2.1 Expectations Operators

The literature on heterogeneous expectation formation is influenced, in part, by the adaptive learning literature (e.g. Evans and Honkapohja, 2001). In this class of models, fully rational expectations are replaced by linear forecasting rules with parameters that are updated by recursive least squares. In this chapter, we imagine different sets of agents who engage in economic forecasting while recognizing there may exist heterogeneity in forecasting rules. Some examples of heterogeneity consistent with our framework include the following: some agents may be rational while others adaptive, as has been examined in a cobweb model by Brock and Hommes (1997) and found empirically relevant in the data in Branch (2004); agents may have different information sets (e.g. Branch, 2007); or, they may use structurally different learning rules as in Honkapohja and Mitra (2006). Our goal is to extend this notion of agents as forecasters to the agents' primitive problem, and to characterize a set of admissible beliefs that facilitates aggregation.

Denote by ˆEτt${\hat{E}}_t^{\tau }$ a (subjective) expectations operator; that is, ${\hat{E}}_t^{\tau }(x_{t+k})$ is the time t expectation of $x_{t+k}$ formed by an agent of type τ. We require that

These assumptions impose regularity conditions consistent with the literature on bounded rationality and they facilitate aggregation in a linear, or linearized environment. Assumption A1 is consistent with reasonable specifications of agent behavior (the forecast of a known quantity should be the known quantity). Assumption A2 requires some continuity in beliefs in the sense that, in a steady state, agents' beliefs will coincide. Assumptions A3 and A4 require expectations to possess some linearity properties. Essentially, linear expectations require agents to incorporate some economic structure into their forecasting model rather than, say, mechanically applying a lag operator to every random variable.

Assumptions A5 restricts agents' expectations so that they satisfy the law of iterated expectations at an individual level. The L.I.E. at the individual level is a reasonable and intuitive assumption: agents should not expect to systematically alter their expectations. In deriving the New Keynesian IS curve, with heterogeneous expectations, Branch and McGough (2009) imposed an additional assumption of the L.I.E. at the aggregate level, essentially ruling out higher order beliefs. In other applications, such as the New Monetarist search model, we explicitly model higher-order beliefs by assuming that an expectation operator consists of two components, a point estimate given by $E_t^{\tau }$ and an uncertainty measure $F_t^{\tau }(\cdot ,\mathrm{\Sigma })$.

2.2 The Economic Environment

Before turning to a discussion of bounded optimality and aggregation of heterogeneous expectations, it is useful to describe a general economic environment that forms the basis for many of the applications in this chapter. The environment is based on a Lucas (1978) asset-pricing model with consumption risk in the form of a frictional goods market characterized by bilateral trading and a limited commitment problem. By opening and closing the frictional market we are able to nest a standard asset-pricing model or a search-based New Monetarist type model, depending on the desired application. We turn to a brief description of the environment, and then develop the analysis as the chapter progresses.

Time is discrete, and each time period is divided into two sub-periods. There are two types of non-storable goods: specialized goods, denoted q, that are produced and consumed in a market that opens during the first sub-period (“DM”); and general goods, denoted x, that are produced and consumed in a competitive market (“CM”) during the second sub-period. There are two types of agents: “buyers” and “sellers”. All agents produce the general good using the same technology that is linear in labor. We assume that labor is traded in the competitive market at a real wage of $w_t$, which, in equilibrium, will equal the real wage in terms of the numeraire good. When we develop results for a standard Lucas asset-pricing model, we assume a perfectly inelastic labor supply so that the model reduces, essentially, to an endowment economy. Thus, the competitive market represents the part of the environment that is standard in macro DSGE models like the New Keynesian model or the Lucas asset-pricing model. In the specialized goods market, though, buyers can consume but not produce the specialized good while sellers can produce but cannot consume. Moreover, trade in these markets is characterized by a limited commitment friction where buyers are unable to commit to repay unsecured debt using their proceeds from producing in the competitive market. Thus, this part of the model captures environments with frictional goods markets where assets such as fiat money, stocks or bonds are used as payment instruments. Some of the most interesting, and recent, applications of heterogeneous expectations occur in such markets. We assume that with probability σ buyers will have preferences for the specialized goods. By setting $\sigma =0$, the economic environment collapses into a standard environment without any real frictions.

There exists a single storable good, an asset a that yields a stochastic payoff $y_t$. This asset could be fiat money, with $y_t=0$, a Lucas tree with a stochastic dividend, or a risk-free bond with a known payoff. Depending on the frictions in the specialized goods market, the asset can be used to smooth consumption or as a liquid asset used in quid pro quo trade in the specialized goods market. That is, the frictional goods market gives a precautionary savings motive to hold the asset to insure against random consumption opportunities for the specialized good. Assume that households have separable preferences over the general good, the specialized good, and leisure.

2.3 Bounded Optimality

We now turn to agent-level decision-making taking the expectations operators as given. To best illustrate the ideas reviewed here, we develop them within the context of a very simple asset-pricing model. We, therefore, take the economic environment described above and shut-down the frictional goods market.

2.4 Aggregating Household Decision Rules

Having derived the individual boundedly rational optimizing behavior of heterogeneous agents, equilibrium requires computing aggregate, or market-clearing consumption/output from which assets can then be priced. Many applications derive an aggregate asset-pricing equation that depends on the aggregate expectations operator. This section illustrates how to aggregate decision rules, while briefly discussing some challenges.

To illustrate, we adopt the Euler equation learning approach. Recall that the behavioral primitives, in the log-linearized economy, are

$\begin{array}{ccc}\hfill c_t^j& \hfill =\hfill & \beta R{\hat{E}}_t^j{c}_{t+1}^j-\beta R{\upsilon }^{-1}{\hat{E}}_t^j{R}_{t+1}\hfill \\ \hfill c_t^j& \hfill =\hfill & \hat{a}R({\hat{a}}_t^j+R_t)-\hat{a}{\hat{a}}_{t+1}^j\equiv {\mathrm{\Omega }}_t^j\hfill \end{array}$

where ${\mathrm{\Omega }}^j$ denotes (end of period) real-wealth. Iterating on the Euler equation, we have

${\mathrm{\Omega }}_t^j={\mathrm{\Omega }}_{\infty }^j-\beta R{\upsilon }^{-1}{\hat{E}}_t^j\sum _{k=1}^{\infty }{\left(\beta R\right)}^{k-1}{R}_{t+k}$

where ${\mathrm{\Omega }}_{\infty }^j={\lim }_{T\to \infty }{\left(\beta R\right)}^T{\hat{E}}_t^j{c}_T$. In the asset-pricing application, $\beta R=1$ and so the Euler equation iterated forward depends on expected limiting wealth. In a model where there is a precautionary savings motivate, for instance in the version of the model with the specialized goods market and a limited commitment friction, it is possible that $R<{\beta }^{-1}$ and ${\mathrm{\Omega }}_{\infty }^j=0$. In this case, aggregation proceeds without any difficulty.

When $\beta R=1$, then additional assumptions on higher-order beliefs are necessary for aggregation. To illustrate this, assume that there are two expectation-types, $j=1,2$. Then, $n{\mathrm{\Omega }}_t^1+(1-n){\mathrm{\Omega }}_t^2={(1-\beta )}^{-1}{R}_t$ in this linearized economy. Then

$\begin{array}{cc}\hfill {(1-\beta )}^{-1}{R}_t& =n{\mathrm{\Omega }}_t^1+(1-n){\mathrm{\Omega }}_t^2\hfill \\ \hfill & =n{\mathrm{\Omega }}_{\infty }^1+(1-n){\mathrm{\Omega }}_{\infty }^2-{\upsilon }^{-1}{\hat{E}}_t\sum _{k⩾1}{R}_{t+k}\hfill \\ \hfill & =[1-{\upsilon }^{-1}(1-\beta )]{\hat{E}}_t{R}_{t+1}+n{\mathrm{\Omega }}_{\infty }^1+(1-n){\mathrm{\Omega }}_{\infty }^2\hfill \\ \hfill & -\hat{E}[n{\mathrm{\Omega }}_{\infty }^1+(1-n){\mathrm{\Omega }}_{\infty }^2]\hfill \end{array}$

where $\hat{E}=n{\hat{E}}^1+(1-n){\hat{E}}^2$. Thus, to have a (linearized) asset-pricing equation that depends on the aggregate expectations operator – i.e. a weighted average of heterogeneous expectations – requires that all agents agree on limiting wealth, an axiom needed for aggregation in the New Keynesian model as shown by Branch and McGough (2009). Alternatively, a model with $\beta R<1$ also facilitates aggregation without difficulty.

3.1 Extrinsic Heterogeneity

An equilibrium with extrinsic heterogeneity takes the distribution of agents across models as fixed and computes equilibrium prices and quantities. In particular, in the asset-pricing application developed in the previous section, each agent of type j, solves for their optimal consumption and asset holdings given their behavioral primitives (e.g. SP-learning, etc.), and then the temporary equilibrium pins down the price. For example, suppose J is a finite index set of types of SP-learning agents who differ only in their time-t beliefs. The asset demand of an agent of type $j\in J$ is then given by

$a_{t+1}^j=a^{SP}(p_t,y_t,a_t^j,{\hat{E}}_t^j{\lambda }_{t+1}^j)$

Let $n_j$ be the measure of type j agents. Given $y_t$, beliefs ${\{{\hat{E}}_t^j{\lambda }_{t+1}^j\}}_{j\in J}$, and asset holdings ${\{a_t^j\}}_{j\in J}$, the equilibrium price is determined by

$1=\sum _{j\in J}{n}_j\cdot a^{SP}(p_t,y_t,a_t^j,{\hat{E}}_t^j{\lambda }_{t+1}^j)$

Note that here we are using the non-linear version of SP-learning. The linearized market-clearing condition would set the sum to zero. Note also that, in this linearized case, an analytic expression for $p_t$ in terms of beliefs, exogenous variables, and the distribution of asset holdings, is available.

3.2 Rationally Heterogeneous Expectations

The theory of rationally heterogeneous expectations, as formulated by Brock and Hommes (1997), holds that expectation formation is a discrete choice from a finite set of predictors.

3.3 Intrinsic Heterogeneity

In the rationally heterogeneous expectations approach, heterogeneity arises because of a cost to using a particular predictor, and for finite intensities of choice, ω, i.e. because of the idiosyncratic preference shocks. In Branch and Evans (2006) an environment is presented where heterogeneity can arise even in the neoclassical case of $\omega \to +\infty$. The framework developed by Branch and Evans is an extension of Brock and Hommes to a stochastic environment where agents select among a set of misspecified forecasting models. In a Misspecification Equilibrium the parameters of the forecasting models, and the distribution of agents across models, are determined jointly as equilibrium objects. Intrinsic Heterogeneity arises if agents are distributed across more than one model even as $\omega \to \infty$. Branch and Evans (2006) developed their results within the context of a cobweb model, though Branch and Evans (2010) extend to an asset-pricing model, and Branch and Evans (2011) show existence of Intrinsic Heterogeneity in a New Keynesian model.

To illustrate the approach, we extend the basic modeling environment to include two different serially correlated shocks, a dividend shock and a share-supply shock. In Branch and Evans (2010) the share-supply shock arises in an OLG model through random variations in population growth, which affects the supply of trees per person. In Branch (2016) variations in the supply can arise in the search-based asset-pricing model outlined in the previous section. Here we will illustrate the approach in the context of the search-based asset-pricing model that, with large enough liquidity services role for the asset in facilitating specialized good consumption, can feature negative feedback as in the Cobweb and New Keynesian economies.

A log-linearization to the asset-pricing equation extended to include a liquidity premium can be written as

${\hat{p}}_t={\alpha }_0{\hat{E}}_t{\hat{y}}_{t+1}+{\alpha }_1{\hat{E}}_t{\hat{p}}_{t+1}+{\alpha }_2{\hat{A}}_t$

for appropriately defined ${\alpha }_k,k=0,1,2$.⁶ The stochastic process $A_t$ is the supply of shares in the Lucas tree, which can be interpreted as asset float resulting from new share issuances, repurchases, and stock splits. Assume that the stochastic processes for dividends and share supply are stationary AR(1) processes, given by

$\begin{array}{ccc}\hfill {\hat{y}}_t& \hfill =\hfill & \rho {\hat{y}}_{t-1}+{\epsilon }_t\hfill \\ \hfill {\hat{A}}_t& \hfill =\hfill & \varphi {\hat{A}}_{t-1}+{\nu }_t\hfill \end{array}$

where $\epsilon ,\nu$ are uncorrelated across time but are potentially correlated with each other. If there is sufficient curvature of the utility function over the specialized good then ${\alpha }_1<0$ and the model exhibits negative feedback. This arises when the liquidity premium is high: if the expected price of the asset is high then it will relax the liquidity constraint and individuals do not need to hold as much of the asset to facilitate specialized-goods consumption, demand for the asset goes down, and price decreases. Alternatively, for smaller liquidity premia then ${\alpha }_1>0$.

In a rational expectations equilibrium, agents would include in their forecasting model, or perceived law of motion (PLM), both dividends and share supply. In Branch and Evans (2006), agents were assumed to select from a set of underparameterized forecasting models. In the present application, the set of forecasting models are:

$\begin{array}{cc}\hfill {\hat{p}}_t=b^1{\hat{y}}_t+{ϵ}_t^1& \Rightarrow {\hat{E}}_t^1{\hat{p}}_{t+1}=b^1\rho {\hat{y}}_t\hfill \\ \hfill {\hat{p}}_t=b^2{\hat{A}}_t+{ϵ}_t^2& \Rightarrow {\hat{E}}_t^2{\hat{p}}_{t+1}=b^2\varphi {\hat{A}}_t\hfill \end{array}$

Underparameterization is motivated by real life decisions encountered by professional forecasters. Often, degrees of freedom limitations lead forecasters to adopt parsimonious models.⁷

Denote by n the fraction of agents who adopt the dividend-only forecasting model, i.e. “model 1”. Plugging in expectations, the actual law of motion can be written as

${\hat{p}}_t={\xi }_0(b_1,n){\hat{y}}_t+{\xi }_1(b_2,n){\hat{A}}_t$

In a Misspecification Equilibrium, agents only select the best-forming statistical models. Thus, in equilibrium we require that the coefficients of the forecasting models are determined by the least-squares projection of price onto the restricted set of regressors, i.e. $b^1,b^2$ satisfy the least-squares orthogonality conditions

(17) $E{\hat{y}}_t(p_t-b^1{\hat{y}}_t)=0$

(18) $E{\hat{A}}_t(p_t-b^2{\hat{A}}_t)=0$

A Restricted Perceptions Equilibrium is a stochastic process for $p_t$, given n, such that $b^1,b^2$ satisfy (17)–(18). In a restricted perceptions equilibrium the agents forecasting models are misspecified but, within the context of their forecasting model, they do not recognize the misspecification. A restricted perceptions equilibrium can be justified in environments where data is slow to reveal the misspecification, a property shared by most time series data.

A Misspecification Equilibrium requires that the agents only select their best performing statistical model. Branch and Evans (2006) endogenize the distribution of agents across forecasting models, as in Brock–Hommes, by assuming they make a discrete choice and the distribution is determined by the MNL-map:

$n=\frac{1}{2}\{\tanh [\frac{\omega }{2}F(n)]+1\}\equiv T_{\omega }(n)$

where $F(n)$ is the relative mean-squared error between predictor 1 and predictor 2 within a restricted perceptions equilibrium for a given n. Notice that the T-map: $T_{\omega }:[0,1]\to [0,1]$ is continuous and so the set of equilibria is indexed by the intensity of choice ω and the properties of the function F.

The set of equilibria depend on the properties of F, and can be characterized in the following way.

Intrinsic Heterogeneity arises when agents do not want to mass on one or the other predictors, which occurs when $F(0)>0,F(1)<0$, i.e. there is always an incentive to deviate from expectations homogeneity. In the Branch–Evans papers it was shown that Intrinsic Heterogeneity requires negative feedback, ${\alpha }_1<0$, which can occur for large liquidity premia in the present environment. In the monetary policy application, negative feedback can arise when the central bank sets its policy rate according to an expectations-based Taylor-type rule with an aggressive response to expectations. Negative feedback implies that the asset pricing model features strategic substitution effects. There can also be multiple equilibria when there are sufficiently strong strategic complementarities, i.e. with ${\alpha }_1>0$. In this case, there are misspecification equilibria with all agents massed onto one model or the other, and a third misspecification equilibria that exhibits Intrinsic Heterogeneity.⁸

4.1 Regime-Switching Returns

Branch and Evans (2010) construct a mean-variance asset-pricing model where stock prices are driven by expected returns, which depend directly on dividends and asset share supply, and indirectly via the self-referential feature of the pricing Euler equation. As in the previous section, it was assumed that traders are distributed across underparameterized forecasting models that depend only on dividends or share supply. It was shown that multiple Misspecification Equilibria exist and a real-time adaptive learning version of the model generates regime-switching returns and volatilities.

The asset-pricing equation in Branch and Evans (2010) was derived from an overlapping generations model with mean-variance preferences. The dynamic structure of OLG models is quite similar to the sub-periods of the Lagos–Wright based model developed in Section 2. The CARA preferences underlying the mean-variance structure lead to a downward sloping demand curve, similar to what emerges in the search-based model where the limited demand is based on the liquidity properties of the asset. Thus, the results in Branch and Evans (2010) are robust to the version of the model with liquidity considerations provided that there are log preferences over the specialized good: see Branch (2016). For ease of exposition, this subsection deviates from the model derived in this chapter and presents the asset-pricing equation derived by Branch and Evans (2010).

The mean-variance asset-pricing model, with stochastic processes for dividends and share supply, has asset demand represented by equations of the form, for an agent with expectations-type j,

$a_{jt}=\frac{1}{\gamma {\hat{\sigma }}^2}{\hat{E}}_t^j(p_{t+1}+y_{t+1}-R{p}_t)$

where γ is the coefficient of absolute risk aversion and ${\hat{\sigma }}^2$ is the (perceived) conditional variance of excess returns. The risk parameter $\hat{\sigma }$ is pinned down as an equilibrium object of the model. As in the previous section, assume that there are two-types of agents who differ based on whether they forecast with a dividends-based or share supply-based model. Then market equilibrium requires that

$n{a}_{1t}+(1-n)a_{2t}=A_t$

where, again, $A_t$ represents asset share supply (or, float). The equilibrium process for price is, then, given by

(19) $p_t=\beta [n{\hat{E}}_t^1{p}_{t+1}+(1-n){\hat{E}}_t^2{p}_{t+1}]+\beta \rho y_t-\beta a{\hat{\sigma }}^2{A}_t$

where $\beta =1/R$. As before, assume that dividends and share supply are represented as stationary AR(1) processes, written in deviation-from-mean form:

$\begin{array}{cc}\hfill y_t& =(1-\rho )y_0+\rho y_{t-1}+{\epsilon }_t\hfill \\ \hfill A_t& =(1-\varphi )A_0+\varphi A_{t-1}+{\nu }_t\hfill \end{array}$

Traders are assumed to form their forecasts by selecting from one of the following misspecified models:

$\begin{array}{cc}\hfill & {\text{PLM}}_1:p_t=b_0^1+b^1{y}_t+{\eta }_t\hfill \\ \hfill & {\text{PLM}}_1:p_t=b_0^2+b^2{A}_t+{\eta }_t\hfill \end{array}$

where ${\eta }_t$ is a perceived white noise error. As before, the distribution of agents, after selecting their predictor, is given by the MNL map:

$T_{\omega }(n)=\frac{1}{2}\{\tanh [\frac{\omega }{2}F(n)]+1\}$

A Misspecification Equilibrium is a fixed point $n^{⁎}$ to the T-map, i.e. $n^{⁎}=T_{\omega }(n^{⁎})$.

Branch and Evans (2010) show theoretically that there can exist multiple misspecification equilibria. An example is given in Fig. 3, which is based on a calibrated version of the model with $\omega \to \infty$. Fig. 3 plots the T-map and $F(n)$ functions, where the T-map crosses the ${45}^{\circ }$ line is a Misspecification Equilibrium. In the calibrated version of the model, there are multiple misspecification equilibria. When all agents use forecast model 1, then it is a best-response for an agent to use forecast model 1 as it delivers a lower mean-square forecast error. Similarly, for model 2. Notice that there is also an interior equilibrium exhibiting Intrinsic Heterogeneity. The right hand panels compute the volatility of returns for a given n. These figures reveal that there exist multiple misspecification equilibria, that differ in mean returns and volatilities. In particular, there exists a low return-low volatility and a high return-high volatility equilibrium.

The set of Misspecification Equilibria depends on the structural parameters of the model. In particular, the degree of risk-aversion – analogously, in the search-based model, the preferences for the differentiated good – plays an important role in determining whether multiple equilibria and whether Intrinsic Heterogeneity can exist. In Fig. 4, Branch and Evans (2010) compute the misspecification equilibria treating the degree of risk-aversion, a, as a bifurcation parameter. For low and high risk aversion there exist unique equilibria with homogeneous expectations. For moderate, and empirically realistic, degrees of risk-aversion there are multiple equilibria, including ones with Intrinsic Heterogeneity.

Branch and Evans (2010) exploited the existence of multiple misspecification equilibria to generate regime-switching returns and volatilities that match those estimated in the data by Guidolin and Timmermann (2007). To generate plausible dynamics, the Misspecification Equilibrium values for the belief parameters $b^j$, and the mean-square errors, were replaced by real-time recursive econometric estimators. Branch and Evans also specified a recursive estimator for the conditional variance ${\hat{\sigma }}^2$. The intensity of choice parameter was also calibrated to a finite value in order to match the Guidolin–Timmermann statistics. Fig. 5 illustrates a typical simulation that exhibits regime-switching returns and volatilities. As the distribution of heterogeneity n evolves in real-time, agents' beliefs fluctuate between neighborhoods of the low return/low volatility and high return/high volatility Misspecification Equilibria. Thus, misspecification and heterogeneity can lead to endogenous recurring regime switching asset returns.

4.2 Bubbles with Rationally Heterogeneous Expectations

Beginning with Brock and Hommes (1998), a large literature has developed studying paths from rationally heterogeneous expectations to excess volatility, bubbles, and crashes in otherwise standard asset-pricing models. Much of this literature is surveyed in Hommes (2013), with special attention to identifying simple forecasting heuristics that are consistent with learning-to-forecast experiments in asset pricing and other experimental economies. To illustrate the main mechanics, we turn now to a version of the model (19), but where agents choose between rational and adaptive expectations.

Consider a non-stochastic version of (19):

$p_t=\mu +\beta [n_{t-1}{\hat{E}}_t^1{p}_{t+1}+(1-n_{t-1}){\hat{E}}_t^2{p}_{t+1}]$

where $\mu =\beta \overline{y}-a{\sigma }^2\overline{A}$, and we assume the following expectations operators:

$\begin{array}{ccc}\hfill {\hat{E}}_t^1{p}_{t+1}& \hfill =\hfill & p_{t+1}\hfill \\ \hfill {\hat{E}}_t^2{p}_t& \hfill =\hfill & \overline{p}+\theta (p_{t-1}-\overline{p})\hfill \\ \hfill {\hat{E}}_t^2{p}_{t+1}& \hfill =\hfill & \overline{p}+{\theta }^2(p_{t-1}-\overline{p})\hfill \end{array}$

Predictor 1 is rational expectations (perfect foresight) and predictor 2 is an adaptive predictor that forecasts future prices as deviations from steady-state. Predictor 2 is an “anchor and adjustment” rule of the type that Hommes (2013) finds fits many experimental subjects' forecasts from the laboratory. In the example of this section, we assume that $\theta >1$, so that agents who use this predictor are strongly extrapolative. Notice also that we have explicitly assumed a time-varying weight on n while preserving the typical $t-1$-dating convention that forecasts made at time t are conditional on all information available through time t. This timing assumption breaks the simultaneity of expectations and equilibrium outcomes that are a defining feature of rational expectations equilibria but are not cognitively consistent in a model of bounded rationality. We also continue to assume that the distribution of predictor selection is given by the MNL map.

Given these assumptions, the actual law of motion for this economy is given by the following pair of non-linear difference equations:

$\begin{array}{ccc}\hfill p_t& \hfill =\hfill & \hat{\mu }+\left(\frac{1}{\beta n_{t-1}}\right)p_{t-1}-\left(\frac{1-n_{t-1}}{n_{t-1}}\right){\theta }^2{p}_{t-2}\hfill \\ \hfill n_t& \hfill =\hfill & \frac{1}{2}\{\tanh \left[\frac{\omega }{2}(U_t^1-U_t^2)\right]+1\}\hfill \end{array}$

where $\hat{\mu }=-{(\beta n)}^{-1}\mu -(1-n_{t-1})(1-{\theta }^2)\overline{p}/n_{t-1}$ and $U_t^1-U_t^2={(p_t-{\hat{E}}_{t-1}^2{p}_t)}^2-C$. For illustrative purposes, we measure predictor fitness as inversely related to the most recent squared forecast error net of the predictor cost, with the cost to using the adaptive predictor normalized to zero.

Fig. 6 plots a typical simulation that exhibits complex price dynamics. This figure was generating assuming $\theta =1.5,C=1,\omega =2.6,\beta =0.95$, and $\mu =1$. The first two panels plot the time paths for price and the fraction of rational agents along the stable attractor. The bottom panel plots the attractor in phase space. Price oscillates around the steady-state value with complex dynamics, featuring both excess volatility and bubbles/crashes. This simple example illustrates how rationally heterogeneous expectations can generate complex asset price dynamics. Other formulations of the predictors can lead to more realistic asset prices with larger bubbles and crashes, see, e.g. Hommes (2013) for examples.

The intuition for the complex dynamics arising from rationally heterogeneous expectations is based on attracting/repelling dynamics of predictor selection along with the stability reversal property discussed earlier. In this particular example, with forward- vs. backward-looking expectations, the steady-state is unstable when all agents adopt either rational or adaptive expectations. When the economy is close to steady-state the two predictors forecast equally well, so agents are unwilling to pay the cost for the rational predictor. This drives the economy away from the steady-state, i.e. the repelling dynamics. As price moves further from steady-state, the perfect foresight predictor's forecasting benefits begin to outweigh the cost, for some agents depending on their preference shocks, which balances the opposing repelling forces and tends to attract the system back towards the steady-state. These attracting/repelling forces combine to produce complex dynamics that can feature endogenous paths to bubbles and crashes.

4.3 Restricted Perceptions and Endogenous Fluctuations

A substantial body of research introduces volatility via fluctuations in expectations, for instance, through self-fulfilling sunspot shocks, news shocks, expectations shocks, sentiment shocks, forecasting add factors, exuberance shocks, etc. Broadly speaking, a drawback to these approaches to introducing expectations shocks, or animal spirits, is that it is difficult to construct instances where individuals and firms can coordinate their expectations. Branch and Zhu (2016), on the other hand, show how forecast model misspecification can expand the set of equilibria including restricted perceptions equilibria that depend on extrinsic noise. Interestingly, equilibria that depend on extrinsic noise do not necessarily exhibit more volatility than equilibria that only depend on fundamental economic variables. This is in contrast with much of the extant literature. Branch and Zhu (2016) show, however, that heterogeneity in forecast model misspecification can lead to more volatile equilibria that depend on extrinsic noise.

To illustrate the approach consider again the simplified asset-pricing model (19) with the share-supply shocks shut down:

$p_t=\beta [n{\hat{E}}_t^1{p}_{t+1}+(1-n){\hat{E}}_t^2{p}_{t+1}]+\gamma y_t$

and $y_t=\rho y_{t-1}+{\epsilon }_t,0<\rho <1$ and ε is i.i.d. $N(0,{\sigma }_{\epsilon }^2)$. Branch and Zhu (2016) follow the behavioral learning equilibria approach developed in Hommes and Zhu (2014), who assume that the exogenous shocks, $y_t$, are unobservable to all agents and so they instead fit autoregressive models to $p_t$, with the coefficients picked to ensure that the first autocorrelation of the data generating process and the approximating models coincide. In Branch and Zhu (2016) agents differ in a subset of their regressors, where the models can be thought of as ARMAX, so that agents may include more or less autoregressive terms, the fundamental exogenous variable y, or even moving average terms. The novelty in Branch and Zhu (2016) is that they also potentially include non-fundamental exogenous variables – loosely speaking, expectations shocks – in their forecasting model. To illustrate, we consider the simplest case in Branch and Zhu (2016):

$\begin{array}{cc}\hfill & {\text{PLM}}_1:p_t=b{p}_{t-1}+d{\eta }_t\hfill \\ \hfill & {\text{PLM}}_2:p_t=c{y}_t+f{\eta }_t\hfill \end{array}$

where ${\eta }_t=\varphi {\eta }_{t-1}+{\nu }_t$. The first predictor fits price to an AR(1) model plus a component common to the second predictor. The second predictor assumes these agents can observe the fundamental shocks, but also includes the common shock. The ${\eta }_t$ can be thought of as all other information, exogenous to the fundamentals of the asset-pricing economy, i.e. a statistical sunspot variable. Whether agents coordinate on the ${\eta }_t$ shock is determined as an equilibrium outcome. With these perceived laws of motion, the actual law of motion can be written as

(20) $p_t={\alpha }_1{p}_{t-1}+{\alpha }_2{y}_t+{\alpha }_3{\eta }_t$

for appropriately defined ${\alpha }_j$, $j=1,2,3$ that depend on the structural model parameters and, importantly, the belief coefficients $b,c,d,f$ and the fraction of agents using model 1, n. Assume that n is an exogenous parameter in the model. Notice from (20) that the two predictors are misspecified: model 1 underparameterizes by excluding the fundamental shocks while model 2 omits lagged prices.

Because the two forecast models are misspecified, the appropriate equilibrium concept is Restricted Perceptions Equilibrium (RPE). Let $X_t^1={(p_{t-1},{\eta }_t)}^{\prime },X_t^2={(y_t,{\eta }_t)}^{\prime }$. Then, in an RPE the beliefs satisfy the least-squares orthogonality conditions:

$\begin{array}{cc}\hfill E{X}_t^1[p_t-(b,d)X_t^1]& =0\hfill \\ \hfill E{X}_t^2[p_t-(c,f)X_t^2]& =0\hfill \end{array}$

If $n=1$ and the RPE value $d=0$ then the equilibrium value of b coincides with the behavioral learning equilibrium in Hommes and Zhu (2014). If $n=0$ then there is a unique RPE value $f=0$ and the value of c is the same as the rational expectations equilibrium coefficient. However, when $n=1$ Branch and Zhu (2016) show that there can also exist non-fundamental RPE with $d\ne 0$. Thus, we focus on when $0<n⩽1$.

Branch and Zhu (2016) show that there always exists fundamental RPE where $d=0$ and $f=0$. In this case, the associated mappings from perceived parameters $(b,c)$ to their least-squares projection values, computed using the implied actual law of motion, i.e. the “T-map” are

$\begin{array}{cc}\hfill b& \to \frac{b^{2}n\beta +\rho }{1+b^{2}n\beta \rho }\hfill \\ \hfill c& \to \frac{1+c(1-n)\beta \rho }{1-b^{2}n\beta \rho }\hfill \end{array}$

The fundamental RPE reveals an important insight for the existence of the non-fundamental RPE. In this simple example, the second perceived law of motion is only misspecified because of $n>0$, i.e. the presence of the agents who do not observe or forecast with the dividend shocks as a regressor. The existence of these agents, though, alters the serial correlation properties of asset prices, which because of the cross-equation restrictions associated to the RPE least-squares orthogonality condition, impacts the regression coefficient c of the type-2 agents. Thus, the misspecification alters the equilibrium weight on the exogenous shocks.

Now consider the case where $d,f\ne 0$. Such equilibria can arise when the feedback effects in the model are strong, i.e. when β is close to one, a condition that will be satisfied in most asset-pricing models. In particular, the T-map components for the coefficients $b,c$ now become

$\begin{array}{cc}\hfill b& \to \frac{1-\beta \varphi }{n\beta (1-{\varphi }^2)}\hfill \\ \hfill c& \to \frac{1}{1-(1-n)\beta \rho -b^{2}n\beta \rho }\hfill \end{array}$

and d is pinned down by a complicated quadratic expression in the structural parameters $n,\beta ,\rho ,\varphi$, and $f\to d/(1-\beta \varphi )$. The intuition for the existence of non-fundamental RPE is clear. The misspecification and heterogeneity introduce serial correlation in a temporary equilibrium sense. The agents optimally attribute some of that serial correlation to the endogenous variables – lagged prices for model 1 and serially correlated dividends for model 2 – and to the extrinsic variable ${\eta }_t$. The specific values of those regression coefficients depend on the distribution of agents across the two forecast models, as well as the stochastic properties of the shocks, i.e. $\rho ,\varphi$. Importantly, the non-fundamental RPE are self-fulfilling equilibria that only arise when expectations play an important role in the pricing equation. There are, therefore, three important ingredients for the existence of these non-fundamental equilibria: misspecification, heterogeneity, and strong expectational feedback.

Fig. 7 illustrates the existence of the full set of RPE, including those depending on the statistical sunspots. This figure contains contour plots for the $(b,c)$ components of the T-map.⁹ The $T_d$ component consists of a horizontal line at $d=0$ and another vertical line at the non-fundamental value for b. The $T_b$ mapping is the parabolic lines in the figure. The figure illustrates the fixed points to the T-map for two different values of n. The $n=1$ T-maps provide the existence of RPE without heterogeneity, while the $n=0.9$ consists of some degree of heterogeneous expectations. Notice that a fundamental RPE exists in both cases and that the RPE value of b are very close to each other for both values of n. For both values of n, there exist two non-fundamental RPE, symmetric around the $d=0$ line. Increasing the degree of heterogeneity shifts the $T_d$ line to the right, implying higher degrees of perceived autocorrelation. Similarly, the $T_b$ shifts away from the $d=0$ line, implying a relatively greater weight placed on the extrinsic noise d.

Branch and Zhu (2016) then ask the question of whether RPE that display dependence on extrinsic noise terms introduce more volatility into asset prices. It turns out that the effect is not obvious, as it depends on the equilibrium effect on all of the RPE belief coefficients. For example, bifurcating the equilibrium from a fundamental RPE to a non-fundamental RPE has a direct effect that increases asset-price volatility through the coordination of agents onto the extrinsic noise. However, there is a potentially competing effect on the equilibrium value for the AR(1) coefficient, b, in agents of type-one's forecasting model. These agents attribute, in part, some of the serial correlation in prices to the extrinsic term, lowering their estimated value of b, which will tend to push down asset-price volatility. Thus, the effect on volatility is a balancing of these two effects. See, for example, a particular numerical case in Branch and Zhu (2016) that is plotted in Fig. 8. This figure plots the ratio of RPE price volatility to the volatility in the rational expectations equilibrium as a function of the degree of heterogeneity, n, for both the fundamental RPE (solid) and non-fundamental RPE (dashed). This particular example is for a very strongly serially correlated process η. Depending on the degree of heterogeneity, the RPE can introduce excess volatility, but it is not necessarily the case. Moreover, for relatively high values of n, that are not too close to homogeneity, the excess volatility that arises from the non-fundamental RPE is greater than in the fundamental RPE. However, interestingly, when $n=1$ the excess volatility is greater in the fundamental RPE, which coincides with the equilibrium identified by Hommes and Zhu (2014). The intuition for these findings can be seen in Fig. 7. The equilibrium with heterogeneous expectations is more volatile than the $n=1$ equilibrium because the autoregressive coefficient b and the weight placed on the sunspot d is relatively higher. Thus, to have statistical sunspots generate more volatility requires a self-reinforcing effect arising from heterogeneous expectations. However, if there is a sufficiently low n ($n<0.85$ in the figure) then the non-fundamental RPE again produces less economic volatility. The reason is that as n decreases the $T_d$-map shifts to the right, and the forecasting models place less weight on both the sunspot $(d,f)$ and the fundamental shock c.

4.4 Related Literature

Much of the literature on asset-pricing in models with heterogeneous expectations was started by the seminal Brock and Hommes (1998), which applied the framework developed in Brock and Hommes (1997) to an asset pricing model where agents select between perfect foresight and a simple adaptive rule that nests extrapolative, contrarian, and trend-chasing expectations. These papers were inspired by LeBaron (1995) and other attempts at building artificial stock markets with heterogeneous, evolving technical trading strategies. Brock and Hommes (1998) demonstrated that rationally heterogeneous expectations could feature “a rational route to randomness” in asset prices for a wide specification of expectations-types. This launched an extensive literature with key contributions by Hommes and co-authors at the CENDEF at the University of Amsterdam. Important extensions include the “large-type limit” introduced by Brock et al. (2005), where agents select from a continuum of predictors, which has a natural Bayesian interpretation.

Importantly, this literature tackled the empirical foundations of heterogeneous expectations beyond their theoretical implications. A paper by Hommes et al. (2005) introduced “learning to forecast” experiments as a way to test for heterogeneous expectations in a simple laboratory environment. In this paper, a simple asset-pricing framework similar to Brock and Hommes (1998) was set in a laboratory where the only decision subjects were required to make was a forecast of the asset price in the next period, which automatically determines their portfolio allocation. Hommes et al. showed that the lab featured asset price dynamics that often converged to cycles around the fundamental price, i.e. bubbles and crashes. Moreover, they found that the best fit to the set of individual prediction strategies was a heterogeneous collection of simple autoregressive forecasting rules. Subsequent experiments across a range of model environments provide further evidence for heterogeneous beliefs and rationally evolving fractions of lab participants distributed across forecasting models. See, for instance, Hommes (2013) and Anufriev et al. (2015) for an extensive review.

There is also a literature that considers heterogeneity and adaptive learning. The seminal Marcet and Sargent (1989) featured a setting where agents had private information that creates for the other agents “hidden state variables,” but differed in which sets of variables were unobservable. They focused on convergence to a limited information rational expectations equilibrium. Subsequent papers, such as Marcet and Sargent (1995), examine where agents fit heterogeneous ARMA models, similar to the class of forecasting models in Branch and Zhu (2016). Guse (2005) studies an environment with agents who have heterogeneous expectations derived from forecasting rules that fit the full set of rational expectations equilibria, i.e. one rule is based on the minimal state variable solution while the other is for an ARMA with sunspot, and derives conditions under which the MSV equilibrium is stable under learning in a general environment. Similarly, Berardi (2007) extends that framework to forward-looking models with a lag that nests the asset-pricing model considered here.

Besides Branch and Evans (2010), there have been other heterogeneous expectations models of adaptive learning and asset-pricing. LeBaron (2012) constructs a model with constant gain learning where agents differ in their gain size, i.e. their robustness to structural change of an unknown form. Markiewicz (2012) considers a form of misspecification equilibrium, where the distribution across predictors is determined by running statistical specification tests based on the Kullback–Leibler information criterion, and applies it to explaining exchange rate puzzles. Nakov and Nuno (2015) apply the learning-from-experience model of Malmendier and Nagel (2016) to an asset-pricing model. Finally, Parke and Waters (2007) construct an evolutionary model of learning that endogenously generates ARCH effects in asset-prices. Elias (2016) includes expectations shocks into an asset-pricing model with heterogeneous learning.

5.1 A Monetary Search Model with Heterogeneous Expectations

We begin by turning the preference shock, interpretable as a search friction, back on. With probability σ a buyer in the specialized good market will meet a seller. In this environment, the solution usually proceeds recursively, taking as given beliefs about the future. To set the stage, we turn to how a recursive solution can be found by starting with solving for equilibrium in the specialized goods sector. Following Branch and McGough (2016) we assume that the terms of trade are determined via take-it-or-leave-it offers made by buyers. If agents in this market are anonymous, buyers will be unable to commit to repaying any credit issued from sellers and all trade must be quid pro quo, i.e. the buyer must present a payment instrument, or equivalently, use an asset as collateral for a loan from the seller until the buyer receives proceeds from producing in the general goods market. We assume the existence of a single liquid asset, fiat money, which is a special case of the general framework where the asset has a zero payment flow, i.e. it is intrinsically worthless.

Buyers will decide on their asset holdings based on the discounted expected payoff of the asset, in terms of the general good, plus any expected liquidity services they provide. The literature has a number of ways of approaching how they decide on these asset holdings and consumption plans, depending on their beliefs about the future. To see this, denote by $V(a)$ the value function for a buyer at the beginning of the period before entering the specialized goods market. It is straightforward to see, using our earlier notation, that

$V_t(a_t)=\underset{{\varphi }_t{q}_t⩽p_t{a}_t/w_t}{\max }\sigma [u(q_t)-p_t{q}_t]+\frac{p_t{a}_t}{w_t}+W_t(0)$

and, the Bellman equation for the agent is

$W(a_t)=\underset{x_t,h_t,a_{t+1}}{\max }v(x_t)-h_t+\beta {\hat{E}}_t{V}_{t+1}(a_{t+1})$

subject to