A Generalized Approach to Indeterminacy in Linear Rational Expectations Models

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A Generalized Approach to Indeterminacy in Linear Rational Expectations Models Francesco Bianchi and Giovanni Nicol`o 2019-033 Please cite this paper as: Bianchi, Francesco, and Giovanni Nicol`o (2019). “A Generalized Approach to Indeterminacy in Linear Rational Expectations Models,” Finance and Economics Discussion Series 2019-033. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2019.033. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

A Generalized Approach to Indeterminacy in Linear Rational Expectations Models (cid:3) Francesco Bianchi Giovanni Nicol(cid:242) Duke University Federal Reserve Board CEPR and NBER 29th April 2019 Abstract We propose a novel approach to deal with the problem of indeterminacy in Linear RationalExpectationsmodels. Themethodconsistsofaugmentingtheoriginalstatespacewith 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 packages without restricting the estimates tothedeterminacyregion. WeapplyourmethodtoestimatetheNew-Keynesianmodelwith rationalbubblesbyGal(cid:237)(2017)overtheperiod1982:Q4until2007:Q3. We(cid:133)ndthatthedata supportthepresenceoftwodegreesofindeterminacy,implyingthatthecentralbankwasnot reacting strongly enough to the bubble component. JEL classi(cid:133)cation: C19, C51, C62, C63. Keywords: Indeterminacy, General Equilibrium, Solution method, Bayesian methods. (cid:3)Theviewsexpressedinthispaperarethoseoftheauthorsanddonotre(cid:135)ecttheviewsoftheFederalReserve Board or the Federal Reserve System. We thank Jonas Arias, Jess Benhabib, Roger Farmer, Fran(cid:231)ois 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. Correspondence: Francesco Bianchi, Duke University, Department of Economics. Email: francesco.bianchi@duke.edu. Giovanni Nicol(cid:242), Board of Governors of the Federal Reserve System. Email: giovanni.nicolo@frb.gov. 1

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 et al. (2018)). Furthermore, 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 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 and Sims(cid:146)(2001) code Gensys. 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 behind our methodology consists of augmenting 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 de(cid:133)ned vector of sunspot shocks. When the Blanchard-Kahn condition for determinacy is satis(cid:133)ed, 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. The law of motion for the endogenous variables is in this case equivalent to the solution obtained using standard solution algorithms (King and Watson (1998), Klein (2000), Sims (2001)). When the modelisindeterminate, theappropriatenumberofappendedautoregressiveprocessesisassumed 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) or, equivalently, Farmer et al. (2015). Our methodology can be used with standard estimation packages such as Dynare. The solution or estimation under indeterminacy is not generally implementable in standard packages. Our method solves this problem by expanding the state space and making sure that in this expanded state space the conditions for determinacy always hold. Thus, our approach allows the researcher to solve and estimate a model under indeterminacy using standard software packages. 2

Our methodology also simpli(cid:133)es the common approach used to deal with indeterminacy. The common procedure requires the researcher to solve the model di⁄erently depending on the area of the parameter space that is being studied. Under indeterminacy, existing methods require to construct the solution ex-post following the seminal contribution of Lubik and Schorfheide (2003) or to rewrite the model based on the existing degree of indeterminacy (Farmer et al. (2015)). In itself, this is not an insurmountable task, but it implies that the researcher interested in a structural estimation of the model 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 re(cid:135)ect the maximum degree of indeterminacy and can therefore be used with no modi(cid:133)cation of the solution approach. Finally, we show that our approach can facilitate the transition between the determinacy and indeterminacy regions of the parameter space. This method works because our auxiliary processes can be used to make more likely a draw that crosses the threshold of determinacy and to keep track of the distance from such threshold. This idea is particularly easy to implement when the threshold of the determinacy region is known. 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 et al. (2000) and Kerr and King (1996), that study the possibility of multiple equilibria as a result of violations of the Taylor Principle in New-Keynesian (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) (cid:133)nd that the reduction of the target in(cid:135)ation rate in the United States also played a key role in explaining the Great Moderation, and Arias et al. (2017) support this (cid:133)nding in the context of a medium-scale model (cid:224) la Christiano et al. (2005). In a similar spirit, Arias (2013) studies the dynamic properties of medium-sized NK models with trend in(cid:135)ation. More recently, Aruoba et al. (2018) study in(cid:135)ation dynamics at the Zero Lower Bound (ZLB) and during an exit from the ZLB. The paper closest to our is Farmer et al. (2015). As explained above, the main di⁄erence 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 et al. (2015) requires us to rewrite the model based on the existing degree of indeterminacy. With respect to Lubik and Schorfheide (2003), the main novelty of our approach is to provide a uni(cid:133)ed approach to study determinacy and indeterminacy of di⁄erent degrees.1 Finally, we deliberately use Dynare in all the examples presented in this paper to show that our method can be combined with standard packages. However, our solution method can be 1Ascarietal.(forthcoming)allowfortemporarilyunstablepaths,whilewerequireallsolutionstobestationary, in line with previous contributions in the literature. 3

combined with more sophisticated estimation techniques such as the ones developed in Herbst and Schorfheide (2015). To show how to use our methodology in practice, we estimate the small-scale NK model of Gal(cid:237) (2017) using Bayesian techniques using U.S. data over the period 1982:Q4 until 2007:Q3. Gal(cid:237)(cid:146)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 (cid:133)nd 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. The remainder of the paper is organized as follows. Section 2 builds the intuition by using a univariate example in the spirit of Lubik and Schorfheide (2004). Section 3 describes the methodology and shows 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 et al. (2015). In Section 4, we provide guidance on how to properly implement our methodology, and suggestions on how it could be used to improve the e¢ ciency of existing estimation algorithms. In Section 5, we apply our theoretical results to estimate NK modelwithrationalbubblesofGal(cid:237)(2017)usingBayesiantechniques. Wepresentourconclusions in Section 6. 2 Building the intuition Before presenting the theoretical results of the paper, this section builds the intuition behind our approach by considering a univariate example similar to the one proposed in Lubik and Schorfheide (2004). While Section 2.1 explains our approach from an analytical perspective, Section 2.2 addresses questions which could arise at the time of its practical implementation. 2.1 A useful example Consider a classical monetary model characterized by the Fisher equation i = E ((cid:25) )+r ; (1) t t t+1 t and the simple Taylor rule i = (cid:30) (cid:25) ; (2) t (cid:25) t 4

where i denotes the nominal interest rate, (cid:25) represents the in(cid:135)ation rate, and (cid:30) > 0 is a t t (cid:25) parameter controlling the response of the nominal interest to in(cid:135)ation. We assume that the real interest rate r is given and described by a mean-zero Gaussian i.i.d. shock.2 To properly specify t the model, we also de(cid:133)ne the one-step ahead forecast error associated with the expectational variable, (cid:25) , as t (cid:17) (cid:25) E ((cid:25) ): (3) t t t 1 t (cid:17) (cid:0) (cid:0) Combining (1) and (2), we obtain the univariate model E ((cid:25) ) = (cid:30) (cid:25) r : (4) t t+1 (cid:25) t t (cid:0) First, we consider the case (cid:30) > 1. Rewriting equation (4), it is clear that this case is associated (cid:25) with the determinate solution, 1 1 (cid:25) = E ((cid:25) )+ r t t t+1 t (cid:30) (cid:30) (cid:25) (cid:25) 1 = r : (5) t (cid:30) (cid:25) where the last equality is obtained by solving the equation forward and recalling the assumptions on r . The strong response of the monetary authority to changes in in(cid:135)ation ((cid:30) > 1) t (cid:25) guarantees that in(cid:135)ation is pinned down as a function of the exogenous real interest r . From a t technical perspective, when (cid:30) > 1 the Blanchard-Kahn condition for uniqueness of a solution is (cid:25) satis(cid:133)ed: The number of explosive roots matches the number of expectational variables, that in this univariate case is one. The second case corresponds to (cid:30) 1. The solution is obtained by combining (4) with (3), and (cid:25) (cid:20) it corresponds to any process that takes the following form (cid:25) = (cid:30) (cid:25) r +(cid:17) : (6) t (cid:25) t 1 t 1 t (cid:0) (cid:0) (cid:0) Thepreviousequationalsoholdsunderdeterminacy, butinthatcasethecentralbank(cid:146)sbehavior induces restrictions on the expectation error (cid:17) . Instead, when the monetary authority does not t respond aggressively enough to changes in in(cid:135)ation ((cid:30) 1), there are multiple solutions for the (cid:25) (cid:20) in(cid:135)ation rate, (cid:25) , each indexed by the expectations that the representative agent holds about t future in(cid:135)ation, (cid:17) . Equivalently, the solution to the univariate model is indeterminate: The t 2Intheclassicalmonetarymodel,therealinterestrateresultsfromtheequilibriuminlaborandgoodsmarket, anditdependsonthetechnologyshocks. Weareconsideringanexogenousprocessforthetechnologyshocks,and therefore we take the process for the real interest rate as given. 5

Blanchard-Kahn solution is not satis(cid:133)ed as there is no explosive root to match the number of expectational variables. The simple model considered here can be solved 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 (cid:133)nds the solution, but also implements the estimation algorithm. Hence, the result 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. The problem that a researcher faces when solving a LRE model under indeterminacy using standard solution algorithms can be easily understood based on the example provided above. Under determinacy, the model already has a su¢ cient number of unstable roots to match the number of expectational variables. However, under indeterminacy, the model is missing one explosive root. Thus, we propose to augment the original state space of the model by appending an independent process which could be either stable or unstable. The key insight consists of choosing this auxiliary processes in a way to deliver the correct solution. When the original model is determinate, the auxiliary process must be stationary so that also the augmented representation satis(cid:133)es the Blanchard-Kahn condition. In this case, the auxiliary process represents a separate block that does not a⁄ect the law of motion of the model variables. When the model is indeterminate, the additional process should however be explosive so that the Blanchard-Kahn condition is satis(cid:133)ed for the augmented system, even if not for the original model. By choosing the auxiliary process in the appropriate way, the solution under determinacy in this expanded state space corresponds to the solution under indeterminacy under theoriginalstatespace. Inwhatfollows,weapplythisintuitiontotheexampleconsideredabove. InSection3, weshowthattheapproachcanbeeasilyextendedtorichermodelstoaccommodate any degree of indeterminacy. Our methodology proposes to solve an augmented system of equations which can be dealt with by using standard solution algorithms such as Sims (2001) under both determinacy and indeterminacy. Consider the following augmented system E ((cid:25) ) = (cid:30) (cid:25) r ; t t+1 (cid:25) t (cid:0) t (7) ( ! t = (cid:11) 1 ! t (cid:0) 1 (cid:0) (cid:23) t +(cid:17) t ; (cid:0) (cid:1) where ! is an independent autoregressive process, (cid:11) [0;2] and (cid:23) is a newly de(cid:133)ned mean-zero t t 2 6

Blanchard-Kahn condition in the augmented representation Unstable Roots B-K condition in Solution augmented model (7) Determinacy (cid:30) > 1 (cid:25) in original model (4) 1< 1 1 Satis(cid:133)ed (cid:25) = 1 r ; ! = (cid:11)! (cid:23) +" (cid:11) t (cid:30) (cid:25) t t t (cid:0) 1 (cid:0) t t 1> 1 2 Not satis(cid:133)ed n - o (cid:11) Indeterminacy (cid:30) 1 (cid:25) (cid:20) in original model (4) 1< 1 0 Not satis(cid:133)ed - (cid:11) 1> 1 1 Satis(cid:133)ed (cid:25) = (cid:30) (cid:25) r +(cid:17) ; ! = 0 (cid:11) f t (cid:25) t (cid:0) 1 (cid:0) t (cid:0) 1 t t g Table 1: The table reports the regions of the parameter space for which the Blanchard-Kahn condition in the augmented representation is satis(cid:133)ed, even when the original model is indeterminate. sunspot shock with standard deviation (cid:27) . (cid:23) Table 1 summarizes the intuition behind our approach. When the original LRE model in (4) is determinate, (cid:30) > 1, the Blanchard-Kahn condition for the augmented representation in (7) is (cid:25) satis(cid:133)ed when 1=(cid:11) < 1. Indeed, for (cid:30) > 1 the original model has the same number of unstable (cid:25) roots as the number of expectational variables. Our methodology thus suggests to append a stable autoregressive process such that 1=(cid:11) < 1. In this case, the method of Sims (2001) delivers the same solution for the endogenous variable (cid:25) as in equation (5) and for the autoregressive t process ! . Importantly, ! is an independent autoregressive process, and its dynamics do not t t impact the endogenous variable (cid:25) . t Considering the case of indeterminacy (i.e. (cid:30) 1), the original model has one expectational (cid:25) (cid:20) variable, but no unstable root, thus violating the Blanchard-Kahn condition. By appending an explosive autoregressive process, the augmented representation that we propose satis(cid:133)es the Blanchard-Kahn conditionand delivers thesame solution asthe oneresulting fromthe methodologyofLubikandSchorfheide(2003)orFarmeretal.(2015)describedbyequation(6). Moreover, stability imposes conditions such that ! is always equal to zero at any time t, and even in this t case, the solution for the endogenous variable does not depend on the appended autoregressive process. Summarizing, the choice of the coe¢ cient 1 should be made as follows. For values of (cid:30) greater (cid:11) (cid:25) than 1, the Blanchard-Kahn condition for the augmented representation is satis(cid:133)ed for values of (cid:11) greater than 1. Conversely, under indeterminacy (i.e. (cid:30) 1) the condition is satis(cid:133)ed (cid:25) (cid:20) 7

when (cid:11) is smaller than 1. The choice of parametrizing the auxiliary process with 1=(cid:11) instead of (cid:11) induces a positive correlation between (cid:30) and (cid:11) that facilitates the implementation of our (cid:25) method when estimating a model. Finally, notethatunderbothdeterminacyandindeterminacy, theexactvalueof1=(cid:11)isirrelevant for the law of motion of (cid:25) . Under determinacy, the auxiliary process ! is stationary, but t t its evolution does not a⁄ect the law of motion of the model variables. Under indeterminacy, ! is always equal to zero. Thus, the introduction of the auxiliary process does not a⁄ect the t propertiesofthesolutioninthetwocases. However, thisprocessservestwoimportantpurposes: It provides the correct number of explosive roots under indeterminacy and creates a mapping between the sunspot shock and the expectation errors. As we will see in Section 3, this result can be generalized and applies to more complicated models with potentially multiple degrees of indeterminacy. 2.2 Choosing (cid:11) Before presenting detailed suggestions for the practical implementation of our method in Section 4, it is useful to provide the intuition for the choice of the parameter (cid:11) in the context of the simple model presented above. First of all, from the discussion above, it should be clear that what matters is only if this parameter is smaller or larger than 1. Its exact value does not a⁄ect the solution for (cid:25) : Thus, if a researcher wants to solve the model only under indeterminacy t (determinacy), it can simply (cid:133)x the parameter to a value smaller (larger) than 1. In this way, standard solution algorithms proceed to solve the model in the augmented state space only when the model under the original state space is characterized by indeterminacy (determinacy). However, aresearchermightwanttoallowforbothdeterminacyandindeterminacywhensolving the model. We consider the following two cases: (1) The analytic condition de(cid:133)ning the region of determinacy are known; (2) The analytic condition de(cid:133)ning the region of determinacy are unknown. We consider the two cases separately. We (cid:133)rst consider the case in which the researcher is able to analytically derive the condition whichde(cid:133)neswhenthemodelisdeterminateorindeterminate. Fortheexampleconsideredinthis section,thiscasecorrespondstoknowingthatwhen(cid:30) 1themodelin(4)isindeterminate. We (cid:25) (cid:20) thus suggest to write the parameter (cid:11) as a function of the parameter (cid:30) so that the augmented (cid:25) representation in (7) always satis(cid:133)es the Blanchard-Kahn condition. In this example, we set (cid:11) (cid:30) . When the original model is determinate ((cid:30) > 1), the appended autoregressive process (cid:25) (cid:25) (cid:17) is stationary because 1=(cid:11) < 1. If the original model is indeterminate ((cid:30) 1), the coe¢ cient (cid:25) (cid:20) 1=(cid:11) is greater than 1 and the appended process is therefore explosive. Hence, when the region of determinacyisknown, theresearchercaneasilychoose(cid:11)suchthattheaugmentedrepresentation 8

always delivers a solution under both determinacy and indeterminacy. Note that in this case (cid:11) is a transformation of (cid:30) and e⁄ectively no auxiliary extra parameters are introduced. (cid:25) There are however instances in which the researcher does not know the exact properties of the determinacy region. In this case, the researcher can start with an arbitrary value of (cid:11) for a given sets of parameters (cid:18). Suppose that the researcher starts with a value less than 1 and (cid:133)nds that the model is indeterminate for the given set of parameters (cid:18). Then, the researcher can just change (cid:11) to a value larger than 1, for example (cid:11) = 1=(cid:11): A similar logic applies to the case 0 with multiple degrees of indeterminacy that we discuss below: If the solution algorithm returns a solution with m degrees of indeterminacy, m explosive auxiliary processes are necessary. 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 ofLREmodelsdescribedinSims(2001), thispaperproposesanaugmentedrepresentationwhich 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 equivalently Farmer et al. (2015). In the following, we generalize the intuition built in the previous section. Consider the following LRE model (cid:0) ((cid:18))X = (cid:0) ((cid:18))X +(cid:9)((cid:18))" +(cid:5)((cid:18))(cid:17) ; (8) 0 t 1 t 1 t t (cid:0) where X Rk is a vector of endogenous variables, " R‘ is a vector of exogenous shocks, t t 2 2 (cid:17) Rp collects the p one-step ahead forecast errors for the expectational variables of the system t 2 and (cid:18) (cid:2) is a vector of parameters. The matrices (cid:0) and (cid:0) are of dimension k k, possibly 0 1 2 (cid:2) singular, and the matrices (cid:9) and (cid:5) are of dimension k ‘ and k p, respectively. Also, we (cid:2) (cid:2) assume E (" ) = 0; and E ((cid:17) ) = 0: t 1 t t 1 t (cid:0) (cid:0) We also de(cid:133)ne the ‘ ‘ matrix (cid:10) ; "" (cid:2) (cid:10) E (" " ); "" t 1 t 0t (cid:17) (cid:0) which represents the covariance matrix of the exogenous shocks. 9

Consider a model whose maximum degree of indeterminacy is denoted by m.3 The proposed methodology appends to the original LRE model in (8) the following system of m equations 1 0 (cid:11)1 ! t = (cid:8)! t 1 +(cid:23) t (cid:17) f;t ; (cid:8) 2 ... 3 (9) (cid:0) (cid:0) (cid:17) 60 1 7 6 (cid:11)m7 4 5 where the vector (cid:17) is a subset of the endogenous shocks and the vectors ! ;(cid:23) ;(cid:17) are of f;t t t f;t dimension m 1. The equations in (9) are autoregressive processes whose innovations are linear (cid:8) (cid:9) (cid:2) combinations of a vector of newly de(cid:133)ned sunspot shocks, (cid:23) , and a subset of forecast errors, t (cid:17) , where E ((cid:23) ) = E ((cid:17) ) = 0. As we will show below, the choice of which expectational f;t t 1 t t 1 f;t (cid:0) (cid:0) errors to include in (9) does not a⁄ect the solution. The intuition behind the proposed methodology works as in the example considered in the previous section. Let m ((cid:18)) denote the actual degree of indeterminacy associated with the parameter (cid:3) vector (cid:18): Under indeterminacy the Blanchard-Kahn condition for the original LRE model in (8) is not satis(cid:133)ed. Given that the system is characterized by m ((cid:18)) degrees of indeterminacy, it (cid:3) is necessary to introduce m ((cid:18)) explosive roots to solve the model using standard solution al- (cid:3) gorithms. In this case, m ((cid:18)) of the diagonal elements of the matrix (cid:8) are assumed to be outside (cid:3) the unit circle (in absolute value), and the augmented representation is therefore determinate because the Blanchard-Kahn condition is now satis(cid:133)ed. On the other hand, under determinacy the (absolute value of the) diagonal elements of the matrix (cid:8) are assumed to be all inside the unit circle, as the number of explosive roots of the original LRE model in (8) already equals the number of expectational variables in the model (m ((cid:18)) = 0). Also, in this case the augmented (cid:3) representation is determinate due to the stability of the appended auxiliary processes. Importantly, as shown for the univariate example in Section 2, the block structure of the proposed methodology guarantees that the autoregressive process, ! , never a⁄ects the solution for the t endogenous variables, X . t Denoting the newly de(cid:133)ned vector of endogenous variables X^ (X ;! ) and the newly de(cid:133)ned t t t 0 (cid:17) vector of exogenous shocks ^" (" ;(cid:23) ), the system in (8) and (9) can be written as t t t 0 (cid:17) (cid:0)^ X^ = (cid:0)^ X^ +(cid:9)^^" +(cid:5)^(cid:17) ; (10) 0 t 1 t 1 t t (cid:0) where 3Denoting by n the minimum number of unstable roots of a LRE model and p the number of one-step ahead forecast errors, the maximum degrees of indeterminacy are de(cid:133)ned as m p n. When the minimum number of (cid:17) (cid:0) unstablerootsofamodelisunknown,thenmcoincideswithnumberofexpectationalvariablesp. Thisrepresents the maximum degree of indeterminacy in any model with p expectational variables. 10

(cid:0) ((cid:18)) 0 (cid:0) ((cid:18)) 0 (cid:9)((cid:18)) 0 (cid:5) ((cid:18)) (cid:5) ((cid:18)) (cid:0)^ 0 ; (cid:0)^ 1 ; (cid:9)^ ; (cid:5)^ n f ; 0 1 (cid:17) " 0 I# (cid:17) " 0 (cid:8)# (cid:17) " 0 I# (cid:17) " 0 I # (cid:0) and without loss of generality the matrix (cid:5) in (8) is partitioned as (cid:5) = [(cid:5) (cid:5) ], where the n f matrices (cid:5) and (cid:5) are respectively of dimension k (p m) and k m.4 n f (cid:2) (cid:0) (cid:2) Section 3.1 and 3.2 show that the augmented representation of the LRE model delivers solutions which under determinacy are equivalent to those obtained using standard solution algorithms, andunderindeterminacytothoseobtainedusingthemethodologyprovidedbyLubikandSchorfheide (2003, 2004) and Farmer et al. (2015). In order to simplify the exposition, when analyzing the case of indeterminacy we assume, without loss of generality, m ((cid:18)) = m. As it will become (cid:3) clear, the case of m ((cid:18)) < m is a special case of what we present below. (cid:3) 3.1 Equivalence under determinacy This section considers the case in which the original LRE is determinate, and shows the equivalence of the solution obtained using the proposed augmented representation with the one from the standard solution method described in Sims (2001). 3.1.1 Canonical solution Consider the LRE model in (8) and reported in the following equation (cid:0) X = (cid:0) X + (cid:9) " + (cid:5) (cid:17) : (11) 0 t 1 t 1 t t k kk 1 k k k (cid:0)1 k ll 1 k pp 1 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) The method described in Sims (2001) delivers a solution, if it exists, by following four steps. First, Sims (2001) shows how to write the model in the form SZ X = TZ X +Q(cid:9)" +Q(cid:5)(cid:17) ; (12) 0 t 0 t 1 t t (cid:0) where (cid:0) = QSZ and (cid:0) = QTZ result from the QZ decomposition of (cid:0) ;(cid:0) , and the k k 0 0 0 1 0 0 0 1 f g (cid:2) matrices Q and Z are orthonormal, upper triangular and possibly complex. Also, the diagonal 4Suppose that (cid:5) (cid:5) 1 (cid:5) 2 (cid:5) 3 . The proposed augmented representation would therefore allow for the k (cid:2) 3(cid:17) h k (cid:2) 1 k (cid:2) 1 k (cid:2) 1 i (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) following three possible alternatives, (cid:5)^ 1 2 3 ; (cid:5)^ 1 2 3 and (cid:5)^ 1 2 3 . In 1 (cid:17) 0 0 1 2 (cid:17) 0 1 0 3 (cid:17) 1 0 0 (cid:20) (cid:0) (cid:21) (cid:20) (cid:0) (cid:21) (cid:20)(cid:0) (cid:21) Appendix B, we show with an analytic example that the alternative representations have a unique mapping that ensures the equivalence among each of them. 11

elements of S and T contain the generalized eigenvalues of (cid:0) ;(cid:0) . 0 1 f g Second, giventhattheQZdecompositionisnotunique, Sims(cid:146)algorithmchoosesadecomposition that orders the equations so that the absolute values of the ratios of the generalized eigenvalues are placed in an increasing order, that is t = s t = s for j > i. jj jj ii ii j j j j (cid:21) j j j j The algorithm then partitions the matrices S; T, Q and Z as S S T T Z Q 11 12 11 12 1 1 S = ; T = ; Z = ; Q = ; 0 " 0 S 22# " 0 T 22# "Z 2# "Q 2# where the (cid:133)rst block corresponds to the system of equations for which t = s 1 and the jj jj j j j j (cid:20) second block groups the equations which are characterized by explosive roots, t = s > 1. jj jj j j j j The third step imposes conditions on the second, explosive block to guarantee the existence of at least one bounded solution. De(cid:133)ning the transformed variables (cid:24) 1;t (cid:24) t Z 0 X t = 2 (k (cid:0) n) (cid:2) 1 3; (cid:17) (cid:24) 2;t 6 n 1 7 6 (cid:2) 7 4 5 where n is the number of explosive roots, and the transformed parameters (cid:9) Q(cid:9); and (cid:5) Q(cid:5); 0 0 (cid:17) (cid:17) the second block can be written ase e (cid:24) = S 1T (cid:24) +S 1((cid:9) " +(cid:5) (cid:17) ): 2;t 2(cid:0)2 22 2;t 1 2(cid:0)2 2 t 2 t (cid:0) As this system of equations contains the explosive rootes of theeoriginal system, then a bounded solution, if it exists, will set (cid:24) = 0 (13) 2;0 n 1 (cid:2) (cid:9) " + (cid:5) (cid:17) = 0; (14) 2 t 2 t n ‘‘ 1 n pp 1 (cid:2) (cid:2) (cid:2) (cid:2) e e where n also denotes the number of equations in (14). A necessary condition for the existence of a solution requires that the number of unstable roots (n) equals the number of expectational 12

variables(p). Inthissection,weareconsideringthesolutionunderdeterminacy,andthisguarantees that there are no degrees of indeterminacy m ((cid:18)) = 0. The su¢ cient condition then requires (cid:3) that the columns of the matrix (cid:5) are linearly independent so that there is at least one bounded 2 solution. In that case, the matrix (cid:5) is a square, non-singular matrix and equation (14) imposes 2 e linear restrictions on the forecast errors, (cid:17) , as a function of the fundamental shocks, " , t t e (cid:17) = (cid:5) 1(cid:9) " : (15) t (cid:0) (cid:0)2 2 t The fourth and last step (cid:133)nds the solution foretheeendogenous variables, X , by combining the t restrictions in (13) and (15) with the system of stable equations in the (cid:133)rst block, (cid:24) = S 1T (cid:24) +S 1((cid:9) " +(cid:5) (cid:17) ) 1;t 1(cid:0)1 11 1;t 1 1(cid:0)1 1 t 1 t (cid:0) = S 1T (cid:24) +S 1 (cid:9) (cid:5) (cid:5) 1(cid:9) " (16) 1(cid:0)1 11 1;t (cid:0) 1 1(cid:0)1 e 1 (cid:0) e1 (cid:0)2 2 t (cid:16) (cid:17) e e e e Using the algorithm by Sims (2001), we can describe the solution under determinacy of the LRE model in (11) with equations (13), (15), and (16). 3.1.2 Augmented representation We now consider the methodology proposed in this paper, and we augment the LRE model in (11) with the following system of m equations 1 0 (cid:11)1 ! t = (cid:8)! t 1 +(cid:23) t (cid:17) f;t ; (cid:8) 2 ... 3 (cid:0) (cid:0) (cid:17) 60 1 7 6 (cid:11)m7 4 5 where (cid:8) is a m m diagonal matrix. As the original model in (11) is determinate, then we (cid:2) assume that all the diagonal elements (cid:11) belong to the interval [1;2]. Therefore, we are appendi ing a system of stable equations, and we show that the solution for the endogenous variables, X , is equivalent to the one found in Section 3.1.1. De(cid:133)ning the augmented vector of endogent ous variables, X^ (X ;! ) and the augmented vector of exogenous shocks ^" (" ;(cid:23) ), the t t t 0 t t t 0 (cid:17) (cid:17) representation that we propose takes the form (cid:0)^ X^ = (cid:0)^ X^ +(cid:9)^^" +(cid:5)^(cid:17) ; (17) 0 t 1 t 1 t t (cid:0) 13

where (cid:0) 0 (cid:0) 0 (cid:9) 0 (cid:5) (cid:5) (cid:0)^ 0 ; (cid:0)^ 1 ; (cid:9)^ ; (cid:5)^ n f ; 0 1 (cid:17) "0 I# (cid:17) "0 (cid:8)# (cid:17) "0 I# (cid:17) " 0 I# (cid:0) and without loss of generality the matrix (cid:5) is partitioned as (cid:5) = [(cid:5) (cid:5) ], where the matrices n f (cid:5) and (cid:5) are respectively of dimension k (p m) and k m. n f (cid:2) (cid:0) (cid:2) We can (cid:133)nd a solution to the augmented representation in (17) by using Sims(cid:146)algorithm. Similarly to the previous section, we follow the four steps which describe the algorithm. First, the solution algorithm performs the QZ decomposition of the matrices (cid:0)^ ;(cid:0)^ and the augmented 0 1 f g representation takes the form S^Z^ X^ = T^Z^ X^ +Q^(cid:9)^^" +Q^(cid:5)^(cid:17) ; (18) 0 t 0 t 1 t t (cid:0) where (cid:0)^ = Q^ S^Z^ and (cid:0)^ = Q^ T^Z^ result from the QZ decomposition of (cid:0)^ ;(cid:0)^ ; and 0 0 0 1 0 0 0 1 f g S 0 S T 0 T Z 0 Q 0 11 12 11 12 1 1 S^ = 0 I 0 ; T^ = 0 (cid:8) 0 ; Z^T = 0 I ; Q^ = 0 I : 2 3 2 3 2 3 2 3 0 0 S 0 0 T Z 0 Q 0 22 22 2 2 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5 Importantly, note that the inner matrices of S^;T^;Z^;Q^ are the same as those which de(cid:133)ne the f g matrices S;T;Z;Q found in the previous section using the canonical solution method. f g Second, the algorithm chooses a QZ decomposition which groups the equations in a stable and an explosive block. Because this section assumes that the original model is determinate and that the diagonal elements of the matrix (cid:8) are within the unit circle, the explosive block corresponds to the third system of equations in (18) which is characterized by explosive roots. Recalling the de(cid:133)nition of the matrices (cid:9)^ and (cid:5)^, the system of equations in the third block is (cid:24) = S 1T (cid:24) +S 1((cid:9) " +(cid:5) (cid:17) ): (19) 2;t 2(cid:0)2 22 2;t 1 2(cid:0)2 2 t 2 t (cid:0) e e The third step imposes conditions to guarantee the existence of a bounded solution. However, the explosive block in (19) is identical to the system of equations found in the previous section. Therefore, the algorithm imposes the same restrictions to guarantee the existence of a bounded solution, that is (cid:24) = 0 (20) 2;0 and as found earlier (cid:17) = (cid:5) 1(cid:9) " : (21) t (cid:0) (cid:0)2 2 t e e 14

Finally, the last step combines these restrictions with the system of equations in the stable block which corresponds to the (cid:133)rst and second systems of equations in (18), (cid:24) = S 1T (cid:24) +S 1 (cid:9) (cid:5) (cid:5) 1(cid:9) " ; (22) 1;t 1(cid:0)1 11 1;t (cid:0) 1 1(cid:0)1 1 (cid:0) 1 (cid:0)2 2 t ! = (cid:8)! +(cid:23) (cid:17) : (cid:16) (cid:17) (23) t t (cid:0) 1 t (cid:0) f;t e e e e Recalling that (cid:24) Z X , the solution in (20) (23) obtained for the augmented representation t 0 t (cid:17) (cid:24) of the LRE model delivers the same solution for the endogenous variables of interest, X , found t in the previous section and de(cid:133)ned in equations (13), (15), and (16). Two remarks should be made when comparing the two solutions. First, as shown in (21), the forecast errors are only a function of the exogenous shocks " , and not of the newly de(cid:133)ned t sunspot shocks, (cid:23) . It is therefore clear that the endogenous variables, X , of the original LRE t t modeldonotrespondtosunspotshockseither, asexpectedunderdeterminacy. Second, (22)and (23) indicate that under determinacy the appended system of equations constitutes a separate block, which does not a⁄ect the dynamics of the endogenous variables, X . Thus, the likelihood t associated with a vector of observables Z that represents a linear transformation of the variables t inX isinvariantwithrespecttothemethodusedtocomputethesolution. Thisstatementholds t because the latent processes do not a⁄ect Z . t 3.2 Equivalence under indeterminacy ThissectionshowstheequivalenceofthesolutionsobtainedforaLREmodelunderindeterminacy using the proposed augmented representation and the methodology of Lubik and Schorfheide (2003, 2004). 3.2.1 Lubik and Schorfheide (2003) As in Section 3.1, we consider the LRE model in (11) and reported below as (24) (cid:0) X = (cid:0) X +(cid:9)" +(cid:5)(cid:17) : (24) 0 t 1 t 1 t t (cid:0) In this section we assume that the model is indeterminate, and we present the method used by LubikandSchorfheide(2003). Theauthorsimplementthe(cid:133)rsttwostepsofthealgorithmbySims (2001) and described in Section 3.1.1.5 They proceed by (cid:133)rst applying the QZ decomposition to the LRE model in (24) and then ordering the resulting system of equations in a stable and 5It is relevant to mention that in this section the matrices obtained from the QZ decomposition and the orderingoftheequationsintoastableandanexplosiveblockdi⁄erfromthoseinSection3.1bothintermsoftheir dimensionality and the elements constituing them. However, we opted to use the same notation for simplicity. 15

an explosive block as de(cid:133)ned in equation (12). However, their approach di⁄ers in the third step when the algorithm imposes restrictions to guarantee the existence of a bounded solution. In particular, the restrictions in (13) and (14) reported below as (25) and (26) require that (cid:24) = 0; (25) 2;0 n 1 (cid:2) (cid:9) " + (cid:5) (cid:17) = 0: (26) 2 t 2 t n ‘‘ 1 n pp 1 (cid:2) (cid:2) (cid:2) (cid:2) e e Nevertheless, it is clear that the system of equation in (26) is indeterminate as the number of forecast errors exceeds the number of explosive roots (p > n). Equivalently, there are less equations (n) than the number of variables to solve for (p). To characterize the full set of solutions to equation (26), Lubik and Schorfheide (2003) decompose the matrix (cid:5)~ using the 2 following singular value decomposition n (cid:5) (cid:2) 2 p (cid:17) n U (cid:2) n h D n (cid:2) 1 n 1 n (cid:2) 0 m i p V (cid:2) 0 p ; e where m represents the degrees of indeterminacy. Given the partition V V 1 V 2 ; p p (cid:17) p n p m (cid:2) (cid:20) (cid:2) (cid:2) (cid:21) equation (26) can be written as D 1 U (cid:9) " + V (cid:17) = 0: (27) 1(cid:0)1 0 2 t 10 t n (cid:2) n n (cid:2) nn (cid:2) ‘‘ (cid:2) 1 n (cid:2) pp (cid:2) 1 e Given that the system is indeterminate, Lubik and Schorfheide (2003) append additional m equations, M " + M (cid:16) = V (cid:17) : (28) t (cid:16) t 20 t m ‘‘ 1 m mm 1 m pp 1 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) The m 1 vector (cid:16) is a set of sufnspot shocks that is assumed to have mean zero, covariance t (cid:2) matrix (cid:10) and to be uncorrelated with the fundamental shocks, " , that is (cid:16)(cid:16) t E[(cid:16) ] = 0; E (cid:16) " = 0; E (cid:16) (cid:16) = (cid:10) : t t 0t t 0t (cid:16)(cid:16) (cid:2) (cid:3) (cid:2) (cid:3) ThematrixM capturesthecorrelationoftheforecasterrors,(cid:17) ,withfundamentals," ,andLubik t t and Schorfheide (2003) choose the normalization M = I . The reason to append the system of (cid:16) m f equations in (28) to the equations in (27) is to exploit the properties of the orthonormal matrix V. Premultiplying the system by the matrix V and recalling that V V = I, the expectational 0 (cid:3) 16

errors can be written as a function of the fundamental shocks, " ; and the sunspot shocks, (cid:16) ; t t (cid:17) = V D 1U (cid:9) + V M " + V (cid:16) : p (cid:2) t 1 (cid:18) (cid:0)p (cid:2) n 1 n (cid:2) 1(cid:0)1 nn (cid:2) 10 nn (cid:2) 2 ‘ p (cid:2) m 2 m (cid:2) ‘ (cid:19) ‘ (cid:2) t 1 p (cid:2) m 2 m (cid:2) t 1 e f More compactly, (cid:17) = V N + V M " + V (cid:16) ; (29) t 1 2 t 2 t p (cid:2) 1 (cid:18) p (cid:2) nn (cid:2) ‘ p (cid:2) mm (cid:2) ‘ (cid:19) ‘ (cid:2) 1 p (cid:2) mm (cid:2) 1 where f N D 1U (cid:9) : n ‘ (cid:17) (cid:0) n 1(cid:0) n 1 n 10 nn 2 ‘ (cid:2) (cid:2) (cid:2) (cid:2) is a function of the parameters of the model. Given thee restriction in (25) and (29), the fourth step in the algorithm combines these equations with the system of stable equations in the (cid:133)rst block as in Section 3.1.1, (cid:24) = S 1T (cid:24) +S 1((cid:9) " +(cid:5) (cid:17) ) 1;t 1(cid:0)1 11 1;t 1 1(cid:0)1 1 t 1 t (cid:0) = S 1T (cid:24) +S 1 (cid:9) +(cid:5) V N +(cid:5) V M " +S 1 (cid:5) V (cid:16) : (30) 1(cid:0)1 11 1;t 1 1(cid:0)1 e 1 e1 1 1 2 t 1(cid:0)1 1 2 t (cid:0) (cid:16) (cid:17) (cid:16) (cid:17) e e e f e Using the method in Lubik and Schorfheide (2003), we can describe the solution for the original LRE model under indeterminacy with equations (25), (29) and (30). 3.2.2 Augmented representation We now consider the augmented representation as in (17) and reported below as (cid:0)^ X^ = (cid:0)^ X^ +(cid:9)^^" +(cid:5)^(cid:17) ; (31) 0 t 1 t 1 t t (cid:0) where X^ (X ;! ), ^" (" ;(cid:23) ) and t t t 0 t t t 0 (cid:17) (cid:17) (cid:0) 0 (cid:0) 0 (cid:9) 0 (cid:5) (cid:5) (cid:0)^ 0 ; (cid:0)^ 1 ; (cid:9)^ ; (cid:5)^ n f : 0 1 (cid:17) "0 I# (cid:17) "0 (cid:8)# (cid:17) "0 I# (cid:17) " 0 I# (cid:0) where the matrix (cid:5) is partitioned as (cid:5) = [(cid:5) (cid:5) ] without loss of generality. n f Thenoveltyofourapproachisthat,givenourrepresentation,wecaneasilyobtainthesolutionby using Sims(cid:146)algorithm even when the original LRE is assumed to be indeterminate. It is enough to assume that the auxiliary processes ! are characterized by explosive roots, or equivalently t that the diagonal elements of the matrix (cid:8) are outside the unit circle. This approach guarantees that the Blanchard-Kahn condition for the augmented representation is satis(cid:133)ed and, given 17

the analytic form that we propose for the auxiliary processes, we show that the solution for the endogenous variables of interest, X , is equivalent to the method of Lubik and Schorfheide t (2003). To show this result, we simply apply the four steps of the algorithm described in Sims (2001) to the proposed augmented representation. First, the QZ decomposition of (31) takes the form S^Z^ X^ = T^Z^ X^ +Q^(cid:9)^^" +Q^(cid:5)^(cid:17) ; (32) 0 t 0 t 1 t t (cid:0) where (cid:0)^ = Q^ S^Z^ and (cid:0)^ = Q^ T^Z^ result from the QZ decomposition6 of (cid:0)^ ;(cid:0)^ and 0 0 0 1 0 0 0 1 f g S S 0 T T 0 Z 0 Q 0 11 12 11 12 1 1 S^ = 0 S 0 ; T^ = 0 T 0 ; Z^T = Z 0 ; Q^ = Q 0 : (33) 2 22 3 2 22 3 2 2 3 2 2 3 0 0 I 0 0 (cid:8) 0 I 0 I 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5 Notethatintheexpressionabovetheauxiliarymatrix(cid:8)isinthelower(explosive)blockbecause of our simplifying assumption that m ((cid:18)) = m: When m ((cid:18)) < m; part of the matrix (cid:8) would (cid:3) (cid:3) belong in the stable block. As mentioned above, we made this simplifying assumption without loss of generality and only to simplify the exposition. Second, the QZ decomposition chosen by the algorithm groups the explosive dynamics of the model in the second and third system of equations in (32), which are reported below as (34) S 0 (cid:24) T 0 (cid:24) Q 0 22 2 = 22 2;t 1 + 2 (cid:9)^^" +(cid:5)^(cid:17) : (34) (cid:0) t t " 0 I#"! t# " 0 (cid:8)#"! t 1# " 0 I# (cid:0) (cid:16) (cid:17) In the third step, the following restrictions are imposed, (cid:24) = 0; (35) 2;0 n 1 (cid:2) ! = 0; (36) 0 m 1 (cid:2) Q 0 2 (cid:9)^^" +(cid:5)^(cid:17) = 0: (37) t t " 0 I# (cid:16) (cid:17) 6Note that the inner matrices of S^;T^;Z^T;Q^ are the same as those which de(cid:133)ne the matrices S;T;ZT;Q f g f g found from the QZ decomposition using the methodology of Lubik and Schorfheide (2003). 18

Recalling the de(cid:133)nition of (cid:9)^ and (cid:5)^ in (31), then equation (37) can be written as (cid:9) 0 (cid:5) (cid:5) 2 n;2 f;2 ^" + (cid:17) = 0; (38) t t " 0 I#(‘+m) 1 " 0 I #p 1 e (cid:2) e e(cid:0) (cid:2) p (‘+m) p p (cid:2) (cid:2) | {z } | {z } where (cid:9) Q(cid:9) and (cid:5) Q(cid:5). Equation (38) shows transparently how the explosive auxiliary 0 0 (cid:17) (cid:17) process that we append in our augmented representation helps to solve the original LRE model e e underindeterminacy. Thesystemofequationsin(38)isdeterminate, asthenumberofequations de(cid:133)ned by the explosive roots of the system equals the number of expectational errors of the model. Thus, the necessary condition for the existence of a bounded solution for the augmented representation is satis(cid:133)ed. Assuming that the columns of the matrix associated with the vector of non-fundamental shocks, (cid:17) , are linearly independent, we can impose linear restrictions on the t forecast errors as a function of the augmented vector of exogenous shocks ^" (" ;(cid:23) ); t t t 0 (cid:17) (cid:5) 1(cid:9) (cid:5) 1(cid:5) (cid:17) = (cid:0)n;2 2 (cid:0)n;2 f;2 ^" : t t (cid:0) " 0 I # e e e (cid:0)e More compactly, (cid:17) = C " +C (cid:23) ; (39) t 1 t 2 t (cid:5) 1(cid:9) (cid:5) 1(cid:5) where C (cid:0)n;2 2 and C (cid:0)n;2 f;2 are a function of the structural parameters of 1 2 (cid:17) (cid:0) " 0 # (cid:17) (cid:0) " I # the model. e e e (cid:0)e The last step of Sims(cid:146)algorithm combines the restrictions in (35), (36) and (39) with the stationary block derived from the QZ decomposition in (32), (cid:24) = S 1T (cid:24) +S 1((cid:9) " +(cid:5) (cid:17) ) 1;t 1(cid:0)1 11 1;t 1 1(cid:0)1 1 t 1 t (cid:0) = S 1T (cid:24) +S 1 (cid:9) +(cid:5) C " +S 1 (cid:5) C (cid:23) : (40) 1(cid:0)1 11 1;t 1 1(cid:0)1 e 1 e1 1 t 1(cid:0)1 1 2 t (cid:0) (cid:16) (cid:17) (cid:16) (cid:17) e e e 3.2.3 Indeterminate equilibria and equivalent characterizations The indeterminate equilibria found using the methodology of Lubik and Schorfheide (2003) are parametrized by two sets of parameters. The (cid:133)rst set is de(cid:133)ned by (cid:18) (cid:2) , where 1 1 2 (cid:18) vec((cid:0) ;(cid:0) ;(cid:9);(cid:10) ) is a vector of structural parameters of the model as well as the co- 1 0 1 "" 0 (cid:17) variance matrix of the exogenous shocks. The second set corresponds to (cid:18) (cid:2) , where 2 2 2 (cid:18) vec (cid:10) ;M 0 is a parameter vector related to the additional equations introduced in 2 (cid:16)(cid:16) (cid:17) (cid:16) (cid:17) f 19

(28) and reported below as (41), M " + M (cid:16) = V (cid:17) : (41) t (cid:16) t 20 t m ‘‘ 1 m mm 1 m pp 1 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) f Given the normalization M = I chosen by Lubik and Schorfheide (2004), equation (41) intro- (cid:16) duces m (m+1)=2 parameters associated with the covariance matrix of the sunspot shocks, (cid:2) (cid:10) ; and additional m ‘ parameters of the matrix M that is related to the covariances between (cid:16)(cid:16) (cid:2) (cid:17) and " . In Appendix A, we show how the normalization chosen by Lubik and Schorfheide t t f (2004) maps one-to-one into a speci(cid:133)c covariance matrix for the exogenous shocks under the methodology proposed in this paper. The characterization of a Lubik-Schorfheide equilibrium is a vector (cid:18)LS (cid:2)LS, where (cid:2)LS is 2 de(cid:133)ned as (cid:2)LS (cid:2) ;(cid:2) : 1 2 (cid:17) f g Similarly, the full characterization of the solutions under indeterminacy using the proposed augmented representation is parametrized by the set of parameters (cid:18) (cid:2) common between the 1 1 2 two methodologies, and the set of additional parameters (cid:18) (cid:2) , where (cid:18) vec((cid:10) ;(cid:10) ). 3 3 3 (cid:23)(cid:23) (cid:23)" 0 2 (cid:17) Using our approach, we also introduce m (m+1)=2 parameters associated with the covariance (cid:2) matrix of the sunspot shocks, (cid:10) ; and m ‘ parameters of the covariances, (cid:10) ; between the (cid:23)(cid:23) (cid:23)" (cid:2) sunspot shock (cid:23) and the exogenous shocks " . A Bianchi-Nicol(cid:242) equilibrium is characterized by t t a parameter vector (cid:18)BN (cid:2)BN, where (cid:2)BN is de(cid:133)ned as 2 (cid:2)BN (cid:2) ;(cid:2) : 1 3 (cid:17) f g Thefollowingtheoremestablishestheequivalencebetweenthecharacterizationsofindeterminate equilibria obtained by using the methodology in Lubik and Schorfheide (2003) and the proposed augmented representation. Theorem 1 Let (cid:18)LS and (cid:18)BN be two alternative parametrizations of an indeterminate equilibrium of the model (cid:0) X = (cid:0) X +(cid:9)" +(cid:5)(cid:17) : 0 t 1 t 1 t t (cid:0) For every BN equilibrium, parametrized by (cid:18)BN, there exists a unique matrix M and a unique matrix (cid:10) such that (cid:18) = vec((cid:10) ;M), and (cid:18) ;(cid:18) (cid:2)LS de(cid:133)nes an equivalent LS equilibrium. (cid:16)(cid:16) 2 (cid:16)(cid:16) 0 1 2 f g 2 f Conversely, for every LS equilibrium, parametrized by (cid:18)LS, there is a unique matrix (cid:10) and a (cid:23)(cid:23) f unique covariance matrix (cid:10) such that (cid:18) = vec((cid:10) ;(cid:10) ), and (cid:18) ;(cid:18) (cid:2)BN de(cid:133)nes an (cid:23)" 3 (cid:23)(cid:23) (cid:23)" 0 1 3 f g 2 equivalent BN equilibrium. 20

Proof. See Appendix A. In the paper Farmer et al. (2015), the authors also show that their characterization of indeterminate equilibria is equivalent to Lubik and Schorfheide (2003). Therefore, the following corollary holds. Corollary 2 Given a parametrization (cid:18)BN of a Bianchi-Nicol(cid:242) indeterminate equilibrium, there exists a unique mapping into the parametrization of an indeterminate equilibrium for Farmer et al. (2015), and vice-versa. Moreover, the following two considerations support Corollary 3 below, which describes a relevant result on the likelihood function of the augmented representation. First, as emphasized in this section, the solution of the model in the augmented state space has a block structure which ensuresthattheevolutionoftheendogenousvariablesinX isnotafunctionoftheautoregressive t processes, ! . Second, note that the appended autoregressive processes in ! only serves the t t purpose of providing the necessary explosive roots under indeterminacy and creating a mapping from the sunspot shocks to the expectational errors. These auxiliary processes are not mapped intotheobservablevariablesthroughthemeasurementequation. Thesetwoconsiderationsimply that the parameters of the matrix (cid:8) introduced with the augmented representation are not identi(cid:133)ed within certain parameter region. The algorithm only requires them to be inside or outside the unit circle. Corollary 3 then follows.7 Corollary 3 Conditional on the existence of a solution, the likelihood function associated with the newly de(cid:133)ned vector of endogenous variables, X^ , does not depend on the additional parat meters included in the augmented representation, (cid:8), and is equivalent to the likelihood function associated with the endogenous variables, X . t While Section 3.1 shows that the augmented representation of the LRE model delivers solutions which under determinacy are equivalent to those obtained using standard solution algorithms, Theorem 1 proves that under indeterminacy the solutions of our methodology are equivalent to those obtained using Lubik and Schorfheide (2003, 2004) and Farmer et al. (2015). This theoretical result is crucial for the application of our methodology to the New-Keynesian (NK) model with rational bubbles of Gal(cid:237) (2017) in Section 5. 7NoticethatCorollary3holdswhentheaugmentedrepresentationhasauniquesolution. Thishappensintwo cases. First,valuesofthestructuralparameters(cid:18) whichguaranteedeterminacyintheoriginalLRE modelshould be combined with values for (cid:11) in the matrix (cid:8) whose absolute value lies within the unit circle. Second, values i of the structural parameters (cid:18) for which the original model is indeterminate should be combined with (absolute) values of (cid:11) outside the unit circle. i 21

4 Estimation In this section, we present some suggestions for the practical implementation of our method with an emphasis on the use of standard software packages such as Dynare. We consider both the model and the data that Lubik and Schorfheide (2004) use to test for indeterminacy in U.S. monetary policy, as it is possible to derive analytically the boundary of the determinacy region and to estimate the model under determinacy and indeterminacy. The model is described by equations (42) (47) below and consists of a dynamic IS curve, (cid:24) y = E (y ) (cid:28) (i E ((cid:25) ))+g ; (42) t t t+1 t t t+1 t (cid:0) (cid:0) a NK Phillips curve, (cid:25) = (cid:12)E ((cid:25) )+(cid:20)(y z ); (43) t t t+1 t t (cid:0) and a Taylor rule, i = (cid:26) i +(1 (cid:26) )( (cid:25) + (y z ))+" : (44) t i t 1 i 1 t 2 t t i;t (cid:0) (cid:0) (cid:0) The demand shock, g , and the supply shock, z , follow univariate AR(1) processes t t g = (cid:26) g +" ; (45) t g t 1 g;t (cid:0) z = (cid:26) z +" ; (46) t z t 1 z;t (cid:0) where the standard deviations of the fundamental shocks " , " and " are denoted by (cid:27) , (cid:27) g;t z;t i;t g z and (cid:27) , respectively. As in Lubik and Schorfheide (2004), we allow for the correlation between i demand and supply shocks, (cid:26) , to be nonzero. The rational expectation forecast errors are gz de(cid:133)ned as (cid:17) y E [y ]; (cid:17) (cid:25) E [(cid:25) ]: (47) y;t t t 1 t (cid:25);t t t 1 t (cid:17) (cid:0) (cid:0) (cid:17) (cid:0) (cid:0) We de(cid:133)ne the vector of endogenous variables as X (y ;(cid:25) ;i ;E (y );E ((cid:25) );g ;z ), the t t t t t t+1 t t+1 t t 0 (cid:17) vectors of fundamental shocks and expectation errors, " t = (" i;t ;" g;t ;" z;t ) 0 ; (cid:17) t = (cid:17) y;t ;(cid:17) (cid:25);t 0 (cid:0) (cid:1) and the vector of parameters (cid:18) = 1 ; 2 ;(cid:26) i ;(cid:12);(cid:20);(cid:28);(cid:26) g ;(cid:26) z ;(cid:27) g ;(cid:27) z ;(cid:27) i ;(cid:26) gz 0: We can therefore represent the model as in the following equation, (cid:0) (cid:1) (cid:0) ((cid:18))X = (cid:0) ((cid:18))X +(cid:9)((cid:18))" +(cid:5)((cid:18))(cid:17) : (48) 0 t 1 t 1 t t (cid:0) 22

The LRE model in (48) is determinate when the following analytic condition is satis(cid:133)ed, (1 (cid:12)) j (cid:3) j (cid:17) 1 + (cid:0) (cid:20) 2 > 1: (49) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) When the model is indeterminate, 0 < (cid:12) (cid:3) 1, the sy(cid:12)stem is characterized by one degree of j j (cid:20) indeterminacy (m = 1) because there are two expectational variables (E (y ) and E ((cid:25) )) t t+1 t t+1 and at most one root outside the unit circle. Thus, to implement our methodology we need to augment the original state space of the model in (48) with the autoregressive process 1 ! = ! +(cid:23) (cid:17) : (50) t (cid:11) t (cid:0) 1 t (cid:0) (cid:25);t (cid:18) (cid:19) where, without loss of generality, we have parameterized the auxiliary process with respect to (cid:17) .8 We then de(cid:133)ne a new vector of endogenous variables X^ (X ;! ) and a newly de(cid:133)ned (cid:25);t t t t 0 (cid:17) vector of exogenous shocks as ^" (" ;(cid:23) ) = (" ;" ;" ;(cid:23) ). The system in (48) and (50) can t t t 0 i;t g;t z;t t 0 (cid:17) now be written as (cid:0)^ X^ = (cid:0)^ X^ +(cid:9)^^" +(cid:5)^(cid:17) : (51) 0 t 1 t 1 t t (cid:0) As in Lubik and Schorfheide (2004), we estimate the model using Bayesian methods using three series as observables: The percentage deviations of (log) real GDP per capita from an HP-trend (y ), the annualized percentage change in the Consumer Price Index for all Urban Consumers obs;t ((cid:25) ), and the annualized Federal Funds Rate (i ). We focus on the data for the pre-Volcker obs;t obs;t period (1960Q1 - 1979Q2) as Lubik and Schorfheide (2004) show that during this period the monetary authority did not respond aggressively enough to changes in in(cid:135)ation, thus not suppressing self-ful(cid:133)lling in(cid:135)ation expectations. We repeat the estimation of the model of Lubik and Schorfheide (2004) by adopting the same prior distributions for the structural parameters. The Bayesian estimation is conducted using conventional Metropolis-Hastings algorithm in Dynare. Priors for the auxiliary parameters. As a (cid:133)rst step, we discuss how we choose the prior distribution for the additional parameters introduced under our methodology. Our augmented representationintroducesthevectorofparameters(cid:11)andparametrizesthecontinuumofequilibria underindeterminacybyintroducingthestandarddeviationofasunspotshockanditscorrelations with the exogenous shocks. Regarding the vector of parameters (cid:11), we can distinguish three cases. Case 1: When the determinacy threshold is known, then (cid:11) can be expressed as a function of the other parameters. 8In Appendix B, we show an analytic example of the unique mapping that exists between the alternative representations that can be considered using our augmented representation. We therefore ensure that the representationsare equivalentup to a transformation ofthe correlationsbetween the exogenousshocksand the forecast error included in the auxiliary process. 23

In this case, there is no need to specify a prior on (cid:11) and the prior probability of determinacy is given by the prior on the parameter vector (cid:18). Case 2: When the threshold is unknown and the researcher writes her own code, she can start with all the roots inside the unit circle for (cid:11) at each draw of (cid:18) and then (cid:135)ip the appropriate number of elements in the vector (cid:11). Thus, even in this case, there is no need to specify a prior on (cid:11) and the prior probability of indeterminacy depends on the prior on the parameter vector (cid:18). Case 3: The researcher is using Dynare and the region of the parameter state is unknown. In this case, we suggest to choose priors that are symmetric between the two regions, i.e. that attach 50% probability to determinacy, and orthogonal with respect to the priors on the other parameters. In what follows, we focus on Case 3 and discuss how to proceed under the assumption that the researcherdoesnotknowtheregionofdeterminacyandmightbeinterestedinusinganestimation package such as Dynare. For a given draw of the structural parameters, the researcher would like to make draws of (cid:11) smaller or greater than1with equal probabilities. In this case, the researcher could use a uniform distribution over the interval [0;2] or any symmetric interval around 1 as a prior distribution.9 Note that when the determinacy region is not known, the e¢ ciency of the algorithm can be improved when the researcher writes her own estimation/solution algorithm. We describe how to improve the e¢ ciency of traditional MCMC algorithms in the subsection (cid:147)E¢ ciency(cid:148)below. The correlations of the sunspot shocks with the exogenous disturbances are crucial parameters that a⁄ect the (cid:133)t of the model. In line with the theoretical results, a given set of correlations under the representation that includes the forecast error for the in(cid:135)ation rate has a unique mapping to (di⁄erent values of) the correlations in the representation with the forecast errors for the output gap, and vice versa. Therefore, in order for the two representations to deliver the same (cid:133)t to the data, a researcher has to leave the correlations unrestricted. One simple option is to set a uniform prior distributions over the interval ( 1;1) for the correlations of the sunspot (cid:0) shocks. This approach guarantees that the (cid:133)t of the model does not depend on which forecast error is included in the auxiliary process.10 This is the approach that we follow in the estimation of the model of Gal(cid:237) (2017) in Section 5. Lubik and Schorfheide (2004) center the prior distributions for the additional parameters introducedintheirrepresentationonvaluesthatminimizethedistancebetweentheimpulseresponses of the model under indeterminacy and determinacy evaluated at the boundary of the region of 9Notethatiftheresearcherwritesherowncode,thepriordistributiondoesnotnecessarilyhavetobecontinuous. A discrete probability distribution that allows to make draws of (cid:11) to be either equal 0:5 or 1:5 could also be speci(cid:133)ed as a prior. 10Similarly, imposing a zero correlation between the sunspot shock and the exogenous shocks under a given representation does not map into an assumption of zero correlation under the alternative representations. In this case, the (cid:133)t of the model di⁄ers across the di⁄erent speci(cid:133)cations. 24

determinacy. Thus, they estimate parameters controlling deviations from this centering point. We showed the mapping between the two representations in Section 3 and we illustrate this equivalence with an analytical example in Appendix B. Given this equivalence, the priors for the correlations between sunspot shocks and structural shocks could also be speci(cid:133)ed in a way to replicate the approach of Lubik and Schorfheide (2004). Speci(cid:133)cally, we could specify a prior on the auxiliary matrices used in Lubik and Schorfheide (2004) and then map the matrices into the correlations used in our approach. However, to easily implement our approach using standard packages, we suggest choosing a (cid:135)at prior. Thus, we do not center our priors on the impulse response at the boundary of the region of determinacy, but still cover this case as equally likely with respect to the others. Convergence. We are interested in showing that the methodology allows a standard estimation algorithm such as the one implemented in Dynare to travel to the correct region of the parameter space. At the same time, we also want to emphasize the importance of conducting standard convergence diagnostics. To achieve these goals, we set the initial parametrization in the (cid:147)wrong(cid:148)region of the parameter space and consider 1,000,000 draws to show that the methodology accommodates the case of determinacy and indeterminacy, as well as to highlight the importance of checking convergence before interpreting the estimation results. Figure 1 reports theposteriordistributionfortheparameter and(cid:11)obtainedforaninitialparametrizationsuch 1 that the Taylor Principle holds (i.e. we set = 2). At (cid:133)rst glance, the posterior distribution 1 of the parameter would appear to be bimodal. This (cid:133)nding is consistent with the fact that 1 the proposed augmented representation allows the Metropolis-Hastings algorithm to visit both regions of the parameter space. At the same time, the posterior distribution for the parameter (cid:11) is very similar to the prior distribution, which is speci(cid:133)ed as a uniform distribution over the interval [0;2]. Such result conveys the same evidence derived from the posterior for because 1 the algorithm explores both regions by considering draws of (cid:11) which are within as well as outside the unit circle. A researcher should then verify the occurrence of either of the following two circumstances. This bimodal distribution could arise because the log-likelihood is highly discontinuous between the two regions. In this case, the algorithm could have jumped towards the region where the peak of the posterior lies, without having spent a signi(cid:133)cant time there. In other words, convergence has not occurred yet. Alternatively, if the log-likelihood function varies smoothly between the two regions of the parameter space, the posterior distribution plotted in Figure 1 could be the result of the algorithm traveling across the two regions multiple times. Wethereforerecommendtheresearchertoanalyzethedrawsoftheparameter(cid:11)whichhavebeen accepted during the MCMC algorithm. By inspecting the behavior of the auxiliary parameter (cid:11), a researcher can detect if the algorithm reached convergence or not. We report the draws 25

Posterior distribution of parameter and (cid:11) 1 5 1.5 4 1 3 2 0.5 1 0 0 0 1 2 3 0 0.5 1 1.5 2 1 (cid:11) Figure 1: Initial parametrization = 2. The grey line represents the prior distribution and the 1 black line is the posterior distribution. that we obtained during our exercise in Figure 2. After approximately 400,000 draws of (cid:11) in the region of determinacy (i.e. outside the unit circle), the algorithm jumps to the indeterminate region and never visits the determinacy region again. Draws of the parameter (cid:11) 2 1.5 1 0.5 0 0 2 4 6 8 10 5 10 Figure 2: Sequence of draws for (cid:11) given an initial parametrization = 2. 1 Thus,Figure1and2suggestthatthealgorithmisinfactabletojumptowardthecorrectregionof the parameter space, but also that convergence has not occurred yet. Therefore, the researcher should repeat the estimation exercise, increase the number of draws, and make sure that the parameter (cid:11) stabilizes on one region of the parameter space. Under di⁄erent circumstances, 26

the researcher could face the second scenario, for which the log-likelihood function transitions smoothly between the two regions. In this case, the parameter (cid:11) would repeatedly transition between the two areas of the parameter space and could be used to infer the probability attached to determinacy. Below we discuss how our methodology can be used to e¢ ciently facilitate such transition. Only (in)determinacy. In some cases, a researcher might want to estimate the model exclusively under determinacy or exclusively under indeterminacy. Our approach easily accommodates this need. If the researcher is only interested in the solution under determinacy, the parameter vector of (cid:11) should be chosen in a way to guarantee stationarity of the auxiliary process (for example, (cid:133)xing all values of the alphas to 2). Furthermore, all parameters that are relevant only under indeterminacy should be (cid:133)xed to zero or any other constant, given that they do not a⁄ect the (cid:133)t of the model under determinacy. If instead the researcher is only interested in estimating themodelunderindeterminacy, theparametersoftheauxiliaryprocesscanbechoseninawayto guarantee that the correct number of explosive roots are provided. In this case, the parameters describing the properties of the sunspot disturbances should also be estimated. Model comparison. A researcher might also be interested in comparing the (cid:133)t of the model under determinacy and under indeterminacy. Model comparison can be conducted by using standard techniques, such as the harmonic mean estimator proposed by Geweke (1999). If the researcher is interested in comparing the same model under determinacy and under indeterminacy, we recommend the following procedure that adapts the approach used by Lubik and Schorfheide (2004): 1. Estimate the model under determinacy by (cid:133)xing the parameter(s) alpha to a value larger than one in a way that the model is solved only under determinacy. Note that in this case allparametersthatpertaintothesolutionunderindeterminacy,suchasthevolatilityofthe sunspot shocks and its correlations with the exogenous shocks, should be restricted to zero (or any other constant). This restriction avoids penalizing the model for extra parameters that do not a⁄ect its (cid:133)t under determinacy. 2. Estimatethemodelunderindeterminacyby(cid:133)xingtheparameter(s)alphatoavaluesmaller than one in a way that the model is solved only under indeterminacy. Note that in this case all parameters that pertain to the solution under indeterminacy, such as the volatility of the sunspot shocks and its correlations, should be estimated. 3. Use standard methods to compare the (cid:133)t of the model under determinacy with the (cid:133)t of the same model under indeterminacy. E¢ ciency. As mentioned above, our method can also be used to make traditional MCMC 27

algorithms more e¢ cient. We have left the discussion of this point last because, unlike the procedures described above, it generally requires the researcher to write its own code. The key ideaisthatinmanycasestheauxiliaryparameter(cid:11)canbeusedtosummarizethedistanceofthe current parameter vector from the threshold separating determinacy and indeterminacy regions. This approach is substantially easier when the partition of the parameter space is known, as in Lubik and Schorfheide (2004). However, the idea can be used even when the region of the parameterspaceisnotknown. Forthesakeofpresentingthegeneralidea, wefocusontheformer case. Asillustratedabove,ourmethodprescribestosettheparameter(cid:11)toavaluesmallerthan1when the original model presents indeterminacy. Therefore, the value of this auxiliary parameter can beconsideredanindicatorvariableforthepresenceofindeterminacyintheoriginalmodel. When determinacy or indeterminacy depends on a large number of parameters, access to this indicator variable can facilitate transition between the two regions of the parameter space. To see why, suppose that indeterminacy depends on k parameters and that the threshold for indeterminacy is known. Then, we can easily obtain a draw for (cid:11) and k 1 of the parameters that control (cid:0) determinacy, check whether the drawn (cid:11) implies determinacy or indeterminacy, and, (cid:133)nally, solve for the k th parameter. Therefore, the probability of jumping between the two regions (cid:0) is controlled by a single parameter. The proposal distribution for this parameter can then be chosen to ensure that once it approaches the threshold, the proposal distribution is such that a jump is more likely. Instead, in the standard approach the k parameters are drawn without consideration of how far the current parameter vector is from the threshold separating the two areas of the parameter space. There are of course many possible ways to choose the proposal distribution for (cid:11) in a way that jumps between the two regions is more likely. One simple way consists of choosing a mixture of normals and then using a standard Metropolis-Hastings algorithm that corrects for the asymmetry in the proposal distribution. In what follows, we present this approach in the context of the model of Lubik and Schorfheide (2004). Let(cid:146)s choose (cid:11) in a way that every draw implies existence and uniqueness of a solution in the augmented parameter space: 1 (cid:12) (cid:11) + (cid:0) (cid:17) 1 (cid:20) 2 In this case, determinacy depends on four parameters, ; ;(cid:20);(cid:12) : However, we know that 1 2 f g whenever (cid:11) > 1, these four parameters are such that determinacy holds, while whenever (cid:11) < 1; the model is in the indeterminacy region. Thus, we could implement an MCMC algorithm in 28

which the proposal distribution draws (cid:11); ;(cid:20);(cid:12) and then derives : 2 1 f g 1 (cid:12) = (cid:11) (cid:0) : 1 (cid:0) (cid:20) 2 Let d (cid:11) (cid:11) = (cid:11) 1 be the distance between the current value for (cid:11) and the boundary (cid:17) (cid:0) (cid:0) of the determinacy region for this auxiliary parameter, (cid:11) 1. Note that when the distance is (cid:17) negative,(cid:11)isbelowthethresholdandthemodelisunderindeterminacy. Supposethatwespecify the proposal distribution to be a mixture of normals: One centered on the current parameter value and one just beyond the threshold of the determinacy region. Speci(cid:133)cally, we assume the following proposal distribution for the proposed draw (cid:11); given the current value, (cid:11) : n e N 1 sign(d)(cid:22) ;(cid:27)2 with probability w (cid:11) (cid:0) c c (cid:24) ( (cid:0) N (cid:11) n ;(cid:27)2 (cid:11) (cid:1) with probability 1 w (cid:0) w = K exp( d K ); w = exp( d K ) e 0 (cid:0) 1 (cid:1) m 2 (cid:0)j j (cid:0)j j (cid:22) = (1 w )(cid:22) +w :5 c m b m (cid:0) (cid:3) (cid:27)2 = (1 w )(cid:27) +w :1 m b m (cid:0) (cid:3) where K is a parameter between 0 and 1 that controls the maximum weight on the auxiliary 0 normal and K > 0 is a parameter controlling the speed with which the weight on the auxili- 1 ary normal goes to zero as the MCMC algorithm gets further from the threshold region. The parameters (cid:22) > 0 and (cid:27) > 0 control the position and shape of the auxiliary normal. These, b b in turn, depend on the parameter K > 0 that makes sure that as the current alpha approaches 2 the threshold of the determinacy region, the location and shape of the auxiliary distribution are adjusted accordingly. The parameter (cid:27) controls the variance for the typical normal proposal (cid:11) distribution centered on the current value of the parameter. Note that the typical Metropolis- Hastings algorithm would have w = 0 implying that the proposal distribution does not vary in response to the distance from the boundary of the determinacy region. Figure 3 presents the proposal distribution for di⁄erent values of the current (cid:11) : To facilitate the n interpretation of the graphs, the top panel plots the proposal distribution for a series of values implyingdeterminacy,whilethelowerpanelconsidersaseriesofvaluesimplyingindeterminacy.11 When (cid:11) is far from the threshold separating the two regions (dotted black line), the proposal n distribution is symmetric. As the current (cid:11) becomes closer to 1, the weight on the auxiliary n normal increases and more and more mass is assigned to drawing a value of (cid:11) that implies a jump between the two regions. Furthermore, as the the current (cid:11) gets closer to 1, the mean of n 11We use these hyperparameters K =:5; K =2; K =10; (cid:22) =:01; (cid:27) =:01: 0 1 2 b b 29

Figure 3: Proposal distribution for di⁄erent values of (cid:11): The proposal distribution is chosen to facilitate crossing the determinacy threshold and is obtained with a mixture of normals. The upper (lower) panel assumes that (cid:11) is currently above (below) the threshold of the determinacy region. the auxiliary normal distribution moves further from the determinacy threshold. To understand how the algorithm helps in crossing the determinacy threshold, we estimate the model of Lubik and Schorfheide (2004) for the post-1982 period using the modi(cid:133)ed Metropolis algorithm involving the parameter (cid:11) and the traditional algorithm that only involves the model parameters and a symmetric proposal distribution. We start the two algorithms 1,000 times by making a draw from the posterior mode. For each iteration, we count the number of draws necessary for the parameters to cross the determinacy threshold for the (cid:133)rst time. We stop when the algorithm has reached 100,000 iterations. Figure 4 reports the distribution for the number of draws necessary to cross the determinacy threshold for the (cid:133)rst time in the two cases. The blue/dark colored bars correspond to the algorithm implemented by drawing values for the auxiliary parameter (cid:11) and then using the value of (cid:11) to obtain the corresponding value of . Instead, the yellow/light colored bars correspond 1 to the traditional algorithm that makes draws for the original parameter space. The distribution is truncated at 100,000 draws. From the graph, it is clear that the modi(cid:133)ed algorithm greatly facilitates crossing the determinacy region. The median value for the number of draws necessary to cross the determinacy region is only 16;555 for the modi(cid:133)ed algorithm. Instead, for the traditional algorithm in 74;2% of the cases the parameters have not crossed the determinacy threshold after 100;000 iterations. Finally, we also verify that the modi(cid:133)ed MCMC algorithm is able to repeatedly jump back and 30

Figure 4: The (cid:133)gure reports the distribution for the number of draws necessary to cross the determinacy threshold for the (cid:133)rst time when using a Metropolis-Hastings algorithm to estimate themodelofLubikandSchorfheide(2004). Twocasesareconsidered. Inthe(cid:133)rstcase(blue/dark coloredbars),thealgorithmisimplementedbydrawingvaluesfortheauxiliaryparameter(cid:11). The value of (cid:11) is then used to obtain the corresponding value of . In the second case (yellow/light (cid:25) colored bars), the algorithm is implemented by drawing directly the parameters of the model. The distribution is truncated at 100,000 draws. forth between the two regions of the parameter space. We then make 2;100;000 draws from the posteriorusingthemodi(cid:133)edalgorithm. We(cid:133)ndthatthealgorithmtransitionsatotalof34times between the two regions. The posterior probability of being under determinacy, computed as the fraction of draws for which (cid:11) > 1, is 98:9%: Therefore, the algorithm is able to explore the entireareaoftheparameterspacedespitethefactthatthedeterminacyregionisoverwhelmingly favored by the data. Whenconductingthesameexerciseforthepre-1979period,wefoundthatthealgorithmwasable to quickly move to the indeterminacy region independently of the starting point. However, once the algorithm had reached such region, it was not able to leave it because of a large discontinuity in the likelihood. This result is important to highlight that while our approach can facilitate the transitionacrossregions,itcannotovercomethefactthatforsomemodelsandsomedatasamples the boundary of the determinacy regions might imply a large discontinuity in the posterior. In this case, jumping between the two regions becomes extremely unlikely, even when a clever proposal distribution is used. In these cases, more recent methods, such as the ones described in Herbst and Schorfheide (2015), can be used to make sure that the entire parameter space is explored. However, for the example considered in this paper, it is worth emphasizing that the conclusions of the analysis are unlikely to change because the lack of jumps between the two 31

regions re(cid:135)ects the fact that the data strongly favor indeterminacy. 5 Monetary Policy and Asset Bubbles In this section, we implement the proposed methodology to estimate the small-scale NK model of Gal(cid:237) (2017) using Bayesian techniques. The model extends a conventional NK model to allow for the existence of rational expectations equilibria with asset price bubbles. Interestingly, the model displays up to two degrees of indeterminacy for realistic parameter values. We estimate the model using U.S. data over the period 1982:Q4 until 2007:Q3, and we consider the case that the U.S. monetary policy aimed at stabilizing the in(cid:135)ation rate and leaning against the bubble. We (cid:133)nd that the strength of such responses 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. In particular, we show that the model speci(cid:133)cation that provides the best (cid:133)t to the data is characterized by two degrees of indeterminacy.12 5.1 The Model The model of Gal(cid:237) (2017) is described by the following equations. First, equation (52) represents a dynamic IS curve y = (cid:8)E (y ) (cid:9)(i E ((cid:25) ))+(cid:2)q ; (52) t t t+1 t t t+1 t (cid:0) (cid:0) where the variables are expressed in deviations from a balanced growth path (henceforth BGP), and the parameters (cid:8);(cid:9);(cid:2) are function of the structural parameters of the model.13 The f g term q denotes the size of an aggregate bubble in the economy (normalized by trend output) t relative to its value along the BGP. The aggregate bubble plays the role of demand shifter and is de(cid:133)ned as q q = b +u ; t t t where b denotes the aggregate value in period t of bubble assets that were already available for t 12In line with our theoretical results, we show that, given the degree of indeterminacy, the estimation delivers the same marginal data densities regardless of which forecast error we include in our representation since there exists a unique mapping among them. 13In particular, (cid:8) (cid:17) (cid:3) (cid:12) (cid:0)v 2 (0;1], (cid:9) (cid:17) (cid:7)(cid:8) 1+ (cid:8) v(cid:13) (1 (1 (cid:0)(cid:12) (cid:8) (cid:13) ) ) , (cid:7) (cid:17) 1 1 (cid:0)(cid:3) (cid:12) (cid:0) (cid:13) v(cid:13) 2 (0;1] and (cid:2) (cid:17) (1 (cid:0) (cid:12)(cid:13) (cid:12) ) (cid:13) (1 (cid:0) v(cid:13)). The (cid:0) (cid:0) parametersarefunctionofthefollowingstructu(cid:16)ralparamete(cid:17)rsofthemodel: i)(cid:13),theconstantprobabilityofeach individualintheOLGmodeltosurvivetothenextperiod;ii)v,theprobabilityofeachindividualtobeemployed in the next period; iii) (cid:12), the discount factor of each individual; iv) (cid:3) 1=(1+r), the steady state stochastic (cid:17) discount factor for one-period ahead payo⁄s derived from a portfolio of securites; v) (cid:0) (1+g), the gross rate of (cid:17) productivity growth. 32

q q trade in period t 1, and u is the value of a new bubble at time t. We assume that u follows t t (cid:0) an exogenous autoregressive process of the form u q = (cid:26) u q +" q ; " q iid N(0;(cid:27)2): t q t 1 t t (cid:24) q (cid:0) Equation (53) de(cid:133)nes the evolution of the value of the asset bubble q as t q = (cid:3)(cid:0)E (b ) q(i E ((cid:25) )); (53) t t t+1 t t t+1 (cid:0) (cid:0) (cid:13)((cid:12) (cid:3)(cid:0)v) where q (cid:0) represents the steady state bubble-to-output ratio, (cid:3) 1=(1+r) is (cid:17) (1 (cid:12)(cid:13))(1 (cid:3)(cid:0)v(cid:13)) (cid:17) (cid:0) (cid:0) the steady state stochastic discount factor for one-period ahead payo⁄s derived from a portfolio of securities and (cid:0) (1+g) is the gross rate of productivity growth. To guarantee that newly (cid:17) created bubbles along the BGP are non-negative, the model requires that (cid:3)(cid:0) = 1+g 1. 1+r (cid:21) Equivalently, it must hold that r g on a BGP characterized by the creation of (non-negative) (cid:20) new asset bubbles. Equation (53) shows how "optimistic" expectations about the future value of the bubble lead to a higher price for those assets today. The model is then closed by the following NK Phillips curve (cid:25) = (cid:3)(cid:0)v(cid:13)E ((cid:25) )+(cid:20)y +us; (54) t t t+1 t t whereus = (cid:26) us +"s and "s iid N(0;(cid:27)2).14 Finally,theconductofmonetarypolicyisdescribed t s t (cid:0) 1 t t (cid:24) s by an interest rate rule of the form, i = (cid:26) i +(1 (cid:26) ) (cid:30) (cid:25) +(cid:30) q +"i; (55) t i t 1 i (cid:25) t q t t (cid:0) (cid:0) (cid:0) (cid:1) according to which monetary policy that displays a certain degree of interest rate inertia, and aims not only at stabilizing in(cid:135)ation, but also at leaning against the bubble.15 The rational expectation forecast errors are de(cid:133)ned as (cid:17) y E [y ]; (cid:17) (cid:25) E [(cid:25) ]; (cid:17) b E [b ]: (56) y;t t t 1 t (cid:25);t t t 1 t b;t t t 1 t (cid:17) (cid:0) (cid:0) (cid:17) (cid:0) (cid:0) (cid:17) (cid:0) (cid:0) Equations (52) (56) describe the equilibrium dynamics of the model economy around a given (cid:24) BGP. We de(cid:133)ne the vector of endogenous variables as X (y ;(cid:25) ;b ;i ;q ;E (y );E ((cid:25) ); t t t t t t t t+1 t t+1 (cid:17) 14In particular, k (cid:17) (1 (cid:0) (cid:18))(1 (cid:0)(cid:18) (cid:3)(cid:0)v(cid:13)(cid:18))(cid:26), where (cid:18) represents the Calvo probability that a (cid:133)rm keeps its price unchanged in any given period and (cid:26) is the elasticity of hours worked. 15We also assume that "i iidN(0;(cid:27)2). t (cid:24) i 33

E (b );u q ;us), and the vectors of fundamental shocks, " , and non-fundamental errors, (cid:17) , as t t+1 t t 0 t t " t " q t ;"s t ;"i t 0; (cid:17) t (cid:17) y;t ;(cid:17) (cid:25);t ;(cid:17) b;t 0: (cid:17) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) The model can therefore be represented as (cid:0) ((cid:18))X = (cid:0) ((cid:18))X +(cid:9)((cid:18))" +(cid:5)((cid:18))(cid:17) ; (57) 0 t 1 t 1 t t (cid:0) where (cid:18) represents the vector of structural parameters of the model. Gal(cid:237) (2017) shows that for realistic parameter values, the model is characterized by up to two degrees of indeterminacy. Therefore, the proposed methodology augments the representation of the model in (57) with two autoregressive processes 1 ! = ! +(cid:23) (cid:17) ; (58) 1;t (cid:11) 1 1;t (cid:0) 1 1;t (cid:0) 1;t (cid:18) (cid:19) 1 ! = ! +(cid:23) (cid:17) ; (59) 2;t (cid:11) 2 2;t (cid:0) 1 2;t (cid:0) 2;t (cid:18) (cid:19) where (cid:17) ;(cid:17) could be any combination consisting of two of the three forecast errors de(cid:133)ned 1;t 2;t f g by the vector (cid:17) t (cid:17) y;t ;(cid:17) (cid:25);t ;(cid:17) b;t 0. Hence, we de(cid:133)ne a new vector of endogenous variables (cid:17) X^ (X ;! ;! ) and a newly de(cid:133)ned vector of exogenous shocks as ^" (" ;(cid:23) ;(cid:23) ). The t t 1;t 2;t 0 (cid:0) (cid:1) t t 1;t 2;t 0 (cid:17) (cid:17) system in (57), (58) and (59) can then be written as (cid:0)^ X^ = (cid:0)^ X^ +(cid:9)^^" +(cid:5)^(cid:17) : 0 t 1 t 1 t t (cid:0) 5.2 Estimation WeestimatethemodeltomatchU.S.dataovertheperiod1982:Q4until2007:Q3. Weconsidera subset of three macroeconomic quarterly time series used in Smets and Wouters (2007) to match the number of exogenous shocks in the model. In particular, we use the growth rate in real GDP, the in(cid:135)ation rate measured by the GDP de(cid:135)ator and the Federal Funds rate. We implement Bayesian techniques, and the measurement equations that relate the macroeconomic data to the endogenous variables of the model are de(cid:133)ned as dlGDP g y y t t t 1 (cid:0) (cid:0) dlP = (cid:25) + (cid:25) ; 2 t 3 2 (cid:3)3 2 t 3 FFR i i 6 t 7 6 (cid:3)7 6 t 7 4 5 4 5 4 5 where dl denotes the percentage change measured as log di⁄erence. 34

WefollowGal(cid:237)(2017)andsetthediscountfactorofeachindividual, (cid:12);to0:998. Weestimatethe remaining structural parameters of the model using Bayesian techniques. We report the prior distributions for the parameters in Table 2. As mentioned when studying equation (53) for the evolution of the value of the asset bubble q , the model requires that the real interest rate, r, t and the growth rate of output, g, satisfy r g to ensure that newly created bubbles along the (cid:20) BGP are non-negative. To guarantee that this inequality holds for each draw of the Metropolis- Hastings algorithm, we express the real interest rate, r, as r = (cid:21)g, where (cid:21) (0;1). We then set 2 the prior for the quarterly growth rate of output, g, as a gamma distribution centered at 0:45, and the prior for (cid:21) as a beta distribution with mean 0:8. These priors imply that the annualized growth rate of output is 1:6% and the annualized real interest rate is approximately 1:3% over the considered period. We center the prior for the employment ratio, (cid:11), to 0:6. Following the calibration in Gal(cid:237) (2017), the prior distribution for the probability that an individual survives to the next period, (cid:13), is centered at 0:996 . The prior for the slope of the New Keynesian Phillips Curve, (cid:20), is set at 0:04, a value chosen for the calibration in Gal(cid:237) (2017) and consistent with an average duration of individual prices of 4 quarters. The parameter describing the response of the monetary authority to changes in in(cid:135)ation, (cid:30) , follows a gamma distribution with mean 1 and standard error 0:4. (cid:25) The response to deviations of the bubble relative to its value along the BGP follows a gamma distribution with mean 0:3 and standard error 0:15. The parameter which governs the degree of interest rate inertia, (cid:26) , follows a beta distribution centered at 0:7. i The priors on the stochastic processes that de(cid:133)ne the fundamental shocks are inverse gamma distributions centered at 0:3 with a standard deviation of 0:15. Finally, when we estimate the model under indeterminacy, we specify uniform prior distributions for both the standard deviations of the non-fundamental shocks (cid:27) where l = (cid:25);y;b , and their correlations with f (cid:23)lg f g the exogenous shocks of the model ’ where j = i;q;s . f (cid:23)l;jg f g We estimate the model in each region of the parameter space: Determinacy, one degree of indeterminacy and two degrees of indeterminacy. When the model is indeterminate, we run the estimation for the di⁄erent combinations consisting of one or two of the forecast errors de(cid:133)ned by the vector (cid:17) t (cid:17) y;t ;(cid:17) (cid:25);t ;(cid:17) b;t 0 depending on the degree of indeterminacy. In line (cid:17) with our theoretical results, we show that, given the degree of indeterminacy, the estimation (cid:0) (cid:1) delivers the same marginal data densities regardless of which forecast error(s) we include in our representation. In Appendix C, we also show that the posterior distributions for the model parameters are equivalent up to a transformation of the correlations between the exogenous shocks and the sunspot disturbances considered in each speci(cid:133)cation.16 16In Appendix B,we show theanalyticalequivalenceamong thedi⁄erentrepresentationsapplying ourmethodology to a bivariate model. 35

Prior distribution for model parameters Name Density Mean Std. Dev. g Gamma 0.45 0.04 (cid:21) Beta 0.80 0.10 (cid:11) Beta 0.60 0.10 100((cid:13) (cid:0) 1 1) Gamma 0.4 0.10 (cid:0) (cid:20) Gamma 0.04 0.005 (cid:25) Gamma 0.9 0.30 (cid:3) i Gamma 1.2 0.30 (cid:3) (cid:30) Gamma 1 0.40 (cid:25) (cid:30) Gamma 0.3 0.10 q (cid:26) Beta 0.70 0.10 i (cid:27) Inv: Gamma 0.30 0.15 q (cid:27) Inv: Gamma 0.30 0.15 s (cid:27) Inv: Gamma 0.30 0.15 i (cid:26) Beta 0.70 0.10 q (cid:26) Beta 0.70 0.10 s (cid:27) Uniform[0;10] 5 2.89 (cid:23)i ’ Uniform[ 1;1] 0 0.57 (cid:23)i;j (cid:0) Table 2: The table reports the prior distribution for the model parameters. Model comparison Speci(cid:133)cation Marginal data densities Indet-2 -72.3 Indet-1 -83.0 Determinacy -158.3 Table 3: The table reports the (log) marginal data densities for each model speci(cid:133)cation. Table 3 reports the (log) marginal data density for each of the model speci(cid:133)cation. We (cid:133)nd that the data favor the speci(cid:133)cation of the model with two degrees of indeterminacy. We attribute thisresulttothestylizednatureofthemodel, andtheobservationthatindeterminatemodelsare consistent with a richer dynamic and stochastic structure. In future work, it would be valuable to study whether the (cid:133)ndings would carry over in the context of a more realistic, mediumscale model that could explain the persistence and volatility in the data without recurring to indeterminate dynamics. Table 4 reports the mean and 90% probability interval of the posterior distribution of the estimated structural parameters. The probability of surviving to the next period, (cid:13), is estimated to be approximately 99%. The posterior of the slope of the NK Phillips curve is 0:039, which in this model is consistent with a probability of 24:1% that a (cid:133)rm keeps its price unchanged in 36

any given period. The steady-state in(cid:135)ation rate and nominal interest rate are about 0:7% and 1:4% on an quarterly basis. We also (cid:133)nd that the strength of the responses of U.S. monetary policy to stabilize the in(cid:135)ation rate and lean against 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 mean of the standard error of the bubble component is 0:28, and larger than the standard deviation of the supply and monetary policy shocks that are estimated to be 0:11 and 0:12, respectively. The data also provide evidence that the bubble shock is less persistent than the supply shock. Finally, we report the standard deviation of the sunspot shocks for the representation that includes the forecast error for the output gap and the in(cid:135)ation rate. The posterior estimates show that the standard error related to forecast errors for the output gap is approximately twice as large as the standard deviation of the sunspot shock associated with the in(cid:135)ation rate. The data also appear to be informative on the correlations of both sunspot shocks with the exogenous shocks of the model. A monetary policy shock is negatively correlated with both sunspot shocks, implying a contemporaneous impact on both in(cid:135)ation and output. A shock due to a new bubble can be interpreted as a demand shifter, and it has no signi(cid:133)cant correlation with unexpectedchangesinexpectationsaboutfuturein(cid:135)ationandeconomicactivity. Asupplyshock has a positive correlation with the sunspot shock associated with in(cid:135)ation, as well as a negative correlation with the sunspot shock for output. These correlations are crucial to interpret the impact that each shock has on the model economy as described next. Figure 5 plots the impulse response of output, in(cid:135)ation and nominal interest rate. We orthogonalize the fundamental shocks using a Cholesky decomposition with the same order as in the plots " ;" ;" . The last two panels report the impulse response functions in which each sunq s i f g spot shock is the most exogenous shock in the Cholesky decomposition. We plot the impulse responsestoaone-standard-deviationshock. Thesolidlinesrepresenttheposteriormeans, while the dashed line correspond to the 90% probability intervals. Considering the estimated correlations reported in Table 4, we observe that a shock due to the creation of a new bubble generates no signi(cid:133)cant e⁄ect on the economy in line with the estimated correlations. A positive supply shock has both in(cid:135)ationary and contractionary e⁄ects on impact. The persistence of the shock on output is then associated to de(cid:135)ationary e⁄ects to which the monetary authority responds by decreasing the nominal interest rate. A monetary policy tightening generates contractionary and de(cid:135)ationary pressures. The persistence of these e⁄ects on the in(cid:135)ation rate then requires the monetary authority to adopt an accommodative stance to stabilize the economy. 37

Posterior distribution for model parameters Mean 90% prob. int. g 0.46 [0.41,0.51] (cid:21) 0.79 [0.64,0.94] (cid:11) 0.59 [0.51,0.68] 100((cid:13) (cid:0) 1 1) 0.44 [0.30,0.58] (cid:0) (cid:20) 0.039 [0.032,0.047] (cid:25) 0.69 [0.38,1.01] (cid:3) i 1.41 [1.07,1.71] (cid:3) (cid:30) 0.35 [0.16,0.53] (cid:25) (cid:30) 0.13 [0.06,0.19] q (cid:26) 0.68 [0.54,0.84] i (cid:27) 0.28 [0.13,0.43] q (cid:27) 0.11 [0.09,0.13] s (cid:27) 0.12 [0.09,0.14] i (cid:26) 0.70 [0.54,0.86] q (cid:26) 0.89 [0.84,0.95] s (cid:27) 0.28 [0.24,0.32] (cid:23)(cid:25) (cid:27) 0.69 [0.60,0.78] (cid:23)y ’ -0.42 [-0.67,-0.16] (cid:23)(cid:25);i ’ 0.07 [-0.43,0.59] (cid:23)(cid:25);q ’ 0.61 [0.48,0.73] (cid:23)(cid:25);s ’ -0.14 [-0.40,0.13] (cid:23)y;i ’ -0.01 [-0.52,0.55] (cid:23)y;q ’ -0.68 [-0.77,-0.59] (cid:23)y;s Table 4: The table reports the posterior distribution of the model parameters under two degrees of indeterminacy (cid:23) ;(cid:23) . (cid:25) y f g 38

The last two panels show the impulse response to the sunspot shocks that we assume to be uncorrelated. In this economy, a positive shock to in(cid:135)ation expectations generates self-ful(cid:133)lling e⁄ects on in(cid:135)ation. Given that in this panel the sunspot shock, " , is assumed to be the most (cid:23)(cid:25) exogenous, economic activity does not respond on impact, while the increase in the nominal interest rate triggers a contractionary e⁄ect in the medium term. Finally, a positive sunspot shock to the expectation about future deviations of output from its trend leads to a rise in economic activity due to its self-ful(cid:133)lling nature. Given that in the last panel we assume that the sunspot shock, " , is the most exogenous, the in(cid:135)ation rate does not respond on impact, (cid:23)y while it is characterized by a mild de(cid:135)ationary e⁄ect in the medium term that leads to a decrease in the nominal interest rate. Table 5 reports the variance decompositions for output, in(cid:135)ation and interest rate. The means and the 90% probability intervals are calculated from the output of the Metropolis-Hastings algorithm. Because the estimated correlations of the two sunspot shocks with the exogenous shocks are nonzero, the reported variance decomposition results from the orthogonalization of the shocks using a Cholesky factorization in which the order of the shocks follows the list in Table 5. The results are in line with those in the literature and in particular with Lubik and Schorfheide (2004). The deviations of output from its trend are mostly explained by supply shocks. In addition to supply-side shocks, (cid:135)uctuations in in(cid:135)ations are also accounted for by unexpected changes in monetary policy. Similar conclusions can be drawn for the decomposition of the nominal interest rate. Interestingly, both sunspot shocks play only a marginal role in explaining business cycle (cid:135)uctuations for each of the three endogenous variables. Variance Decomposition Output dev. from trend In(cid:135)ation Interest rate Mean 90% prob. int. Mean 90% prob. int. Mean 90% prob. int. " 0.68 [0.37,0.93] 0.27 [0.01,0.57] 0.27 [0.01,0.56] s " 0.22 [0.02,0.45] 0.61 [0.37,0.85] 0.60 [0.36,0.85] i " 0.02 [0.01,0.03] 0.02 [0.01,0.04] 0.02 [0.01,0.04] q " 0.07 [0.01,0.14] 0.08 [0.01,0.21] 0.09 [0.01,0.21] (cid:23)y " 0.01 [0.001,0.02] 0.02 [0.01,0.03] 0.02 [0.01,0.03] (cid:23)(cid:25) Table 5: The table reports the means and 90-percent probability intervals for the unconditional variance decomposition. Since the estimated correlations with the two sunspot shocks are nonzero, the decomposition of the orthogonalized shocks via Cholesky decomposition follows the order of the shocks as listed in the Table. 39

Output Inflation Interest Rate 0.4 0.2 0.3 0.2 0 0.2 0.1 0.2 0 0 0.4 0.1 5 10 15 20 5 10 15 20 5 10 15 20 0.2 0.1 0.4 0 0.2 0.6 0.2 0.3 0.4 0.8 0.4 0.5 5 10 15 20 5 10 15 20 5 10 15 20 0.1 0 0.2 0.1 0.2 0 0.2 0.3 0.3 0.2 5 10 15 20 5 10 15 20 5 10 15 20 0 0.2 0.2 0.05 0.15 0.15 0.1 0.1 0.1 0.05 0.05 5 10 15 20 5 10 15 20 5 10 15 20 0.4 0 0.05 0.3 0.1 0.1 0.2 0.15 0.1 0.2 0.2 5 10 15 20 5 10 15 20 5 10 15 20 Figure 5: The (cid:133)gures plot the posterior means (solid lines) and 90-percent probability intervals (dashed lines) for the impulse responses of output, in(cid:135)ation and nominal interest rate to a shock of one standard deviation for each orthogonalized disturbance using a Cholesky decomposition with the same order as in the plots. 40

6 Conclusions Inthispaper,weproposeageneralizedapproachtosolveandestimateLREmodelsovertheentire 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. When a LRE model is characterized by m degrees of indeterminacy, our approach augments it by appending m autoregressive processes whose innovations are linear combinations of a subset of endogenous shocks and a vector of newly de(cid:133)ned sunspot shocks. We show that the solution for the resulting augmented representation embeds both the solution which is 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 et al. (2015). We apply our methodology to estimate the small-scale NK model of Gal(cid:237) (2017) using Bayesian techniques. Gal(cid:237)(cid:146)s model extends a conventional NK model to allow for the existence of rational bubbles. Aninterestingaspectofthemodelisthatitdisplaysuptotwodegreesofindeterminacy for realistic parameter values. We estimate the model using U.S. data over the period 1982:Q4 until 2007:Q3. Using Bayesian model comparison we (cid:133)nd 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. One caveat to note, however, is that the model of Gal(cid:237)(2017)isquitestylized, buttheresultsareintriguingandmeritfurtherexplorationinfuture research. 41

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7 Appendix 7.1 Appendix A We prove the equivalence between the parametrization of the Lubik-Schorfheide indeterminate equilibrium (cid:18)LS (cid:2)LS and the Bianchi-Nicol(cid:242) equilibrium parametrized by (cid:18)BN (cid:2)BN: In 2 2 particular,weshowthatthereisauniquemappingbetweenthelinearrestrictionsimposedineach of the two methodologies on the forecast errors to guarantee the existence of at least a bounded solution. As shown in Section 3.2.1, the method by Lubik and Schorfheide (2003) imposes the following restrictions on the non-fundamental shocks, (cid:17) , as a function of the exogenous shocks, t " ; and the sunspot shocks introduced in their speci(cid:133)cation, (cid:16) ; t t (cid:17) = V N + V M " + V (cid:16) : (60) t 1 2 t 2 t p 1 0p nn ‘ p mm ‘1 ‘ 1 p mm 1 (cid:2) (cid:2) (cid:2) (cid:2) m (cid:2) ‘ (cid:2) (cid:2) (cid:2) @ f(cid:2) A Using the methodology proposed in this paper, Section 3.2.2 shows that the restrictions on the non-fundamental shocks, (cid:17) , as a function of the exogenous shocks, " ; and the sunspot shocks, t t v ; are t (cid:17) = C " + C (cid:23) ; (61) t 1 t 2 t p 1 p ‘‘ 1 p mm 1 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) where (cid:5) 1(cid:9) (cid:5) 1(cid:5) C (cid:0)n;2 2 and C (cid:0)n;2 f;2 : 1 2 (cid:17) (cid:0) " 0 # (cid:17) (cid:0) " I # e e e (cid:0)e Post-multiplying equation (60) and (61) by " and taking expectations on both sides, 0t (cid:10) = V N (cid:10) + V M (cid:10) ; (cid:17)" 1 "" 2 "" p (cid:2) l p (cid:2) nn (cid:2) ‘‘ (cid:2) l p (cid:2) mm (cid:2) ‘ ‘ (cid:2) l (cid:10) (cid:17)" = C 1 (cid:10) "" + C 2 (cid:10) (cid:23)" f p l p ‘‘ l p mm l (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) Pre-multiplying by V and equating the equations, 20 M (cid:10) = V C V V N (cid:10) + V C (cid:10) : (62) "" 20 1 20 1 "" 20 2 (cid:23)" m ‘ ‘ l m pp ‘ (cid:0) m pp nn ‘! ‘ l m pp mm l (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) f Using the properties of the vec operator, the following result holds vec(M) = ((cid:10) I ) 1 I V C V V N vec((cid:10) )+ I V C vec((cid:10) ) : (63) "" m (cid:0) l 20 1 20 1 "" l 20 2 (cid:23)" (cid:10) (cid:10) (cid:0) (cid:10) (m ‘) 1 (m ‘) (m ‘) " (m ‘) ‘2 ‘2 1 (m ‘) (m ‘) (m ‘) 1 # (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:0) (cid:2) (cid:2) (cid:1)(cid:3) (cid:2) (cid:0) (cid:2) (cid:2) (cid:2)(cid:1) (cid:2) (cid:2) f 44

Equation (63) is the (cid:133)rst relevant equation to show the mapping between the representation in Lubik and Schorfheide (2003) and our representation. For a given variance-covariance matrix of the exogenous shocks, (cid:10) , that is common between the two representations, equation (63) "" tells us that the covariance structure, (cid:10) , of the sunspot shock in our representation with the (cid:23)" exogenous shocks has a unique mapping to the matrix, M, in Lubik and Schorfheide (2003). Clearly, equation (62) can also be used to derive the mapping from their representation to our f method. We now show how to derive the mapping between the variance-covariance matrix, (cid:10) , of the (cid:23)(cid:23) sunspotshocksinourrepresentationtothevariance-covariancematrix,(cid:10) ,ofthesunspotshocks (cid:16)(cid:16) in Lubik and Schorfheide (2003). Considering again equation (60) and (61), we post-multiply by (cid:16) and take expectations on both sides, 0t (cid:10) = V (cid:10) ; (cid:17)(cid:16) 2 (cid:16)(cid:16) p m p mm m (cid:2) (cid:2) (cid:2) (cid:10) = C (cid:10) (cid:17)(cid:16) 2 (cid:23)(cid:16) p m p mm m (cid:2) (cid:2) (cid:2) Pre-multiplying both equations by V and equating them, 20 (cid:10) (cid:16)(cid:16) = (cid:10) (cid:16)(cid:23) V 20 C 2 0: (64) m m m m m m (cid:2) (cid:2) (cid:0) (cid:2) (cid:1) Finally, to obtain an expression for (cid:10) , we post-multiply equation (60) and (61) by (cid:23) and (cid:16)(cid:23) 0t taking expectations (cid:10) = V N + V M (cid:10) + V (cid:10) ; (cid:17)(cid:23) 1 2 "(cid:23) 2 (cid:16)(cid:23) p (cid:2) m (cid:18) p (cid:2) nn (cid:2) ‘ p (cid:2) mm (cid:2) ‘ (cid:19) ‘ (cid:2) m p (cid:2) mm (cid:2) m (cid:10) = C (cid:10) + C (cid:10) (cid:17)(cid:23) 1 "(cid:23) 2 (cid:23)(cid:23)f p m p ‘‘ m p mm m (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) Pre-multiplying both equations by V and solving for (cid:10) ; 20 (cid:16)(cid:23) (cid:10) = V C V V N M (cid:10) + V C (cid:10) : (65) (cid:16)(cid:23) 20 1 20 1 "(cid:23) 20 2 (cid:23)(cid:23) m (cid:2) m m (cid:2) pp (cid:2) ‘ (cid:0) m (cid:2) pp (cid:2) nn (cid:2) ‘(cid:0)m (cid:2) ‘!‘ (cid:2) m (cid:0) m (cid:2) m (cid:1) m (cid:2) m f Post-multiplying (65) by (V C ) and using (64), then 20 2 0 m m (cid:2) (cid:10) (cid:16)(cid:16) = V 20 C 1 V 20 V 1 N M (cid:10) "(cid:23) V 20 C 2 0+ V 20 C 2 (cid:10) (cid:23)(cid:23) V 20 C 2 0: (66) m (cid:2) m m (cid:2) pp (cid:2) ‘ (cid:0) m (cid:2) pp (cid:2) nn (cid:2) ‘(cid:0)m (cid:2) ‘!‘ (cid:2) m (cid:0) m (cid:2) m (cid:1) (cid:0) m (cid:2) m (cid:1) m (cid:2) m (cid:0) m (cid:2) m (cid:1) f 45

Therefore, equation (66) de(cid:133)nes the mapping between the variance-covariance matrix, (cid:10) , of (cid:23)(cid:23) the sunspot shocks in our representation to the variance-covariance matrix, (cid:10) , of the sunspot (cid:16)(cid:16) shocksinLubikandSchorfheide(2003). Togetherwithequation(63), weshowthatthisequation de(cid:133)nes the one-to-one mapping between the parametrization in Lubik and Schorfheide (cid:2);(cid:2)LS f g and the parametrization in Bianchi-Nicol(cid:242) (cid:2);(cid:2)BN : f g 7.2 Appendix B InthisAppendix,weprovideananalyticalexampletoshowtheequivalencebetweenthesolutions for an indeterminate LRE model using two alternative methodologies: Lubik and Schorfheide (2003) and our proposed method. In particular, we consider the following simple model 1 1 y = E (y )+ E (x )+" (67) t t t+1 t t+1 t (cid:18) (cid:18) y y 1 x = E (x ) (68) t t t+1 (cid:18) x where " iid N(0;(cid:27)2) and the corresponding forecast errors are denoted as t " (cid:24) (cid:17) y E (y ) (69) y;t t t 1 t (cid:17) (cid:0) (cid:0) (cid:17) x E (x ) (70) x;t t t 1 t (cid:17) (cid:0) (cid:0) 7.2.1 Lubik and Schorfheide (2003) The LRE model in (67) (70) can be written in the following matrix form (cid:24) (cid:0) S = (cid:0) S +(cid:9)" +(cid:5)(cid:17) ; (71) 0 t 1 t 1 t t (cid:0) where S (y ;x ;E (y );E (x )) and (cid:17) ((cid:17) ;(cid:17) ). t t t t t+1 t t+1 0 t y;t x;t 0 (cid:17) (cid:17) As the matrix (cid:0) is non-singular, the LRE model in (71) can be written as 0 S = (cid:0) S +(cid:9) " +(cid:5) (cid:17) ; (72) t (cid:3)1 t 1 (cid:3) t (cid:3) t (cid:0) where (cid:0) (cid:0) 1(cid:0) = 0 A ; (cid:5) (cid:0) 1(cid:5) = A (cid:3)1 (cid:17) (cid:0)0 1 4 (cid:2) 2 4 (cid:2) 2 (cid:3) (cid:17) (cid:0)0 4 (cid:2) 2 h i 46

0 1 0 (cid:9) (cid:0) 1(cid:9) = 2 0 3 ; A = 20 1 3 (cid:3) (cid:17) (cid:0)0 6(cid:0) (cid:18) x7 4 (cid:2) 2 6 (cid:18) y (cid:0) (cid:18) x7 6 7 6 7 6 0 7 60 (cid:18) x 7 6 7 6 7 4 5 4 5 Note that equation (72) corresponds to equation (20) in Lubik and Schorfheide (2004). We now show how to solve the model and obtain equation (26) in Lubik and Schorfheide (2004). Applying the Jordan decomposition, the matrix (cid:0) can be decomposed as (cid:0) J(cid:3)J 1, where (cid:3)1 (cid:3)1 (cid:17) (cid:0) the elements of the diagonal matrix (cid:3) denote the roots of the system 0 0 0 0 20 0 0 03 (cid:3) 11 0 (cid:3) = : (cid:17) 6 0 0 (cid:18) x 0 7 " 0 (cid:18) y# 6 7 60 0 0 (cid:18) y7 6 7 4 5 Assuming without loss of generality that (cid:18) 1 and (cid:18) > 1, the system in (72) is indeterminx y j j (cid:20) j j ate because the number of expectational variables, E (y );E (x ) , exceeds the number of t t+1 t t+1 f g explosive roots, (cid:18) : De(cid:133)ning the vector w J 1S ; the model can be represented as y t (cid:0) t (cid:17) w (cid:3) 0 w (cid:9)~ (cid:5)~ 1;t 11 1;t 1 1 1 w t (cid:17) "w 2;t# = " 0 (cid:18) y#"w 2;t (cid:0) 1# + "(cid:9)~ 2# " t + "(cid:5)~ 2# (cid:17) t ; (73) (cid:0) where the (cid:133)rst block denotes the stationary block of the system and the second block is unstable. TheadoptionofSims(cid:146)(2002)code, Gensys, tosolvethismodelisnotappropriateasitdealswith determinate models. After having obtained the representation in (73), Gensys would construct a matrix (cid:8) such that premultiplying the system by a matrix [I (cid:8)] would eliminate the e⁄ect (cid:0) of non-fundamental shocks. Equivalently, the matrix has to satisfy the condition (cid:5)~ [I (cid:8)] 1 = (cid:5)~ (cid:8)(cid:5)~ = 0: (74) (cid:0) "(cid:5)~ 2# 1 (cid:0) 2 Under determinacy, the matrix (cid:5)~ is square and, assuming that it is also non-singular17, it is 2 1 possible to solve for (cid:8) = (cid:5)~ (cid:5)~ (cid:0) . 1 2 The approach in Lubik and(cid:16)Scho(cid:17)rfheide (2003) modi(cid:133)es this intuition to account for the indeterminacy that characterizes the model in (73). Under indeterminacy, the matrix (cid:5)~ is a vector 2 17Note that Gensys obtains the matrix (cid:8) even when the matrix (cid:5)~ is singular by applying a singular value 2 decomposition. 47

with more columns than rows, implying that it is not possible to obtain a matrix (cid:8) that satis(cid:133)es the above condition in (74). Nevertheless, Lubik and Schorfheide (2003) apply a singular value decomposition (SVD) to the matrix (cid:5)~ to obtain 2 D 0 V (cid:5)~ UDV = U U 11 :10 = U D V ; (75) 2 0 :1 :2 :1 11 :10 (cid:17) " 0 0#"V :20# h i where D is a diagonal matrix and U and V are orthonormal matrices. In this particular 11 example, the matrix to decompose is (cid:5)~ = a b , where a (cid:18) and b (cid:18) (cid:18) =((cid:18) (cid:18) ), and 2 y x y x y (cid:17) (cid:0) (cid:17) (cid:0) (cid:0) the resulting SVD is h i a b (cid:5)~ UDV = 1 d 0 d d ; (76) 2 (cid:17) 0 b a " # h i d (cid:0)d where d pa2+b2. Lubik and Schorfheide (2003) then proceed by de(cid:133)ning the matrix (cid:8) as (cid:17) 0 0 0 0 a 1 (cid:8) = (cid:5)~ V d 1U = 0 0 d = 0 0 ; 1 :1 (cid:0) :01 2 3 b d 2 3 " # 0 (cid:18) d 0 (cid:18) b (cid:0) (cid:1) 6 x 7 6 xd27 4 5 4 5 and premultiply the system in (73) by the following matrices I (cid:8) w I (cid:8) (cid:3) 0 w 1;t 11 1;t 1 (cid:0) = (cid:0) (cid:0) + "0 1 #"w 2;t# "0 0 #" 0 (cid:18) y#"w 2;t 1# (cid:0) I (cid:8) (cid:9)~ I (cid:8) (cid:5)~ 1 1 + (cid:0) " + (cid:0) (cid:17) ; (77) "0 0 #"(cid:9)~ 2# t "0 0 #"(cid:5)~ 2# t =0 6 | {z } wherethesecondblockrepresentstheconstraintthatguaranteestheboundednessofthesolution, b w = 0 E (y ) = E (x ): (78) 2;t t t+1 t t+1 () (cid:0)a Importantly, given that the model is indeterminate, the last term in equation (77) di⁄ers from zero and therefore non-fundamental disturbances a⁄ect the model dynamics. Solving (77) for the endogenous variables, S , the system takes the form t S = (cid:0)~ S +(cid:9)~ " +(cid:5)~ (cid:17) ; (79) t (cid:3)1 t 1 (cid:3) t (cid:3) t (cid:0) 48

where 1 (cid:0)~ 0 B ; (cid:9)~ a 22 b=a 3 ; (cid:3)1 (cid:17) h 4 (cid:2) 2 4 (cid:2) 2 i (cid:3) (cid:17) (cid:16) d (cid:17) 6 6 (cid:0) (cid:18) x (b=a)2 7 7 6 (cid:18) x b=a 7 6 7 4 5 b2=d2 b(1 b2=d2) (cid:0)a (cid:0) (cid:5)~ B = 2 (cid:0) (cid:0) ab=d (cid:1) 2 (1 (cid:0) b2=d2) 3 : (cid:3) (cid:17) 4 (cid:2) 2 6 (cid:18) x b2=d2 (cid:0) (cid:18) xa b(1 (cid:0) b2=d2) 7 6 6 (cid:18) (cid:0)x ab=d2 (cid:1) (cid:18) x (1 b2=d2) 7 7 6(cid:0) (cid:0) 7 4 5 The last step that Lubik and Schorfheide (2003) implement is to express the forecast errors as a function of the fundamental shock, " , and a sunspot shock, (cid:16) , as t t (cid:17) = V D 1U (cid:9)~ " +V M v " +M (cid:16) ; (80) t (cid:0) :1 1(cid:0)1 :01 2 t :2 t (cid:16) t (cid:18) (cid:19) where V = b a . Combining (79) with (80) and normalizing M = 1, the solution to the :20 d (cid:0)d (cid:16) LRE model ish 18 i S = (cid:0)~ S +(cid:9)~ " +(cid:5)~ V M v " +(cid:16) : (81) t (cid:3)1 t 1 (cid:3) t (cid:3) :2 t t (cid:0) (cid:18) (cid:19) This solution can be equivalently written in a form that explicitly includes the boundedness condition in (78) for which w = 0 and therefore E (y ) = bE (x ). Recalling that 2;t t t+1 (cid:0)a t t+1 S = (y ;x ;E (y );E (x )), the dynamics of the solution in (81) are now expressed as a t t t t t+1 t t+1 0 function of only one state variable, b=a (cid:0) S = 2 1 3 E (x )+(cid:9)~ " +(cid:5)~ V M v " +(cid:16) t t 1 t (cid:3) t (cid:3) :2 t t 6(cid:0) (cid:18) x b=a 7 (cid:0) (cid:18) (cid:19) 6 7 6 (cid:18) x 7 6 7 4 5 (cid:18)x 1 (cid:18)x ((cid:18)y (cid:18)x) ((cid:18)y (cid:18)x) = 2 6 6 6 ((cid:18)y (cid:18) (cid:18) (cid:0) (cid:0) 1 2 x (cid:18)x) 3 7 7 7 E t (cid:0) 1 (x t )+ d (cid:18)2 y 2 2 6 6 6 (cid:0) ( ( (cid:18) (cid:18) x x (cid:18) (cid:18) (cid:0) (cid:0) x 2 x (cid:18) (cid:18) 3 x (cid:18) y y ) )2 3 7 7 7 " t + (cid:18) d y 2 6 6 6 ((cid:18)y (cid:18) (cid:18) (cid:0) (cid:0) 1 2 x (cid:18)x) 3 7 7 7 (cid:18) M v " t +(cid:16) t (cid:19) ; (82) 6 4 x 7 5 6 6 ((cid:18)x (cid:0) (cid:18)y) 7 7 6 4 x 7 5 4 5 18Note that the term (cid:0) (cid:5)~ (cid:3) V :1 D 1(cid:0)1 1U :01 (cid:9) v 2 " t always equals to zero since (cid:5)~ (cid:3)V :1 =0 by the properties of the orthonormal matrix V: (cid:18) (cid:19) (cid:16) (cid:17) 49

where d = (cid:18)2 +((cid:18) (cid:18) )2=((cid:18) (cid:18) )2. y x y x y (cid:0) q 7.2.2 Our proposed methodology We now provide the derivation of the solution for the LRE model in (71) and reported below in equation (83) using the methodology proposed in this paper (cid:0) S = (cid:0) S +(cid:9)" +(cid:5)(cid:17) : (83) 0 t 1 t 1 t t (cid:0) The methodology consists of appending the following equation to the original LRE model 1 ! = ! +(cid:23) (cid:17) ; t (cid:11) t (cid:0) 1 x;t (cid:0) x;t wherev denotesanewlyde(cid:133)nedsunspotshockandwithoutlossofgenerality(cid:11) (cid:18) . Denoting t x (cid:17) j j the newly de(cid:133)ned vector of endogenous variables S^ (S ;! ) = (y ;x ;E (y );E (x );! ), t t t 0 t t t t+1 t t+1 t 0 (cid:17) and the newly de(cid:133)ned vector of exogenous shocks ^"x (" ;(cid:23) ), the augmented representation t t x;t 0 (cid:17) of the LRE model is (cid:0)^ S^ = (cid:0)^ S^ +(cid:9)^^"x+(cid:5)^(cid:17) : (84) 0 t 1 t 1 t t (cid:0) Pre-multiplying the system in (84) by (cid:0)^ 1, we obtain (cid:0)0 S^ = (cid:0)^ S^ +(cid:9)^ ^"x+(cid:5)^ (cid:17) ; (85) t (cid:3)1 t 1 (cid:3) t (cid:3) t (cid:0) where (cid:0) 0 (cid:9) 0 (cid:5) (cid:3)1 4 1 (cid:3) 4 1 (cid:3)4 2 (cid:0)^ (cid:2) ; (cid:9)^ (cid:2) ; (cid:5)^ (cid:2) : (cid:3)1 2 3 (cid:3) 2 3 (cid:3) 2 3 (cid:17) (cid:17) (cid:17) 0 1 0 1 0 1 6 1 (cid:2) 4 (cid:11) 7 6 (cid:0) 7 6 7 4 5 4 5 4 5 and the matrices (cid:0) ;(cid:9) ;(cid:5) are the same as those found in (72). Applying the Jordan decomf (cid:3)1 (cid:3) (cid:3) g position, the matrix (cid:0)^ can be decomposed as (cid:0)^ J^(cid:3)^J^ 1, where the elements of the diagonal (cid:3)1 (cid:3)1 (cid:17) (cid:0) matrix (cid:3)^ denote the roots of the system 0 0 0 0 0 0 0 0 0 0 2 3 (cid:3) 0 (cid:3) 0 (cid:3)^ = 0 0 (cid:18) 0 0 = 11 : (cid:17) "0 (cid:11) 1 # 6 60 0 0 x (cid:18) 0 7 7 " 0 (cid:3) 22# 6 y 7 6 7 60 0 0 0 17 6 (cid:11)7 4 5 50

Assuming as in the previous section that (cid:18) 1 and (cid:18) > 1, then 1=(cid:11) = 1= (cid:18) > 1 and x y x j j (cid:20) j j j j (cid:18) 0 y the diagonal elements of the matrix (cid:3) = correspond to the explosive roots of the 22 "0 1=(cid:11)# system. While the original system in (83) is indeterminate, the augmented representation in (84) is determinate as the number of expectational variables, E (y );E (x ) , equals the number t t+1 t t+1 f g of explosive roots, (cid:18) ;1=(cid:11) : De(cid:133)ning the vector w^ J^ 1S^; the model can be represented as y t (cid:0) t f g (cid:17) w^ (cid:3) 0 w^ (cid:9)^ (cid:5)^ w^ t (cid:17) "w^ 1 2 ; ; t t# = " 0 11 (cid:3) 22#"w^ 1 2 ; ; t t (cid:0) 1 1# + "(cid:9)^ (cid:3)1 (cid:3)2 (cid:3) (cid:3)# ^"x t + "(cid:5)^ (cid:3)2 (cid:3)1 ;(cid:3) (cid:3) x# (cid:17) t ; (86) (cid:0) where the (cid:133)rst block is stationary. Given that the second block is unstable, the following two conditions have to be imposed to guarantee the boundedness of the solution. First, the linear combination of the endogenous variables, w^ , is set to zero, 2;t E (y ) = bE (x ) w^ = 0 t t+1 (cid:0)a t t+1 (87) 2;t () ( ! t = 0 Second, the linear combination of fundamental and non-fundamental shocks also has to equal zero. Therefore, the non-fundamental shocks, (cid:17) , become a function of the augmented vector of t exogenous shocks, ^"x, t 1 1 (cid:18)x " (cid:17) = (cid:5)^ (cid:0) (cid:9)^ ^"x (cid:17) = (cid:0)(cid:18)x (cid:18)y t (88) t (cid:3)2;(cid:3)x (cid:3)2(cid:3) t t (cid:0) (cid:0) () "0 1 #"(cid:23) x;t# (cid:16) (cid:17) Considering equation (86), it is relevant to point out that the matrix (cid:5)^ di⁄ers from the (cid:3)2;(cid:3)x corresponding matrix for the representation in which we incorporate the forecast error, (cid:17) , y;t de(cid:133)ned as (cid:5)^ , (cid:3)2;(cid:3)y (cid:18) (cid:18)x(cid:18)y (cid:18) (cid:18)x(cid:18)y (cid:5)^ y (cid:18)x (cid:18)y (cid:5)^ y (cid:18)x (cid:18)y : (cid:3)2;(cid:3)x (cid:0) (cid:3)2;(cid:3)y (cid:0) (cid:17) "0 1 # (cid:17) " 1 0 # (cid:0) (cid:0) Therefore, whentheauxiliaryprocessiswrittenasafunctionofthenon-fundamentalshock,(cid:17) , y;t the restriction imposed on (cid:17) to guarantee the boundedness of the solution also di⁄ers from the t one found in (88) 1 0 1 " (cid:17) = (cid:5)^ (cid:0) (cid:9)^ ^" y (cid:17) = t (89) t (cid:0) (cid:16) (cid:3)2;(cid:3)y (cid:17) (cid:3)2(cid:3) t () t " (cid:18)x (cid:18) (cid:0) x (cid:18)y (cid:0) (cid:18)x (cid:18) (cid:0) x (cid:18)y #"(cid:23) y;t# Importantly, from equations (88) and (89) it is possible to establish a relationship that links the 51

two non-fundamental disturbances (cid:23) ;(cid:23) and the exogenous shock " , x;t y;t t f g (cid:18) (cid:18) (cid:18) (cid:18) x y x y (cid:23) = (cid:0) " (cid:0) (cid:23) : (90) x;t t y;t (cid:18) (cid:0) (cid:18) x x We show below that equations (87) and (90) are crucial for the equivalence between the augmented representations that include di⁄erent non-fundamental shocks in the auxiliary processes that our methodology proposes. Theaugmentedmodelin(86)isdeterminateasthesecondblockhastwoexplosiverootstomatch the two expectational variables of the model. It is therefore possible to apply the approach in Sims(cid:146)(2002) to construct a matrix (cid:8)^ such that premultiplying the system by a matrix [I (cid:8)^ ] x x (cid:0) would eliminate the e⁄ect of non-fundamental shocks. Equivalently, the matrix has to satisfy the condition (cid:5)^ [I (cid:8)^ ] (cid:3)1(cid:3) = (cid:5)^ (cid:8)^ (cid:5)^ = 0: (91) (cid:0) x "(cid:5)^ (cid:3)2;(cid:3)x# (cid:3)1(cid:3) (cid:0) x (cid:3)2(cid:3);x Importantly, the matrix (cid:5)^ is square under determinacy and, assuming that it is also non- (cid:3)2;(cid:3)x 1 singular19, it is possible to solve for (cid:8)^ = (cid:5)^ (cid:5)^ (cid:0) . x (cid:3)1(cid:3) (cid:3)2;(cid:3)x To solve the model, the system in (86) is then(cid:16)prem(cid:17)ultiplied by the following matrices I (cid:8)^ w^ I (cid:8)^ (cid:3) 0 w^ x 1;t x 11 1;t 1 (cid:0) = (cid:0) (cid:0) + "0 I #"w^ 2;t# "0 0 #" 0 (cid:3) 22#"w^ 2;t 1# (cid:0) I (cid:8)^ (cid:9)^ I (cid:8)^ (cid:5)^ + (cid:0) x (cid:3)1(cid:3) ^"x+ (cid:0) x (cid:3)1(cid:3) (cid:17) ; (92) "0 0 #"(cid:9)^ (cid:3)2(cid:3)# t "0 0 #"(cid:5)^ (cid:3)2;(cid:3)x# t =0 | {z } wherethesecondblockrepresentstheconstraintthatguaranteestheboundednessofthesolution, w^ = 0. Importantly, the augmented representation is determinate, and the last term of the 2;t system in (92) equals zero. Nevertheless, the non-fundamental disturbance, (cid:23) , a⁄ects the x;t dynamics of the original model through vector of exogenous shocks, ^"x (" ;(cid:23) ). Solving (91) t t x;t 0 (cid:17) for the endogenous variables, S^ (S ;! ) = (y ;x ;E (y );E (x );! ), the system takes t t t 0 t t t t+1 t t+1 t 0 (cid:17) 19Note that Gensys obtains the matrix (cid:8)^ even when the matrix (cid:5)^ (cid:3)2(cid:3) is singular by applying a singular value decomposition. 52

the form S^ = (cid:0)^ S^ +(cid:9)^ ^"x t (cid:3)1(cid:3) t 1 (cid:3)S(cid:3) t (cid:0) (cid:18)x 1 (cid:18)x ((cid:18)y (cid:18)x) ((cid:18)y (cid:18)x) (cid:0) (cid:0) 2 1 3 0 2 1 3 2 3 = 6 6 6 6 ((cid:18)y (cid:18) (cid:18) (cid:0) 2 x x (cid:18)x)7 7 7 7 E t (cid:0) 1 (x t )+ 6 6 6 0 0 7 7 7 " t + 6 6 6 6 ((cid:18)y (cid:18) (cid:18) (cid:0) 2 x x (cid:18)x)7 7 7 7 (cid:23) x;t : (93) 6 7 6 6 0 7 7 607 6 6 0 7 7 6 7 6 7 6 7 4 5 4 5 4 5 Finally, to rewrite the reduced-form solution for the augmented representation that includes the non-fundamental shock, (cid:17) , in the auxiliary process, we recall equations (87) and (90) that we y;t report below in equations (94) and (95) E (y ) = (cid:18)x E (x ) w^ = 0 t t+1 (cid:0)(cid:18)x (cid:18)y t t+1 (94) 2;t (cid:0) () ( ! t = 0 (cid:18) (cid:18) (cid:18) (cid:18) x y x y (cid:23) = (cid:0) " (cid:0) (cid:23) (95) x;t t y;t (cid:18) (cid:0) (cid:18) x x Using the above equations, we can rewrite the system in (93) as y 1 0 1 t 2 x t 3 = 2 (cid:18)y (cid:18) (cid:0) x (cid:18)x 3 E t 1 (y t )+ 2 (cid:18)x (cid:18) (cid:0) x (cid:18)y 3 " t + 2 (cid:0) (cid:18)x (cid:18) (cid:0) x (cid:18)y 3 (cid:23) y;t : (96) 6 E t (y t+1 ) 7 6 (cid:18) x 7 (cid:0) 6 (cid:0) (cid:18) x 7 6 (cid:18) x 7 6 7 6 7 6 7 6 7 6E t (x t+1 )7 6(cid:18) y (cid:18) x7 6(cid:18) x (cid:18) y7 6 ((cid:18) x (cid:18) y )7 6 7 6 (cid:0) 7 6 (cid:0) 7 6(cid:0) (cid:0) 7 4 5 4 5 4 5 4 5 7.2.3 Equivalence of methodologies In this section, we show the equivalence of the representations obtained using the two methodologies. In equation (97) below, we report the solution for the endogenous variables, S = t (y ;x ;E (y );E (x )), using the methodology of Lubik and Schorfheide (2003), t t t t+1 t t+1 0 y (cid:18)x 1 (cid:18)x t ((cid:18)y (cid:18)x) ((cid:18)y (cid:18)x) 2 6 6 6 6 4 E E t t ( ( x x y t t t + + 1 1 ) ) 3 7 7 7 7 5 = 2 6 6 6 6 4 ((cid:18)y (cid:18) (cid:18) (cid:0) (cid:0) 1 2 x x (cid:18)x) 3 7 7 7 7 5 E t (cid:0) 1 (x t )+ d (cid:18)2 y 2 2 6 6 6 6 6 (cid:0) ( ( ( (cid:18) (cid:18) (cid:18) x x x (cid:18) (cid:18) (cid:0) (cid:0) (cid:0) x 2 x (cid:18) (cid:18) (cid:18) 3 x (cid:18) y y y ) ) )2 3 7 7 7 7 7 " t + (cid:18) d y 2 6 6 6 6 4 ((cid:18)y (cid:18) (cid:18) (cid:0) (cid:0) 1 2 x x (cid:18)x) 3 7 7 7 7 5 (cid:18) M v " t +(cid:16) t (cid:19) ; (97) 4 5 53

where d = (cid:18)2 +((cid:18) (cid:18) )2=((cid:18) (cid:18) )2. We now report in equation (98) below the solution using y x y x y (cid:0) our methodqology when we include the forecast error, (cid:17) , in the auxiliary process20 x;t y (cid:18)x 1 (cid:18)x t ((cid:18)y (cid:18)x) ((cid:18)y (cid:18)x) (cid:0) (cid:0) 2 x t 3 2 1 3 203 2 1 3 = E (x )+ " + (cid:23) : (98) 6 6 E t (y t+1 ) 7 7 6 6 ((cid:18)y (cid:18) (cid:0) 2 x (cid:18)x)7 7 t (cid:0) 1 t 6 6 0 7 7 t 6 6 ((cid:18)y (cid:18) (cid:0) 2 x (cid:18)x)7 7 x;t 6E t (x t+1 )7 6 (cid:18) x 7 607 6 (cid:18) x 7 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5 To show the equivalence between the two representations, we need to recall the restrictions that each methodology imposed on the forecast errors, (cid:17) , as a function of the exogenous shock, " , t t and the additional sunspot shock. Following Lubik and Schorfheide (2003), we derived that (cid:17) = V D 1U (cid:9)~ " +V M v " +M (cid:16) ; t (cid:0) :1 1(cid:0)1 :01 2 t :2 t (cid:16) t (cid:18) (cid:19) V a b where we know that V = :10 = d d , D = d = pa2+b2, U = 1, (cid:9)~ = a = (cid:18) and 0 "V :20# " d b (cid:0) a d # 11 1 2 (cid:0) y b = (cid:18) (cid:18) =((cid:18) (cid:18) ): Therefore, normalizing M = 1, we obtain x y x y (cid:16) (cid:0) (cid:0) a a b v (cid:17) = d " + d M" +(cid:16) t b d t a t t " # " # d (cid:0)d (cid:18) (cid:19) = (cid:18)2 y 1 + (cid:18) y (cid:0)((cid:18)x (cid:18)x (cid:18)y) M v " + (cid:18) y (cid:0)((cid:18)x (cid:18)x (cid:18)y) (cid:16) : (99) ( d2 " (cid:18)x # d " 1 (cid:0) # ) t d " 1 (cid:0) # t ((cid:18)x (cid:18)y) (cid:0) Similarly, from the derivation using our methodology, we know that 1 1 (cid:18)x " (cid:17) = (cid:5)^ (cid:0) (cid:9)^ ^"x (cid:17) = (cid:0)(cid:18)x (cid:18)y t (100) t (cid:3)2;(cid:3)x (cid:3)2(cid:3) t t (cid:0) (cid:0) () "0 1 #"(cid:23) x;t# (cid:16) (cid:17) Comparing equations (99) and (100), we also point out that the sunspot shock introduced in our representation, (cid:23) , has a clear interpretation: It is always equivalent to the forecast error x;t that is included in the auxiliary process. On the contrary, the sunspot shock, (cid:16) , in Lubik and t Schorfheide(2003)hasamorecomplexinterpretationandtheauthorsprovideaformalargument to consider it as a trigger of belief shocks that lead to a revision of the forecasts. 20Notice that the last equation of the solution in (93) shows that the auxiliary variable is such that ! = 0 t and therefore we report here the solution for the endogeonus variables of interest S . Also, we showed earlier t the equivalence with the representations that includes the forecast error, (cid:17) , obtained using the methodology of y;t Bianchi and Nicol(cid:242) (2018). 54

We then combine equations (99) and (100) to establish the following relationship (cid:18)2 y (cid:18) x (cid:18) y v (cid:18) y (cid:23) = + M " + (cid:16) : (101) x;t d2((cid:18) (cid:18) ) d t d t " x y # (cid:0) Plugging this relationship in the solution in equation (98) obtained using our methodology, we derive the solution in (97) derived using the methodology of Lubik and Schorfheide (2004). This result shows that any parametrization in Lubik and Schorfheide (2004) has a unique mapping to v our representation. In particular, we now consider the parametrization M = M ((cid:18))+M, where (cid:3) M is centered at 0 and M ((cid:18)) is found by minimizing the distance between the impulse response (cid:3) functions under determinacy and indeterminacy at the boundary of the determinacy region. We can therefore write equation (101) as (cid:23) = (cid:13) (M ((cid:18)))" +(cid:13) (cid:16) ; (102) x;t " (cid:3) t (cid:16) t where (cid:13) (M ((cid:18))) (cid:18)2 y (cid:18)x + (cid:18)yM ((cid:18)) and (cid:13) (cid:18)y. Given a parametrization M ((cid:18));(cid:27) " (cid:3) (cid:17) d2((cid:18)x (cid:18)y) d (cid:3) (cid:16) (cid:17) d f (cid:3) (cid:16) g (cid:0) and the normalizatiohn E[" t (cid:16) t ] = 0 in Lubikiand Schorfheide (2004), we derive the corresponding variance and covariance terms of the non-fundamental shock, (cid:23) , introduced in our approach as x;t (cid:27)2 (M ((cid:18))) = (cid:13)2(M ((cid:18)))(cid:27)2+(cid:13)2(cid:27)2 (103) (cid:23)x (cid:3) " (cid:3) " (cid:16) (cid:16) (cid:27) (M ((cid:18))) = (cid:13) (M ((cid:18)))(cid:27)2 (104) ";(cid:23)x (cid:3) " (cid:3) " The variance-covariance matrix of the shocks ^"x = " ;(cid:23) can be written as t t x;t 0 f g (cid:27)2 (cid:27) (M ((cid:18))) (cid:10) ^"x(M (cid:3) ((cid:18))) (cid:17) "(cid:27) ";(cid:23)x (M " (cid:3) ((cid:18))) (cid:27) " 2 (cid:23) ;(cid:23) x x (M (cid:3) (cid:3) ((cid:18))) # : (105) Implementing a Cholesky decomposition, the shocks ^"x = " ;vx can be written as t t t 0 f g (cid:27) 0 " " u ^"x t = "(cid:23) x t ;t# = L(M (cid:3) ((cid:18)))u t (cid:17) 2(cid:27)";(cid:23)x ( (cid:27) M " (cid:3) ((cid:18))) (cid:27)2 (cid:23)x (M (cid:3) ((cid:18))) (cid:0) (cid:27)";(cid:23)x ( (cid:27) M " (cid:3) ((cid:18))) 2 3 "u 1 2 ; ; t t# ; (106) r 4 (cid:16) (cid:17) 5 where Var(u ) = I and E(u ) = 0. Finally, the parametrization in Lubik and Schorfheide (2004) t t 55

can be mapped to the solution we obtained in equation (98) as y (cid:18)x 1 (cid:18)x t ((cid:18)y (cid:18)x) ((cid:18)y (cid:18)x) (cid:0) (cid:0) (cid:27) 0 2 x t 3 2 1 3 2 0 1 3 " u 1;t 6 6 6E E t t ( ( x y t t + + 1 1 ) ) 7 7 7 = 6 6 6 ((cid:18)y (cid:18) (cid:18) (cid:0) 2 x x (cid:18)x)7 7 7 E t (cid:0) 1 (x t )+ 6 6 6 0 0 ((cid:18)y (cid:18) (cid:18) (cid:0) 2 x x (cid:18)x)7 7 7 2 4 (cid:27)";(cid:23)x ( (cid:27) M " (cid:3) ((cid:18))) r (cid:27)2 (cid:23)x (M (cid:3) ((cid:18))) (cid:0) (cid:16) (cid:27)";(cid:23)x ( (cid:27) M " (cid:3) ((cid:18))) (cid:17) 2 3 5 "u 2;t# : 6 7 6 7 6 7 4 5 4 5 4 5 (107) 56

7.3 Appendix C Posterior distribution for model parameters (2-degrees of indeterminacy) (cid:23) =(cid:23) ;(cid:23) = (cid:23) (cid:23) =(cid:23) ;(cid:23) = (cid:23) (cid:23) =(cid:23) ;(cid:23) = (cid:23) 1 (cid:25) 2 y 1 (cid:25) 2 b 1 y 2 b f g f g f g Mean 90% prob. int. Mean 90% prob. int. Mean 90% prob. int. g 0.49 [0.45,0.54] 0.47 [0.42,0.52] 0.46 [0.42,0.51] (cid:21) 0.96 [0.93,0.99] 0.80 [0.65,0.94] 0.80 [0.67,0.94] (cid:11) 0.66 [0.60,0.72] 0.60 [0.52,0.69] 0.60 [0.52,0.68] 100((cid:13) (cid:0) 1 1) 0.50 [0.34,0.65] 0.46 [0.31,0.61] 0.45 [0.31,0.59] (cid:0) (cid:20) 0.040 [0.032,0.048] 0.042 [0.034,0.051] 0.039 [0.032,0.047] (cid:25) 0.68 [0.36,1.00] 0.70 [0.38,1.01] 0.72 [0.37,1.01] (cid:3) i 1.43 [1.09,1.75] 1.41 [1.09,1.74] 1.42 [1.08,1.74] (cid:3) (cid:30) 0.31 [0.14,0.48] 0.33 [0.17,0.50] 0.30 [0.14,0.46] (cid:25) (cid:30) 0.08 [0.04,0.12] 0.10 [0.05,0.15] 0.16 [0.08,0.23] q (cid:26) 0.75 [0.61,0.88] 0.67 [0.51,0.82] 0.62 [0.45,0.79] i (cid:27) 0.29 [0.15,0.44] 0.26 [0.13,0.38] 0.51 [0.19,0.85] q (cid:27) 0.11 [0.09,0.13] 0.11 [0.09,0.12] 0.11 [0.09,0.12] s (cid:27) 0.10 [0.08,0.12] 0.10 [0.09,0.12] 0.11 [0.09,0.13] i (cid:26) 0.94 [0.91,0.97] 0.68 [0.51,0.84] 0.68 [0.54,0.81] q (cid:26) 0.89 [0.83,0.94] 0.90 [0.85,0.94] 0.87 [0.81,0.92] s (cid:27) 0.28 [0.24,0.32] 0.27 [0.23,0.31] 0.70 [0.61,0.78] (cid:23)1 (cid:27) 0.69 [0.60,0.78] 2.59 [1.12,4.14] 1.74 [0.73,2.76] (cid:23)2 ’ -0.42 [-0.67,-0.16] -0.27 [-0.55,0.01] 0.08 [-0.15,0.34] (cid:23)1;i ’ 0.07 [-0.43,0.59] 0.11 [-0.45,0.67] 0.60 [0.40,0.81] (cid:23)1;q ’ 0.61 [0.48,0.73] 0.61 [0.48,0.73] -0.58 [-0.71,-0.44] (cid:23)1;s ’ -0.14 [-0.40,0.13] -0.53 [-0.75,-0.31] -0.72 [-0.93,-0.51] (cid:23)2;i ’ -0.01 [-0.52,0.55] -0.11 [-0.51,0.27] -0.30 [-0.63,0.03] (cid:23)2;q ’ -0.68 [-0.77,-0.59] -0.67 [-0.80,-0.55] -0.39 [-0.63,-0.18] (cid:23)2;s MDD -72.3 -73.0 -75.1 Table 6: Posterior distributions of the estimated model with two degrees of indeterminacy. 57

Posterior distribution for model parameters (1-degree of indeterminacy) (cid:23) =(cid:23) (cid:23) = (cid:23) (cid:23) =(cid:23) 1 b 2 (cid:25) 1 y f g f g f g Mean 90% prob. int. Mean 90% prob. int. Mean 90% prob. int. g 0.49 [0.45,0.54] 0.50 [0.45,0.54] 0.50 [0.45,0.54] (cid:21) 0.96 [0.93,0.99] 0.97 [0.94,0.99] 0.97 [0.94,0.99] (cid:11) 0.66 [0.60,0.72] 0.66 [0.60,0.72] 0.67 [0.61,0.73] 100((cid:13) (cid:0) 1 1) 0.50 [0.34,0.65] 0.47 [0.34,0.60] 0.50 [0.34,0.66] (cid:0) (cid:20) 0.040 [0.032,0.048] 0.043 [0.033,0.049] 0.040 [0.032,0.048] (cid:25) 0.68 [0.36,1.00] 0.71 [0.39,1.04] 0.70 [0.38,1.00] (cid:3) i 1.43 [1.09,1.75] 1.44 [1.13,1.77] 1.43 [1.11,1.74] (cid:3) (cid:30) 0.31 [0.14,0.48] 0.33 [0.14,0.50] 0.30 [0.12,0.45] (cid:25) (cid:30) 0.08 [0.04,0.12] 0.09 [0.04,0.14] 0.10 [0.04,0.16] q (cid:26) 0.75 [0.61,0.88] 0.74 [0.61,0.88] 0.79 [0.66,0.92] R (cid:27) 0.29 [0.15,0.44] 0.30 [0.14,0.46] 0.32 [0.14,0.48] q (cid:27) 0.11 [0.09,0.13] 0.11 [0.09,0.14] 0.11 [0.09,0.13] s (cid:27) 0.10 [0.08,0.12] 0.10 [0.09,0.12] 0.10 [0.08,0.12] i (cid:26) 0.94 [0.91,0.97] 0.93 [0.90,0.97] 0.92 [0.88,0.96] q (cid:26) 0.89 [0.83,0.94] 0.89 [0.84,0.94] 0.89 [0.84,0.94] s (cid:27) 6.11 [3.44,9.38] 0.28 [0.23,0.31] 0.74 [0.63,0.85] (cid:23)1 ’ -0.41 [-0.65,-0.19] -0.47 [-0.81,-0.12] -0.46 [-0.78,-0.10] (cid:23)1;i ’ -0.70 [-0.84,-0.56] -0.52 [-0.70,-0.33] 0.01 [-0.27,0.29] (cid:23)1;q ’ -0.46 [-0.57,-0.35] 0.46 [0.26,0.64] -0.64 [-0.76,-0.53] (cid:23)1;s MDD -83.2 -84.2 -83.0 Table 7: Posterior distributions of the estimated model with one degree of indeterminacy. 58

Posterior distribution for model parameters (determinacy) Mean 90% prob. int. g 0.40 [0.35,0.44] (cid:21) 0.98 [0.97,0.99] (cid:11) 0.69 [0.66,0.71] 100((cid:13) (cid:0) 1 1) 0.46 [0.41,0.51] (cid:0) (cid:20) 0.050 [0.042,0.058] (cid:25) 0.60 [0.34,0.85] (cid:3) i 1.38 [1.12,1.65] (cid:3) (cid:30) 1.58 [1.46,1.74] (cid:25) (cid:30) 0.03 [0.01,0.04] q (cid:26) 0.78 [0.74,0.82] i (cid:27) 0.23 [0.13,0.33] q (cid:27) 0.08 [0.07,0.09] s (cid:27) 0.20 [0.17,0.23] i (cid:26) 0.95 [0.94,0.97] q (cid:26) 0.94 [0.90,0.97] s MDD -158.3 Table 8: Posterior distributions of the estimated model under determinacy. 59