Learning and Macroeconomics
Annual Review of Economics
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Expectations play a central role in modern macroeconomic theories. The econometric learning approach models economic agents as forming expectations by estimating and updating forecasting models in real time. The learning approach provides a stability test for rational expectations and a selection criterion in models with multiple equilibria. In addition, learning provides new dynamics if older data are discounted, if models are misspecified, or if agents choose between competing models. This paper describes the expectational stability (E-stability) principle and the stochastic approximation tools used to assess equilibria under learning. Applications of learning to a number of areas are reviewed, including the design of monetary and fiscal policy, business cycles, self-fulfilling prophecies, hyperinflation, liquidity traps, and asset prices.
Expectations play a central role in modern macroeconomics. Economic agents are assumed to be dynamic optimizers whose current economic decisions are the first stage of a dynamic plan. Thus, households must be concerned with expected future incomes, employment, inflation, and taxes, as well as the expected trajectories of the stock market and the housing market. Firms must forecast the level of future product demand, wage costs, productivity levels, and foreign exchange rates. Monetary and fiscal policy makers must forecast inflation and aggregate economic activity and consider both the direct impact of their policies and the indirect effect of policy rules on private-sector expectations.
Macroeconomic models can be summarized as a reduced-form multivariate dynamic system,
 | (1) |
. The precise information set available to economic agents for forming expectations will depend on the specific model, and in some cases yt will also depend on forecasts of contemporaneous variables.Since the work of Muth (1961), Lucas (1972), and Sargent (1973), the benchmark model of expectation formation in macroeconomics has been rational expectations (RE). This posits, for both private agents and policy makers, that expectations are equal to the true statistical conditional expectations of the unknown random variables. RE is a very strong assumption because it implicitly assumes knowledge of the correct form of the model, knowledge of all parameters, and knowledge that other agents are rational, as well as the knowledge that other agents know that other agents are rational, etc.
The learning theory approach in macroeconomics argues that although RE is the natural benchmark, it is implausibly strong. We need a more realistic model of rationality, which may, however, be consistent with agents eventually learning to have RE. A natural criterion for a model of rationality is the cognitive consistency principle, which holds that economic agents should be assumed to be about as smart as (good) economists. This still leaves open various possibilities because we could choose to model households and firms as economic theorists or, alternatively, model them as econometricians.1 The adaptive or econometric learning approach, which is our principal focus, takes the latter viewpoint, arguing that economists, when they forecast future economic aggregates, usually do so using time-series econometric techniques.2 This seems particularly natural because neither private agents nor economists at central banks know the true model. Instead, economists formulate and estimate models. These models are re-estimated and possibly reformulated as new data become available. Economists themselves engage in processes of learning about the economy.
The econometric learning approach to expectation formation leads to several distinct roles for learning in macroeconomics. Closest to the RE view, econometric learning can be viewed as a stability test for RE equilibria (REE): Under what circumstances will least squares (LS) or closely related econometric learning schemes converge asymptotically to RE? The stability analysis can also be used as a selection device when there are multiple REE. This is of particular interest when the REE include sunspot equilibria or cycles that can be viewed as self-fulfilling prophecies. Can economic agents, using econometric forecasting rules updated over time in accordance with LS, converge over time to nonfundamental solutions such as sunspot equilibria? These questions can be examined using the expectational stability (E-stability) tool. According to the E-stability principle, the local stability of an REE under LS-type learning rules can be determined via a differential equation that is often straightforward to compute.
The econometric learning approach also generates additional insights for macroeconomic theory and economic policy. For example, the reality that econometricians sometimes use misspecified models suggests that we should consider agents using misspecified econometric forecasting models. There is then the possibility of convergence to restricted perceptions equilibria, in which the agents are doing the best they can, given their misspecified models. Another consideration is that, even if agents converge asymptotically to an REE, the economy will deviate from this equilibrium during the learning transition. Furthermore, if agents use constant-gain (discounted) LS, which weights more recent data more heavily, then convergence will be to a stochastic process near the REE, rather than to the REE itself. In some cases, this can have major implications in applications or for economic policy. Additional learning dynamics arise when one allows for heterogeneous expectations, due to agents using either different learning rules or differing forecasting models.
In this paper, we survey the main tools of macroeconomic learning theory in Section 2 and then consider a range of applications in Section 3. The applications examined in detail include monetary policy, business cycles, and asset prices. The section on learning and monetary policy describes the implications for optimal policy if agents use constant-gain learning, recent empirical work on inflation dynamics, and results on the stability of alternative interest-rate rules in New Keynesian (NK) models, including both Taylor-type rules and rules aiming to implement optimal policy. In examining business cycles under learning, we first consider the stability of the REE under LS learning, in the standard real business cycle (RBC) model, and then examine the stability of sunspot equilibria in RBC-type models with distortions. We then turn to the stability of sunspot equilibria in NK and in cash-in-advance (CA) models. Next, we summarize the implications of learning in models that analyze exceptional phenomena, namely hyperinflation and liquidity traps. The final section reviews some applications to asset pricing, specifically to stock-price returns and to exchange rates.
We develop the basic ideas of econometric learning using a simple linear model,
 | (2) |
Here pt is a scalar endogenous variables, wt−1 is a vector of exogenous observable variables, and nt is an unobservable random shock. The key assumption is that the expectations of economic agents, 
, are not necessarily rational, because the agents do not know all the structural parameters. Expectations are instead formed as forecasts from an estimated model and observations wt−1. The parameters of the forecasting model are estimated through use of past data and are updated over time. For simplicity, all agents are assumed to have the same expectations. (We discuss heterogeneous expectations below.)
As a benchmark, we note that for the model given by Equation 2 the unique REE is
 |
. Two well-known economic examples lead to the reduced-form model given by Equation 2.Example 1 (Lucas aggregate supply model). A simple version of the Lucas islands model, presented by Lucas (1973), consists of the aggregate supply function
 |
 |
 |
 |
Thus, mt responds to past shocks to velocity. Here 
, and ζt are white noise shocks. The reduced form of the model is of the form shown in Equation 2 with 
and 
Example 2 (Muth market model). Demand and supply functions are
 |
Assume that wt is white noise with Ewt = 0, 
. With market clearing dt = st, we obtain Equation 2 as the reduced form with 
, 
, 
, and 
. Note that α<0 for 
.
2.1.1. Econometric learning. We now develop the formal details of econometric learning. There are two key building blocks to learning. First, agents' beliefs are described by means of a forecasting model. Agents are assumed to use a perceived law of motion (PLM),
 |
Second, we need to describe how agents obtain estimates for the parameters in the PLM. It is postulated that agents use the most popular estimation method, LS. Thus, agents estimate a and b by recursive least squares (RLS) from past data 
and they forecast using the estimated model
 |
Here at−1 and 
denote the estimated parameter values obtained using data up to the period t − 1. Given the forecasts, the economy attains a temporary equilibrium in period t. Alternatively, defining 
and 
, the actual law of motion (ALM),
 |
Formally, RLS estimation is given by
 | (3) |
 | (4) |
Making the shift 
and defining 
, RLS formally becomes a stochastic recursive algorithm (SRA), as was first shown by Marcet & Sargent (1989). There are general methods for analyzing the dynamics of SRAs, which we outline in Section 2.2. In particular, conditions for convergence of SRA are given by local stability conditions of an associated ordinary differential equation (ODE). For the RLS algorithm, the ODE takes the forms
 | (5) |
 | (6) |
Some papers in the literature employ the stochastic gradient (also known as least mean squares) algorithm in place of LS. In the current setting, the gradient algorithm and its associated ODE take the form
 |
Generalized stochastic gradient (GSG) algorithms are well motivated when agents allow for parameter drift or model uncertainty. See Evans et al. (2009) for a discussion of GSG algorithms and references to the literature.
2.1.2. Expectational stability. Inspecting the differential Equations 5 and 6, we observe that 
in the second equation. Thus, the local stability of the fixed point for the whole ODE is determined by local stability under the “small” ODE,
 | (7) |
Note that the unique REE for model 2 is the fixed point of the system (Equation 7). We say that a fixed point 
is expectationally stable (E-stable) if it is locally stable under the small ODE (Equation 7), as defined in Evans (1989) and Evans & Honkapohja (1992). In economic terms, the small ODE is partial adjustment in virtual time τ. The relationship between RLS learning and E-stability is highlighted by the following result.
Proposition: The economy converges to the REE under RLS learning iff the REE is E-stable. The latter occurs iff α < 1.
For further details of the outlined steps and a proof of the result for the model 2, see chapter 2 of Evans & Honkapohja (2001). The result that E-stability of REE gives the conditions for (local) convergence of RLS and related learning schemes is quite general, and it holds for a wide variety of models, as discussed in Evans & Honkapohja (2001).
Demonstrations of convergence of RLS learning, and additional approximation results, are available through stochastic approximation techniques.
2.2.1. Decreasing-gain algorithms. A general form of SRA is given by
 | (8) |
Although in our example Xt follows an exogenous process, this is not at all essential. In particular, Xt can be permitted to follow a vector autoregression (VAR) with parameters that may depend on θt−1.The stochastic approximation approach associates an ODE with the SRA,
 | (9) |
 | (10) |
is the stochastic process for Xt obtained by holding θt−1 at the fixed value θt−1 = θ [thus, 
if Xt does not depend on θt−1], and E denotes the expectation of 
, for θ fixed, taken over the invariant distribution of the stochastic process 
For the RLS algorithm (Equations 3 and 4), with notation 
the associated ODE is
 |
The stochastic approximation results show that the behavior of the SRA is well approximated by the behavior of the associated ODE for large t. In particular, possible limit points of the SRA correspond to locally stable equilibria of the ODE. We have the following results:
Under suitable assumptions, if 
is a locally stable equilibrium point of the ODE, then 
is a possible point of convergence of the SRA. If 
is not a locally stable equilibrium point of the ODE, then 
is not a possible point of convergence of the SRA; that is, 
with probability zero.
The precise theorems are complex in detail. First, there are various ways to formalize the positive convergence result (when 
is a locally stable equilibrium point). In certain cases, when there is a unique solution and under the SRA the ODE is globally stable, 
with probability one from any starting point. When there are multiple equilibria, such a strong result will not be possible. If one artificially constrains θt to an appropriate neighborhood of a locally stable equilibrium 
(using a so-called projection facility), one can still obtain convergence with probability one. Other versions of local stability results are also available.
Second, a careful statement is required of the technical assumptions under which the convergence conditions obtain. There are three broad classes of assumptions: (a) regularity assumptions on Q, (b) conditions on the rate at which 
and (c) assumptions on the properties of the stochastic process followed by Xt. For details of assumptions and precise statements, see part II of Evans & Honkapohja (2001).
2.2.2. Constant-gain least-squares mean dynamics and escape dynamics. Under constant gain, we replace γt in Equation 8 by a constant 
. For example, in the RLS equations for φt and Rt given above, t−1 is replaced by a small constant γ. We rewrite the SRA as
 |
. Several types of results are available in this setting. See, in particular, chapter 7 of Evans & Honkapohja (2001), Cho et al. (2002), and Williams (2009).First, one can obtain the mean dynamics of 
for the limiting case of small gains, that is, for γ > 0 sufficiently small. Defining h(θ) as in Equation 10, we consider the solution to the associated ODE (Equation 9). Let 
denote the solution to Equation 9 for the initial condition 
. We often refer to τ as notional time. Consider a fixed notional time 
and a fixed compact set 
. Assume that 
for all 
. The main result is that the solution 
approximates the mean dynamics of θt over 
. Define
 |
Thus, 
is a continuous-time interpolation of the realization 
of the SRA. It can be shown that as γ → 0, the normalized random variables 
over 
converge weakly to the solution U(τ) of the stochastic differential equation
 |
. The solution is described in Evans & Honkapohja (2001), chapter 7.4. In particular, the solution satisfies 
. It follows that for 
, the mean dynamics of the SRA for γ > 0 small can be approximated via 
. Specifically, the mean dynamics of the SRA satisfy  |
. The solution to the stochastic differential equation in 
can also be used to approximate 
for small γ, and this approximation often takes a simple form as γt becomes large.The other class of results that are available for constant-gain algorithms are so-called escape dynamics based on large deviation theory. Although as γ → 0 the solution to the above stochastic differential equation provides a good approximation to the distribution of 
, with positive probability there will be large deviations from the mean dynamics, and over long stretches of time these unusual escape dynamics may be of considerable practical interest, as argued in Sargent (1999), Cho et al. (2002), and Williams (2009). Through use of large deviation tools, it is possible to compute the direction of the paths that are most likely to leave a specified neighborhood of the mean dynamics and thus provide useful information on these escape dynamics.3
Thus far, we have considered linear or linearized models. One attractive feature of linear models is that for them the class of possible REE can be described explicitly. For some models, such as model 2, a unique stationary solution exists. The economy or model is then said to be determinate. If there are multiple nonexplosive solutions, the model is said to be indeterminate.
Consider the simple forward-looking linear model
 | (11) |
and indeterminate if 
. The fundamental REE is 
. Writing 
, where under RE 
, substituting into Equation 11, and rearranging we get  |
Thus, for any stationary 
this defines a stationary stochastic process for 
when 
.
Next, consider the nonlinear forward-looking model
 | (12) |
A steady state 
is defined by the equation 
, whereas a deterministic k-period cycle 
is defined by the equations 
for 
and 
. A stationary sunspot equilibrium (SSE) is a stochastic REE in which agents' expectations depend on an extraneous random variable that has no fundamental significance for the economy. A widely discussed case of SSEs takes the form of a finite Markov chain. Consider a finite Markov chain st that can take values 
with transition probabilities πij, where 
. A k − tuple 
is an SSE with transition probabilities πij if
 |
Figures 1 and 2 illustrate these different types of REE in the nonlinear model (Equation 12).
 |
Figure 1 Two-cycle and steady state. |
 |
Figure 2 Stationary sunspot equilibrium (SSE) and multiple steady states. |
Global and local determinacy and indeterminacy for the nonlinear model are discussed in, for instance, Chiappori & Guesnerie (1991). We use model 12 to discuss learning as a selection criterion when there are multiple REE.
Evans & Honkapohja (1995) develop E-stability conditions for steady states and cycles and show that the relationship between convergence of LS learning and E-stability continues to hold for models such as Equation 12.5 For sunspot equilibria, the first result concerning convergence of learning to SSE was obtained by Woodford (1990) in the context of a specific model. SSEs near deterministic equilibria are of special interest in many contexts. Local stability results for models of type 12 were developed by Evans & Honkapohja (1994) and Evans & Honkapohja (2003b). They showed that
| 1. | E-stable SSEs exist near a pair of distinct steady states iff both steady states are E-stable, | ||||
| 2. | E-stable SSEs exist near an E-stable deterministic cycle, and | ||||
| 3. | E-stable SSEs exist near a single steady state iff | ||||
These results help to select among REE when multiple equilibria exist. Learning can therefore be a useful selection criterion, even if it does not necessarily select a unique solution.
A variety of selection results also exist for linear models (see, for instance, Evans & Honkapohja 1992 and part III of Evans & Honkapohja 2001). For the basic forward-looking model (Equation 11), it is easy to check that the E-stability condition is α < 1. It follows that determinacy is sufficient but not necessary for E-stability of the fundamental solution. This result has been significantly generalized by McCallum (2007). However, it should be noted that the result depends on the timing of information (see Bullard & Eusepi 2008).
We close this section with a brief general discussion of heterogeneous expectations and dynamic predictor selection.
2.4.1. Heterogeneous expectations. The preceding discussion has assumed homogeneous expectations for analytical convenience. In practice, heterogeneous expectations can be a major concern. In some models the presence of heterogeneous expectations does not have major effects on stability conditions, as first suggested by Evans & Honkapohja (1996) and substantially generalized by Giannitsarou (2003). However, Honkapohja & Mitra (2006) showed that interaction of structural and expectational heterogeneity can make the conditions for convergence of learning significantly more stringent than those obtained under homogeneous expectations.
Consider a forward-looking model with S types of agents,
 |
 |
 |
Define 
and 
, 
Agents are assumed to have PLMs
 |
The resulting ALM takes the form
 |
Consider mixed RLS/SG learning in which initial conditions, learning rules and gains can vary across agent types. Specifically, agent types 
use RLS, and types 
use SG learning rules, employing the algorithms given in Section 2.1.1 but with t−1 replaced by type-specific gain sequences γi,t. The γi,t are assumed to satisfy 
, where the decreasing and positive sequence γt satisfies natural conditions (see Honkapohja & Mitra 2006).
In the case of mixed RLS/SG learning, stability is determined by
 |
The generalized E-stability condition is stricter than usual E-stability (or SG-stability), and heterogeneous speeds of learning δi can affect convergence. However, when k = 1, if (a) the aggregate economy is E-stable, (b) the parameters Ai have the same sign, and (c) all agents use either RLS or SG learning rules, then the economy converges to the MSV REE for all 
and Mz.2.4.2. Dynamic predictor selection. Another natural way to introduce heterogeneity is to assume that agents choose from a finite set of forecasting models, with the proportions of agents using the different models at each point in time determined endogenously. Brock & Hommes (1997) postulate that each predictor has a fitness measure associated with it, based on past performance and on the cost of using that predictor, and that relative fitness determines the proportion of agents selecting each predictor.
Brock & Hommes (1997) study the resulting adaptively rational expectations dynamics for the standard “cobweb” model with two predictors: rational and naive. Because the model is nonstochastic, RE is equivalent to perfect foresight. Demand is assumed to be linear: 
. Firms have a quadratic cost function 
with supply curve 
. The perfect foresight predictor 
costs C ≥ 0, whereas the naive predictor 
is free. Letting n1t and n2t denote the proportion of agents using perfect foresight and naive predictors, respectively, market equilibrium at t is given by
 |
The main fitness measure examined is realized profit last period, less predictor cost. Because profits are 
, fitness at t is
 |
The proportions of agents using the two predictors are given by the multinomial logit ratios
 |
Here β measures the intensity with which agents choose predictors with higher fitness. For 
, agents choose only the fittest predictors.
These equations define the adaptively rational equilibrium dynamics. The system has a unique steady state. Brock & Hommes (1997) focus on the case b/B > 1, in which the model is locally unstable under naive expectations. If C > 0, the dynamics depend crucially on β. Above a critical value β1, the steady state is an unstable saddle point. Stable two-cycle and higher-order cycles, the coexistence of low periodic attractors, and chaotic attractors appear as β increases. Intuitively, near the steady state it pays to use the cheap predictor. Doing so pushes the economy away from the steady state, where agents then use the costly, stabilizing predictor. For high enough intensity of choice β, this tension leads to local instability and complex global dynamics.
The dynamic selector framework has been extended to incorporate stochastic features and econometric learning. Branch & Evans (2006) consider a stochastic cobweb model, driven by two exogenous shocks, with agents choosing between two equally costly models, each of which omits one of the variables. They formulate the notion of a misspecification equilibrium (ME). In contrast to the REE, there are cases of intrinsic heterogeneity, in which both predictors are used, even with 
. The stability of this ME under real-time learning and dynamic predictor selection is also examined. Branch & Evans (2007) consider a Lucas-type monetary model with positive expectational feedback. This setting can have two ME, in which agents coordinate on either of the two models. Inflation and output can then exhibit regime switching or parameter drift, in line with much macroeconometric evidence.
Analysis of monetary policy rules from the learning viewpoint has recently become a very popular research topic.6 We now discuss some key aspects of this rapidly growing literature.
3.1.1. New classical model with constant gain. Orphanides & Williams (2005b) (OW) use a simple two-equation macro model to show that constant-gain learning by private agents has major implications for economic policy. Their model is based on a New Classical expectations-augmented Phillips curve with inertia,
 | (13) |
is the rate of inflation over this period expected at time t, yt+1 is the level of the output gap in t + 1, and et+1 is a white noise inflation shock. 
represents intrinsic inflation persistence. We assume 
.The other equation is an aggregate demand relation that embodies a lagged policy effect,
 |
Policy makers have a target inflation rate π*. Their instrument is xt, and they are assumed to follow a rule of the form
 | (14) |
The policy makers aim to choose θ optimally given their loss function
 |
parameterizing the relative weight on inflation versus output stabilization. Under RE, the optimal choice takes the form 
where θP is increasing in its arguments, and inflation follows a first-order autoregressive [AR(1)] process. Under LS learning, private agents estimate the PLM  |
. The REE can be shown to be E-stable, so under decreasing gain, LS learning would converge to the REE.With constant-gain LS (which OW term perpetual learning), estimates 
no longer fully converge to the REE, but instead to a stochastic process. If the gain parameter κ > 0 is very small, then estimators will be close to the REE values for most of the time with high probability, and output and inflation will be near their REE paths. Nonetheless, small plausible values such as κ = 0.05 can lead to very different outcomes in the calibrations OW consider. Using simulations, OW find that (a) the standard deviations of 
and 
are large even though forecast performance remains good, (b) there is a substantial increase in the persistence of inflation, compared to the REE, as measured by the AR(1) coefficient for πt, and (c) the policy trade-off between standard deviations σπ and σy shifts out substantially and sometimes in a nonmonotonic way.
Under perpetual learning by private agents, if policy makers keep to the same class of rules, then they should choose a different θ than under RE. One key finding is that the naive policy choice θ = θP can be strictly inefficient when agents are learning. In general, policy should be more hawkish; that is, under learning the monetary authorities should pick θ > θP.7 Finally, following a sequence of unanticipated inflation shocks, inflation “doves” can do very poorly, with expectations deviating substantially from RE. The intuition for these results is that a more hawkish policy helps to keep inflation expectations 
in line, specifically, closer to RE values.
3.1.2. The rise and fall of inflation. Several recent papers have argued that the learning approach plays a central role in the historical explanation of the rise and fall of U.S. inflation over the 1960–1990 period. Sargent (1999) and Cho et al. (2002) emphasize the role of policy maker learning. They argue that if monetary policy makers attempt to implement optimal policy while estimating and updating the coefficients of a misspecified Phillips curve, there will be both periods of inefficiently high inflation and occasional escapes to low inflation. Sargent et al. (2006) estimate a version of this model. They find that shocks in the 1970s led the monetary authority to perceive a trade-off between inflation and unemployment, leading to high inflation, and that subsequent changed beliefs about this trade-off account for the conquest of U.S. inflation during the Volker period.
Primiceri (2006) makes a related argument, emphasizing both (a) policy maker learning about both the Phillips curve parameters and the aggregate demand relationship, and (b) uncertainty about the unobserved natural rate of unemployment, 
. The great inflation of the 1970s initially resulted from a combination of underestimates of both 
and the persistence of inflation. This also led policy makers to underestimate the impact of unemployment on inflation until estimates of the perceived trade-off between inflation and unemployment changed during the Volker period.
Other empirical accounts of the period that emphasize learning include (a) Bullard & Eusepi (2005), which examines the implications of policy maker learning about the growth rate of potential output, (b) Orphanides & Williams (2005a), which underscores both private-agent learning and policy maker misestimates of the natural rate of unemployment, and (c) Cogley & Sargent (2005), which develops a historical account of inflation policy emphasizing Bayesian model averaging and learning by policy makers uncertain about the true economic model.
3.1.3. New Keynesian models and policy rules. The NK model is currently the most widely used vehicle for studying monetary policy. The NK model is a dynamic stochastic general equilibrium model with a representative consumer and price rigidity modeled using monopolistic competition with constraints on price setting.8
We directly employ the log-linearized version of the NK model. The aggregate demand and supply curves summarize private-sector behavior. The simplest version of the NK model takes the form
 | (15) |
 | (16) |
allows for nonrational expectations. The aggregate demand (IS) curve (Equation 15) is obtained by log-linearizing the consumer's Euler equation and employing the goods market-clearing condition, so that the equation is expressed in terms of the output gap. The aggregate supply (AS or NK Phillips) curve (Equation 16) is derived as a linearization of the firms' optimality condition under the price-setting constraint.The model is completed by specifying an interest-rate rule for monetary policy of, for instance, the contemporaneous or forward-looking Taylor-type form proposed by Taylor (1993):
 |
The form of the policy rules affects the determinacy and learnability properties of the NK model. Multiplicity of equilibria or expectational instability of equilibrium under learning means that there can be undesirable fluctuations in the economy. To avoid this possibility, good policy should focus on interest-rate rules that deliver stability under learning and determinacy.
For Taylor rules, Bullard & Mitra (2002) showed the following.
| 1. | The standard Taylor rule | ||||
| 2. | The forward-looking rule | ||||
Taylor rules do not usually describe optimal policy in the NK model. Optimal monetary policy under learning has been considered by Evans & Honkapohja (2003c) under discretion and Evans & Honkapohja (2006) under commitment (using the timeless perspective described in Woodford 2003). The FOC for the timeless-perspective optimum (often termed the targeting rule) is
 | (17) |
There are different ways for attempting to implement optimal monetary policy. One natural formulation is to solve Equations 16 and 17 for the optimal REE for 
and to insert this solution into Equation 15 to obtain a fundamentals-based rule of the form 
. (There are also hybrid rules in which it responds to deviations from the targeting rule.) Another implementation solves Equations 15–17, given expectations, for an expectations-based rule of the form 
, where RE has not been imposed on private-sector expectations. We have the following.
Proposition: Optimal rules based only on fundamentals lead to E-instability and (often) to indeterminacy. Optimal expectations-based rules deliver both E-stability and determinacy.
This proposition is based on the formulation presented in Equations 15 and 16, which presumes that agents make decisions based on their subjective one-step-ahead Euler equations.9 Under boundedly rational learning, there are alternative approaches to modeling individual agents' behavior for infinitely lived households.10 In chapter 10 of Evans & Honkapohja (2001), the learning framework was kept close to the RE reduced-form setup, with agents' decisions formulated directly using their Euler equations. An alternative approach assumes that agents base current decisions on their forecasts of the infinite sequence of prices and exogenous variables relevant to their decisions. Preston (2005, 2006) analyzes modifications to the preceding analysis when it is assumed that agents use the latter approach.
Our discussion in this section covers only the beginning of what is already a large and growing literature. Further aspects of policy design in the NK model include lack of observability of expectations and other variables, imperfect knowledge of structural parameters for optimal policy rules, implications of constant-gain learning, and extensions of the NK model to open economies, models with capital, and supply-side channels of monetary policy transmission (see Evans & Honkapohja 2009 for a discussion of these and other topics, with references).
3.2.1. The basic real business cycle model. RBC models have been widely discussed since the 1980s (e.g., see Cooley 1995). The basic RBC model describes an infinite-horizon, representative-agent economy with flexible prices and perfect competition. In such economies the competitive equilibrium is Pareto efficient, so that computing the equilibrium can be done by solving the corresponding planning problem.
The social planner maximizes
 |
 |
Here Ct and Kt denote consumption and capital, respectively, St is a productivity shock, and Vt is an i.i.d. innovation with mean one. The log of St thus follows an AR(1) process and ρ captures the persistence of the technology shocks.
This planning problem does not have an explicit solution, but the dynamics of the economy can be described through the use of a linearization around a nonstochastic steady state. (See section 10.4 of Evans & Honkapohja 2001 for formal details.) Defining detrended variables 
etc., the first-order optimality conditions are transformed to equations with asymptotically stationary variables and a unique steady state. Log-linearizing around the steady state, defining the variables 
, 
, 
and 
and introducing vector notation 
give the model the standard form
 | (18) |
It is well known that the basic RBC model is determinate (saddle-point stable) and that the unique solution has the VAR(1) form
 | (19) |
. To check the E-stability of this solution one treats Equation 19 with general values for a, b, c as the PLM. The ALM can then be computed in a standard way. It is possible to develop general E-stability conditions for models of the form of Equation 18 and to check their validity numerically. For the Farmer (1999) parameter values, the RE solution is E-stable. The analysis of RLS learning can be done in a standard way, provided the dimension of shocks is increased to avoid an exact linear relationship among contemporaneous consumption, capital, and the technology shock.3.2.2. Applications and extensions of the real business cycle model. Williams (2004) explores further features of the RBC model dynamics under learning. Using simulations, he shows that the dynamics under RE and learning are not very different unless agents need to estimate structural aspects as well as the reduced-form PLM parameters. Huang et al. (2008) focus on the role of misspecified beliefs and suggest that these can substantially amplify the fluctuations due to technology shocks in the standard RBC model.11
Other papers on learning and business cycle dynamics include Van Nieuwerburgh & Veldkamp (2006) and Eusepi & Preston (2008). The former formulates a model of Bayesian learning about productivity and suggests that the resulting model can explain the sharp downturns that are an empirical characteristic of business cycles. The latter paper introduces the notion of infinite-horizon decision rules to RBC models and argues that variants of the model under learning can resolve some of the empirical difficulties of RE models with business cycles.
Giannitsarou (2006) extends the basic RBC model to include government spending financed by capital and labor taxes. Her objective is to compare the transitional dynamics of the model under RE and under RLS learning when an unanticipated reduction in the capital tax displaces the steady-state equilibrium. Under RE there is the usual saddle-path adjustment: Consumption jumps instantaneously, and the economy monotonically converges to the new steady state. In contrast, the nature of dynamics under learning depends on the nature of technology shocks near the time of tax change. With negative shocks, the adjustment shows a delayed response in economic activity, although eventually the dynamics approximate the saddle-path dynamics. In contrast, under positive technology shocks, the learning and RE adjustment paths are nearly identical. This is an important finding, as tax reductions are often carried out in bad times.
3.2.3. Sunspot fluctuations. When the RBC model is generalized to include externalities, monopolistic competition, or other distortions, it is possible for the steady state to be indeterminate, that is, to possess multiple solutions, including a dependence on sunspots, in a neighborhood of the steady state. Examples are the models of Farmer & Guo (1994), Benhabib & Farmer (1996), and Schmitt-Grohe & Uribe (1997). This line of research suggests SSEs as a possible model of the business cycle.
Are these SSEs stable under learning? This issue was initially studied in chapter 10.5 of Evans & Honkapohja (2001), where it was found that the SSEs in the Farmer-Guo calibrated model are not stable under learning. The stability of SSEs in these models was examined further in Evans & McGough (2005a) and Duffy & Xiao (2007). In general, the stability of SSEs can depend both on their parametric representations and on the precise information set available to agents. However, for this class of models both Evans & McGough (2005a) and Duffy & Xiao (2007) obtain predominantly negative results. Duffy and Xiao further argue that empirically plausible adjustment dynamics rule out stable sunspots in this class of models. Evans and McGough do find some cases of stable sunspots in the case of common-factor representations of SSEs, when the information sets of private agents include contemporaneous aggregate endogenous variables. Stable SSEs of this type arise only in small-parameter regions and are sensitive to the information set, but they do arise for some plausible calibrations of the Schmitt-Grohe & Uribe (1997) model. On balance, the existing results challenge future researchers to design versions of RBC-type models that exhibit robustly stable SSEs.
Turning to other models, we have already seen that stable SSEs (i.e., stable under learning) have been shown to exist by Woodford (1990) in a monetary overlapping-generations model. Stable SSEs have also been obtained by Howitt & McAfee (1992) in a model with search externalities and by Evans et al. (1998) in an endogenous growth model. We briefly discuss some positive results for the standard NK model, introduced in Section 3.1.3, and for a CA model.
It is well known that in NK models indeterminacy and existence of SSEs arise for some policy parameters 
. The stability of SSEs under learning is examined in Honkapohja & Mitra (2004) and Evans & McGough (2005b). Writing the model in bivariate form
 |
 |
, and noisy Markov SSEs can be represented as  |
In many cases with indeterminacy, SSEs in the NK model are not stable under learning. For example, if 
with 
, there is indeterminacy and SSEs exist, but they are never stable under learning. However, there do exist cases of stable SSEs (in some representations) for the forward-looking rule 
, with 
sufficiently large and χπ not too small. This was demonstrated for noisy finite-state Markov SSEs in Honkapohja & Mitra (2004). The result was generalized by Evans & McGough (2005b), who show that the noisy Markov SSEs are special cases of common-factor representations in which the sunspot takes an AR(1) form. That is, st can be replaced by a sunspot 
, where ɛt is an exogenous martingale difference sequence and γ satisfies a so-called resonant-frequency condition. (For finite-state Markov processes, the resonant frequency corresponds to specific transition probabilities.)
Stable SSEs can also arise in representative-agent CA models when the government deficit is at least partially financed by seigniorage. Evans et al. (2007) consider a standard representative-agent model with cash goods, credit goods, and variable labor supply. There is no capital, but agents can hold assets in the form of money or bonds, and there is a CA constraint for purchases of cash goods. Government spending gt is assumed to be an exogenous i.i.d. process, and in the simplest version of the model gt is entirely financed by seigniorage.
There are two regimes, depending on the magnitude of the elasticity of intertemporal substitution (ITS). When ITS is high, there are two steady states, with differing inflation rates. This is a CA version of the hyperinflation model, which is discussed in the next section. When ITS is low, there is a single steady state, and with sufficiently low ITS the steady state is indeterminate. Evans et al. (2007) show that in this case there are finite-state Markov sunspot equilibria that are stable under learning. These stable SSEs exhibit random variations over time in inflation and output in response to the extraneous sunspot variable. Evans et al. (2007) also examine the impact of changes in fiscal policy: A sufficient reduction in the mean of gt, or a sufficient increase in taxes will in many cases eliminate the SSEs.
3.3.1. Hyperinflations. Using the seigniorage model of inflation, Marcet & Nicolini (2003) aim to provide a unified theory to explain the empirical regularities of Latin American hyperinflation experienced by many countries in the 1980s. The model is based on the linear money demand equation
 |
 |
Rewriting this equation as 
, setting 
, and assuming dt = d, we get
 |
Under perfect foresight, there are two steady states, 
, provided d ≥ 0 is not too large. There is also a continuum of perfect foresight paths converging to βH. Some early theorists suggested that these paths might provide an explanation for actual hyperinflation episodes. The learning approach provides a different perspective.
Consider now the situation under adaptive learning. Suppose the PLM is that the inflation process is perceived to be a steady state, namely 
, where ηt is perceived white noise. For this PLM, expectations are 
all t, and the corresponding ALM is
 |
The map 
corresponds in Figure 3 to the part of 
that lies below the value βU. Under steady-state learning, agents estimate β based on past average inflation, that is,
 | (20) |
 |
Figure 3 Inflation as a function of expected inflation. |
This is a recursive algorithm for the average inflation rate, which is equivalent to LS regression on a constant.12
Stability under this learning rule is governed by the E-stability differential equation
 |
Because 
and 
, βL is E-stable, and therefore locally stable under learning, whereas βH is not. This can be seen from Figure 3.
Marcet & Nicolini (MN) extend the preceding model to an open-economy setting. They assume price flexibility with purchasing power parity (PPP), so that 
, where 
is the exogenous foreign price of goods. A CA constraint for local currency generates the money demand as in the basic model. The variable dt is assumed to be i.i.d. There are two exchange-rate regimes. In the floating regime, the economy behaves just like the closed-economy model, with PPP determining the price of foreign currency. In the exchange-rate rule (ERR) regime, the government buys or sells foreign exchange as needed to guarantee 
The government imposes ERR if the inflation rate would otherwise exceed 
, the maximum acceptable level.
MN argue that under RE the model cannot properly explain the main stylized facts of hyperinflation, and that a learning formulation is more successful. They use a variation of learning rule (Equation 20) in which t is replaced by αt, where 
if 
falls below some bound, and otherwise 
; that is, a constant gain is used. The qualitative features of the model are approximated by the system 
, where
 |
Figure 3 describes the dynamics of system. There is a stable region, consisting of values of β below the unstable high-inflation steady state βH, and an unstable region that lies above it. This gives rise to very natural recurring hyperinflation dynamics: Starting from βL, occasionally a sequence of random shocks may push βt into the unstable region, at which point the gain is revised upward to 
, and inflation follows an explosive path until it is stabilized by ERR. Then the process begins again.
The model with learning has useful policy implications. ERR is valuable as a way of ending hyperinflation if the economy enters the explosive regime. However, a higher E(dt) makes average inflation higher and the frequency of hyperinflations greater. This indicates the importance of the orthodox policy of reducing deficits as a way of minimizing the likelihood of hyperinflation paths.
3.3.2. Liquidity traps and deflationary spirals. Deflation and liquidity traps have at times been a concern. As we have seen, in contemporaneous Taylor rules, interest rates should respond to the inflation rate more than one-for-one in order to ensure determinacy and stability under learning near the target inflation rate. However, as emphasized by Benhabib et al. (2001), if one considers the interest-rate rule globally, the requirement that net nominal interest rates must be nonnegative implies that the rule must be nonlinear and also, for any continuous rule, the existence of a second steady state at a lower (possibly negative) inflation rate. This is illustrated in Figure 4. The Fisher equation R = π/β, depicted in the figure, is obtained from the usual Euler equation for consumption in a steady state. Here R stands for the interest rate factor (the net interest rate is R − 1), 
stands for the inflation factor (π − 1 is the net inflation rate), π* denotes the intended steady state, πL is the unintended steady state, and πL may correspond to either a very low positive or a negative net inflation rate, i.e., deflation. Benhabib et al. (2001) show that under RE, there is a continuum of “liquidity trap” paths that converge on πL. The pure RE analysis thus suggests a serious risk of the economy following these liquidity trap paths.
 |
Figure 4 Multiple steady states with a global Taylor rule. Shown is the interest-rate policy |
What happens under learning? Evans & Honkapohja (2005) analyzed a flexible-price perfect competition model. We showed that deflationary paths are possible, but that the real risks, under learning, were paths in which inflation slipped below πL and then continued to fall further. For this flexible-price model, we showed that this could be avoided by a switch to an aggressive money supply rule at low inflation rates.
Evans et al. (2008) reconsider the issues in an NK model with sticky prices, due to adjustment costs, and deviations of output from flexible-price levels. Monetary policy follows a global Taylor rule, as above. Fiscal policy is standard, including exogenous government purchases gt and a Ricardian tax policy that depends on real debt level. The model equations are nonlinear, and the nonlinearity in its analysis under learning is retained. The key equations are
 |
The first equation is the nonlinear NK Phillips curve, and the second equation is the IS curve. There are also money- and debt-evolution equations.
There are two stochastic steady states at πL and πH. If the random shocks are i.i.d., then steady-state learning is appropriate for both ce and πe, specifically
 |
 |
Figure 5 Expectation dynamics under normal policy. The star indicates the intended steady state, which is locally stable under learning. The dashed curve gives the boundary of the stable region, and the arrow shows an unstable path leading to deflation and stagnation. |
For the intuition, suppose that we are initially near the πL steady state and that we consider a small drop in πe. With fixed R this would lead through the IS curve to lower c and thus, through the Phillips curve, to lower π. Because only small reductions in R are possible given the global Taylor rule, the reduction in c and π cannot be offset. The falls in realized c and π lead, under learning, to reductions in ce and πe, and this sets in motion the deflationary spiral.
Thus, large adverse shocks to expectations or structural changes can set in motion unstable downward paths. Can policy be altered to avoid deflationary spiral? Evans et al. (2008) show that it can. The recommended policy is to set a minimum inflation threshold 
, where 
. The authorities would follow normal monetary and fiscal policy, provided that it delivers 
. However, if πt threatens to fall below 
, then aggressive policies would be implemented to ensure that 
: Interest rates would be reduced, if necessary to near the zero lower bound R = 1, and if this is not sufficient, then government purchases gt would be increased as required. It can be shown that these policies can indeed ensure 
always under learning and that they can lead to global stability of the intended steady state at φ*. Perhaps surprisingly, it is essential to have an inflation threshold. Use of an output threshold to trigger aggressive polices will not always avoid deflationary spirals.
Asset pricing is another area of recent focus in the learning literature. The potential for adaptive learning to generate new phenomena for asset prices was already apparent in the early work of Timmermann (1993, 1996). Consider the standard risk-neutral asset-pricing framework
 | (21) |
is the discount factor, assumed constant. Assume also that dt is an exogenous stochastic process, such as  | (22) |
and 
Under RE the fundamentals solution is  | (23) |
Iterating Equation 21 forward, one obtains the present value formula
 | (24) |
There are a number of empirical puzzles in asset pricing based on this model, including excess volatility of stock prices and predictability of stock returns. A potentially simple and appealing explanation is that traders do not have a priori knowledge of the parameters of the 
process. The equilibrium price process under learning is generated, as usual, assuming that the parameter estimates are updated over time as new data become available.
There are two natural ways to model stock prices under learning, depending on whether we want to treat Equation 24 or 21 as the key equation that determines pt, given expectations. If traders are “fundamentalists,” then price will be set in accordance with Equation 24, based on forecasts 
where 
and 
are the time t estimates of μ and σ2, respectively. Such price setting leads to
 |
This approach is investigated by Timmermann (1993, 1996) under the name present-value learning.
An alternative approach is to assume that pt is determined by the expected rate of return over the coming period, in accordance with Equation 21. Traders would then also estimate a model for pt, e.g., 
Using estimates at, λt, 
, and 
, traders form forecasts 
and 
, with pt determined by Equation 21. This self-referential learning approach was also studied by Timmermann (1996).
More recent work on learning and stock prices has extended the basic framework in several directions. Brock & Hommes (1998) introduce heterogeneous expectations using the dynamic predictor selection methodology discussed above. Branch & Evans (2008) and Adam et al. (2008) focus on self-referential learning.
Adam et al. (2008) use the consumption-based version of model 21. Forecasts are given by 
, where bt is updated according to
 |
Branch & Evans (2008) examine learning within a mean-variance linear portfolio model with a risk-free asset-paying fixed rate of return 
and a risky stock. The supply of the stock is exogenous and random. Demand depends positively on expected excess returns but negatively on risk, that is, expected conditional variance of returns. Equating supply and demand yields
 |
is the time t estimate of the conditional variance. The dividend process is assumed exogenous and known, and forecasts 
are set at the true conditional expectation. The variable 
is generated from estimates of the price process  |
There is a fundamental REE with fixed parameters 
, which is stable under LS learning. Interesting dynamics arise if agents instead estimate the parameters 
using constant-gain LS learning. There are occasional “escapes” to nonfundamental random-walk behavior of asset prices, in which estimates are close to 
. In this regime, there is bubblelike behavior in pt. However, subsequent revisions in risk estimates eventually lead to crashes back to fundamentals values. Thus, learning about both returns and risk can lead to recurrent bubbles and crashes.
Exchange rate dynamics also exhibit a number of puzzles that learning models may resolve. Chakraborty & Evans (2008) focus on the forward-premium puzzle, arguing that this anomaly can be explained by learning. Let st be the log of the price of foreign currency. The reduced form model for st is
 |
The REE is 
, where 
. Agents estimate the parameter b by constant-gain LS and make forecasts 
, where bt is their time t estimate of b. For small gains, bt converges to a stochastic process distributed in a small neighborhood of 
. Surprisingly, this implies a strong downward bias in the coefficients of the usual forward-premium regression for the realistic case of ρ near one. This can explain the small or even negative empirical estimates of the coefficient.Another, potentially complementary, approach to exchange-rate modeling is based on dynamic predictor selection (see De Grauwe & Grimaldi 2006). Further applications of learning to exchange rates include Kasa (2004), Kim (2008), Mark (2007), and Markiewicz (2008).
The adaptive learning approach to macroeconomics treats economic agents—firms, households, and policy makers—as econometricians when modeling how they make forecasts. Macroeconomic models under learning, in which estimated forecast rules are updated in accordance with LS or other statistical rules, can be analyzed with stochastic approximation techniques, and E-stability provides a key tool for analyzing the dynamics under LS learning. Applications of learning in macroeconomics have expanded rapidly in recent years, and the learning approach has provided novel insights into central issues of monetary policy, business cycles, and asset pricing.
There are a number of current research areas that, because of space constraints, our review has not covered, but which are likely to become more prominent. Empirical work on forecasts, based on survey data, experiments, and indirect measures from asset markets, will help to assess the alternative models of learning and expectations formation (for recent papers, see Branch 2004, Adam 2007, Pfajfar & Santoro 2007). In addition to the empirical topics covered in our review, there are applications of learning to several other areas. One new example is dynamic stochastic general equilibrium models (see Milani 2007, Slobodyan & Wouters 2007). Alternatives to econometric approaches to learning include those based on genetic algorithm learning and evolutionary dynamics (e.g., see Arifovic 2000, Georges & Wallace 2007).
There are also a number of unresolved conceptual questions that merit further investigation. One set of issues revolves around heterogeneous expectations, model selection, and Bayesian model averaging. Heterogeneity in forecasting and multiple models are clearly evident in reality, but this contrasts with Bayesian approaches that suggest eventual convergence to a single model. Another major area concerns the use of structural information and forward-looking reasoning in forecasting by private agents. The econometric approach to learning currently emphasizes forecasting using reduced-form models, but in many cases it would be natural for agents to incorporate structural knowledge into their forecasting.
Figure 1 Two-cycle and steady state.
...Figures 1 and 2 illustrate these different types of REE in the nonlinear model (Equation 12)....
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