Learnability of an equilibrium with private information

Journal of Economic Dynamics & Control 59 (2015) 58–74

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Learnability of an equilibrium with private information Ryuichi Nakagawa n Faculty of Economics, Kansai University, 3-3-35 Yamate Suita, Osaka 564-8680, Japan

a r t i c l e in f o

abstract

Article history: Received 3 June 2014 Received in revised form 25 June 2015 Accepted 27 June 2015 Available online 16 July 2015

This paper investigates the learnability of an equilibrium with private information. Agents of each type have their own private information about an exogenous variable and conduct adaptive learning with a heterogeneously misspecified perceived laws of motion (PLM) that includes only this variable. The paper shows that the existence of private information has a nonnegative impact on the learnability of the equilibrium; that is, the condition for learnability is unaffected or relaxed by heterogeneity and/or misspecification in PLMs caused by private information. In a New Keynesian model with private information about fundamental shocks, the learnability of the equilibrium is ensured by the Taylor principle of monetary policy. The paper also confirms that these results hold true not only in the presence of private information, but also in a variety of informational structures. & 2015 Elsevier B.V. All rights reserved.

JEL: C62 D83 E52 Keywords: Adaptive learning Private information Heterogeneous misspecification Taylor principle

1. Introduction Rational expectations are based on an unlikely assumption that agents have perfect knowledge about the structure of the economy. Recent research has incorporated the concept of adaptive learning as an alternative framework in which agents formulate their forecasts by estimating econometric models (that is, perceived laws of motion, PLMs) (see Bray, 1982; Evans and Honkapohja, 2001). The issue to be addressed in this framework is whether agents' forecasts converge around an equilibrium; that is, whether the equilibrium is learnable, and a number of papers have investigated macroeconomic policies ensuring the learnability of the equilibrium (e.g., Bullard and Mitra, 2002; Evans and Honkapohja , 2003a,b). Learnability has been analyzed in different structures of information sets of economic variables held by agents. Evans and Honkapohja (2001) provide a benchmark framework where each agent has the full information set of economic variables to form a correctly specified PLM including all relevant variables (hereafter, CS learning). Branch (2004) and Guse (2008) focus on the restricted perceptions equilibrium (hereafter, RPE) where agents' information sets are limited so that they are constrained to form underparameterized PLMs excluding unobservable variables. Adam et al. (2006) and Berardi (2007) introduce heterogeneity in agents' information sets, where a fraction of agents have limited information sets while other agents have full information sets.

n

Tel.: þ81 6 6368 0590; fax: þ 81 6 6339 7704. E-mail address: [email protected] URL: http://www2.itc.kansai-u.ac.jp/  ryu-naka

http://dx.doi.org/10.1016/j.jedc.2015.06.010 0165-1889/& 2015 Elsevier B.V. All rights reserved.

R. Nakagawa / Journal of Economic Dynamics & Control 59 (2015) 58–74

59

This paper investigates the learnability of an equilibrium in the presence of private information. Agents have private information about exogenous variables, which makes agents' information sets limited and heterogeneous.1 When agents make forecasts, each type of agent is constrained to form a heterogeneously misspecified PLM that includes only her own observable variable (hereafter, HM learning). In this economy, there emerges a generalization of the RPE, which we call a heterogeneous misspecification equilibrium (hereafter, HME). The paper clarifies the learnability of the HME and investigates whether and how the learnability is affected by the existence of private information. Next, in a basic New Keynesian (NK) macroeconomic model with private information about fundamental shocks, this paper provides learnability conditions imposed on monetary policy with a Taylor-type interest rate rule. The result of the paper is that the existence of private information has a nonnegative impact on the learnability of an equilibrium; that is, the condition for learnability is unaffected or relaxed by the existence of private information. Specifically, this impact is nondecreasing in the degrees of heterogeneity and misspecification in PLMs caused by private information. The paper also confirms that these results hold true not only in the presence of private information, but also in a variety of informational structures considered in the literature. Next, in the NK model, the learnability condition under HM learning imposed on monetary policy is not more restrictive than the condition under CS learning, that is, the Taylor principle—to raise nominal interest rates more than one-for-one in response to an increase in inflation (Taylor, 1993). Calibrations show that the impact of private information is economically significant in the NK model, and that if lagged endogenous variables are included in PLMs as public information, the impact is reduced, but remains significant. These results are intuitive as limited and/or heterogeneous information about fundamentals have been widely recognized to cause the reduced and slow adjustment of the economy (e.g., Calvo, 1983; Veldkamp, 2005), which makes it easy to learn the dynamics of the economy. This mechanism is consistent with the learning literature that explains the observed persistence of the economy using heterogeneity and/or misspecification in PLMs (e.g., Adam, 2007; Slobodyan and Wouters, 2012; Hommes and Zhu, 2014). Our results reinforce the Bullard and Mitra (2002)'s finding (that is, the Taylor principle is a sufficient condition for the learnability of an equilibrium) to be robust to the existence of private information, and they also complement the Honkapohja and Mitra (2006)'s finding that heterogeneity does not make learnability conditions more restrictive. This paper is closely related to the learning literature regarding the presence of private information. Marcet and Sargent (1989a) establish learning schemes for an equilibrium with private information. Branch and McGough (2011) study business cycle dynamics under adaptive learning with private information. Heinemann (2009) considers the existence of private noisy signals of economic variables. It has not, however, been investigated whether/how the private information that continuously makes agents' PLMs heterogeneously misspecified has an impact on learnability. Our analysis is also related to the literature on heterogeneity and misspecification in learning. Adam et al. (2006) and Berardi (2007, 2009) consider heterogeneous and misspecified learning where a fraction of agents form underparameterized PLMs, while other agents form correctly specified PLMs. Their models may be interpreted as allowing the partial existence of private information, but they do not include a plausible case where each type of agent has her own private information.2 Meanwhile, the heterogeneously misspecified PLMs in this paper are also considered in the context of the literature introducing dynamic predictor selection, which was originally established by Brock and Hommes (1997). Branch and Evans (2006, 2007), for example, allow agents to choose among a list of heterogeneously misspecified PLMs. However, their model premises homogeneity in agents' information sets so that all agents have the same list of PLMs, and this homogeneity conflicts with the existence of private information that causes intrinsic heterogeneity in information sets and constrains each agent to hold a specific heterogeneously misspecified PLM. Thus, the HM learning analyzed in this paper describes adaptive learning with private information more suitably. The paper is structured as follows. The next section presents our model and provides a benchmark analysis about learning without private information. Section 3 introduces learning with private information and investigates the dynamics of the HME. Section 4 examines the impact of private information on the learnability of an equilibrium. Section 5 evaluates numerically the impact in a basic NK model. Finally, we present our conclusions. 2. Model 2.1. Setup We establish the general form of the multivariate linear expectations model. The economy is represented by two vector equations: yt ¼ Aþ BEnt yt þ 1 þ Cwt ;

ð1Þ

wt ¼ Φwt  1 þvt :

ð2Þ

1 For example, a preference shock possessed by a household continues to be observable only for this household (see Allen and Gale, 2004). In financial markets, the profitability of a borrower tends to be observable only by this borrower (see Stiglitz and Weiss, 1981). Branch (2007) provides evidence that information sets are limited and heterogeneous using the Michigan survey of inflation expectations. Bovi (2013) uses the survey data of the European Commission to show that heterogeneous beliefs are persistent in UK citizens. 2 Honkapohja and Mitra (2004b) and Muto (2011) consider that the private sector and the central bank specify mutually different PLMs. However, their models include not only the heterogeneity in PLMs but also structural heterogeneity, and hence the effect of the former heterogeneity is not identified.

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R. Nakagawa / Journal of Economic Dynamics & Control 59 (2015) 58–74

The equations represent the dynamics of the endogenous variables and the evolution of the exogenous variables. The economy has m endogenous variables and n exogenous variables. yt is an m  1 vector of endogenous variables at time t. wt ¼ ðw1t ; …; wnt Þ0 is an n  1 vector of autoregressive exogenous variables. The standard deviation of wit for each i A f1; …; ng
 is defined by σ ii 40, and the correlation matrix of wt is defined by Γ  ρij , where ρij ¼ ρji and ρij A ½0; 1 denotes the 1 r i;j r n

correlation between wi and wj for each i; j A f1; …; ng. vt is an n  1 vector of fundamental shocks with means of zero that drive the stochastic process of wt.3 Matrix A is an m  1 vector of constant terms. B is an m  m coefficient matrix of Ent yt þ 1 . C

is an m  n coefficient matrix of wt, and Φ is an n  n matrix of coefficients of wt. Ent is the operator of the aggregate expectation of yt þ 1 at time t, which is not necessarily rational under adaptive learning. The model is purely forward-looking, but it can be applied to the analysis of models with lagged endogenous variables yt  1 . For ease of calculation, we impose regularity assumptions on these parameters in Appendix A. In particular, Φ is assumed   to be a diagonal and nonnegative matrix whose diagonal elements exist in the interval ½0; 1Þ: Φ  diag φi 1 r i r n where 0 r φi o 1 for each i. In addition, Γ is assumed to be a nonnegative matrix, in which 0 r ρij r1 for each i; j A f1; …; ng. These assumptions are not crucial for our analysis, as most stationary linear models in the literature can be transformed to satisfy these conditions. 2.2. CS learning Before considering the existence of private information, let us review the benchmark of adaptive learning with a full information set to make it easier later to highlight the impact of private information. If agents could formulate rational expectations, the system (1) and (2) has a nonexplosive fundamental REE, which takes the form of a minimal state variable (hereafter, MSV) solution (see McCallum, 2004): yt ¼ a þ cwt þ εt ;

ð3Þ

where a is an m  1 vector of constant terms, c is an m  n coefficient matrix for wt, and εt is an m  1 vector of error terms that are perceived to be white noise. Using the method of undetermined coefficients, the solution ða; c Þ is uniquely obtained as a ¼ ðI m  BÞ  1 A;

ð4Þ

c ¼ Bc Φ þC;

ð5Þ

where Im is an m-dimensional identity matrix. Note that the constant terms vector a corresponds to the steady state of the fundamental REE. If agents do not have enough knowledge to develop rational expectations, agents might adopt adaptive learning using all available data to formulate their forecast Ent yt þ 1 . If all economic variables up to time t, fys ; ws gts ¼ 1 , are observable for all agents, they can estimate a correctly specified PLM of the form of the MSV solution (3) (that is, CS learning). Using the estimated parameters ða; cÞ, agents formulate Ent yt þ 1 ¼ a þ cΦwt , which is incorporated into Eq. (1) and yields the actual law of motion (hereafter, ALM) of the economy, yt ¼ ðA þ BaÞ þ ðBcΦ þ C Þwt .4 Evans and Honkapohja (2001, Chapter 6) show that the global convergence of ða; cÞ to the REE in real-time learning through least-squares techniques is governed by the associated ordinary differential equation (hereafter, ODE): d ða; cÞ ¼ T ða; cÞ  ða; cÞ; dτ

ð6Þ

where T ða; cÞ  ðT a ðaÞ; T c ðcÞÞ ¼ ðA þ Ba; BcΦ þ C Þ is the mapping from the PLM ða; cÞ to the ALM T ða; cÞ, and τ denotes notional time. If the ODE is globally asymptotically stable, then ða; cÞ globally converges to the solution (4) and (5), meaning that the fundamental REE is found to be learnable under CS learning. 3. HM learning In what follows, we consider adaptive learning by agents with private information and find the dynamics of an equilibrium under HM learning. 3.1. Private information First, we introduce private information about exogenous variables wt, such that each agent has access to information on only a subset of those variables (see Marcet and Sargent, 1989a). 3

Note that exogenous variables with nonzero and heterogeneous means can be transformed to the form (2). Forming Ent yt þ 1 using an information set including current variables is considered by Honkapohja and Mitra (2006); McCallum (2007); Ellison and Pearlman (2011), and Bullard and Eusepi (2014). Evans and Honkapohja (2001) indicate that excluding those variables from information sets might provide different conclusions about the learnability of an equilibrium, but the analysis of alternative frameworks is omitted here because of limited space. 4

R. Nakagawa / Journal of Economic Dynamics & Control 59 (2015) 58–74

Assumption 1. For each iA f1; …; ng, the evolution of an exogenous variable fwis gts ¼ 1 is observable for the proportion agents (hereafter, type i) and unobservable for agents of other types.

61 1 n

of

For analytical tractability, the population of each type is assumed to be the same at 1n, but the distribution of the populations of different types does not affect our results (see Section 4.4). Assumption 1 implies that agents of type i recognize the stochastic characteristics of wit and do not recognize the correlations fρij gnja i and the number n of exogenous variables. In this situation, the agent of type i has the set of public and private information fys ; wis gts ¼ 1 , which is limited and different from the information sets held by other types in terms of fwit gni¼ 1 .5 This type of private information describes a feature of idiosyncratic fundamental shocks at a microeconomic level: for example, a preference shock possessed by a household (see Allen and Gale, 2004) and the profitability of a borrower in a financial market (see Stiglitz and Weiss, 1981); those shocks are likely to continue to be observable only for specific agents. Under Assumption 1, the number n of exogenous variables can be considered the degree of limitation in the information set of each agent. If n ¼1, each information set is reduced to the full information set in Section 2.2. The larger the n is, the more limited each information set is relative to the full one. Thus, the number n represents not only the number of privately observable variables but also the degree of limitation of each information set.   In addition, for each i; j, the correlation ρij of the two exogenous variables wit ; wjt can be considered the degree of wjt wit homogeneity in the information sets of types i and j. If ρij ¼ 1 (and hence φi ¼ φj and σ ii ¼ σ jj ), both information sets are perfectly homogeneous, as if both types observed the same variable. The smaller the ρij is, the more heterogeneous are the information sets. Thus, 1  ρij is the degree of the heterogeneity in the information sets of types i and j. Although the degree of heterogeneity in the information sets of all types cannot be represented by a single scalar measure, we will say that the degree of heterogeneity in the information sets of all types increases if ρij falls for at least one ði; jÞ and falls or remains the same for every ði; jÞ. 3.2. Heterogeneous misspecification equilibrium Next, we consider adaptive learning with the limited and heterogeneous information sets. In contrast to agents with the full information set in Section 2.2, the agent of type i with the information set fys ; wis gts ¼ 1 is constrained to estimate a heterogeneously misspecified PLM: yt ¼ ai þ ci wit þ εit ;

ð7Þ

which underparameterizes the MSV solution (3) and differs from the PLMs of other types (that is, HM learning). Parameters ai and ci are m  1 vectors of constant terms and coefficients, and εit is an m  1 vector of error terms that are perceived to be white noise.6 When each type of agent specifies a PLM of the form (7), the degree of misspecification in the PLM of each type is specified one-to-one by the degree of limitation of each information set (that is, the number n), and the degree of heterogeneity in the PLMs of types i and j is specified one-to-one by the degree of heterogeneity of information sets of both types (that is, 1  ρij ). Later, the impact of private information on learnability will be identified by examining the effects of misspecification n and heterogeneity 1  ρij for each i; j in the PLMs on the learnability conditions.7 0 The agent estimates the parameter matrix ϕi  ðai ; ci Þ using a recursive least-squares (hereafter, RLS) method, but does not recognize the misspecification in the PLM (7) as wit and εit are orthogonalized. Finally, using the PLM (7) and the estimated ϕi, the agent formulates the forecast Enit yt þ 1 as Enit yt þ 1 ¼ ai þ ci φi wit ;

ð8Þ

n

where Eit is the operator of expectations formed by type i at time t. The aggregate forecast Ent yt þ 1 in Eq. (1) is determined by the aggregation of the forecasts of all types of the form (8). For n simplicity, let us assume that forecasts of different types fEnit yt þ 1 gi ¼ 1 have equal contributions to the dynamics of the n 8 economy. Then, Et yt þ 1 is formulated by Ent yt þ 1 ¼ a þ cΦwt ; n

ð9Þ Pn

where Et is the operator of the average of heterogeneous forecasts, and a  i ¼ 1 ai is the average of the constant term vectors for all types, and c  1nðc1 ; …; cn Þ is an m  n matrix that combines the coefficients fci gni¼ 1 of the PLMs of the form (7) for all types and multiplies them by 1n. 1 n

5 Note that Marcet and Sargent (1989a) consider the private information of not only exogenous variables but also endogenous variables. They also consider the existence of hidden state variables that are unobservable by all agents. Our assumption is designed to focus on clarifying the impact of private information on the learnability of an equilibrium. 6 Agents might include the vector of lagged endogenous variables yt  1 in the PLM as public information. This case will be discussed in Sections 4.3 and 5.3.2. On the other hand, we do not consider that agents include sunspot variables in their PLMs. The learnability of sunspot equilibria under HM learning is discussed in Conclusions as a future work. 7 The literature measures the degree of heterogeneity in PLMs by the proportions of different agents who specify different forms of PLMs, instead of correlations between variables in PLMs (e.g., Branch and Evans, 2006; Berardi, 2007). However, their heterogeneity can be reproduced in our model (see Section 4.4). n 8 Note that the proportions of contributions of fEnit yt þ 1 gi ¼ 1 to Ent yt þ 1 do not affect our results (see Section 4.4).

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The ALM of the economy depends upon Ent yt þ 1 in Eq. (9). Substituting Eq. (9) into the system (1) and (2), the ALM is determined by yt ¼ ðA þ BaÞ þ ðBcΦ þ C Þwt :

ð10Þ

The stability of the equilibrium attainable under HM learning is subject to whether the aggregate parameters

ϕ0  ða; cÞ converge to bounded values; the dynamics of ϕ is established by the real-time learning processes of agents of n all types for fϕi gi ¼ 1 . If the PLM (7) were correctly specified, the E-stability principle of Evans and Honkapohja (2001,

Chapter 2) would hold; that is, the convergence of parameters in real-time learning would be characterized by the ODE that could be made by a mapping from the PLM (7) to the ALM (10). Under HM learning, where the PLMs are n underparameterized, the principle does not necessarily hold, and the convergence of fϕi gi ¼ 1 is inferred from the n stochastic recursive algorithms of fϕi gi ¼ 1 formulated by the PLMs of the form (7) for all i and the ALM (10). As a n result, we find that the global convergence of fϕi gi ¼ 1 is governed by the following associated ODE for aggregate parameters ða; cÞ: d ða; cÞ ¼ ðT a ðaÞ; T c ðcÞÞ  ða; cÞ; dτ where

ð11Þ

τ denotes notional time and

T a ðaÞ  A þ Ba; T c ðcÞ  ðBcΦ þ C Þ

 1 Ψ ; n

Ψ  diagðσ ii Þ1 r i r n  Γ  diagðσ ii Þ1r1 i r n : The derivation of the ODE is in Appendix B. The mapping ðT a ðaÞ; T c ðcÞÞ provides the coefficients of the forecasts of yt updated by agents of all types using their limited and heterogeneous information sets. The ODE of a in Eq. (11) is equivalent to each ODE of fai gni¼ 1 as a is an arithmetic average of fai gni¼ 1 which have the same form of ODEs. The ODE of c in Eq. (11) is a combination of the ODEs of fci gni¼ 1 The effects of misspecification n and heterogeneity 1  ρij for each i; j on the convergence of the aggregate parameters ða; cÞ are realized through 1nΨ in T c ðcÞ. Thus, the ODE (11) includes the benchmark ODE (6) under CS learning as follows: Proposition 1. The ODE (11) under HM learning is equivalent to the ODE (6) under CS learning 1. if n ¼1 (no misspecification) or 2. if ρij ¼ 1 for all i; jA f1; …; ng (no heterogeneity). The first part is trivial, and the second part is proved by the fact that if ρij ¼ 1 for all i; j (and hence φi ¼ φj and wσ iiit ¼ σ jjjt for all   i; j), then 1nΨ wt ¼ wt , which makes the mapping T c ðcÞ under HM learning equivalent to the mapping for c under CS learning in Eq. (6). The second part means that if exogenous variables are perfectly correlated, misspecification in PLMs at an individual level has no effect on the dynamics of the economy under adaptive learning. This is intuitive because in this situation, an exogenous variable included in each PLM represents unobservable exogenous variables not included in the PLM as if there were no misspecification in the PLM. An equilibrium under HM learning is attained if the ODE is globally asymptotically stable such that parameter estimates ða; cÞ converge to the fixed point ða; c Þ of the ODE: w

a ¼ ðI m  BÞ  1 A;

ð12Þ

 1 c ¼ ðBc Φ þ C Þ Ψ ; n

ð13Þ

where the fixed point a corresponds to the steady state of the equilibrium. Let us call the equilibrium with the fixed point (12) and (13) the heterogeneous misspecification equilibrium (HME), which exists uniquely through part 1 of Assumption 2 in Appendix A: 1

n Definition 1. The HME is a stationary stochastic process for fyt g1 t ¼ 0 following the system (1) and (2) given that fE t yt þ 1 gt ¼ 0 n 0 is the average of the forecasts formed by the PLMs of the form (7) for all i with the parameters fϕi ¼ ðai ; ci Þgi ¼ 1 determined at the fixed point (12) and (13) of the ODE (11).

3.3. Observable steady state We also provide the ODE when the steady state a of the HME is observable for all agents. An observable steady state is a popular assumption in recent macroeconomic studies. In particular, a nonlinear macroeconomic model tends to be loglinearized around a steady state (or a point close to the steady state) by assuming the steady state to be observable (e.g., Bullard and Mitra, 2002; Mitra et al., 2013). If the steady state a is observable for agents of all types, agents of type i for each i

R. Nakagawa / Journal of Economic Dynamics & Control 59 (2015) 58–74

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can estimate a PLM with ai fixed at a: yt ¼ a þ ci wit þ εit ;

ð14Þ

which is equivalent to the PLM excluding the constant term: y~ t ¼ ci wit þ εit where y~ t  yt  a (see Slobodyan and Wouters, 2012).9 n 0 In this situation, the global convergence of fϕi ¼ ða; ci Þgi ¼ 1 of all types is subject to only the c part of the ODE (11). If the ODE is globally asymptotically stable, matrix c converges to the fixed point given by Eq. (13). Regardless of whether the steady state a is observable or not, the HME (12) and (13) is a unique equilibrium under HM learning.

4. Learnability of HME To investigate the impact of private information, we examine the effects of heterogeneity and misspecification of the PLMs on the learnability conditions of the HME separately. We also confirm that our results hold true not only in the presence of private information, but also in a variety of informational structures. For simplicity of exposition, we introduce the notation λ½X  as the largest value of the real parts of the eigenvalues of the matrix X. 4.1. Learnability conditions If the steady state a is unobservable, the learnability of the HME is ensured by the stability of the ODE (11). According to Evans and Honkapohja (2001, Section 6.6), the ODE is globally asymptotically stable if and only if their Jacobians, DðT a ðaÞ  aÞ ¼ B I m ;   0 1  B I mn ; DðT c ðcÞ  cÞ ¼ Φ Ψ n

ð15Þ ð16Þ h  0 1

i

 B o 1, respectively.10 If the steady state a is have all eigenvalues with negative real parts; that is, λ½B o 1 and λ Φ nΨ observable, the stability is governed solely by the c's ODE hof iis globally asymptotically stable if and only if Eq.  (11),  which 0 (16) has all eigenvalues with negative real parts; that is, λ Φ 1nΨ  B o1. 1  Matrix Φ nΨ in the stability conditions has the following characteristics: Lemma 1. For each n Z 1 and i; j A f1; …; ng,   1. all eigenvalues of Φ 1nΨ are real and exist in the interval ½0; 1Þ; 1 dλ½ΦðnΨ Þ 2. Z0; dρ ij

3.

dλ½Φð1nΨ Þ Z0. dφi

The proof of Lemma 1 is in Appendix C. Therefore, the learnability conditions of the HME are summarized as follows: Proposition When the state a is unobservable (respectively, observable), the HME is learnable if and only if
h 2. i steady   0 λ½B o1 λ Φ 1nΨ  B o1 . h  0 i

   B o1 (but not vice versa). The Note that part 1 of Lemma 1 means 0 r λ Φ 1nΨ o1 so that if λ½B o1, then λ Φ 1nΨ HME has a feature found in the literature on CS learning (e.g., Bullard and Mitra, 2002): the learnability condition of the HME is more restrictive when the steady state a is unobservable than when it is observable. 4.2. Effect of HM learning We clarify the effects of heterogeneity and misspecification in HM learning on learnability. Hereafter, our discussions are restricted to the case of λ½B 40, otherwise the HME is always learnable. Corollary 1. When the steady state a is unobservable, the learnability condition of the HME under HM learning is independent of the degree of heterogeneity 1  ρij in the PLMs of types i and j for any i; jA f1; …; ng and the degree of misspecification n in the PLMs of all types, and is identical to the learnability condition under CS learning. 9 Note that if the means of fwit gni¼ 1 are nonzero and/or heterogeneous, the fixed points of the constant terms of the PLMs of different types should be different from the steady state of yt and/or heterogeneous. However, this case never violates our analytical results about the learnability. 10 Actually, the ODE can also be asymptotically stable if the Jacobians have one or more zero real parts of eigenvalues, which are ruled out in this paper as nongeneric cases.

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The corollary means that HM learning has no effect on learnability under the observable steady state. Proposition 2 implies that when each agent has to estimate the constant term of the PLM, the learnability of the HME is solely determined by the stability condition of the constant term, λ½B o1, which is identical to the condition under CS learning. Corollary 2. When the steady state a is observable, the learnability condition of the HME under HM learning is not more restrictive than the learnability condition under CS learning. Furthermore, for any n Z1, the learnability condition under HM learning is unaffected or more relaxed by an increase in the degree of heterogeneity 1  ρij in the PLMs of types i and j for each i; j A f1; …; ng or a reduction in the autocorrelation φi for each i. The proof is simple. Proposition 2 indicates that when a is observable and λ½B 4 0, the condition for learnability becomes

 

  more restrictive as λ Φ 1nΨ increases. According to part 2 of Lemma 1, λ Φ 1nΨ is weakly increased as ρij for each i; j

1  increases, and λ Φ nΨ ¼ λ½Φ under the perfect correlations fρij ¼ 1gni;j ¼ 1 or n ¼1, where HM learning is reduced to CS learning (Proposition 1). In addition, the effects of fφi gni¼ 1 are found in part 3 of Lemma 1.11 The nonnegative effect of heterogeneity results from the reductions in the extent of the updating of the parameters fci gni¼ 1 in the PLMs. Suppose that an exogenous variable wit evolves. Without heterogeneity, all types of agents update fci gni¼ 1 to the same degree. In contrast, in the presence of heterogeneity because of the imperfect correlations of fwit gni¼ 1 , the types except type i do not update fcj gnja i to the same degree as the updating of ci. These reductions make the aggregate forecast Ent yt þ 1 and the economy yt less responsive to the evolution of wt so that they produce a positive feedback that helps the parameters to converge to the bounded values. This effect arises if a is observable for at least one agent, and has the potential to increase with the degree of heterogeneity in the PLMs.12 Corollary 2 also means a nonnegative effect of misspecification on learnability. The effect of misspecification also results from the reductions in the extent of the updating of fci gni¼ 1 . Without misspecification (n ¼1), agents update fci gni¼ 1 in response to the evolution of each variable. In the presence of misspecification (n 4 1), agents update only if their observable variables evolve.13 Note that in contrast to the degree of heterogeneity, the relationship between the degree of misspecification and the learnability condition under HM learning depends upon the stochastic characteristics of the exogenous variables. Remark 1. When the steady state a is observable, there can in general be a non-monotonic relationship between the degree of misspecification n and the learnability condition under HM learning. However, if σ ii ¼ σ 40 and ρij ¼ ρ A ½0; 1 and φi ¼ φ A ½0; 1Þ for any i; j A f1; ⋯; ng and i a j, then the learnability condition is unaffected or more relaxed by an increase in the degree n.

    dλ½Φð1nΨ Þ Note that given these conditions, λ Φ 1nΨ ¼ φn 1 þ ðn 1Þρ , and r 0 with the equality iff ρ ¼ 1 or φ ¼ 0. Thus, the dn degree of misspecification has the same effect as that of the degree of heterogeneity on the learnability condition under HM learning when exogenous variables have similar stochastic characteristics, which are typical of, for example, idiosyncratic shocks held by similar economic agents. 4.3. Discussion Our results indicate that the existence of private information about exogenous variables provides a nonnegative impact on the learnability of an equilibrium. In particular, the impact is likely to be increased as the limitation and/or heterogeneity of information sets held by agents are deepened. It provides a policy implication that even if fundamental variables of the economy become privately observable for agents, the government may continue stabilization policies under CS learning as if there were no private information. The impact of private information is economically intuitive by a wide recognition that the limitation and/or heterogeneity of information about fundamentals cause the reduced and slow adjustment of the economy. For example, the seminal Calvo sticky pricing in the New Keynesian macroeconomic model is based on the assumption that firms receive the information relevant to optimal prices occasionally and heterogeneously (Calvo, 1983).14 Veldkamp (2005) demonstrates that the economic boom following a recession becomes slow if less information about the state of the economy is generated in the recession.15 This mechanism works here in a manner that private information causes the reduced extent of the 11

The effects of the autocorrelations of exogenous variables are also found in Marcet and Sargent (1989a) and Bullard and Mitra (2002). In a different framework of heterogeneous learning, Berardi (2007) finds numerically that heterogeneity in PLMs is able to provide a positive effect on learnability. 13 Note that even if n-1 such that each wit has a negligible impact on the economy, fci gni¼ 1 do not necessarily converge to zero, but continue to be updated because fwit gni¼ 1 are correlated. 14 The stickiness of the Calvo pricing also has a nonnegative effect on learnability (see Section 5). Note that the scarcity of information received by firms is represented by the inverse of the notation κ in the forward-looking Phillips curve (18). 15 The abundance and homogeneity of information are also found to increase or speed up the economic adjustment; for example, the sudden crash seen in a good time (Veldkamp, 2005), the quick price-adjustment processes caused by investigative herding (Froot et al., 1992; Hirshleifer et al., 1994). 12

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updating of agents' beliefs and the reduced adjustment of the economy.16 This reduction makes it easy for agents to learn the dynamics of the economy. One might think that the impact of private information should disappear if agents include the lagged endogenous variable yt  1 in their PLMs as public information. In this paper, agents of all types are assumed to have the information fys gts ¼11 , which reflects past evolutions of unobservable exogenous variables, and hence including yt  1 into PLMs seems to cover the disadvantage of updating in the presence of private information. However, the evolutions of unobservable exogenous variables cannot be perfectly identified as long as the number of unobservable exogenous variables for each agent is greater than that of observable endogenous variables; that is, n  1 4 m (see Levine et al., 2012). Hence, the impact of private information is likely to remain even if yt  1 is included in the PLMs. In Section 5.3.2, this case will be numerically illustrated in a NK model. 4.4. Connections to other informational frameworks We confirm that our analytical results hold true not only in the presence of private information, but also in a variety of informational structures that are limited and/or heterogeneous in terms of fwit gni¼ 1 . This is because those structures can be reproduced under Assumption 1 by accommodating the characteristics of fwit gni¼ 1 : that is, n, fρij gni;j ¼ 1 , and fφi ; σ ii gni¼ 1 .17 Let us reproduce, for example, a limited and homogeneous information set considered in the RPE where there exists an exogenous variable unobservable by any agent (e.g, Evans and Honkapohja, 2001, Section 13.1.1). In our model, suppose that there are a large number of exogenous variables (n C 1) such that the population of each type is infinitesimal (1n C 0), and such that an exogenous variable w1t is imperfectly correlated with the other variables fwit gni¼ 2 (fρ1i o1gni¼ 2 ), while the other variables are perfectly correlated (fρij ¼ 1gni;j ¼ 2 ). The HME in this structure is asymptotically equivalent to the RPE where in the presence of two exogenous variables fwit g2i ¼ 1 , each agent has the information of only w2t and forms a misspecified PLM: yt ¼ a þ c2 w2t .18 See Appendix D for other examples. This fact provides the benefit of being able to compare the learnability condition of the equilibrium with private information with the learnability conditions of equilibria with other informational structures: for example, the equilibrium with the full information set (benchmark) and the above RPE. The learnability condition of the HME is unaffected or more relaxed as ρij for each i; j decreases (Corollary 2). The full information set is reproduced by fρij ¼ 1gni;j ¼ 1 , and the limited information set of the RPE is reproduced by fρ1i o 1gni¼ 2 and fρij ¼ 1gni;j ¼ 2 , while private information needs more imperfect correlations than the RPE does. Thus, the learnability conditions of these equilibria are ordered as HME with private info: r RPEr Benchmark; ðunrestrictiveÞ

ðrestrictiveÞ

that is, the condition of the HME with private information is not more restrictive than that of the RPE, which is not more restrictive than that of the benchmark equilibrium. 5. Numerical example In this section, we examine the learnability of the HME in a basic NK macroeconomic model.19 The NK model is a benchmark macroeconomic framework for establishing DSGE models and analyzing optimal monetary policy.20 We obtain the constraints that must be imposed on monetary policy rules to ensure the learnability of the HME. Next, we demonstrate numerically the impact of private information on learnability and the effect of including yt  1 in the PLMs. 5.1. NK model We consider a basic NK model with a monetary policy shock   xt ¼  α it  Ent π t þ 1 þEnt xt þ 1 ;

π t ¼ κ xt þ βEnt π t þ 1 :

ϵt along the lines of Woodford (2003): ð17Þ ð18Þ

The model has three endogenous variables: output gap xt, inflation rate πt, and nominal interest rate it. Eq. (17) is a loglinearized intertemporal Euler equation that is derived from the households' optimal choice of consumption. Eq. (18) is a forward-looking Phillips curve that is derived from the optimizing behavior of monopolistically competitive firms with 16 Similar mechanisms are found in the learning literature that explains the observed persistence of the economy using heterogeneity and/or misspecification in PLMs (e.g., Adam, 2007; Branch and McGough, 2011; Slobodyan and Wouters, 2012; Guse, 2014; Hommes and Zhu, 2014). 17 Note that information sets that are limited and/or heterogeneous in terms of endogenous variables yt cannot be reproduced in our model. See Berardi (2007) to consider heterogeneous PLMs in terms of endogenous variables. h  0 i 18 The learnability condition of the above RPE with the observable steady state is obtained using Proposition 2 as limn-1 λ Φ 1nΨ  B ¼ φ2 λ½B o 1, which corresponds to Evans and Honkapohja (2001, Section 13.1.1)'s model with the observable steady state and AR(1) exogenous variables assumed. 19 Learnability in this section means the local stability under adaptive learning, because a NK model is a log-linearization of a nonlinear model. 20 See Bullard and Mitra (2002), Evans and Honkapohja (2003a,b,2006), Honkapohja and Mitra (2004a, 2005), Adam (2007), Berardi (2009), and Branch and McGough (2009) for an analysis of optimal monetary policy that ensures the learnability of an equilibrium.

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Table 1 Learnability conditions under CS and HM learning. Learning type

Unobservable SS

Observable SS

CS

  κ ϕπ  1 þ ϕx ð1  βÞ 40

HM

  κ ϕπ  1 þ ϕx ð1  βÞ 40

  ð1  λc Þð1 βλc Þ κ ϕπ  λc þ ϕx ð1  βλc Þ 4  α 



 1  λh 1  βλh κ ϕπ  λh þ ϕx 1  βλh 4  α

  Note: The derivations of the learnability conditions are summarized in Appendix E. λc  λ½Φ, λh  λ Φ 1nΨ , and 0 r λh r λc o 1.

Calvo price setting. α 4 0, κ 4 0, and 0 r β o 1 are assumed. The central bank adopts a Taylor-type contemporaneous data interest rate rule: it ¼ ϕπ π t þ ϕx xt þ ϵt ;

ð19Þ

where ϕπ and ϕx are the policy parameters controlled by the central bank, and they are assumed to be nonnegative, and ϵt is a monetary policy shock. To consider the impact of private information in the NK model, we assume that ϵt is the aggregation of idiosyncratic P monetary policy shocks: ϵt  ni¼ 1 ϵit , which stem from, for example, idiosyncratic preference shocks held by the central bank staff. The shock ϵit for each i follows an AR(1) process: ϵit ¼ φi ϵi;t  1 þvit , where 0 r φi o 1 and the disturbance term vit has a zero mean. The correlation of ϵit and ϵjt is ρij Z 0 for each i; j. In the absence of private information, ϵit for all i is observable for all agents, whereas in the presence of private information,   the shock is observable for 1=n of agents and unobservable for other agents. The aggregate forecasts Ent xt þ 1 ; Ent π t þ 1 are the averages of the forecasts of all types  n  n f Eit xt þ 1 ; Enit π t þ 1 gi ¼ 1 . Replacing the aggregate forecast with the average of heterogeneous forecasts in the NK model is justified by following the assumptions given by Branch and McGough (2009), who develop a version of the NK model extended to include two types of agents who are identical except in the way they form expectations. In our model, because of the absence of idiosyncratic shocks in the Euler equation and the Phillips curve, the optimal decision rules of different agents underlying the two equations are identical except in the way of forming forecasts. Therefore, our model with heterogeneous forecasts shares the same form as the NK model with homogeneous forecasts.21

5.2. Learnability conditions   Let us obtain learnability conditions imposed on monetary policy parameters ϕπ ; ϕx in the interest rate rule (19). The learnability conditions of equilibria under CS and HM learning are provided in Table 1, the derivation of which is shown in Appendix E. The conditions under the unobservable and observable steady states are given for each type of learning,   respectively. For ease of comparison, Fig. 1 illustrates the domains of ϕπ ; ϕx that satisfy the learnability conditions given in Table 1. The table shows that if the steady state is unobservable, the learnability condition under HM learning is exactly the same as the condition under CS learning:     κ ϕπ  1 þ ϕx 1  β 40; ð20Þ which is identical to the Taylor principle that ensures the determinacy of the REE (see Bullard and Mitra, 2002). The figure demonstrates that if the steady state is observable, the learnability conditions of equilibria under CS and HM learning are not more restrictive than the Taylor principle. In addition, the learnability condition under HM learning is not more restrictive than the condition under CS learning. Proposition 3. In the NK model (17) and (18) with a Taylor-type nominal interest rate rule (19), the Taylor principle (20) is the sufficient condition for the learnability of an equilibrium under adaptive learning, regardless of the existence of private information and regardless of the observability of the steady state. The proposition reinforces the importance of the Taylor principle in monetary policy. Bullard and Mitra (2002) and Honkapohja and Mitra (2004a) find that with the contemporaneous data rule (19), the Taylor principle is a sufficient condition for the learnability of the REE under CS learning. Guse (2008) confirms that their result is robust even if agents' learning is misspecified. Proposition 3 emphasizes that the sufficiency of the Taylor principle remains true even if agents' learning is heterogeneously misspecified because of private information. 21

See Branch and Evans (2011) and Muto (2011) following the same methodology.

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Fig. 1. Parameter domains of learnability conditions.

5.3. Calibrations Next, we numerically evaluate the impact of private information on the learnability of an equilibrium. Furthermore, we examine whether this impact is robust if agents include the lagged endogenous variable yt  1 in the PLMs.

5.3.1. Effect of HM learning   We calibrate the parameter domains of ϕπ ; ϕx satisfying the learnability conditions under different degrees of heterogeneity and misspecification in the PLMs separately. We focus on the HME with the observable steady state, which produces the effect of HM learning, and exogenous variables with the same stochastic characteristics: ρij ¼ ρ A ½0; 1Þ, and φi ¼ φ A ½0; 1Þ for all i; j and i a j. For robustness, we consider two cases of structural parameters: (a) α ¼ 1, κ ¼ 0:3, and β ¼ 0:99 (Clarida et al., 2000); (b) α ¼ 1=0:157, κ ¼ 0:024, and β ¼ 0:99 (Woodford, 1999). For both cases, we set the autocorrelations φ of the shocks to be equal to 0.9 (Milani, 2008). Fig. 2 shows the parameter domains satisfying the learnability conditions under different degrees of heterogeneity in the PLMs. In each panel, the boundary lines of the domain ensuring the learnability of the equilibrium under CS learning and the domain satisfying the Taylor principle are also plotted for comparison. The domain under ρ ¼ 1:0 corresponds to the learnability condition under CS learning. We consider the existence of 10 idiosyncratic shocks (n ¼10). In both examples of the structural parameters, the learnability domains under HM learning are significantly enlarged as the degree of   heterogeneity 1  ρ increases. When ρ ¼ 0:6, the domains cover all positive values of ϕπ ; ϕx .22 Fig. 3 shows the effect of misspecification in the PLMs. We set ρ ¼ 0:8 and focus on two degrees of misspecification n A f10; 100g. In both examples of the structural parameters, the learnability domains are enlarged by misspecification, while the domain is nearly unchanged by the increase in the degree of misspecification.23 The above results imply that the impact of private information on learnability is economically significant. In particular, if information sets are highly heterogeneous, the learnability of the equilibrium is likely to be ensured under a wide range of policy parameters. In such a situation, the central bank would not need to consider the learnability constraints. On the other hand, the impact of private information is not significantly changed by the degree of the limitation of each information set. 22 Fidrmuc and Korhonen (2003, Table 2) find the correlations of demand shocks between euro countries in the 1990s to be 0.65 at the highest. Those values might be seen as the proxies for the average of the correlations of idiosyncratic shocks at an individual level.

  23 Note that in Proposition 2, if n ¼ f10; 100; þ 1g, then λ Φ 1nΨ ¼ f0:738; 0:721; 0:72g.

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Fig. 2. Heterogeneity and learnability conditions (with the observable steady state). Note: Dashed and dotted lines represent the boundary lines of the parameter domain satisfying the learnability condition under CS learning and the domain satisfying the Taylor principle, respectively.

5.3.2. Existence of public information yt  1 Variations in yt  1 reflect past evolutions of not only observable but also unobservable exogenous variables. If agents have the information fys gts ¼11 , they may utilize yt  1 as public information for updating their forecasts.24 We consider a type i agent's PLM that includes yt  1 into the PLM (14): yt ¼ a þbi yt  1 þci wit þ εit ;

ð21Þ

where coefficient bi is the m  m matrix of coefficients on yt  1 . Under HM learning with Eq. (21) for all i, the learnability of the HME might be similar to the learnability of the equilibrium under CS learning. Fig. 4 shows parameter domains satisfying the learnability condition of the HME with the observable steady state in case (a) of the structural parameters.25 Compared with the results in Fig. 2, the introduction of yt  1 into the PLMs reduces the learnability domains of the HME because yt  1 contributes to the updating of the parameters. However, the domain of the HME continues to be larger than the domain of the equilibrium under CS learning; that is, the extent of the updating of Ent yt þ 1 under HM learning remains reduced. This is because as long as n  1 4 m (that is, the number of unobservable exogenous variables for each agent is greater than that of observable endogenous variables), agents cannot perfectly identify the evolution of the unobservable variables by observing yt  1 . Therefore, the nonnegative impact of private information on learnability remains significant even if agents utilize the public information.26 This result implies that the features of the HME hold true for the equilibrium where each agent specifies a VAR(1) PLM, yt ¼ a þ byt  1 þ εt , which has been considered in the literature as a possible PLM when agents have only data for endogenous variables (e.g., Marcet and Sargent, 1989a; Slobodyan and Wouters, 2012; Hommes and Zhu, 2014). In our model, the VAR(1) PLM is reproduced by n C 1 and fρij ¼ 0gni;j ¼ 1 , where the PLM (21) for each i is asymptotically equivalent to the VAR(1) PLM because the evolution of each exogenous variable makes a negligible contribution to forecasting the dynamics of the economy so that the parameters fci gni¼ 1 are estimated to be zero asymptotically. Thus, the learnability 24 Grossman and Stiglitz (1980) assume that agents infer from market prices the unobservable information of other agents. Marcet and Sargent (1989a) consider that firms in different industries, each of which has private information about its own capital stock, include the prices of outputs of all industries into their PLMs as public information in order to forecast the future prices of their outputs. 25 Here, projection facilities are employed in the real-time learning process to keep agents' forecasts bounded. First, we assume that the eigenvalues of n fbi gi ¼ 1 are strictly inside the unit circle so that agents' forecasts fEit yt þ s g1 s ¼ 1 for all i are asymptotically stationary (see Evans and Honkapohja, 2001, n Chapter 10). Second, we also assume that the diagonal elements of fbi gi ¼ 1 are strictly inside the range ð  1; 1Þ (see Mitra et al., 2013). When projection facilities are applied in this paper, parameter estimates are reset to their previous estimates (see Marcet and Sargent, 1989a). We find that there exists a unique equilibrium, that is, a unique set of parameters satisfying these constraints. 26 The results of case (b) are omitted for reasons of space, but give similar results.

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Fig. 3. Misspecification and learnability conditions (with the observable steady state). Note: Details are as in Fig. 2.

condition for an equilibrium with a VAR(1) PLM is not more restrictive than the condition for the equilibrium under CS learning. 6. Conclusions This paper has investigated the learnability of an equilibrium with private information, which makes agents' information sets limited and heterogeneous. In the real economy, there might exist economic variables observable by some agents and unobservable by other agents. In such a situation, agents are constrained to form heterogeneously misspecified PLMs, and there exists a heterogeneous misspecification equilibrium (HME). The paper finds that the existence of private information about exogenous variables has a nonnegative impact on the learnability of an equilibrium. The learnability condition of the HME is unaffected or more relaxed as the degrees of heterogeneity and misspecification in PLMs caused by private information are increased. In a basic NK model with private information about fundamental shocks, the paper finds that the central bank should follow the Taylor principle for learnability regardless of whether or not there exists private information. The numerical analysis indicates that the impact of private information on learnability is economically significant, and this impact is found to be robust to including lagged endogenous variables into the PLMs as public information. These results are applicable to other frameworks of adaptive learning with misspecified and/or heterogeneous PLMs. Potential issues remain for future research. This paper has focused on the fundamental HME, while the learnability of sunspot HMEs has not yet been investigated. Sunspot equilibria have already been found to be learnable in several macroeconomic models (e.g., Woodford, 1990; Evans and McGough, 2005). On the other hand, Duffy and Xiao (2007) emphasize that those equilibria are not learnable in standard RBC models with empirically plausible values of structural parameters. Although McGough et al. (2013) discover that sunspot equilibria are learnable and empirically plausible in nonconvex RBC models, the existence of private information considered in this paper raises an alternative possibility of such sunspot equilibria. Heterogeneous misspecification in adaptive learning may be investigated as a possible ingredient of the persistence of economic fluctuations. Adam (2007), Milani (2008), and Slobodyan and Wouters (2012) find that adaptive learning prolongs the response of an economy to fundamental shocks in NK DSGE models. This paper finds that heterogeneous misspecification in learning makes an aggregate forecast less responsive to exogenous variables. Thus, the responses of the HME to fundamental shocks are expected to be more persistent than the responses of an equilibrium under CS learning. In addition, heterogeneous misspecification in learning may be applied to the analysis of the impact of financial frictions on the macroeconomy. Assenza and Berardi (2009) consider the impact of bankruptcy in a credit economy where borrowers

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Fig. 4. Heterogeneity and learnability conditions (with the observable steady state and the PLMs with yt  1 ). Note: Details are as in Fig. 2.

and lenders form heterogeneous expectations. While their heterogeneity stems from different gain parameters in learning algorithms, this paper considers the other type of heterogeneity in terms of different forecasting models, which might be prevailing in financial markets where private information exists. Thus, HM learning might well describe the dynamics of a credit economy in the framework of adaptive learning.

Acknowledgments The author thanks the editor James Bullard and two anonymous referees for their valuable suggestions, which have improved the manuscript considerably. The author is also grateful to George W. Evans, Bruce McGough, Ichiro Muto and participants at the University of Oregon Macro Group, the Econometric Society, the European Economic Association, and the Royal Economic Society for their helpful comments and suggestions. An earlier version of this paper was circulated under the title “Learnability of Heterogeneous Misspecification Equilibrium.” For financial support, the author thanks JSPS KAKENHI (Grant Nos. 20730139, 23530234, 15K03372), Murata Science Foundation, Nomura Foundation for Academic Promotion, and Zengin Foundation for Studies on Economics and Finance. Appendix A. Regularity assumptions

Assumption 2. 1. detðI m BÞ a 0 and detðI mn  Φ  BÞ a0. 2. Φ is a diagonal and nonnegative matrix whose diagonal elements exist in the interval ½0; 1Þ. 3. Γ is a nonnegative matrix, in which 0 r ρij r1 for each i; jA f1; …; ng. The first part of the assumption avoids the possibility that a nonexplosive fundamental REE could be indeterminate (see Honkapohja and Mitra, 2006, Proposition 1). The diagonal representation of Φ in the second part simplifies the analysis by equating the eigenvalues of Φ with its diagonal elements existing in the interval ½0; 1Þ. Note that this assumption is not crucial for our analysis, because even if Φ were originally nondiagonal, Eq. (2) could be transformed to an equation that includes a diagonal autoregressive matrix by premultiplying Eq. (2) by the n  n matrix formed from the eigenvectors of Φ. The diagonal elements in the interval ½0; 1Þ ensure the stationarity of wt. Neither is the third part of the assumption crucial for our analysis because any linear model can be transformed to the system with Γ Z 0nn . For example, if any ρ ij is negative in an original model, this negative correlation can be transformed to be positive by changing the sign of w i (or wj) and redefining the correlation between  wi and wj as ρij Z0. Applying this transformation to all negative correlations, the original model is transformed to the system with Γ Z 0nn . Appendix B. Derivations of ODE under HM learning The ODE under HM learning is obtained by accommodating the global convergence of the ODE associated with an RPE in Evans and Honkapohja (2001, Section 13.1.1). Agent i for each i A f1; …; ng forms Enit yt þ 1 by using real-time learning with the PLM (7) and the information set fys ; wis gts ¼ 1 . We assume the t-dating of expectations considered by Evans and Honkapohja n t1 (2001, Chapter 10): coefficient parameters  ϕit at time t are estimated with past data until time 0t, fys ; wis gs ¼ 1 , and Eit yt þ 1 is formed with ϕit and the current data yt ; wit . The estimates of the coefficient parameters ϕit ¼ ðait ; cit Þ are given by the    0 0 least-squares projection of yt  1 on z0i;t  1 ¼ 1; wi;t  1 : Ezi;t  1 yt  1  ϕit zi;t  1 ¼ 02m . Then, the updating rule of ϕit is

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shown by the RLS representation:
0 ϕit ¼ ϕi;t  1 þ t  1 Rit 1 zi;t  1 yt  1  ϕ0i;t  1 zi;t  1 ;

ðB:1Þ


 Rit ¼ Ri;t  1 þ t  1 zi;t  1 z0i;t  1  Ri;t  1 ;

ðB:2Þ

Pt 1

0 where Rit ¼ t s ¼ 1 zi;s  1 zi;s  1 , which is the updating of the matrix of the second moment of zit. The stochastic recursive algorithm (SRA) for ϕit for each i is obtained by substituting the ALM (10) into Eq. (B.1): i0  h 
ϕit ¼ ϕi;t  1 þ t  1 Rit 1 zi;t  1 1 w1;t  1 ⋯ wn;t  1 D0;t  1 D1;t  1 ⋯ Dn;t  1  ai;t  1 ci;tþ  1 ;

P where we denote D0t  A þ Bat , at  1n ni¼ 1 ait as the average of the constant term vectors for all types, Dit  1nBcit φi þ C i for   each i A f1; …; ng, Ci as the i-th column of matrix C in Eq. (1), and citþ  0mði  1Þ ; cit ; 0mðn  iÞ as an m  n matrix in which the columns, except cit, are zero vectors. To obtain the ODEs for ϕi associated with the SRA, we have to calculate the unconditional expectations of the updating terms in the SRA. The convergence of the SRA is analyzed by Marcet and Sargent (1989b) in the stochastic approximation approach, which is also introduced by Evans and Honkapohja (2001, Chapter 6). Denote the operator E as the expectation of variables for ϕi fixed, taken over the invariant distributions of wt. Then, by letting Ezi z0j ¼ limt-1 Ezit z0jt for each i; j A f1; …; ng, the unconditional expectation of the updating term in Eq. (B.1) is transformed to  

 0 ERi 1 zi;t  1 1 w1;t  1 ⋯ wn;t  1 ðD0 D1 ⋯ Dn Þ  ai ciþ 0 11 !0 n
 X 1  1@ 0 0 0 AA 0 @ ¼ Ri Ezi;t  1 zi;t  1 ½ðD0 Di Þ  ðai ci Þ þ E wj;t  1 Dj  wi;t  1 Di wi;t  1 j¼1 0 11 0 01m
 B n B X    C CC Ezi;t  1 z0i;t  1 ½ðD0 Di Þ  ðai ci Þ0 þ B ¼ Ri 1 B @ Ewi;t  1 wj;t  1 D0j  Ewi;t  1 wi;t  1 D0i A C @ A j¼1

0

11 01m
B B CC n X   1  C : ½ðD0 Di Þ  ðai ci Þ0 þ B ¼ Ri 1 Ezi;t  1 z0i;t  1 B @ Ewi;t  1 wi;t  1 Ewi;t  1 wj;t  1 D0  D0 A C @ A 0

j

i

j¼1

In addition, the expectation of the updating term in Eq. (B.2) is given by Ezi z0i Ri : Hence, the ODEs for

ϕi and Ri associated with the SRA are obtained as

  dϕi 0 0 ¼ Ri 1 Ezi z0i T i ðai ; ci Þ  ϕi ; dτ

ðB:3Þ

dRi ¼ Ezi z0i Ri ; dτ

ðB:4Þ

where

0

T i ðai ; ci Þ  @D0

n X

1 Dj ωij ω

 1A : ii

j¼1

A scalar ωij denotes the covariance of wi and wj; ωij  σ ii ρij σ jj for each i; j. Furthermore, because Ri and Ezi z0i in Eq. (B.4) are   0 asymptotically equal, Ri 1 Ezi z0i in Eq. (B.3) globally converges to unity. Hence, the stability of the ODE for ϕi ¼ ðai ; ci Þ in Eq. (B.3) is determined by smaller differential equations: dai ¼ D0  ai ; dτ

ðB:5Þ

n X dci ¼ Dj ωij ωii 1 ci : dτ j¼1

ðB:6Þ

In the same manner, smaller ODEs for the parameters fϕj gj a i are obtained. The ODEs (B.5) and (B.6) for all i are represented by the ODEs for the aggregate parameters ða; cÞ in Eq. (9). First, the ODEs for all ais have the same form, and a is an arithmetic average of all ais. Then, the convergence property of a is equivalent to that of ai for each i; the ODEs for all ais are represented by a single ODE for a that has the same form as that for ai : n

da ¼ T a ðaÞ a; dτ

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where T a ðaÞ  D0 ¼ A þ Ba: Next, the ODEs for all cis are represented by a single ODE for the aggregate parameter c. If the ODEs (B.6) for all i are multiplied by 1n and combined in a single m  n matrix, the single ODE for c is obtained by dc ¼ T c ðcÞ c; dτ where

0 1 n n X X 1 1  1  1 T c ðcÞ  @ Dω ω ⋯ Dω ω A n j ¼ 1 j 1j 11 n j ¼ 1 j nj nn  1 ¼ ðBcΦ þ C Þ Ψ ; n

and

0

1 B 1 B ω21 ω11 Ψ B B ⋮ @

1

1 1 ω12 ω22 ⋯ ω1n ωnn 1 C 1 ⋯ ω2n ωnn C

⋮

⋱

⋮

1 1 ωn1 ω11 ωn2 ω22 ⋯

1

C C A

¼ diag ðσ ii Þ1 r i r n  Γ  diagðσ ii Þ1r1 i r n : The derivation is complete. Appendix C. Proof of Lemma 1 C.1. Proof of Part 1   To find that the eigenvalues of Φ 1nΨ are real and exist in the interval ½0; 1Þ, we use Trenkler (1995)'s lemma. Lemma 2. If X and Y are nonnegative definite matrices of the same dimension, then eigenvalues of XY are real and nonnegative, and they are zero if and only if XY ¼ 0. Proof. If X is nonnegative definite, it may be written as X ¼ PP 0 for some matrix P. Then, the eigenvalues of XY ¼ PP 0 Y are those of P 0 YP (plus possibly some zeros). P 0 YP is obviously a nonnegative definite matrix. It follows that all eigenvalues of XY are nonnegative, and that they are zero iff XY ¼0. The proof is complete.□   As Φ and 1nΨ are nonnegative definite matrices by parts 2 and 3 of Assumption 2, Lemma 2 yields eigenvalues of Φ 1nΨ that are real and nonnegative. Notice that the diagonal elements of ΦΨ are equal to the diagonal elements of Φ; then,      tr Φ 1nΨ o 1. Therefore, all eigenvalues of Φ 1nΨ are real and exist in the interval ½0; 1Þ. The proof is complete. C.2. Proof of Parts 2 and 3 According to the Perron–Frobenius Theorem (see Berman, A., Plemmons, R., Nonnegative Matrices in the Mathematical Sciences, Academic Press, 1979, p. 27), Kolotilina (Kolotilina, L.Y., “Bounds for the Perron root, Singularity/Nonsingularity Conditions, and Eigenvalue Inclusion Sets,” Numerical Algorithms, vol. 42, no. 3–4, 2006, pp. 247–280) shows the monotonicity property of the Perron root (Theorem 2.1): “Let A and B be nonnegative matrices of order n Z 1 and let dðΦð1Ψ ÞÞ dðΦð1Ψ ÞÞ dλ½Φð1nΨ Þ dλ½Φð1nΨ Þ A Z B. Then, λ½A Z λ½B.” In our paper, dρn Z 0 and dφn Z 0 for each i; j A f1; …; ng; thus Z0 and Z 0 for dρ dφ ij

i

ij

i

each i; j. The proof is complete. Appendix D. Reproductions of different informational structures: examples D.1. Different populations of different types of agents As well as the symmetric populations at 1n assumed in Section 3.1, different populations of different types of agents are reproduced under Assumption 1. For example, let us reproduce the existence of two types of agents who have populations 23 and 13, respectively. Suppose the existence of three exogenous variables fwit g3i ¼ 1 , each of which is privately observable for 13 of agents, and the perfect correlation of two of three variables (e.g., w1 and w2). Then, the population of agents detecting the

R. Nakagawa / Journal of Economic Dynamics & Control 59 (2015) 58–74

73

evolutions of w1 and w2 is 23, and the population of agents observing the evolution of w3 is 13. This environment is equivalent to the existence of two types of agents with the populations of 23 and 13, respectively. D.2. Overlapping information sets The information sets that partly overlap with each other in terms of fwit gni¼ 1 are equivalent to the case of the existence of highly correlated exogenous variables under Assumption 1. D.3. Asymmetric information sets The asymmetric information sets of a full information set and a limited information set are reproduced under Assumption 1. Suppose that there exist two exogenous variables fw1t ; w2t g with ρ12 ¼ 0 and σσ 22 C0. In this case, the 11   evolution of yt is governed only by the evolution of w1t, and because Et w2t yt C 0, agents of type 2 are unable to forecast the evolution of yt by observing the evolution of w2t . This structure is equivalent to the case where the agent of type 1 has full information on the economy and the agent of type 2 has a limited information set. Appendix E. Derivations of learnability conditions Before proceeding, we provide a lemma that will be used to obtain the learnability conditions of equilibria under different learning rules. A simpler derivation under CS learning is shown in Bullard and Mitra (2002). Lemma 3. Define an n  n matrix X whose eigenvalues are all real and exist in the interval ½0; 1Þ. Given α Z0, κ Z 0, 0 r β o1,
  1
 1α x αϕπ ϕπ Z 0, ϕx Z 0, and B ¼ 1 þαϕ κ 1 0 β , then the real parts of eigenvalues of X  B  I 2n are all negative if and only if        1  λ½X  1  βλ½X  κ ϕπ  λ½X  þ ϕx 1  βλ½X  4  : ðE:1Þ

α

Proof. Define the eigenvalues of X as 0 r χ i o 1 for each i A f1; …; ng and the eigenvalues of B as δj for each j A f1; 2g. Then, 0 r λ½X  o1, and the eigenvalues of X  B are given by χ i δj for each i; j. First, we show λ½B Z 0 by calculating the characteristic equation of B: qðxÞ ¼ x2 þ p1 x þ p0 , where p0 ¼ 1 þ καϕβ

π þ αϕx

4 0 and p1 ¼ 

1 þ β þ κα þ αβϕx o0. 1 þ καϕπ þ αϕx

According to the

Routh Theorem (see Alpha C. Chiang, Fundamental Methods of Mathematical Economics: Second Edition, McGraw-Hill, 1974), the eigenvalues of B have all negative real parts; that is, λ½B o 0, if and only if p1 and p1 0 are all positive, that is, p1 4 0 1 p0

and p1 p0 40. The above qðxÞ violates these conditions; therefore, λ½B Z 0. Here, let us prove Lemma 3. Because λ½B Z 0 and



 λ½X  Z 0, λ½X  B ¼ λ½X λ½B ¼ λ λ½X B , and hence λ½X  B I2n  ¼ λ λ½X B I 2 . Thus, the real parts of the eigenvalues of X  B  I 2n are all negative if and only if the eigenvalues of λ½X B  I 2 have all negative real parts. The characteristic equation of λ½X B I2 is qðxÞ ¼ x2 þ p1 x þp0 , where         1  λ½X  1  βλ½X  þ α κ ϕπ  λ½X  þ ϕx 1  βλ½X  p0 ¼ ; 1 þ καϕπ þ αϕx         1  λ½X  þ 1  βλ½X  þ κα 2ϕπ  λ½X  þ αϕx 2  βλ½X  : p1 ¼ 1 þ καϕπ þ αϕx Note that p1 ¼ p0 þ

ð1  βλ½X Þ þ αðκϕπ þ ϕx Þ 1 þ καϕπ þ αϕx

; then p1 4 p0 . The eigenvalues of λ½X B I 2 have all negative real parts if and only if

p1 40 and p1 p0 4 0. As p1 4 p0 , the necessary and sufficient condition is given solely by p0 4 0, that is, Eq. (E.1). The proof is complete.□ Substituting Eq. (19) into Eqs. (17) and (18), the NK model is transformed into the form of the system (1) and (2) with
  1
 1α x αϕπ yt ¼ ðxt ; π t Þ0 , wt ¼ ðϵ1t ; …; ϵnt Þ0 , and B ¼ 1 þαϕ 0β . κ 1 Under the unobservable steady state a, the HME (12) and (13) is globally asymptotically stable if and only if the Jacobian (15) has all eigenvalues with negative real parts. In the NK model of the above form, m¼ 2, the eigenvalues of the Jacobian are equal to ½  those of I n  B I 2n . This case corresponds to X ¼I n in Lemma  3. As λ X ¼ 1, the sufficient and necessary condition for the stability of the equilibrium is obtained by κ ϕπ  1 þ ϕx 1  β 4 0, which is also the learnability condition under CS learning. In the same manner, under the observable steady state, the HME is stable if and only if the Jacobian (16) has all   0

  h eigenvalues with negative real parts. This case corresponds to X ¼ Φ 1nΨ in Lemma 3. If we define λ  λ Φ 1nΨ , the    

 h h 1λ 1  βλ h h 0 . Under CS learning, X ¼ Φ . Then, by defining stability condition is provided by κ ϕπ  λ þ ϕx 1  βλ 4  α c c     λc  λ½Φ, the stability condition is obtained by κ ϕπ  λc þ ϕx 1  βλc 4  ð1  λ Þαð1  βλ Þ. Note that 0 r λh r λc o1.

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References Adam, K., 2007. Experimental evidence on the persistence of output and inflation. Econ. J. 117, 603–636. Adam, K., Evans, G.W., Honkapohja, S., 2006. Are hyperinflation paths learnable? J. Econ. Dyn. Control 30, 2725–2748. Allen, F., Gale, D., 2004. Financial intermediaries and markets. Econometrica 72, 1023–1061. Assenza, T., Berardi, M., 2009. Learning in a credit economy. J. Econ. Dyn. Control 33, 1159–1169. Berardi, M., 2007. Heterogeneity and misspecifications in learning. J. Econ. Dyn. Control 31, 3203–3227. Berardi, M., 2009. Monetary policy with heterogeneous and misspecified expectations. J. Money Credit Bank 41, 79–100. Bovi, M., 2013. Are the representative agent’s beliefs based on efficient econometric models? J Econ. Dyn. Control 37, 633–648. Branch, W.A., 2004. Restricted perceptions equilibria and learning in macroeconomics. In: Colander, D. (Ed.), Post Walrasian Macroeconomics: Beyond the Dynamic Stochastic General Equilibrium Model, Cambridge University Press, New York, pp. 135–160. Branch, W.A., 2007. Sticky information and model uncertainty in survey data on inflation expectations. J. Econ. Dyn. Control 31, 245–276. Branch, W.A., Evans, G.W., 2006. Intrinsic heterogeneity in expectation formation. J. Econ. Theory 127, 264–295. Branch, W.A., Evans, G.W., 2007. Model uncertainty and endogenous volatility. Rev. Econ. Dyn. 10, 207–237. Branch, W.A., Evans, G.W., 2011. Monetary policy and heterogeneous expectations. Econ. Theory 47, 365–393. Branch, W.A., McGough, B., 2009. A New Keynesian model with heterogeneous expectations. J. Econ. Dyn. Control 33, 1036–1051. Branch, W.A., McGough, B., 2011. Business cycle amplification with heterogeneous expectations. Econ. Theory 47, 395–421. Bray, M., 1982. Learning, estimation, and the stability of rational expectations. J. Econ. Theory 26, 318–339. Brock, W.A., Hommes, C.H., 1997. A rational route to randomeness. Econometrica 65, 1059–1095. Bullard, J., Eusepi, S., 2014. When does determinacy imply expectational stability? Int. Econ. Rev. 55, 1–22. Bullard, J., Mitra, K., 2002. Learning about monetary policy rules. J. Monet. Econ. 49, 1105–1129. Calvo, G.A., 1983. Staggered prices in a utility-maximizing framework. J. Monet. Econ. 12, 383–398. Clarida, R., Gali, J., Gertler, M., 2000. Monetary policy rules and macroeconomic stability: evidence and some theory. Q. J. Econ. 115, 147–180. Duffy, J., Xiao, W., 2007. Instability of sunspot equilibria in real business cycle models under adaptive learning. J. Monet. Econ. 54, 879–903. Ellison, M., Pearlman, J., 2011. Saddlepath learning. J. Econ. Theory 146, 1500–1519. Evans, G.W., Honkapohja, S., 2001. Learning and Expectations in Macroeconomics. Princeton University Press, Princeton, Oxford. Evans, G.W., Honkapohja, S., 2003a. Adaptive learning and monetary policy design. J. Money Credit Bank 35, 1045–1072. Evans, G.W., Honkapohja, S., 2003b. Expectations and the stability problem for optimal monetary policies. Rev. Econ. Stud. 70, 807–824. Evans, G.W., Honkapohja, S., 2006. Monetary policy, expectations and commitment. Scand. J. Econ. 108, 15–38. Evans, G.W., McGough, B., 2005. Stable sunspot solutions in models with predetermined variables. J. Econ. Dyn. Control 29, 601–625. Fidrmuc, J., Korhonen, I., 2003. Similarity of supply and demand shocks between the euro area and the CEECs. Econ. Syst. 27, 313–334. Froot, K., Scharfstein, D.S., Stein, J.C., 1992. Herd on the street: informational inefficiencies in a market with short-term speculation. J. Finance 47, 1461–1484. Grossman, S.J., Stiglitz, J.E., 1980. On the impossibility of informationally efficient markets. Am. Econ. Rev. 70, 393–408. Guse, E.A., 2008. Learning in a misspecified multivariate self-referential linear stochastic model. J. Econ. Dyn. Control 32, 1517–1542. Guse, E.A., 2014. Adaptive learning, endogenous uncertainty, and asymmetric dynamics. J. Econ. Dyn. Control 40, 355–373. Heinemann, M., 2009. E-stability and stability of adaptive learning in models with private information. J. Econ. Dyn. Control 33, 2001–2014. Hirshleifer, D., Subrahmanyam, A., Titman, S., 1994. Security analysis and trading patterns when some investors receive information before others. J. Finance 49, 1665–1698. Hommes, C., Zhu, M., 2014. Behavioral learning equilibria. J. Econ. Theory 150, 778–814. Honkapohja, S., Mitra, K., 2004a. Are non-fundamental equilibria learnable in models of monetary policy? J. Monet. Econ. 51, 1743–1770. Honkapohja, S., Mitra, K., 2004b. Monetary policy with internal central bank forecasting: a case of heterogenous information. In: Widgren, M. (Ed.), From National Gain to Global Markets, Essays in Honour of Paavo Okko on His 60th Birthday, Taloustieto, Helsinki, pp. 99–108. Honkapohja, S., Mitra, K., 2005. Performance of monetary policy with internal central bank forecasting. J. Econ. Dyn. Control 29, 627–658. Honkapohja, S., Mitra, K., 2006. Learning stability in economies with heterogeneous agents. Rev. Econ. Dyn. 9, 284–309. Levine, P., Pearlman, J., Perendia, G., Yang, B., 2012. Endogenous persistence in an estimated DSGE model under imperfect information. Econ. J. 122, 1287–1312. Marcet, A., Sargent, T.J., 1989a. Convergence of least-squares learning in environments with hidden state variables and private information. J. Polit. Econ. 97, 1306–1322. Marcet, A., Sargent, T.J., 1989b. Convergence of least-squares learning mechanisms in self-referential linear stochastic models. J. Econ. Theory 48, 337–368. McCallum, B.T., 2004. On the relationship between determinate and MSV solutions in linear RE models. Econ. Lett. 84, 55–60. McCallum, B.T., 2007. E-stability vis-a-vis determinacy results for a broad class of linear rational expectations models. J. Econ. Dyn. Control 31, 1376–1391. McGough, B., Meng, Q., Xue, J., 2013. Expectational stability of sunspot equilibria in non-convex economies. J. Econ. Dyn. Control 37, 1126–1141. Milani, F., 2008. Learning, monetary policy rules, and macroeconomic stability. J. Econ. Dyn. Control 32, 3148–3165. Mitra, K., Evans, G.W., Honkapohja, S., 2013. Policy change and learning in the RBC model. J. Econ. Dyn. Control 37, 1947–1971. Muto, I., 2011. Monetary policy and learning from the central bank's forecast. J. Econ. Dyn. Control 35, 52–66. Slobodyan, S., Wouters, R., 2012. Learning in an estimated medium-scale DSGE model. J. Econ. Dyn. Control 36, 26–46. Stiglitz, J.E., Weiss, A., 1981. Credit rationing in markets with imperfect information. Am. Econ. Rev. 71, 393–410. Taylor, J.B., 1993. Discretion versus policy rules in practice. Carnegie-Rochester Conf. Ser. Public Policy 39, 195–214. Trenkler, G., 1995. Eigenvalues of the product of nonnegative definite matrices. Econom. Theory 11, 808. Veldkamp, L.L., 2005. Slow boom, sudden crash. J. Econ. Theory 124, 230–257. Woodford, M., 1990. Learning to believe in sunspots. Econometrica 58, 277–307. Woodford, M., 1999. Optimal Monetary Policy Inertia. Manchester School (67S1), 1–35. Woodford, M., 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press, Princeton, Oxford.