bubble-free policy feedback rules

Journal of Economic Theory 144 (2009) 1521–1559

www.elsevier.com/locate/jet

Bubble-free policy feedback rules ✩

Olivier Loisel a,b,∗,1

a Banque de France 41-1422, 39 rue Croix-des-Petits-Champs, 75049 Paris Cédex 01, Franceb CEPREMAP, Paris, France

Received 5 September 2007; final version received 6 August 2008; accepted 22 November 2008

Available online 31 December 2008

Abstract

We consider a broad class of linear dynamic stochastic rational-expectations models made of a finitenumber N of structural equations for N + 1 endogenous variables and to be closed by one policy feedbackrule. We design, for any model of this class and any stationary VARMA solution of that model, a “bubble-free” policy feedback rule ensuring that this solution is not only the unique stationary solution of the closedmodel, but also its unique solution. We apply these results to locally linearisable models of the monetarytransmission mechanism and obtain interest-rate rules that not only ensure the local determinacy of thetargeted equilibrium in the neighbourhood of the steady state considered, but also prevent the economyfrom gradually leaving this neighbourhood.© 2008 Elsevier Inc. All rights reserved.

JEL classification: E52; E61

Keywords: Linear dynamic rational-expectations models; Policy feedback rules; Rational bubbles; Saddle-pathproperty; Interest-rate rules; Local determinacy; Global determinacy

✩ Several earlier versions of this paper have circulated, the last of which was entitled “Bubble-free interest-rate rules.”* Address for correspondence: Banque de France 41-1422, 39 rue Croix-des-Petits-Champs, 75049 Paris Cédex 01,

France. Fax: +33 1 42 92 62 92.E-mail address: [email protected].

1 The views expressed in this paper are those of the author and should not be interpreted as reflecting those of theBanque de France.

0022-0531/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2008.11.011

1522 O. Loisel / Journal of Economic Theory 144 (2009) 1521–1559

0. Introduction

It is well known that linear dynamic rational-expectations models with at least one non-predetermined variable2 admit an infinity of solutions, which are either stationary or non-stationary. For convenience, let us note m the number of unstable eigenvalues, i.e. eigenvaluesof modulus higher than or equal to one, and n > 0 the number of non-predetermined variablesof such a model. As shown by Blanchard and Kahn [8], provided that the exogenous fundamen-tal shocks are stationary and a certain rank condition is satisfied (as is typically the case), themodel admits an infinity of stationary solutions and no non-stationary solution when 0 = m < n;an infinity of stationary solutions and an infinity of non-stationary solutions when 0 < m < n;a unique stationary solution and an infinity of non-stationary solutions when 0 < m = n; and nostationary solution and an infinity of non-stationary solutions when 0 < n < m.

When such a model features a policy feedback rule, the latter is usually designed so as toensure that the model admits a unique stationary solution (0 < m = n). This eliminates stationarysunspot equilibria, but paves the way for a large variety of rational bubbles first described byBlanchard [7].3 Alternatively, the policy feedback rule is sometimes designed so as to ensure thatthe model admits no non-stationary solution (0 = m < n), as Currie and Levine’s [14, Chap. 4]“overstable feedback rules.” That rules out rational bubbles, but leaves the door open to stationarysunspot equilibria. To sum up, a policy-maker having to choose a policy feedback rule amongthose designed in the existing literature and seeking to avoid both stationary sunspot equilibriaand rational bubbles faces a dilemma.

The basic reason why she faces this dilemma is that the existing literature designs policyfeedback rules to control the number of unstable eigenvalues (m) of the model, but not the numberof its non-predetermined variables (n), which remains strictly positive. By contrast, this paperdesigns policy feedback rules that enable the policy-maker to have the cake and eat it by settingboth m and n equal to zero, so that the model admits one unique solution and this solution isstationary. We call them bubble-free because not only do they make the model admit a uniquestationary solution, as do most of the policy feedback rules designed in the literature, but theyalso eliminate all rational bubbles.

2 Throughout the paper, we adopt Buiter’s [9] definition of predetermined and non-predetermined variables in lineardynamic rational-expectations models. According to this definition, a current endogenous variable is predetermined if itcan be expressed as a (possibly degenerate) function of only past endogenous variables and current and past exogenousshocks, and non-predetermined otherwise.

3 In this case, there is no assurance that the private sector will coordinate on the unique stationary solution. Sugges-tively, the solution then selected by McCallum’s [22] minimal state variable criterion can be a rational bubble, as pointedout by McCallum [23]. The latter moreover criticizes the widespread practice of then selecting the unique stationarysolution in the following way: “the stability criterion [. . .] is, to a significant extent, self-defeating. For the criterion isprecisely that the selected solution path must be non-explosive—dynamically stable—under the natural presumption thatexogenous driving variables (such as shocks and policy instruments) are non-explosive. Yet one important objective ofdynamic economic analysis is to determine whether particular hypothetical policy rules—or institutional arrangements—would lead to desirable economic performance, which will usually require stability. Or, to express the point somewhatdifferently, the purpose of a theoretical analysis will often be to determine the conditions under which a system will bedynamically stable or unstable. But, obviously, the adoption of the stability criterion for selection among solutions wouldbe logically incompatible with use of the models’ solution to determine if (or under what conditions) instability would beforthcoming. To the extent, then, that this objective of analysis is important, the stability criterion is inherently unsuitable.One cannot use a model to determine whether property A would be forthcoming, if the model includes a requirementthat A must not obtain.”

O. Loisel / Journal of Economic Theory 144 (2009) 1521–1559 1523

The way bubble-free policy feedback rules manage to remove all non-predetermined variablesfrom the model (n = 0) consists in partly mimicking the structural equations so as to disconnectthe current variables from the private sector’s expectations of future variables. As a consequence,some of their coefficients are tied to the structural parameters by equality constraints, rather thanby inequality constraints as is typically the case for conventional policy feedback rules (i.e. rulesimplying 0 < m = n, instead of 0 = m = n). This naturally raises the question of what happenswhen the policy-maker has imperfect knowledge of the structural parameters and accordinglyfollows a policy feedback rule that is close to, but does not exactly coincide with a bubble-free policy feedback rule. Answering this question leads us to show that any bubble-free policyfeedback rule can be understood as the limit of a sequence of conventional policy feedbackrules that uses the structural equations as a lever to drive the modulus of the model’s unstableeigenvalues towards infinity.

Not only do bubble-free policy feedback rules manage to make the model admit a uniquesolution and to make this solution stationary, but they are also able to select any given candidatestationary VARMA solution. More precisely, we consider a generic framework that encompassesa broad class of linear dynamic stochastic rational-expectations models. These models are madeof a finite number N of structural equations for N + 1 endogenous variables and are to be closedby one policy feedback rule. We design, for any model of this class and any stationary VARMAsolution of that model, a bubble-free policy feedback rule ensuring that this solution is the uniquesolution of the closed model.

We apply these results to interest-rate rules in locally linearisable models of the monetarytransmission mechanism. Whether interest-rate rules lead to equilibrium determinacy or indeter-minacy is an old topic. Sargent and Wallace [25] first noted that rules expressing the nominalinterest rate as a function only of exogenous shocks led to multiple stationary solutions in an adhoc linear dynamic stochastic rational-expectations model of the monetary transmission mecha-nism. McCallum [21] then pointed out that a unique stationary solution could be obtained, in asimilar model, by making the rule express the nominal interest rate as a function of the endoge-nous variables.

Today’s most common practice to design monetary policy—and, more generally, today’s dom-inant way of thinking monetary policy, i.e. Woodford’s [27] neo-Wicksellian theory—builds onMcCallum’s [21] finding. Indeed, this practice consists in choosing, in a locally linearisabledynamic stochastic rational-expectations model of the monetary transmission mechanism, aninterest-rate rule such that the system of equations linearised in the neighbourhood of the steadystate considered admits a unique stationary solution. Such an interest-rate rule, whose locallylinearised form is often a Taylor rule satisfying the Taylor principle, enables the central bank topreclude the kind of macroeconomic fluctuations that, according to Clarida, Galí and Gertler [12]and Lubik and Schorfheide [20], may have occurred in the U.S. before 1979.

However, as first shown by Benhabib, Schmitt-Grohé and Uribe [3], these interest-rate rulescan be consistent with equilibrium trajectories that originate from the neighbourhood of thesteady state considered and gradually leave this neighbourhood. Some of them, for instance,eventually fall into the neighbourhood of another steady state interpreted as the liquidity trap,as arguably did the Japanese economy in the 1990s–2000s. Initially, while gradually leaving theneighbourhood of the steady state considered, these equilibrium trajectories are closely relatedto the trajectories of rational bubbles in the fictitious linear model that corresponds to the lo-cally linearised model. For this reason, the formula of bubble-free policy feedback rules enablesus to obtain locally linearised interest-rate rules that not only ensure the local determinacy ofthe targeted equilibrium in the neighbourhood of the steady state considered, but also prevent


the economy from gradually leaving this neighbourhood. Because we design bubble-free pol-icy feedback rules in a generic linear framework, we are able to obtain these interest-rate rulesfor a broad class of locally linearisable dynamic stochastic rational-expectations models of themonetary transmission mechanism.

The remaining of the paper is organized as follows. Sections 1 and 2 provide examples ofbubble-free policy feedback rules in simple linear models. These results are generalized in Sec-tions 3, 4 and 5, which respectively present a generic linear framework, design bubble-free policyfeedback rules in this framework and study some of their properties. Section 6 applies these gen-eral results to locally linearisable models. Section 7 discusses more specifically their applicationto locally linearisable models of the monetary transmission mechanism. This discussion is illus-trated in Section 8 with a particular model. Section 9 compares our monetary policy proposalwith others made in the literature. We then conclude and provide a technical appendix.

1. The simplest illustration

This section designs a bubble-free policy feedback rule in the simplest relevant linear model.It shows how this rule manages, unlike conventional and overstable policy feedback rules, toremove all non-predetermined variables from the model. It also shows that this rule can be viewedas the limit of a sequence of conventional rules that use the structural equation as a lever to drivethe modulus of the system’s unstable eigenvalue towards infinity.

1.1. Model

Consider an economy with a private sector and a policy-maker. At each date t ∈ Z, the pri-vate sector chooses the endogenous variable yt ∈ R and the policy-maker the policy instrumentzt ∈ R. The private sector’s behaviour is summarized by the single following structural equation:

yt = αEt {yt+1} + zt , (1)

where α is a parameter such that 0 < α < 1 and Et {.} denotes the rational-expectations operatorconditionally on the past values {yt−k, zt−k}k�1 of the endogenous variables and the current andpast values of the exogenous sunspot shocks. The policy-maker wants the economy to follow thepath (yt , zt ) = (0,0) for all t ∈ Z, which we call the “targeted path.” Her behaviour is summa-rized by a policy feedback rule. For convenience, we call a rule “conventional” when it makesthe model admit a unique stationary solution and an infinity of non-stationary solutions, and“overstable” (after Currie and Levine [14, Chap. 4]) when it makes the model admit an infinityof stationary solutions and no non-stationary solutions. We consider three parametric families ofpolicy feedback rules in turn.

1.2. Some conventional and some overstable rules

The first family of rules is made of the rules of type

zt = kyt (2)

with 0 � k < 1. The system made of (1) and (2) for t ∈ Z is equivalent to the system made of

(1 − k)yt = αEt {yt+1} (3)

for t ∈ Z, which determines the dynamics of y, and (2) for t ∈ Z, which residually determinesthe dynamics of z. Now, (3) can trivially be written in Blanchard and Kahn’s [8] form with one


non-predetermined variable and one eigenvalue, equal to 1−kα

. Therefore, if 0 � k � 1 − α thenrule (2) is conventional: the system admits a unique stationary solution, which coincides withthe targeted path, and an infinity of non-stationary solutions. Alternatively, if 1 − α < k < 1 thenrule (2) is overstable: the system admits an infinity of stationary solutions, including one thatcoincides with the targeted path, and no non-stationary solution. In both cases, the targeted pathis thus only one out of an infinity of solutions. The basic reason for this outcome is that whateverthe value of k, the system has one non-predetermined variable.

1.3. A bubble-free rule

The second family of rules is limited to the rule

zt = −αEt {yt+1}. (4)

The system made of (1) and (4) for t ∈ Z is equivalent to the system made of yt = 0 and (4)for t ∈ Z and hence to the system made of yt = 0 and zt = 0 for t ∈ Z. The system has then nonon-predetermined variable and therefore a unique solution, which coincides with the targetedpath. Rule (4) manages to remove all non-predetermined variables from the system by partlymimicking the structural equation (1) so as to disconnect yt from Et {yt+1}. We call it “bubble-free” because not only does it make the system admit the targeted path as its unique stationarysolution, as does rule (2) with 0 � k � 1 − α, but it also eliminates all the rational bubbles firstdescribed by Blanchard [7].

1.4. A sequence of conventional rules

The third family of rules is made of the rules of type

zt = −αEt {yt+1} (5)

where 0 < α < 1 and α �= α. The system made of (1) and (5) for t ∈ Z is equivalent to the systemmade of

yt = (α − α)Et {yt+1} (6)

for t ∈ Z, which determines the dynamics of y, and (5) for t ∈ Z, which residually determinesthe dynamics of z. Now, (6) can trivially be written in Blanchard and Kahn’s [8] form withone non-predetermined variable and one eigenvalue, equal to 1

α−α. Since | 1

α−α| > 1, rule (5) is

conventional: the system admits a unique stationary solution, which coincides with the targetedpath, and an infinity of non-stationary solutions. Besides, as α −→ α, (5) tends towards (4)and | 1

α−α| −→ +∞. Bubble-free rule (4) can therefore be viewed as the limit of a sequence of

conventional rules of type (5) that use the structural equation (1) as a lever to drive the modulusof the system’s unstable eigenvalue towards infinity.

1.5. Discussion

It is useful, for reasons of expositional clarity, to start with this example that shows the exis-tence of bubble-free policy feedback rules and presents their working mechanism in the simplestrelevant linear model, which has only one structural equation. However, this example provideslittle clue to whether they exist and, if they do, how they work in a model that has more than onestructural equation featuring expectational terms. This is the role of the next section.


2. A “New Keynesian” illustration

This section conducts the same kind of analysis as the previous section in a linear model withtwo structural equations that both feature expectational terms. We choose to consider the linearmodel that corresponds to the locally linearised New Keynesian model, because conventionalinterest-rate rules (typically Taylor rules satisfying the Taylor principle) are well known in thismodel. Let us stress however that we deal here with an ad hoc globally linear model, whoseendogenous variables are allowed to take any real-number value, and not with the non-linear butlocally linearisable New Keynesian model itself, whose case is touched upon in Subsection 7.1.

2.1. Model

Consider again an economy with a private sector and a policy-maker. At each date t ∈ Z,the private sector chooses the endogenous variables xt ∈ R and πt ∈ R and the policy-makerthe policy instrument it ∈ R. The private sector’s behaviour is summarized by the following twostructural equations:

xt = Et {xt+1} − σ(it − Et {πt+1} − rn

t

), (7)

πt = βEt {πt+1} + κxt , (8)

where β , κ and σ are three parameters such that 0 < β < 1, κ > 0 and σ > 0, rnt is an exoge-

nous shock and Et {.} denotes the rational-expectations operator conditionally on the past values{xt−k,πt−k, it−k}k�1 of the endogenous variables, the current and past values {rn

t−k}k�0 of thefundamental shock and the current and past values of the exogenous sunspot shocks. The policy-maker wants the economy to follow the path (xt ,πt , it ) = (0,0, rn

t ) for all t ∈ Z, which we callthe “targeted path.” Her behaviour is summarized by a policy feedback rule. We consider againthree parametric families of policy feedback rules in turn.

2.2. Some conventional and some overstable rules

The first family of rules is made of the rules of type

it = rnt + φππt + φxxt (9)

with (φπ ,φx) ∈ R2 and 1 + σφx + κσφπ �= 0. The system made of (7), (8) and (9) for t ∈ Z is

equivalent to the system made of[1 + σφx σφπ

−κ 1

][xt

πt

]=[

1 σ

0 β

][Et {xt+1}Et {πt+1}

](10)

for t ∈ Z, which determines the dynamics of x and π , and (9) for t ∈ Z, which residually de-termines the dynamics of i. Now, (10) can trivially be written in a Blanchard and Kahn’s [8]form that has, whatever the value taken by (φπ ,φx), two non-predetermined variables and twoeigenvalues. As a consequence, if (φπ ,φx) is chosen so that these two eigenvalues are unstable,then rule (9) is conventional: the system admits a unique stationary solution, which coincideswith the targeted path, and an infinity of non-stationary solutions. Alternatively, if (φπ ,φx) ischosen so that these two eigenvalues are stable, then rule (9) is overstable: the system admits aninfinity of stationary solutions, including one that coincides with the targeted path, and no non-stationary solution. In both cases, the targeted path is thus only one out of an infinity of solutions.Again, the basic reason for this outcome is that whatever the value of (φπ ,φx), the system hastwo non-predetermined variables.


2.3. Some bubble-free rules

The second family of rules is made of the rules of type

it = rnt + Et {πt+1} + ψπt + 1

σEt {Δxt+1} (11)

where Δ denotes the first-difference operator and ψ ∈ R∗. The system made of (7), (8) and (11)

for t ∈ Z is equivalent to the system made of xt = 0, πt = 0 and it = 0 for t ∈ Z. Indeed, thelatter straightforwardly implies the former, and the former conversely implies the latter since thereplacement of it in (7) by the right-hand side of (11) leads to πt = 0, as the terms in Et {πt+1},Et {xt+1} and xt cancel each other out; the same reasoning conducted one period ahead leads toEt {πt+1} = 0; the replacement of Et {πt+1} and πt in (8) by 0 then leads to xt = 0; the samereasoning conducted one period ahead leads to Et {xt+1} = 0; finally, (7) or (11) then leads toit = rn

t .The system has then no non-predetermined variable and hence a unique solution, which co-

incides with the targeted path. Rules of type (11) therefore qualify as bubble-free rules. As clearfrom the reasoning above, they manage to remove all non-predetermined variables from thesystem by partly mimicking the structural equation (7) so as to disconnect πt from Et {xt+1},Et {πt+1} and xt and thus to pin down the dynamics of π uniquely. The unique path for π thenimplies, via the structural equation (8), a unique path for x. Finally, the unique paths for π and x

imply, via the structural equation (7) or rule (11), a unique path for i.

2.4. A sequence of conventional rules

The third family of rules is made of the rules of type

it = rnt + Et {πt+1} + ψπt + 1

σEt {Δxt+1} (12)

where σ > 0 and σ �= σ . The system made of (7), (8) and (12) for t ∈ Z is equivalent to thesystem made of[

σ−σσ

σψ

−κ 1

][xt

πt

]=[

σ−σσ

00 β

][Et {xt+1}Et {πt+1}

](13)

for t ∈ Z, which determines the dynamics of x and π , and (12) for t ∈ Z, which residuallydetermines the dynamics of i. Now, provided that σ−σ

σ�= κσψ , (13) can easily be written in

Blanchard and Kahn’s [8] form[Et {xt+1}Et {πt+1}

]=[

1 σψ σσ−σ

− κβ

1β

][xt

πt

]with two non-predetermined variables and two eigenvalues. These eigenvalues are the roots ofthe second-order polynomial

X2 −(

1 + 1

β

)X + κσψ

β

σ

σ − σ.

Provided that σ is sufficiently close to σ , their modulus is higher than one and rule (12) istherefore conventional: the system admits a unique stationary solution, which coincides withthe targeted path, and an infinity of non-stationary solutions. Moreover, as σ −→ σ , (12) tendstowards (11) and the modulus of the eigenvalues tends towards infinity. As a consequence,


bubble-free rule (11) can be viewed as the limit of a sequence of conventional rules of type(12) that use the structural equations as a lever to drive the modulus of the system’s unstableeigenvalues towards infinity.

2.5. Discussion

This example extends the previous one by showing the existence of bubble-free policy feed-back rules and presenting their working mechanism in a well-known framework with two struc-tural equations that both feature expectational terms. It remains, however, specific in many ways.For instance, the simplicity of the working mechanism of bubble-free rule (11) is partly due tothe fact that the structural equation (8) makes the path for x directly recoverable from the pathfor π . The example therefore provides little clue to whether there exists a bubble-free policyfeedback rule and, if there exists one, how it works when a term αEt {xt+1} is artificially addedto the right-hand side of the structural equation (8), leaving the other structural equation (7) un-changed. It provides also little clue to whether there exists a bubble-free policy feedback ruleand, if there exists one, how it works when there are more than two structural equations featuringexpectational terms, when the policy instrument appears in several structural equations and/orwhen the policy instrument appears several times (say, as a current variable and as a lagged vari-able) in the same structural equation. This is the role of the next three sections, which generalizethe results obtained so far in specific examples.

3. A generic linear model

This section presents the generic linear dynamic stochastic rational-expectations model thatwe consider in Sections 4 and 5 for the design and study of bubble-free policy feedback rules.

3.1. Overview

Consider a linear model whose agents are a private sector and a policy-maker. This modelhas N + 1 endogenous scalar variables, where N ∈ N∗.4 Each of these endogenous variablesis allowed to take any real-number value. Only one of them, called the policy instrument, isdirectly controlled by the policy-maker. But this restriction is without any loss in generalitysince, in the case of several policy instruments, the policy-maker could always exogenize all butone. The equilibrium conditions of the model consist in N + 1 time-invariant linear equationsthat can be decomposed into N structural equations, which entirely describe the private sector’sbehaviour, and one policy feedback rule, which entirely describes the policy-maker’s behaviour.Time being discrete, indexed by t ∈ Z, let zt denote the value of the policy instrument chosenby the policy-maker at date t , Yt the N -dimension vector made of the values of the endogenousvariables chosen by the private sector at date t , L the lag operator, ξ t an M-dimension vectorof exogenous fundamental shocks, with M ∈ N, and Et {.} the rational-expectations operatorconditionally on the past values {Yt−k, zt−k}k�1 of the endogenous variables, the current andpast values {ξ t−k}k�0 of the fundamental shocks as well as the current and past values of theexogenous sunspot shocks.5

4 In the following, we sometimes use for convenience notations that implicitly assume N � 2. In such cases, the readershould easily infer the notation rigorously adapted to the case N = 1.

5 This definition of Et {.}, together with the structural equations and the policy feedback rules considered, will implyin particular that ∀t ∈ Z, Et {Yt } = Yt and Et {zt } = zt .


3.2. Structural equations

The N structural equations are written as follows6:

Et

{A(L)Yt + B(L)zt

}+ C(L)ξ t = 0 (14)

with

A(L)(N×N)

≡na∑

k=−ma

AkLk, B(L)

(N×1)

≡nb∑

k=−mb

BkLk and C(L)

(N×M)

≡nc∑

k=0

CkLk,

where (ma,mb,na, nb, nc) ∈ N5 and all Ak , Bk and Ck have real numbers as elements.

Let ei denote, for all i ∈ {1, . . . ,N}, the N -element vector whose ith element is equal toone and whose other elements are equal to zero. Let IA denote the set of i ∈ {1, . . . ,N} suchthat e′

iA(L) �= 0 and IB the set of i ∈ {1, . . . ,N} such that e′iB(L) �= 0. Let ma

i ≡ −min[k ∈{−ma, . . . , na}, e′

iAk �= 0] for i ∈ IA and mbi ≡ −min[k ∈ {−mb, . . . , nb}, e′

iBk �= 0] for i ∈ IB .Lastly, let us note, when IA = {1, . . . ,N},

A(L) ≡⎡⎣ e′

1Lma

1

...

e′NLma

N

⎤⎦A(L).

We make the following two assumptions:

Assumption 1.

(i) IA = {1, . . . ,N};(ii) ∀i ∈ {1, . . . ,N}, ma

i � 0;(iii) A(0) is invertible.

Assumption 2.

(i) 1 ∈ IB ;(ii) mb

1 � 0;(iii) ∀i ∈ IB � {1}, ma

i − mbi > max(0,ma

1 − mb1).

Moreover, let Δi(X) ∈ R[X] for i ∈ {1, . . . ,N + 1} denote the determinant of the N × N

matrix obtained by removing its ith column from N × (N + 1) matrix

Xmax(na,nb)[A(X−1) B

(X−1)],

and let D(X) ∈ R[X] denote the greatest common divisor, defined up to a non-zero multiplicativescalar, of all non-zero Δi(X) for i ∈ {1, . . . ,N + 1}. We make the following additional assump-tion:

6 As is well known, any linear dynamic rational-expectations model with as many equations as endogenous variablescan be rewritten in an equivalent form with only one lag and one lead. But the aim of this paper compels us to considerthe structural equations and the policy feedback rule separately and therefore forbids us to start from the commonly usedexpression with only one lag and one lead for either the structural equations or the policy feedback rule.


Assumption 3. All roots of D(X) have their modulus strictly lower than one.

In particular, this assumption forbids the structural equations to be of the kindEt {Lk A(L−1)e′

iYt } = 0, where k ∈ N, i ∈ {1, . . . ,N} and A(X) ∈ R[X] has one root of modulushigher than or equal to one.

These assumptions will enable us to design bubble-free policy feedback rules in the nextsection. Indeed, more precisely, these rules will use Assumptions 1, 2(i) and 2(ii) to pin downuniquely Yt as a function of Yt−1−k , zt−max(0,ma

1−mb1+1)−k and ξ t−k for k � 0, then Assump-

tion 2(iii) to pin down uniquely zt as a function of Yt−1−k , zt−1−k and ξ t−k for k � 0, and finallyAssumption 3 to ensure that the unique solution thus obtained for Yt and zt is stationary. To ourknowledge, the structural equations of existing linear dynamic stochastic rational-expectationsmodels are typically written in a form of type (14) that satisfies Assumptions 1 and 3. We viewAssumption 2 as slightly more restrictive and will justify it in Subsection 7.1 in the context of lo-cally linearisable dynamic stochastic rational-expectations models of the monetary transmissionmechanism.7

3.3. Policy feedback rule

The policy feedback rule is to be chosen from the set of rules that can be written as follows:

Et

{F(L)Yt

}+ G(L)zt + H(L)ξ t = 0 (15)

with

F(L)(1×N)

≡nf∑

k=−mf

FkLk, G(L) ≡

ng∑k=0

gkLk and H(L)

(1×M)

≡nh∑

k=0

HkLk,

where (mf ,nf ,ng,nh) ∈ N4, all gk are real numbers, g0 �= 0 and all Fk , Hk have real numbersas elements. Such rules qualify as what the existing literature calls “instrument rules” since theirzt -coefficient g0 is non-zero.

3.4. Exogenous fundamental shocks

Each exogenous fundamental shock is assumed to follow a centered stationary autoregressiveprocess of finite order8:

D(L)ξ t = εt with D(L)(M×M)

≡

⎡⎢⎢⎢⎣D1(L) 0 . . . 0

0. . .

. . ....

.... . .

. . . 00 . . . 0 DM(L)

⎤⎥⎥⎥⎦7 That said, we conjecture that all the propositions of the paper would still hold if Assumption 2(iii) were relaxed and

replaced by ∀i ∈ IB � {1}, mai

− mbi

> ma1 − mb

1. We assume that ∀i ∈ IB � {1}, mai

− mbi

> 0 for convenience only, tokeep the proofs of these propositions simple.

8 This assumption is not restrictive in the sense that if each element of ξ t followed instead a centered stationary finite-order ARMA process, then C(L)ξ t and H(L)ξ t could easily be rewritten in the form C∗(L)ξ∗

t and H∗(L)ξ∗t , where

C∗(L) and H∗(L) satisfy the same conditions as C(L) and H(L) respectively and each element of ξ∗t follows a centered

stationary autoregressive process of finite order.


and Di(L) ≡ ∑nd

k=0 di,kLk for 1 � i � M , where nd ∈ N, all di,k are real numbers, D(0) is

invertible, all the eigenvalues of D(L) are of modulus strictly lower than one and εt is a centeredwhite noise vector whose M elements are linearly independent from each other.

3.5. Targeted path

We assume that the policy-maker seeks to implement a given sequence {Yt , zt }t∈Z that can bewritten in the form[

Yt

zt

]= S(L)

[Yt

zt

]+ T(L)εt (16)

with

S(L)((N+1)×(N+1))

≡ns∑

k=1

SkLk and T(L)

((N+1)×M)

≡nt∑

k=0

TkLk

where ns ∈ N∗, nt ∈ N, all Sk and Tk have real numbers as elements, all eigenvalues of the

system I − S(L) are of modulus strictly lower than one, with I denoting the (N + 1) × (N + 1)

identity matrix, and S(L) and T(L) are such that the sequence {Yt , zt }t∈Z defined by (16) satisfies(14) for t ∈ Z. That is, the policy-maker wants the endogenous variables to follow a stationarycentered finite-order VARMA process that is consistent with the structural equations and does notinvolve white noises other than those driving the stochastic process of the fundamental shocks.This targeted path can, but need not, result from the maximization of some quadratic objectivefunction subject to (14), as discussed in greater detail in Subsection 7.1.

4. Design of bubble-free policy feedback rules

This section designs policy feedback rules of type (15) such that the model made of (14) andone of these rules admits a unique solution and this solution coincides with the targeted path(16).

4.1. Their general form

Let us adopt the convention∑v

i=u{.} = 0 for u > v. Consider the policy feedback rules of thefollowing form:

Et

{Lmb

1 e′1A(L)Yt

}+ Lmb1 e′

1B(L)zt + Lmb1 e′

1C(L)ξ t + e′1O(L)D(L)ξ t

+N∑

i=2

Lmb

1+∑ij=2 ma

j[e′iA(L)Yt + e′

iB(L)zt + e′iC(L)ξ t + L−ma

i e′iO(L)D(L)ξt

]+ Lmb

1+∑Ni=2 ma

i[P(L)Yt + Q(L)zt + R(L)D(L)ξ t

]= 0, (17)

where O(L), P(L), Q(L) and R(L) satisfy the following conditions:


Condition 1.

O(L)(N×M)

≡

⎡⎢⎢⎢⎢⎣∑mb

1−1k=0 O1,kL

k∑ma2−1

k=0 O2,kLk

...∑maN−1

k=0 ON,kLk

⎤⎥⎥⎥⎥⎦ ,

where all Oi,k have real numbers as elements.

Condition 2.

P(L)(1×N)

≡np∑k=0

PkLk,

where np ∈ N and all Pk have real numbers as elements, and

Ω(N×N)

≡

⎡⎢⎢⎣P0

e′2A(0)

...

e′N A(0)

⎤⎥⎥⎦is invertible.

Condition 3. Q(L) ≡∑nq

k=max(0,ma1−mb

1+1)qkL

k , where nq ∈ N and all qk are real numbers.

Condition 4. The system[A(L) B(L)

P(L) Q(L)

](18)

has all its eigenvalues of modulus strictly lower than one.

Condition 5.

R(L)(1×M)

≡nr∑

k=0

RkLk,

where nr ∈ N and all Rk have real numbers as elements.

Two points are worth mentioning at this stage about rules of type (17) satisfying Conditions 1to 5. First, they belong to the class of rules (15), in particular because their zt -coefficient e′

1B−mb1

is non-zero. Second, there exist such rules. The latter assertion will come as a consequence ofProposition 2 below. We choose not to prove it until then because its proof is both not straight-forward and very close to the proof of Proposition 2. Let us just note here that the main difficultyin proving this assertion is, of course, to show that there exist P(L) and Q(L) satisfying Con-ditions 2, 3 and 4. To show this, the proof of Proposition 2 will resort to Assumption 3 and thegeneralized identity of Bezout (for Condition 4), the Euclidian division (for Condition 3) andAssumption 1(iii) (for Condition 2).


4.2. Their property of ensuring a unique and stationary solution

We first show that the system made of (14) and any rule of type (17) satisfying Conditions 1to 5 (if such a rule exists) admits a unique solution and that this solution is stationary:

Proposition 1. Whatever O(L), P(L), Q(L), R(L) satisfying Conditions 1 to 5, the system madeof (14) and (17) at all dates t ∈ Z admits a unique solution {Yt , zt }t∈Z and this solution isstationary.

Proof. Cf. Appendix A.1. �As made clear in Appendix A.1, rules of type (17) satisfying Conditions 1 to 5 (if they

exist) achieve the existence and uniqueness of the solution {Yt , zt } by disconnecting Yt fromEt {Yt+k} and Et {zt+k} for k � 1, thus pinning down Yt uniquely, and by making zt uniquelyrecoverable from Yt . Let us briefly elaborate on these two points. First, these rules manage todisconnect Yt from Et {Yt+k} and Et {zt+k} for k � 1 by partly “mimicking” the structural equa-tions. More precisely, the expectation at date t of one of these rules taken at date t + mb

1 hasthe same forward-looking part as the first structural equation, so that subtracting one from theother leads to a backward-looking equation (i.e. an equation without expectational terms); sim-ilarly, the expectation at date t of this backward-looking equation taken at date t + ma

2 has thesame forward-looking part as the second structural equation, and so on. The resulting N -equationsystem is backward-looking and, because the rule of type (17) considered satisfies Condition 2,non-degenerate. As a consequence, this system pins down Yt uniquely. Second, although theserules express zt as a function of Et {Yt+j } for j ∈ {1, . . . ,ma

1 −mb1} when ma

1 > mb1, they make zt

uniquely recoverable from Yt thanks to Assumption 2(iii) and because they satisfy Condition 3.Finally, the basic reason why the unique solution is stationary is, of course, that these rules satisfyCondition 4.

4.3. Their ability to select any given stationary VARMA path

We then show not only that there exist rules of type (17) satisfying Conditions 1 to 5, but alsothat any targeted path of type (16) satisfying (14) can be uniquely implemented by a suitablychosen policy feedback rule of type (17) satisfying Conditions 1 to 5:

Proposition 2. For any sequence {Yt , zt }t∈Z of type (16) that satisfies (14) for all t ∈ Z, thereexist O(L), P(L), Q(L) and R(L) satisfying Conditions 1 to 5 and such that {Yt , zt }t∈Z is theunique solution of the system made of (14) and (17) at all dates t ∈ Z.

Proof. Cf. Appendix A.2. �Given Proposition 1, the proof of Proposition 2 amounts of course to find some O(L), P(L),

Q(L) and R(L) that satisfy Conditions 1 to 5 and are such that the given targeted path (16) issimply one solution of the system made of (14) and (17) for all t ∈ Z. To do this, Appendix A.2proceeds in several steps. The first step consists in choosing, from (14) and (16), an O(L) thatsatisfies Condition 1. The second and third steps consist in using the generalized identity ofBezout and the Euclidian division together with Assumptions 1(iii) and 3 to choose some P(L)

and Q(L) that satisfy Conditions 2, 3 and 4 and are such that the eigenvalues of I − S(L) are


also eigenvalues of the system made of (14) and (17). And the fourth step consists in usingCramer’s rule to residually choose, from (14), (16) and the previous steps, an R(L) that satisfiesCondition 5.

5. Other properties of bubble-free policy feedback rules

This section studies three additional properties of the bubble-free policy feedback rules de-signed in the previous section. These properties will prove useful in the context of locallylinearisable models considered in the next section.

5.1. The limit of a sequence of conventional policy feedback rules

This subsection shows that any of the bubble-free policy feedback rules designed in the pre-vious section can be understood as the limit of a sequence of conventional policy feedback rulesthat use the structural equations as a lever to drive the modulus of the system’s unstable eigen-values towards infinity. Let us first define the metric d by

(X1(L)(N1×N2)

, X2(L)(N1×N2)

) ≡(

nx∑k=−mx

X1,kLk,

nx∑k=−mx

X2,kLk

)→ d

(X1(L),X2(L)

)≡ sup

−mx�k�nx

[max

1�i�N1

(max

1�j�N2

∣∣e′1,i (X1,k − X2,k)e2,j

∣∣)],where (mx,nx) ∈ N

2, (N1,N2) ∈ N∗2, all X1,k , X2,k have real numbers as elements and, for

h ∈ {1,2} and l ∈ {1, . . . ,Nh}, eh,l is the Nh-element vector whose lth element is equal to oneand whose other elements are equal to zero. Let us then consider a given targeted path of type(16) satisfying (14), and a rule of type (17), noted (R), that satisfies Conditions 1 to 5 and isconsistent with this targeted path. Let us also rewrite this targeted path as[

Yt

zt

]= X(L)εt with X(L)

((N+1)×M)

≡+∞∑k=0

XkLk, (19)

where all Xk have real numbers as elements. Lastly, let (Rε ) denote the rule corresponding to theexpression (17) of rule (R) where some exogenous disturbances, each of them randomly drawnfrom a continuous probability distribution supported on a bounded interval including zero,9 areadded to the elements of Ak for −ma � k � na , Bk for −mb � k � nb, Ck for 0 � k � nc, O1,k

for 0 � k � mb1 − 1, Oi,k for 2 � i � N and 0 � k � ma

i − 1, Pk for 0 � k � np , Rk for 0 �k � nr , to qk for max(0,ma

1 − mb1 + 1) � k � nq and to di,k for 1 � i � M and 0 � k � nd , and

let ε denote the maximal length of these distribution-supporting intervals. We get the followingproposition:

Proposition 3. ∀μ ∈ ]1;+∞[, ∀δ ∈ ]0;+∞[, ∃ε ∈ ]0;+∞[, ∀ε ∈ ]0; ε], with probability one,

9 This continuous-probability-distribution assumption enables us to disregard degenerate cases as they are of measurezero.


(i) the system made of (14) and (Rε) has no eigenvalue whose modulus is between 1 and μ;(ii) this system admits a unique stationary solution, which we note[

Yt

zt

]= Xε(L)εt with Xε(L)

((N+1)×M)

≡+∞∑k=0

Xε,kLk, (20)

where all Xε,k have real numbers as elements; and(iii) d(Xε(L),X(L)) < δ.

Proof. Cf. Appendix A.3. �Proposition 3, which generalizes the results obtained in Subsections 1.4 and 2.4, says in

essence that bubble-free policy feedback rule (R) can be viewed as the limit, as ε −→ 0, of asequence of conventional rules of type (Rε ) along which: (i) the modulus of the unstable eigen-values of the system made of (14) and each of these rules tends towards infinity; and (ii) thissystem’s unique stationary solution tends towards the unique solution of the system made of (14)and (R). As made clear in Appendix A.3, these conventional rules manage to drive the modulusof the system’s unstable eigenvalues towards infinity by using the structural equations as a lever.

This analysis can also, of course, be interpreted as a robustness analysis aiming to relax theassumption that the policy-maker has perfect knowledge of the values of the structural param-eters, i.e. the coefficients featuring in the structural equations (14). In this case, the exogenousdisturbances added to the elements of matrices Ak , Bk , Ck and to parameters di,k should be in-terpreted as “measurement errors” on the values of the structural parameters, while those addedto the elements of matrices Oi,k , Pk , Rk and to parameters qk can be interpreted as the conse-quence of these measurement errors. This robustness analysis is particularly welcome as someof the coefficients of rules of type (17) are tightly tied to the structural parameters by equal-ity constraints. By contrast, all the coefficients of conventional policy feedback rules are looselytied to the structural parameters by inequality constraints, as exemplified by the well-known Tay-lor principle. From this point of view, Proposition 3 says that, provided that the policy-maker’sknowledge of the values of the structural parameters is sufficiently accurate, the application ofthe formula of bubble-free policy feedback rules leads to conventional policy feedback rules thatmake the modulus of the system’s unstable eigenvalues very large. The conclusion of the paperwill briefly comment on this result.

5.2. Time needed by bubble-free policy feedback rules to be effective

For any system of equations (S), let L(S) denote the system obtained by applying operatorL on both the left- and the right-hand sides of each equation of (S). The following propositionrepresents a variation on Proposition 2:

Proposition 4. For any sequence {Yi , zi}i∈Z of type (16) that satisfies (14) for all i ∈ Z, thereexist O(L), P(L), Q(L) and R(L) satisfying Conditions 1 to 5 and such that, whatever t ∈ Z,{Yt−i , zt−i}i∈N is the unique solution of the system made of Lj (14) for all j � −τ +mb

1 and Lk

(17) for all k � −τ , where τ ≡ max(ma1,mb

1) +∑Ni=2 ma

i .

Proof. Straightforward from Appendix A.1 and Proposition 2. �


Proposition 4 says in essence that, at each date t ∈ Z, there exists a unique solution for thecurrent endogenous variables Yt and zt (which coincides with the value taken by these endoge-nous variables on the targeted path at this date) provided that the structural equations and thebubble-free policy feedback rule hold until, respectively, dates t + τ − mb

1 and t + τ included.In other words, the time needed by bubble-free policy feedback rules to be effective is equal to τ

periods. The fact that this time is equal to the sum of the lengths of the forward-looking parts ofthe structural equations is, naturally, a direct consequence of the working mechanism of bubble-free policy feedback rules, explained in Subsection 4.2 and shown formally in Appendix A.1.Simple illustrations of Proposition 4 can be found in Subsection 1.3, where τ = 1 and mb

1 = 0,that is to say that the system made of (1) and (4) taken at dates t to t + 1 pins down (yt , zt )

uniquely; and in Subsection 2.3, where τ = 2 and mb1 = 0, that is to say that the system made

of (7), (8) and (11) taken at dates t to t + 2 pins down (πt , xt , it ) uniquely. Finally, it is worthnoting that one straightforward implication of Proposition 4 is that bubble-free policy feedbackrules remain effective in ensuring a unique solution even when the policy-maker cannot committo forever following a given policy feedback rule, provided that she can commit at each date tofollowing a given policy feedback rule during the next τ periods.

5.3. Effect of initial conditions

We have so far considered the case where the policy-maker follows a bubble-free policy feed-back rule at all dates t ∈ Z. The alternative case where she follows such a rule from a given datet0 onwards, with given bounded initial conditions Yt0−k and zt0−k for k ∈ N

∗, is then easily dealtwith. Indeed, as straightforward from Appendix A.1 and Proposition 2, whatever these boundedinitial conditions, the system made of the structural equations and the bubble-free policy feed-back rule for all t � t0 admits a unique solution {Yt , zt }t�t0 and, since this system admits nounstable eigenvalues (due to Condition 4), that solution asymptotically converges, in the senseof the Euclidian metric, whatever the realization of the exogenous fundamental shocks, towardsthe targeted path.

6. Application to locally linearisable models

This section shows that the application of the formula of bubble-free policy feedback rules tolocally linearisable dynamic stochastic rational-expectations models leads to locally linearisedpolicy feedback rules that not only ensure the local determinacy of the targeted equilibrium inthe neighbourhood of the steady state considered, but also prevent the economy from graduallyleaving this neighbourhood.

6.1. A generic locally linearisable model

Consider a non-linear model whose agents are a private sector and a policy-maker. This modelhas N +1 endogenous scalar variables, where N ∈ N

∗. Only one of them, noted Z and called thepolicy instrument, is directly controlled by the policy-maker. (Again, this restriction is withoutany loss in generality since, in the case of several policy instruments, the policy-maker couldalways exogenize all but one.) One of the model’s equilibrium conditions is a potentially non-linear, time-invariant policy feedback rule, which describes the policy-maker’s behaviour. Let Z

denote the set of admissible values for Z. Time being discrete, indexed by t ∈ Z, we assumethat the model’s equilibrium conditions other than the policy feedback rule are such that there


exists Z∗ ∈ Z such that, if the policy-maker followed rule Zt = Z∗ at all dates t ∈ Z and if therewere no exogenous fundamental shocks, then the model would admit an equilibrium path notedE∗ along which all endogenous variables are constant over time. We restrict the choice for thepolicy feedback rule to the set of equations that are satisfied when all endogenous variables taketheir E∗ values and all exogenous fundamental shocks take the value zero. As a consequence,whatever the rule chosen from this set, E∗ is a steady state of the model.

Among the model’s equilibrium conditions (other than the policy feedback rule) are N po-tentially non-linear, time-invariant structural equations, which describe the private sector’s be-haviour. We assume that these structural equations are linearisable in the neighbourhood of E∗and that their locally linearised version can be written in a form (14) that satisfies Assumptions 1,2 and 3, where zt denotes the deviation of Zt from Z∗ and Yt the N -dimension vector made ofthe deviations of the other endogenous variables at date t from their E∗ values. The exogenousfundamental shocks featuring in (14) are assumed to follow the stochastic process describedin Subsection 3.4 with the additional requirement that εt have a bounded probability distribu-tion: there exists a positive real number ζ such that ∀t ∈ Z, whatever the realization of εt and∀i ∈ {1, . . . ,M}, the absolute value of the ith element of εt is strictly lower than ζ . Moreover,we assume that the (potentially non-linear) structural equations involve a finite number of leadsof the endogenous variables, like for instance the Euler equation. As a consequence, not onlydoes their locally linearised form involve a finite number of leads of the endogenous variables,but it also represents a correct first-order local approximation of the true structural equations ata given date even when the endogenous variables leave the neighbourhood of E∗ at some laterdate, provided that this date is sufficiently distant in the future. This can be formally written as

Et

{A(L)Yt

}+ Et

{B(L)zt

}+ C(L)ξ t +⎡⎣ o1[Ξ t (m

a1,mb

1), ζ ]...

oN [Ξ t (maN,mb

N), ζ ]

⎤⎦= 0 (21)

where, ∀(t, k1, k2) ∈ Z3, Ξ t (k1, k2) ≡ {Yt+k1−j , zt+k2−j }j∈N and, ∀i ∈ {1, . . . ,N}, oi is a scalar

function such that, for any sequence s and any real number x, |oi (s,x)|�s −→ 0 and |oi (s,x)|

|x| −→ 0as �s ∼ |x| −→ 0, �. denoting a given norm on sequences.

Similarly, we further restrict the set of admissible (potentially non-linear) policy feedbackrules to the rules that: (i) involve a finite number of leads of the endogenous variables; (ii) arelinearisable in the neighbourhood of E∗; and (iii) are such that their locally linearised versioncan be written in a form (15). Again, this can be formally written as

Et

{F(L)Yt

}+ G(L)zt + H(L)ξ t + oN+1[Ξ t

(mf ,0

), ζ]= 0

where oN+1 is a scalar function such that, for any sequence s and any real number x,|oN+1(s,x)|

�s −→ 0 and |oN+1(s,x)||x| −→ 0 as �s ∼ |x| −→ 0. Moreover, we assume that the targeted

path: (i) satisfies the structural equations at all dates; (ii) is linearisable in the neighbourhood ofE∗; and (iii) is such that its locally linearised version can be written in a form (16). This impliesin particular that the targeted path can be written as[

Yt

zt

]= S(L)

[Yt

zt

]+ T(L)εt + p(ζ ) (22)

where p is an (N + 1) × 1 matrix function such that, for any real number x, ‖p(x)‖|x| −→ 0 as

|x| −→ 0, ‖.‖ denoting a given norm on matrices. Finally, we assume that the model’s equilib-rium conditions other than the structural equations and the policy feedback rule (such as, for


instance, transversality conditions) are necessarily satisfied when all endogenous variables con-stantly remain sufficiently close to E∗.

6.2. Application of the formula of bubble-free policy feedback rules

Consider the policy feedback rules whose locally linearised version can be written in the form(17):

Et

{Lmb

1 e′1A(L)Yt

}+ Lmb1 e′

1B(L)zt + Lmb1 e′

1C(L)ξ t + e′1O(L)D(L)ξ t

+N∑

i=2

Lmb

1+∑ij=2 ma

j[e′iA(L)Yt + e′

iB(L)zt + e′iC(L)ξ t + L−ma

i e′iO(L)D(L)ξt

]+ Lmb

1+∑Ni=2 ma


]+ q[Ξ t (m

a1 − mb

1,0), ζ]= 0, (23)

where O(L), P(L), Q(L) and R(L) satisfy Conditions 1 to 5 and q is a scalar function such that,for any sequence s and any real number x, |q(s,x)|

�s −→ 0 and |q(s,x)||x| −→ 0 as �s ∼ |x| −→ 0.

We get the following proposition, which makes use of the property of bubble-free policy feedbackrules described in Proposition 4:

Proposition 5. Whatever the targeted path of type (22) that satisfies (21) for all t ∈ Z and what-ever the scalar function q such that, for any sequence s and any real number x, |q(s,x)|

�s −→ 0

and |q(s,x)||x| −→ 0 as �s ∼ |x| −→ 0, there exist O(L), P(L), Q(L) and R(L) satisfying Condi-

tions 1 to 5 and a (N + 1) × 1 matrix function r such that:

(i) the system made of (21) and (23) for all t ∈ Z is equivalent to the system[Yt

zt

]= S(L)

[Yt

zt

]+ T(L)εt + r

[Ξ t

(τ + ma

1 − mb1, τ), ζ]

for all t ∈ Z, where τ is defined in Proposition 4; and(ii) for any sequence s and any real number x, ‖r(s,x)‖

�s −→ 0 and ‖r(s,x)‖|x| −→ 0 as �s ∼

|x| −→ 0.

Proof. Straightforward from Appendix A.1 and Proposition 2. �Proposition 5 says in essence that, under a suitably chosen policy feedback rule of type (23),

at each date t ∈ Z, if the endogenous variables Y and z are expected to remain close at the firstorder to their E∗ values during the next τ + ma

1 − mb1 and τ periods respectively, whatever the

realization of the exogenous shocks occurring during that time, and if ζ is close at the first orderto zero, then Yt and zt are close at the second order to their corresponding values on the targetedpath. This implies that if, at each date t ∈ Z such that Yt and zt are close at the second order totheir corresponding values on the targeted path, Y and z are expected to remain close at the firstorder to their E∗ values during the next τ + ma

1 − mb1 and τ periods respectively, whatever the

realization of the exogenous shocks occurring during that time, and if ζ is close at the first orderto zero, then the path followed by Y and z constantly remains close at the second order to thetargeted path.

In the case where there are no exogenous fundamental shocks (which is the case typicallyconsidered in the literature, shortly reviewed in Subsection 7.2 below, on equilibrium trajectories


gradually leaving the neighbourhood of the steady state considered under standard interest-raterules), we have ζ = 0, the targeted path is E∗ and the previous statement becomes: if, at eachdate t ∈ Z such that Yt and zt are close at the second order to their E∗ values, Y and z areexpected to remain close at the first order to their E∗ values during the next τ + ma

1 − mb1 and τ

periods respectively, whatever the realization of the exogenous sunspot shocks occurring duringthat time, then the path followed by Y and z constantly remains close at the second order to E∗.It is in this sense that rules of type (23) can then be said to prevent the economy from graduallyleaving the neighbourhood of E∗. In the alternative case where some exogenous fundamentalshocks do exist, since ζ is close at the first order to zero, the targeted path is close at the firstorder to E∗ and it is therefore in a weaker sense that rules of type (23) can then be said to preventthe economy from gradually leaving the neighbourhood of E∗.

6.3. Robustness to the policy-maker’s imperfect knowledge

Let us now consider: (i) a given targeted path of type (22) that satisfies (21); (ii) a given scalarfunction q such that, for any sequence s and any real number x, |q(s,x)|

�s −→ 0 and |q(s,x)||x| −→ 0

as �s ∼ |x| −→ 0; and (iii) a given rule, noted (R), among the possibly several rules of type(23), whose existence is established by Proposition 5, that correspond to this targeted path andthat function q . Similarly as in Subsection 5.1, let (Rε ) denote the rule corresponding to theexpression (23) of rule (R) where some exogenous disturbances, each of them randomly drawnfrom a continuous probability distribution supported on a bounded interval including zero, areadded to the elements of matrices Ak , Bk , Ck , Oi,k , Pk , Rk and to parameters qk and di,k , andlet ε denote the maximal length of these distribution-supporting intervals. We get the follow-ing proposition, which extends Proposition 5 to the case where the policy-maker has imperfectknowledge of the values of the structural parameters:

Proposition 6. There exists an (N + 1) × 1 matrix function r such that:

(i) the system made of (21) and (Rε) for all t ∈ Z is equivalent to the system[Yt

zt

]= S(L)

[Yt

zt

]+ T(L)εt + r

[Ξ t

(τ + ma

1 − mb1, τ), ζ, ε

]for all t ∈ Z, where τ is defined in Proposition 4; and

(ii) for any sequence s and any real numbers x1 and x2, ‖r(s,x1,x2)‖�s −→ 0, ‖r(s,x1,x2)‖|x1| −→ 0 and‖r(s,x1,x2)‖|x2| −→ 0 as �s ∼ |x1| ∼ |x2| −→ 0.

Proof. Straightforward from Proposition 2 and Appendix A.3. �As a consequence, provided that ε is close at the first order to zero, rule (Rε ) prevents the

economy from gradually leaving the neighbourhood of E∗ in the same sense as do rules of type(23).

7. The case of monetary policy models

The previous results can be directly applied to a wide range of monetary policy models, sincetoday’s most common way of modelling monetary policy is through an interest-rate rule in alocally linearisable dynamic stochastic rational-expectations model of the monetary transmission


mechanism that, as we argue below, is typically encompassed by our generic framework. Whatmakes their application to monetary policy models particularly relevant is that, as we argue next,there are some good reasons to think that, in these models, under standard interest-rate rules,there typically exist equilibrium trajectories gradually leaving the neighbourhood of the steadystate considered.

7.1. Discussion of some assumptions

We first briefly discuss some assumptions made in our generic framework presented in Sec-tions 3 and 6. We argue that they are little restrictive in a monetary policy context in the sensethat they are satisfied by many existing locally linearisable models of the monetary transmissionmechanism.

First, we make Assumptions 1, 2 and 3 on the structural equations. To our knowledge, thelocally linearised structural equations of existing models of the monetary transmission mecha-nism are typically written in a form of type (14) that satisfies Assumptions 1 and 3. In addition,this form often features the short-term nominal interest rate only as a current variable (i.e.mb = nb = 0), for instance in the Euler equation, the Tobin’s Q equation or the uncoveredinterest-rate parity. In this case, the locally linearised structural equations can easily be rewrittenin an equivalent form of type (14) satisfying not only Assumptions 1 and 3, but also Assump-tion 2, by re-ordering these equations so that 1 ∈ IB and ∀i ∈ IB � {1}, ma

i � ma1 and by replacing

e′i (14) by e′

iB0e′1(14) − e′

1B0e′i (14) for all those i ∈ IB � {1} such that ma

i = ma1, where, for any

N -equation system (S), e′i (S) denotes the ith equation of (S).

Second, we assume that the policy feedback rule can be forward-looking, i.e. that it can ex-press the current policy instrument as a function of the private sector’s current expectations offuture variables. This assumption requires that the policy-maker can observe these expectations.It is a crucial assumption because, when ma

1 > mb1, the bubble-free policy feedback rules designed

in Section 4 are forward-looking. But it is a standard assumption in the literature on interest-raterules, and for a good reason: central banks regularly conduct surveys to learn about the privatesector’s expectations of future macroeconomic developments and, at a higher frequency (veryhigh if need be), extract these expectations from the price of financial instruments.

Third, we assume that the targeted path takes the form of a stationary VARMA process thatdoes not involve white noises other than those driving the stochastic process of the fundamentalshocks. Today’s most common choice for the targeted path in a locally linearisable model of themonetary transmission mechanism is the path for the endogenous variables that maximizes thesecond-order approximation of social welfare in the neighbourhood of the steady state consid-ered subject to the first-order approximation of the structural equations in this neighbourhood.(As is well known, this path coincides with the first-order approximation of the social-welfare-maximizing equilibrium in this neighbourhood only when the steady state is sufficiently littledistorted.) When this maximization programme is solved from Woodford’s [27, Chap. 8] time-less perspective, as is usually the case, the targeted path is typically a stationary VARMA processthat does not involve white noises other than those driving the stochastic process of the funda-mental shocks.

Fourth, we assume that the non-linear structural equations involve a finite number of leads ofthe endogenous variables, like for instance the Euler equation. To our knowledge, this assump-tion is satisfied by all the models used in the literature (shortly reviewed in the next subsection)that shows the existence of equilibrium paths gradually leaving the neighbourhood of the steadystate considered. That said, this assumption might be suspected at first sight not to be satisfied


by models with a Calvo-type price-setting mechanism, such as the New Keynesian model andits extensions known as dynamic stochastic general equilibrium models. Indeed, these modelsfeature a non-linear price-setting equation that involves an infinite number of leads of the en-dogenous variables, even though the locally linearised version of this equation—known as theNew Keynesian Phillips curve—involves a finite number of leads of the endogenous variables.However, as first shown by Schmitt-Grohé and Uribe [26], this non-linear price-setting equationcan actually be rewritten, with a change of variables, in an equivalent form that involves a fi-nite number of leads of the endogenous variables. As a consequence, even these models can beconsidered satisfying this assumption, which we therefore view as little restrictive.

7.2. Equilibria gradually leaving the neighbourhood of the steady state

We then justify the usefulness of interest-rate rules that prevent the economy from gradu-ally leaving the neighbourhood of the steady state considered by shortly reviewing the literaturethat shows the existence of equilibrium paths gradually leaving this neighbourhood, under com-monly considered interest-rate rules, in locally linearisable rational-expectations models of themonetary transmission mechanism.

Today’s most common practice to design monetary policy in such models consists in choos-ing an interest-rate rule such that the system of equations linearised in the neighbourhood of thesteady state considered admits a unique stationary solution. However, Benhabib, Schmitt-Grohéand Uribe [3–6], Christiano and Rostagno [11] and Benhabib and Eusepi [2] have shown that,with this kind of interest-rate rule, there could exist equilibrium trajectories originating arbitrar-ily close to the steady state considered and gradually leaving its neighbourhood to eventuallyconverge towards a deterministic cycle or another steady state interpreted as the liquidity trap.In Benhabib, Schmitt-Grohé and Uribe’s [3,6] framework, in particular, these equilibria exist forempirically plausible parameterizations and are robust to wide parameter perturbations. More-over, Benhabib, Schmitt-Grohé and Uribe [3,5,6] provide one reason to suspect that the existenceof such equilibria is not peculiar to their specific frameworks. Indeed, they point out that when-ever the interest-rate rule respects the zero nominal interest-rate lower bound and makes theinterest rate react positively and, at the steady state considered, more than one-to-one to the in-flation rate, there typically exist equilibrium trajectories originating arbitrarily close to the steadystate considered and gradually leaving its neighbourhood to eventually converge towards anothersteady state at which the inflation rate is lower and the interest rate reacts less than one-to-one tothe inflation rate. As forcefully argued by Cochrane [13], there is no good reason to ignore theseequilibria.

As shown in the previous section, the interest-rate rules that we propose do eliminate suchequilibria whatever the type of non-linearity at work outside the neighbourhood of the steadystate considered. How they manage to do it can also be easily understood with the local stablemanifold theorem for discrete-time dynamic systems (Kuznetsov [19, Chap. 2, Theorem 2.3]).Indeed, this theorem straightforwardly implies that, provided that exogenous shocks are smallenough (as is the case in all the papers mentioned above, which consider no exogenous shocks inorder to focus on perfect-foresight equilibria), a necessary condition for such equilibria to existis that the system of equations linearised in the neighbourhood of the steady state consideredadmit at least one unstable eigenvalue. Therefore, by removing all unstable eigenvalues fromthis system, the interest-rate rules that we propose prevent the economy from gradually leavingthe neighbourhood of the steady state considered. Moreover, by removing all non-predetermined


variables from this system and thus making this system satisfy Blanchard and Kahn’s [8] condi-tions, they manage to ensure the existence and uniqueness of a local equilibrium.

Interestingly, Benhabib, Schmitt-Grohé and Uribe [3,5,6] show the existence of such equi-libria only when the interest-rate rule ensures a unique equilibrium in the neighbourhood of thesteady state considered. Actually, given the set of interest-rate rules that they consider whenshowing the existence of such equilibria, that there exists a unique equilibrium in this neighbour-hood happens to be a necessary condition for the existence of such equilibria in their frameworks.Indeed, the interest-rate rules that they consider have two characteristics. First, unlike the interest-rate rules that we propose, they naturally belong to parametric families of interest-rate rules thatare not designed to control the number of non-predetermined variables of the locally linearisedsystem. Therefore, this number of non-predetermined variables remains strictly positive what-ever the values taken by the coefficients of the rule considered. Second, they are “active” inthe neighbourhood of the steady state considered, i.e. they make the nominal interest rate reactmore than one-to-one to the inflation rate in this neighbourhood. Now, in their frameworks, anactive interest-rate rule happens to be necessarily, to use our terminology, either conventionalor overstable for the locally linearised system. As a consequence, if the rule considered is con-ventional, that is to say if the locally linearised system has as many unstable eigenvalues asnon-predetermined variables,10 then the corresponding linear model admits a unique stationarysolution, but an infinity of non-stationary solutions. In this case, there is a unique equilibriumin the neighbourhood of the steady state considered, but the economy can gradually leave thisneighbourhood. Alternatively, if the rule considered is overstable, that is to say if the locallylinearised system has no unstable eigenvalue, then the corresponding linear model admits nonon-stationary solution, but an infinity of stationary solutions. In that case, the economy cannotgradually leave the neighbourhood of the steady state considered, but there are an infinity ofequilibria in this neighbourhood. In other words, the central bank then faces exactly the samekind of dilemma as the one mentioned in the introduction.

8. Illustration with a monetary policy model

This section applies the formula of bubble-free policy feedback rules to one of the locallylinearisable dynamic rational-expectations models that Benhabib and Eusepi [2] use to show theexistence, under standard interest-rate rules, of equilibrium paths gradually leaving the neigh-bourhood of the steady state considered. More precisely, they show the existence, in this model,under those rules, of perfect-foresight equilibrium trajectories originating arbitrarily close to thesteady state considered and gradually leaving its neighbourhood to eventually converge towardsa deterministic cycle.

8.1. Initial form of the locally linearised structural equations

This model with capital and convex adjustment costs of changing prices (à la Rotemberg) hasfour dynamic structural equations, whose locally linearised form is written as follows:

10 Benhabib, Schmitt-Grohé and Uribe’s [3,5,6] analysis is actually conducted in a continuous-time context, where whatmatters is how many eigenvalues of a given matrix have their real part positive. We adapt it here to our discrete-timecontext, where what matters is how many eigenvalues of a given matrix have their modulus higher than one.


ct = Et {ct+1} − 1

σ

(rt − Et {πt+1}

),

Et {st+1} = μ1(rt − Et {πt+1}

)+ μ2Et {ct+1},πt = βEt {πt+1} + ξst and

kt+1 = λ1kt + λ2st − λ3ct ,

where β , ξ , λ1, λ2, λ3, μ1, μ2 and σ are parameters11 such that 0 < β < 1, ξ > 0, λ1 > 1, λ2 > 0,λ3 > 0, μ1 > 0, μ2 > 0, σ > 0; rt , ct , st , πt and kt denote respectively the deviations of the short-term nominal interest rate, the consumption level, the real marginal cost, the inflation rate andthe capital stock at date t from their values at the steady state considered; and Et {.} representsthe rational-expectations operator conditionally on the past values of the endogenous variablesas well as the current and past values of the exogenous sunspot shocks. The first equation is theEuler equation, the second one an arbitrage condition equating real return on bonds with realreturn on capital, the third one a Phillips curve (whose reduced form is identical to that obtainedunder a Calvo-type price-setting assumption), and the fourth equation (derived by Carlstrom andFuerst [10] in a similar model) the rewritten goods market clearing condition. The other structuralequations are static and can be used to uniquely recover the other endogenous variables from(rt , ct , st , πt , kt ).

8.2. Rewritten form of the locally linearised structural equations

We first rewrite this system of locally linearised structural equations in an equivalent form oftype (14) that satisfies Assumptions 1, 2 and 3. To that aim, we introduce the notation ut ≡ kt+1(as kt+1 is chosen at date t) and we use the first equation to remove rt from the second equation,for the system to satisfy Assumption 2(iii). The resulting equivalent system

Et

⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎣

L−1 − 1 0 L−1

σ0

(μ1 + μ2σ

)L−1 − μ1−L−1

σ0 0

0 −ξ βL−1 − 1 0λ3 −λ2 0 1 − λ1L

⎤⎥⎥⎦⎡⎢⎣

ct

stπt

ut

⎤⎥⎦⎫⎪⎪⎬⎪⎪⎭+

⎡⎢⎣−1σ000

⎤⎥⎦ rt = 0

is of type (14) and satisfies Assumptions 1 and 2. The computation of

Δ1(X) =

∣∣∣∣∣∣∣∣0 X2

σ0 −X

σ−X2

σ0 0 0

−ξX X(βX − 1) 0 0−λ2X 0 X − λ1 0

∣∣∣∣∣∣∣∣=−X4

σ 2(βX − 1)(X − λ1)

shows that if some roots of D(X) are of modulus greater than or equal to one, then these rootsare 1

βand/or λ1. The computation of Δ2(X), Δ3(X), Δ4(X) and Δ5(X) then easily shows that

a necessary and sufficient condition for Assumption 3 to be satisfied is

λ1 �= 1

βand λ1(λ2μ1σ + λ2μ2 − λ3) �= λ2μ1σ.

We assume that this condition is met—hardly a restrictive assumption, since the set of values forthe deep structural parameters such that this condition is not met is of measure zero. Finally, we

11 We use the same notations β , ξ and σ as Benhabib and Eusepi [2] and we introduce the new notations λ1, λ2, λ3,μ1 and μ2 for convenience.


assume that the central bank seeks to implement the path (rt , ct , st , πt , kt ) = (0,0,0,0,0) for allt ∈ Z, i.e. that it wants the economy to remain constantly at the steady state considered.

8.3. Application of the formula of bubble-free policy feedback rules

We then apply the formula of bubble-free policy feedback rules to this system and get, forinstance, the following locally linearised interest-rate rule:

rt = σEt {ct+1} + Et {πt+1} − σ

μ1σ + μ2st − μ1σ

2

μ1σ + μ2ct−1 + πt−1

+ ct−2 +(

a − 1

β

)πt−2 + ξ

βst−2 + bct−3 + cut−3

where

a ≡ −1

(1 − βλ1)ξ [(1 − β)μ1σ + μ2] ,

b ≡ μ1σ

1 − βλ1

[1

(1 − β)μ1σ + μ2− βλ2

1λ2

λ1(λ2μ1σ + λ2μ2 − λ3) − λ2μ1σ

]and

c ≡ βλ31

(1 − βλ1)[λ1(λ2μ1σ + λ2μ2 − λ3) − λ2μ1σ ]are well defined given the restrictions imposed on the parameters. The same reasoning as inAppendix A.1, applied to this particular case, shows that the system made of the locally linearisedstructural equations and this locally linearised interest-rate rule for all t ∈ Z is equivalent to thesystem made of

M1[ ct st πt ut ]′ = M2[ ct−1 st−1 πt−1 ut−1 ]′

and this locally linearised interest-rate rule for all t ∈ Z, where

M1 ≡⎡⎢⎣

0 0 1 01 −σ

μ1σ+μ20 0

λ3 −λ2 0 11 0 a 0

⎤⎥⎦ and M2 ≡

⎡⎢⎢⎣0 −ξ

β1β

0μ1σ

μ1σ+μ20 0 0

0 0 0 λ1b 0 0 c

⎤⎥⎥⎦ .

Now, M1 is invertible and, for simplicity, a, b and c have been chosen such that all eigenvaluesof M−1

1 M2 are zero, i.e. such that M−11 M2 is a nilpotent matrix. These eigenvalues are therefore

of modulus strictly lower than one and, provided that the economy originate at time t −→ −∞from the neighbourhood of the steady state considered, the system admits the unique solution(ct , st , πt , kt ) = (0,0,0,0) for all t ∈ Z. We then residually obtain rt = 0 for all t ∈ Z fromeither the first locally linearised structural equation or the locally linearised interest-rate rule. Inthis model τ = 3 and mb

1 = 0 so that, loosely speaking, this locally linearised interest-rate ruleprevents the economy from leaving the neighbourhood of the steady state considered in morethan three periods.


8.4. Discussion

It is by showing the existence, in this model, under standard interest-rate rules, of a supercriti-cal Neimark–Sacker bifurcation,12 as a certain structural parameter goes through a critical value,that Benhabib and Eusepi [2] show the existence of perfect-foresight equilibrium trajectoriesoriginating arbitrarily close to the steady state considered and gradually leaving its neighbour-hood to eventually converge towards a deterministic cycle. Under the interest-rate rules that wepropose, by contrast, there can be no Neimark–Sacker bifurcation. Indeed, by definition, such abifurcation occurs only when a pair of eigenvalues of the locally linearised system with a non-zero imaginary part go through the unit circle as a certain structural parameter goes through acritical value; now, the coefficients of the rules that we propose move in response to any changein any structural parameter so as to keep the modulus of all this system’s eigenvalues strictlylower than one.

This is not to say that our rules eliminate deterministic-cycle solutions to the non-linear sys-tem of equations: they may not. Indeed, there will typically exist a Neimark–Sacker bifurcationas a given structural parameter moves away from its “initial value” while the coefficients ofthe rule are arbitrarily imposed not to adjust accordingly. This suggests that there can exist adeterministic-cycle solution to the non-linear system of equations for the initial value of this pa-rameter, especially if the bifurcation value of this parameter is close to its initial value. However,in that case, since all the locally linearised system’s eigenvalues are of modulus strictly lowerthan one for the initial value of this parameter, the Neimark–Sacker bifurcation will necessarilybe subcritical, that is to say that the deterministic cycle will necessarily be unstable: any equilib-rium trajectory originating in a sufficiently small neighbourhood of the steady state consideredand not leaving it abruptly (as opposed to gradually) will converge towards this steady state, nottowards that cycle.

9. Comparison with other monetary policy proposals

This section briefly compares our monetary policy proposal to others that prevent the econ-omy from gradually leaving the neighbourhood of the steady state considered or ensure globalequilibrium determinacy.

9.1. The most common proposal

The most common proposal, made notably by Benhabib, Schmitt-Grohé and Uribe [5,6],Christiano and Rostagno [11] and Woodford [27, Chap. 2], is in the spirit of Obstfeld and Ro-goff’s [24] fractional-backing mechanism to rule out speculative hyperinflations. It consists inswitching from an interest-rate rule ensuring local equilibrium determinacy to another rule suchas a money growth rate peg (possibly accompanied by a non-Ricardian fiscal policy) when theeconomy goes outside a specified neighbourhood of the steady state considered. However, asargued by Green [17] and more forcefully by Cochrane [13], the credibility of these two-tiermonetary policies, and consequently their effectiveness in preventing in the first place the econ-omy from gradually leaving the neighbourhood of the steady state considered, cannot be taken

12 The Neimark–Sacker bifurcation (Kuznetsov [19, Chap. 4]) can be viewed as the “equivalent”, for discrete-timedynamic systems, of the Andronov–Hopf bifurcation of continuous-time dynamic systems (Kuznetsov [19, Chap. 3]).


for granted, essentially because they are typically very costly out of equilibrium when the econ-omy is off track. The interest-rate rules that we put forward are likely to be more credible fortwo reasons: (i) they act while the economy is still in the neighbourhood of the steady stateconsidered; (ii) they are effective provided that, at each date, the central bank can commit to fol-lowing a given interest-rate rule during the next τ periods. They represent therefore a particularlyinteresting alternative to these two-tier monetary policies.

9.2. The proposals closest to ours

The proposals closest to ours are those of Grandmont [15] and Adão, Correia and Teles [1].Grandmont [15] considers a simple non-linear overlapping-generations rational-expectationsmodel that admits multiple non-local equilibria under standard money-growth rules and designsmoney-growth rules that ensure global equilibrium determinacy in this model, under a stabil-ity condition (corresponding to our Assumption 3) constraining the set of values that some keystructural parameters can take.13 Adão, Correia and Teles [1] design interest-rate rules ensuringglobal equilibrium determinacy in various simple non-linear rational-expectations models of themonetary transmission mechanism, under the assumptions that the nominal interest rate can benegative out of equilibrium and that the central bank has perfect knowledge of the values of somekey structural parameters. Although the monetary policy rules designed in these two papers areglobal and non-linear, their working mechanism is similar to the working mechanism of our rulesapplied to the specific models considered in these papers, as our generic framework encompassesthese locally linearisable models.14

The main advantage of these two papers over ours is, of course, that their rules ensure globalequilibrium determinacy in some non-linear models, while in such models our rules only preventthe economy from gradually leaving the neighbourhood of the steady state considered, in additionto ensuring equilibrium determinacy within this neighbourhood. Conversely, the main advantageof our paper over theirs is that our rules are designed in a generic framework, while their rulesare designed in models that are specific in that they have only one forward-looking structuralequation and this structural equation features the monetary policy instrument. In the modelsconsidered by Adão, Correia and Teles [1], for instance, this structural equation is the Eulerequation, which is the only one to feature the nominal interest rate and features it only as acurrent variable. Therefore, the working mechanism of their interest-rate rules is in effect limitedto that of bubble-free policy feedback rules in the specific, simple case considered in Section 1.

In addition, our paper has the following two advantages over Adão, Correia and Teles [1].First, their global equilibrium determinacy result is obtained only under the assumption thatthe nominal interest rate can be negative out of equilibrium, while our interest-rate rules remaineffective in preventing the economy from gradually leaving the neighbourhood of the steady stateconsidered even in the presence of a zero lower bound on the nominal interest rate, provided ofcourse that the nominal interest rate is strictly positive at the steady state considered. Second,both our rules and theirs have coefficients that are tied to some key structural parameters byequality constraints, but our rules are shown to be effective in preventing the economy fromgradually leaving the neighbourhood of the steady state considered even when the central bank

13 Grandmont [16] shows that fiscal and public-expenditure policies can be designed in this model, independently ofmonetary policy (i.e. independently of the money-growth rule considered), to ensure that this stability condition is metwhatever the values taken by the structural parameters.14 Our work and Adão, Correia and Teles’ [1] work were conducted independently from each other.


has imperfect knowledge of the values of these structural parameters, while their rules are notshown to ensure global equilibrium determinacy in that case.15

10. Conclusion

This paper aims to give a new insight into the design of policy feedback rules in linear dynamicstochastic rational-expectations models and interest-rate rules in locally linearisable rational-expectations models of the monetary transmission mechanism.

The literature on policy feedback rules in linear dynamic stochastic rational-expectationsmodels has so far mostly focused on rules ensuring that the model admits a unique stationarysolution at the cost of an infinity of non-stationary solutions. This unique stationary solution isoften called the saddle path after its geometric representation. By contrast, we design rules en-suring that the model admits not only a unique stationary solution, but also no non-stationarysolutions. In effect, these rules turn the saddle into a necklace. Moreover, they are able to selectany given candidate stationary VARMA solution. We apply these results to locally linearisablemodels of the monetary transmission mechanism and obtain interest-rate rules that not only en-sure the local determinacy of the targeted equilibrium in the neighbourhood of the steady stateconsidered, but also prevent the economy from gradually leaving this neighbourhood.

The main limitation of these interest-rate rules may be that they do not prevent the economyfrom leaving abruptly, as opposed to gradually, the neighbourhood of the steady state considered.Similarly, the bubble-free policy feedback rules that we design in linear models do not eliminaterational bubbles when the policy-maker has imperfect knowledge of the structural parameters,they only make them fast-growing (and at the limit, as the policy-maker’s knowledge becomesperfect, infinitely fast-growing) by using the structural equations as a lever on the private sector’sexpectations. This raises the question of how likely the private sector would coordinate on a pathabruptly leaving the neighbourhood of the steady state considered or on a fast-growing rationalbubble, when it could alternatively coordinate on, respectively, the locally unique equilibrium orthe unique stationary solution. This is a question that has no easy answer, notably because thereis yet no single widely accepted game-theoretic foundation of rational expectations, and that weleave for future research.

Acknowledgments

I am grateful in particular to Klaus Adam, Lawrence Christiano, Daniel Cohen, Harris Del-las, Jordi Galí, Marc Giannoni, Jean-Michel Grandmont, Michel Juillard, Hubert Kempf, GuyLaroque, Philippe Martin, Pedro Teles and two anonymous referees, for their comments at vari-ous stages of this long project. The usual disclaimer applies.

Appendix

For any system of equations (S), let Et {(S)} (respectively L(S)) denote the system obtainedby applying operator Et (respectively L) on both the left- and the right-hand sides of each equa-tion of (S). For any non-zero integers n and p, any n × p matrix K and any p-equation system

15 By extension, Grandmont’s [15] rules could also be viewed as having coefficients that are tied to some key “structuralparameters” by equality constraints. However, in his framework, theory restricts the set of admissible values for each ofthese structural parameters to a singleton. As a consequence, the assumption that the central bank has perfect knowledgeof the values of these structural parameters is justified.


(S), let K(S) denote the n-equation system obtained by applying K to both the p-element vectormade of the left-hand sides of the equations of (S) and the p-element vector made of their right-hand sides—so that, in particular, for any N -equation system (S) and any j ∈ {1, . . . ,N}, e′

j (S)

denotes the j th equation of (S). For any polynomial H(X) ∈ R[X], let dH denote the degree ofH(X). Let |.| denote the determinant operator. Lastly, for any scalar λ, any non-zero integers n

and p and any n × p matrix K, let λK denote the product of the n × n matrix whose diagonalelements are all equal to λ and whose non-diagonal elements are all equal to 0 by matrix K, andlet Kλ denote the product of matrix K by the p ×p matrix whose diagonal elements are all equalto λ and whose non-diagonal elements are all equal to 0 (note that λK = Kλ).

A.1. Proof of Proposition 1

Consider a given t ∈ Z. Assumption 2(ii) implies that the substraction of e′1(14) from

Et {L−mb1 (17)} leads to Eq. (

−→1 ):

N∑i=2

e′iL∑i

j=2 maj[A(L)Yt + B(L)zt + C(L)ξ t + L−ma

i O(L)D(L)ξ t

]+ L

∑Ni=2 ma


]= 0. (−→1 )

Similarly, ∀k ∈ {2, . . . ,N}, Eq. (−→k ) can be derived from equation (

−−−−→k − 1) by substracting e′

k(14)

from Et {L−mak (

−−−−→k − 1)}:

N∑i=k+1

e′iL∑i

j=k+1 maj[A(L)Yt + B(L)zt + C(L)ξ t + L−ma

i O(L)D(L)ξ t

]+ L

∑Ni=k+1 ma


]= 0 (−→k )

and in particular

P(L)Yt + Q(L)zt + R(L)D(L)ξ t = 0. (−→N )

∀k ∈ {2, . . . ,N}, the substraction of Lmak (

−→k ) from (

−−−−→k − 1) leads to Eq. (

←−k ):

e′k

[Lma

k[A(L)Yt + B(L)zt + C(L)ξ t

]+ O(L)D(L)ξ t

]= 0. (←−k )

Equations−→N and

←−k for k ∈ {2, . . . ,N} together can be re-written as follows:

U(L)Yt + V(L)zt + W(L)ξ t = 0 (A.1)

with

U(L)(N×N)

=

⎡⎢⎢⎣P(L)

e′2A(L)

...

e′N A(L)

⎤⎥⎥⎦≡nu∑

k=0

UkLk,

V(L)(N×1)

=

⎡⎢⎢⎣Q(L)

e′2L

ma2 B(L)

...′ ma

⎤⎥⎥⎦≡nv∑

k=max(0,ma1−mb

1+1)

VkLk

eNL N B(L)


due to Assumption 2(iii) and Condition 3, and

W(L)(N×M)

=

⎡⎢⎢⎣R(L)D(L)

e′2[Lma

2 C(L) + O(L)D(L)]...

e′N [Lma

N C(L) + O(L)D(L)]

⎤⎥⎥⎦≡nw∑k=0

WkLk,

where (nu,nv, nw) ∈ N3 and all Uk , Vk , Wk have real numbers as elements. Since U(0) = Ω

is invertible due to Condition 2, (A.1) can be used to express Yt as a function of Yt−1−k ,zt−max(0,ma

1−mb1+1)−k and ξ t−k for k � 0. If ma

1 � mb1, this expression can be used to sequen-

tially replace Et {Yt+j } for j ∈ {0, . . . ,ma1 − mb

1} in (17) and thus get zt as a function of Yt−1−k ,zt−1−k and ξ t−k for k � 0. Alternatively, if ma

1 < mb1 then (17) directly expresses zt as a function

of Yt−1−k , zt−1−k and ξ t−k for k � 0. In both cases, the system thus obtained, made of thisequation for zt and (A.1) for Yt , is backward-looking and non-degenerate and hence uniquelydetermines Yt and zt as a function of Yt−1−k , zt−1−k and ξ t−k for k � 0. This system, noted(S), and the systems Lk(S) for all k ∈ Z

∗ that can be similarly obtained, then uniquely determineYt−k and zt−k for k ∈ Z as functions of ξ t−k−j for j � 0. Given that, as can be readily checked,the system made of Lk(S) for all k ∈ Z is equivalent to the original system made of Lk(14) andLk(17) for all k ∈ Z, we conclude that the latter admits one unique solution for {Yt−j , zt−j }j∈N.This solution is stationary because the eigenvalues of the system made of Lk(14) and Lk(17) forall k ∈ Z are those of (18) which, given Condition 4, are stable.


For the sake of expositional clarity, we omit the expression “for all t ∈ Z” throughout thisappendix. We prove Proposition 2 by showing that, for any (16) satisfying (14), there exist O(L),P(L), Q(L) and R(L) satisfying Conditions 1 to 5 and such that (16) implies (14) and (17) andtherefore, following Proposition 1, such that (16) is the unique solution of the system made of(14) and (17). To that aim, suppose that a given (16) holds that satisfies (14).

Step 1. Then, (14) holds as well. Moreover, there exists O(L) satisfying Condition 1 and suchthat ⎡⎢⎢⎢⎣

Et {e′1L

mb1 A(L)Yt }

e′2L

ma2 A(L)Yt

...

e′NLma

N A(L)Yt

⎤⎥⎥⎥⎦+

⎡⎢⎢⎢⎣e′

1Lmb

1 B(L)

e′2L

ma2 B(L)

...

e′NLma

N B(L)

⎤⎥⎥⎥⎦ zt

+

⎡⎢⎢⎢⎣e′

1[Lmb1 C(L) + O(L)D(L)]

e′2[Lma

2 C(L) + O(L)D(L)]...

e′N [Lma

N C(L) + O(L)D(L)]

⎤⎥⎥⎥⎦ ξ t = 0. (A.2)

Step 2. The generalized identity of Bezout implies that there exists (U1(X), . . . , UN+1(X)) ∈R[X]N+1 such that

N+1∑Ui (X)Δi(X) = D(X). (A.3)

i=1


Let Θ(X) ∈ R[X] denote the polynomial, defined up to a non-zero multiplicative scalar, whichhas the same roots (whose modulus is strictly lower than one) with the same multiplicity asthe eigenvalues of the system I − S(L) corresponding to the autoregressive part of the targetedstationary VARMA process (16). Let Z(X) ∈ R[X] be a given polynomial that: (i) has all itsroots of modulus strictly lower than one; and (ii) is such that Θ(X) is a divisor of Z(X)D(X).Given that Assumption 1(iii) implies the existence of I ∈ {1, . . . ,N} such that⎡⎢⎢⎢⎣

e′I

e′2A(0)

...

e′N A(0)

⎤⎥⎥⎥⎦is invertible, let n ∈ N be such that

n � 2dΔI− dD + max

[max

i∈{1,...,N}(dUi

), dUN+1 + max(−1,ma

1 − mb1

)]− dZ .

Let Q(X) ∈ R[X] and R(X) ∈ R[X] be respectively the quotient and the remainder of theEuclidian division of XnZ(X) by ΔI (X), i.e. the unique polynomials such that XnZ(X) =ΔI (X)Q(X) + R(X) with dR < dΔI

. Multiplying the left-hand side and the right-hand side of(A.3) by R(X), we obtain

R(X)

N+1∑i=1

Ui (X)Δi(X) = R(X)D(X)

and therefore

N+1∑i=1i �=I

[R(X)Ui (X)

]Δi(X) + [R(X)UI (X) + Q(X)D(X)

]ΔI (X) = XnZ(X)D(X).

Let us note Pi (X) ≡ R(X)Ui (X) for i ∈ {1, . . . ,N + 1} � {I } and PI (X) ≡ R(X)UI (X) +Q(X)D(X). Given that⎧⎪⎪⎨⎪⎪⎩

n � 2dΔI− dD + max

[max

i∈{1,...,N}(dUi

), dUN+1 + max(−1,ma

1 − mb1

)]− dZ

n = dΔI+ dQ − dZ

dΔI> dR

�⇒ dQ + dD > dR + max[

maxi∈{1,...,N}

(dUi), dUN+1 + max

(−1,ma1 − mb

1

)]

�⇒

⎧⎪⎨⎪⎩dQ + dD > dR + dUI

dQ + dD > dR + maxi∈{1,...,N}�{I }

(dUi)

dQ + dD > dUN+1 + max(−1,ma

1 − mb1

)�⇒

⎧⎨⎩dPI

= dQ + dDdPI

> dPifor i ∈ {1, . . . ,N} � {I }

dPI� dPN+1 + max

(0,ma

1 − mb1 + 1

),

the choice of


P(L) =N∑

i=1

(−1)N+1−iLdPI Pi

(L−1)e′

i and Q(L) = LdPI PN+1(L−1)

satisfies Conditions 2 and 3.

Step 3. The non-zero eigenvalues of (18) are those of the system

Ψ (L)((N+1)×(N+1))

≡nψ∑

k=−mψ

Ψ kLk =

⎡⎢⎢⎢⎢⎢⎢⎣Lmb

1 e′1A(L) Lmb

1 e′1B(L)

Lma2 e′

2A(L) Lma2 e′

2B(L)...

...

LmaN e′

NA(L) LmaN e′

N B(L)

P(L) Q(L)

⎤⎥⎥⎥⎥⎥⎥⎦ ,

where (mψ,nψ) ∈ N2 and all Ψ k have real numbers as elements. If mb

1 > ma1 then mψ = 0 and

Ψ 0 =

⎡⎢⎢⎢⎢⎢⎣0 · · · 0 e′

1B−mb1

e′2A(0) 0

......

e′N A(0) 0

P0 Q(0)

⎤⎥⎥⎥⎥⎥⎦is invertible (since Condition 2 is satisfied). Therefore, according to a standard matricial result oftime series analysis (cf. e.g. Hamilton [18, Chap. 10, Prop. 10.1]), the eigenvalues of Ψ (L) arethe roots of polynomial |Xnψ

Ψ (X−1)| ∈ R[X]. Alternatively, if mb1 � ma

1 then the eigenvaluesof Ψ (L) are those of the system that is obtained by using the last N lines of Ψ (L) (given thatCondition 2 is satisfied) to sequentially remove the terms in Lk for k ∈ {−mψ, . . . ,0} from the

first line of Ψ (L). This system, noted Λ(L), is of the form∑nψ

k=0 ΛkLk , where all Λk have real

numbers as elements and

Λ0 =

⎡⎢⎢⎢⎢⎢⎣0 · · · 0 e′

1B−mb1

e′2A(0) 0

......

e′N A(0) 0

P0 Q(0)

⎤⎥⎥⎥⎥⎥⎦is invertible. Therefore, according to a standard matricial result of time series analysis (cf. e.g.Hamilton [18, Chap. 10, Prop. 10.1]), the eigenvalues of Λ(L)—and hence those of Ψ (L)—arethe roots of polynomial |Xnψ

Λ(X−1)|, which is equal to polynomial |XnψΨ (X−1)|. To sum

up, whether mb1 > ma

1 or mb1 � ma

1 , the non-zero eigenvalues of (18) are the non-zero roots of

polynomial |XnψΨ (X−1)|, that is to say those of

N∑i=1

[(−1)N+1−iP

(X−1)ei

]Δi(X) + [Q(X−1)]ΔN+1(X)

and hence, by construction of P(L) and Q(L), those of Z(X)D(X). By definition of Z(X) andgiven Assumption 3, all non-zero roots of Z(X)D(X) are of modulus strictly lower than one, sothat the P(L) and Q(L) constructed at Step 2 satisfy Condition 4.


Step 4. (16) implies that there exists a unique

R(L)(1×M)

≡+∞∑k=0

RkLk,

where all Rk have real numbers as elements, such that

P(L)Yt + Q(L)zt + R(L)εt = 0 (A.4)

where P(L) and Q(L) are the ones constructed at Step 2. If mb1 > ma

1 then multiplying both (A.4)

and (A.2) by D(L) ≡∏Mi=1 Di(L) leads to

D(L)P(L)Yt + D(L)Q(L)zt + D(L)R(L)εt = 0 and (A.5)

D(L)

⎡⎢⎢⎢⎣e′

1Lmb

1 A(L) e′1L

mb1 B(L)

e′2L

ma2 A(L) e′

2Lma

2 B(L)

......

e′NLma

N A(L) e′NLma

N B(L)

⎤⎥⎥⎥⎦[

Yt

zt

]+

⎡⎢⎢⎢⎣e′

1Lmb

1 C(L)

e′2L

ma2 C(L)

...

e′NLma

N C(L)

⎤⎥⎥⎥⎦

×

⎡⎢⎢⎢⎣∏M

i=2 Di(L) 0 · · · 0

0. . .

. . ....

.... . .

. . . 00 · · · 0

∏M−1i=1 Di(L)

⎤⎥⎥⎥⎦εt + D(L)O(L)εt = 0 (A.6)

since, as a scalar, D(L) is such that D(L)K = KD(L) for any matrix K. The system madeof (A.5) and (A.6) is backward-looking (since ma

1 > mb1) and non-degenerate (since D(0) =

|D(0)| �= 0 and |Ψ 0| �= 0). Cramer’s rule then implies that there exist (n1, . . . , nN+1) ∈ NN+1

with ni � dΔifor i ∈ {1, . . . ,N + 1} and Υ 1(L)

((N+1)×M)

≡∑nυ1k=0 Υ 1,kL

k , where nυ1 ∈ N and all

Υ 1,k have real numbers as elements, such that this system can be rewritten

D(L)LdZ +dD Z(L−1)D

(L−1)[Yt

zt

]

= Υ 1(L)εt + D(L)

⎡⎣ Ln1Δ1(L−1)R(L)εt...

LnN+1ΔN+1(L−1)R(L)εt

⎤⎦ (A.7)

given Step 3. But Cramer’s rule also implies that there exists Υ 2(L)((N+1)×M)

≡∑nυ2k=0 Υ 2,kL

k , where

nυ2 ∈ N and all Υ 2,k have real numbers as elements, such that the targeted stationary VARMAprocess (16) can be rewritten

LdΘ Θ(L−1)

[Yt

zt

]= Υ 2(L)εt ,

which implies

D(L)LdZ +dD Z(L−1)D

(L−1)[Yt

zt

]= D(L)

LdZ +dD Z(L−1)D(L−1)

LdΘ Θ(L−1)Υ 2(L)εt (A.8)

where XdZ +dD Z (X−1)D(X−1)

XdΘ Θ(X−1)∈ R[X] by definition of Z(X). Given that ΔN+1(X) �= 0 due to

Assumption 1(iii), the identification of (A.7) with (A.8) shows that ∃nr ∈ N, ∀k > nr , Rk = 0,


so that R(L) satisfies Condition 5. Alternatively, if mb1 � ma

1 then a similar reasoning (based onΛ(L) instead of Ψ (L), along the lines of Step 3) shows that R(L) satisfies Condition 5 again. Tosum up, whether mb

1 > ma1 or mb

1 � ma1 , there exists R(L) satisfying Condition 5 and such that

(A.4) holds. Finally, Steps 1 to 4 together imply that (14) holds and that there exist O(L), P(L),Q(L) and R(L) satisfying Conditions 1 to 5 and such that (17) holds. Following Proposition 1,we then conclude that (16) is the unique solution of the system made of (14) and (17).


We proceed in three steps: first, we show that, with probability one, the system made of (14)and (Rε ) can be written in Blanchard and Kahn’s [8] form; second, we show that this systemadmits a unique stationary solution and has no eigenvalue whose modulus is between 1 and με ,with με −→ +∞ as ε −→ 0; third, we show that d(X(L), Xε(L)) −→ 0 as ε −→ 0.

Step 1. Consider a given system (S) of type (14). For the sake of clarity, we deal first withthe case ma

1 � mb1 and then with the case ma

1 > mb1. Suppose therefore that ma

1 � mb1. Re-

place Et {zt+mb1} in e′

1(S) by its expression in Et {L−mb1 (Rε)}; if ma

1 < mb1, in which case (Rε )

is backward-looking, then replace sequentially Et {zt+mb1−k} for k ∈ {1, . . . ,mb

1 − ma1} (if they

appear) in the resulting equation by their expressions in Et {Lk−mb1(Rε)}; note (Eε ) the resulting

equation. Consider

(Sε) ≡

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩(Eε)

e′2(S)

...

e′N(S)

and Aε(L) ≡⎡⎣ e′

1Lma

1

...

e′NLma

N

⎤⎦ Aε(L)

where Aε(L) is defined by writing (Sε ) in the form Et {Aε(L)Yt + Bε(L)zt } + Cε(L)ξ t = 0.Given that the probability distributions of the exogenous additive disturbances are assumed to

be continuous and given Assumption 1(iii), Aε(0) is invertible with probability one.16 Let usrewrite (Sε) step by step and keep for simplicity the same notation (Sε) at each step. Re-orderthe lines of (Sε) so that ma

1 � · · · � maN . Let K ∈ {1, . . . ,N} and {i1, . . . , iK } ∈ {1, . . . ,N}K

be such that ma1 = · · · = ma

i1> ma

i1+1 = · · · = mai2

> · · · > maiK−1+1 = · · · = ma

iK= ma

N . Re-

order the elements of Yt and accordingly the columns of Aε(L) so that ∀i ∈ {1, . . . ,N − 1}, the(N − i)× (N − i) matrix noted Mε,i obtained by removing the first i lines and the first i columns

from Aε(0) is invertible, this re-ordering being made possible by the fact that Aε(0) is invertible.Replace e′

i (Sε) by

e′iAε(0)−1Et

⎡⎢⎢⎣1 0 · · · 0

0 Lma2−ma

1. . .

......

. . .. . . 0

0 · · · 0 LmaN−ma

1

⎤⎥⎥⎦ (Sε)

16 In the remaining of the proof, for simplicity, we may sometimes drop the expression “with probability one.”


for i ∈ {1, . . . , i1}. If K = 1 then replace sequentially Et {zt+mai1

−k} for k ∈ {1, . . . ,mai1} (if they

appear) in (Sε) by their expressions in Et {Lk−mai1 (Rε)}. Given Assumption 2(iii), this operation

removes all variables of type Et {zt+k} with k ∈ N from the system. The resulting system (Sε)

is equivalent to the original one and together with (Rε ) can easily be written in Blanchard andKahn’s [8] form with i1m

ai1

=∑Ni=1 ma

i non-predetermined variables. Otherwise (i.e. if K � 2),

let us set k = 1. Replace Et {zt+mai1

−k} (if it appears) in e′i (Sε) for i ∈ {1, . . . , i1} by its expression

in Et {Lk−mai1 (Rε)}. Then, replace Et {e′

iYt+mai1

−k} for i ∈ {i1 + 1, . . . ,N} (if they appear) in

e′i (Sε) for i ∈ {1, . . . , i1} by their expression in

M−1ε,i

⎡⎢⎢⎢⎢⎢⎣0 0 · · · 0 L

mai1+1−ma

i1+k

0 · · · 0

0. . .

. . .... 0

. . .. . .

......

. . .. . . 0

.... . .

. . . 0

0 · · · 0 0 0 · · · 0 Lma

N−mai1

+k

⎤⎥⎥⎥⎥⎥⎦ (Sε).

If mai1

> mai2

+ 1 then repeat these last two steps sequentially for k ∈ {2, . . . ,mai1

−mai2}. Proceed

in a similar way as previously to transform e′i (Sε) for i ∈ {i1 + 1, . . . , i2}, then (if K � 3) e′

i (Sε)

for i ∈ {i2 + 1, . . . , i3} and so on up to e′i (Sε) for i ∈ {iK−1 + 1, . . . , iK }. The final system (Sε) is

equivalent to the initial one and together with (Rε ) can easily be written in Blanchard and Kahn’s[8] form with

∑Kj=1 ijm

aij

=∑Ni=1 ma

i non-predetermined variables. Suppose now, alternatively,

that ma1 > mb

1. Rewriting the system made of (Sε ) and (Rε ) in a similar way as above enables

us to put it in Blanchard and Kahn’s [8] form with∑N

i=1 mai non-predetermined variables, even

though (Rε ) is then forward-looking, because mai − mb

i > ma1 − mb

1 for i ∈ IB � {1} due toAssumption 2(iii) and because the only variable of type Et {zt+k} with k ∈ N appearing in thesystem made of the rewritten (Sε ) and (Rε ) is zt in (Rε ). In both cases (i.e. whether ma

1 � mb1 or

ma1 > mb

1), since the initial system made of (S) and (Rε ) is equivalent to the final system madeof (Sε ) and (Rε ), we have thus shown that, with probability one, the system made of (S) and(Rε ) can be written in Blanchard and Kahn’s [8] form with m ≡∑N

i=1 mai non-predetermined

variables.

Step 2. For any system or equation (x), let (x) denote the perfect-foresight deterministic formof (x). The same reasoning as the one conducted at the beginning of Appendix A.1, this time

starting from (Eε) instead of (−→1 ) and using (Rε ) instead of (R), leads to an equation (

−→Nε),

corresponding to equation (−→N) in Appendix A.1, such that (

−→Nε) is of the form

Pε(L)Yt + Qε(L)zt = 0

with Pε(L)(1×N)

≡np∑

k=−m

Pε,kLk and Qε(L) ≡

nq∑k=−m+1+max(ma

1−mb1,0)

qε,kLk,

where (np, nq) ∈ N2, all Pε,k have real numbers as elements, all qε,k are real numbers, Pε,−m =e′ Aε(0), d(Pε(L),P(L)) −→ 0 and d(Qε(L),Q(L)) −→ 0 as ε −→ 0. The non-zero eigenval-
1


ues of the system made of (S) and (Rε ) are those of the system made of (S) and (Rε ) which in

turn are those of the system made of (S) and (−→Nε). The latter system can be rewritten

Γ ε,1(L)

[Yt

zt

]= 0

with

Γ ε,1(L) ≡

⎡⎢⎢⎢⎢⎢⎣LmPε(L) LmQε(L)

e′1L

max(ma1 ,mb

1)A(L) e′1L

max(ma1 ,mb

1)B(L)

e′2L

ma2 A(L) e′

2Lma

2 B(L)

......

e′NLma

N A(L) e′NLma

N B(L)

⎤⎥⎥⎥⎥⎥⎦≡nγ1∑k=0

Γ ε,1,kLk

where nγ1 ∈ N and all Γ ε,1,k have real numbers as elements. Let us define

J1(N×(N+1))

≡

⎡⎢⎢⎣1 0 0 · · · 0

0 0 1. . .

......

. . .. . .

. . . 00 · · · 0 0 1

⎤⎥⎥⎦ , J2((N+1)×N)

≡

⎡⎢⎢⎢⎢⎣1 0 · · · 0

0. . .

. . ....

.... . .

. . . 00 · · · 0 10 · · · · · · 0

⎤⎥⎥⎥⎥⎦ ,

J3((N+1)×1)

≡

⎡⎢⎢⎣0...

01

⎤⎥⎥⎦ and J4(N×(N+1))

≡

⎡⎢⎢⎣0 1 0 · · · 0

0. . .

. . .. . .

......

. . .. . .

. . . 00 · · · 0 0 1

⎤⎥⎥⎦ .

If ma1 � mb

1 then replace sequentially Yt−k for k ∈ {0, . . . ,ma1 − mb

1} in the second line ofΓ ε,1(L)[Y′

t zt ]′ = 0 by its expression in (J1Γ ε,1,0J2)−1J1Γ ε,1(L)Lk[Y′

t zt ]′ = 0, given that

|J1Γ ε,1,0J2| = |Aε(0)| �= 0, and note Γ ε,2(L)[Y′t zt ]′ = 0 the resulting system, with Γ ε,2(L) ≡∑nγ2

k=0 Γ ε,2,kLk where nγ2 ∈ N and all Γ ε,2,k have real numbers as elements (Γ ε,2(L) = Γ ε,1(L)

if ma1 < mb

1). Given that J1Γ ε,1,kJ3 = 0 for k ∈ {0, . . . ,max(ma1 − mb

1,0)} due to Assump-tion 2(iii), we have

Γ ε,2,0 =

⎡⎢⎢⎢⎢⎢⎢⎣e′

1Aε(0) 0

0 e′1B−mb

1

e′2Aε(0) 0...

...

e′NAε(0) 0

⎤⎥⎥⎥⎥⎥⎥⎦ .

Since Aε(0) is invertible, Γ ε,2,0 is invertible as well so that, according to a standard matricial re-sult of time series analysis (cf. e.g. Hamilton [18, Chap. 10, Prop. 10.1]), the non-zero eigenvaluesof Γ ε,2(L), which are those of Γ ε,1(L), are the roots of polynomial Eε(X) ≡ |Xnγ2 Γ ε,2(X

−1)| ∈R[X]. Now, Eε(X) = Eε,1(X) + Eε,2(X) where

Eε,1(X) ≡∣∣∣∣∣ Xnγ2−m

∑−1k=−m Pε,kX

−k Xnγ2−m∑−1

k=−m qε,kX−k

J Xnγ2 Γ (X−1)

∣∣∣∣∣ and
4 ε,2


Eε,2(X) ≡∣∣∣∣∣ Xnγ2−m

∑np

k=0 Pε,kX−k Xnγ2−m

∑nq

k=0 qε,kX−k

J4Xnγ2 Γ ε,2(X

−1)

∣∣∣∣∣ .If m = 0, then Eε,1(X) = 0. Otherwise, the degree of Eε,1(X) is equal to nγ2(N + 1), since thecoefficient of Xnγ2 (N+1) in Eε,1(X) is |Γ ε,2,0| �= 0. For ε sufficiently close to 0, the degree ofEε,2(X) is equal to nγ2(N + 1) − m, since the coefficient of Xnγ2 (N+1)−m in Eε,2(X) is∣∣∣∣∣∣∣∣∣∣∣

Pε,0 qε,00 e′

1B−mb1

e′2A(0) 0

......

e′N A(0) 0

∣∣∣∣∣∣∣∣∣∣∣−→ (−1)N+1e′

1B−mb1|Ω| �= 0 as ε −→ 0.

Let us note xε,1, . . . , xε,nγ2 (N+1) the roots of Eε(X), ranked first by increasing modulus(i.e. |xε,1| � · · · � |xε,nγ2 (N+1)|) and second by increasing complex argument (i.e. if ∃i ∈{1, . . . , nγ2(N + 1) − 1}, |xε,i | = |xε,i+1|, then ϕ(xε,i) � ϕ(xε,i+1), where ϕ : C −→[0;2π[denotes the complex argument function). Similarly, let us note x1, . . . , xn the non-zero eigen-values of system (18) ranked first by increasing modulus and second by increasing complexargument, which are all of modulus strictly lower than one since (R) satisfies Condition 4. SinceEε,1(X) −→ 0 as ε −→ 0, we have

(xε,1, . . . , xε,nγ2 (N+1)−m) −→ (0, . . . ,0, x1, . . . , xn) as ε −→ 0

and

∀k ∈ {0, . . . ,m − 1}, |xε,nγ2 (N+1)−k| −→ +∞ as ε −→ 0,

which implies: (i) that the system made of (14) and (Rε ) has no eigenvalue whose modulus isbetween 1 and με , with με −→ +∞ as ε −→ 0; (ii) given Step 1, that this system admits eitherone or zero stationary solution, depending on whether Blanchard and Kahn’s [8] rank conditionis satisfied or not. Since Assumption 3 and the continuous-probability-distribution assumptiontogether ensure that this rank condition is satisfied with probability one, we further get that thissystem admits one unique stationary solution.

Step 3. Let us write equations (−→k ) for k ∈ {1, . . . ,N}, obtained in Appendix A.1, in the form

−→U(L)Yt + −→

V(L)zt + −→W(L)ξ t = 0

with

−→U(L)(N×N)

≡n

−→u∑k=0

−→UkL

k,−→V(L)(N×1)

≡n

−→v∑k=max(0,ma

1−mb1+1)

−→VkL

k and−→W(L)(N×M)

≡n

−→w∑k=0

−→WkL

k,

where (n−→u , n

−→v , n−→w) ∈ N

3 and all−→Uk ,

−→Vk ,

−→Wk have real numbers as elements. For all (i, j) ∈

{1, . . . ,N}2 such that i � j , let ηi,j be defined by ηi,j = 1 if ∀k ∈ {i, . . . , j}, mak = 0 and ηi,j = 0

otherwise. We then have

−→U(0) =

⎡⎢⎢⎢⎢⎢⎣η2,N 1 η2,2 · · · η2,N−1

... 0 1. . .

......

.... . .

. . . ηN−1,N−1ηN,N 0 · · · 0 1

⎤⎥⎥⎥⎥⎥⎦U(0),

1 0 · · · · · · 0


so that since U(0) = Ω is invertible,−→U(0) is invertible as well. In this case, the same reasoning

as the one conducted at the end of Appendix A.1, this time using−→U(L),

−→V(L) and

−→W(L) instead

of U(L), V(L) and W(L), leads to a system of the form

Et

{Λ1(L)

[Yt

zt

]+ Λ2(L)ξ t

}= 0 (A.9)

with

Λ1(L)(N+1)×(N+1)

≡nλ1∑k=0

Λ1,kLk and Λ2(L)

(N+1)×M

≡nλ2∑k=0

Λ2,kLk,

where (nλ1 , nλ2) ∈ N2, all Λ1,k , Λ2,k have real numbers as elements, Λ1,0 is invertible and alleigenvalues of Λ1(L) are of modulus strictly lower than one. Since (A.9) is equivalent to thesystem made of (S) and (R), (19) is the unique solution of (A.9).

Similarly, let us follow the same reasoning as the one conducted at the beginning of Ap-pendix A.1, this time starting from (Eε) instead of (

−→1 ) and using (Rε ) instead of (R), to get

Eqs. (−→2 ε) to (

−→Nε) corresponding to Eqs. (

−→2 ) to (

−→N) in Appendix A.1. Eqs. (Eε ) and (

−→k ε) for

k ∈ {2, . . . ,N} can then be re-written in the form

Et

{−→Uε(L)Yt + −→

Vε(L)zt + −→Wε(L)ξ t

}= 0

with

−→Uε(L)(N×N)

≡n

−→u∑k=−m

−→u

−→Uε,kL

k,−→Vε(L)(N×1)

≡n

−→v∑k=−m

−→v

−→Vε,kL

k and

−→Wε(L)(N×M)

≡n

−→w∑k=−m

−→w

−→Wε,kL

k,

where (m−→u ,m

−→v ,m−→w,n

−→u , n−→v , n

−→w) ∈ N6, all

−→Uε,k ,

−→Vε,k ,

−→Wε,k have real numbers as elements,

d(−→Uε(L),

−→U(L)) −→ 0, d(

−→Vε(L),

−→V(L)) −→ 0 and d(

−→Wε(L),

−→W(L)) −→ 0 as ε −→ 0. Given

that−→Uε(0) −→ −→

U(0) as ε −→ 0,−→Uε(0) is invertible for ε sufficiently small, so that the same

reasoning as the one conducted at the end of Appendix A.1 leads to a system of the form

Et

{Λε,1(L)

[Yt

zt

]+ Λε,2(L)ξ t

}= 0 (A.10)

with

Λε,1(L)(N+1)×(N+1)

≡nλ1∑

k=−mλ1

Λε,1,kLk and Λε,2(L)

(N+1)×M

≡nλ2∑

k=−mλ2

Λε,2,kLk,

where (mλ1 ,mλ2 , nλ1, nλ2) ∈ N4, all Λε,1,k , Λε,2,k have real numbers as elements, d(Λε,1(L),

Λ1(L)) −→ 0 and d(Λε,2(L),Λ2(L)) −→ 0 as ε −→ 0. Since (A.10) is implied by the systemmade of (S) and (Rε ), (20) is one solution of (A.10).


Finally, let us note (θ1, . . . , θnθ ), where

nθ ≡ (ma + na + 1)N2 + (mb + nb + 1

)N + (nc + 1

)N2 + (nd + 1

)M,

the list (in a given order) of the true structural parameters, i.e. the elements of Ak for −ma �k � na , Bk for −mb � k � nb and Ck for 0 � k � nc and di,k for 1 � i � M and 0 � k � nd .Similarly, let us note (θε,1, . . . , θε,nθ ) the list (in the corresponding order) of the measured struc-tural parameters, and let us consider a given infinite sequence of (θε,1, . . . , θε,nθ ) convergingtowards (θ1, . . . , θnθ ). This sequence corresponds to a unique sequence of ε converging to-wards zero and a unique sequence of Xε(L). If Xε,0 did not converge towards X0 along thissequence of Xε(L), then there would exist a strictly positive real number ω0 and an extractedsequence of (θε,1, . . . , θε,nθ ) such that ‖Xε,0 − X0‖ � ω0 for every element of the correspond-ing extracted sequence of Xε(L), where ‖.‖ denotes a given norm on matrices. From (A.9)and (A.10) it is easy to see, but tedious to show formally, that for any element of this ex-tracted sequence of (θε,1, . . . , θε,nθ ) sufficiently close to (θ1, . . . , θnθ ) there would then exista strictly increasing sequence extracted from the sequence (‖Xε,k‖)k∈N corresponding to thiselement, which is impossible given that Xε,k −→ 0 as k −→ +∞, so that we conclude thatXε,0 −→ X0 along the sequence of Xε(L) considered. By the same reasoning we obtain that∀k ∈ N, if (Xε,0, . . . , Xε,k) −→ (X0, . . . ,Xk) along the sequence of Xε(L) considered, thenXε,k+1 −→ Xk+1 along this sequence. By recurrence on k ∈ N we therefore conclude that∀k ∈ N, Xε,k −→ Xk along this sequence. Given that there exists (p, q) ∈ N

2 such that everyelement of the sequence of Xε(L) considered is the Wold form of a VARMA(p, q) process withp � p and q � q , as implied by Blanchard and Kahn’s [8] results in our context, this “sim-ply continuous” convergence (∀k ∈ N, Xε,k −→ Xk) implies in turn the “absolutely continuous”convergence d(Xε(L),X(L)) −→ 0 along the sequence of Xε(L) considered.

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bubble-free policy feedback rules

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