An estimated new keynesian DSGE model with heterogeneous expectations

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An Estimated New Keynesian DSGE Model with

Heterogeneous Expectations

J.P. Witteman

Thesis MSc Economics

Monetary Policy and Banking

Supervisor: dr. C.A. Stoltenberg

University of Amsterdam

2015

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Statement of Originality

This document is written by Student Johan Petrus Witteman who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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ii

Abstract. Recently, several authors have questioned the validity of the Rational Ex-pectations Hypothesis as a cornerstone assumption of contemporary macroeconomic models. In response, several authors have derived New Keynesian DSGE models subject to a Heterogeneous Expectations Hypothesis. These models suggest that the presence of agents with non-rational expectations may have significant implications for the proper conduct of monetary policy. This thesis estimates one such model to assess its empirical merits and implications for monetary policy. The estimation reveals that the estimated fraction of rational agents is close to unity and that the parameter estimates of the Heterogeneous Expectations model differ negligibly from the standard Rational Expec-tations estimates. Moreover, judged by the posterior model probability the Rational Expectations model is preferred over the Heterogeneous Expectations alternative.

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Introduction

The Rational Expectations Hypothesis (REH) is a cornerstone assumption in con-temporary macroeconomic models. It proposes that in forming expectations economic agents use all available relevant information efficiently so that expectations are not sys-tematically biased and thus correct on average with strictly random errors. In terms of an economic model, the implication typically considered is that expectations are model-consistent; that is, the expectations of economic agents coincide with the moments of the true, objective probability distributions of economic outcomes or events. It should be clear that this hypothesis attributes substantial knowledge and information processing capacities to economic decision makers. Agents must fully understand their economic environment, including structural parameters, the distributions and moments of shock processes, and the behaviour of other agents in order to derive objective probability dis-tributions of aggregate outcomes on basis of which they can form Rational Expectations (RE).

This leads some authors to criticise the REH arguing that it is overly optimistic to assume that economic decision makers fully understand the economic environment and use all information efficiently. A commonly posited line of thought is that even econometricians struggle with the inference of economically relevant objective probabil-ity distributions and that it thus seems like a stretch to assume that for some reason economic agents have overcome the problems of econometric inference. The quip is that to assume RE is to assume that apparently everybody has RE except for the profes-sional economist. Such criticism is furthermore near ubiquitously supported by empirical studies on expectation formation. Using data on output, inflation and unemployment expectations from the Michigan Survey of Consumer Attitudes (MSCA), the Livington Survey, and the Survey of Professional Forecasters (SPF), Mankiw et al. (2004), Carroll (2003), and Souleles (2004) find little support for the REH. These studies corroborate earlier work on for instance foreign exchange rate expectations as proxied by survey data by Frankel and Froot (1987), Ito (1990), and Takagi (1991). Such evidence leads Branch (2004) to conclude that the failure of the REH to account for survey-data on expectation formation is amply documented. Instead, these studies typically find that expectations are best characterised by pervasive heterogeneity. A similar picture appears in the labo-ratory experiments on expectation formation by Hommes (2011) and Pfajfar and Zakelj (2014) amongst others.

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1. INTRODUCTION 2

Motivated by this evidence Massaro (2012), Branch and McGough (2009), and Branch and McGough (2010) derive New Keynesian (NK) Dynamic Stochastic General Equilib-rium (DSGE) models under a Heterogeneous Expectations Hypothesis (HEH). These models typically assume that a certain fraction of agents has RE and that the remaining agents employ other forecasting methods which describe their expectations. Analysis of the determinacy properties of these models suggests that the presence of non-rational agents may have significant implications for the proper conduct of monetary policy in the sense that conventional monetary policy responses may induce explosive or indeter-minate dynamics. These authors stress that given these results monetary authorities should take bounded rationality into account in designing monetary policy. These con-clusions are broadly in line with the literature on learning and monetary policy design. Evans and Honkapohja (2003) for instance find that under adaptive learning the Rational Expectations Equilibrium (REE) may not be stable for all monetary policy rules.

Despite these concerns a formal empirical assessment of these Heterogeneous Expec-tations (HE) models and their implications is lacking in the literature. This thesis aims to fill this gap by estimating a HE DSGE model. The main question of interest is whether the model parameter estimates suggest a strong degree of non-rationality in expectation formation thus substantiating the concerns for monetary policy. As a secondary aim this thesis also evaluates the overall fit of the HE DSGE model and compares it to a RE DSGE benchmark.

The model considered in this thesis is in large part due to Massaro (2012). Although the model relaxes the assumption that all economic decision makers form their expec-tations according to the REH, the assumption that agents behave optimally given their (potentially non-rational) expectations is maintained. This results in an analytically tractable model that collapses to a standard textbook NK DSGE model if homogeneous RE are imposed. The heterogeneity of expectations is modelled though a combination of discrete and continuous choice models. In the resulting HE benchmark model a frac-tion of agents then has RE whereas the remaining fracfrac-tion of agents uses rule-of-thumb forecasting rules as the basis of their expectations. The model then can be expressed strictly in terms of RE so that otherwise standard solution and estimations methods can be applied. The model is estimated on US data on output, inflation, and the nominal interest rate between Q1-1966 and Q4-2004. The estimation reveals that the inclusion of non-rational agents in a basic NK DSGE model has little significant qualitative and quantitative effects on model parameter estimates, that the estimated degree of agents with RE is close to unity, and that the data prefers the REH benchmark over the HEH benchmark specification judged both by the posterior model likelihood and posterior model probability. The estimates are robust with respect to the calibrated parameters, with respect to the sampling periods and to alterations in the assumed way in which non-rational agents forecast economic variables. In short, this thesis reports little sup-port for the empirical relevance of heterogeneity in expectation formation. These findings

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1. INTRODUCTION 3

appear to suggest that the concerns voiced by e.g. Massaro (2012), Branch and McGough (2009), and Branch and McGough (2010) with respect to the conduct of monetary policy may be overstated. More generally, despite concerns over the validity of the REH, the models subject to this hypothesis appear to be better empirically supported than the here considered alternative. In general, this too can be thought to alleviate concerns due to potential heterogeneity of expectations.

The remainder of this thesis is organised as follows. First, Chapter 2 presents a general HE DSGE model. Chapter 3 then describes the specification of non-rational expectation formation and presents the model fully parametrised with respect to expectation forma-tion. Next, Chapter 4 presents the estimation of the model. First a REH benchmark is considered and second the HE model. This chapter also reports several reestimated models to probe the robustness of the parameter estimates. Lastly, Chapter 5 concludes.

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CHAPTER 2

The Heterogeneous Expectations Model

This chapter presents a basic HE NK DSGE model. In essence the model is a textbook NK DSGE model derived under the assumption that agents have arbitrary, possibly non-rational expectations. Given these arbitrary expectations agents behave optimally. This setup is sufficiently general to encompass the standard RE DSGE model as a limiting case. The model is in large part due to Massaro (2012) but augmented with three structural shock processes to facilitate estimation in Chapter 4. The shock processes are chosen to ensure that the final reduced form equations are all subject to one shock process. The remainder of this chapter is organised as follows. The first section deals with the households’ problems. Second, the firms’ problems are discussed. Lastly, after a brief note on the specification of monetary policy I aggregate and log-linearize the model to obtain an estimable reduced form system.

2.1. Households

I consider a continuum of uniformly distributed infinitely lived households on the unit interval indexed i ∈ [0, 1]. All households maximise anticipated discounted instantaneous utility over an infinite horizon subject to a sequence of anticipated real flow budget constraints. Formulating the problem as an anticipated utility problem implies that households solve their optimisation problems holding their expectations fixed for all future periods. Households derive utility from consumption and leisure, supply homogeneous labor to monopolistically competitive firms, earn wages and dividend income, and have access to an interest-bearing one-period security that is in zero-net supply to transfer wealth intertemporally. Lastly, I abstract from the role of the government or real money balances. In summary, households solve:

max

{ci,t,li,t,bi,t}∞t=0

e Ei,t ∞ X s=t βs−t c 1−σ i,s 1 − σ − χ l_i,s1+γ 1 + γ ! s.t. e_Ei,t ∞ X s=t

βs−t wsli,s+ eb,s−1Rs−1πs−1bi,s−1+ ds− ci,s− bi,s = 0

ln eb,t= ρbln eb,t−1+ εb,t

εb,t∼ N (0, σε2b),

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where β ∈ (0, 1] is the subjective discount factor, ci,t is a composite consumption

good, σ is the inverse of the intertemporal elasticity of substitution, li,t is household

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2.1. HOUSEHOLDS 5

labor supply, γ is the inverse of the real wage elasticity of labor supply, wt is the real

wage, eb,t−1 is an AR(1) risk-premium shock, Rt is the gross nominal interest, πt is the

gross inflation rate, bi,t is the real household bond portfolio, and dt is the households’

dividend income. The composite consumption good and the aggregate price index are defined by the usual Dixit-Stiglitz CES aggregators,

(2) ci,t = Z 1 0 ci,t(j) η−1 η dj η η−1 and Pt= Z 1 0 Pt(j)1−ηdj 1−η1 ,

where η is the price elasticity of demand for good j, Pt is the aggregate price index

and Pt(j) the price of good j.

Note that all households have the same utility function and budget constraint and thus only differ in the way they form expectations. The main implication is that households with identical expectations will make identical choices. The fraction of households using expectations of type τ can thus be denoted nh

τ.

The households’ problem now is to choose a set of sequences {ci,t, li,t, bi,t}∞t=0 that is

a solution to system (1). Under the REH this solution is characterised by the first-order conditions of the dynamic program and an associated transversality condition. However, this is not necessarily the case under the HEH. The main equation of interest is the standard consumption Euler-equation:

(3) c−σ_i,t = β e_Ei,teb,tRtπ−1t+1c −σ i,t+1

given that the remaining first-order conditions are independent of expectations and the transversality condition can be assumed to be satisfied regardless of expectations; that is, it must either be satisfied ex-post or it must be assumed that although agents do not have rational expectations they are not allowed to believe that they can or will borrow indefinitely.

One solution commonly suggested by the learning literature and in the context of the HEH by Branch and McGough (2009) is to take the subjective Euler equation as a behavioural primitive and to impose the transversality condition ex-post. These authors argue that it is unreasonable to model agents as not equating the marginal benefits and costs of present consumption versus future consumption and as accumulating assets faster than they are discounted. However, Preston (2005) and by extension Massaro (2012) argue that this setup does not describe optimal household behaviour because it fails to account for initial household wealth. Instead they argue that agents’ optimal plans must also satisfy the anticipated intertemporal budget constraint. Preston (2005) furthermore argues that this approach also yields a more internally consistent model and requires less restrictive assumption on agent behaviour. Massaro (2012) follows this line of thought as does this thesis. The main implication is that optimal agent behavior is characterised by expectations of economic conditions over an infinite horizon. This is a direct consequence

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2.1. HOUSEHOLDS 6

of the fact that agents only know their own objectives and constraints and do not have a complete model of determination of aggregate variables. Given that from the perspective of the agent there is no aggregate law of iterated expectations their behavioural rules do not reduce to the standard equations in which only one-period ahead forecasts matter for optimal agent behaviour.

In the present context the derivation of the consumption equation then proceeds as follows. First iterate forward the one-period real flow budget constraint to obtain the intertemporal budget constraint:

e Ei,t ∞ X s=t Qt,sci,s= eEi,t ∞ X s=t Qt,s{wsli,s+ ds} + eb,t−1Rt−1πt−1bi,t−1 Qt,s= s Y k=t e−1_b,k−1R−1_k−1πk e−1_b,t−1R−1_t−1πt ⇐⇒        1 if s = t s Y k=t+1 e−1_b,k−1R−1_k−1πk if s ≥ t + 1 lim s→∞Qt,sbi,s+1= 0 (4)

Next, given that agents solve anticipated utility problems, (3) is iterated forward up to time s and solved backwards to time t to find:

(5) ci,s= eEi,t s Y k=t βkeb,kRkπ_k+1−1 (βeb,sRs)−1πs+1 !1_σ ci,t

Next, recall that the condition for optimal intratemporal substitution is independent of expectations and given by:

(6) c−σ_i,t wt = χlγi,t

Now substituting (6) and (5) in (4) yields:

(7) e Ei,t ∞ X s=t Qt,s s Y k=t βkeb,kRkπk+1−1 (βeb,sRs)−1πs+1 !1_σ ci,t = e_Ei,t ∞ X s=t Qt,s  w 1+γ γ s 1 χ 1_γ s Y k=t βkeb,kRkπ_k+1−1 (βeb,sRs)−1πs+1 !−_γ1 c− σ γ i,t   + e_Ei,t ∞ X s=t Qt,sds+ eb,t−1Rt−1π−1t bi,t−1

This is the HEH equation. It is the non-linear version of the consumption-equation in Massaro (2012). It can be likened to a standard consumption model. Observe that present consumption is decreasing in the expected course of the real interest rate,

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2.2. FIRMS 7

increasing in the expected stream of wage and dividend income, and increasing in present bond holdings.

2.2. Firms

I consider a continuum of uniformly distributed infinitely lived monopolistically com-petitive firms on the unit interval producing differentiated goods j ∈ [0, 1]. Households own the firms. Firms produce good j by employing a homogeneous labor input and a constant returns to scale technology that is subject to an aggregate AR(1) productivity shock,

yt(j) = ea,tlt(j)

where ln ea,t = ρaln ea,t−1+ εa,t

and εa i.i.d.

∼ N (0, σ2 εa)

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Firms hire labor on a perfectly competitive labor market. Firms maximise profits by minimising the costs of their labor inputs and by setting prices for their output subject to a Calvo staggered price-setting constraint. As a result, every period firms only can re-optimise prices with a probability (1 − ω) ∈ [0, 1]. The evolution of the aggregate price index is thus given by:

(9) P_t1−η = ωP_t−11−η+ (1 − ω)P_t∗ 1−η

where P_t∗ is the mean re-optimised price at time t. The solution the the firms’ cost minimisation problems,

(10) min

lt(j)

wtlt(j) + ϕt(j) (yt(j) − ea,tlt(j)) ,

is standard given that it is independent of expectations. The implication is also that marginal costs equate across firms:

(11) ϕt(j) = wt ea,t =⇒ ϕt = wt ea,t

This implies furthermore that in setting prices firms are identical except for their expectations and that as a result firms with identical expectations set identical prices. The fraction of firms using expectations of type τ can thus be denoted nf

τ.

The firms’ pricing decisions are then the solutions to the firms’ profit maximisation problems: (12) max {Pt(j)}∞t=0 e Ej,t ∞ X s=t ωs−t∆s P_t(j) Ps − ϕs P_t(j) Ps −η ys

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2.4. AGGREGATE DYNAMICS 8

and given by:

(13) P ∗ j,t Pt = η η − 1 e Ej,tP ∞ s=tω s−t_∆ s Ps Pt η ϕsys e Ej,tP ∞ s=tωs−t∆s Ps Pt η−1 ys

where ∆t is the stochastic discount factor. Given that households own the firms it is

justifiable that firms discount profits in the same way households would. The implication is furthermore that nh

τ = nfτ = nτ.

2.3. Monetary Policy

To minimise the number of estimable parameters, I assume the monetary authority to follow a simple Taylor-type monetary policy rule:

Rt = πtφπer,t where ln er,t = ρaln er,t−1+ εr,t and εr i.i.d. ∼ N (0, σ2 εr) (14)

This is a slight deviation from Massaro (2012) who assumes that the central bank not only responds to inflation but also to output. As noted in the first section of this chapter, bonds are assumed to be in zero net-supply and the government has no expenditures so that market clearing requires

(15) yt= ct

The total amount of dividends transferred to the households is

(16) dt = yt− wtlt

2.4. Aggregate Dynamics

In order to obtain an estimable reduced form system of aggregate variables I consider a first-order log-linear approximation in the neighbourhood of a non-stochastic zero-inflationary steady state. The resulting model is identical to Massaro (2012) except for the specification of monetary policy and shock processes.

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2.4.1. Investment-Savings Curve. To derive the HEH IS Curve, first log-linearize the consumption equation to find:

γ + (1 − η−1_{) σ} γ ˆ ci,t = (1 − β) (1 − η−1_{) (1 + γ)} γ e Ei,t ∞ X s=t βs−twˆs + (1 − β)η−1Ee_i,t ∞ X s=t βs−tdˆs − γ + (1 − η −1_{) σ} γσ β e_Ei,t ∞ X s=t βs−t ˆRs− ˆπs+1+ ˆeb,t + (1 − β)β−1¯bi,t−1 (17)

Details of this log-linearization are given in Appendix A. Hatted variables denote percentage deviations from the non-stochastic zero-inflationary steady state. Barred vari-ables denote absolute deviations from the non-stochastic zero-inflationary steady state.

Next, integrating (17) over i and noting that R_iˆci,t = ˆct,

R

iEei,t = eEt, and

R

i¯bi,t = 0,

and collecting terms yields:

ˆ ct = (1 − β)(η − 1)(1 + γ) η(γ + σ) − σ Eet ∞ X s=t βs−twˆs+ (1 − β)γ η(γ + σ) − σEet ∞ X s=t βs−tdˆs −β σEet ∞ X s=t βs−t ˆRs− ˆπs+1+ ˆeb,t (18)

Next, the conditions for optimal intratemporal substitution, market clearing, produc-tion and total dividends accruing to households can be log-linearized to yield:

ˆ wt= σˆct+ γˆlt ˆ yt= ˆct= ˆlt+ ˆea,t ˆ dt= η ˆyt− (η − 1)(ˆlt+ ˆwt), (19)

The substitution of this last set of equations yields the the HEH IS Curve:

(20) yˆt= (1 − β)eEt ∞ X s=t βs−tyˆs− β σEet ∞ X s=t βs−t ˆRs− ˆπs+1+ eb,t

These substitution are perfectly consistent given that these equations must hold every period. Note that (20) can be reduced to the standard REH IS Curve by imposing e_Et= Et

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2.4.2. Phillips Curve. To derive the HEH Phillips Curve I log-linearize the firms’ pricing equation, the aggregate price index and the definition of marginal costs to find:

ˆ pi,t = ωβ eEi,t ∞ X s=t (ωβ)s−tπˆs+1+ (1 − ωβ)eEi,t ∞ X s=t (ωβ)s−tϕˆs ˆ πt= 1 − ω ω pˆt ˆ ϕt= ˆwt− ˆea,t (21)

where ˆpi,t = ˆPi,t − ˆPt. Now integrating (21) over i, noting that

R

ipˆi,t = ˆpt, ˆwt =

σˆct+ γˆlt, and ˆyt = ˆct= ˆlt+ ˆea,t straightforward algebra yields the HE Phillips Curve:

ˆ πt = (1 − ω)β eEt ∞ X s=t (ωβ)s−tˆπs+1+ κeEt ∞ X s=t (ωβ)s−tyˆs− κ(1 + γ) (σ + γ)Eet ∞ X s=t (ωβ)s−tˆea,s, κ = (1 − ω)(1 − ωβ)(σ + γ) ω (22)

Note that (22) can be reduced to the standard Phillips Curve by imposing e_Et = Et

and expressing it recursively so that only one-period ahead forecasts matter. 2.4.3. Monetary Policy. The monetary policy equation is approximated by:

(23) Rˆt= φππˆt+ ˆer,t

This equation together with (20) and (22) defines a system of equations in ˆy, ˆπ, and ˆ

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CHAPTER 3

Expectation Formation

The model derived in the previous chapter is general with respect to exactly how ex-pectations are formed. The operator e_Ei,t only denotes arbitrary, potentially non-rational

expectations. This chapter presents the specification of non-rational expectation for-mation. I follow Massaro (2012) and assume that agents can choose to have rational expectations at some cost C or use freely available rule-of-thumb forecasting rules to guide their expectations. The resulting model then is a convex combination of these two classes of predictors. I will first discuss the distribution of expectations and second the fraction of non-rational agents. I then express the general model presented in the previous chapter solely in terms of rational expectations.

3.1. The Distribution of Expectations

Agents’ expectations are represented in belief space Θ and are parametrized by belief parameter θ. Each θi,t ∈ Θ fully characterises the expectations of agent i at time t and

hence determines the agent’s behaviour. I follow Massaro (2012) and in what follows consider θi,t a constant; that is, agents employ constant forecasting rules. Allowing for

differences between agents every period there will be a distribution of expectations. Thus, θi,t ∼ ψt(θ), where ψt(θ) is the probability density function of expectations. From a

modelling point of view θi,t then becomes a random variable. Note moreover that this

specification is quite general in the sense that any expectation formation rule that yields a point predictor can be thought to be an element of Θ.

To provide some structure I follow e.g. Massaro (2012) and assume the choice of predictor to be subject to a fitness criterion. Agents thus evaluate their choice of rule-of-thumb rule based on past performance allowing them to learn from past mistakes. The distribution of expectations then coevolves with the past performance of predictors and observed past outcomes through a continuous choice model over an evolutionary fitness criterion: ψt(θ) = v(θ) exp(δcFt−1(θ)) R Θv(ϑ) exp(δcFt−1(ϑ))dϑ Ft−1(θ) = −(θ − xt−1)2, x ∈ {y, π, R} (24)

where v(θ) is an opportunity function that puts various weights on various parts of Θ and δc is the intensity of choice that captures how strongly agents respond to differences

in predictor performances. To keep in line with Massaro (2012), the fitness measure

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3.2. THE FRACTION OF RATIONAL AGENTS 12

F is parametrised as minus the past squared forecast error. This can be argued to be furthermore justified by the empirical work of Branch (2004) that based on survey data on inflation expectations reports evidence of agents switching between various forecasting rules depending on this criterion, and the laboratory experiments of Hommes (2011) that yield similar findings.

The distribution of expectations can now be characterised by the moments of ψt(θ).

As in Massaro (2012) I assume that v(θ) = 1, which implies that agents initially weigh parameter values equally. Next, given that the denominator of ψt(θ) is independent of θ:

(25) ψt(θ) ∝ exp(−δc(θ − xt−1)2)

Next, recall that the probability density function of a normal distribution is propor-tional to the exponential of a quadratic equation and note that Ft−1(θ) is quadratic in θ.

It hence follows that ψt(θ) ∼ N (µt, σt2). Straightforward comparison of exponents then

yields: (26) −δcθ2+ 2δcxt−1θ − δcx2t−1= − 1 2σ2 t θ2 + 1 σ2 t µtθ − 1 2σ2 t µ2_t =⇒ µt= xt−1, σt2 = 1 2δc

The remainder of the distribution is accounted for by the denominator of the con-tinuous choice model. The dynamics of these moments now characterise ψt(θ). In the

absence of dependence between agents for a sufficiently large economy the law of large numbers applies and the average expectation ¯θ in Θ will converge to the mean of ψt(θ)

for all t which is xt−1, x ∈ {y, π, R}.

3.2. The Fraction of Rational Agents

Given the evolution of the distribution of predictors, I assume the decision to have rational expectations or not is given by a discrete choice model. The assumption is that agents are capable of obtaining rational expectations but only through some effort. At the beginning of every period agents can choose to gather and process sufficient information in order to derive the probability distributions that allows them to have RE. This, however, comes at some cost C. If they choose to not have rational expectations agents resort to using freely available heuristics to forecast economic outcomes. Given these two classes of predictors, agents determine the utility of each class of predictors based on an evaluation of past performance and the costs associated with each class according to a discrete choice model: (27) nτ,t = exp(δdV (Fτ,t−1)) PT τ =1exp(δdV (Fτ,t−1)) , τ ∈ T

where T denotes the set of possible expectation formation strategies and V (Fτ,t−1)

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3.3. THE ASSOCIATED RATIONAL EXPECTATIONS REPRESENTATION 13

associated with each class of predictors is parametrised as the squared forecast error after accounting for shocks minus the costs of that predictor. Thus, let the RE predictor have an associated (effort) cost C and let the heuristics-predictor have no cost and let τ = RE and T = {RE, θ}. Then,

(28) nRE,t=

exp(−δdC)

exp(−δdC) + exp(−δdUθx,t−1)

Given that the RE predictor has no systematic bias but comes at a cost, the utility of that predictor is simply the product of the intensity of choice of the discrete choice model and the cost C. Regarding the class of heuristics predictors, the forecast-error of θx, x ∈ {y, π, R} over the distribution is given:

Fθx,t−1 = X x Z (θx,t−1 − ˆxt−1)2ψt−1(θx) dθx =X x 1 2δc,x (29)

Observing that this latter equation is in fact a constant and recalling that the heuris-tics predictors have no cost, the utility of the heurisheuris-tics forecast is simply the sum of variances of θx. It then follows that:

(30) nRE = exp(−δdC) exp(−δdC) + exp −δdP_x 1 2δc,x

That is, the fraction of rational agents is a constant depending on the intensity of choice of the discrete choice model, the cost associated with obtaining rational expecta-tions and the variance of the distribution of non-rational expectaexpecta-tions.

3.3. The Associated Rational Expectations Representation

The previous section allows for the explicit expectational parametrisation of the HEH IS and Phillips Curves. Bisecting the HEH function into a rational and a non-rational part yields: ˆ yt = nRE (1 − β)Et ∞ X s=t βs−tyˆs− β σEt ∞ X s=t βs−t ˆRs− ˆπs+1+ ˆeb,t ! + Z 1 nRE (1 − β)e_Ei,t ∞ X s=t βs−tyˆs− β σeEi,t ∞ X s=t βs−t ˆRs− ˆπs+1+ ˆeb,t ! di Doing the same for the Phillips Curve yields:

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3.3. THE ASSOCIATED RATIONAL EXPECTATIONS REPRESENTATION 14 ˆ πt= nRE (1 − ω)βEt ∞ X s=t (ωβ)s−tπˆs+1+ κEt ∞ X s=t (ωβ)s−tyˆs− κ(1 + γ) (σ + γ) Et ∞ X s=t (ωβ)s−tˆea,s ! + Z 1 nRE (1 − ω)β e_Ei,t ∞ X s=t (ωβ)s−tπˆs+1+ κeEi,t ∞ X s=t (ωβ)s−tyˆs− κ(1 + γ) (σ + γ) Eei,t ∞ X s=t (ωβ)s−teˆa,s ! di The rational parts of these equations can be expressed recursively. The non-rational parts can be rewritten using the results obtained in the previous section. First, note that given the fact that agents solve anticipated utility problems and employ constant predictors: (31) Ee_i,t ∞ X s=t βs−txˆs= 1 1 − β ˆ θi,x,t, x ∈ {y, π, R}

Second, by the analysis of the previous section the integral reduces to:

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Z 1

nRE

ˆ

θi,x,tdi = (1 − nRE)ˆµx,t = (1 − nRE)ˆxt−1, x ∈ {y, π, R}

Note that these equations are only descriptive with respect to what non-rational agents predict, not with respect to what they observe contemporaneously. Note moreover that non-rational agents are not assumed to forecast future shocks. Regarding contempora-neous observations I assume that non-rational agents observe contemporacontempora-neous shocks. Furthermore following Massaro (2012) and a large part of the learning literature, it is assumed that the non-rational agents observe the current interest rates. The conjunction of the preceding yields an Associated Rational Expectations Representation (ARER) of the general HEH DSGE model. In the following estimation this model will be considered the HEH benchmark.

ˆ yt= nRE Etyˆt+1− 1 σ ˆRt− Etπˆt+1 + (1 − nRE) ˆ yt−1− β (1 − β)σ (1 − β) ˆRt+ β ˆRt−1− ˆπt−1 − 1 σeˆb,t ˆ πt= nRE(βEtˆπt+1+ κˆyt) + (1 − nRE) (1 − ω)β 1 − βω πˆt−1+ κ 1 − βωyˆt−1 − κ(1 + γ) σ + γ eˆa,t ˆ Rt= φπˆπt+ ˆer,t (33)

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CHAPTER 4

Estimation

In this chapter I estimate the HE DSGE model derived in the previous chapters. Given that the model can be reduced to an ARER model otherwise standard methods for solving and estimating DSGE models can be applied. I estimate the model using quarterly data on US output, inflation and the nominal interest rate from Q1-1966 to Q4-2004. The estimation reveals that the fraction of rational agents is close to one and that empirically the REH benchmark model is preferred over the HEH alternative. Moreover, the model parameter estimates align closely over both models. These findings appear to suggest little empirical relevance of the HEH hypothesis in the context of a NK DSGE model and that the concerns voiced with respect to the conduct of monetary policy in this context may be overstated.

Output is measured as the real seasonally adjusted gross domestic product (GDP) in 2009 dollars. Inflation is measured as the change in the Consumer Price Index (CPI) for urban households. The gross nominal interest rate is measured by the effective federal funds rate. To match these series to the model however, several transformations are in order. First, the logarithm of the GDP series is detrended using a one-sided Hodrick-Prescott (HP) filter. A similar procedure is employed on the series for inflation. Lastly, given that the effective federal funds rate is reported in annualised terms, it first is approximated in quarterly terms before its logarithm is detrended using the one-sided HP filter. A one-sided HP filter was employed due to the non-causal nature of the two-sided HP filter (see Stock and Watson (1999) for a discussion). The MATLAB implementation of the one-sided HP-filter is due to Alexander Meyer-Gohde and available through IDEAS RePEc.

In all estimations the initial posterior mode is obtained using a two-step procedure that first maximises the posterior likelihood using a CMA-ES algorithm and second using a Monte-Carlo (MC) optimiser. This proved the most reliable way of finding a well-behaved initial posterior mode. The MC optimiser furthermore sets the Metropolis-Hastings (MH) scale-parameter so that the acceptance ratio of the MH chains is roughly between 30 and 40 percent. With one exception, the final reported estimates are obtained by 2 MH-chains of 2.5 million draws each, the first 1.25 million of which are discarded as burn-in.

In what follows I first present a RE benchmark and then proceed with the estimation of the HE benchmark alternative. After a brief note on model comparison I continue to probe the sensitivity of the model parameter estimates with respect to the calibrated

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4.1. RATIONAL EXPECTATIONS BENCHMARK 16

parameters, the sampling period and the proposed rule of non-rational expectation for-mation.

4.1. Rational Expectations Benchmark

To obtain a benchmark model I first estimate the model under homogeneous RE; that is, the parameter nRE is calibrated to one. As noted in the preceding chapters, the model

then collapses to a standard NK DSGE model.

4.1.1. Prior Distribution. For the rational expectations benchmark I use the same priors as in Smets and Wouters (2007). I calibrate two parameters given well known identification issues in small-scale NK DSGE models. The calibrated parameters are β = 0.99 and γ = 1. Below, I check for the robustness of the estimation with respect to the second parameter. The inverse of the intertemporal elasticity of substitution σ is assumed to be distributed normally with a mean of 1.5 and a standard-error of 0.37. The Calvo price-stickyness parameter ω is assumed to follow a beta-distribution with a mean of 0.5 and a standard-error of 0.1. The monetary policy response coefficient φπ

is assumed to be normally distributed with a mean of 1.5 and a standard error of 0.25. All the auto-correlation coefficients of the shock processes are assumed to be distributed according to a beta probability density function with mean 0.5 and a standard-error of 0.2. Lastly, all exogenous shocks are assumed to follow an inverse gamma distribution with a mean of 0.1 and a standard error of 2.

4.1.2. Posterior Distribution. The posterior estimates of the RE benchmark model are presented in the table below under M1. The first column reports the prior

distribu-tions, the second column the posterior mean and the third colum the 90 percent credible interval. The log data density L is 1758.166. All parameters are fully identified by the model and the data. The majority of the posterior estimates are in line with the posterior estimates reported by Smets and Wouters (2007). However, the inverse of the elastic-ity of intertemporal substitution and the Calvo parameter are (markedly) higher than typically estimated. It should be noted, however, that more conventional estimates for these parameters– e.g ω ∼ 0.66 – can be obtained by shutting down one of the shock processes and estimating the model on two instead of three data-series.1 Estimating the model in this fashion however results in a failure of the rank conditions necessary for full identification of the parameters. Second, some specifications in Smets and Wouters (2007) that probe the sensitivity of their model with respect to the parameters relating to capital as a factor of production yield estimates closer to the estimates generated by M1. It therefore stands to reason that the inclusion of capital in the present model would

align estimates closer. Supposing this is in fact the case the present deviation of the M1

estimates from the Smets and Wouters (2007) posterior distributions should be no real

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4.2. HETEROGENEOUS EXPECTATIONS BENCHMARK 17

M₁ L = 1758.166 M₂ L = 1749.750 Prior Post. Mean HCI Prior Post. Mean HCI β 0.99 0.99 0.99 0.99 0.99 0.99 σ N (1.5, 0.37) 3.283 [2.879,3.708] N (1.5, 0.37) 3.242 [2.826,3.677] γ 1 1 1 1 1 1 ω β(0.5, 0.1) 0.857 [0.837,0.879] β(0.5, 0.1) 0.834 [0.810,0.861] φπ N (1.5, 0.25) 2.750 [2.521,2.984] N (1.05, 0.25) 2.547 [2.444,2.640] nRE 1 1 1 U (0, 1) 0.978 [0.967,0.989] ρa β(0.5, 0.2) 0.983 [0.973,0.993] β(0.5, 0.2) 0.957 [0.923,0.989] ρr β(0.5, 0.2) 0.876 [0.855,0.897] β(0.5, 0.2) 0.737 [0.668,0.806] ρb β(0.5, 0.2) 0.408 [0.333,0.482] β(0.5, 0.2) 0.452 [0.370,0.532] εa Γ−1(0.1, 2) 0.015 [0.013,0.017] Γ−1(0.1, 2) 0.017 [0.014,0.019] εr Γ−1(0.1, 2) 0.012 [0.012,0.013] Γ−1(0.1, 2) 0.012 [0.012,0.013] εb Γ−1(0.1, 2) 0.015 [0.012,0.017] Γ−1(0.1, 2) 0.016 [0.013,0.019]

concern. I the remainder of this thesis I therefore choose to estimate all models on three data-series and to employ the estimates presented under M1 as the REH baseline.

4.2. Heterogeneous Expectations Benchmark

The Heterogeneous Expectations benchmark considered here is the model derived in the previous chapters. The estimated ARER model is the last system listed in Chapter 3.

4.2.1. Prior Distributions. The prior distribution for this estimation is identical to that of the rational expectations benchmark with the exception of the fraction of rational agents and the monetary policy coefficient. I take an agnostic view with respect to the fraction of rational agents and assume a uniform prior distribution on the unit interval. The resulting prior mean of 0.5 also aligns with what some authors consider a likely degree of rationality. Massaro (2012) for instance argues 0.4-0.6 to be an appropriate plausible range of heterogeneity in the context of the very model estimated here. Estimates by Gali and Gertler (1999) of a NK Phillips Curve with fractions of forward- (rational) and backward-looking firms are however suggestive of a higher degree of rationality of between 0.6 and 0.8. Laboratory experiments of Pfajfar and Zakelj (2014) however suggest a degree of rationality as low as 0.4. Given this wide range of suggestions and the few concrete studies with respect to such a fraction a uniform prior seems justified.

The specification of the monetary policy coefficient takes a little more care. Indeed, the theoretical work of e.g. Massaro (2012) and Branch and McGough (2009) illustrates clearly that even the presence of a small number of non-rational agents may significantly alter the determinacy properties of the model. In the present context the implication is that the usual necessary condition φπ > 1 for a unique stable solution may not yield

a determinate equilibrium. To find a prior for which the model is stable and unique at the prior means I perform a grid search on the (nRE, φπ) tuples of the bifurcations

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script augmented with the Simpy and Numpy libraries for scientific computing. The bifurcations are listed in Appendix B. The grid considered is nRE ∈ [0, 1] ∪ φπ ∈ [0, 2].

The grid is constructed using 1500 evenly spaced points per unit-interval per axis. Given that the model has two forward-looking variables a unique saddle-path solution requires exactly two eigenvalues outside of the unit disk.

The output of this grid-search is plotted in the figure below. The various curves in the figure delineate the areas for nRE and φπ that produce various combinations of multiple

real and complex eigenvalues. The large top-right area is the only area that produces a determinate solution. The figure serves several purposes. First, it stresses the results obtained by Massaro (2012), Branch and McGough (2009), and Branch and McGough (2010) showing that for some low degrees of rationality typical monetary policy responses– e.g. φπ = 1.5–could result in unstable dynamics. In the present context they would result

in explosive solutions. Second, the figure shows that in principle the central bank can always ensure a unique stable equilibrium as long as it is aware of the degree of non-rationality. Although the area corresponding to a determinate solution is shrinking for lower values of nRE, there always exists a (nRE, φπ) tuple that yields a unique saddle-path

stable solution given the prior means. Third, the figure shows that the usual necessary condition φπ > 1 still applies but that there is a boundary condition to φπ > 1 conditional

on nRE. Lastly, the figure allows the selection of a prior mean for φπ > 1 that results

in a stable solution at the prior mode so that the initial posterior likelihood can be maximised and the model can be estimated. The selected prior distribution then is that φπ is assumed to be distributed normally with a mean of 1.05 and a standard-error of

0.25. It should be stressed that this choice of prior is purely mechanical and in no way reflects any factual prior belief of the distribution and moments of this parameter. What should furthermore be stressed here is that as a result the tacit conceptual assumption is that the monetary authority is aware of the distribution of expectations and therefore always picks a policy response coefficient that stabilises the economy after the realisation of a shock and a single unique solution to the model exists. The question then becomes given that the assumption of determinate equilibria is maintained, to what degree is the chosen prior degree of non-rationality of expectation formation supported by the data?

4.2.2. Posterior Distributions. The posterior distributions of the baseline HEH model are listed in the table above under header M2. Similar to the RE model all

parameters in the model are fully identified. Interestingly, they are in fact identified more strongly in M2 than in M1. Perhaps surprisingly, the parameter estimates are

virtually identical to the REH baseline and the estimated fraction of agents with RE is in the neighbourhood of 0.98 implying that a mere 2 percent of agents should be classified as having non-rational expectations. This is a striking result. First, it is very much at odds with the micro-evidence presented by studies based on survey-data and laboratory experiments on expectation formation. These studies typically hint at much larger degrees of non-rationality. Apparently the conclusions of these studies on the distribution of

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expectations carry little weight if one tries to match such a distribution to observed economic outcomes. What is striking moreover is the fact that the data apparently is quite informative with respect to nRE given the small 90 percent credible interval,

suggesting there is little uncertainty with respect to the estimated degree of rationality in expectations. This cannot easily be reconciled with the current empirical literature either that finds strongly varying plausible degrees of non-rationality in expectations. This is what partially informed the use of a uniform prior. In light of the posterior this choice of prior almost seems unfounded. The posterior distribution of nRE in fact has so

little variance that it almost is hard to speak of a distribution at all. Third, the limited importance of non-rationality in expectations is striking because the non-rationality of expectations is modelled as essentially backward-looking behaviour. It is well known that RE DSGE models augmented with backward-looking features like habit-formation in consumption or inflation indexation typically perform better than their purely forward-looking counterparts. Somewhat mechanical reasoning would then suggest that this would also be the case through backward-looking expectations. However, judging by the log data density the HE benchmark actually performs worse and parameter estimates are

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4.4. ROBUSTNESS 20

unaffected. The M2 estimates then carry two important implications. First, with respect

to the concerns voiced for the conduct of monetary policy the parameter estimates seems to suggest that these concerns from an empirical point of view are overstated. That is, supposing that economic equilibria are determinate, these equilibria are associated with a high estimated degree of rationality in expectations. Second, despite concerns over the validity of the REH, models based on this assumption appear to account better for the data than a HEH alternative.

4.3. Model Comparison

Judged by the log data density the RE model M1 appears to be preferred to the

HE model M2. Although the log data density can be used to compare models it is

sensitive with respect to the choice of priors. In the current comparison M1 has a very

tight prior for nRE—namely E[nRE] = 1 with zero variance—whereas M2 has a very

loose prior—namely E[nRE] = 0.5 over a uniform distribution on the unit interval. To

correct for this I calculate the posterior model probability that M1 is preferred to M2

based on the Partial Bayes Factor. It is argued that this method for model comparison is less sensitive to the use of different priors (O’Hagan, 1995; Sims, 2003). For a recent application of this methodology in the context of a DSGE model see e.g. Kriwoluzky and Stoltenberg (2015). In short, the posterior model probability of M1 compared to M2 is

given by: (34) p(M1|D, D0) = p(D|M1) p(D0_|M 1) π(M1) p(D|M1) p(D0_|M 1) π(M1) + p(D|M 2) p(D0_|M 2) π(M2)

where p(·) is a conditional probability function, D is the data sample, D0 is a partition of the data usually referred to as the training sample, and π(·) denotes the prior model probability.

I employ a sample of the data between 1966:Q2-1979:Q2 as a training sample. This period is commonly referred to as the Great Inflation. This sample is also employed as a robustness check below. In comparing models I assign equal prior probabilities to both models. Straightforward computation then yield a posterior model probability of model M1 compared to M2 of ∼ 0.999 indicating that M1 is likely to be preferred to M2.

4.4. Robustness

This section probes to what extent the model parameter estimates are sensitive with respect to the the calibrated parameters, the sampling periods and the specification of non-rational expectation formation.

4.4.1. Calibrated Parameters. In the preceding estimations γ was calibrated to unity. To probe for the robustness of the estimates with respect to this calibration, I

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4.4. ROBUSTNESS 21

calibrate γ to 2 and reestimate models M1 and M2. All remaining prior distributions

remain unchanged. The reestimated models M3 and M4 are listed in Appendix C1. The

reestimation of the models reveals little substantial qualitative or quantitative differences with the estimates under γ = 1. The model likelihoods are somewhat depressed however. The estimated degree of rationality in expectations is of comparable magnitude across specifications.

4.4.2. Sampling Periods. To probe the robustness of the model with respect to the sampling period the baseline REH and HEH models are reestimated first on data between 1966:Q2-1979:Q2 and second on data between 1984:Q1-2004:Q4. The choice of sampling periods is motivated by Smets and Wouters (2007). These periods are typically dubbed the ’Great Inflation’ and ’Great Moderation’ respectively. The results are presented in in Appendix C2. Models M5 and M6 are the REH model subsample estimates of the

’Great Inflation’ and ’Great Moderation’ respectively. Models M7 and M8 are the HEH

counterparts.

By and large, the within subsample estimates align quite closely with little quantita-tive differences between models M5 and M7, and models M6 and M8. Across samples,

there are some differences in parameter estimates along the lines of those reported in Smets and Wouters (2007). Notable are for instance the estimated shock process vari-ances, which are estimated markedly higher in the Great Inflation period compared to the Great Moderation period in both model specifications. The main parameter of interest in this thesis however, nRE, is estimated to be remarkably stable across both periods albeit

somewhat lower than in the HE benchmark model M2.

4.4.3. Expectation Formation. The model estimated here due to Massaro (2012) has assumed that non-rational agents use heuristics to forecast economic variables and that such rule-of-thumb behaviour can be described by constant expectations. In effect, this boils down that agents forming expectations ’naively’. However, another popular ap-proach is to assume agents form expectations according to some adaptive or extrapolative mechanism based on past observations. This typically boils down to agents employing basic deterministic AR(1) forecasts–see e.g. Branch and McGough (2009). As a final robustness check I forgo the analysis of Chapter 3 and simply impose such expectation formation rules ad-hoc. The reason for this final check is twofold. First, it is a common way to model bounded rational expectations. Secondly from a more technical perspective the previous estimates seem to suggest that the coefficients implied by the HEH do not allow for apt fit with the data–i.e. these coefficients may be too large or too small. Allow-ing for more adaptive behaviour provides a simple scalAllow-ing mechanism, albeit mechanical and introduced ad-hoc.

Model M9 assumes that non-rational expectations are made as ρxt−1, x = {y, π, R}.

Model M10is slightly more sophisticated and assumes that these forecasts are represented

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4.4. ROBUSTNESS 22

distribution with a mean of one and a variance of 0.25. The prior mean of ρx in M10 is

assumed to be unity for all x with a normal distribution with variance 0.25. This choice of prior is motivated by the analysis of Branch and McGough (2009) who consider these parameters slightly above and slightly below unity in their determinacy analysis. In this case these parameters are estimated in this neighborhood.

The model estimates are reported in Appendix C3. The estimates generated by M9

are highly comparable to the posterior distributions of the HE baseline model M2. The

main parameter of interest nRE is negligibly different from the estimate in M2. The

autocorrelation coefficient on the non-rational agents’ expectants is close the unity thus mimicking naive expectations. The last model considered here, M10, is slightly more

sophisticated in the sense that all forecasted variables are assumed to carry their own autocorrelation coefficients. This estimation reveals several interesting things. First and foremost, the estimated degree of agents with rational expectations is once again high. Although estimated slightly lower than in the HE benchmark, qualitatively the implications are highly similar. Second the fit of this model is markedly better and the estimated autocorrelation coefficients on the forecasted variables differ substantially. This suggests that apparently adaptive or extrapolative expectations better describe the patterns of economic outcomes than the naive expectations used in the HE baseline. Given the high estimated degree of rationality however, it seems more likely that the effect is induced through a mechanical refinement of the backward-looking component than anything else.

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CHAPTER 5

Conclusion

This thesis has assessed whether the concerns voiced with respect to the conduct of monetary policy due to heterogeneity in expectation formation have strong empirical support. The main question of interest has been whether the degree of estimated non-rationality in expectation formation is sufficient to induce limitations to monetary policy responses. The secondary aim has been to assess the support for HE DSGE models in general. An estimation of the HE DSGE model due to Massaro (2012) has revealed only minor differences in model parameter estimates compared to a REH benchmark and that–perhaps surprisingly–the estimated fraction of agents with rational expectations is close to one. Moreover, the REH model in general appears to be preferred over the HEH alternative. These findings suggest that concerns with respect to the conduct of monetary policy may be overstated.

These findings are striking for several reasons. First, given the micro-evidence pro-vided by survey-data and laboratory experiments on expectation formation it is surprising to find so little evidence supporting non-rational expectations in a DSGE model. Ap-parently such micro-evidence does not translate well to a macroeconomic context based on observed economic outcomes. What drives this discrepancy has not been studied here but should be looked into in future research. Second, the obtained estimates are surprising given the fact that the non-rationality of expectations has been modelled as essentially backward-looking behavior. Although DSGE models augmented with various backward-looking components like habit-formation or inflation-indexation typically per-form much better than their strictly forward-looking counterparts, the estimations that have been presented above suggest that backward-looking behavior due to differences in expectation formation does little to account for the observed patterns in the data. This would tentatively suggest that the coefficients on the backward-looking terms implied by the derivation of the model under a HEH simply cannot be reconciled with observed economic outcomes.

Although the model parameter estimates have proven to be quite robust in the present setup, a few notes nevertheless apply. First, the parametrization of expectation formation may be too simplistic; that is, non-rational agents may have been supposed to be too non-rational. Although the parametrization considered is well grounded in the literature, future work would nevertheless do well to consider a greater multitude of forecasting strategies and fitness criteria. On the other hand though, although the parametriza-tion considered has been on the lower end in terms of sophisticaparametriza-tion, it may well be at the upper end in terms of the implications of non-rationality in expectation formation.

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5. CONCLUSION 24

Conceptually, the more knowledge and information processing capacities are endowed to agents, the closer their expectations should align with the RE predictor. It may therefore be argued that the present parametrization entails an upper bound in terms of implica-tions of a HEH in a NK DSGE model. Second, the model considered here only is a small scale model that abstracts from the role of government, capital, and money amongst others. It also has not included features like inflation-indexation that are typical of the current state-of-the-art DSGE models. It therefore stands to reason that some parame-ter estimates may change (substantially) if reestimated in a larger framework. How this would affect the estimated degree of rationality is unclear and should be assessed in future work. A highly interesting question would be to what extent controlling for conventional backward-looking components allows any effect of non-rationality in expectation forma-tion to persist. Third, the fracforma-tion of raforma-tional agents has been designed to be a model parameter and not a variable. A possibility completely avoided in this thesis is that fluctuations of the degree of rationality may have implications for parameter estimates and thereby monetary policy. Even though the robustness checks with respect to the sampling period suggest that this fraction is fairly constant over the medium-run, there may still be important short-run effects. Such effects have not been studies in this thesis, but should be in future research. Fourth, the estimations have only revealed that under the assumption of determinate equilibria such equilibria are associated with a high degree of rationality. A possible extension of this thesis lies in allowing for indeterminate dy-namics in equilibrium. Concretely, the extension lies in the fact that the model presented here could be reestimated using an estimation methodology due to Lubik and Schorfheide (2004) and more recently applied by Kriwoluzky and Stoltenberg (2015), that allows to obtain parameter estimates even if multiple bounded solutions exist. To what extent such a change in methodology affects the model estimates and thereby the conclusions drawn in this thesis remains an open question. Lastly, although this thesis documents the fact that determinate equilibria are associated with high fractions of rational agents it provides no insight into why exactly that is the case. Future work should try to account for the here documented high fraction of rational agents.

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Bibliography

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Expectations’. Journal of Economic Dynamics and Control 33 (5), 1036–1051.

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Mankiw, N. G., R. Reis, and J. Wolfers (2004). ’Disagreement About Inflation Expec-tations. In M. Gertler and K. Rogoff (Eds.), NBER Macroeconomics Annual 2003, Volume 18, pp. 209–270. National Bureau of Economic Research.

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APPENDIX A

Notes on the Log-Linearization of the Consumption-Equation

Regarding the steady state, note that at the steady state expectations no longer matter and that households have identical preferences so that the steady state is similar to the standard case. Normalizing χ = η−1_η , this then implies R = β−1, y = l = c, w = η−1_η , and d = η−1.

Next, take the natural logarithm of both sides of the consumption-equation:

ln   eEi,t ∞ X s=t Qt,s s Y k=t βkeb,kRkπk+1−1 (βeb,sRs)−1πs+1 !_σ1 ci,t   = ln        e Ei,t ∞ X s=t Qt,s  w 1+γ γ s 1 χ _γ1 s Y k=t βkeb,kRkπk+1−1 (βeb,sRs)−1πs+1 !−1_γ c− σ γ i,t   + e_Ei,t ∞ X s=t Qt,sds+ eb,t−1Rt−1π−1t bi,t−1        Recall that: Qt,s= s Y k=t e−1_b,k−1R−1_k−1πk e−1_b,t−1R−1_t−1πt

Next, take a first-order Taylor-approximation around a zero-inflationary steady state. For clarity, note the left-hand side of the equation can be rewritten as:

ln          e Ei,tci,t          e−1_b,t−1R−1_t−1πt e−1_b,t−1R−1_t−1πt βeb,tRtπt+1−1β −1 e−1_b,tR−1_t π_t+1−1 1 σ +e −1 b,t−1R −1 t−1πte−1b,tR −1 t πt+1 e−1_b,t−1R−1_t−1πt βeb,tRtπt+1−1βeb,t+1Rt+1πt+2−1β −1 e−1_b,t+1R−1_t+1π_t+2−1 1 σ + . . .                  

Note that this expression evaluated at the non-stochastic zero-inflationary steady state yields: ln ci 1 + R−1+ R−2+ . . . = ln ci 1 + β + β2+ . . . = ln ci 1 − β

The first-order Taylor-approximation of the left-hand side of the equation then reads:

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A. NOTES ON THE LOG-LINEARIZATION OF THE CONSUMPTION-EQUATION 28 ln ci 1 − β +1 − β ci ci,t− ci 1 − β +1 − β ci ∂ ∂Rs R,π,ci,eb (Rs− R) + ∂ ∂πs+1 R,π,ci,eb (πs+1− π) + ∂ ∂eb,s R,π,ci,eb (eb,s− eb) !

where the derivatives of the internal functions are still to be determined. Considering the derivative with respect to the interest rate yields:

∂ ∂Rs R,π,ci (Rs− R) = ci             β 1 σ − 1 β1+ 1 σ − 1 β2+ 1 σ − 1 β3+ . . . (Rt− R) +β 1 σ − 1 β2+ 1 σ − 1 β3+ 1 σ − 1 β4+ . . . (Rt+1− R) +β 1 σ − 1 β3+ 1 σ − 1 β4+ 1 σ − 1 β5+ . . . (Rt+2− R) + . . .             = ci           1 σβ 1 _β0_{+ β}1_{+ β}2 _{+ β}3_{+ . . .} _ˆ Rt− β1 β0+ β1+ β2+ β3+ . . . _ˆ Rt +1 σβ 2 _β0_{+ β}1_{+ β}2 _{+ β}3_{+ . . .} _ˆ Rt+1− β2 β0+ β1+ β2+ β3 + . . . _ˆ Rt+1 +1 σβ 3 _β0_{+ β}1_{+ β}2 _{+ β}3_{+ . . .} _ˆ Rt+2− β3 β0+ β1+ β2+ β3 + . . . _ˆ Rt+2 + . . .           = ci β (1 − β)σEei,t ∞ X s=t βs−tRˆs− ci β (1 − β)Eei,t ∞ X s=t βs−tRˆs

The derivatives with respect to the internal functions of π and eb are obtained in

similar fashion. The first-order Taylor-approximation of the left-hand side of the equation then reads: ln ci 1 − β + ˆci,t+ β σEei,t ∞ X s=t βs−t ˆRs− ˆπs+1+ ˆeb,s − β e_Ei,t ∞ X s=t βs−t ˆRs− ˆπs+1+ ˆeb,s

Turning to the right-hand side of the equation, note that evaluated at the non-stochastic zero-inflationary steady state it reduces to:

ln 1 1 − β w 1+γ γ 1 χ 1_γ c− σ γ i + η −1 ! + β−1bi !

Temporarily, denoting the right-hand side evaluated at the steady state as A, the right-hand side can be approximated by:

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A. NOTES ON THE LOG-LINEARIZATION OF THE CONSUMPTION-EQUATION 29 A − 1 A w 1+γ γ 1 χ _γ1 c− σ γ i ! β (1 − β)γEei,t ∞ X s=t βs−t ˆRs− ˆπs+1+ êb,s ! − 1 A w 1+γ γ 1 χ _γ1 c− σ γ i ! β (1 − β)Eei,t ∞ X s=t βs−t ˆRs− ˆπs+1+ êb,s ! + 1 A −η −1 β 1 − βEei,t ∞ X s=t βs−t ˆRs− ˆπs+1+ êb,s ! + 1 A w 1+γ γ 1 χ _γ1 c− σ γ i ! 1 + γ γ Eei,t ∞ X s=t βs−tws ! + 1 A w 1+γ γ 1 χ _γ1 c− σ γ i ! −σ γ ˆ ci,t 1 − β + 1 Aη −1 e Ei,t ∞ X s=t βs−tdˆs + 1 A β−1bi ˆRt−1− ˆπt+ êb,t−1 + β−1¯bi,t−1

Now exploiting the steady state relationship

ci− η−1− (1 − β)β−1 = w 1+γ γ 1 χ 1_γ c−σγ

and noting steady state values for all households, tedious algebra yields the log-linearized consumption equation.

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APPENDIX B

Bifurcations

The characteristic equation implied by the HE benchmark ARER model is: P(λ) = λ4_{+ c}

3λ3+ c2λ2+ c1λ + c0 = 0

The flip and fold bifurcations are easily obtained by substitution of λ = 1 and λ = −1. These bifurcations correspond to the positive and negative real eigenvalues respectively going in or out of the unit disk.

Bf old= 1 + c3+ c2+ c1+ c0 = 0

Bf lip= 1 − c3+ c2− c1+ c0 = 0

With respect to the complex roots note that all coefficients of the characteristic equa-tions are real-valued so that by the complex conjugate root theorem all complex roots show up as conjugate pairs. Thus, the bifurcations associated with complex roots follow from (λ1, λ2) = (eiα, e−iα).

The relations between the coefficients of the characteristic polynomial and its roots are:

c3 = −λ1− λ2− λ3− λ4

c2 = λ1λ2+ λ1λ3+ λ1λ4+ λ2λ3+ λ2λ4+ λ3λ4

c1 = −λ1λ2λ3− λ2λ3λ4− λ1λ3λ4− λ1λ2λ4

c0 = λ1λ2λ3λ4

Given that (λ1, λ2) = (eiα, e−iα), first c3 = −eiα − e−iα − λ3− λ4, and second their

product will be one so that c0 = λ3λ4. Exploiting these relations then yields:

c2 = 1 + c0 + (−c3− eiα− e−iα)(eiα+ e−iα) = 1 + c0+ 2 cos(α)(−c3− 2 cos(α))

c1 = (1 − c0)(eiα + e−iα) + c3 = (1 − c0)2 cos(α) + c3

Combining these last two equations yields the bifurcations associated with pairs of complex eigenvalues as:

Bns= 1 + c0− c2− c3(c1− c3) 1 − c0 − c1− c3 1 − c0 2 = 0 such that (c1−c3) 2(1−c0) ≤ 1.

The plot of the bifurcations was generated based on the three B-equations using a Python script augmented with the Simpy and Numpy libraries for scientific computing.

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APPENDIX C

Robustness

C.1. Calibrated Parameters

M1 L = 1758.166 M2 L = 1749.750

Prior Post. Mean HCI Prior Post. Mean HCI β 0.99 0.99 0.99 0.99 0.99 0.99 σ N (1.5, 0.37) 3.283 [2.879,3.708] N (1.5, 0.37) 3.242 [2.826,3.677] γ 1 1 1 1 1 1 ω β(0.5, 0.1) 0.857 [0.837,0.879] β(0.5, 0.1) 0.834 [0.810,0.861] φπ N (1.5, 0.25) 2.750 [2.521,2.984] N (1.05, 0.25) 2.547 [2.444,2.640] nRE 1 1 1 U (0, 1) 0.978 [0.967,0.989] ρa β(0.5, 0.2) 0.983 [0.973,0.993] β(0.5, 0.2) 0.957 [0.923,0.989] ρr β(0.5, 0.2) 0.876 [0.855,0.897] β(0.5, 0.2) 0.737 [0.668,0.806] ρb β(0.5, 0.2) 0.408 [0.333,0.482] β(0.5, 0.2) 0.452 [0.370,0.532] εa Γ−1(0.1, 2) 0.015 [0.013,0.017] Γ−1(0.1, 2) 0.017 [0.014,0.019] εr Γ−1(0.1, 2) 0.012 [0.012,0.013] Γ−1(0.1, 2) 0.012 [0.012,0.013] εb Γ−1(0.1, 2) 0.015 [0.012,0.017] Γ−1(0.1, 2) 0.016 [0.013,0.019] M₃ L = 1750.210 M₄ L = 1744.221 Prior Post. Mean HCI Prior Post. Mean HCI β 0.99 0.99 0.99 0.99 0.99 0.99 σ N (1.5, 0.37) 3.3454 [2.968,3.771] N (1.5, 0.37) 3.294 [2.882,3.725] γ 2 2 2 2 2 2 ω β(0.5, 0.1) 0.871 [0.853,0.890] β(0.5, 0.1) 0.853 [0.831,0.770] φπ N (1.5, 0.25) 2.748 [2.522,2.990] N (1.05, 0.25) 2.546 [2.442,3.264] nRE 1 1 1 U (0, 1) 0.976 [0.965,0.987] ρa β(0.5, 0.2) 0.986 [0.978,0.995] β(0.5, 0.2) 0.960 [0.931,0.990] ρr β(0.5, 0.2) 0.875 [0.854,0.896] β(0.5, 0.2) 0.731 [0.664,0.800] ρb β(0.5, 0.2) 0.413 [0.338,0.486] β(0.5, 0.2) 0.462 [0.383,0.543] εa Γ−1(0.1, 2) 0.013 [0.012,0.015] Γ−1(0.1, 2) 0.015 [0.013,0.016] εr Γ−1(0.1, 2) 0.012 [0.012,0.013] Γ−1(0.1, 2) 0.012 [0.012,0.012] εb Γ−1(0.1, 2) 0.015 [0.012,0.017] Γ−1(0.1, 2) 0.016 [0.012,0.020] 31

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C.2. SAMPLING PERIODS 32

C.2. Sampling Periods

M5 L = 556.013 M6 L = 962.150

Prior Post. Mean HCI Prior Post. Mean HCI β 0.99 0.99 0.99 0.99 0.99 0.99 σ N (1.5, 0.37) 2.493 [2.020,2.963] N (1.5, 0.37) 3.112 [2.657,3.575] γ 1 1 1 1 1 1 ω β(0.5, 0.1) 0.841 [0.792,0.883] β(0.5, 0.1) 0.922 [0.910,0.933] φπ N (1.5, 0.25) 2.497 [2.186,2.811] N (1.5, 0.25) 2.383 [2.098,2.671] nRE 1 1 1 1 1 1 ρa β(0.5, 0.2) 0.964 [0.963,0.997] β(0.5, 0.2) 0.679 [0.607,0.752] ρr β(0.5, 0.2) 0.866 [0.824,0.903] β(0.5, 0.2) 0.913 [0.913,0.937] ρb β(0.5, 0.2) 0.262 [0.111,0.406] β(0.5, 0.2) 0.318 [0.318,0.514] εa Γ−1(0.1, 2) 0.018 [0.015,0.022] Γ−1(0.1, 2) 0.016 [0.013, 0.019] εr Γ−1(0.1, 2) 0.013 [0.012,0.015] Γ−1(0.1, 2) 0.012 [0.012, 0.013] εb Γ−1(0.1, 2) 0.020 [0.015,0.025] Γ−1(0.1, 2) 0.018 [0.018, 0.028] M₇ L = 559.175 M₈ L = 963.840 Prior Post. Mean HCI Prior Post. Mean HCI β 0.99 0.99 0.99 0.99 0.99 0.99 σ N (1.5, 0.37) 2.472 [2.011,2.9322] N (1.5, 0.37) 3.179 [2.740,3.636] γ 1 1 1 1 1 1 ω β(0.5, 0.1) 0.848 [0.809,0.887] β(0.5, 0.1) 0.887 [0.851,0.921] φπ N (1.05, 0.25) 2.298 [2.047,2.572] N (1.05, 0.25) 2.462 [2.295,2.640] nRE U (0, 1) 0.933 [0.905,0.963] U (0, 1) 0.937 [0.897,0.980] ρa β(0.5, 0.2) 0.877 [0.787,0.972] β(0.5, 0.2) 0.939 [0.891,0.991] ρr β(0.5, 0.2) 0.651 [0.536,0.771] β(0.5, 0.2) 0.782 [0.724,0.844] ρb β(0.5, 0.2) 0.394 [0.246,0.540] β(0.5, 0.2) 0.482 [0.352,0.609] εa Γ−1(0.1, 2) 0.021 [0.016,0.026] Γ−1(0.1, 2) 0.015 [0.012,0.018] εr Γ−1(0.1, 2) 0.013 [0.012,0.014] Γ−1(0.1, 2) 0.013 [0.012,0.013] εb Γ−1(0.1, 2) 0.027 [0.019,0.035] Γ−1(0.1, 2) 0.026 [0.017,0.034]

Contrary to all other models, there parameter estimates of M5 did not converge

smoothly at 2.5 million draws per MCMH chain. In order to achieve convergence, the number of draws had to be scaled up to 5 million per chain.

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C.3. EXPECTATION FORMATION 33

C.3. Expectation Formation

M1 L = 1758.166 M2 L = 1749.750

Prior Post. Mean HCI Prior Post. Mean HCI β 0.99 0.99 0.99 0.99 0.99 0.99 σ N (1.5, 0.37) 3.283 [2.879,3.708] N (1.5, 0.37) 3.242 [2.826,3.677] γ 1 1 1 1 1 1 ω β(0.5, 0.1) 0.857 [0.837,0.879] β(0.5, 0.1) 0.834 [0.810,0.861] φπ N (1.5, 0.25) 2.750 [2.521,2.984] N (1.05, 0.25) 2.547 [2.444,2.640] nRE 1 1 1 U (0, 1) 0.978 [0.967,0.989] ρa β(0.5, 0.2) 0.983 [0.973,0.993] β(0.5, 0.2) 0.957 [0.923,0.989] ρr β(0.5, 0.2) 0.876 [0.855,0.897] β(0.5, 0.2) 0.737 [0.668,0.806] ρb β(0.5, 0.2) 0.408 [0.333,0.482] β(0.5, 0.2) 0.452 [0.370,0.532] εa Γ−1(0.1, 2) 0.015 [0.013,0.017] Γ−1(0.1, 2) 0.017 [0.014,0.019] εr Γ−1(0.1, 2) 0.012 [0.012,0.013] Γ−1(0.1, 2) 0.012 [0.012,0.013] εb Γ−1(0.1, 2) 0.015 [0.012,0.017] Γ−1(0.1, 2) 0.016 [0.013,0.019] M₉ L = 1744.190 M₁₀ L = 1772.849 Prior Post. Mean HCI Prior Post. Mean HCI β 0.99 0.99 0.99 0.99 0.99 0.99 σ N (1.5, 0.37) 3.291 [2.885,3.731] N (1.5, 0.37) 3.014 [2.579,3.433] γ 1 1 1 1 1 1 ω β(0.5, 0.1) 0.853 [0.830,0.877] β(0.5, 0.1) 0.867 [0.841,0.894] φπ N (1.05, 0.25) 2.547 [2.445,2.640] N (1.05, 0.25) 2.498 [2.364,2.640] nRE U (0, 1) 0.976 [0.963,0.989] U (0, 1) 0.952 [0.936,0.968] ρ N (1, 0.25) 1.090 [0.729,1.450] ρy N (1, 0.25) 1.175 [0.804,1.550] ρπ N (1, 0.25) 0.295 [0.109,0.483] ρR N (1, 0.25) 1.258 [0.903,1.608] ρa β(0.5, 0.2) 0.959 [0.929,0.990] β(0.5, 0.2) 0.892 [0.836,0.950] ρr β(0.5, 0.2) 0.726 [0.655,0.796] β(0.5, 0.2) 0.933 [0.886,0.983] ρb β(0.5, 0.2) 0.464 [0.382,0.550] β(0.5, 0.2) 0.746 [0.667,0.824] εa Γ−1(0.1, 2) 0.015 [0.013,0.017] Γ−1(0.1, 2) 0.020 [0.016,0.022] εr Γ−1(0.1, 2) 0.012 [0.012,0.013] Γ−1(0.1, 2) 0.012 [0.012,0.013] εb Γ−1(0.1, 2) 0.016 [0.013,0.020] Γ−1(0.1, 2) 0.017 [0.014,0.020]

An estimated new keynesian DSGE model with heterogeneous expectations