents employ a simple moving-average forecast rule for the price-rent ratio that places a large weight on the most recent observation. These two versions of the model can be distinguished by their predictions for the correlation between expected future returns on housing and the price-rent ratio. Only the moving-average model predicts a positive correlation such that agents tend to expect high future returns when prices are high relative to fundamentals-a feature that is consistent with a wide variety of survey evidence from real estate and stock markets.

... A promising general strategy to tackle the ``wilderness of bounded rationality'' may be to consider models with simple (linear) heuristics. For example, in a recent paper Hommes and Zhu (2014) follow this route and propose behavioural learning equilibria, where a homogeneous representative agent uses the best univariate linear forecasting rule in a higher dimensional linear world. The simple one-dimensional forecasting rule is mis-specified, but the best rule within this simple class is obtained by pinning down the two parameters by observable quantities: the sample average and the first-order sample autocorrelation of the rule must coincide with empirical observations. ...

... This leads to an equilibrium concept different from RE, with a representative agent using a simple one-dimensional linear heuristic in a higher dimensional linear world that correctly forecasts the first two moments of the distribution. Hommes and Zhu (2014) show that these equilibria typically exhibit much more persistence and excess volatility than RE, consistent with empirical data. Similar to this research strategy, one could allow for heterogeneity, e.g., as in the Brock-Hommes framework, with a class of simple (linear) strategies with endogenous (nonlinear) switching based upon relative performance. ...

... A closelyrelated concept is the "restricted perceptions equilibrium" described by Evans and Honkopohja (2001, Chapter 13). For other applications of consistent expectations to asset pricing or in ‡ation, see Sögner and Mitlöhner (2002), Branch and McGough (2005), Evans and Ramey (2006), Lansing (2009Lansing ( , 2010, and Hommes and Zhu (2014). 2 For evidence of variability in estimated UIP slope coe¢ cients, see Bansal (1997), Flood and Rose (2002), Baillie and Chang (2011), Baillie and Cho (2014), and Ding and Ma (2013). ...

... We now de…ne a "consistent expectations equilibrium"along the lines of Hommes and Sorger (1998) and Hommes and Zhu (2014). Speci…cally, the parameter in the PLM (12) is pinned down using the moments of observable data. ...

... A closely-related concept is the "restricted perceptions equilibrium" described by Evans and Honkapohja (2001, Chapter 13). For other applications of consistent expectations to asset pricing or inflation, see Branch and McGough (2005), Evans and Ramey (2006), Hommes and Zhu (2014), Lansing (2009Lansing ( , 2010, and Sögner and Mitlöhner (2002). 2 For evidence of variability in estimated UIP slope coefficients, see Baillie and Chang (2011), Baillie and Cho (2014), Bansal (1997), Ding and Ma (2013), and Flood and Rose (2002). 3 Lansing (2010) employs a similar random walk plus fundamentals subjective forecast rule in a standard Lucas-type asset pricing model to account for numerous quantitative features of long-run U.S. stock market data. ...

... We now define a "consistent expectations equilibrium" along the lines of Hommes and Sorger (1998) and Hommes and Zhu (2014). Specifically, the parameter α in the PLM (12) is pinned down using the moments of observable data. ...

... The use of simple low-order autoregressive rules to forecast has also been documented in laboratory experiments with human subjects (e.g., Assenza et al. 2014). Hommes and Zhu (2014) apply the BLE concept in the simplest class of models, where the actual law of motion of the economy is a one-dimensional linear stochastic process driven by exogenous AR(1) shocks. 9 Two important applications of this framework are an asset pricing model driven by AR(1) dividends and an NKPC with inflation driven by an AR(1) process for marginal costs. ...

... and show that there exists at least one nonzero BLE ( α ⁎ , β ⁎ ) with α ⁎ = ¯ π ⁎ (i.e., the sample average equals REE inflation) and β ⁎ a fixed point of the autocorrelation map F ( β ) in (15). Hommes and Zhu (2014) also show that when F ′ ( β ⁎ ) < 1 the E-stability principle holds for the sample autocorrelation (SAC) learning process to learn the optimal parameters α ⁎ and β ⁎ . The time-varying parameters are given by the sample average ...

This survey discusses behavioral and experimental macroeconomics, emphasizing a complex systems perspective. The economy consists of boundedly rational heterogeneous agents who do not fully understand their complex environment and use simple decision heuristics. Central to our survey is the question of under which conditions a complex macro-system of interacting agents may or may not coordinate on the rational equilibrium outcome. A general finding is that under positive expectations feedback (strategic complementarity)—where optimistic (pessimistic) expectations can cause a boom (bust)—coordination failures are quite common. The economy is then rather unstable, and persistent aggregate fluctuations arise strongly amplified by coordination on trend-following behavior leading to (almost-)self-fulfilling equilibria. Heterogeneous expectations and heuristics switching models match this observed micro and macro behavior surprisingly well. We also discuss policy implications of this coordination failure on the perfectly rational aggregate outcome and how policy can help to manage the self-organization process of a complex economic system. (JEL C63, C90, D91, E12, E71, G12)

... where α t and β t are determined through sample autocorrelation (SAC-)learning (Hommes & Sorger 1998, Hommes & Zhu 2014: ...

... This form of learning fits with a large literature on so-called misspecification equilibria under learning, e.g.Branch (2006),Branch & Evans (2006),Hommes & Zhu (2014). ...

This essay surveys some of my work on expectations, learning and bounded rationality within the classical cobweb model following early inspiring ideas from Carl Chiarella. In particular, I focus on the role of nonlinear dynamics, learning and heterogeneity within the cobweb framework and how price fluctuations in the cobweb theory fit with observations of individual and aggregate behaviour from laboratory experiments with human subjects.

... A second argument for considering less rational agents is behavioral-based. Hommes and Zhu (2012) argue that even when agents do observe aggregate fundamental shocks, they may not incorporate them into their forecasting functions. The reason is that agents may fail to understand the underlining mechanism of the economy, and that these shocks are the driving forces. ...

A landmark result in the optimal monetary policy design literature is that fundamental-based interest rate rules invariably lead to rational expectations equilibria (REE) that are not stable under adaptive learning. In this paper, we make a novel information assumption that private agents cannot observe aggregate fundamental shocks, and use simple linear forecasting rules for learning. We find that with fundamental-based rules, there exist limited information equilibria that are stable under learning. Moreover, there are multiple equilibria. Learning can be used as a selection tool to identify a unique equilibrium.

... Note that the excess-volatility puzzle can be solved in a setting with asymmetric information, like the stationary Kyle model, if assumptions such as risk-neutrality and/or perfect structural knowledge are relaxed. The accuracy of the microfounded model will increase if, for example, risk-aversion [22,23] and/or learning dynamics [24][25][26][27] are introduced. We look forward to explore this research direction. ...

We compare the predictions of the stationary Kyle model, a microfounded multi-step linear price impact model in which market prices forecast fundamentals through information encoded in the order flow, with those of the propagator model, a purely data-driven model in which trades mechanically impact prices with a time-decaying kernel. We find that, remarkably, both models predict the exact same price dynamics at high frequency, due to the emergence of universality at small time scales. On the other hand, we find those models to disagree on the overall strength of the impact function by a quantity that we are able to relate to the amount of excess-volatility in the market. We reveal a crossover between a high-frequency regime in which the market reacts sub-linearly to the signed order flow, to a low-frequency regime in which prices respond linearly to order flow imbalances. Overall, we reconcile results from the literature on market microstructure (sub-linearity in the price response to traded volumes) with those relating to macroeconomically relevant timescales (in which a linear relation is typically assumed).

... According to Branch (2006), one way to move beyond rational expectations is to assume that our perception of how the economy evolves over time is mis-specified and agents are unable to detect this. Hommes and Zhu (2014) introduce a behavioral learning equilibrium. Agents are assumed to follow a simple mis-specified forecasting model such as a first-order univariate or multivariate auto-regressive process whose parameters are learned adaptively over time using collected data. ...

While modeling macroeconomic interactions, post-Keynesians propose rationales to verbally motivate the choice of behavioral equations. This informal approach to microfoundation results in inconsistencies and fuzzy arguments. The rationales for different behavioral rules are mutually inconsistent, require strong and nontransparent assumptions, or refer to highly endogenous variables that are not part of the model. The postulated behavioral rules are invariant to endogenous changes in the microenvironment, whereas the rationales imply that they adjust endogenously. The prevailing assumption of purely backward-looking expectations is neither theoretically nor empirically satisfying. The article concludes that revisiting the issue of microfoundation within the post-Keynesian framework may be a rewarding line of research. Furthermore, post-Keynesians should be open to various microfoundations as long as models feature the core of post-Keynesian theory—the principle of effective demand.

... For other applications of the SAC learning algorithm, seeLansing (2009Lansing ( , 2010 andHommes and Zhu (2014). ...

... Such models can foster new links between theorists with great modeling tools and experimenters with equally useful tools, selecting empirically relevant behaviors in the two cases of indeterminacy, through multiple equilibria or multiple bounded rational reasoning procedures. There is already such a small but growing and fruitful interaction in macroeconomics (see survey by Duffy, 2016), and on experiments featuring beauty contest elements in particular (e.g., Benhabib et al., 2016, or in new Keynesian models as in Pfajfar and Zakelj, 2011, Hommes and Zhu, 2014, Mauersberger, 2016. ...

... We have exploited this flexibility to study the effect of heterogeneous information sets in a learning environment. However, this flexibility can also be used to introduce different assumptions on the objective functions, e.g., by introducing behavioral biases, or different expectation formation mechanism, by assuming different PLMs (e.g., Hommes and Zhu, 2014) or heuristic switching behavior (e.g., Dilaver et al., 2016). Future research should introduce more heterogeneity, e.g, income inequality, input-output networks and financial systems. ...

The aim of this paper is to bridge macro agent-based models with mainstream macroeconomic models by agentifying the baseline New Keynesian DSGE model. The model features multiple, boundedly rational, optimizing agents and is analyzed through numerical simulations. We exploit the flexibility of agent-based modeling to explore the effect of dispersed information on the learning process and on macroeconomic outcomes. We find that with dispersed information monetary and fiscal policy acquire the role of public signals.

... Next we illustrate what happens if the endogenous persistence from unanchored expectations interacts with exogenous persistence from auto-correlated shocks. The larger volatility and persistence that we find in our model with bounded rational-ity is in line with findings of other learning models such as in Williams (2004, 2007) and Hommes and Zhu (2014 ...

We study the possibility of (almost) self-fulfilling waves of optimism and pessimism and self-fulfilling liquidity traps in a New Keynesian model with a continuum of heterogeneous expectations. In particular, all agents choose, based on past forecasting performance, expectation values out of a distribution around the targets of the central bank. This framework allows us to explicitly model the “anchoring” of expectations as the variance of this distribution of possible expectation values. We find that when the zero lower bound on the nominal interest rate is not binding, adequate monetary policy can prevent waves of optimism and pessimism and exclude near unit root dynamics, even when expectations are unanchored. However, as shocks bring the economy to a situation with a binding zero lower bound, there is a danger of a long lasting self-fulfilling liquidity trap that can take the form of a deflationary spiral. This can be prevented if expectations are strongly enough anchored to the targets, or if the inflation target is high enough.

... The literature on crashes includes papers on herding behavior ( Lux, 1995 ), speculation ( Lei et al., 2001 ), arbitrage ( Abreu and Brunnermeier, 2003 ), short-selling restriction ( Haruvy and Noussair, 2006 ), leverage ( Friedman and Abraham, 2009 ), behavior biases ( Baghestanian et al., 2015 ), information asymmetry ( Veldkamp, 2005;Oechssler et al., 2011;Sutter et al., 2012 ), learning ( Branch and Evans, 2011;Hommes and Zhu, 2014;Agliari et al., 2016;Hommes and Veld, 2017 ), and communications ( Hong et al., 20 04;20 05;Sornette and Zhou, 2006;Zhou and Sornette, 2007;Harras and Sornette, 2011 ). Among these researches, the most commonly used approaches are experimental asset market design and agent-based simulation. ...

In this paper, we study how agents’ social learning behavior influences market volatility and how the flash crash emerges. We build a model of order-driven market, in which agents use a combination of four components to form their return anticipations: a social learning component, a chartist component, a fundamentalist component and a noise induced component. By numerical simulations, we find that social learning plays a double-edged role in market volatilities. On the one hand, social learning plays a role in reducing total price volatilities and stabilizing the market. On the other hand, in some excitable regimes, social learning instead acts as the critical factor contributing to a flash crash. More interestingly, the lower volatility associated with social learning in the stable regime is crucial to give birth to a flash crash. In addition, we do some robust analyses on the roles of social learning by running the model under many different parameter settings. With the increase of the social learning innate parameter, both the average draw-down and draw-up, the average spread and the average gap show a downward trend. Meanwhile, the tail exponents for the draw-downs and draw-ups also show a downward trend, confirming that social learning plays a double-edged role. The chartist belief can stabilize the market when the social influence is not so trusted, while the chartist belief transfers to contribute to the market instability when the social influence is sufficiently trusted. The fundamentalist belief shows quite opposite impacts in respect to the chartist belief. Finally, we summarize typical return and volatility patterns before a flash crash, which will give some inspirations to regulators and investors.

... Many examples, however, have been provided where learning does not settle down to RE, but to non-rational equilibria, explaining high persistence and excess volatility, as, e.g., in the learning equilibria in Bullard (1994) or the self-fulfilling mistakes in Grandmont (1998). A behavioral learning approach based on simplicity and parsimony has been advocated by so-called Restricted Perception Equilibria (Branch 2006;Hommes and Zhu (2013)). Agents base their expectations on simple forecasting heuristics, such as an AR(1) rule, with the parameters pinned down by simple consistency requirements between beliefs and market realizations, for example, based on intuitive and observable quantities such as the mean and the first-order autocorrelation. ...

We discuss recent work on bounded rationality and learning in relation to Soros' principle of reflexivity and stress the empirical importance of non-rational, almost self-fulfilling equilibria in positive feedback systems. As an empirical example, we discuss a behavioral asset pricing model with heterogeneous expectations. Bubble and crash dynamics is triggered by shocks to fundamentals and amplified by agents switching endogenously between a mean-reverting fundamental rule and a trend-following rule, based upon their relative performance. We also discuss learning-to-forecast laboratory experiments, showing that in positive feedback systems individuals coordinate expectations on non-rational, almost self-fulfilling equilibria with persistent price fluctuations very different from rational equilibria. Economic policy analysis may benefit enormously by focusing on efficiency and welfare gains in correcting mispricing along almost self-fulfilling equilibria.

... Hommes and Zhu (2014) show the possibility of multiple RPE arising when dynamics are underparamerized. Branch et al. (2017) find that stable "sunspot RPE" exist in linear models for which no sunspot REE exist. ...

We introduce a new class of solutions to nonlinear forward-looking models called near-rational sunspot equilibria (NRSE). NRSE are natural nonlinear extensions of the usual sunspot equilibria associated with the linearized version of the economy, and are near-rational in that agents use the optimal linear forecasting model when forming expectations. Generic results for existence and stability under learning are established. NRSE in indeterminate nonlinear models are found to be stable under learning provided that the corresponding linearized model's minimal state variable solution is E-stable. NRSE are readily computable, and our results make it possible to use the standard linear tools to search for stable NRSE. We illustrate our results using a canonical nonlinear New Keynesian model.

... In line with e.g. Williams (2005, 2007), Milani (2007), Slobodyan and Wouters (2012) and Hommes and Zhu (2014), I find that the presence of backward-looking agents adds persistence. In both expectations-driven and mixed liquidity traps, this persistence amplification becomes so severe for larger fractions of backward-looking agents that the economy never recovers. ...

I study liquidity traps in a model where agents have heterogeneous expectations and finite planning horizons. Backward-looking agents base their expectations on past observations, while forward-looking agents have fully rational expectations. Liquidity traps that are fully or partly driven by expectations can arise due to pessimism of backward-looking agents. Only when planning horizons are finite, these liquidity traps can be of longer duration without ending up in a deflationary spiral. I further find that fiscal stimulus in the form of an increase in government spending or a cut in consumption taxes can be very effective in mitigating the liquidity trap. A feedback mechanism of heterogeneous expectations causes fiscal multipliers to be the largest when the majority of agents is backward-looking but there also is a considerable fraction of agents that are forward-looking. Labor tax cuts are always deflationary and are not an effective tool in a liquidity trap.

... Bullard et al. (2008) consider judgment in monetary policy when the forecasting model used by policymakers is a loworder VAR. Hommes and Zhu (2014) show that multiple RPE can arise with underparameterized dynamics. ...

We demonstrate existence, and stability under adaptive learning, of restricted perceptions equilibria in a nonlinear cobweb model.

This paper investigates the learnability of an equilibrium with private information. Agents of each type have their own private information about an exogenous variable and conduct adaptive learning with a heterogeneously misspecified perceived laws of motion (PLM) that includes only this variable. The paper shows that the existence of private information has a nonnegative impact on the learnability of the equilibrium; that is, the condition for learnability is unaffected or relaxed by heterogeneity and/or misspecification in PLMs caused by private information. In a New Keynesian model with private information about fundamental shocks, the learnability of the equilibrium is ensured by the Taylor principle of monetary policy. The paper also confirms that these results hold true not only in the presence of private information, but also in a variety of informational structures.

Standard rational expectations (RE) models with an occasionally binding zero lower bound (ZLB) constraint either admit no solutions (incoherence) or multiple solutions (incompleteness). This paper shows that deviations from full-information RE mitigate concerns about incoherence and incompleteness. Models with no RE equilibria admit self-confirming equilibria involving the use of simple mis-specified forecasting models. Completeness and coherence is restored if expectations are adaptive or if agents are less forward-looking due to some information or behavioral friction. In the case of incompleteness, the E-stability criterion selects an equilibrium.

This essay links some of my own work on expectations, learning and bounded rationality to the inspiring ideas of Jean-Michel Grandmont. In particular, my work on consistent expectations and behavioral learning equilibria may be seen as formalizations of JMG’s ideas of self-fulfilling mistakes. Some of our learning-to-forecast laboratory experiments with human subjects have also been strongly influenced by JMG’s ideas. Key features of self-fulfilling mistakes are multiple equilibria, excess volatility and persistence amplification.

Agent‐based computational economics (ACE) has been used for tackling major research questions in macroeconomics for at least two decades. This growing field positions itself as an alternative to dynamic stochastic general equilibrium (DSGE) models. In this paper, we provide a much needed review and synthesis of this literature and recent attempts to incorporate insights from ACE into DSGE models. We first review the arguments raised against DSGE in the macroeconomic ACE (macro ACE) literature, and then review existing macro ACE models, their explanatory power and empirical performance. We then turn to the literature on behavioural New Keynesian models that attempts to synthesize these two approaches to macroeconomic modelling by incorporating insights of ACE into DSGE modelling. Finally, we provide a thorough description of the internally rational New Keynesian model, and discuss how this promising line of research can progress.

This paper proposes a financial accelerator framework to study the effects of heterogeneous and bounded rational expectations on macroeconomic dynamics. The paper examines the fluctuations effects departing from the rational expectations hypothesis in order to understand if there are significant implications on macroeconomic volatility and policy prescriptions. The findings suggest that macroeconomic stability and inflation dynamics depend on the chosen set of forecasting rules, as well as on the monetary policy adopted. The model shows that no monetary policy is able to quickly stabilize the system, as some fluctuations persist. Central banks face a trade-off between macro-volatility and speed of convergence to the steady state. This result offers some ground for fiscal policies aiming to prompt system stability. In addition, the analysis reveals a counterintuitive result confirming the “less-is-more” effect: increasing the decision-making and computational abilities of the agents may not lead the system to converge to the preferable steady state.

Agents can learn from financial markets to predict macroeconomic outcomes, and learning dynamics can feed back into both the macroeconomy and financial markets. This paper builds on the adaptive learning (AL) model of [Slobodyan, S. and R. Wouters (2012a) American Economic Journal: Macroeconomics 4, 65–101.] by introducing the term structure of interest rates. This extension enables term structure information to fully characterize agents’ expectations in real time. This feature addresses an imperfect information issue neglected in the related AL literature. The term structure of interest rates results in a strong channel of persistence driven by multi-period forecasting. Including the term structure in the AL model results in a model fit similar to that obtained in the rational expectation (RE) version of the model, but it greatly reduces the importance of other endogenous sources of aggregate persistence such as price and wage stickiness and the elasticity of the cost of adjusting capital. The model estimated also shows that term premium innovations are a major source of persistent fluctuations in nominal variables under AL. This stands in sharp contrast to the lack of transmission of term premium shocks to the macroeconomy under REs.

We study the existence of Stochastic Consistent Expectations Equilibria (SCEE) in linear Markov regime switching models. A SCEE exists when the model-implied mean and first order autocorrelation coincide with those predicted by the agents via misspecified forecasting rules. For a simple regime-switching monetary policy model, the parametric space where at least one SCEE exists is rather wide, and may extend well beyond the rational expectations equilibrium determinacy frontier. Misspecified expectations combined with regime-switching yield a strong endogenous amplification mechanism that help generate the near unit root dynamics for inflation observed in the U.S. before the Great Moderation.
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The notion of consistent expectations equilibrium is extended to economies that are described by a nonlinear stochastic system. Agents in the model do not know the nonlinear law of motion and use a simple linear forecasting rule to form their expectations. Along a stochastic consistent expectations equilibrium (SCEE), these expectations are correct in a linear statistical sense, i.e., the unconditional mean and autocovariances of the actual (but unknown) nonlinear stochastic process coincide with those of the linear stochastic process on which the agents base their beliefs. In general, the linear forecasts do not coincide with the true conditional expectation, but an SCEE is an ‘approximate rational expectations equilibrium’ in the sense that forecasting errors are unbiased and uncorrelated. Adaptive learning of SCEE is studied in an overlapping generations framework.

Since the 1970's Rational Expectations (RE) has become the dominant paradigm in macroeconomics. One reason for its popularity is the consistency it imposes between beliefs and outcomes. Under RE agents'subjective probability distribution coincides with the true distribution for the economy. Not surprisingly, a large literature objects to RE on the grounds that it requires agents possess unreasonable information and cognitive abilities. Instead many researchers (e. g. (Evans and Honkapohja 2001)) replace RE with an econometric forecasting model and ask under what conditions the forecasts converge to RE. The validity of RE is not just theoretical curiosum, (Branch 2004) and (Carroll 2003) demonstrate persistent heterogeneity in survey data on inflation expectations. Such phenomena can not be explained by RE models. Papers such as (Brock and Hommes 1997) generate heterogeneity by assuming agents make conscious choices between costly predictor functions, thereby, hypothesizing that deviations from rationality come from a weighing of benefits and costs.

The notion of consistent expectations equilibrium is extended to economies that are described by a nonlinear stochastic system. Agents in the model do not know the nonlinear law of motion and use a simple linear forecasting rule to form their expectations. Along a stochastic consistent expectations equilibrium (SCEE), these expectations are correct in a linear statistical sense, i.e., the unconditional mean and autocovariances of the actual (but unknown) nonlinear stochastic process coincide with those of the linear stochastic process on which the agents base their beliefs. In general, the linear forecasts do not coincide with the true conditional expectation, but an SCEE is an ‘approximate rational expectations equilibrium’ in the sense that forecasting errors are unbiased and uncorrelated. Adaptive learning of SCEE is studied in an overlapping generations framework.

We study an investment model in which agents have the wrong beliefs about the dynamic properties of fundamentals. Specifically, we assume that agents underestimate the rate of mean reversion. The model exhibits the following six properties: (i) Beliefs are excessively optimistic in good times and excessively pessimistic in bad times. (ii) Asset prices are too volatile. (iii) Excess returns are negatively autocorrelated. (iv) High levels of corporate profits predict negative future excess returns. (v) Real economic activity is excessively volatile; the economy experiences amplified investment cycles. (vi) Corporate profits are positively autocorrelated in the short run and negatively autocorrelated in the medium run. The paper provides an illustrative model of animal spirits, amplified business cycles, and excess volatility.

A large body of empirical evidence suggests that beliefs systematically deviate from perfect rationality. Much of the evidence implies that economic agents tend to form forecasts that are excessively influenced by recent changes. We present a parsimonious quasi-rational model that we call natural expectations, which falls between rational expectations and (naïve) intuitive expectations. (Intuitive expectations are formed by running growth regressions with a limited number of right-hand-side variables, and this leads to excessively extrapolative beliefs in certain classes of environments). Natural expectations overstate the long-run persistence of economic shocks. In other words, agents with natural expectations turn out to form beliefs that don't sufficiently account for the fact that good times (or bad times) won't last forever. We embed natural expectations in a simple dynamic macroeconomic model and compare the simulated properties of the model to the available empirical evidence. The model's predictions match many patterns observed in macroeconomic and financial time series, such as high volatility of asset prices, predictable up-and-down cycles in equity returns, and a negative relationship between current consumption growth and future equity returns.

This paper demonstrates that an asset pricing model with least-squares learning can lead to bubbles and crashes as endogenous responses to the fundamentals driving asset prices. When agents are risk-averse they need to make forecasts of the conditional variance of a stock's return. Recursive updating of both the conditional variance and the expected return implies several mechanisms through which learning impacts stock prices. Extended periods of excess volatility, bubbles, and crashes arise with a frequency that depends on the extent to which past data is discounted. A central role is played by changes over time in agents' estimates of risk. (JEL D81, D83, E32, G01, G12)

We present a decision theoretic framework in which agents are learning about market behavior and that provides microfoundations for models of adaptive learning. Agents are ‘internally rational’, i.e., maximize discounted expected utility under uncertainty given dynamically consistent subjective beliefs about the future, but agents may not be ‘externally rational’, i.e., may not know the true stochastic process for payoff relevant variables beyond their control. This includes future market outcomes and fundamentals. We apply this approach to a simple asset pricing model and show that the equilibrium stock price is then determined by investorsʼ expectations of the price and dividend in the next period, rather than by expectations of the discounted sum of dividends. As a result, learning about price behavior affects market outcomes, while learning about the discounted sum of dividends is irrelevant for equilibrium prices. Stock prices equal the discounted sum of dividends only after making very strong assumptions about agentsʼ market knowledge.

This article advocates a theory of expectation formation that incorporates many of the central motivations of behavioral finance theory while retaining much of the discipline of the rational expectations approach. We provide a framework in which agents, in an asset pricing model, underparameterize their forecasting model in a spirit similar to Hong, Stein, and Yu (2007) and Barberis, Shleifer, and Vishny (1998), except that the parameters of the forecasting model and the choice of predictor are determined jointly in equilibrium. We show that multiple equilibria can exist even if agents choose only models that maximize (risk-adjusted) expected profits. A real-time learning formulation yields endogenous switching between equilibria. We demonstrate that a real-time learning version of the model, calibrated to U.S. stock data, is capable of reproducing regime-switching returns and volatilities, as recently identified by Guidolin and Timmermann (2007). The Author 2010. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org., Oxford University Press.

This paper demonstrates that the behavior of the conventional Phelps-Taylor model of overlapping wage contracts stands in stark contrast with important features of U.S. macro data for inflation and output. In particular, the Phelps-Taylor specification implies far too little inflation persistence. The authors present a new contracting model, in which agents are concerned with relative real wages, that is data-consistent. In a specification that nests both models, the authors resoundingly reject the conventional contracting model but cannot reject the new contracting model. Copyright 1995, the President and Fellows of Harvard College and the Massachusetts Institute of Technology.

Price fluctuations under adaptive learning in renewable resource markets such as fisheries are examined. Optimal fishery management with logistic fish population growth implies a backward-bending, discounted supply curve for bioeconomic equilibrium sustained yield. Higher discount rates bend supply backward more to generate multiple steady-state rational expectations equilibria. Under bounded rationality, adaptive learning of a linear forecasting rule generates steady-state, two-cycle as well as chaotic consistent expectations equilibria, which are self-fulfilling in sample average and autocorrelations. The possibility of learning to believe in chaos is robust and even enhanced by dynamic noise.

We consider a class of nonlinear dynamic economic models inwhich the actual realizations of a certain variable (e.g., price) dependon the agents expectations about this variable. We define a consistentexpectations equilibrium (CEE) by the property that the sample averageand the sample autocorrelations of the realizations of the actual lawof motion equal the average and the autocorrelations of the perceived lawof motion. Along a CEE agent s expectations are thus self-fulfilling interms of the observable sample average and sample autocorrelations.Restricting ourselves to the case in which beliefs are described byan AR(1) process, we study existence and stability of three differenttypes of CEE: steady-state, two-cycle, and chaotic. We illustrate howthese equilibria can emerge in the nonlinear cobweb model. Learningdynamics based on sample average and sample autocorrelations areintroduced and stability of CEE under this learning process isinvestigated.

This paper attempts to identify, in a framework deliberately stripped of unnecessary technicalities, some of the basic reasons why adaptive learning may or may not lead to stability and convergence to self-fulfilling expectations in large socioeconomic systems where no agent, or collection of agents, can act to manipulate macroeconomic outcomes. It is shown that if agents are somewhat uncertain about the local stability of the system, and are accordingly ready to extrapolate a large range of regularities (trends) that may show up in past small deviations from equilibrium, including divergent ones, the learning dynamics is locally divergent. On the other hand, if agents are fairly sure of the local stability of the system, and extrapolate only convergent trends out of small past deviations from equilibrium, one may get local stability. This "uncertainty principle" does show up in a wide variety of contexts: smooth or discontinuous, finite or infinite memory learning rules, error learning, recursive least squares, Bayesian learning.

The way in which individual expectations shape aggregate macroeconomic variables is crucial for the transmission and effectiveness of monetary policy. We study the individual expectations formation process and the interaction with monetary policy, within a standard New Keynesian model, by means of laboratory experiments with human subjects. Three aggregate outcomes are observed: convergence to some equilibrium level, persistent oscillatory behavior and oscillatory convergence. We fit a heterogeneous expectations model with a performance-based evolutionary selection among heterogeneous forecasting heuristics to the experimental data. A simple heterogeneous expectations switching model fits individual learning as well as aggregate macro behavior and outperforms homogeneous expectations benchmarks. Moreover, in accordance to theoretical results in the literature on monetary policy, we find that an interest rate rule that reacts more than point for point to inflation has some stabilizing effects on inflation in our experimental economies, although convergence can be slow in presence of evolutionary learning.

A crucial challenge for economists is figuring out how people interpret the world and form expectations that will likely influence their economic activity. Inflation, asset prices, exchange rates, investment, and consumption are just some of the economic variables that are largely explained by expectations. Here George Evans and Seppo Honkapohja bring new explanatory power to a variety of expectation formation models by focusing on the learning factor. Whereas the rational expectations paradigm offers the prevailing method to determining expectations, it assumes very theoretical knowledge on the part of economic actors. Evans and Honkapohja contribute to a growing body of research positing that households and firms learn by making forecasts using observed data, updating their forecast rules over time in response to errors. This book is the first systematic development of the new statistical learning approach. Depending on the particular economic structure, the economy may converge to a standard rational-expectations or a "rational bubble" solution, or exhibit persistent learning dynamics. The learning approach also provides tools to assess the importance of new models with expectational indeterminacy, in which expectations are an independent cause of macroeconomic fluctuations. Moreover, learning dynamics provide a theory for the evolution of expectations and selection between alternative equilibria, with implications for business cycles, asset price volatility, and policy. This book provides an authoritative treatment of this emerging field, developing the analytical techniques in detail and using them to synthesize and extend existing research.

Do laboratory subjects correctly perceive the dynamics of a mean-reverting time series? In our experiment, subjects receive historical data and make forecasts at different horizons. The time series process that we use features short-run momentum and long-run partial mean reversion. Half of the subjects see a version of this process in which the momentum and partial mean reversion unfold over 10 periods ('fast'), while the other subjects see a version with dynamics that unfold over 50 periods ('slow'). Typical subjects recognize most of the mean reversion of the fast process and none of the mean reversion of the slow process.

In order to explain fairly simply how expectations are formed, we advance the hypothesis that they are essentially the same as the predictions of the relevant economic theory. In particular, the hypothesis asserts that the economy generally does not waste information, and that expectations depend specifically on the structure of the entire system. Methods of analysis, which are appropriate under special conditions, are described in the context of an isolated market with a fixed production lag. The interpretative value of the hypothesis is illustrated by introducing commodity speculation into the system.

How does an economy behave if (1) fundamentals are truly humpshaped, exhibiting momentum in the short run and partial mean reversion in the long run, and (2) agents do not know that fundamentals are hump- shaped and base their beliefs on parsimonious models that they fit to the available data? A class of parsimonious models leads to qualitatively similar biases and generates empirically observed patterns in asset prices and macroeconomic dynamics. First, parsimonious models will robustly pick up the short- term momentum in fundamentals but will generally fail to fully capture the long- run mean reversion. Beliefs will therefore be characterized by endogenous extrapolation bias and procyclical excess optimism. Second, asset prices will be highly volatile and exhibit partial mean reversion-that is, overreaction. Excess returns will be negatively predicted by lagged excess returns, P/E ratios, and consumption growth. Third, real economic activity will have amplified cycles. For example, consumption growth will be negatively auto- correlated in the medium run. Fourth, the equity premium will be large. Agents will perceive that equities are very risky when in fact long- run equity returns will co- vary only weakly with long- run consumption growth. If agents had rational expectations, the equity premium would be close to zero. Fifth, sophisticated agents-that is, those who are assumed to know the true model-will hold far more equity than investors who use parsimonious models. Moreover, sophisticated agents will follow a countercyclical asset allocation policy. These predictedeffects are qualitatively confirmed in US data.

Analytical expectational stability results are obtained for both Euler-equation and infinite-horizon adaptive learning in a simple stochastic growth model. The rational expectations equilibrium is stable under both types of learning, though there are important differences in the learning dynamics.

The article presents a temporary equilibrium framework for macroeconomic analysis that allows for a wide range of possible specifications of expectations but reduces to a standard new Keynesian model in the limiting case of rational expectations. This common framework is then used to contrast the assumptions and implications of several different ways of relaxing the assumption of rational expectations. As an illustration of the method, the implications of alternative assumptions for the selection of a monetary policy rule are discussed. Other issues treated include the conditions required for Ricardian equivalence and for existence of a deflation trap.

Using laboratory experiments within a New Keynesian macro framework, we explore the formation of inflation expectations and its interaction with monetary policy design. The central question in this paper is how to design monetary policy in the environment characterized by heterogeneous expectations. Rules that use actual rather than forecasted inflation produce lower inflation variability and alleviate expectational cycles. Degree of responsiveness to deviations of inflation from its target in the Taylor rule produces nonlinear effects on inflation variability. We also provide considerable support for the existence of heterogeneity of inflation expectations and show that a significant proportion of subjects are rational in our experiment. However, most subjects rather than using a single model they tend to switch between alternative models.

The way in which individual expectations shape aggregate macroeconomic variables is crucial for the transmission and effectiveness of monetary policy. We study the individual expectations formation process and the interaction with monetary policy, within a standard New Keynesian model, by means of laboratory experiments with human subjects. We find that a more aggressive monetary policy that sets the interest rate more than point for point in response to inflation stabilizes inflation in our experimental economies. We use a simple model of individual learning, with a performance-based evolutionary selection among heterogeneous forecasting heuristics, to explain coordination of individual expectations and aggregate macro behavior observed in the laboratory experiments. Three aggregate outcomes are observed: convergence to some equilibrium level, persistent oscillatory behaviour and oscillatory convergence. A simple heterogeneous expectations switching model fits individual learning as well as aggregate outcomes and outperforms homogeneous expectations benchmarks.

This article derives a general class of intrinsic rational bubble solutions in a Lucas-type asset pricing model. I show that the rational bubble component of the price-dividend ratio can evolve as a geometric random walk without drift, such that the mean of the bubble growth rate is zero. Driftless bubbles are part of a continuum of equilibrium solutions that satisfy a period-by-period no-arbitrage condition. I also derive a near-rational solution in which the agent's forecast rule is under-parameterised. The near-rational solution generates intermittent bubbles and other behaviour that is quantitatively similar to that observed in long-run US stock market data. Copyright (C) The Author(s). Journal compilation (C) Royal Economic Society 2010.

We consider a standard two generations version of the overlapping generations model. Agents predict inflation rates on the basis of a perceived law of motion, which is estimated by running a linear regression on past inflation rates. We introduce the notion of beliefs equilibrium. At such an equilibrium: (i) the perceived law of motion is such that it fits the time series of inflation rates best, and (ii) this time series of actual inflation rates is generated when agents use that perceived law of motion. We show that beliefs might converge, although inflation rates keep on fluctuating. These fluctuations are consistent with the limit belief.

In this article we investigate the question whether the highly demanding informative requirements of rational expectations models are necessary to derive equilibria within capital market models. In this analysis agents are only provided with publicly available information such as prices and dividends. Nevertheless, we require that agents should behave like econometricians. Additionally, we skip the assumption of rational expectations models that agents know the implied actual law of motion of the system. By these assumptions, the stock market can be considered as a Hommes–Sorger consistent expectations model. We show the existence of consistent expectations equilibria with myopic agents and independent identically distributed dividends. The only CEE is the rational expectations equilibrium. In the simulation part we demonstrate how the steady-state CEE can be derived by means of sample autocorrelation learning. Thus, we are able to derive a stock market equilibrium with less demanding requirements, where this equilibrium is equal to the rational expectations equilibrium.

Monetary DGSE models under rational expectations typically require large degrees of features as habit formation in consumption and inflation indexation to match the inertia of macroeconomic variables.This paper presents an estimated model that departs from rational expectations and nests learning by economic agents, habits, and indexation. Bayesian methods facilitate the joint estimation of the learning gain coefficient together with the ‘deep’ parameters of the economy.The empirical results show that when learning replaces rational expectations, the estimated degrees of habits and indexation drop closer to zero, suggesting that persistence arises in the model economy mainly from expectations and learning.

This paper surveys learning-to-forecast experiments (LtFEs) with human subjects to test theories of expectations and learning. Subjects must repeatedly forecast a market price, whose realization is an aggregation of individual expectations. Emphasis is given to how individual forecasting rules interact at the micro-level and which structure they cocreate at the aggregate, macro-level. In particular, we focus on the question wether the evidence from laboratory experiments is consistent with heterogeneous expectations.

We provide evidence on the fit of the New Phillips Curve (NPC) for the Euro area over the period 1970–1998, and use it as a tool to compare the characteristics of European inflation dynamics with those observed in the U.S. We also analyze the factors underlying inflation inertia by examining the cyclical behavior of marginal costs, as well as that of its two main components, namely, labor productivity and real wages. Some of the findings can be summarized as follows: (a) the NPC fits Euro area data very well, possibly better than U.S. data, (b) the degree of price stickiness implied by the estimates is substantial, but in line with survey evidence and U.S. estimates, (c) inflation dynamics in the Euro area appear to have a stronger forward-looking component (i.e., less inertia) than in the U.S., (d) labor market frictions, as manifested in the behavior of the wage markup, appear to have played a key role in shaping the behavior of marginal costs and, consequently, inflation in Europe.

The evolution of many economic variables is affected by expectations that economic agents have with respect to the future development of these variables. We show, by means of laboratory experiments, that market behavior depends to a large extent on whether realized market prices respond positively or negatively to average price expectations. In the case of negative expectations feedback, as in commodity markets, prices converge quickly to their equilibrium value, confirming the rational expectations hypothesis. In the case of positive expectations feedback, as is typical for speculative asset markets, large fluctuations in realized prices and persistent deviations from the benchmark fundamental price are likely. We estimate individual forecasting rules and investigate how these explain the differences in aggregate market outcomes.

This paper investigates the dynamics in a simple present discounted value asset pricing model with heterogeneous beliefs. Agents choose from a finite set of predictors of future prices of a risky asset and revise their ‘beliefs’ in each period in a boundedly rational way, according to a ‘fitness measure’ such as past realized profits. Price fluctuations are thus driven by an evolutionary dynamics between different expectation schemes (‘rational animal spirits’). Using a mixture of local bifurcation theory and numerical methods, we investigate possible bifurcation routes to complicated asset price dynamics. In particular, we present numerical evidence of strange, chaotic attractors when the intensity of choice to switch prediction strategies is high.

This paper generalizes existence results on first-order Stochastic Consistent Expectations Equilibria (SCEE) obtained by Hommes et al. (Learning to Believe in Linearity in an Unknown Nonlinear Stochastic Economy, 2002). We present a stochastic non-linear self-referential model in which expectations are based on linear perceptions. In an SCEE the sample mean and correlation coefficients of the true and perceived processes coincide. We provide conditions on the non-linear maps governing the stochastic process that are sufficient to establish existence of SCEE. Our approach defines a map that takes linear perceptions to actual outcomes in such a way that fixed points of this map are SCEE; by establishing existence of fixed points, we are able to demonstrate existence of SCEE. Stability of SCEE under real-time learning is analyzed numerically.

We introduce the concept of Misspecification Equilibrium to dynamic macroeconomics. Agents choose between a list of misspecified econometric models and base their selection on relative forecast performance. A Misspecification Equilibrium is a stochastic process in which agents forecast optimally given their choices, with forecast model parameters and predictor proportions endogenously determined. Under appropriate conditions, the Misspecification Equilibrium will exhibit Intrinsic Heterogeneity, in which all predictors are used at all times, even in the neoclassical limit in which only the most successful predictors are used. This equilibrium is attainable under least-squares learning and dynamic predictor selection based on average profits.

This paper characterizes equilibrium asset prices under adaptive, rational and Bayesian learning schemes in a model where dividends evolve on a binomial lattice. The properties of equilibrium stock and bond prices under learning are shown to differ significantly. Learning causes the discount factor and risk-neutral probability measure to become path-dependent and introduces serial correlation and volatility clustering in stock returns. We also derive conditions under which the expected value and volatility of stock prices will be higher under learning than under full information. Finally, we investigate restrictions on prior beliefs under which Bayesian and rational learning lead to identical prices and show how the results can be generalized to more complex settings where dividends follow either multi-state i.i.d. distributions or multi-state Markov chains.

This paper shows how to use an apparatus of A. Marcet and the author [see J. Econ. Theory 48, No.2, 337-368 (1989; Zbl 0672.90023); J. Polit. Econ. 97, 1306-1322 (1989)] to compute equilibria of a class of models described by R. M. Townsend [ibid. 91, 546-588 (1983)]. An equilibrium is modelled in which agents are extracting signals from observations on endogenous variables. By modelling agents’ perceived laws of motion as vector ARMA processes, the equilibrium of the model can be formulated as the fixed point of an operator that maps perceived laws of motion into statistically optimal estimators of those laws of motion.

We study how the use of judgment or "add-factors" in forecasting may disturb the set of equilibrium outcomes when agents learn by using recursive methods. We isolate conditions under which new phenomena, which we call exuberance equilibria, can exist in a standard self-referential environment. Local indeterminacy is not a requirement for existence. We construct a simple asset-pricing example and find that exuberance equilibria, when they exist, can be extremely volatile relative to fundamental equilibria.

Within a New Keynesian model, we incorporate bounded rationality at the individual agent level, and we determine restrictions on expectations operators sufficient to imply aggregate IS and AS relations of the same functional form as those under rationality. This result provides dual implications: the strong nature of the restrictions required to achieve aggregation serve as a caution to researchers--imposing heterogeneous expectations at an aggregate level may be ill-advised; on the other hand, accepting the necessary restrictions provides for tractable analysis of expectations heterogeneity. As an example, we consider a case where a fraction of agents are rational and the remainder are adaptive, and find specifications that are determinate under rationality may possess multiple equilibria in case of expectations heterogeneity.

Two of the most discussed anomalies in the financial literature are the predictability of excess returns and the excess volatility
of stock prices. Learning effects on stock price dynamics are an intuitive candidate to explain these empirical findings:
estimation uncertainty may increase volatility of stock prices and an estimate of the dividend growth rate that is, say, lower
than the “true” value tends to increase the dividend yield and capital gain. Simulations of learning effects in a present
value model confirm that learning may help to explain excess volatility and predictability of stock returns.

This paper introduces a form of boundedly-rational inflation expectations in the New Keynesian Phillips curve. The representative agent is assumed to behave as an econometrician, employing a time series model for inflation that allows for both permanent and temporary shocks. The near-unity coefficient on expected inflation in the Phillips curve causes the agent's perception of a unit root in inflation to become close to self-fulfilling. In a "consistent expectations equilibrium," the value of the Kalman gain parameter in the agent's forecast rule is pinned down using the observed autocorrelation of inflation changes. The forecast errors observed by the agent are close to white noise, making it difficult for the agent to detect a misspecification of the forecast rule. I show that this simple model of inflation expectations can generate time-varying persistence and volatility that is broadly similar to that observed in long-run U.S. data. Model-based values for expected inflation track well with movements in survey-based measures of U.S. expected inflation. In numerical simulations, the model can generate pronounced low-frequency swings in the level of inflation that are driven solely by expectational feedback, not by changes in monetary policy. (Copyright: Elsevier)

This paper identifies two channels through which the economy can generate endogenous inflation and output volatility, an empirical regularity, by introducing model uncertainty into a Lucas-type monetary model. The equilibrium path of inflation depends on agents' expectations and a vector of exogenous random variables. Following Branch and Evans (2006) agents are assumed to underparameterize their forecasting models. A Misspecification Equilibrium arises when beliefs are optimal, given the misspecification, and predictor proportions are based on relative forecast performance. We show that there may exist multiple Misspecification Equilibria, a subset of which are stable under least squares learning and dynamic predictor selection. The dual channels of least squares parameter updating and dynamic predictor selection combine to generate regime switching and endogenous volatility. (Copyright: Elsevier)

A fundamentals based monetary policy rule, which would be the optimal monetary policy without commitment when private agents
have perfectly rational expectations, is unstable if in fact these agents follow standard adaptive learning rules. This problem
can be overcome if private expectations are observed and suitably incorporated into the policy maker's optimal rule. These
strong results extend to the case in which there is simultaneous learning by the policy maker and the private agents. Our
findings show the importance of conditioning policy appropriately, not just on fundamentals, but also directly on observed
household and firm expectations.

We investigate expectation formation in a controlled experimental environment. Subjects are asked to predict the price in
a standard asset pricing model. They do not have knowledge of the underlying market equilibrium equations, but they know all
past realized prices and their own predictions. Aggregate demand for the risky asset depends upon the forecasts of the participants.
The realized price is then obtained from market equilibrium with feedback from six individual expectations. Realized prices
differ significantly from fundamental values and typically exhibit oscillations around, or slow convergence to, this fundamental.
In all groups participants coordinate on a common prediction strategy.

In the conventional view of inflation, the New Keynesian Phillips curve (NKPC) captures most of the persistence in inflation. The sources of persistence are twofold. First, the "driving process" for inflation is quite persistent, and the NKPC implies that inflation must "inherit" this persistence. Second, backward-looking or indexing behavior imparts some "intrinsic" persistence to inflation. This paper shows that, in practice, inflation in the NKPC inherits very little of the persistence of the driving process, and it is intrinsic persistence that constitutes the dominant source of persistence. The reasons are that, first, the coefficient on the driving process is small, and, second, the shock that disturbs the NKPC is large.

This paper considers the implications of an important source of model misspecification for the design of monetary policy rules: the assumed manner of expectations formation. In the model considered here, private agents seek to maximize their objectives subject to standard constraints and the restriction of using an econometric model to make inferences about future uncertainty. Because agents solve a multiperiod decision problem, their actions depend on forecasts of macroeconomic conditions many periods into the future, unlike the analysis of Bullard and Mitra (2002) and Evans and Honkapohja (2002). A Taylor rule ensures convergence to the rational expectations equilibrium associated with this policy if the so-called Taylor principle is satisfied. This suggests the Taylor rule to be desirable from the point of view of eliminating instability due to self-fulfilling expectations.

I modify the uniform-price auction rules in allowing the seller to ration bidders. This allows me to provide a strategic foundation for underpricing when the seller has an interest in ownership dispersion. Moreover, many of the so-called "collusive-seeming" equilibria disappear.

This paper employs the Hopf bifurcation theorem to prove the existence of complicated equilibrium trajectories under least squares learning in a standard version of the overlapping generations model. The periodic and quasiperiodic learning equilibria exist when the locally unique perfect foresight equilibrium is the monetary steady state, and thus are induced by the introduction of learning alone. Learning equilibria can be stable or unstable depending on higher order derivatives of the underlying utility function not specified by economic theory; examples of both attracting and repelling invariant dosed curves are provided. This research confirms the intuition of some previous authors, who have suggested that stationary equilibrium trajectories under learning may differ from those under rational expectations. I would like to thank Robert Becker and members of my dissertation committee for helpful comments on this and related work. Suggestions made by Michael Woodford materially improved this paper. I also thank Albert Marcet, Mark Salmon, George Evans, Seppo Honkapohja, and participants at the June 1991 Meetings of the Society for Economic Dynamics and Control in Capri, Italy, and the May 1991 Midwest Mathematical Economics meeting at Northwestern University, for helpful discussions. All errors are the author's responsibility.

Expectations of the future play a large role in macroeconomics. The rational expectations assumption, which is commonly used in the literature, provides an important benchmark, but may be too strong for some applications. This paper reviews some recent research that has emphasized methods for analyzing models of learning, in which expectations are not initially rational but which may become rational eventually provided certain conditions are met. Many of the applications are in the context of popular models of monetary policy. The goal of the paper is to provide a largely nontechnical survey of some, but not all, of this work and to point out connections to some related research.