A generalized approach to indeterminacy in linear rational expectations models

Bianchi, Francesco; Nicolò, Giovanni Article A generalized approach to indeterminacy in linear rational expectations models Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Bianchi, Francesco; Nicolò, Giovanni (2021) : A generalized approach to indeterminacy in linear rational expectations models, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 12, Iss. 3, pp. 843-868, https://doi.org/10.3982/QE949 This Version is available at: https://hdl.handle.net/10419/253563 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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 12 (2021), 843–868 1759-7331/20210843 A generalized approach to indeterminacy in linear rational expectations models Francesco Bianchi Duke University, CEPR, and NBER Giovanni Nicolò Federal Reserve Board We propose a novel approach to deal with the problem of indeterminacy in linear rational expectations models. The method consists of augmenting the original state space with a set of auxiliary exogenous equations to provide the adequate number of explosive roots in presence of indeterminacy. The solution in this expanded state space, if it exists, is always determinate, and is identical to the indeterminate solution of the original model. The proposed approach accommodates determinacy and any degree of indeterminacy, and it can be implemented even when the boundaries of the determinacy region are unknown. Thus, the researcher can estimate the model using standard software packages without restricting the estimates to the determinacy region. We combine our solution method with a novel hybrid Metropolis–Hastings algorithm to estimate the New– Keynesian model with rational bubbles by Galí (2021) over the period 1982:Q4– 2007:Q3. We find that the data support the presence of two degrees of indeterminacy, implying that the central bank was not reacting strongly enough to the bubble component. Keywords. Indeterminacy, general equilibrium, solution method, Bayesian methods. JEL classification. C19, C51, C62, C63. 1. Introduction Sunspot shocks and multiple equilibria have been at the center of economic thinking at least since the seminal work of Cass and Shell (1983), Farmer and Guo (1994), and Farmer and Guo (1995). The zero lower bound has brought renovated interest to the problem of indeterminacy (Aruoba, Cuba-Borda, and Schorfheide (2018)). Furthermore, Francesco Bianchi: [email protected] Giovanni Nicolò: [email protected] The views expressed in this paper are those of the authors and do not reflect the views of the Federal Reserve Board or the Federal Reserve System. We thank Jonas Arias, Jess Benhabib, Roger Farmer, Francois Geerolf, Frank Schorfheide, and all participants at UCLA seminars, NBER Multiple Equilibria and Financial Crises Conference, CEPR-IMFS New Methods for Macroeconomic Modeling, Model Comparison and Policy Analysis Conference, Federal Reserve Bank of St. Louis, Society of Economic Dynamics, 12th Dynare Conference, 2017 NBER-NSF conference on Bayesian Inference in Econometrics and Statistics. We would also like to thank four anonymous referees for helpful feedback and suggestions. ©2021 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE949 844 Bianchi and Nicolò Quantitative Economics 12 (2021) in many of the Linear Rational Expectation (LRE) models used to study the properties of the macroeconomy the possibility of multiple equilibria arises for some parameter values, but not for others. This paper proposes a novel approach to solve LRE models that easily accommodates both the case of determinacy and indeterminacy. As a result, the proposed methodology can be used to easily solve and estimate a LRE model that could potentially be characterized by multiplicity of equilibria. Our approach is implementable even when the analytic conditions for determinacy or the degrees of indeterminacy are unknown. Importantly, the proposed method can be easily implemented to study indeterminacy in standard software packages, such as Dynare (Adjemian et al., 2020)andSims’(2001)codeGensys. To understand how our approach works, it is useful to recall the conditions for determinacy as stated by Blanchard and Kahn (1980). Indeterminacy arises when the parameter values are such that the number of explosive roots is smaller than the number of non-predetermined variables. The key idea of our method is to augment the original model by appending additional autoregressive processes that can be used to provide the missing explosive roots. The innovations of these exogenous processes are assumed to be linear combinations of a subset of the forecast errors associated with the expectational variables of the model and a newly defined vector of sunspot shocks. When the Blanchard–Kahn condition for determinacy is satisfied, all the roots of the auxiliary autoregressive processes are assumed to be within the unit circle and the auxiliary process is irrelevant for the dynamics of the model. In this case, the law of motion for the endogenous variables is equivalent to the solution obtained using standard solution algorithms (King and Watson (1998); Klein (2000); Sims (2001)). When the model is indeterminate, the appropriate number of appended autoregressive processes is assumed to be explosive. For example, if there are two degrees of indeterminacy, two of the auxiliary processes are assumed to be explosive. The solution that we obtain for the endogenous variables is equivalent to the one obtained with the methodology of Lubik and Schorfheide (2003)orFarmer, Khramov, and Nicolò (2015). Our methodology simplifies the common approach used to deal with indeterminacy. The common procedure requires the researcher to solve the model differently depending on the area of the parameter space that is being studied. Under indeterminacy, existing methods require to either construct the solution ex post following the seminal contribution of Lubik and Schorfheide (2003) or rewrite the model based on the existing degree of indeterminacy (Farmer, Khramov, and Nicolò (2015)). In itself, this is not an insurmountable task, but it implies that the researcher cannot simply use standard solution methods or software packages. What is more, if the researcher is interested in a structural estimation of the model, she would need to write the estimation codes and not just the solution codes. Our proposed method only requires the researcher to augment the original system of equations to reflect the maximum degree of indeterminacy and can therefore be used with standard solution approaches and estimation packages such as Dynare. Our method can also be combined with sophisticated Bayesian techniques to facilitate the transition between the determinacy and indeterminacy regions of the parameter space. One possibility would be to adopt the sequential Monte Carlo algorithm Quantitative Economics 12 (2021) A generalized approach to indeterminacy 845 of Herbst and Schorfheide (2015b)asinHirose, Kurozumi, and Zandweghe (2020)and Ettmeier and Kriwoluzky (2020). In this paper, we instead propose an hybrid Metropolis– Hastings algorithm that builds on a specific example developed in An and Schorfheide (2007) following Giordani, Kohn, and Strid (2010). The algorithm combines the standard Metropolis–Hastings random walk algorithm with a Markov Chain Monte Carlo (MCMC) algorithm in which the proposal distribution is based on a mixture of normals centered on the different posterior modes. This algorithm guarantees that the chain quickly moves to the region of the parameter space with the highest posterior and allows to visit local peaks with more frequency. We combine our solution method with the proposed hybrid Metropolis–Hastings algorithm to estimate the small-scale New–Keynesian (NK) model of Galí (2021) using Bayesian techniques on U.S. data over the period 1982:Q4 until 2007:Q3. Galí’s model extends a conventional NK model to allow for the existence of rational bubbles. An interesting aspect of the model is that it displays up to two degrees of indeterminacy for realistic parameter values. We find that the data support the version of the model with two degrees of indeterminacy, implying that the central bank was not reacting strongly enough to the bubble component. Importantly, we show that the combination of our method with the hybrid algorithm ensures a significantly faster transition to the region of the parameter space with the best model fit relative to a standard Metropolis–Hastings random walk algorithm. Furthermore, in Section 6, we reconsider the NK model of Lubik and Schorfheide (2004) to show that the proposed algorithm also ensures a more efficient exploration of the parameter space when regions of the parameters characterized by different degrees of indeterminacy present a more similar fit by increasing significantly the frequency with which the different regions are visited. Our work is related to the vast literature that studies the role of indeterminacy in explaining the evolution of the macroeconomy. Prominent examples in the monetary policy literature include the work of Clarida, Galí, and Gertler (2000)andKerr and King (1996) that study the possibility of multiple equilibria as a result of violations of the Taylor Principle in NK models. Applying the methods developed in Lubik and Schorfheide (2003) to the canonical NK model, Lubik and Schorfheide (2004) test for indeterminacy in U.S. monetary policy. Using a calibrated small-scale model, Coibon and Gorodnichenko (2011) find that the reduction of the target inflation rate in the United States also played a key role in explaining the Great Moderation, and Arias et al. (2020)support this finding in the context of a medium-scale model àlaChristiano, Eichenbaum, and Evans (2005). More recently, Aruoba, Cuba-Borda, and Schorfheide (2018)studied inflation dynamics at the Zero Lower Bound (ZLB) and during an exit from the ZLB. The paper closest to our is Farmer, Khramov, and Nicolò (2015). The main difference between the two approaches is that our method accommodates the case of both determinacy and indeterminacy while considering the same augmented system of equations. Instead, the method proposed by Farmer, Khramov, and Nicolò (2015) requires rewriting the model based on the existing degree of indeterminacy. With respect to Lubik and Schorfheide (2003), our theoretical results show the characterization of the full set of indeterminate equilibria is equivalent between the two methods. Our novel approach provides a unified methodology to study determinacy 846 Bianchi and Nicolò Quantitative Economics 12 (2021) and indeterminacy of different degrees.1 Lubik and Schorfheide (2004) proposed a baseline solution that minimizes the distance between the impulse responses of the model under indeterminacy and determinacy evaluated at the boundary of the region of determinacy. Given the equivalence between our methods, their baseline solution can be mapped into our representation. However, as explained in Section 2.2, our method naturally suggests a baseline solution that sets the correlations of the fundamental disturbances with the sunspot shocks to zero. Such identification can be equivalently thought as an assumption that fundamental shocks do not have a contemporaneous impact on the variables whose sunspot shocks are considered as drivers of the economy. This identification strategy is reminiscent of the zero restrictions often used in the Structural VAR (SVAR) literature. Importantly, the baseline solution is just meant to provide an intuitive and simple-to-implement benchmark and does not impose a constraint on a researcher who wants to consider alternative solutions. The remainder of the paper is organized as follows. Section 2builds the intuition by using a univariate example in the spirit of Lubik and Schorfheide (2004)anddiscusses how to construct the baseline solution in our method. In Section 3,wepresent the methodology and show that the augmented representation of the LRE model delivers solutions, which under determinacy are equivalent to those obtained using standard solution algorithms, and under indeterminacy to those obtained using the methodology provided by Lubik and Schorfheide (2003, 2004) and Farmer, Khramov, and Nicolò (2015). In Section 4, we describe the hybrid algorithm proposed to facilitate the estimation of models with different degrees of indeterminacy. In Section 5, we apply our method and the proposed hybrid algorithm to estimate the NK model with rational bubbles of Galí (2021). Section 6discusses the advantages of using the proposed hybrid algorithm when estimating models, such as Lubik and Schorfheide (2004), in which the fit across regions of determinacy is similar. We present our conclusions in Section 7.Appendices A–D can be found in the Online Supplementary Material, (Bianchi and Nicolò (2021)). 2. Building the intuition In this section, we consider a univariate example in the spirit of Lubik and Schorfheide (2004) to provide the intuition behind our solution method and to explain how to construct the associated baseline solution. 2.1 A useful example Consider a classical monetary model characterized by the Fisher equation, it= Et(πt+1)+rt, and the simple Taylor rule, it=φππt,whereitdenotes the nominal interest rate, πtrepresents the inflation rate, and φπ>0is a parameter controlling the response of the nominal interest to inflation. We assume that the real interest rate, rt, is given and 1Ascari, Bonomolo, and Lopes (2019) allowed for temporarily unstable paths, while we require all solutions to be stationary, in line with previous contributions in the literature. Quantitative Economics 12 (2021) A generalized approach to indeterminacy 847 described by a mean-zero Gaussian i.i.d. shock.2To properly specify the model, we also define the one-step ahead forecast error associated with the expectational variable, πt, as ηt≡πt−Et−1(πt). Combining the Fisher equation and the simple Taylor rule, we obtain the univariate model Et(πt+1)=φππt−rt(1) Any solution to (1) satisfies πt=φππt−1−rt−1+ηt(2) First, we consider the case φπ>1. Solving (2) forward and recalling the assumptions on rt, it is clear that this case is associated with the determinate solution πt=1 φπ rtηt=1 φπ rt(3) The strong response of the monetary authority to changes in inflation (φπ>1) guarantees that inflation is pinned down as a function of the exogenous real interest rate rt. From a technical perspective, when φπ>1the Blanchard–Kahn condition for uniqueness of a solution is satisfied: The number of explosive roots matches the number of expectational variables, that in this univariate case is one. The second case corresponds to φπ≤1. The solution corresponds to any process that takes the form in (2). Such solution also holds under determinacy, but in that case the central bank’s behavior induces restrictions on the expectation error ηtas a function of the exogenous shock, rt. Instead, when the monetary authority does not respond aggressively enough to changes in inflation (φπ≤1), there are multiple solutions for the inflation rate, πt, each indexed by the expectations that the representative agent holds about future inflation, ηt. Equivalently, the solution to the univariate model is indeterminate: The Blanchard–Kahn solution is not satisfied as there is no explosive root to match the number of expectational variables.3Finally, the fundamental shock rthas a contemporaneous effect on inflation only to the extent that it affects the expectational error ηt. The simple model considered here can be solved with pencil and paper. However, when considering richer models with multiple endogenous variables, indeterminacy represents a challenge from a methodological and computational perspective. Standard software packages such as Dynare do not allow for indeterminacy. Of course, a researcher could in principle code an estimation algorithm herself, following the methods outlined in Lubik and Schorfheide (2004). However, this approach requires a substantial amount of time and technical skills. The researcher would need to write a code that not only finds the solution, but also implements the estimation algorithm. Hence, the result 2In the classical monetary model, the real interest rate results from the equilibrium in labor and goods market, and it depends on the technology shocks. We are considering an exogenous process for the technology shocks and, therefore, we take the process for the real interest rate as given. 3To ensure boundedness, the indeterminate solution requires the forecast error, ηt, to be any covariancestationary martingale difference process. 848 Bianchi and Nicolò Quantitative Economics 12 (2021) Table 1. Blanchard–Kahn condition in the augmented representation. Unstable Roots B-K Condition in Augmented Model (4) Solution Determinacy φπ>1in original model (1) 1 α<11 Satisfied πt=1 φπrtη t=1 φπrt ωt=αωt−1−νt+ηt 1 α>12 Not satisfied – Indeterminacy φπ≤1in original model (1) 1 α<10 Not satisfied – 1 α>11 Satisfied πt=φππt−1−rt−1+νt ηt=νtω t=0 Note: The table reports the regions of the parameter space for which the Blanchard–Kahn condition in the augmented representation is satisfied, even when the original model is indeterminate. is that in practice most of the papers simply rule out the possibility of indeterminacy, even if the model at hand could in principle allow for such a feature. To alleviate these issues, our methodology consists of augmenting the original state space of the model by appending an auxiliary process, which could be either stable or unstable ⎧ ⎪ ⎨ ⎪ ⎩ Et(πt+1)=φππt−rt ωt=1 αωt−1−νt+ηt(4) where ωtis an independent autoregressive process, α∈[02]and νtis a newly defined mean-zero sunspot shock with standard deviation σνand correlation ρνr. Table 1summarizes the intuition behind our approach.4When the original LRE model in (1) is determinate (φπ>1), the auxiliary process must be stationary (1/α < 1), so that the augmented representation in (4) satisfies the Blanchard–Kahn condition.5 In this case, the method of Sims (2001) delivers the same solution for the endogenous variable πtas in equation (3). Importantly, ωtrepresents a separate block and does not impact the endogenous variable πt. Considering the case of indeterminacy (φπ≤1), the original model has one expectational variable, but no unstable root, thus violating the Blanchard–Kahn condition. By appending an explosive autoregressive process (1/α > 1), the solution in this expanded state space is determinate as the Blanchard–Kahn condition is satisfied for the augmented system, even if not for the original model. In particular, the solution for the endogenous variable πtcorresponds to the one in (2) resulting from the methodology of Lubik and Schorfheide (2003)orFarmer, Khramov, and Nicolò (2015). Moreover, stability imposes conditions such that ωtis always equal to zero at any time t, thus requiring to 4We refer the reader to Appendix D for detailed suggestions on the practical implementation of our method. 5The choice of parametrizing the auxiliary process with 1/α instead of αinduces a positive correlation between φπand αthat facilitates the implementation of our method when estimating a model. Quantitative Economics 12 (2021) A generalized approach to indeterminacy 849 impose ω0equal to zero and ηt=νt. Importantly, even in this case (φπ≤1), the solution for the endogenous variable does not depend on the appended autoregressive process. Under this scenario, shocks to the real interest rate have a contemporaneous effect on inflation only through their effect on the sunspot shock, as we explain in more detail below. Finally, in both cases, the auxiliary process ωtconstitutes a separate block that is not mapped into an observable variable. The process ωtonly serves the purpose of providing the necessary explosive roots under indeterminacy and creating a mapping from the sunspot shock to the expectational error. Therefore, the law of motion of the endogenous variables is invariant with respect to the adoption of our method to solve the model. 2.2 Baseline solution Our augmented representation parametrizes the continuum of equilibria under indeterminacy by introducing the standard deviation of the sunspot shock included in the auxiliary processes, σν, and its correlation, ρνr, with the fundamental shock. In this section, we propose a baseline solution that arises naturally in the context of our solution method. Lubik and Schorfheide (2004) proposed a baseline solution that minimizes the distance between the impulse response functions of the model under indeterminacy and determinacy evaluated at the boundary of the region of determinacy. In Section 3,our theoretical results show the equivalence between our methods and, therefore, the possibility of mapping their baseline solution into our representation. However, it is not always immediate to construct the baseline solution proposed by Lubik and Schorfheide (2004), given that the boundaries of the determinacy region are often unknown. In our approach, the baseline solution restricts to zero the correlation ρνr, implying no contemporaneous impact of the fundamental shock, rt, on inflation, πt.InTable1, the indeterminate solution is such that the expectational variable, πt, is predetermined and its contemporaneous deviations from its steady state are only due to the sunspot shock, νt. In other words, the fundamental shock rtcan affect πtonly if it affects the sunspot shock νt. Thus, it seems natural to choose as baseline solution the one associated with no correlation between the sunspot shock and the fundamental shock, rt.Such identification strategy equivalently implies that the fundamental shock does not have a contemporaneous impact on the inflation rate and is therefore reminiscent of the zero restrictions often used in the SVAR literature. The baseline solution represents a useful benchmark, but it is not restrictive: All alternative solutions can be obtained by allowing the correlation between the fundamental and sunspot shock to be different from zero. 3. Methodology We now present the main contribution of the paper generalizing the intuition provided above to a multivariate model with potentially multiple degrees of indeterminacy. Given the general class of LRE models described in Sims (2001), this paper proposes an 850 Bianchi and Nicolò Quantitative Economics 12 (2021) augmented representation, which embeds the solution for the model under both determinacy and indeterminacy. In particular, the augmented representation of the LRE model delivers solutions which under determinacy are equivalent to those obtained using standard solution algorithms, and under indeterminacy to those obtained using the methodology provided by Lubik and Schorfheide (2003, 2004) or Farmer, Khramov, and Nicolò (2015). Consider the following LRE model: 0(θ)Xt=1(θ)Xt−1+Ψ(θ)εt+Π(θ)ηt(5) where Xt∈Rkis a vector of endogenous variables, εt∈Ris a vector of exogenous shocks, ηt∈Rpcollects the pone-step ahead forecast errors for the expectational variables of the system and θ≡vec(01ΨΩεε)∈Θis a vector of structural parameters of the model as well as the covariance matrix of the exogenous shocks. The matrices 0and 1are of dimension k×k, possibly singular, and the matrices Ψand Πare of dimension k×and k×p, respectively. Also, we assume Et−1(εt)=Et−1(ηt)=0. We also define the ×matrix Ωεε ≡Et−1(εtε t)to represent the covariance matrix of the exogenous shocks. Consider a model whose maximum degree of indeterminacy is denoted by m.6The proposed methodology appends to the original LRE model in (5) the following system of mequations: ωt=ωt−1+νt−ηft ≡⎡ ⎢ ⎣ α−1 10  0α−1 m ⎤ ⎥ ⎦(6) where the vector ηft is a subset of the endogenous shocks and the vectors {ωtνtηft } are of dimension m×1.Theequationsin(6) are autoregressive processes whose innovations are linear combinations of a vector of newly defined sunspot shocks, νt,anda subset of forecast errors, ηft ,whereEt−1(νt)=Et−1(ηft )=0.Aswewillshowbelow, the choice of which expectational errors to include in (6) does not affect the solution. The intuition behind the proposed methodology works as in the example considered in the previous section. Let m∗(θ) denote the actual degree of indeterminacy associated with the parameter vector θ. Under indeterminacy the Blanchard–Kahn condition for the original LRE model in (5) is not satisfied. Given that the system is characterized by m∗(θ) degrees of indeterminacy, it is necessary to introduce m∗(θ) explosive roots to solve the model using standard solution algorithms. In this case, m∗(θ) of the diagonal elements of the matrix are assumed to be outside the unit circle (in absolute value), and the augmented representation is therefore determinate because the Blanchard–Kahn condition is now satisfied. On the other hand, under determinacy the 6Denoting by nthe minimum number of unstable roots of a LRE model and pthe number of one-step ahead forecast errors, the maximum degrees of indeterminacy are defined as m≡p−n. When the minimum number of unstable roots of a model is unknown, then mcoincides with number of expectational variables p. This represents the maximum degree of indeterminacy in any model with pexpectational variables. Quantitative Economics 12 (2021) A generalized approach to indeterminacy 857 the model.10 The term qtdenotes the size of an aggregate bubble in the economy (normalized by trend output) relative to its value along the BGP. The aggregate bubble plays the role of demand shifter and is defined as qt=bt+uq t(10) where btdenotes the aggregate value in period tof bubble assets that were already available for trade in period t−1,anduq tis the value of a new bubble at time t. We assume that uq tfollows an exogenous autoregressive process of the form uq t=ρquq t−1+εq t,where εq t iid ∼N(0σ2 q).Equation(11) defines the evolution of the value of the asset bubble qtas qt=ΛEt(bt+1)−qit−Et(πt+1)(11) where q≡γ(β−Λv) (1−βγ)(1−Λvγ) represents the steady state bubble-to-output ratio, Λ≡1/(1+ r) is the steady state stochastic discount factor for one-period ahead payoffs derived from a portfolio of securities and ≡(1+g) is the gross rate of productivity growth. Equation (11) shows how “optimistic” expectations about the future value of the bubble lead to a higher price for the assets today. As shown in Galí (2021), the existence of a BGP with positive asset bubbles requires that Λv < β.Inaddition,toguaranteethat newly created bubbles are nonnegative along the BGP, the model requires that Λ ≥1. Equivalently, these two conditions imply that there exists a continuum of bubbly BGPs indexed by a real interest rate in the range v/β −1<r≤g. The model features a NK Phillips curve πt=ΛvγEt(πt+1)+κyt+us t(12) where us t=ρsus t−1+εs tand εs t iid ∼N(0σ2 s).11 Note that both the dynamic IS curve in (9) and the NK Phillips curve in (12) nest the standard NK model when the probability of death and retirement approach zero (i.e., {vγ}→1), given that Λ =βalong a balanced growth path under the assumption of an infinitely-lived representative consumer with log utility. Finally, the conduct of monetary policy follows an interest rate rule that features some inertia and aims not only at stabilizing inflation, but also at leaning against the bubble: it=ρiit−1+(1−ρi)(φππt+φqqt)+εi t(13) 10ThedynamicIScurvein(9) combines equations (30)∼(34) in Galí (2021) such that the parameters ≡ Λv β∈(01],Ψ≡Υ(1+vγ(1−) (1−βγ) ),Υ≡1−βγ 1−Λvγ ∈(01], and Θ≡(1−βγ)(1−vγ) βγ are function of the following structural parameters: (i) γ, the constant probability of each individual in the OLG model to survive to the next period; (ii) v, the probability of each individual to be employed in the next period; (iii) β,thediscount factor of each individual; (iv) Λ≡1/(1+r), the steady state stochastic discount factor for one-period ahead payoffs derived from a portfolio of securities; (v) ≡(1+g), the gross rate of productivity growth. 11In particular, k≡(1−θ)(1−Λvγθ) θρ,whereθrepresents the Calvo probability that a firm keeps its price unchanged in any given period and ρis the elasticity of hours worked. 858 Bianchi and Nicolò Quantitative Economics 12 (2021) where εi t iid ∼N(0σ2 i). Note that, because of the absence of a trade-off between stabilization of inflation and output gap, the “divine coincidence” still holds in this modified version of the standard NKmodel, and the macroeconomic impact of bubble fluctuations mainly work through aggregate demand. Equations (9)∼(13) describe the equilibrium dynamics of the model economy around a given BGP. We define the vector of variables Xt≡(ytπtbtitqtEt(yt+1) Et(πt+1)Et(bt+1)uq tus t), the vector of fundamental shocks, εt≡(εq tεs tεi t),andthe vector of nonfundamental errors, ηt≡(ηytηπtηbt), where the rational expectation forecast errors are defined as ηxt ≡xt−Et−1[xt]and x={yπb}. AsshowninSection5.3 below, the model of Galí (2021) is characterized by up to two degrees of indeterminacy. Therefore, the proposed methodology augments the model by appending two autoregressive processes ωjt =α−1 jωjt−1+νjt −ηjtj={12}(14) where {η1tη2t}could be any combination consisting of two of the three forecast errors defined by the vector ηt≡(ηytηπtηbt). Hence, defining a new vector of endogenous variables ˆ Xt≡(Xtω1tω2t)and a newly defined vector of exogenous shocks as ˆεt≡ (εtν1tν2t), the system can then be written in canonical form. 5.2 Data and priors We estimate the model to match U.S. data over the period 1982:Q4 until 2007:Q3. We consider three of the macroeconomic quarterly time series used in Smets and Wouters (2007): the growth rate in real GDP, measured as the log change in real GDP, inflation, measured by the log change in the GDP deflator, and the Federal Funds rate. The measurement equations that relate the macroeconomic data to the endogenous variables are defined as ⎡ ⎢ ⎣ log(GDPt) log(Pt) FFRt ⎤ ⎥ ⎦=⎡ ⎢ ⎣ g π∗ r+π∗⎤ ⎥ ⎦+⎡ ⎢ ⎣ yt−yt−1 πt it ⎤ ⎥ ⎦ Table 2reports the prior distributions for the parameters. We calibrate three parameters to guarantee identification. Following Galí (2021), we calibrate the discount factor of each individual, β,to0998 and the probability of surviving to the next period, γ,to 0996. As mentioned when studying equation (11) describing the evolution of the value of the asset bubble qt, the model requires that the real interest rate, r,andthegrowthrate of output, g,satisfyr≤gto ensure that newly created bubbles along the BGP are nonnegative. To ensure that this inequality holds for each draw of the Metropolis–Hastings algorithm, we express the real interest rate, r,asr=λug,whereλu∈(01]defines the ratio between the real interest rate and its upper bound, the rate of output growth. We then calibrate λuto 0.925 and set the prior for the quarterly growth rate of output, g,as a gamma distribution centered at 045. These assumptions imply an annualized growth rate of output of 18% and real interest rate of approximately 165% over the considered period. Quantitative Economics 12 (2021) A generalized approach to indeterminacy 859 Table 2. Prior and posterior distributions of model parameters. Posteriors Priors Mean 90% prob. int. Density Mean Std. Dev. 100(λ−1 l−1)0028 [00190038]Gamma 004 001 κ003 [00300047]Gamma 004 0005 g048 [043053]Gamma 045 004 π∗091 [047148]Gamma 09030 φπ037 [018065]Gamma 1040 φq004 [002009]Gamma 005 002 ρi049 [023076]Beta 050 020 σq119 [058219]Inv. Gam. 100 050 σs011 [009014]Inv. Gam. 030 015 σi012 [010015]Inv. Gam. 030 015 ρq076 [056091]Beta 070 010 ρs087 [077093]Beta 070 010 σνπ029 [025034]U[010]5289 σνy070 [062081]U[010]5289 ϕνπi −060 [−077−034]U[−11]0057 ϕνπq 023 [−025060]U[−11]0057 ϕνπs 053 [034067]U[−11]0057 ϕνyi −040 [−069−007]U[−11]0057 ϕνyq 005 [−043054]U[−11]0057 ϕνys −054 [−070−034]U[−11]0057 ϕνπνy022 [004040]U[−11]0057 Note: The table reports the prior and posterior distributions under two degrees of indeterminacy {νπνy}. As previously discussed, the existence of a BGP with positive asset bubbles requires that v<β/Λ,wherevis the probability that an individual remains “active” by supplying labor and managing the firm, as opposed to “retiring” with probability (1−v). Hence, we express such probability as v=λlβ/Λ,whereλl∈(01). We then center the gamma prior distribution for the term 100(λ−1 l−1)such that the probability of remaining “active,” v,is09973, therefore coinciding with the calibration in Galí (2021). The resulting range of admissible BGPs is indexed by an (annualized) real interest rate r∈(v/β −1g]=(15%18%]. The prior for the slope of the NK Phillips Curve, κ, is set at 004, a value consistent with an average duration of individual prices of 4 quarters in this model. The parameter describing the response of the monetary authority to changes in inflation, φπ, follows a gamma distribution with mean 1and standard error 04. The response to deviations of the bubble relative to its value along the BGP, φq, follows a gamma distribution with mean 005 and standard error 002, corresponding to a region of the parameter space with up to two degrees of indeterminacy. Finally, the autoregressive parameter of the interest rate rule, ρi, follows a beta distribution with mean 05and standard error 02. The prior distribution of the supply and monetary policy shocks are inverse gamma centered at 03with a standard deviation of 015. The inverse gamma prior for the shock 860 Bianchi and Nicolò Quantitative Economics 12 (2021) associated with the creation of a new bubble, εq t, is more agnostic and centered at 1with standard deviation 05. Finally, when we estimate the model under indeterminacy, we specify uniform prior distributions for the standard deviations of the nonfundamental shocks {σνl}where l={π y b}, and their correlations with both the exogenous shocks of the model {ϕνlj}where j={i q s}and between them {ϕνl1νl2}where {l1l2}={π y b}. Importantly, to ensure that the covariance matrix is always positive definite, the joint prior for the covariance matrix is effectively truncated by rejecting the parameter draws that violate this condition. 5.3 Results The hybrid algorithm and the best-fitting model specification We estimate the model using the hybrid algorithm described in Section 4. Using different starting values and applying a numerical optimization procedure, we find the conditional posterior mode in each region of the parameter space: Determinacy, one degree of indeterminacy, and two degrees of indeterminacy. We then use those posterior modes to construct the proposal distribution as in step 6of the hybrid algorithm. In what follows, we consider the specification in which the two auxiliary processes in (14) include the nonfundamental errors associated with inflation and output, {η1tη2t}={ηπt ηyt}. When the algorithm draws a vector of structural parameters θsuch that the model is indeterminate of degree 1, we set only α1to a value within the unit circle in a way that only the nonfundamental shock ηπt is redefined as fundamental. When the algorithm draws θsuch that the model is indeterminate of degree 2, we set both α1and α2to a value within the unit circle such that both nonfundamental shocks {ηπt ηyt}are redefined as fundamental. In Appendix B, we show that the estimation delivers the same posterior distributions of the model parameters regardless of which forecast errors we include in our representation.12 Starting the algorithm at values in each region of the parameter space, we find that the data favor the specification of the model with two degrees of indeterminacy. The algorithm quickly moves to that region of the parameter space and never leaves. This can be explained inspecting the log-posterior mode of the different regions of the parameters space: −3093 with two degrees of indeterminacy, −4494 with one degree of indeterminacy, and −11966 with determinacy. These large differences could in principle represent a problem for traditional MCMC algorithms. In fact, the adoption of the hybrid algorithm (“Mixture”) proposed in Section 4considerably speeds up the transition to the region with two degrees of indeterminacy relative to a standard Metropolis– Hasting random walk algorithm (“Random walk”). To illustrate how the Mixture algorithm helps to ensure a more efficient convergence to the region of the parameter space with the highest posterior, we estimate the model using both algorithms. We simulate 2000 chains by making an initial draw around the posterior mode of either the region of determinacy or indeterminacy of degree 1.For 12As explained in Section 3, the posterior distributions of the model parameters are equivalent up to a transformation of the correlations between the exogenous shocks and the sunspot disturbances considered in each specification. Quantitative Economics 12 (2021) A generalized approach to indeterminacy 861 Figure 1. Distribution of number of draws necessary to cross to the indeterminacy-2 region. each iteration, we count the number of draws necessary for the parameters to cross the two-degree indeterminacy threshold for the first time. If the transition has not occurred after 100,000 iterations, we stop and record this upper bound. Using both the Mixture and the Random walk algorithms, Figure 1reports the histogram of the number of draws necessary to jump to the region of two degrees of indeterminacy when starting from the region of determinacy (left panel) or indeterminacy of degree 1(right panel). Table 3 reports the corresponding summary statistics of the plotted distributions. Both algorithms eventually move to the region of the parameter space that contains the global peak. Once they reach such region, the chains do not jump back to explore the other regions where the fit of the model at the local peaks is substantially worse. However, the Mixture algorithm ensures a considerably faster switch to the region of the parameter space with the global peak.13 For starting values in the region of one-degree indeterminacy, the median number of draws necessary for the Mixture algorithm is nearly 15 times smaller than the corresponding statistic for the Random walk algorithm. Moreover, when we experimented with alternative versions of the model that imposed restrictions on some model parameters, we found that, for the traditional algorithm, the parameters had not crossed the two-degree indeterminacy threshold even after 100,000 iterations. The convergence to one region of the parameter space can be easily checked tracking the behavior of the auxiliary parameters {α1α2}. Both auxiliary parameters are outside the unit circle when the algorithms converge to the region with two degrees of indeterminacy. Across all simulations, when starting from one of the alternative regions, the auxiliary parameters {α1α2}eventually jump and after that they never take values within the unit circle again. In Section 6, we use the model of Lubik and Schorfheide (2004) to show that when the determinate and indeterminate regions of the parameter space present a more similar fit, the hybrid algorithm facilitates the transition between them. We show that when using the hybrid algorithm, the auxiliary parameter frequently 13In Appendix B, Table SII reports the Raftery–Lewis diagnostics for each parameter chain in Galí (2021). Using the hybrid algorithm, all the model parameters quickly converge. 862 Bianchi and Nicolò Quantitative Economics 12 (2021) Table 3. Summary statistics for the distribution of number of draws necessary to cross to the indeterminacy-2 region. Starting value Algorithm Mean Median 5% 95% Determinacy Mixture 89 17 2 165 Random walk 118 72 6 390 Indeterminacy-1 Mixture 90 21 2 201 Random walk 1294 874 48 3822 Note: The table reports summary statistics for the distribution of the number of draws necessary to cross to the region with two degrees of indeterminacy for different starting values and using both the hybrid algorithm (“Mixture”) and the standard Metropolis–Hastings random walk algorithm (“Random walk”). jumps between values within and outside of the unit circle, carrying valuable information about the probability attached to determinacy and ensuring a faster convergence with respect to the traditional random walk algorithm. Parameter estimates and impulse responses Table 2also reports the mean and 90% probability interval of the posterior distribution of the estimated structural parameters. The posterior mean of the slope of the NK Phillips curve is 0038, which in this model is consistent with a probability of roughly 25% that a firm keeps its price unchanged in any given period. The steady-state quarterly growth rate of output, g,isabout048% and the resulting real interest rate, r,is044%. The posterior mean for the inflation rate, π∗, is about 36% on an annual basis. The strength of the responses of U.S. monetary policy to inflation and the bubble was not enough to guarantee a stabilization of the U.S. economy and to avoid that unexpected changes in expectations could drive U.S. business cycles. The posterior mean of the term λlis 09997 such that probability of remaining “active” is v=λlβ/Λ =0997. The mean of the standard error of the bubble component is 119, and larger than that of the supply and monetary policy shocks estimated to be 011 and 012, respectively. The data also provide evidence that the bubble shock is nearly as persistent as the supply shock. The posterior estimate of the standard error related to forecast errors for the output gap is roughly twice as large as that of the sunspot shock associated with the inflation rate. As discussed next, the estimates of the correlations between the sunspot and exogenous shocks of the model and those between the two sunspot shocks are crucial to interpret the contemporaneous impact of each shock and the associated impulse responses. Figure 2plots the impulse response functions of output, inflation, the FFR, and the Real FFR to a one-standard-deviation fundamental and sunspot shocks.14 The solid lines represent the posterior median, while the dark and light shaded areas correspond to the 68% and 90% probability intervals, respectively. In line with the estimated correlations reported in Table 2, we observe that a shock due to the creation of a new bubble generates a positive contemporaneous and persistent effect on inflation. Moreover, the 14The real FFR corresponds to the difference between the FFR and the one-period ahead inflation expectation. Quantitative Economics 12 (2021) A generalized approach to indeterminacy 863 Figure 2. Bayesian impulse response functions. value of asset bubbles is higher over time due to its increasing expected value. The monetary authority responds to these deviations by increasing the nominal FFR, causing a slight slowdown in economic activity over the period shown. The estimated correlations of the supply shock with the sunspot shocks induce both an inflationary and contractionary effects on impact. The persistence of the shock on output is then associated to deflationary effects to which the monetary authority responds by decreasing the FFR. As indicated in Table 2, a monetary policy shock is negatively correlated with both sunspot shocks, implying a negative contemporaneous impact on both inflation and output. The contemporaneous effect of the shock on the FFR combines the downward impact on both inflation and the bubble component with the upward effect of the monetary policy shock itself. The initial positive monetary policy shock is almost perfectly offset by the endogenous component, with the result that the nominal FFR remains close to zero despite the initial shock. However, the Real FFR still increases in response to the positive monetary policy shock, as illustrated in the last column of Figure 2.Thus,apositive monetary policy shock is still contractionary despite the equilibrium behavior of the nominal FFR. The rise in the Real FFR combined with the decrease in expectations about the future value of the bubble generates a downward contemporaneous impact on the value of the asset today (equation (11) in the model). The persistence of these effects on the real economy and the inflation rate requires the monetary authority to adopt an accommodative stance. The behavior of the nominal FFR is reminiscent of an example presented in Galí (2008) in the context of a prototypical purely forward-looking three- 864 Bianchi and Nicolò Quantitative Economics 12 (2021) equation NK model. Galí (2008) showed that if a monetary policy shock has very large effects on the macroeconomy, the monetary policy interest rate can decline in equilibrium, despite the monetary policy shock being positive. The last two panels show the impulse responses to the sunspot shocks, that we assume to be correlated with each other. A positive shock to inflation expectations, νπ, generates a contemporaneous rise in inflation. The Real FFR decreases due to an increase of inflation expectations that more than compensates for the rise in the FFR. The drop in the Real FFR and higher expectations about the future value of the bubble explain the rise in economic activity and a higher current value of the bubble, consistent with a more restrictive monetary policy stance. Finally, a positive sunspot shock to output expectations leads to a rise in economic activity. Such increase is consistent with higher aggregate fundamental wealth in the economy and a lower current and expected value of the bubble.15 Given that the Cholesky decomposition assumes that the output sunspot shock, νy, is orthogonal to the inflation sunspot shock, νπ, a one-standard deviation shock to νyhas no contemporaneous effect on inflation, while it generates mild deflationary effects in the medium term due to a decrease in inflation expectations. On impact, monetary policy responds to the decrease in the current value of the asset bubble and subsequently adopts a more accommodative stance also induced by the deflationary pressure. 6. Advantages of the hybrid algorithm In this section, we discuss the advantages of using the hybrid estimation algorithm for the case in which the fit of the model at the global peak is only marginally better than that at a local peak belonging to a different region of indeterminacy. To provide an example of this case, we estimate the three-equation NK model in Lubik and Schorfheide (2004) over the “post-1982” period using our solution method and hybrid algorithm.16 The parameter space is characterized by a region of determinacy and a region of indeterminacy of degree 1. To solve the model under indeterminacy, we augment it by appending only one auxiliary process. The global peak in the determinate region marginally outperforms the local peak in the region of one-degree indeterminacy. Based on the marginal data density under determinacy (−23681) and indeterminacy (−23684), the resulting posterior probability of determinacy is 505% based on the modified harmonic mean estimator of Geweke (1999). When estimating the model in Lubik and Schorfheide (2004) using the Mixture algorithm, Figure 3plots the resulting bimodal distribution of the parameter ψπthat governs the response of the monetary authority to deviations of the inflation rate from its target. To cautiously verify the converge of the model parameters, we assign a value of 1 to a dummy variable for each draw of structural parameters in the determinate region (α>1) and a value of 0 for draws in the indeterminate region (α≤1). Figure 4reports 15The model in Galí (2021)includestheequationyt=(1−βν)(qt+xt),wherextdenotes aggregate fundamental wealth (i.e., discounted sum of current and future income expected to accrue to currently alive consumers) normalized by trend output. 16We refer the reader to Lubik and Schorfheide (2004) for a detailed description of the standard model. Quantitative Economics 12 (2021) A generalized approach to indeterminacy 865 Figure 3. Posterior distribution of ψπin Lubik and Schorfheide (2004). values of the dummy equal to 1 (dark areas) and evidently shows that, while both algorithms visit the two regions, the Mixture algorithm jumps between them much more frequently ensuring an efficient exploration of the parameter space. In particular, using the hybrid algorithm, the parameter αis greater than 1in 5145% of the accepted draws, therefore containing relevant information related to the probability of determinacy. Finally, in Appendix C, Table SIII reports the Raftery–Lewis diagnostics for each parameter using the two algorithms: The Mixture algorithm cuts the number of draws required for convergence in half.17 7. Conclusions In this paper, we propose a generalized approach to solve and estimate LRE models over the entire parameter space. Our approach accommodates both cases of determinacy and indeterminacy and it does not require the researcher to know the analytic conditions describing the region of determinacy or the degrees of indeterminacy. Figure 4. Jumps between determinacy and indeterminacy in Lubik and Schorfheide (2004). 17We target the 5% quantile, with 1% precision, and 90% probability. 866 Bianchi and Nicolò Quantitative Economics 12 (2021) When a LRE model is characterized by mdegrees of indeterminacy, our approach appends mautoregressive processes whose innovations are linear combinations of a subset of endogenous shocks and a vector of newly defined sunspot shocks. We show that the solution for the resulting augmented representation embeds both the solution obtained under determinacy using standard solution methods and that delivered by solving the model under indeterminacy using the approach of Lubik and Schorfheide (2003) and equivalently Farmer, Khramov, and Nicolò (2015). We pair our solution method with an hybrid MCMC algorithm to estimate the smallscale NK model of Galí (2021). Galí’s model extends a conventional NK model by allowing for the existence of rational bubbles and is characterized by up to two degrees of indeterminacy for realistic parameter values. We estimate the model using U.S. data over the period 1982:Q4–2007:Q3. We find that the data support the version of the model with two degrees of indeterminacy, implying that the central bank was not reacting strongly enough to the bubble component. Finally, we show that our MCMC hybrid algorithm facilitates the transition to the correct area of the posterior and repeated jumps between local peaks of the posterior when these are close in value. Relative to the standard random walk Metropolis–Hastings algorithm, the hybrid algorithm minimizes the possibility of remaining stuck in an area of the posterior characterized by a local peak and substantially improves the speed of convergence. References Adjemian, S., H. Bastani, M. Juillard, F. Karamé, J. Maih, F. Mihoubi, W. Mutschler, G. Perendia, J. Pfeifer, M. Ratto, and S. Villemot (2020), “Dynare: Reference manual version 4. Dynare Working Papers, 1.” https://EconPapers.repec.org/RePEc:cpm:dynare: 001.[844] An, S. and F. Schorfheide (2007), “Bayesian analysis of DSGE models.” Econometric Reviews, 26 (2–4), 113–172. [845,854,855] Arias, J. E., G. Ascari, N. Branzoli, and E. Castelnuovo (2020), “Positive trend inflation and determinacy in a medium-sized new Keynesian model.” International Journal of Central Banking, 16 (3), 51–94. [845] Aruoba, S. B., P. Cuba-Borda, and F. Schorfheide (2018), “Macroeconomic dynamics near the ZLB: A tale of two countries.” Review of Economic Studies, 85, 87–118. [843,845] Ascari, G., P. Bonomolo, and H. F. Lopes (2019), “Walk on the wild side: Temporarily unstable paths and multiplicative sunspots.” The American Economic Review, 109 (5), 1805–1842. [846] Bianchi, F. and G. Nicolò (2021), “Supplement to ‘A generalized approach to indeterminacy in linear rational expectations models’.” Quantitative Economics Supplemental Material, 12, https://doi.org/10.3982/QE949.[846,851] Blanchard, O. J. and C. M. Kahn (1980), “The solution of linear difference models under rational expectations.” Econometrica, 48, 1305–1313. [844]