Social Learning and Monetary Policy at the Effective Lower Bound

This paper develops a model that jointly accounts for the missing disinflation in the wake of the Great Recession and the subsequently observed inflation-less recovery. The key mechanism works through heterogeneous expectations that may durably lose their anchorage to the central bank (CB)'s target and coordinate on particularly persistent below-target paths. The welfare cost associated with persistent low inflation may be reduced if the CB announces to the agents its target or its own inflation forecasts, as communication helps coordinate expectations. However, the CB may lose its credibility whenever its announcements become decoupled from actual inflation.


Introduction
The Great Recession in the US and Europe and the ensuing monetary policy reactions have given way to a 'new normal' in economic conditions: interest rates have remained at historically low levels. This situation is particularly acute in Europe, where interest rates have remained at the effective lower bound (ELB) ever since. Yet, no substantial changes in the price levels have been recorded, neither in the wake of the downturn -despite the severity of the recession -nor along the recent output growth episode, which then resembles an inflation-less recovery. Meanwhile, inflation expectations have remained consistently below target, as depicted in Figure 1, which puts at risk the long-run anchorage of expectations. Low inflationary pressures pushed a number of major central banks (CBs) to further ease monetary policy, before the COVID-19 pandemic brought interest rates further down to their ELB. This low-inflation narrative is hard to unfold within the standard macroeconomic model -namely the New Keynesian (NK) class of models -for at least two reasons. First, zero interest rates generate implausible macroeconomic dynamics in those models. Under rational expectations (RE), the dynamics are indeterminate at the ELB (Benhabib et al. 2001), which implies excess volatility in inflation that is clearly at odds with the recent experience. This puzzle is clearly visible from survey data, which have been lying in the indeterminacy region of the inflation-output state space since the financial crisis, as depicted in Figure 2. Replacing RE by boundedly rational and learning agents induces diverging deflationary spirals at the ELB, which does not match the current situation either (Evans et al. 2008  of exogenous floors on deflation-motivated e.g. by a subsistence production level ensured by the government (Evans et al. 2020).
Second, the assumption of complete information and common beliefs leaves little room for expectations to be persistently off the target and play any autonomous role in driving business cycles. In those models, recessive episodes are typically generated by exogenous and persistent technology or financial shocks. 1 Not only does this conception of expectations conflict with the empirical evidence of unanchored and dispersed forecasts, 2 but it also does not leave any room for monetary policy to influence or coordinate heterogeneous private expectations through communication.
1 There are some recent exceptions, e.g. Angeletos et al. (2018), who investigate the role of strategic uncertainty in the presence of heterogeneous information within a general equilibrium model. However, those authors use a real business cycle (RBC) model, which implies that monetary policy is left out.
2 See, inter alia, Mankiw et al. (2003) in survey data from professional forecasters; and Branch (2004) from households. Coibion et al. (2019) show that more than half of the surveyed firms and households do not know the value of the Fed inflation target. One-yearahead households' inflation expectations are on average 1.5 percentage points (p.p.) above the target, and the cross-sectional dispersion reaches up to 3 p.p. (Coibion et al. 2020). 2008Q1-now (Since the financial crisis) Indeterminacy Notes: The shaded area denotes the region of the state space that violates the Blanchard-Kahn determinacy condition under RE and leads to diverging deflationary spirals under recursive learning. The white area denotes the determinate region of the state space, which is also the basin of attraction of the target under recursive learning. Data on expectations are taken from the SPF. The output gap is computed using a linear trend. Calibration of the NK model is taken from Galí (2015). Therefore, the main contribution of this paper is to address these challenges by developing a model in which time-varying heterogeneity in expectations endogenously produces ELB dynamics so as to account for the recent economic experience. The use of heterogeneity and learning in agents' forecasts is not anecdotal given the above-mentioned large literature documenting pervasive heterogeneity in real-world expectations.
We develop a micro-founded NK model featuring inflation and output dynamics to which we add a parsimonious evolutionary learning process that specifies the dynamics of expectations and nests the RE homogeneous agent benchmark. This latter feature, together with the sole use of white noise fundamental shocks, isolates learning as the only source of persistence in the endogenous variables and allows us to identify the amplifier role of expectations in driving business cycles. In our model, agents use steady-state learning, i.e. they form beliefs about the long-run values of inflation and output, which easily translates into the issue of expectation anchorage.
Specifically, we choose a social learning (SL) process. Our choice is motivated by the parsimony of this class of learning models, their ability to match experimental findings and the evolutionary role of heterogeneity in the adaptation of the agents. In these models, agents collectively adapt to an everchanging environment in which their own expectations contribute to shape the macroeconomic variables that they are trying to forecast. This feature is well suited to self-referential economic systems such as standard macroeconomic models. SL expectations also find an intuitive interpretation that is reminiscent of the idea of epidemiological expectations where 'expert forecasts' only gradually diffuse across the entire population (Carroll 2003).
In a novel effort within the related literature, 3 we take our stylized model to the data and show that it is able to jointly replicate ten salient business cycle moments from the Survey of Professional Forecasters (SPF) and the main US macroeconomic time series, including the frequency of ELB episodes, major dimensions of heterogeneity in expectations and a substantial share of the persistence in output and inflation data. This empirical exercise is already a remarkable result given the parsimony of the model. Our empirical exercise adds to the literature (i) an estimation routine of a non-linear model under heterogeneous expectations and (ii) an empirical discipline device to learning models by offering estimated values to the learning parameters for which there 3 Del Negro & Eusepi (2011) attempt to replicate expectation data with RE models. Milani (2007) fits an adaptive learning NK model to macroeconomic time series only. Closer to our contribution, Slobodyan & Wouters (2012a,b) estimate an NK model on both macroeconomic and expectation times series. However, the authors use exogenous autocorrelated shocks on expectations to reproduce the observed persistence in the data.

are no observable counterparts. 4
A second major contribution is to show that our model endogenously produces stable dynamics at the ELB. Those stable dynamics correspond to inflation-less recoveries as recently experienced, i.e. inflation persists for an extended period of time below its target, the ELB binds, but output expands.Hence, our simple framework jointly accounts for the missing deflation in the wake of the crisis and the missing inflation in the wake of the recovery.
In our model, recessive dynamics arise endogenously when agents coordinate on pessimistic expectations following a series of adverse fundamental shocks.
From there, the transition back to the target can be particularly long if expectations have become unanchored and, per their self-fulfilling nature, nurture the bust. Hence, we offer a reading of the recent economic experience as a long-lasting coordination of agents on pessimistic expectations rather than as the result of persistent and exogenous shocks. Given that our model nests the RE homogeneous-agent benchmark, we interpret the dispersion of expectations as a friction and quantify the ensuing welfare loss with respect to the RE outcome. We find that heterogeneous expectations entail a consumption loss of almost 3.3% with respect to the RE allocation. From there, a natural follow-up analysis is to introduce an additional monetary policy instrument, namely CB communication, and investigate whether it may offset the costs of forecast dispersion. To address 4 Hence, our empirical exercise does not aim to compare the matching abilities of the SL model regarding macroeconomic time series with those of an RE counterpart. For a meaningful comparison, the SL model would need to compete with an RE version of the model with sunspot dynamics at the ELB. While certainly an interesting exercise, it is beyond the scope of the present paper. this question, we exploit the flexibility of the SL model, which enables us to integrate CB communication into the learning process of the agents. From two simple communication examples, we show the critical role of a strong credibility for the CB's announcements to reshape expectations. The CB may lose credibility whenever the announcements become decoupled from the actual realizations of inflation. Moreover, accurate but below-target inflation forecasts may turn self-defeating. In light of these observations, we discuss recent policy debates, such as the forward-guidance puzzle or the adoption of (temporary) higher inflation targets.  Aruoba et al. 2017, Arifovic et al. 2018, Lansing 2020. However, the coordination mechanism generating liquidity traps in our model is fundamentally different from the one used in the above-cited contributions.

Related literature
In the context of our model, agents never coordinate on the low-inflation steady state, nor do they contemplate the possibility of a regime switching between the two steady states. The target is the only stable equilibrium under SL and expectations always remain within its basin of attraction, which is shown to be larger under SL than the determinacy region under RE. As a result of a series of adverse fundamental shocks, SL agents may 'pick up' a downward trend and their expectations may travel to regions of that basin from where inflation and output gaps stagnate below their target and the convergence back to target takes a very long time. This is because in those regions of the state space, the ZLB binds and the pessimistic expectations are self-defeating per the self-fulfilling nature of the expectations in the NK model.
In particular, while we borrow from Arifovic et al. (2013Arifovic et al. ( , 2018 a similar SL mechanism to model expectations, our work differs substantially. Importantly, those two theoretical contributions study the asymptotic stability of the NK model under SL, while we focus on the short-term fluctuations arising from the interplay between fundamental shocks and learning dynamics and their empirical performances. Arifovic et al. (2018) interpret liquidity trap episodes as the coordination of expectations on the low inflation steady-state that is stable under their learning mechanism. By contrast, our agents have a finite memory and our empirical calibration differs from theirs, which does not allow us to generalize their result to our setup. In fact, in our model, the low-inflation state is unstable under SL as it belongs to the basin of attraction of the target: if expectations shift on the low-inflation state, they will eventually converge back to the target, but after a considerable amount of time.
The rest of the paper proceeds as follows. In Section 2, we develop the model; the estimation is presented in Section 3; the dynamic properties of the model are analyzed in Section 4; Section 5 discusses CB communication; and Section 6 concludes.

The model
We first describe the building blocks of the model, then discuss the solution under the RE benchmark and finally explain our implementation under SL.

A piecewise linear New Keynesian model
Our model builds on the workhorse three-equation NK model. All variables below are expressed in deviation from their steady-state level that corresponds to the target of the CB.
The first equation, the IS curve, describes aggregate demand: where y t is the output gap, ı t the nominal interest rate set by the CB, π t the deviation of the inflation rate from the target (hence, ı t − E t π t+1 is the real in-8 terest rate), σ > 0 the inter-temporal elasticity of substitution of consumption (based on a CRRA utility function), and E * j,t the (possibly boundedly rational) expectation operator based on information available at time t. The subscript j is introduced to suggest the possibility of heterogeneous expectations, where each agent-type j = 1, ..., N forms her own expectation (with N the number of agent-types). 6 g is an exogenous real disturbance.
Second, the forward-looking NK Phillips curve summarizes the supply side: where 0 < β < 1 represents the discount factor, κ > 0 a composite parameter capturing the sensitivity of inflation to the output gap and u t an exogenous cost-push shock.
Monetary policy implements a flexible inflation-targeting regime subject to the ELB constraint, which results in the following non-linear Taylor rule: where φ π and φ y are, respectively, the reaction coefficients to the inflation and the output gaps, and r ≡ π T +ρ the steady-state level of interest rate associated 6 We follow here most of the learning literature and introduce heterogeneity in the reduced-form models rather than in the micro-foundations (see, inter alia, Bullard & Mitra (2002), Arifovic et al. (2013), Hommes & Lustenhouwer (2019)). We are well aware of the conceptual limitation of this approach. Nonetheless, while the complications of the alternative are clear (see e.g. Woodford (2013)), the benefits in terms of qualitative results remain uncertain. For instance, in an asset-pricing model, Adam & Marcet (2011) show that under a sophisticated form of adaptive learning, the infinite-horizon pricing equation reduces to a myopic mean-variance equation. Bearing in mind those caveats, we proceed within the reduced-form model.

9
with the inflation target π T and the households' discount rate ρ ≡ − log(β).
The forward-looking rule translates the emphasis of CBs on expectations as contemporaneous variables are not instantaneously observable.
We now solve the model under the RE benchmark and then detail how we introduce SL in the expectation formation process of the agents.

The model under rational expectations
In this section, we consider RE and impose E * j,t (·) = E(· | I t ) to be the RE operator given the information set I t common to all agents in period t. We solve for the Minimal State Variable (MSV) solution using the method of undetermined coefficients (see Appendix A.1).
It is well known that the ELB introduces a non-linearity in the Taylor rule and generates an additional deflationary steady-state next to the target (Benhabib et al. 2001). Hence, expressing the model in reduced form is challenged by this non-linearity, and we need to disentangle two pieces, one around the target and one when the ELB is binding. 7 A short digression through the one-dimensional Fisherian model easily illustrates this configuration. Figure 3 displays inflation and interest rate dynamics, abstracting from the production side: the inflation target corresponds 7 We follow here the related NK literature and impose the ELB constraint in the loglinearized model around the targeted steady-state to describe the dynamics around the low inflation state, see, inter alia, Guerrieri & Iacoviello (2015). This method gives a secondbest estimate of the dynamics around the deflationary steady-state. A first-best would be to log-linearize the model around this second steady-state but would result in an MSV solution involving extra additional state variables (Ascari & Sbordone 2014) and, hence, additional coefficients to learn under SL (see Section 2.3). However, the benefits in terms of qualitative results are unlikely to outweigh the costs of such a complication of the learning process of the agents.
Coming back to the two-dimensional model, we have to specify a process for the exogenous shocks. In the rest of the paper, we consider white noise shocks only, so g and u are non-observable i.i.d. processes. In this case, the MSV solution boils down to a noisy constant without persistence. The presence of a floor on the nominal rate makes this solution piece-wise linear: where the first case is the law of motion when the ELB is not binding (denoted by a 'T' superscript) and the second case when the ELB is binding (denoted by a 'elb' superscript). The exact expression of the matrix coefficients can be found in Appendix A.1. Note that as variables are expressed in deviation from their steady-state values at the target, we have a T = (0 0) . We now introduce expectations under SL.

Expectations under social learning
Under SL, we relax the assumption of homogeneous agents endowed with RE and consider instead a population J of N heterogeneous and interacting agents, indexed by j = 1, · · · , N . We now define E * j,t (·) = E SL j (· | I j,t ) to be the expectation operator under SL given the information set I j,t available in period t to agent j. The information set is agent-specific as it contains, besides the history of past inflation and output gaps up until period t − 1, the current and past individual forecasts that need not be shared with the whole population.
Individual forecasting rules Following Arifovic et al. (2013Arifovic et al. ( , 2018, we assume that agents are endowed with a forecasting rule that involves the same variables as the MSV solution. The form of the rule is the same across agents, but with agent-specific coefficients that they revise over time. In any period t, each agent j is therefore entirely described by a two-component forecast [a y j,t , a π j,t ] and her expectations read as: 12 Those forecast values find an appealing interpretation. In the absence of shocks, they correspond to her long-run output and inflation gap forecasts. In Under learning, the model is solved sequentially so as to obtain a temporary equilibrium in each period, which makes it straightforward to account for the non-linearity induced by the ELB. Figure 4 summarizes the sequence of events within a period under SL. Let us now detail each step. Arifovic et al. (2013Arifovic et al. ( , 2018, individual expectations (5) are aggregated using the arithmetic mean as:

Aggregation of individual forecasts Following
8 In the sequel, we denote by Ω such an indicator of expectation anchorage. Specifically, we use the average squared distance of individual expectations to zero: Ω Eπ  We now detail how SL agents form their individual forecasts. Specifically, this class of learning models utilizes two operators.

Mutation
The first one is a stochastic innovation process, or mutation, that allows for a constant exploration of the state space outside the existing population of forecasts. In each period, each agent's forecasts are modified by an idiosyncratic shock with exogenously given probabilities. Her output gap forecast is modified with probability µ y and her inflation gap forecast with probability µ π . In short, her forecasts of any variable x = {y, π} is modified in any period as follows: with ι j,t an idiosyncratic random draw from a standard normal distribution with standard deviation ξ x . The larger parameters ξ x , the wider the neighborhood to be explored around the existing forecasts. Mutation can be interpreted as an innovation, a trial-and-error process or a control error in the computation of the corresponding expectations.

Tournament and computation of forecasting performances
The second operator, the tournament, is the selection force of the learning process and allows better-performing forecasts to spread among the population at the expense of lower-performing ones. Forecast performance is evaluated using the forecast errors over the whole past history of the economy given the stochastic nature of the environment (see Branch & Evans 2007).
To each forecast a x j,t , x = {y, π}, of each agent j is assigned a so-called fitness, computed as: The terms y t−τ −a y j,t and π t−τ −a π j,t correspond, respectively, to the output and inflation gap forecast errors that agent j would have made in period t − τ − 1, had she used her current forecasts a y j,t and a π j,t to predict the output and inflation gaps in period t − τ . The smaller the forecast errors, the higher the fitness.
Parameter ρ x ∈ [0, 1] (for x = y, π) represents memory. In the nested case where ρ x = 0, the fitness of each forecast is completely determined by the forecast error on the most recent observable data. For any 0 < ρ x ≤ 1, all past forecast errors impact the fitness but with exponentially declining weights while, for ρ x = 1, all past errors have an equal weight in the computation of the fitness. This memory parameter allows the agents to discriminate between a one-time lucky draw and persistently good forecasting performances.
In the tournament, agents are randomly paired (the number of agents is conveniently chosen even), their fitness on inflation and output gap forecasts are each compared and the agent with the lowest fitness copies the forecast of the other. There are two separate tournaments: one for inflation gap forecasts {a π j,t } j∈J and one for output gap forecasts {a y j,t } j∈J . 9 Formally, for each pair of agents k, l ∈ J, k = l, with individual forecasts a x k,t and a x l,t (x ∈ {π, y}), the tournament operates an imitation of the more successful forecasts as follows: The tournament occurs after the mutation operator in order to screen out bad-performing forecasts stemming from mutation. This allows the model to be less sensitive to the parameter values tuning mutation than if mutation were to take place after the tournament selection, and all newly created forecasts were to determine aggregate expectations without consideration of their performances. This way, the mutation process can be more frequent and of wider amplitude so as to allow for a faster adaptation of the agents to new macroeconomic conditions, while limiting the amount of noise introduced by the SL algorithm.

Simulation protocol
We study the dynamics of the model using numerical simulations. Throughout the rest of the paper, we proceed as described in Arifovic et al. (2013Arifovic et al. ( , 2018. We generate a history of 100 periods along the law of motion of the economy around the target (see Eq. (4)) and introduce a population of SL agents in t = 100. Their initial forecasts are drawn from the same support as the one used in the mutation process, i.e. from a normal distribution with standard deviation ξ x , x = π, y. The first 100 periods are used to provide the agents with a history of past inflation and output gaps in order to compute the fitness of their newly introduced forecasts. In the simulation exercises in the next section, we vary the initial average of the normal distribution to tune the degree of pessimism in the economy. The further below zero the initial average forecasts are, the more pessimistic views the agents hold about future inflation and output gaps.
Finally, it is important to recognize that the RE representative agent benchmark is nested in our heterogeneous-agent model: as soon as the inflation and output gap expectations of all agents are initialized at the targeted values and mutation is switched off (i.e. ξ y , ξ π = 0), the dynamics boil down to the RE benchmark. Under SL, our model involves a few parameters, namely the probabilities of mutation, the sizes of those mutations and the memory of the fitness function. We now detail how we estimate those parameter values.

Estimation under social learning
We jointly estimate the learning parameters and the structural parameters of the model. We first describe our choice and construction of the dataset, then discuss our estimation method and, finally, present the results.

Dataset
Macroeconomic US time series for output, price index and nominal rates are taken from the FRED database. Forecast data come from the SPF of the To make the dataset stationary, we divide output by both the working age population and the price index. In order to obtain a measurement of the output gap, we compute the percentage deviations of the resulting output time series from its linear trend. The inflation rate is measured by the growth rate of the GDP deflator.
As heterogeneity in expectations is essential to the dynamics of the SL model, we construct an empirical measure of that heterogeneity in the survey data. We use the cross-sectional dispersion of the individual forecasts, measured by the standard deviation of the individual forecasts among all participants in each period, to obtain time series of forecasts' heterogeneity.

Estimation method
With those data at hand, we proceed by matching the statistics from empirical moments with their simulated counterparts under SL. We provide technical details of our estimation method in Appendix B. In short, we use the Simulated Moments Method (SMM), which provides a rigorous basis for evaluating whether the model is able to replicate salient business cycle properties. 10 To avoid identification issues, we take the number of estimated parameters to be equal to the number of matched moments. Hence, we first reduce the number of dimensions of the matching problem and calibrate some of the parameters, namely the monetary policy and the preference parameters, as is standard in the related macroeconomic literature, and the number of agents (see Table 1).
We are left with four structural parameters from the NK model, namely the size of the fundamental shocks σ g and σ u , the slope of the NK Phillips curve (parameter κ) and the natural rate r. As we have calibrated the value of the discount factor β (see Table 1), we estimate the value of the inflation trend over the period considered, which uniquely determines the value ofr. 11 As for the SL parameters, we need not estimate common values for the inflation and the output gap expectation processes as the two tournaments are separated and the two time series are likely to behave differently and exhibit different properties, both in reality and in the model. For instance, estimating inflation and output gap-specific memory parameters ρ π and ρ y may translate the fact that agents can learn that one variable may display more persistence than the other. Hence, we estimate six learning parameters, namely the mutation sizes and frequencies ξ x and µ x as well as the memory of the fitness measures ρ x for We now discuss the mapping between those parameters and the empirical moments to match. First, the standard deviations of the shocks σ g and σ u naturally capture the empirical volatility of output and inflation. Second, the inflation trend π aims to match the ELB probability. To see why, recall that a higher natural rater mechanically decreases the probability of hitting the ELB, as the latter is defined as ı t = −r, which is strictly decreasing in the value  of the inflation target. Finally, the slope of the Phillips curve κ determines the correlation between the output and inflation gaps per Eq. (2).
As for the SL parameters, the memories of the fitness function ρ y and ρ π tune the sluggishness of the expectations because they determine the weights on recent versus past forecast errors in the computation of the forecasting performances. The higher ρ y and ρ π , the longer the memory of the agents, the less reactive the learning process to recent errors and the more sluggish the expectations. As sluggishness in expectations is the only source of persistence in the model once we consider i.i.d. shocks, parameters ρ y and ρ π are matched with the autocorrelation of, respectively, the output and the inflation gaps.
The remaining four learning parameters control the mutation processes that are the source of the pervasive heterogeneity in expectations in the SL model. We understandably use those parameters to match four moments characterizing heterogeneity in the SPF data: the average dispersion of the output and the inflation gap forecasts over the time period considered, denoted respectively by ∆ Ey and ∆ Eπ , and their first-order autocorrelations, denoted by ρ(∆ Ey t , ∆ Ey t−1 ) and ρ(∆ Eπ t , ∆ Eπ t−1 ). In line with intuition, sensitivity analyses of the objective function of the matching problem with respect to those learning parameters have reported the following associations: the mutation sizes ξ y and ξ π capture a substantial share of the empirical dispersion of output and inflation gap forecasts, while the mutation frequencies µ y and µ π match most of their autocorrelation.
Finally, in the same vein as Ruge Murcia (2007) Table 3). Table 2 reports the matched moments and their empirical counterparts (in p.p.). Table 3 gives the corresponding estimated values of the parameters.

Estimation results
It is first striking to see that the simple two-dimensional model accounts for a substantial share of all ten moments. For half of them, the simulated moments even fall within the confidence interval of their empirical counterparts, which means that our model replicates those moments fully. We succeed in capturing not all, but a non-negligible part, of the persistence in macroeconomic variables with a model that employs only white-noise shocks. 12 We shed further light on the source of that persistence in Section 4.1, but at that stage, we can state that learning acts as an endogenous propagation mechanism that 12 Matching all the persistence would not be a realistic or desirable objective: it is unlikely that all macroeconomic persistence stems from learning in expectations and our model ignores all other fundamental sources of persistence in the economy. We rather provide a measure of the share of the persistence that could be attributed to learning.

Moments Empirical Matched moments
Empirical    estimates. For instance, the estimated (yearly) inflation trend is 3.4%, which nicely falls into the range between the average inflation rate over the time span considered that includes the 1970s (4.3%) and the Fed inflation target that was adopted later (2%). Next, given the calibrated discount factor β, the implied value for the (yearly) natural interest rate is 5.45%, which is close to the average federal funds rate over the sample (namely 5.2%).
As for the estimated values of the mutation parameters of SL, we can see that they are all in line with the values usually employed in numerical simulations in the related literature (Arifovic et al. 2013). The estimated values of ρ y and ρ π imply that agents' memory is bounded, 13 which is highlighted by experimental evidence (Anufriev & Hommes 2012) and empirical estimates from micro data (Malmendier & Nagel 2016).
We conclude with our first major contribution: our parsimonious model is able to jointly and accurately reproduce ten salient features of macroeconomic time series and survey data, including the ELB duration and the pervasive heterogeneity in forecasts, while using plausible parameter values. We now proceed to the analysis of the underlying propagation mechanism in the model induced by SL.

Dynamics under social learning
This section first analyzes the stability properties of the targeted steady state under SL. To unravel the dynamics of expectations at the ELB, we analyze one transitory path to the target as an illustration. Next, we systematically compare the business cycles properties under SL and RE and assess the welfare loss entailed by heterogeneous expectations with respect to the RE representative agent benchmark.

Stability analysis
We examine here the asymptotic behavior of the model over the entire state space of the endogenous variables (π,ŷ), as utilized in the introduction (see, again, Fig. 2). We proceed through Monte Carlo simulations. Figure  A wider stability region of the target under SL than under recursive learning is due to a key difference between the two expectation formation mechanisms. SL expectations are heterogeneous at any point in time. Among that diversity, only the forecasts that deliver the lowest forecast errors over the past 14 Per consequence, in our model, the low-inflation state is unstable under SL as it belongs to the basin of attraction of the target: if expectations shift on the low-inflation state, they will eventually converge back to the target. Hence, the stability result in Arifovic et al. (2018), that is obtained under infinite memory in the fitness function, does not generalize to our setup where agents discard past observations. Intuitively, the difference in dynamics is akin to the differences observed in the adaptive learning literature between constant and decreasing gain algorithms. Notes: See explanations at the end of Section 2.3. We perform 1,600,000 Monte-Carlo simulations over 1000 periods. The targeted steady state is denoted by the green dot, and the deflationary steady state by the red one. The ELB frontier (yellow dashed line) is the locus of points for which −r = φ π π + φ y y: on the left-hand side, the ELB binds. The stable manifold associated with the saddle low inflation steady-state (red line) is computed under recursive learning and corresponds to the stable eigenvector of B elb : on the left-hand side, the model is indeterminate under RE and E-unstable. The empty area represents pairs of expectation values for which the model diverges along a deflationary spiral. We define convergence as -convergence, i.e. inflation and output respectively enter and do not exit the neighborhood [− π , π ] and [− y , y ] with { π , y } = {0.1%, 0.5%}. Results are robust to tighter convergence criteria. Left: The darker, the higher the probability to converge back to the steady state. Right: The darker, the faster the convergence back to the steady state.  Fig. 5 (albeit below the target), they deliver lower forecast errors than the more pessimistic ones when it comes to forecasting on average over the whole history, which includes pre-shock dynamics. Hence, those less pessimistic forecasts spread out and steer the economy back to the target.
By contrast, an adaptive learning algorithm is not concerned with alternative forecasting solutions. A single forecast in the indeterminacy region would result in a negative forecast error, i.e. realized inflation and output gaps decline even further below their expected values as they diverge in that region of the state space. This negative forecast error causes agents to revise down their expectations even further, which eventually drives the economy along a deflationary spiral. Yet, our model may also lead to self-sustaining deflationary spirals when shifts in expectations are large enough to throw the entire population of strategies beyond the stable manifold. However, for this to happen, as shown by the white area in Figure 5a, the one-time shift in expectations has to be implausibly large in light of where the actual data lie, as depicted by Fig. 2. Another related interesting observation is given by Figure 5b. Using the same state space as Figure 5a, the figure reports the speed of convergence to the target for each pair of initial average expectations. The darker the area, the faster the convergence. It is striking to see that the closer expectations to the targeted steady-state, the faster the convergence. In general, there is a locus of points spiraling around the target where convergence is fast, which is consistent with the complex eigenvalues associated with that steady state.
Most interestingly, the area in the southwest side from the target, beyond the stability frontier, is depicted in light gray. This means that for those severely pessimistic inflation and output gap expectations, the model under SL does converge back to the target, but does so at a particularly slow speed.
This area is beyond the ELB frontier (yellow dashed line), which indicates that the ELB is binding yet the model does not diverge along a depressive downward spiral.
Those observations show that our model can produce persistent but nondiverging episodes at the ELB, and heterogeneity in expectations plays an essential role in generating those dynamics. To shed more light on these dynamics, we now focus on a single expectational shock as simulated for Fig. 5 and study how it propagates in the model.   Finally, it is interesting to note that our model reproduces another stylized fact discussed in Mankiw et al. (2003): a recession is associated with an 16 Admittedly, the number of periods before convergence back on target appears implausibly large but the model does a good job once one bears in mind that the only policy in our simple model is a Taylor rule constrained by the ELB; hence, our model abstracts from many empirically relevant dimensions of policy that would be likely to play a role in fostering the recovery. The simple structure of the model depicts inflation as almost entirely expectation-driven. It also ignores many other empirically relevant determinants of inflation which could also entail a quicker inflation response. Lastly, as explained above, this exercise is only meant for illustrative purposes, not to match any empirical counterpart in recent history. Hence, one should refrain from drawing an explicit time interpretation.

Illustration of persistent dynamics at the ELB
increase in the dispersion of forecasts among agents or, in other words, the level of disagreement between agents -in our estimated model, the correlation between output gap and output gap forecast dispersion is in fact significant and reaches -0.34. Indeed, Figure 6 reports how the dispersion of individual expectations spikes in the aftermath of the shock. The rise in forecast dispersion does not last: this is because the selection pressure of the SL algorithm pushes the agents to adapt to the 'new normal' in the aftermath of the shock.
The level of heterogeneity between agents then returns to its long-run value, which is dictated by the size of the mutations.
We conclude that our simple model offers a stylized representation of the From an allocation perspective, the coordination of expectations on large and persistent recessive paths leaves out the economy into second-best equilibria with respect to the benchmark representative-agent model under RE. 17 Hence, SL expectations can be envisioned as a friction with respect to the RE representative-agent allocation, which may imply a substantial welfare cost, as we now demonstrate.

Welfare cost of social learning expectations
To evaluate this cost, we use the welfare function, which has become the main microfounded criterion, to compare alternative policy regimes. Following Woodford (2002), we consider a second-order approximation of this criterion and use the unconditional mean to express this criterion in terms of inflation and output volatility. The detailed derivations and explicit forms are deferred to Appendix A.3. The corresponding welfare function reads as: whereW is the steady-state level of welfare and λ π and λ y are, respectively, the elasticities of the loss function with respect to the variance of the inflation gap E [π 2 t ] and the output gap E [ŷ 2 t ]. It is straightforward to notice that macroeconomic volatility reduces the welfare of households.
While in representative-agent models the loss function is unique, it may be expressed in an agent-specific manner in a heterogeneous-agent framework.
Since the aggregation of agents is performed within the linearized model, we proceed in the same way with the welfare function and linearize it up to the second order. The welfare criterion provides a metric to compare macroeconomic performances under SL and under RE. Comparing these two allocations results in a measurement of the welfare cost of expectation miscoordination, which can be expressed in permanent consumption equivalents (Lucas 2003).
Using a standard no-arbitrage condition between the SL and the RE allocations, the fraction of consumption λ that SL households are willing to pay to 33 live in an RE world solves the following conditions on utility streams:

Central bank communication
We first introduce a simple form of CB communication to develop intuition on its anchoring effect on expectations and then discuss how these insights may inform a broader range of topical communication policy debates.

Modeling communication under SL
We represent communication as an announcement, which we denote by A CB t , made by the CB at the end of any period t. In the model, this is an announcement about inflation in the next period (t + 1). We focus on inflation because it is the main objective under an inflation targeting regime.
To introduce the CB announcements into the SL algorithm, we follow Arifovic et al. (2019), albeit in a simpler game. Besides her output and inflation gap forecasts (a y j,t and a π j,t ), each agent j now carries a probability ψ j,t ∈ [0, 1] of incorporating the CB announcement into her inflation forecast in any period t. If she does so, her inflation forecast in t + 1 is simply the CB announcement.
Conversely, with a probability 1 − ψ j,t , she ignores the announcement and sets her inflation forecast equal to her forecast a π j,t as before. The determination of her output gap forecasts remains unchanged and equal to a y j,t .
Formally, in the presence of announcements, the expectation formation process of the agents given by (5) is modified as: The communication-augmented inflation forecast {(ψ j,t , a π j,t )} j∈J undergoes the same mutation and tournament processes as the output gap forecast a y j,t (see Section 2.3). 18 The only difference from the algorithm used so far lies in the computation of the fitness of inflation forecasts, where Eq. (8) is modified as follows: where the first (resp. second) term now corresponds to the discounted sum of squared forecast errors had the agent followed (resp. ignored) the announcements of the CB.
The probabilities {ψ j } can be easily interpreted as the credibility of the announcements. If agents following the announcements (i.e. agents with a relatively high value of ψ j ) have lower forecast errors than agents ignoring the announcements (i.e. agents with a relatively low value of ψ j ), the following strategy shall spread among agents, which means that the average value

Two simple communication examples
We consider the following two communication examples under SL.
The CB announces the inflation target We then have A CB t = 0 (as the model is written in deviations from steady state). It should be noted that the target corresponds to the RE inflation forecasts in our simple model. The announcement of the CB is therefore consistent with the conduct of monetary policy under RE. Hence, the inflation target is redundant information to RE agents, but this piece of information may play a non-trivial role under SL.

The CB announces its own inflation forecasts for the next period
We assume that the policy authority estimates a commonly used VAR forecasting model that is recursively updated with new observations in each period (see Eusepi & Preston (2010) for a similar assumption). Note that assuming VAR forecasting amounts to assuming that the CB is aware of agents being boundedly rational and, therefore, includes past realizations of the endogenous variables in its forecasting model to account for the propagation mechanism induced by learning. Indeed, such a forecasting model would be misspecified should the agents have RE and, hence, the economy evolve according to the MSV solution. In this second communication scenario, the announcement of the CB is therefore consistent with the conduct of monetary policy under SL. 19 We now develop intuitions on how communication affects agents' expectations under SL. First, Table 5 compares the business cycles statistics of the model under RE and SL -for ease of reading, the first two columns recall the statistics in Table 4 -and under the two communication scenarios, i.e. when the target and the inflation forecasts are announced.
The first three rows of Table 5 indicate that communication significantly improves macroeconomic stabilization with respect to the baseline SL model: the volatility of inflation decreases by more than 50% and the risk of ELB episodes drops considerably. A look at the next four lines of Table 5 reveals that not only are expectations better coordinated (i.e. disagreement between agents is reduced) in the presence than in the absence of communication, but coordination occurs around the CB objectives (i.e. expectations are better anchored at the target).
19 The MSV solution under SL is a complicated and non-linear function of all the states in the system, including those pertaining to the SL process, and an explicit form is not available. We claim that the best the CB can do in such an environment is to estimate the law of motion of the economy with an atheoretical model, such as a VAR. We 8 lags, in line with the memory of the agents that is implied by the estimated value of the fitness memory on inflation (see, again, Table 3) but results are robust to more or fewer lags and to assuming a decreasing gain.    announcing forecasts, the credibility loss results from the inaccuracy of the announced forecasts, as the pessimistic shock is unexpected -to see that, look at the discrepancy between the plunging inflation and the near-target announcements immediately after the shock (Fig. 7a vs. 7g). In both cases, the forecasting performances of the followers deteriorate and a large fraction of the agents stop following the CB's announcements.
This credibility loss leads us to the second conclusion: in our model, agents need to 'see it to believe it'. In other words, if the CB's announcements are decoupled from the actual inflation dynamics, they lose their anchoring power on expectations.
Next, as time goes forward, the CB, by updating its model, provides more accurate forecasts -to see that, notice the similarity between the announcements and actual inflation some periods after the shock -and regains credibility. In parallel, this coordination on the forecast announcements leads to a reduction in expectation heterogeneity -to see that, notice the drop in inflation forecast dispersion (Fig. 7c) as credibility increases (Fig. 7h). By contrast, if announcing the target, the CB only regains its credibility once inflation has converged back to the target, which may take a considerable amount of time, as discussed in Section 4.
Yet, announcing forecasts is not a panacea: it also accentuates the downturn. Indeed, inflation dives deeper, the ELB binds for a longer period (Fig.   7j) and output overshoots further (Figs. 7d-7e) than when the CB announces its target. This observation illustrates an important pitfall of communication: by extrapolating the bust, the announced forecasts may turn self-defeating per the self-fulfilling nature of expectations and contribute to driving expectations away from the target. This striking effect is illustrated in the three graphs of the average inflation forecasts (Fig. 7b), the CB forecast announcements (Fig.   7g) and the actual inflation (Fig. 7a) which all almost overlap.  (Woodford 2007).

A broader policy perspective
One good example where IFT has turned particularly useful is the case of transition economies upon adoption of an inflation-targeting regime; see, e.g, Clinton et al. (2017) for the case of Czech Republic. The CB aims to bring inflation to the newly announced target and anchor expectations at this target.
It does so by announcing inflation forecasts that gradually converge to the target in an attempt to coordinate expectations on these forecasts and gradually steer inflation and inflation expectations towards the target. Perhaps closer to our current experience, in a low-inflation environment, our communication exercise shows how IFT allows the CB to coordinate inflation expectations despite the indeterminacy generated by the neutralization of the interest rate feedback at the ELB.
Let us now consider another recent policy discussion, namely the 'forwardguidance puzzle' (Carlstrom et al. 2012): under RE, any CB announcement about the future is immediately incorporated into agents' expectations and optimal decisions and triggers dramatic effects right from the time of the announcement, while empirical evidence contradicts such a strong effect; see, inter alia, (Del Negro et al. 2012, Campbell et al. 2016. Based on our model, imperfect credibility may play a central role in explaining this puzzle. 20 If agents need to 'see it first to believe it', announcements that are at odds with the actual inflation dynamics have a milder effect on expectations and hence actual decisions than under the full credibility assumption underlying RE.
One more example of a topical policy debate that can be informed by our work is the recent inflation targeting reviews undertaken by a number of major CBs. As persistently low inflationary pressures, exacerbated by the ELB con- if agents, accustomed to a decade of low inflation, need first to see higher inflation to revise upward their expectations. Those considerations reinforce the rationale for intensifying the CBs' efforts to communicate beyond market participants and reach the general public, as most recently emphasized by a number of major CBs.

Conclusion
This paper develops a model that features expectation-driven business cycles. The key mechanism works through heterogeneous expectations that may lose their anchorage to the target and persistently coordinate on below-target paths, which triggers prolonged ELB episodes. Heterogeneous expectations are introduced via an SL process into an otherwise standard two-equation macroeconomic model with a constrained Taylor rule. Our model nests the RE representative agent benchmark. In particular, we use white noise fundamental shocks to identify the propagation mechanism stemming from expectations in the formation of business cycles.
Our first contribution is to bring such a model to the data and estimate jointly its fundamental and learning parameters by matching moments from both US inflation and output gaps and the SPF. Our parsimonious model is able to account for ten stylized facts, including properties related to heterogeneity in forecasts, persistence in macroeconomic variables and endogenous occurrence of ELB episodes.
We then analyze the dynamics of the model and show that the basin of attraction of the target under SL is larger than the determinacy region under RE. In the context of our model, ELB episodes are episodes where expectations have coordinated on pessimistic outlooks following a series of adverse fundamental shocks and have visited regions of that basin from where the transition back on the target does occur but at a particularly slow pace. Our second major contribution is then to provide a framework that can account for the recent inflation dynamics that are challenging to capture in standard simple macroeconomic models. In particular, our model accounts for the 'missing disinflation' along the Great Recession per its stable but below-target dynamics and extensive ELB episodes. It also accounts for the 'inflation-less recovery' per consequence of the combination of unanchored inflation expectations that put downward pressure on inflation and the boosting effect of low interest rates on output.
Finally, we extent our model to illustrate how CB communication may influence expectations. In our model, the credibility of the announcements is not a priori granted but rather follows the same evolutionary process as the forecasts of the agents. From two simple examples, we show that this endogenous credibility plays a central role in reshaping expectations: in our model, agents need to 'see it to believe it'. Moreover, pessimistic inflation forecasts may turn self-defeating per the self-fulfilling nature of inflation expectations.
From those observations, we discuss broader policy implications to shed light on recent debates such as the forward-guidance puzzle or inform topical policy proposals such as temporary higher inflation targets.
Our model offers a simple framework that yet opens up the possibility for 45 analyzing a rich set of monetary policy alternatives. As for our estimation routine, it may be applied to a wide range of standard workhorse models that could then be explored under heterogeneous expectations. Those research avenues are left for further work.

A.1 Solution under rational expectations
We solve the model under RE using the method of undetermined coefficients (with and without the ELB).
(1) provides the reduced-form expression of the log-linearized model: with the two endogenous variables z t = ( y t π t ) ; matrices χ g = (1 κ) and χ u = (0 1) are related to the shocks g and u while α and B are related, respectively, to the steady-state values and the forward-looking variables. The values of α and B depend on the steady-state considered. Given Eq. (13), the general form of the MSV solution reads as: where the coefficient values in matrices a, c and d depend on whether the ELB is binding or not. Taking expectations based on an AR(1) specification of the stochastic processes g and u with autoregressive coefficients ρ g and ρ u ∈ (0, 1) yields (assuming for now that shocks are observable in t): Inserting Eq. (15) back into (13) uniquely identifies the MSV solution as: with a = (I − B) −1 α, c = (I − Bρ g ) −1 χ g and d = (I − Bρ u ) −1 χ u , which makes clear that the coefficient values of matrices B and α depend on the steady-state considered.
First, we consider the REE at the targeted steady-state that we denote by a star superscript. We insert the specification of the Taylor rule (3) when the ELB is not binding, i.e. ı t = φ π E t π t+1 + φ y E t y t+1 into (1) and obtain the The MSV-REE solution at the target is then given by: (17) Similarly, when the ELB is binding, the monetary policy rule reads as ı t = −r. Inserting this expression back into Eq. 1, the REE at the ELB, which we denote with a elb superscript, is described by: If shocks are i.i.d., ρ g = ρ u = 0, Eq. (15) reduces to an intercept a, with a T = (I − B T ) −1 α T at the target and a elb = (I − B elb ) −1 α elb at the ELB.
By contrast, the REE at the ELB (18) is indeterminate under RE and unstable under learning. To see that, notice that the characteristic polynomial of B elb is β + λ(−1 − β − κσ −1 ) + λ 2 = 0 ⇔ a 0 + a 1 λ + λ 2 . For both eigenvalues to be within the unit circle and the REE to be determinate, we need | a 0 |< 1 and | a 1 |< 1 + a 0 . The first condition always holds as β < 1 but the second is always violated as σ −1 κ > 0. Therefore, the deflationary state is indeterminate under RE and features multiple equilibria. 21 Furthermore, the determinant of B elb − I (I being the identity matrix) is −σ −1 κ < 0, which implies that one eigenvalue of B elb − I has negative real part and one has positive real part (equivalently, one eigenvalue of B elb is lower than one, the other is not). Therefore, under learning, the deflationary steady-state is unstable and is a saddle.
Under our calibration, the REE values at the ELB are a elb = (−0.007 − 0.013) , and the two eigenvalues of B elb are real and equal λ − i = 0.906 < 1 and λ + i = 1.099 > 1.

A.3 Welfare criterion
In this section, we develop the approximation of the welfare criterion.

A.3.1 The welfare in terms of output and prices
Before approximating the welfare function, we first rewrite the welfare function by expressing the utility function in terms of output and price equivalents.
Recall that: with In the absence of physical capital, the resource constraint reduces to: where aggregate production is the sum of the production of each variety i of goods in the economy. This equation allows us to substitute output for consumption in the utility function. Using a constant-return-to-scale production function allows us to further substitute output for the hours worked. We then have y it = h it and, at the aggregate, Y t = H t . Let us now aggregate the optimal demands for each variety i: where > 1 is the elasticity of substitution between differentiated types of goods i, p it is the price of variety i and P t the aggregate price level of all varieties in the economy. Market clearing imposes (22) to be equal to (21). In Eq. (22), the term 1 0 p it Pt − di is the price dispersion across varieties i induced by price stickiness, which we rewrite as ∆ t . To summarize, market clearing implies: Substituting consumption C t and labor H t into the utility function using expressions (21) and (23), we have: where χ = ( − 1) / is the inverse of the markup in the economy. The price dispersion ∆ t is hard to interpret and has no observable counterpart in macroeconomic time series. Following Woodford (2003), we now express price dispersion in term of inflation.

A.3.2 Price dispersion
The price dispersion is induced by the Calvo probability θ that constrains firms in updating their price. Following Schmitt-Grohé & Uribe (2004), we can rewrite the price dispersion as: whereπ is the rate of inflation at the steady-state. Now that we have an expression for the price dispersion, we need to replace the optimal price P * t /P t from the previous expression by the inflation rate. To do so, we use the aggregation condition on prices of constrained firms, and firms that can update their price approximated using the law of large numbers: Dividing by P 1− t , the price index becomes: From the latter expression, the relative optimal price P * t Pt is a non-linear function of past inflation: Combining Eq. 29 and Eq. 30, the price dispersion term ∆ t may be expressed in terms of inflation: In this latter expression, ∆ t is a function of inflation rates in t and t − 1 as well as previous dispersion ∆ t−1 . It is not possible to obtain a closed-form solution of the price dispersion as a function of inflation. However, up to second order, the variance is unconditional and we may express the variance of the price dispersion as a function of inflation (see Woodford (2003)).

B Technical appendix: estimation strategy (for online publication)
We proceed by following the related literature on the estimation of macroeconomic models under RE, but we need to adapt the method to a model under SL because SL introduces an additional non-linearity and an additional source of stochasticity into the model. This section explains how we do so.
Together with the specification of the SL process, the model -described in Eq. 1 to 3 -can be expressed in the following compact form: where z t is the set of endogenous variables, ε t the set of i.i.d. shocks and f Θ (·) the equations of the model using calibration Θ. First, we partition the parameters Θ into two sets: the first set contains mainly monetary policy and preferences parameters, which we calibrate following the literature as given in Table 1. The second set, θ ∈ Θ, contains parameters that we estimate by minimizing the distance between simulated and empirical moments.
To do so, the first difference that we need to address between an RE and an SL specification of the model is the issue of stochastic replications. An RE model only involves stochasticity in the exogenous shocks ( in Eq. (45)). Hence, the standard way to proceed is to exploit the asymptotic properties of Monte Carlo methods: a high number S of time series (also called 'chains') of shocks is drawn and the average over the whole series is used to obtain an unconditional measure of the simulated moments.
By contrast, the SL model introduces an additional source of stochasticity via the mutation processes and the random pairing in the tournament selection. We therefore have to apply Monte Carlo methods on each chain of shocks s ∈ S to obtain a representative behavior of the model under SL for any given chain s of shocks.
Specifically, we draw S = 100 chains of shocks u and g at the beginning of the estimation procedure and keep them unchanged. For each of the 100 chains, we run 100 Monte Carlo simulations of the model under SL and only retain one 'representative' simulation prior to the estimation exercise. To select this representative simulation, we choose the one for which the squared distances of inflation and output gaps to their median values over the 100 replications is the smallest. 22 Therefore, for each chain of shocks, we retain only one simulation. We do so for each of the S = 100 series of shocks. This has the advantage of considerably reducing the computational cost of the estimation and is analogous to choosing the stable root in the policy function of an RE model, as commonly used in the standard related literature.
Let us now define m T (x t ) , a p × 1 vector of moments calculated using stationary and ergodic real data x t of sample size T , and m s,τ x θ t , the modelgenerated counterpart based on artificial seriesx θ t of size τ generated using the set of parameters θ and Eq. (45). In our case, p = 10 as we match 10 moments. Our estimation procedure aims to minimize the distance between those two sets of moments.
To do so, the two sets of data must have similar properties. In particular, it is best practice to consider ergodic simulated data and same-sized samples. We then use T = 200 (quarters) given the time span considered in the SPF data. As for the ergodicity, we note that RE models are usually ergodic, i.e. the generated moments remain of similar magnitude across different chains and different starting values, but nothing guarantees that such a property holds a priori under SL. Prior to estimating the model, we have checked that it is the case if we use T = τ = 200. Yet, we have observed that for six chains of shocks out of the 100 used, the model may be subject to unstable recessive paths. We therefore discard those six chains from the computation of the moments.
Note that nothing imposes a priori that the number of moments to match p equal the dimension of θ (i.e. the number of parameters to estimate). However, it is seen as best-practice in the SMM literature to require so in order to rule out over-identification or under-identification issues (Hansen 1982, McFadden 1989. Hence, we adopt this restriction. Finally, we follow Ruge Murcia (2007) and include priors on the distribution of the parameter values. We denote by P (θ) those priors into the objective function. We use techniques from Bayesian econometrics based on the optimization of a log-likelihood function. We log-linearize the contribution of the priors so that the objective function is the sum of the squared distance of the moments plus the sum of the log-priors. The resulting SMM estimator is then defined as: where m T (x t ) − m s,τ x θ t is the distance vector between the observed and the median. However, as we estimate a two-dimensional model (inflation and output gaps), our procedure provides a way to approximate the median simulation.
simulated moments that we seek to minimize, as explained above, and W is the weighting matrix. Hence, the matrix product in Eq. (46) provides the sum of the squares of the residuals between the observed and matched moments. The second term ΞP (θ) introduces a penalty into the objective function when the estimated values range outside their prior distributions. We set the weight on those priors to Ξ = 0.001. This small value is necessary when using the SMM method because this method relies on a small number of observations (i.e. a small sample of the moments of the time series), which magnifies the contribution of the prior information to the objective function. Specifically, we define a prior for κ that is in line with empirical results by Smets & Wouters (2007) with a Beta distribution of 0.05 mean and 0.1 standard deviation. As for the SL parameters, we impose a Beta distribution for the mutation probabilities µ x and the fitness persistence ρ x , x = {π, y} with a prior mean and standard deviation in line with the values used in the SL literature (Arifovic et al. 2013). We further impose to the sizes of mutation ξ x a positive support with a diffuse prior through an inverse gamma distribution with mean 0.1 and standard deviation of 5. These are the prior values reported in Table 3.
We solve Eq. (46) using the CMAES optimization algorithm of Hansen et al. (2003). The CMAES algorithm is a global estimation strategy that has the ability to deal with large-scale optimization problems and avoid local mimima. This algorithm provides an accurate measure of the Hessian matrix, even in the presence of bound restrictions and priors for control variables, as is the case in Eq. (46). Specifically, learning the covariance matrix in the CMAES is analogous to learning the inverse Hessian matrix in a gradientbased, local optimization method such as the quasi-Newton method.