Monetary policy switching and indeterminacy

Barthélémy, Jean; Marx, Magali Article Monetary policy switching and indeterminacy Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Barthélémy, Jean; Marx, Magali (2019) : Monetary policy switching and indeterminacy, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 10, Iss. 1, pp. 353-385, https://doi.org/10.3982/QE673 This Version is available at: https://hdl.handle.net/10419/217145 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 10 (2019), 353–385 1759-7331/20190353 Monetary policy switching and indeterminacy Jean Barthélemy Banque de France and Department of Economics, Sciences Po Magali Marx Banque de France This paper determines conditions for the existence of a unique rational expectations equilibrium—determinacy—in a monetary policy switching economy. We depart from the existing literature by providing such conditions considering all bounded equilibria. We then apply these conditions to a new Keynesian model with switching Taylor rules. First, deviation from the Taylor principle in one regime does not necessarily cause indeterminacy. Second, very different responses to inflation may trigger indeterminacy even if both regimes satisfy the Taylor principle. Determinacy thus results from the adequacy between monetary regimes rather than the determinacy of each of them taken in isolation. Keywords. Markov-switching, indeterminacy, monetary policy. JEL classification. E31, E43, E52. 1. Introduction Good monetary policy should prevent indeterminacy, i.e. the existence of multiple stable equilibria. Without a policy tool to coordinate expectations on a particular equilibrium an economy experiencing indeterminacy may respond to nonfundamental sunspot disturbances, and hence, may be affected by extrinsic volatility. Since limiting inflation volatility is a widely-accepted objective for monetary policy, extrinsic volatility that is incapable of being controlled is undesirable from a policy perspective. If monetary authorities set the nominal interest rate as a state-contingent rule with constant parameters as suggested by Taylor (1993), preventing indeterminacy requires the nominal interest rate to adjust by more than one-for-one in response to inflation. This condition is known as the Taylor principle. Jean Barthélemy: [email protected] Magali Marx: [email protected] We thank Eric Leeper, Dan Waggoner and Tao Zha for stimulating discussions. We are grateful to Klaus Adam, Pamfili Antipa, Robert Barsky, Jess Benhabib, Francesco Bianchi, Vincent Bignon, Florin Bilbiie, Regis Breton, Braz Camargo, Carlos Carvalho, Seonghoon Cho, Lawrence Christiano, Nicolas Coeurdacier, Filippo Ferroni, Gaetano Gaballo, Jordi Gali, Christian Hellwig, Michel Juillard, Raphaël Jungers, Bruce McGough, Eric Mengus, Leonardo Melosi, Christopher Sims, François Velde, Toni Yates, and Michael Woodford for their very helpful comments on this paper. We are also grateful to seminar participants at the 2012 and 2013 Conference on Computing in Economics and Finance, the 2013 T2M conference, the 2013 EEA-ESEM conference, the 5th Joint French Macro Workshop, the CREI, the Bristol University and Sciences Po. Finally, we are indebted to Melissa Mundell for her careful reading of the paper. The views expressed in this paper do not necessarily reflect the opinion of the Banque de France. All remaining errors are ours. ©2019 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE673 354 Barthélemy and Marx Quantitative Economics 10 (2019) Monetary policy, however, does not necessarily follow a constant-parameter rule.1 Many empirical works (for instance, Clarida, Galí, and Gertler (2000), Lubik and Schorfheide (2004), Bianchi (2013)) document the existence of monetary policy switching in the post-World War II US economy. As a result, economic agents should internalize the possibility of future policy switches when forming their expectations. Since determinacy depends on economic agents’ expectations, regime switching affects conditions of stability and, therefore, requires an update of the Taylor principle. In a regime switching environment, history-dependent equilibria may emerge making determinacy dependent on restrictions of the class of equilibria. As noted by Farmer, Waggoner, and Zha (2010), equilibria of a regime switching model can depend on all past regimes. Previous literature (Davig and Leeper (2007), Farmer, Waggoner, and Zha (2009b), Cho (2016)) however restricts admissible equilibria by imposing some restrictions in the way equilibria depend on past regimes. In this paper, we characterize stable equilibria when the economy faces regime switching without restrictive assumptions related to the class of equilibria, and, especially, without excluding equilibria dependent on past regimes. In particular, we study determinacy in the context of a new Keynesian economy experiencing switching between multiple monetary policy regimes, described as periods for which the interest rate obeys a constant-parameter Taylor rule. Our findings are fourfold. First, we provide a necessary and sufficient determinacy condition for forwardlooking rational expectations models with parameters following a Markov process. This condition requires that the sequence of matrices products dependent on future regimes trajectories converges to a value below one. Yet, in general, this limit cannot be computed analytically as it requires keeping track of an infinite number of trajectories. Furthermore, we provide tools to apply our theoretical result in commonly used models. First, we extend our results to models with predetermined variables. Second, we provide for an algorithm that checks determinacy in less than 1 second in most of parameters configurations. We establish that the efficiency of the algorithm depends on the norm that is used and we greatly boost up the determinacy checking by choosing an adequate norm.2 Second, we settle a controversy in the literature related to conditions of determinacy (Davig and Leeper (2007), Farmer, Waggoner, and Zha (2010), Davig and Leeper (2010)). Using a new Keynesian model with monetary policy switching, we show that imposing that equilibria depend on a limited number of past regimes leads to an underestimation of the indeterminacy region. All of the existing literature implicitly or explicitly restricts the class of equilibria; see, for instance, Davig and Leeper (2007), Farmer, Waggoner, and 1There are at least three reasons to believe that the parameters of the monetary policy rule may vary over time. First, if monetary policy is optimal, any structural change in the economy should result in a change in monetary policy. Second, the monetary policy rule stems from multiple beliefs regarding the structure of the economy, the role of monetary policy and monetary policy transmission mechanisms. All these beliefs may change over time according to empirical as well as theoretical advances in macroeconomics. Third, the rule captures economic preferences, which have little reason to be stable over time. Governors of major central banks are chosen by the government according to the latter’s own preferences, and hence depend on political cycles. 2All our algorithms are available in an additional separated file. Quantitative Economics 10 (2019) Monetary policy switching and indeterminacy 355 Zha (2009b), Cho (2016), or Foerster, Rubio-Ramírez, Waggoner, and Zha (2016), providing a necessary but not sufficient condition for indeterminacy.3To our knowledge, this paper is the first to provide determinacy conditions for the whole class of equilibria. Third, we apply our results to a two-monetary policy regime economy and we prove that the Taylor principle is neither a necessary nor a sufficient condition for determinacy. Indeed, the Taylor principle is not necessary. One of the regimes can violate the Taylor principle without triggering indeterminacy if monetary policy responds sufficiently (but not too much) to inflation in the other regime. We hence resurrect one of the main findings of Davig and Leeper (2007). Compared to this paper, such policy configurations however appear less often and strong departures from the Taylor principle are not allowed. The Taylor principle is not sufficient either. Indeterminacy can emerge even if both regimes adhere to the Taylor principle. Why does the Taylor principle fail? In a purely forward-looking model, there is always a bounded equilibrium which is the unique equilibrium associated with zero expectations. Indeterminacy thus appears if, and only if, another equilibrium with non-zero expectations exists. As we focus on bounded equilibria, these nonzero expectations have to be consistent with a stable expectations path. For instance, in an economy without regime switching, the Taylor principle guarantees that any nonzero expectations will eventually diverge, and hence are not admissible. Expectations, however, diverge only asymptotically. In a finite-time horizon, expectations may converge in one direction while diverging in others. These directions may change from one regime to another. Regime switching may thus induce converging expectations. More concretely, indeterminacy arises when the policy response to inflation changes dramatically from one regime to another. In the monetary regime that reacts the most strongly against inflation, the central banker provokes a large recession to stabilize inflation in case of positive inflation expectations. In finite-time horizon, it means that inflation expectations diverge while output gap expectations converge. In the other regime, the reaction of the central bank to inflation may not ensure that inflation expectations diverge for large output gap expectations. We thus identify cases in which the direction of the convergence of expectations switches from one regime to another. This succession of “incompatible” local behaviors eventually allows for a nonzero stable expectations path and leads to indeterminacy. Fourth, we show under which conditions the US Great Inflation in the 1970s could have been caused by indeterminacy. Clarida, Galí, and Gertler (2000) famously suggest that the great volatility in the 1970s was the consequence of a violation of the Taylor principle. However, they rely on subsample estimations that do not take into account expectations of regime switching. We calibrate a new Keynesian model following Lubik and Schorfheide (2004) and check determinacy for different transition probabilities. We 3When a unique stable equilibrium among all possible equilibria exists, the unique stable equilibrium depends only on current shocks and the current regime. Therefore, the unique stable equilibrium always belongs to the classes of equilibria considered in these papers. However, for certain configurations of policy parameters, such restrictions lead to conclude with determinacy while multiple stable equilibria exist. 356 Barthélemy and Marx Quantitative Economics 10 (2019) find that indeterminacy requires a highly persistent violation of the Taylor principle, that is, the probability of remaining in this regime should be greater than 094. Such persistence is consistent with the historical duration of the Great Inflation as well as estimated parameters in the literature. From a technical side, our stability concept is boundedness and departs from some recent contributions (Farmer, Waggoner, and Zha (2009b), Cho (2016), Foerster et al. (2016)) which favor the mean square stability concept. We adopt this concept for two main reasons: first, it is the most common concept in the rational expectations literature, second, it is consistent with an underlying nonlinear model and a perturbation approach (Barthélemy and Marx (2017)). Of course, the choice of the stability concept matters for determinacy. However, we believe that our main argument—equilibria can depend on past regimes and following, that determinacy conditions depend on the exact class of equilibria—does not depend on the precise definition of stability. Extending our results to other stability concepts would be a natural avenue for future research. The remainder of the paper is organized as follows. In Section 2,weprovidefora simple new Keynesian model with monetary policy switching that we use throughout the paper to illustrate our results and we depict the restrictions on the solution space, done in the literature, and their consequences. We then turn to a general class of models in Section 3. We provide for a necessary and sufficient determinacy condition that we complement with an efficient algorithm to check determinacy in practice. In Section 4, we demonstrate that the definition of the solution space is crucial when dealing with regime switching and we provide examples of sunspot equilibria. We illustrate our results with two applications. First, we show the limits of the Taylor principle when monetary policy’s reaction to inflation switches between different values in Section 5. Second, we demonstrate that the US Great inflation could only have resulted from a violation from the Taylor principle if economic agents were convinced that this violation was sufficiently long lasting in Section 6. Finally, we draw conclusions in Section 7. 2. Monetary policy switching In this section, we present a new Keynesian model with monetary policy switching that we will use throughout the paper and we recall standard determinacy conditions in the absence of regime switching. Then we review existing findings in the literature and show that the definition of the solution space is critical for determinacy conditions. 2.1 The model We consider a log-linearized new Keynesian model following Clarida, Galí, and Gertler (2000)andWoodford (2003) in which private decisions satisfy: yt=Etyt+1−σrt−Etπt+1−rn t(1) πt=βEtπt+1+κyt+ut(2) Quantitative Economics 10 (2019) Monetary policy switching and indeterminacy 357 where variables yt,πt,andrtare respectively the output gap, inflation (in log), and the nominal interest rate (in deviation around a certain steady state). The operator Etdenotes expectations at time t.Equation(1) is an IS curve that links the output gap to all future ex ante real interest rates and future and current shocks, rn t. Parameter σmeasures risk aversion. Equation (2) is a new Keynesian Phillips curve linking inflation to all future marginal costs summarized by the output gap. Parameter κmeasures the degree of nominal rigidities while βstands for the discount factor. Shock utdenotes a cost-push shock translating the Phillips curve. We define a monetary policy regime, denoted by st∈{12}, as a period during which monetary policy obeys a Taylor rule. We suppose that regime stfollows a Markov proccess characterized by a transition probability matrix P; the probability of switching from regime ito regime jis denoted pij . The current policy regime is known to private agents while future regimes are not. In regime st, the monetary authority sets the nominal interest rate following: rt=ρstrt−1+(1−ρst)αstπt+γstyt+εr t(3) where the parameters αstand γstmeasure the sensitivity of the interest rate to inflation and to the output-gap in each regime. The parameter ρststands for the inertia in monetary policy decisions. For exposition purposes, we first consider the case ρst=0and then show in Section 3.3 how to extend the results in a more general set up. The shock εr tcaptures the unsystematic part of monetary policy. Finally, we assume that shocks are bounded4and, without loss of generality, are i.i.d. and zero mean. Then the question is to determine conditions ensuring the existence of a unique bounded equilibrium.5We say that this economy is determinate if it admits a unique bounded equilibrium. Otherwise, it is indeterminate. By plugging the monetary policy rule, equation (3), into the IS curve, equation (1), we end up with a system of two forward-looking equations simultaneously determining inflation and the output gap: stzt=Etzt+1+Ωεt(4) where the column vector ztdenotes endogenous variables, [πtyt]and the column vector εtdenotes the shocks, [utεr trn t]. The matrices stand Ωwhich gather the parameters of the model are given by st=1/β −κ/β σ(αst−1/β) 1+σγst+κσ/βΩ=1/β 00 −σ/β −σσ  Since shocks are uncorrelated, it seems natural to look for a solution with zero expectations. This is the case for the fundamental equilibrium, denoted by subscript F 4Boundedness of shocks is required since we are considering bounded equilibria. 5As proved below, there is always at least one bounded equilibrium satisfying the model when there is no backward-looking component, ρst. Otherwise, the model may have no bounded equilibrium. 358 Barthélemy and Marx Quantitative Economics 10 (2019) anddefinedaszF t=−1 stΩεt, which means yF t=−σαstut−rn t+εr t 1+σγst+σαstκand πF t=κyF t+ut(5) The question is then whether or not this equilibrium is unique. 2.2 In the absence of regime switching We start with studying the case without regime switching (1=2=). The bounded equilibrium (πF tyF t)is the only one consistent with zero expectations. It is therefore natural to investigate whether expectations can be different from zero. Following Sims (2002), we thus analyze expectations of inflation and the output gap, which we denote by the column vector ze t=Etzt+1. We also introduce the associated forecast error, ξt+1=zt+1−ze t. Finally, it is convenient to rewrite equation (4)asfollows: ze t+1=ze t−ξt+1−Ωεt+1(6) The equilibrium, zF t, is the unique bounded equilibrium if it is impossible to find non-zero stable expectations satisfying equation (6). Suppose that at period 0,expectations are different from zero. Then, depending on the eigenvalues of , expectations will explode or implode. If all eigenvalues of are greater than one, then expectations diverge. As we rule out unbounded equilibrium, having nonzero expectations in the first place is inconsistent with a stable equilibrium. This proves that zF tis the only stable solution of the model. Thus, determinacy ultimately depends on the lowest eigenvalue of this matrix. This condition is equivalent to: α+1−β κγ>1(see Woodford (2003)). If monetary authorities do not respond to the output gap, monetary policy can prevent indeterminacy by increasing the nominal interest rate by more than one-for-one in response to inflation. Through such a policy, monetary authorities guarantee that nonzero expectations diverge asymptotically. This result is known as the Taylor principle. The question we want to address in this paper is how these conditions evolve in the context of a regime switching monetary policy. 2.3 Theroleofsolutionspaceintheliterature Davig and Leeper (2007) provide determinacy conditions for regime switching models assuming that equilibria only depend on the current regime and shocks (see Branch, Davig, and McGough (2007), for a discussion). By denoting by M0, the set of all bounded equilibria satisfying this property, and by M=(P ⊗1n)×diag(12),Davig and Leeper’s result can be restated as follows: there exists a unique bounded equilibrium in M0if, and only if, the spectral radius of M, that is, the largest eigenvalue in absolute value, is strictly less than one. If it is not the case, then any bounded equilibrium in M0can be put into Quantitative Economics 10 (2019) Monetary policy switching and indeterminacy 359 the following form: zt=zF t+Vstwtand wt=Jwwt−1+ξt(7) with, ξtbeing any bounded zero mean process (Etξt+1=0) independent of current and past regimes. The proof as well as the definition of matrices Jwand Vstcan be found in Appendix A. Based on this result, Davig and Leeper (2007) prove that one monetary policy regime can fail to satisfy the Taylor principle without endangering the overall determinacy as long as the other regime is sufficiently frequent, long-lasting, and that the reaction to inflation in the other regime is sufficiently great. They call this latter result the long run Taylor principle. However, their theoretical results do not hold when considering more general solution spaces and thereby cast doubts about the validity of the long run Taylor principle. Farmer, Waggoner, and Zha (2010) have noticed, for example, that when α1=3,γ1=0, α2=092,γ2=0,p11 =08,andp22 =095,6bounded equilibria exist outside M0on top of the fundamental solution. They give the following example: zt=zF tand wt=0if st=1 zt=zF t+Vw tand wt=wt−1+Mξtif st=2 where matrices ,V,andMare given in Farmer, Waggoner, and Zha (2010)andξtis once again an i.i.d. zero-mean shock. The reason why the Davig and Leeper (2007) result does not rule out this additional bounded equilibrium is that the latter depends on past regimes and, therefore, does not belong to M0. The additional part wtof the above equilibrium depends on past wt−1 which implicitly depends on all past regimes, while in equation (7), this additional part does not depend on past regimes. In fact, Farmer, Waggoner, and Zha (2011) prove that any equilibrium can be written as follows: zt=zF t+Vstwt(8) wt=φst−1stwt−1+V stξt(9) where Vstand φst−1stare regime dependent matrices and ξtis an arbitrary zero mean process. The sunspot component ξtmay depend on past regimes in a sophisticated way, such that checking the stability of the process wtis hard. To circumvent this issue, the literature (Farmer, Waggoner, and Zha (2011), Cho (2016), for instance) supposes that the sunspot part ξtdoes not depend on past shocks or regimes. There is, however, no clearcut reason to make such an assumption. Indeed, the process ξtmay depend on past regimes and we show in Section 4that its structure affects the stability of the process wt. In the next section, we establish the determinacy conditions which are valid whatever the structure of the process ξtand, therefore, do not depend on a particular restriction to the solution space. 6Other parameters are calibrated as follows: β=099,σ=1, and κ=017. 360 Barthélemy and Marx Quantitative Economics 10 (2019) 3. Determinacy conditions In this section, we first present the class of models we deal with (Section 3.1). Second, we derive determinacy conditions for purely forward-looking regime switching models without assuming restrictions on the solution space (Section 3.2). We then extend our findings to regime switching models with backward-looking components (Section 3.3). Finally, we provide a concrete and simple application and describe algorithms that check our determinacy conditions efficiently (Section 3.4). 3.1 The class of models Most micro-founded macroeconomic models may be summarized by a system of nonlinear equations involving structural parameters governing economic agents’ preferences, technology, market structures, and economic policies. Allowing these parameters to switch over time results in nonlinear regime-switching models. When shocks are small enough, the stability of this class of models can be checked by studying the determinacy of linear regime-switching models as demonstrated by Barthélemy and Marx (2017). In this paper, we focus on linear models of the following form: AstEtzt+1+Bstzt+Cstzt−1+Dstεt=0(10) where index tdenotes time and belongs to {−∞∞},vectorztis a (n ×1)real vector of endogenous variables, vector εtis a (p×1)real vector of exogenous shocks, and index stindicates the current regime, in {1N}. For any index i∈{1N},matricesAi,Bi, and Ciare (n ×n) real matrices. We assume that matrices Aiand Biare invertible. The vector of shocks εtis assumed to be i.i.d. and bounded. Finally, we assume that regimes follow a Markov-chain with constant transition probabilities: ∀(i j) ∈{1N}2Pr(st=j|st−1=i) =pij (11) where the scalar pij is between 0and 1and naturally sums to 1over regimes j.Wethus assume that transition probabilities are constant over time and over state. We consider that an equilibrium is stable if it is bounded. Precisely, we assume that Fis a bounded set such that εt∈F, and we denote by εt={εtε−∞}and st={sts−∞}the history of shocks and regimes. We define a stable equilibrium as follows. Definition 1. A stable equilibrium is a function zon {1N}∞×F∞, satisfying the model (10)andsuchthat z∞=sup ztεt zstεt <∞(12) We denote by Bthe set of all the bounded functions on {1N}∞×F∞.ThesetB, with the norm · ∞defined in equation (12) is a Banach space. Following the literature (Blanchard and Kahn (1980), Lubik and Schorfheide (2004)), we say that the model/the economy is determinate if there exists a unique stable equilibrium. For purely forward-looking models (Cst=0), when the model is not determinate Quantitative Economics 10 (2019) Monetary policy switching and indeterminacy 367 Table 1. Illustration of Proposition 3for different policy parameters (α1α2). (α1α2)(r 1r2)ρ({R})ν({R})Case of Proposition 3 (0225)(16−04i −12+05i)     09     071 (16+04i −12−05i) 0907 (−21−03)4622 (11)(−01−11)1952 (16−11i 04+09i)     12     083 (16+11i 04−09i) 1208 (21)(1526)     05     051 (−01−11)3388 (4404)3415 (2502)(−00−60)1955 4887 (0546)     04     122 (0546)6327 Note: See the text for detailed explanation of each column. Each group of lines correspond to a pair of policy parameter (α1α2). We surround the values of ρ({R})and ν({R})allowing to conclude in which case of Proposition 3we are. Case 1 corresponds to a region of parameters that is consistent with a unique equilibrium (determinacy), Case 2 with multiple equilibria (indeterminacy), and Case 3 with no equilibria. In this univariate example, ρ({R})=max{|μ1| |r1||μ2| |r2|}and ν({R})=1−p11 |r1|+p11 |r2|. remains in state 2 (p22 =0). The matrices {Rh}in Proposition 3are scalars r1and r2, solutions to the two quadratic equations: p11r1+(1−p11)r2r1−α1r1−μ1=0 r1r2=α2r2+μ2 Depending on the roots of these equations, Proposition 3characterizes the equilibria. Table 1illustrate how to concretely apply Proposition 3for four values of the pair (α1α2). We calibrate other parameters to μ1=02,μ2=12,p11 =05. The first column reports the value of the pair (α1α2); the second column reports all the roots of the polynomial described above (r1r2); the third column and the fourth column report the computation of the joint spectral radius ρ({R})and the limit ν({R})for each root; finally, the last column concludes based on Proposition 3. Notice that because the model is univariate, we can easily compute the joint spectral radius as well as the limit. Figure 7in the Appendix reports the different regions of determinacy, indeterminacy, and no-equilibria with respect to the pair (α1α2). This graph shows that we never find the fourth case of Proposition 3in this simple example. Boosting the convergence process Except very special cases such as the univariate environment presented above, computing the limit involved in Proposition 1may be timeconsuming. The speed of convergence of the sequence ukis sensitive to the choice of the matricial norm ·. Though there is no known method to find the best matricial norm, we propose a choice of norm that proves to be efficient in most applications. From any matricial norm, ·, we can derive many norms by changing the basis of the vectorial 368 Barthélemy and Marx Quantitative Economics 10 (2019) space. We denote by · Qsuch a norm: AQ= Q−1AQ (25) where Qis an arbitrary invertible matrix. Let define Q∗ kthe optimal matrix minimizing ukfor an arbitrary k>0with respect to matrix Q. We then define the associated sequence u kas follows: u k= (i1ik)∈{1N}k pi1i2···pik−1k −1 i1−1 i2···−1 ik Q∗ k1/k  We now prove that this new sequence converges toward the same limit as the initial sequence ukand can thus be used in practice to check determinacy of Markov switching rational expectations models. Proposition 5. The sequence (u∗ k)and the sequence (uk)admit the same limits. Proof. Let us consider the sequence u∗ kdefined when considering the matricial norm · Q∗ kfor all k. It is evident in view of Proposition 2, that this sequence is above the limit. Furthermore, this sequence is always smaller than uk(the sequence associated to Q=1n), which converges to νaccording to Proposition 1. As a consequence, ukalso converges to ν. Proposition 5gives the best norm among the subset of norms generated by a basis change in the vectorial space. While this optimization is potentially costly for large k, it is really powerful in enhancing the accuracy of the approximation of the limit ν. Thus we recommend finding first the best matricial norm for small k(let say k=5) in order to find a matrix that is convenient for the considered problem and then iterate on kwithout changing the matricial norm anymore. This method is both accurate and fast in most applications. Let us now show the practicality of Proposition 5in the monetary policy switching model presented in Section 2when the Taylor rule does not incorporate backwardlooking component (ρst=0). In Figure 1, we plot the determinacy region for parameters calibrated as in Davig and Leeper (2007). The determinacy region is significantly smaller than in their paper explaining why Farmer, Waggoner, and Zha (2010) can find policy parameters satisfying their determinacy conditions but consistent with multiple bounded equilibria. We discuss economic consequences of this determinacy frontier in Sections 5 and 6. The choice of the norm appears critical for the accuracy of the approximation of the limit. Figure 2reports the values of ukfor different choices of norms with respect to the computing time (in log scaled). The black line with diamonds plots the sequence when we choose the optimal transition matrix Q∗ kfor all k. Compared to other norms, this is the most precise. However, the minimization is computationally costly.10 In practice, we 10This minimization can be computed analytically in some circumstances (depending on the chosen norm). But the gain to compute it numerically appears small as the analytical result is difficult to manipulate. Quantitative Economics 10 (2019) Monetary policy switching and indeterminacy 369 Figure 1. Determinacy condition and policy parameters. Note: The white area depicts the determinacy region with respect to policy parameters α1and α2. Other parameters are calibrated as follows: γ1=0,p11 =08,γ2=0,p22 =095,κ=017,β=099,andσ=1. The approximation of the limit νis computed by computing u20 with the optimized norm · Q∗ 15 . Figure 2. Speed of convergence, accuracy, and norm choice. Note:They-axis represents ukfor different chosen matricial norms for k<16.Thex-axis reports time in seconds (log-10 axis). The black line reports matricial norm induced by the vector 2-norm. The grey plain line with crosses (dotted line, plain line) reports convergence for norms induced by the vector ∞-norm, induced by the vector 1-norm and the standard Frobenius norm, respectively. Along the black line with crosses, we plot the sequence ukwhen we choose the basis change matrix Q∗ 5, that is, optimized for u5, and along the black line with diamonds, when we choose the optimal matrix Q∗ kfor all k as in Proposition 5. Policy parameters are calibrated as follows: in regime 1, α1=14,γ1=0,and p11 =08while α2=097,γ1=0,andp11 =095. Programs are launched on an Intel(R) Core(TM) i7-4770 CPU 34GHz machine using Matlab R2014a. 370 Barthélemy and Marx Quantitative Economics 10 (2019) recommend computing first the optimized norm, · Q∗ 5and then iterating to compute the sequence of uk. This choice leads to an approximated value of the limit similar to the optimal matrix choice but without implying such a large computational cost. In the calibration used for the figure, determinacy can be settled in less than a second. We thus propose the following algorithm. Algorithm 1. Determinacy conditions for purely forward-looking models: 1. Choose a norm (among norms induced by a vectorial norm). 2. Compute the optimal basis change matrix Q∗ 5(this can be done by using exponential matrices as a basis of the unitary matrix). 3. Compute ukfor the matricial norm · Q∗ kfor k>5. 4. Stop if ukis lower than one or if the increment from kto k+1is negligible compared to the distance to 1otherwise iterate step 3 for a larger k. 5. If uk<1, the model is determinate, indeterminate otherwise. This algorithm is available in the Online Supplemental Material (Barthélemy and Marx (2019)) and describes the computations required to get the black line with crosses in Figure 2. In most cases, we found that the norm induced by the vector 1-norm is the one providing the best speed/accuracy trade-off; however, we cannot reject that for some economic problems, other norms should be preferred. General algorithm Let us now present our proposed algorithm to solve and check determinacy of regime switching models when introducing backward-looking components. We closely follow the three steps identified in Section 3.3. The first step is to find a set of matrices solving (18); the second step consists in checking the stability of this set; the last step is to check the determinacy conditions of the forward-looking model associated with this set of matrices. The algorithm is as follows: Algorithm 2. Determinacy conditions for general models 1. Find matrices {Rst}satisfying equation (18). To do so, one can use the forward iteration procedure developed by Cho (2016). 2. Compute the joint spectral radius of this set of matrices, ρ({R}).Weusetheprogram developed by Vankeerberghen, Hendrickx, and Jungers (2014)whichprovidesupper and lower bounds of the joint spectral radius efficiently. 3. Compute the limit ν({R})using Algorithm 1. 4. Conclude: (a) If ρ({R})<1and ν({R})<1, the model is determinate and the stable equilibrium is given by equation (21). (b) If ρ({R})<1and ν({R})>1, the model admits many stable equilibria. (c) If ρ({R})>1and ν({R})<1, the model admits no stable equilibria. Quantitative Economics 10 (2019) Monetary policy switching and indeterminacy 371 (d) If ρ({R})>1and ν({R})>1return to step 1 and try to find another solution to the matricial equation (18) by using another solving techniques (Foerster et al. (2016), Maih (2015), for instance). The Matlab codes of Algorithm 2are available in the Online Supplemental Material. It is worth noticing that, in the economic models we have studied up to now, we have never encountered case (d). We use this algorithm in Section 6. 4. Classes of equilibria and indeterminacy In the absence of regime switching, restriction of the stochastic properties of the equilibria, such as their correlation with past fundamental shocks, does not affect stability. By contrast, in a regime-switching economy, the stability of equilibria depends on their co-movement with past regimes. By ruling out some classes of equilibria, the existing literature thus underestimates the size of the indeterminacy region. In this section, we construct some sunspot equilibria that have never been described before and that appear when Proposition 2is not verified. This illustrates the sensitivity of determinacy conditions to restrictions related to the class of equilibria and argues in favor of an intrinsic determinacy condition as in Proposition 5that does not arbitrarily restrict the solution space. For the sake of simplicity, we only consider purely forwardlooking models (13), the extension to predetermined variables in Section 3.3 can be used to construct sunspots for more general models. 4.1 Sunspot equilibria Here, we generalize the approach by Farmer, Waggoner, and Zha (2011). We focus on the class of equilibria, Mq, that depend on lagged variables and regimes of this form:11 zt=zF t+Vt(st−q···st)wt wt=φt(st−qst)wt−q+ξt where Vt(·)is a time-varying matrices depending on qlast regimes and that belongs to a single vector every qregimes. The scalar φt(·)is time-varying and the product of φ(·) along a trajectory of qsuccessive regimes is smaller than one. The sunspot shock ξtis a zero mean i.i.d. shock. Finally, we assume that wtis zero for t≤0.Thistypeofsolution corresponds to recursive solutions which cyclically belong to the same one-dimensional space and are exponentially decreasing on it. We provide determinacy conditions for this particular subspace, Mq, in Proposition 6. We call a nonfundamental solution in Mqa sunspot equilibrium of order q. 11Class of equilibria considered by Davig and Leeper (2007), Farmer, Waggoner, and Zha (2011), or Cho (2016) can be put into these forms. 372 Barthélemy and Marx Quantitative Economics 10 (2019) Proposition 6. If for a certain integer q,there exist Nq+1real numbers in the open unit disk,α(i0iq),such that the highest eigenvalue of the matrix  (i1iq−1)∈{1N}q−1 pii1pi1i2···piq−1j−1 i···−1 iq−1α(i i1iq−1j)(ij) (26) is larger than 1,then there exist multiple bounded solutions in Mq. This proposition leads then to an algorithm easy to implement. First, we fix an integer q(not too large). Then equation (26) defines a function of 2q+1real numbers in the open unit disk, for which we compute the maximum. When qincreases, the cost of computation is exponentially increasing. We prove this proposition by constructing a continuum of stable equilibria in Appendix G. For a given length, q, the equilibria we consider depend on the qpast regimes, shocks and endogenous variables. These equilibria belong to a fixed span every qperiods, and are exponentially decreasing. We notice that, when qincreases, the size of the state variables needed to describe the equilibrium increases. As part of our reasoning for this proposition, let us first consider the special case of a specific q-regime trajectory, (k0kq−1k0), which is sufficient to generate a largerthan-one eigenvalue of the matrix defined by equation (26).12 In this case, there exists a vector usuch that pk0k1pk1k2···pkq−1k0−1 k0···−1 kq−1u=λu,withλ>1. Suppose that the economy is initially in regime k0and that expectations ze tbelongs to the unstable eigenvector, u. In addition, suppose that all the future expectations are zero except along the regimes’ trajectory (k0kq−1k0kq−1k0). Then, according to equation (16), the expectations decrease exponentially every qperiods. Hence, economic agents can form nonzero expectations consistent with converging expectations. Therefore, the economy is indeterminate. More generally, Proposition 6proves that indeterminacy may arise in a more general context where such a trajectory does not exist. It may happen when a combination of trajectories leads to a larger-than-one eigenvalue. In this case, nonzero expectations are consistent with converging expectations in many different regimes trajectories. A generalization of this proposition was recently advanced by Ogura and Jungers (2014). Basically, these authors refine our results by replacing the weights, α(i i1 iq−1j),informula(26), with particular unitary matrices.13 The classes of equilibria that we consider in Proposition 6encompass those considered in the literature (Davig and Leeper (2007), Farmer, Waggoner, and Zha (2009b), Cho (2016), Foerster et al. (2016)). The equilibria put forward after Farmer, Waggoner, and Zha (2009b) correspond to the specific case of q=1.Whenq=0, we find back the regime dependent solution space of Davig and Leeper (2007). By focusing on smaller classes of equilibria, the literature underestimates the size of the indeterminacy region. 12This corresponds to α(i0iq)=δk0i0···δk0iqin equation (26), where δij =1when i=j. 13In their construction, the vector of endogenous variables belongs to a multidimensional space every q periods. Quantitative Economics 10 (2019) Monetary policy switching and indeterminacy 373 In turn, determinacy condition put forward in Section 3takes into account all kinds of sunspot equilibria.14 Proposition 6also allows us to link our approach to the one with an initial condition and a starting date, helping interpreting indeterminacy in the context of regime switching. When Proposition 6proves that the model is indeterminate, Lemma 4 gives examples of distinct stable equilibria solution of the model for any date greater than an initial date t0that we can reinterpret as an initial condition (t0=0). According to this lemma, the model admits at least two different equilibria: •A fundamental solution z1 t=zF t •A nonfundamental solution z2 t=zF t+wt,wherewtis given in Lemma 4, Appendix G: it depends on wt−1and a sunspot component ξtindependent of the history of regimes st. By assuming that wt=0for all t≤0and w0=V(s 0)ξ0with ξ0being any real-valued scalar, we see that the two solutions only differ from date-0 and onwards because of the sunspot component ξ0. The proposition is thus able to show that whenever there is indeterminacy, there exists multiple initial conditions (w0=0and w0= 0)atleastinone regime. Interestingly, indeterminacy does not necessarily mean that there are multiple initial conditions for any initial regime, in contrast with the no regime switching case, in which when the model is indeterminate there are always multiple initial conditions.15 4.2 Illustration in the monetary regime switching We apply Proposition 6to analyze the relationship between determinacy conditions and the restriction of the class of equilibria. Figure 3displays different determinacy frontiers depending on the solution space: the whole solution space (the plain line), the regime dependent equilibria, M0(the line with crosses), and the sunspot equilibria of order q built in Proposition 6(the dashed lines). The sufficient indeterminacy frontier nears the determinacy condition computed as in Section 3.2 when the order of the sunspot equilibria qincreases. The diminution of the determinacy region when going from q=0to q=1, that is to say when adopting the solution space of Farmer, Waggoner, and Zha (2011), is substantial. However, pushing to a higher order is not negligible either, implicitly proving the existence of multiple solutions of order larger than 1 even if there exists a unique bounded equilibrium in M1. The existing literature increasingly adopts the restriction first proposed by Farmer, Waggoner, and Zha (2009b) consisting of M1and sometimes referred to as minimum state variables solutions. Restricting the class of equilibria results in underestimating 14It is worth noting that some history dependent equilibria do not belong to Mqwith a finite q. 15To get a simple intuition of this difference, consider a two-regime model with absorbing states. Suppose that the first regime corresponds to a determinate model (when taken in isolation) while the second corresponds to an indeterminate model (when taken in isolation). According to our definitions, we will say that the regime switching model is indeterminate. But if the economy is initially in the first regime, there exists a unique initial condition (and a unique path of the economy), while multiple initial conditions if the economy is initially in the second regime. 374 Barthélemy and Marx Quantitative Economics 10 (2019) Figure 3. Existence of sunspots of order q.Note: The thick line displays the determinacy frontier (constructed as in Figure 1), dashed lines correspond to sufficient indeterminacy conditions given by Proposition 6for sunspot equilibria of order q. The line with crosses represents the Davig and Leeper (2007) determinacy condition. Probabilities are set to p11 =08and p22 =095. the size of the indeterminacy region. In Section 5, we provide economic results that are valid for the broadest solution space. 5. The Taylor principle with regime switching In this section, we discuss the usefulness of the Taylor principle as a guideline for monetary policy in a context of regime switching. We put forward two main results. First, we revisit one of the main findings of the paper by Davig and Leeper (2007). They show that determinacy is not necessarily generated by a deviation from the Taylor principle if this deviation is small and short lasting. We prove that the result remains valid even if we consider all bounded equilibria. Thus, the criticism by Farmer, Waggoner, and Zha (2010) does not invalidate this result even if the occurrence of such policy configurations is substantially smaller than what was predicted by the long run Taylor principle initially put forward. Second, we show that indeterminacy can arise even if the two monetary policy regimes satisfy the Taylor principle. This finding has also been noted by Foerster (2016) and Cho (2016) in separate and independent works. However, we find that such configurations arise in more cases than what is put forward in these papers, because we do not restrict the solution space. Finally, we give intuitions on why such policy configurations lead to indeterminacy and draw some policy conclusions. In this section, we assume that all the nonpolicy parameters are unchanged across regimes. For policy parameters, we assume that the interest rate only reacts to inflation. This assumption allows for an economic interpretation of our determinacy results without loss of generality. Quantitative Economics 10 (2019) Monetary policy switching and indeterminacy 375 5.1 The Taylor principle is neither necessary. .. Result 1. A regime-switching economy may be determinate even if one of the regimes does not satisfy the Taylor principle. Monetary policy may deviate from the Taylor principle moderately (α1=098)but relatively persistently (p11 =095) without implying indeterminacy if monetary policy reacts sufficiently to inflation in the other regime (for instance, if α2=15)asshownin Figure 1. While the first regime is not active enough to ensure determinacy on its own, expectations of a switch towards a more active regime are enough to anchor expectations, and hence to rule out indeterminacy. Thus, expectations of a more aggressive monetary policy may be effective in guaranteeing macroeconomic stability. This result definitively proves the Davig and Leeper (2007) claim that “a unique bounded equilibrium does not require the Taylor principle to hold in every period”. In this paper, the authors obtain this result by restricting the solution space. The counterexample given in Farmer, Waggoner, and Zha (2010) could suggest that this result does not hold when considering a broader solution space. We prove that Davig and Leeper (2007)’s result holds even if we consider the whole class of equilibria, and hence, is not an artifact due to the specific solution space they consider. However, compared to the paper by Davig and Leeper (2007), the occurrence of such policy configurations is less frequent than initially conjectured as displayed in Figure 1. Only small and brief deviations from the Taylor principle do not endanger determinacy issues. 5.2 ...Norsufficient Result 2. An economy may suffer from indeterminacy even if the two monetary policy regimes satisfy the Taylor principle. We identify such a configuration when the two monetary policies share the same rule—the nominal interest rate reacts proportionally to inflation in both regimes—but with different intensities. The two regimes satisfy the Taylor principle. In the first regime, the less active one, the central bank reacts moderately to inflation (α1=101). In the second, the more active regime, the central bank reacts (extremely) strongly to inflation (α2=6). Proposition 6proves that the economy is indeterminate when monetary policy switches between prolonged periods of the less active regime (p11 =095)andshortlasting periods of the more active regime (p22 =05). Figure 8in the Appendix plots the ten largest contributions to sequence uk,which measures the convergence of expectations. The larger the contribution, the more converging the expectations along the regimes trajectory. Unsurprisingly, a prolonged less active monetary policy regime significantly contributes to an increase in this measure of stability. Indeed, in this regime, monetary policy causes expectations to diverge but only slowly as the response to inflation is weak in this regime. This regimes trajectory is not enough however to explain indeterminacy by itself as the less active regime satisfies the Taylor principle, and hence, induces determinacy when taken in isolation. The other 376 Barthélemy and Marx Quantitative Economics 10 (2019) Figure 4. Impact of a positive output gap expectation in a new Keynesian model with Markov-switching monetary policy. Note: The two figures report the dynamics of the output gap (left) and inflation (right) expectations to an expectation of an increase in the output gap in 17 periods. We relate the four largest contributors to u16 (see Figure 8). The bold line represents the trajectories of inflation and the output gap conditional on staying in the less active regime (regime 1). Along the plain line with crosses (the dashed line with diamonds and the dashed line with squares), we plot the trajectories when the economy is in the more active regime during one (two and three, resp.) period (regime 2) and in the less active regime afterward (regime 1). Policy parameters and probabilities are set to α1=101,γ1=0,andp11 =095 and α2=6,γ2=0,and p22 =05. greatest contributions correspond to the alternation between a short period of the more active regime and a protracted period of the less active regime. We explain in Figure 4 why these regimes trajectories explain indeterminacy. Indeterminacy arises when expectations can be nonzero without generating diverging expectations. Figure 4reports the responses of the output gap and inflation to a 17-quarter-ahead expectation conditional on the future regimes path. We plot the responses to an expected increase in the output gap (Etyt+17 =1while Etπt+17 =0)in17 quarters. If the model is determinate such expectations should necessarily lead to small expectations in the first period (when weighted by the regime trajectory’s probability) as nonzero expectations always diverge in a determinate economy. Along the line without markers, we plot the expectations dynamics if the economy remains in the less active regime. In this regime, a positive expectation of the output gap in the remote future leads to inflationary pressure with the new Keynesian Phillips curve explaining the downward dynamics of inflation expectations. On the other hand, these inflation expectations lead to a moderate increase in the real rate that contributes to a slightly moderate output gap through the IS-curve. This means that output gap expectations increase up to the final period. 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