model reference adaptive expectations in markov-switching economies

Economic Modelling 32 (2013) 551–559

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Economic Modelling

j ourna l homepage: www.e lsev ie r .com/ locate /ecmod

Model reference adaptive expectations in Markov-switching economies

Francesco Carravetta a, Marco M. Sorge b,⁎a Istituto di Analisi dei Sistemi ed Informatica del CNR, Roma, Italyb University of Napoli “Federico II” and CSEF, Via Cinthia C.U. Monte S. Angelo, 80126 Napoli, Italy

⁎ Corresponding author. Tel.: +39 081675311; fax: +E-mail addresses: [email protected] (F. Carrav

[email protected] (M.M. Sorge).

0264-9993/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.econmod.2013.02.033

a b s t r a c t
a r t i c l e i n f o
Article history:Accepted 21 February 2013Available online xxxx

Keywords:Rational expectationsMarkov-switching dynamic systemsDynamic programmingTime-varying Kalman filter

This paper offers a theory of model reference adaptive beliefs as a selection device in Markov-switchingeconomies under equilibrium indeterminacy. Consistent with the classical rational choice paradigm, ourtheory requires that endogenous expectations be replaced with a general-measurable function of the ob-servable states of the model, to be determined optimally. This forecasting function is derived as theregime-independent feedback control minimizing the mean-square deviation of the equilibrium pathfrom the corresponding perfect-foresight state motion (the reference model). We show that model refer-ence adaptive expectations always generate a rational expectations equilibrium, irrespective of the pres-ence of nonlinearities and/or imperfect information. Under equilibrium indeterminacy, this forecastingmechanism enforces the unique mean-square stable solution producing nearly perfect-foresight dynamics.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Since the early work of Muth (1961) and Lucas (1972), rational ex-pectations (RE) have been widely adopted as the benchmark model ofexpectation formation in macroeconomics. It essentially reduces tothe assumption that the prediction made by the forecaster, conditionalon all the information available at the time the prediction is made, beconsistent with the forecast model derived from the underlying eco-nomic structure.

The RE hypothesis has been severely criticized on the ground of itsimplausibly strong implications. In particular, it is silent about theprocess by which economic agents translate current information intooptimal forecasts, which are simply assumed not systematically differ-ent from equilibrium outcomes (Lucas and Prescott, 1971). In this re-spect, the learning approach has provided a natural interpretation ofRE as asymptotic outcomes of a well-specified learning process, inwhich boundedly rational agents engage to form estimates of the true(unknown) economy they act in (e.g. Evans and Honkapohja, 2001;Marcet and Sargent, 1989). According to this approach, the agents' sub-jective beliefs are derived as forecasts from an estimationmodel, whichcombines a parametric description of the model dynamics (the per-ceived law of motion) and an estimation algorithm (least squares orBayesian updating).

The aim of this paper is to present an optimization-based theory ofexpectations formation as a selection device in multiple equilibriummodel economies. According to this theory, forward-looking agentsformulate (and coordinate upon) subjective beliefs on the basis of a

39 0817663540.etta),

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well-specified dynamic criterion, which goes beyond the static mini-mum mean-square error one – the one at the core of the RE paradigm.This approach, which traces back to Basar (1989), complies with abroader definition of rationally formed expectations, according towhich agents are endowedwith a general-measurable forecasting func-tion, which is to be chosen optimally given the available information(the measurement) on the state(s) of the underlying model economy.

More specifically, we think of unobservable subjective expectationsas the aggregate decision of the economic agents as to their best esti-mate of some future economic state(s), to be taken on the basis of amodel reference adaptive criterion. Subjective expectations are then de-rived from an adaptive forecastingmodel inwhich the actualmodel dy-namics are forced to track the evolution the system would have if theagents were able to form an exact prediction of the future realizationsof the endogenous variables. Under imperfect state observability, theagents are not able to form subjective forecasts on the basis of perceivedlaws of motion that are consistent with the RE solution(s). In this view,imperfectly informed agents are thought of as adaptively revising their(best) estimate of the future variables governing the dynamics of theeconomic system as new observations are generated.

We describe our approach in the context of forward-looking stochas-tic systemswhich introduce regime-switching and imperfect informationinto the standard linear model of Evans and Honkapohja (2001).1 As amain contribution, we show that model reference adaptive expectationsalways generate a rational expectations equilibrium, irrespective of thepresence of nonlinearities and/or imperfect information. When themodel equilibrium is indeterminate, this rational forecasting mechanismis able to pin down the unique mean-square stable solution producing

1 For the linear case, the reader is referred to Carravetta and Sorge (2010).

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552 F. Carravetta, M.M. Sorge / Economic Modelling 32 (2013) 551–559

nearly perfect-foresight dynamics. In fact, the adaptive forecasting (be-lief) function is able to recover all the conditional (rational) expectationsterms entering the canonical MSRE systems, and thus selects a particular(history-dependent) RE equilibrium in the solution set. As emphasizedby Farmer (2002), multiple equilibriummodels create room for adaptiverules, or belief functions, to select a particular equilibrium. Specificationsof beliefs are restricted by the requirement that, in a stationary environ-ment, the forecasting function should not generate systematic forecasterrors. While Farmer (2002) puts forward a quite ad hoc mechanismfor the formation of expectations, the model reference adaptive rulepresented in this paper is the outcome of a well-defined dynamicoptimization/estimation problem, which always fulfills the rationalityrequirement. Moreover, and differently from Farmer (2002), it doesnot require that the agent possess a priori knowledge of the structureof the model solution.

The paper is organized as follows. Section 2 briefly reviews the relat-ed literature. In Section 3 we present the basic framework of analysis,and describe our approach to expectations formation. In Section 4, westudy the model dynamics under model reference adaptive expecta-tions, and also discuss the issue of equilibrium stability. While the anal-ysis here is for an abstract macroeconomic system, an illustrativeapplication to the New Keynesian framework is provided in Section 5.Section 6 concludes.

2 Pearlman et al. (1986) were the first to address the partial information issue in REmodels in the fixed regime setting. While generalized to regime-switching frame-works, our analysis also departs from Pearlman et al. (1986) in that it develops anadaptive control technique which jointly exploits Kalman filtering and stochastic con-trol theory. Furthermore, we allow the observation process to be regime-dependent.See also, Lungu et al. (2008), Shibayama (2011) for a different approach to solutionof linear RE systems with imperfect information.

3 Lagged expectations can be associated with different microfoundations, like thepresence of staggered-price setting under past information (e.g. Woodford, 2003), in-formation stickiness (e.g. Mankiw and Reis, 2002) or imperfect information in mone-tary policy-making (e.g. McCallum and Nelson, 2000).

2. Related literature

Our analysis clearly relates to several lines of research on the processof expectations formation in forward-looking models. The well-knownlearning approach in macroeconomics focuses on the way systematicforecasting biases are eliminated over time (e.g. Evans and Honkapohja,2001;Marcet and Sargent, 1989). Specifically, the adaptive learning liter-ature endows boundedly rational agents with a forecasting model – theperceived law of motion of the economy – which can be an arbitraryfunction of past endogenous and past and current exogenous variables,and has to be optimally parameterized based on newdata and observable(past) forecast errors. RE equilibria are thus regarded as asymptotic out-comes of this learning process, whenever conditions for convergence ofagents' beliefs to the equilibrium values hold. Our analysis differs fromthe learning literature in two crucial respects. First, it posits thatforward-looking agents form their beliefs via a dynamic optimizationcriterion within a correctly specified (parametric) forecasting rule;however, under unobserved state variables, the agents can no longerbe thought of as employing a forecasting model which is consistentin form with any of the RE solutions. Second, we are not concernedwith asymptotic convergence properties, as subjective forecasts formedaccording to our adaptive approach always coincide – as functions ofthe common measurable space – with RE, even in finite horizon econo-mies. Though methodologically related, our method also differs fromthe Bayesian learning literature (e.g. Bullard and Suda, 2008; McGough,2003), as these studies typically assume that agents employfiltering tech-niques to update estimates of (possibly time-varying) parameters withinnot fully rational forecasting functions. Rather, our approach posits thatforward-looking agents update their (best) estimate of the (hidden) vari-ables governing themodel dynamics as newdata is released, when only alimited information set – themeasurement process – is available to them.

From a computational perspective, our work is also related to therecent and increasing literature on Markov-switching rational expec-tations (MSRE) models, that is stochastic difference systems in whichthe parameters governing the dynamic behavior of the equations arefunctions of a discrete-state Markov chain. Since able to account forparameter instability and yield quantitatively different responses ofmacroeconomic variables to fundamental shocks from those impliedby fixed regime models, MSRE systems have recently been advocatedto investigate the role of regime-switching monetary policy in NewKeynesian frameworks (e.g. Davig and Leeper, 2007) or rather to

gauge the effects of uncertainty over structural parameters governingthe optimal behavior of rational agents (e.g. Liu et al., 2009).

From a technical viewpoint, regime dependency engenders struc-tural nonlinearities which preclude applicability of standard solutiontools for linear RE systems, such as Blanchard and Kahn (1980)'s, KingandWatson (2002)'s and Sims (2002)'s. In this respect, a number of au-thors have been interested in deriving determinacy (local uniqueness)conditions for RE equilibria to MSRE models. In their seminal contribu-tion to the generalization of the Taylor principle, Davig and Leeper(2007) study how regime-switching alters the determinacy propertiesof RE solutions and provide analytical restrictions on monetary policybehavior to ensure (local) uniqueness of the equilibrium path. Thenonlinearity problem is addressed by introducing a two-step solutionmethod that consists of studying an augmented system which is linearin fictitious variables, the latter coinciding with the actual ones in someof the regimes, and then using the solution to the linear representationin order to construct solutions for the original nonlinear system.

From a more general perspective, Farmer et al. (2009, 2011) haveprovided a series of characterization results for the set of minimalstate variable (MSV) solutions as well as the full set of RE equilibria –

also sunspot ones – toMSRE frameworks, which satisfy a suitable stabil-ity concept. Their approach rests on expanding the state-space of theunderlying stochastic system and to focus on an equivalent model inthe expanded space that features state-invariant parameters. Further-more, Farmer et al. (2009) demonstrate an equivalence property be-tween determinacy for MSRE models and mean-square stability in aclass of Markov jump autoregressive systems.

Our approach to the analysis of regime-switching models differsfrom the mentioned studies in several respects. First, unlike other ap-proaches which posit the presence of RE, we rather focus on a differentmodel for expectations formation. The latter is explicitly designed tohandle dynamic models with an imperfect information structure. Inmany macroeconomic environments, in fact, variables of interest aretypically observed with some delay, and only lagged values of themcan be exploited for the formation of expectations (e.g. Collard andDellas, 2004; Mankiw and Reis, 2002). Also, observability of some rele-vant economic variables, like factor productivity, can be plagued bymeasurement errors. From a different perspective, economic policy istypically conducted under substantial uncertainty about the state ofthe economy and the timing of structural disturbances (e.g. Svenssonand Woodford, 2003). Intuitively, a fundamental problem of inferencearises, as agents must gather information on unobserved variablesfrom observed ones.2 Under regime-switching, the possibility of futurechanges in the model's structure is also crucial to the determination ofagents' expectations. In fact, the problem of learning about future (un-certain) Markov regimes cannot be disentangled from that of filteringunobservable (current and expected) variables. As a consequence,methods generally employed to handle learning in models withregime-switching exogenous drifts (e.g. Schorfheide, 2005) cannot beadapted to studyMSREmodelswith imperfect information. The presentpaper offers some new insights on these issues for a given class ofregime-switching systems, namely those involving lagged expecta-tions.3

4 See also Evans and McGough (2005) and Guse (2010) for a different analysis ofoverspecified forecasting models employing lagged data and (possibly) sunspots,which may well take the form of the perfect-foresight reference model.

5 Our method is conceived to handle noisily observed regime-switching frame-works. However, as will appear clear in the following, it also applies to full informationmodels, i.e. those in which Φs tð Þ ¼ I;∀s tð Þ∈S and wt = 0, ∀ t a.s..

6 The initial conditions x; s0ð Þ are taken to be independent random variables.

553F. Carravetta, M.M. Sorge / Economic Modelling 32 (2013) 551–559

Secondly, available methods for solving MSRE models generallyimpose a priori stability requirements on the solution set to weak-en the equilibrium multiplicity issue. Operationally, since theagents' expectations in RE frameworks are obtained by iteratingthe system forward, (asymptotic) stationarity is needed for thisprocess to be well-defined (e.g. Farmer et al., 2009). While equilib-rium stability is usually enforced by the existence of transversalityconditions in the underlying (infinite horizon) dynamic economicframeworks, there exist models for which no such boundary condi-tions arise or rather, though present, they do not serve as necessaryoptimality requirements (e.g. Driskill, 2006; Halkin, 1974). By pro-viding a readily computable expression of the RE components bothin finite and infinite horizon model representations, our approachneed not invoke approximation hypotheses or stability conceptsto solve forward the system, for only initial conditions knowledgeis required.

3. Environment

3.1. Model reference adaptive expectations

We illustrate the notion of model reference adaptive expectationsby means of a simple example. Let xt be a (scalar) square-integrablerandom variable defined on a properly filtered probability space,which evolves according to the following expectational differencemodel:

xt ¼ αE t−ixtþ1 þ vt ; α∈R; vt ∼N 0;1ð Þ ðiÞ

where E t−i captures subjective beliefs of the one-step ahead value of thevariable xt, conditional on information available up to time t-i, i = 0,1,….Since expectations are mostly unobservable, models incorporating be-liefs such as (i) must be specified together with an assumption abouttheir formation. According to the RE hypothesis, subjective beliefs coin-cide with the true (equilibrium) statistical expected value, conditionalon available information, i.e.:

E t−ixtþ1 ¼ arg minu I t−ið Þ

E xtþ1−u I t−ið Þ� �2 ðiiÞ

Hence, for any given information set I t−i, the conditional expectationE xtþ1 I t−ij Þ�

is the best predictor of xt + 1 among all square-integrablefunctions of I t−i. The RE hypothesis is nonetheless silent with respectto the process by which economic agents translate current informationinto optimal forecasts of actual equilibrium outcomes. Following Basar(1989), we maintain that the starting point for any rational choice foru I t−ið Þ should be a criterion such as (ii), and not directly the conditionalexpectation itself. In fact, since dynamic models involve multiplestages, forward-looking economic agents may well be thought of asnot restricting to (ii) for a particular t, but rather as trying to mini-mize its cumulative over all t of interest, subject to the actualmodel dynamics:

minu I t−ið Þf g

XTt¼t0

E xtþ1−u I t−ið Þ� �2

s:t: xt ¼ αu I t−ið Þ þ vt

ðiiiÞ

where [t0, T] is the (not necessarily finite) time interval and predic-tion errors at all stages are equally valued. As shown in Basar(1989), the solution to (iii) need not coincide with (ii), and hencenot always generates an RE equilibrium (i.e. a process satisfying (i)).

Remarkably, a dynamic decision process like (iii) does not re-quire that economic agents possess a priori knowledge of the struc-ture of the model solution itself. In the same spirit, building uponCarravetta and Sorge (2010), we submit a strong alternative to

both (ii) and (iii) by introducing the notion ofmodel reference adap-tive expectations (MRAE), i.e. expectations which are computed via adynamic criterion of the form:

minu I t−ið Þf g

XTt¼t0

E xtþ1−x�tþ1� �2

s:t: xt ¼ αu I t−ið Þ þ vt ; x�t ¼ g x�t−1; vt

� � ðivÞ

where xt∗ is a referencemodel, x∗ t − 1 := σ(xt − k∗ ,k ≤ t − 1) and vt :=

σ(vt − k,k ≤ t) are the observed histories of xt∗ and vt. The referencemodel is chosen as the stochastic equation describing the perfect-foresight behavior of the underlying REmodel: according to (iv), the ac-tual process for xt is forced to track the evolution the systemwould haveif agentswere able to forman exact prediction of – i.e., perfectly foresee –the future realizations of the endogenous variable xt.

Two remarks are in order. First, the MRAE approach does not re-quire that the perfect-foresight model – the reference one – be anRE equilibrium. Hence, it is generally well-suited even for RE modelswhich do not admit such a solution. Second, the MRAE approach gainsparticular appeal in indeterminate equilibrium economies, i.e. econo-mies which possess multiple stable equilibria. In this context, theperfect-foresight model represents an ideal (stable) world in whichthe future behavior of variables of interests is perfectly forecastable(as a function of the observables) in advance, and no role is played byextrinsic uncertainty (sunspots). Hence, it has an inherent value as areference model in economies populated by forward-looking agents.4

3.2. The class of models

Let Ω;F ;Pð Þ be a given probability space, and —F :¼F t : t ¼ 0; 1;…f g a filtration of F . Let also s(t) index the stateof an ergodic Markov chain, with s tð Þ∈S :¼ 1;…; Sf g and:

P s t þ 1ð Þ ¼ jjF tf g ¼ P s t þ 1ð Þ ¼ jjs tð Þf g ¼ ps tð Þj

with pi,j ≥ 0 for i; j∈S and ∑j = 1S pij = 1 for each i∈S.

We study an abstract forward-looking, regime-switching macro-economic system with past expectations on both current and futurestates, given by:

xt ¼ Πs t−1ð ÞE t−1xt þ Γ−1s t−1ð ÞE t−1xtþ1 þ Γ−1

s t−1ð ÞΨs t−1ð Þvt ; x0 ¼ x ð1Þ

yt ¼ Φs tð Þxt þwt ð2Þ

where xt is a real n-dimensional vector of random variables of eco-nomic interest, yt is a real l-dimensional vector of observables, andthe state error vt∈Rn, the measurement noise wt∈Rl and the initialstate x∈Rn are zero-mean white Gaussian processes. With no lossof generality,5 the covariances of the unobserved structural distur-bance and of the measurement noise are normalized to the In × n

and Il × l identity matrices respectively, whereas x has covariance P0.Πs(t), Γs(t),Ψs(t) andΦs(t) are conformable matrices holding the coeffi-cients of the underlying economic model; we let Γs(t) be invertible forall s tð Þ∈S, as in Farmer et al. (2009).6

The term E t−1 denotes subjective beliefs held by economic agents,which are formed on the basis of the set F t−1. Under imperfect infor-mation, not all the variables in xt or the structural disturbances vt aredirectly (or perfectly) observed. We let F t consist of the σ-algebra


generated by {yk,sk;k ≤ t}, that is the t-dated information set includesthe complete filtrations generated by the measurement Eq. (2) andthe Markov state realizations. We thus allow for observable shifts inmodes solely, as in most of the literature on regime-switching REmodels (e.g., Davig and Leeper, 2007; Farmer et al., 2009).7 Accord-ingly, while the current values of parameters are known, futureones are uncertain. As a working assumption, we also require that x,vt, wt and st be mutually independent at all steps.

Under the RE hypothesis, E t−1xtþk ¼ E xtþk F t−1j Þ; k ¼ 0;1;∀t∈Nð .In this case, any locally integrable process {xt, yt} which, for fixed ini-tial conditions and under both finite and infinite horizon representa-tions, satisfies Eqs. (1)–(2), is called an RE equilibrium. The followingcharacterization result holds:

Lemma 1. Let Et−1xtþk ¼ E xtþk F t−1j Þ; k ¼ 0; 1ð in Eqs. (1)–(2). Thenthe model admits an infinity of solutions, which are non-parameterizable.

Proof. See Appendix A. □

In general, the non-uniqueness issue can beweakened – yet often notremoved – by forcing the solution to be stable (e.g. Farmer et al., 2009).From a different perspective, the learning approach can be exploited tocheck whether outcomes from recursive learning are locally convergentto RE equilibria. This paper is rather concernedwith a different theory ofbeliefs as an equilibrium selection device, according to which agents'subjective beliefs are derived from a forecasting function – measurablewith respect to the available information F t – which adapts the actualsystem evolution to a reference model, chosen as the perfect-foresightregime-switching version of Eqs. (1)–(2):

x�tþ1 ¼ Γs t−1ð Þ I−Πs t−1ð Þh i

x�t−Ψs t−1ð Þvt ; x�0 ¼ x; x�−1 ¼ 0 ð3Þ

y�t ¼ Φs tð Þx�t þwt ð4Þ

Remarkably, the process (3)–(4) is not a solution to Eqs. (1)–(2)under RE, irrespective of the model's determinacy properties.8

4. The solution under MRAE

According to the MRAE approach, subjective beliefs in Eq. (1) arereplaced with a specific (F t -measurable) forecasting function ut, tobe determined optimally. This is equivalent to considering the follow-ing causal system:9

xtþ1 ¼ ut þ Γ−1s tð ÞΨs tð Þvtþ1; x0 ¼ x ð5Þ


Let us introduce:

et :¼ xt−x�t ; z0

t :¼ e0

t x�0t x

�0tþ1

� �

As stated above, MRAE are represented by the input sequenceu = {ut}t ∈ T,T = [0,T] ⊂ IN, ut ∈ Ut – with Ut denoting the space of

7 For theoretical work dealing with unobserved current regimes, see, among others,Andolfatto and Gomme (2003), Leeper and Zha (2003) and Davig (2004).

8 This feature does not depend on the nonlinearities introduced by regime-switching, nor is related to the presence of imperfect information. In fact, it relies ex-clusively on the lagged expectations structure of the model and the stochastic proper-ties of the structural disturbances vt (see Appendix A).

9 In fact, a simple implication of Lemma 1 is that E xt F t−1j Þð can be regarded to as theonly relevant equilibrium RE term in Eq. (1) (see Appendix A).

all square-integrable F t-measurable random vectors – which minimizesthe objective functional:

J uð Þ ¼ EXTþ1

t¼0

e0

tet ¼ EXTþ1

t¼0

z0

tMzt ðOFÞ

under the following state-space recursive constraints10:

ztþ1 ¼ As tð Þzt þ But þ Cs tð Þvtþ1; z0 ¼ z ð7Þ

yt ¼ Φs tð Þzt þwt ð8Þ

whereM consists of the identitymatrix In × n as first block on themain di-agonal and 0's elsewhere. Expression (8) can be properly used as the ob-servation equation for the augmented regime-switching system (7) inwhich the first n entries of the state vector zt describe the evolution ofthe deviation from the perfect-foresight dynamics.

The solution to the problem (OF)-(7)-(8) is derived as an optimalMarkov jump feedback controller in conjunction with the minimummean-square error estimate (MMSE) obtained by a time-varyingKalman filter. We indeed show that a separation principle holds forthe system at issue – i.e., the optimal input sequence depends on theobserved state only through the optimal estimate of the latter. In theclassical literature on Markov jump linear quadratic (MJLQ) problems(e.g. Costa et al., 2005), it has been shown that the solution of suchproblems engenders a twofold set of coupled Riccati equations,each associated to the filtering and control programs respectively.Since these equations cannot be represented as a single higher-dimensional Riccati equation, structural concepts and algorithmsfrom the classical linear theory are not directly applicable to Markovjump systems. While further requirements are generally needed forthe existence of a steady-state solution to the coupled Riccati equa-tions (e.g. Abou-Kandil et al., 1995; Blair and Sworder, 1975; Chizecket al., 1986), we show that, for the setting at issue, the Riccati gainadmits a simple time-invariant and state-independent representation,both in finite and infinite horizon problems of the form (OF)-(7)-(8).The following result clarifies this statement:

Lemma 2. Given the system (5)–(6), the input sequence ut :¼ utf gwhich produces for any t = 0,1,… the mean-square minimum deviationfrom the regime-switching perfect-foresight state motion (3), is in theform:

ut ¼ x�tþ1jt ð9Þ

where the optimal estimate x�tþ1jt :¼ 00 Ið Þ0E zt jF tð Þ is obtained recursivelyvia a time-varying Kalman filter.

Proof. See Appendix C. □The estimator of the one-step ahead perfect-foresight state, x�tþ1jt , is

mean-square optimal with respect to the σ-algebra generated by theactual measurement process (2).11 Our central claim rests on showingthat, for any t and the input (9), the optimal prediction estimates ofthe perfect-foresight states xt

∗ following the regime-switching law ofmotion (3), given the measurement (y0,…,yt) and the σ-algebraσ(st) = (s0,…,st), are equal to those relative to the actual states xt inEq. (5), generated by Eq. (9):

10 See Appendix B.11 This result, presented in Carravetta and Sorge (2010), improves upon De Santis etal. (1993)'s filtering approach to solution of linear RE models, as their MMSE estimatorrather exploits the fictitious observations yt

∗ according to Eq. (4), which are not avail-able. Moreover, De Santis et al. (1993) only find an approximate solution to the original(linear) RE model.


Proposition 1. Let x = (xt),y = (yt) be the solution of Eqs. (5)–(6)under the control law (9). Then, for any t and Markov state s(t) =i ∈ {1,2,…,S}, it holds:

xtþkjt ¼ x�tþkjt ; k ¼ 1;2;… ð10Þ

Proof. It follows readily from Lemma 2 and the independence be-tween vt and st. □

Remarkably, given the perfect-foresight dynamics 3 it is easilyverified that:

E xtþ1jF t−1� � ¼ Γs t−1ð Þ I−Πs t−1ð Þ

� �ut−1

which provides, in conjunction with Proposition 1, a simple relationbetween the optimal forecasting function (9) and the conditional (ra-tional) expectation terms featuring in Eq. (1). In other words, theunobservable expectational components of the noncausal model (1)can always be estimated on the causal perfect-foresight state motion(3), fed by the actual observations (2), by means of a suitable Kalmanfiltering technique.

As a consequence, the solution x = (xt),y = (yt) of Eqs. (5)–(6)forced by ut is an RE equilibrium for the imperfect informationMarkov-switching models (1)–(2). In other words, both in finite and in-finite horizon model representations, there always exists an equilibriumprocess {xt} - that is, a stochastic sequence of (functions of) states and ob-servables in F t fulfilling the noncausal regime-switching model (1)–(2)under RE –which is computable via a causal regime-switching (control-lable) system of the form (5)–(6), when the optimal forecasting functionis set to the optimal Markov jump feedback control law (9).12

We emphasize two fundamental implications of our analysis. First,the optimal forecasting function ut has an adaptive structure: newforecasts are formed as linear combinations of past forecasts and esti-mation errors, where the adaptation parameter is given by thetime-varying Kalman gain.13 This is consistent with the notion of gen-eralized adaptive expectations as put forward in Farmer (2002). In fact,since the optimal forecasting function always coincides with RE, ourwork can be thought of as providing new insights into the deep con-nections between the two mechanisms of expectations formation.Second, the equilibrium process under Eq. (9) is not in the form ofthe minimum state variable (MSV) solution as in Farmer et al.(2009, 2011). Rather, given the recursive representation of RE emerg-ing from Eq. (9), the nearly perfect-foresight equilibrium defined inthis paper is history-dependent, as its distribution exhibits depen-dence on the entire history of regimes (s0,…,st − 1) and measurement(y0,…,yt − 1).14

Finally, we characterize the mean-square stability properties ofthe nearly perfect-foresight solution, whose evolution is given by:

xtþ1 ¼ x�tþ1jt þ Γ−1s tð ÞΨs tð Þvtþ1 ð11Þ

by the following:

12 While the model considered in the paper involves past expectations of the currentand the one-step ahead states, the algorithm can be easily generalized to stochasticsystems with any kind of lagged expectations on current and future states, as shownin Appendix D.13 See Appendix C.14 Through the observations {yt}, the history-dependent RE equilibrium clearly de-pends on the samples (v0,…,vt − 1),(w0,…,wt) and the initial condition x0. See alsoBranch et al. (2007) for an investigation of history-dependent equilibria in MSREmodels under adaptive learning.

Proposition 2. The nearly perfect-foresight equilibrium (11) underthe measurement (2) is mean-square stable if and only if:

rσ Λ ið Þ<1; i ∈ S ð12Þ

where rσ(Λi) := maxk{|λk|} and λk is an eigenvalue of the probability

weighted matrix Λ i :¼ pjiΓ i I−Πið Þ⊗Γ i I−Πið Þh i

; i; j ∈ S.

Proof. See Appendix E. □

Remarkably, the nearly perfect-foresight solution, while existingfor any model in the form of Eqs. (1)–(2) under RE, is mean-squarestable only under equilibrium indeterminacy. We can thus interpretthe MRAE approach as an optimization-based equilibrium selectiondevice for indeterminate model economies under regime-switching.

5. A New Keynesian example

In this Section, we provide an illustrative application of the MRAE ap-proachby considering a simple versionof theprototypicalNewKeynesianmonetary model with information lags (e.g. Woodford, 2003).15 The po-tential existence of lags in the information available to forward-lookingagents is an important constituent of optimal policy-making, as in thiscase expectations are predetermined with respect to policy. This in turnimplies that the ability of monetary authorities to influence con-sumption and/or investment decisions of private agents may be se-verely constrained.

The linearized equilibrium representation of the model is de-scribed by the following equations:

gt ¼ E gtþ1jF t−1� �

−σE it−πtþ1jF t−1� �þ v1;t; σ > 0 ð13Þ

πt ¼ βE πtþ1jF t−1� �þ κE gt jF t−1ð Þ þ v2;t; 0<β<1; κ > 0 ð14Þ

it ¼ μs t−1ð ÞE πt F t−1j Þð ð15Þ

whereF t stands for the t-dated information set, gt denotes the outputgap, πt is the inflation rate and it is the nominal interest rate.16

Eqs. (13)–(14) represent the private sector block of the model. Thefirst equation is the dynamic IS equation that arises from the consumers'optimization problemwith planning lags. The second equation is the ag-gregate supply function, as derived from firms' optimal price-settingunder past information (Woodford, 2003). The monetary authority im-pinges on the economy's dynamics via the interest rule (15). Arguably,informational uncertainty, in the form of partial data observability ormeasurement errors, is likely to surround the actual design of monetarypolicies. Following McCallum and Nelson (2000) we assume that themonetary authority does not observe the current level of inflation andhence exploits the best forecast available, on the basis of past informa-tion. The policy rule is a function of a two-state Markov chain withindex s(t) ∈ {1,2}, which reflects the past regime in place. As in Farmeret al. (2009), we assume that monetary policy can be either passive(|μ1| b 1), or active (|μ2| > 1). Thematrix P = [pij], for i,j ∈ {1,2}, collectsthe probabilities of transition from one regime to the other.

The economy is perturbed by two fundamental shocks, i.e. an ag-gregate demand shock v1,t and a cost-push shock v2,t, which are as-sumed to be zero-mean white Gaussian processes with unitaryvariances. For the purpose of the analysis, we assume that only a

15 Information lags are easily introduced in the basic New Keynesian model by as-suming that aggregate consumption or investment decisions entail planning lags andare thus based on past information.16 All variables are expressed as log-deviations from a unique non-stochastic steadystate.


noisy measure πt ¼ πt þwt of inflation is available to agents whenforming expectations, with wt ∼ N(0,1). The system (13)–(15) withimperfect information can then be written as:

xt ¼ Πs t−1ð ÞE xt jF t−1ð Þ þ Γ−1E xtþ1jF t−1� �þ vt ð16Þ

yt ¼ xt þ 0 1ð Þ0wt ð17Þ

where:

x0

t ¼ gt πtð Þ; v0

t ¼ v1;t v2;t� �

Πs t−1ð Þ ¼0 −σμs t−1ð Þk 0

� �; Γ−1 ¼ 1 σ

0 β

� �

System (16)–(17) is thus in the form of Eqs. (1)–(2) under the REhypothesis, with F t−1 :¼ σ yk; sk; k≤ t−1ð Þ. Under the MRAE criteri-on, given the reference (perfect-foresight) model:

x�tþ1 ¼ Γ I−Πs t−1ð Þ� �

x�t−Γvt ð18Þ

y�t ¼ x�t þ 0 1ð Þ′wt ð19Þ

a causal system for z′t ¼ xt−x�tð Þ′; x�′t ; x�′

tþ1

� �, as well as a linear obser-

vation equation relating yt to zt, can be constructed by following thelines of Section 4. We can then exploit Lemma 2 for the design ofthe optimal forecast ut−1 :¼ E x�t jF t−1ð Þ as a function of the observa-tions (y0,…,yt − 1),(s0,…,st − 1), which yields the following equilibri-um output and price for the imperfect information MSRE model(13)–(15):

gt ¼ Ω11g�t−1jt−2 þΩ12π

�t−1jt−2

þ �K t−2�Φs t−2ð Þηt−2 þwt−2

� �h i1þ v1;t ð20Þ

πt ¼ Ω21g�t−1jt−2 þΩ22π

�t−1jt−2

þ �K t−2�Φs t−2ð Þηt−2 þwt−2

� �h i2þ v2;t ð21Þ

where Ωij denotes the ij-th entry of the matrix Ω :¼ Γ I−Πs t−2ð Þh i

;K t

is the precomputable filter gain and η :¼ zt−zt is the estimationerror.17

6. Conclusion

In this paper, we describe a model reference adaptive approach toequilibrium selection in indeterminate Markov-switching economies.Under this approach, the evolution of the actual model is adapted to thecorresponding regime-switchingperfect-foresight state behavior (the ref-erencemodel); the resulting statemotion is a history-dependent solutionto theMSRE system for the optimal forecasting function features the samestructure of the (unobservable) expectations component. Under equilibri-um indeterminacy, this mechanism enforces the unique stable solutionwith nearly perfect-foresight dynamic properties. The next research ques-tion is to ask how these findings would translate into fully microfoundedDSGE frameworks.

Acknowledgments

Wewish to thankMichael P. Evers, Vijay Gupta, Thomas Hintermaier,Gernot Müller, an anonymous referee and the workshop participants atthe Universities of Bielefeld, Bonn and Salerno for their valuable com-ments and discussions. Any errors are our own. The usual disclaimerapplies.

17 Here, [ξ]i denotes the i-th row of a given vector ξ.

Appendix A. Proof of Lemma 1

Define the endogenous (Ft−adapted) revision processθt :¼ E xtþ1 F tj Þ−E xtþ1 F t−1j Þðð , and notice that, given themodel dynamics,any RE equilibrium must satisfy:

E xtþ1jF t−1� � ¼ Γs t−1ð Þ I−Πs t−1ð Þ

� �E xt F t−1j Þð

The full set of RE equilibria (reduced form) can then be written inthe form:

xtþ1 ¼ λt þ Γ−1s tð ÞΨs tð Þvtþ1

λtþ1 ¼ Γs tð Þ I−Πs tð Þ� �

λt þ θtþ1

yt ¼ Φs tð Þxt þwt

where λt is any non-fundamental component other than the funda-mental (MSV) solution xt

MSV = Γs(t − 1)−1 Ψs(t − 1)vt, and the entries of

the n-dimensional vector process θt are arbitrary martingale differ-ences with respect to F t (non-parameterizable infinity ofsolutions).

Remarkably, the process (3) is not an RE solution. In fact, any REequilibrium necessarily satisfies the recursive equation:

xt ¼ Πs t−1ð Þλt−1 þ Γ−1s t−1ð Þ λt−θtð Þ þ Γ−1

s t−1ð ÞΨs t−1ð Þvt ð22Þ

and hence existence of a perfect-foresight solution would require:

Ψjvtþ1 ¼ −ΓjΓiðΠi−IÞΓ−1j Ψjvt a:s: ∀t∈N

where j ¼ s tð Þ; i ¼ s t−1ð Þ; i; j ∈ S. This has probability zero in oursetting.

Appendix B. Definition of matrices for the augmented system(7)–(8)

The matrices As(t), Bs(t), Cs(t) and Φs tð Þ appearing in Eqs. (7)–(8)have the following block structure:

As tð Þ ¼0 0 −I0 0 I0 0 Γs tð Þ I−Πs tð Þ

h i0@

1A; B ¼

I00

0@

1A

Cs tð Þ ¼Γ−1s tð ÞΨs tð Þ

0−Ψs tð Þ

0@

1A; Φs tð Þ ¼ Φs tð Þ Φs tð Þ 0

� �

where 0 denotes a conformable zero matrix.

Appendix C. Proof of Lemma 2

To save notation, let us define ut+ := [ut', ut + 1

' ,…,uT' ]', and let ξt :=[ξ0' ,…,ξt']' denote a sequence of random vectors ξ0,…,ξt. The σ-algebragenerated by ξ0,…,ξt, namely σ(ξt), will be for simplicity identifiedwith ξt.

We first derive the conditional expectations for the augmentedstate vector zt. This is accomplished by employing a time-varyingKalman filter for the state-space system (7)–(8). Indeed, the objectiveis to identify at every time step t, an estimate zt that minimizes themean-squared error covariance:

Pt ¼ E zt−ztð Þ zt−ztð Þ0st��

A potential issue lies in that the noise provides information aboutthe state since the regime-switchingmatrices multiplying the two de-pend on the same underlying Markov state. However, as long as the


current realization of the Markov chain is observable, the state vari-able zt and the noise vt become independent. Likewise, though thenoise turns correlated, conditioned on the current state estimateand the Markov state, the next period noise remains (conditionally)zero-mean.

Since the estimator at time t has access to observations (y0,…,yt)and the Markov state values (s0,…,st), the optimal linear MMSE filter-ing estimate E zt F tj Þð is obtained from a time-varying (sample path)Kalman filter (e.g. Chizeck and Ji, 1988). Let st ¼ i∈S be the Markovstate observed at time t, then:

zt ¼ ztjt−1 þ K t yt−Φi ztjt−1

� �; z0 ¼ E z0ð Þ ð23Þ

�K t ¼ Ptjt−1Φ�i I þ �Φ iPtjt−1

�Φ0

i

� �†

ztþ1jt ¼ Aizt þ But

Pt ¼ Ptjt−1−�K tCiPtjt−1

Ptþ1jt :¼ E ztþ1−ztþ1jt� �

ztþ1−ztþ1jt� �0

jst� �

¼ AiPtA0

i þ CiC0

i

ð24Þ

where P0 = cov(z0,z0|s0).Using the measurement Eqs. (8), (23) rewrites:

ztþ1 ¼ Aizt þ ButK t Φi zt−ztð Þ þwt

� �which along with Eq. (7) yields the equation of the estimation errorηt :¼ zt−zt:

ηtþ1 ¼ Ai−K tΦ i

� �ηt þ Civtþ1−K twt ð25Þ

from which we observe that ηt is independent of ut.We turn now to the Markov jump LQG problem described by

(OF)-(7)-(8). Let us define the cost-to-go at t:

Jt uþt ;F t

� �¼ E

XTþ1

h¼t

zThMzhjF t

( )ð26Þ

and the optimal cost-to-go (at t):

J�t F tð Þ ¼ minu∈U

Jt uþt ;F t

� �; ð27Þ

where U readily follows from the above defined Ut, and the min istaken samplewise with respect to F t . Finally denote:

uþt� ¼ argmin

u∈UJt uþ

t ;F t

� �ð28Þ

Straightforward computation yields the following recursive rela-tion between the optimal cost-to-go functionals (27):

J�t F tð Þ ¼ E z0

tMzt jF t

n oþmin

ut

E J�tþ1 F tþ1� �jF t

ð29Þ

Going backwards, at the last stage one has:

uþ�0 ¼ argmin

u∈UJ0 uþ

0 ;F 0

� �

hence a fortiori:

uþ�0 ¼ argmin

u∈UE J0 uþ

0 ;F 0

� �n o¼ argmin

u∈UJ uð Þ

which delivers the desired solution.

As to the initial stage, we need J�T F Tð Þ, which requires us to solvefor:

u�T ¼ argmin

uTJT uT ;F Tð Þ ¼ argmin

uTE z

0

TMzT þ z0

Tþ1MzTþ1jF T

n oð30Þ

and then to substitute it into the functional:

J�T FTð Þ ¼ JT u�T ;F Tð Þ ¼ Efz0

TMzT þ z0

Tþ1MzTþ1 F Tj g¼ Efz0

TMzT þ z0

TA0

s Tð ÞMAs Tð ÞzT þ u�0T B

0MBu�

Tgþ 2z

0

TA0

s tð ÞMBu�T þ v

0

Tþ1C0

s Tð ÞMCs Tð ÞvTþ1 F Tj gð31Þ

where it has been used the independence of zT, s (T) from vT + 1,which implies:

E z0

TA0

s Tð ÞMCs Tð ÞvTþ1jF T

n o¼ E z

0

TA0

s Tð ÞMCs Tð ÞE vTþ1� �jF T

n o¼ 0 ð32Þ

as well as:

E u0

TB0MCs Tð ÞvTþ1jF T

n o¼ E u

0

TB0MCs Tð ÞE vTþ1

� �jF T

n o¼ 0 ð33Þ

by the independence of sT, yT, hence of uT≡uT F Tð Þ, from vT + 1.Noting that uT only affects the quadratic form of zT + 1 in Eq. (30),

thus using the system equation, it holds:

u�T ¼ argmin

uTE z

0

Tþ1MzTþ1jF T

n oð34Þ

Using Eqs. (32), (33), and noting that uT does not affect the qua-dratic terms in zT and vT + 1, we obtain:

u�T ¼ arg minuT

E u0

TB0M BuT þ 2z

0

TA0

s tð ÞM BuT jF T

n o¼ arg minuT

u0

TB0M BuT þ 2z

0

TA0

s tð ÞM BuT

� �

Setting to zero the derivative with respect to uT of the positivequadratic functional in the above equation yields:

u�T ¼ − B′MB

� �−1B′MAs Tð ÞzT ð35Þ

and substituting Eq. (35) into Eq. (31), the following expression of theoptimal cost at time T obtains:

J�T FTð Þ ¼ E z0

TKTzT þ zT−zTð Þ0LT zT−zTð Þ þ v

0

Tþ1C0

s Tð ÞMCs Tð ÞvTþ1jF T

n oð36Þ

where:

LT ¼ A0

s Tð ÞMAs Tð Þ ð37Þ

KT ¼ M−LT þ A0

s Tð ÞMAs Tð Þ ¼ M ð38Þ

Now, the DPA (29) for t = T-1 implies:

u�T−1 ¼ argminuT−1

E J�T F Tð ÞjF T−1f g¼ argminuT−1

E z0

TKTzT jF T−1

n o;

¼ argminuT−1E z

0

TE KTð ÞzT jF T−1

n o ð39Þ

where the second equality comes from being the estimation errorzt−ztð Þ not affected by ut, and the third one from the independencebetween zT ;F T−1 and s(T). Eqs. (34) and (39) display the recursive


representation of the problem at hand, thereby the following generalcharacterization holds for the optimal control:

u�t ¼ argmin

utE z

0

tE Ktð Þzt jF t−1

n o

whose value is given by:

u�t ¼ − B′E Ktð ÞB

� �−1B′E Ktð ÞAs tð Þzt ð40Þ

where the gain Kt solves the recursive equations:

Lt ¼ A0

s tð ÞE Ktþ1� �

B B0E Ktþ1� �

B� �−1

B0E Ktþ1� �

As tð Þ ð41Þ

Kt ¼ E Ktþ1� �

−Lt þ A0

s tð ÞE Ktþ1� �

As tð Þ; KTþ1 ¼ M ð42Þ

As M is a square, idempotent matrix, from Eq. (38) it follows thatKt = M for all periods t = 1,⋯,T and states s(t) ∈ S.

Finally, by substitution of Kt in Eq. (40) we derive18:

ut :¼ u�t ¼ x�tþ1jt ð43Þ

Insofar as the expression for the feedback matrices does not de-pend on the finite horizon T, it yields the optimal control law for allthe LQG control problems in the (OF)-(7)-(8) form for any T = 0,1,….

Appendix D. General multivariate RE model withlagged expectations

Consider the general multivariate RE model featuring lagged ex-pectations on both current and future states, under imperfect infor-mation19:

xtþ1 ¼XHh¼0

XKk¼0

Et−hΓk;s t−hð Þxtþ1þk þΩs tð Þvtþ1 ð44Þ


where, for notational convenience, Et-h is used to denote the conditionalexpectations E ⋅ F t−hj Þð , h = 0,…,H. The matrices Γk,s(t − h) and Ωs(t)

collect the model parameters. The corresponding perfect-foresightmodel is:

x�tþ1 ¼XHh¼0

XKk¼0

Γk;s t−hð Þx�tþ1þk þΩs tð Þvtþ1 ð46Þ

y�t ¼ Φs tð Þx�t þwt ð47Þ

Define:

Λk;s t−ið Þ :¼ ∑hΓk;s t−hð Þ; k∈ 0; 1;…;Kf g

18 The third entry of z t is E x�tþ1 F tj Þ�.

19 The model (1)–(2) under RE corresponds to the case where H = 0, K = 1, Γ0,s(t) = Πs(t), Γ1,s(t) = Γs(t)−1 and Ωs(t) = Γs(t)−1Ψs(t).

and assume ΛK,s(t − i) invertible. The augmented system (zt, yt) is easilywritten as:

ztþ1 ¼ A s tð Þ;s t−ið Þð Þzt þ But þ C s tð Þ;s t−ið Þð Þvtþ1 ð48Þ

yt ¼ Φs tð Þzt þwt ð49Þ

where:

z0

t :¼ e0

t x�0t x�

0tþ1 … x�

0tþK−1

� �

and:

A s tð Þ;s t−ið Þð Þ ¼

0 0 −I 0 0½ �0 0 I 0 0½ �0 0 0 0 0½ �⋮ ⋮ ⋮ ⋮ ⋮0 0 0 0½ � I0 0 Λ−1

K;s t−ið Þ I−Λ0;s t−ið Þh i

… Λ−1K;s t−ið ÞΛK−1;s t−ið Þ

0BBBBBBB@

1CCCCCCCA

B ¼I00½ �0

0BB@

1CCA; C s tð Þ;s t−ið Þð Þ ¼

Ωs tð Þ00½ �

−Λ−1K;s t−ið ÞΩs tð Þ

0BB@

1CCA

Φs tð Þ ¼ Φs tð Þ Φs tð Þ 0½ ��

where [0] denotes a (conformable) zero block. The optimal feedbackcontroller for the corresponding objective functional (OF) is in theform:

ut ¼ − B′MB

� �−1B′MA s tð Þ;s t−ið Þð Þzt ð50Þ

from which it readily follows that ut ¼ x�tþ1 tj . Since the RE system(44) can be written as:

xtþ1 ¼ Etxtþ1 þΩs tð Þvtþ1

Etxtþ1 ¼XHh¼0

XKk¼0

Et−hΓk;s t−hð Þxtþ1þk

and given the perfect-foresight dynamics (46) the optimal forecast-ing function ut will deliver a nearly perfect-foresight equilibriumpath for it.

Appendix E. Proof of Proposition 2

Let us recall the dynamic equation describing the behavior of thenearly perfect-foresight solution:

xtþ1 ¼ x�tþ1jt þ Γ−1s tð ÞΨs tð Þvtþ1

with:

x�tþ1jt ¼ Γs t−1ð Þ I−Πs t−1ð Þ� �

x�tjt−1 þ K t−1Φs t−1ð Þηt−1 þ K t−1wt−1 ð51Þ

As s(t) takes values in a finite set, the noise covariance is uniformlybounded with respect to t, i.e. there exists L∈R such that:

XSi¼1

∥Γ−1i ΨiΨ

0

iΓ0−1i ∥P s tð Þ ¼ if g≤L<þ ∞; ∀t ð52Þ

Hence, the process xt in Eq. (11) is mean-square stable if and onlyif x�tjt−1 is mean-square stable. Consider first the homogeneous part of


Eq. (51). Then the process x�tjt−1 is mean-square stable if and only if(e.g. Costa et al., 2005):

rσ pjiΓ i I−Πið Þ⊗Γ i I−Πið Þh i� �

<1; i; j ∈ S

For the non-homogeneous system, being ηt orthogonal to x�tþ1jt ,and the measurement noise wt independent of xt and the σ-algebra{yk,k ≤ t}, we obtain:

γ�t ¼ E Γs t−1ð Þ I−Πs t−1ð Þ

� �γ�t−1 I−Πs t−1ð Þ

� �0

Γ0

s t−1ð Þ

� �þ

E K t−1K0

t−1

� �þ E K t−1Φs t−1ð ÞPt−1Φ

0

s t−1ð ÞK0

t−1

� � ð53Þ

where Pt ¼ E Ptð Þ, and γ�t :¼ E x�tþ1jt x

�′tþ1jt

� �. Since

XSi¼1

∥ΦiΦ′i∥P s tð Þ ¼ if g≤L<þ ∞; ∀t ð54Þ

is always verified, and:

∥Pt∥≤L<þ ∞; ∀t ð55Þ

obtains20 if rσ pjiΓ i I−Πið Þh i� �

<1; i; j ∈ S,21 it follows that require-

ment (12) fully characterizes the mean-square stability of the nearlyperfect-foresight equilibrium.

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