Refining linear rational expectations models and equilibria

Refining linear rational expectations models and equilibria

Journal of Macroeconomics 46 (2015) 160–169 Contents lists available at ScienceDirect Journal of Macroeconomics journal homepage: www.elsevier.com/l...

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Journal of Macroeconomics 46 (2015) 160–169

Contents lists available at ScienceDirect

Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro

Refining linear rational expectations models and equilibria Seonghoon Cho a,∗, Bennett T. McCallum b,c,1 a b c

School of Economics, Yonsei University, 50 Yonsei-ro, Seodaemun-Gu, Seoul 120-749, Republic of Korea Tepper School 256, Carnegie Mellon University, Pittsburgh, PA 15213, USA National Bureau of Economic Research, Cambridge, MA 02138, USA

a r t i c l e

i n f o

Article history: Received 30 August 2014 Accepted 18 September 2015 Available online 9 October 2015 JEL Classification: C6 D8 E3

a b s t r a c t This paper develops the case for forward convergence as a model refinement scheme for linear rational expectations models and an associated no-bubble condition as a solution selection criterion. We relate these two concepts to determinacy and characterize the complete set of economically relevant rational expectations solutions to the linear rational expectations models under determinacy and indeterminacy. Our results show (1) why a determinate solution is economically cogent in most, but not all, cases, and (2) that those models that are not forward-convergent have no economically relevant solutions. © 2015 Elsevier Inc. All rights reserved.

Keywords: Determinacy Forward convergence No-bubble condition Rational expectations

1. Introduction Recent macroeconomics literature has utilized the concept of determinacy as a primary criterion for characterizing the economic properties of a given rational expectations model and its solutions. While some researchers argue that determinacy is necessary and sufficient for a model to be economically relevant, others argue that multiple rational expectations solutions can be admissible as well in some cases of indeterminacy. Thus models and their solutions are not dismissed as implausible simply because they are indeterminate. Furthermore, determinacy is a criterion purely based on the number of stable solutions. Therefore, determinacy alone does not automatically warrant an economic plausibility of a given model and its determinate solution, as has been argued by Bullard and Mitra (2002), Cho and McCallum (2009), Honkapohja and Mitra (2004), and others.2 Several solution selection criteria have been proposed to narrow down the set of relevant equilibria in indeterminate models



Corresponding author. Tel.: +82 2 2123 2470; Fax:+82 2 393 1158. E-mail addresses: [email protected], [email protected] (S. Cho), [email protected] (B.T. McCallum). 1 Tel.: +1 412 268 2347; Fax: +1 412 268 6830. 2 Here the word “determinacy” is being used in the sense that is standard in monetary economics and frequently utilized elsewhere, namely, to designate a model specification in which there exists only one rational expectations solution that is dynamically stable - literally a “single stable solution”. (McCallum, 2012) has argued that this terminology is highly inappropriate, however, since the traditional meaning of the word “determinate” is that the model at hand clearly points to a single relevant solution. Thus the usage in question proceeds as if the single stable solution requirement was equivalent to the desired condition - i.e., that the model at hand provides a unique prediction as to the behavior of the (model) economy. A unique prediction is what “determinacy” is supposed to mean, however, so it is unsatisfactory for this word to be used as a synonym for the single stable solution condition, especially since there are examples in which it is clearly inappropriate. In the present paper we retain that usage, nevertheless, in order to facilitate communication. http://dx.doi.org/10.1016/j.jmacro.2015.09.004 0164-0704/© 2015 Elsevier Inc. All rights reserved.

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including, for example, the minimum state variable (MSV) criterion of McCallum (1983), McCallum (2007), the expectational stability criterion of Evans and Honkapohja (2001), and a fairly recent proposal by Driskill (2006). This extent of disagreement over the role of determinacy may have been a consequence of the absence of any step for refining “models” to begin with. Here we propose a model refinement scheme for a general class of linear models, together with a solution selection criterion, and characterize the complete set of economically plausible equilibria under determinacy and indeterminacy on the ground of our criterion. In particular, we adopt the forward convergence and no-bubble conditions proposed by Cho and Moreno (2011), and relate these refinement schemes to determinacy by means of characterization results reported by McCallum (2007). In their study of hyperinflation and monetary reform, Flood and Garber (1980) introduce the notion of “process consistency” and suggest that it is an essential characteristic of any model variable that pretends to serve as money. Specifically, in the context of a Cagan-type monetary model, they solve the model forward and argue that any “process inconsistent” money supply that explodes eventually will be rejected by the public because it does not provide a finite solution for the price level. In this setting, process consistency simply amounts to the case that a rational expectations model can be solved forward and they argue that any reasonable model should possess this minimal economic characteristic. (Blanchard and Kahn, 1980) also consider this principle in a class of models with lagged endogenous variables. When they derive the fundamental solutions, they state at the beginning of their Section 2 as “we also require that expectations of Xt (predetermined) and P t (non-predetermined) do not explode,… This in particular rules out “bubbles” of the sort considered by Flood and Garber (1980).” However, their requirement is an assumption and thus it is not examined because they did not solve the model forward explicitly. This method of solving rational expectations models forward had not been developed and applied to more general models with lagged variables until Cho and Moreno (2011) developed the forward method for such models. Their forward convergence property, which we propose as a model refinement, amounts to a generalization of process consistency and here we argue that, indeed, any model that fails to satisfy the forward convergence condition has no cogent rational expectations solution. The Cho and Moreno methodology also provides a solution selection criterion known as no-bubble condition in the class of the models that satisfy the forward convergence, and yields a well-known forward (forward-looking) solution in the sense of Blanchard (1979). When a model is solved forward, there remains a term involving the expectation of future endogenous variables, often called “a bubble term”. A non-zero expectational term implies that agents’ decision on current endogenous variables is influenced by this expectational effect in the future. It is well-known that the expectational term is not zero when evaluated with any non-fundamental solution, which depends not only on the state variables of the underlying model but also other variables outside the model. For this reason, this solution is often referred to as a bubble or sunspot solution. In contrast, a common understanding was that the expectational term should be zero when it is evaluated with a fundamental solution that depends only on the state variables of the model. Therefore, the absence of the expectational effect is a natural requirement for an equilibrium path to be characterized by the history of the state variables. Indeed, the literature has adopt this notion to select an equilibrium, and the resulting solution has been named as the forward solution. For example, using a simple asset bubble model, (Evans and Honkapohja, 2001) (pp. 220–221) actually define a fundamental solution by assuming that the bubble term is zero and show that the forward solution is the true equilibrium in their model. The transversality condition in infinite horizon models is also equivalent to the no-bubble condition in that the present discount value of capital stock should be zero. The contribution of Cho and Moreno (2011) is that general models with lagged endogenous variables can also be solved forward and this assumption can be verified by the forward method. Their key result is that the no-bubble condition – zero expectational term – holds only for one fundamental solution and that all other fundamental solutions fail to satisfy it, even though they are technically referred to as “bubble-free”. The solution that satisfies the no-bubble condition is named as the forward solution following the literature. This condition has been recently applied to find an equilibrium for a New-Keynesian model with bond pricing in Campbell et al. (2014). Accordingly, we contend that the no-bubble condition constitutes a relevant solution refinement scheme. Cho and Moreno (2011), however, do not relate their refinement schemes to determinacy. Accordingly, it is our task here to derive this relationship, drawing on results of McCallum (2007), and to characterize all the rational expectations equilibria in relation to determinate and indeterminate models. In the process, we will also extend one of McCallum’s results so as to apply to non-fundamental, as well as fundamental, solutions. The relationship between determinacy and our refinement schemes derived in this paper has three important implications. First, a determinate solution satisfies the no-bubble condition, thus coincides with the forward solution in the vast majority of macroeconomic models. As mentioned above, determinacy has been treated as a property for a well-defined economic model to possess in the literature although it is purely a statistical criterion. Therefore, our result provides an economic explanation for determinacy. Second, there does exist some cases, though rare, in which a determinate solution does not coincide with the forward solution. We show, through an example, that the determinate solution is hardly acceptable as an equilibrium in Section 5. Third, in the case of indeterminacy, we identify the conditions under which the forward solution does not exist. This class of models is ruled out on the ground of our model refinement, the forward convergence condition in the spirit of Flood and Garber (1980). The paper is organized as follows. Section 2 presents a general class of linear rational expectations models and characterizes the set of rational expectations equilibria. In Section 3, necessary and sufficient conditions for determinacy are stated. In section 4, we formally define the concept of forward convergence and study the relation between determinacy and forward convergence. Section 5 classifies rational expectations models with these two properties and characterizes the full set of equilibria. In Section 6, we apply our methodology to an example based on a standard New-Keynesian model. Section 7 concludes.

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2. Linear rational expectations models and rational expectations equilibria 2.1. The model We consider general linear rational expectations models of the form:

xt = AEt xt+1 + Bxt−1 + Czt , zt = Rzt−1 + et ,

(1)

where xt is an n × 1 vector of endogenous variables and zt is an l × 1 vector of exogenous variables, with R a stable l × l matrix, and et is an l × 1 vector of i.i.d. and mean zero shock processes. Also, Et (·) is mathematical expectation operator conditional on the information set available at time t.3 The linear model (1) that we have in mind is a local linear approximation around the steady state of the underlying dynamic stochastic general equilibrium model. Hence, we assume that the steady state is well-defined and known to all economic agents, and we present the model in terms of the deviations of the endogenous variables from their steady states. 2.2. Classes of solutions Any process xt that is consistent with the model (1 ) is a solution to the model. We decompose it into two components:

xt = xtFUN + wt ,

(2)

xtFUN

component.4

xtFUN ,

where is a fundamental solution and wt is a non-fundamental Note that for any given there is the corresponding class of wt . That is, the process of wt is restricted by a particular xtFUN . We discuss the two classes in detail. A. Fundamental solutions Any fundamental solution has the following form.

xt = xt−1 +  zt ,

(3)

where (,  ) must satisfy the following conditions:

 = (I − A)−1 B,

(4a)

 = (I − A)−1C + F  R,

(4b)

where F is given by:

F = (I − A)−1 A.

(5)

There are in general multiple solutions for  (thus  as well).5 But the number of fundamental solutions, which is the same as the number of  satisfying (4a), is finite as we explain in Section 4. B. Non-fundamental solutions The class of non-fundamental solutions has the following form:

xt = xt−1 +  zt + wt , where wt is an arbitrary process

(6) satisfying6

wt = F Et [wt+1 ],

(7)

where it is important to note that F is restricted in (5) by a particular . For each F, the following proposition characterizes the complete set of solutions to Eq. (7), thereby extending the result of McCallum (2007), which refers only to fundamental solutions. Proposition 1. Let m be the rank of the matrix F. Any real-valued solution wt to Eq. (7) can then be written as:

wt = wt−1 + V ηt ,

(8)

3 (McCallum, 2007) shows that any model of the class studied by King and Watson (1998) and Klein (2000) - which admits any finite number of lags, expectational leads, and lags of expectational leads - can be written in the form (1). 4 A fundamental solution is one that includes no extraneous state variables. This concept differs from McCallum (1983) minimum state variable (MSV) solution, however, in that the MSV solution is in all cases uniquely defined whereas there may be multiple fundamental solutions. 5 When there are no predetermined variables (B = 0n×n ), then  = 0n×n and F = A, implying that the fundamental solution is unique if it exists. 6 To see this, forward Eq. (6) one period ahead and plug it into (1) as follows:

xt = A(xt +  Rzt ) + AEt wt+1 + Bxt−1 + Czt . Rearranging this equation yields:

xt = (I − A)−1 Bxt−1 + [(I − A)−1C + F  R]zt + (I − A)−1 AEt wt+1 = xt−1 +  zt + F Et wt+1 , where ,  and F are given by Eqs. (4) and (5).

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where  is an n × n matrix such that  = V V  , V is an n × k matrix with 0 ≤ k ≤ m ≤ n such that its columns form a subset of an orthonormal basis associated with the inverses of non-zero eigenvalues of F,  is a k × k block-diagonal matrix such that V = FV , and ηt is an arbitrary k × 1 stochastic vector such that Et ηt+1 = 0n×1 . The indeterminate solution associated with (8) is given by:

xt = ( + )xt−1 − xt−2 +  zt −  zt−1 + V ηt .

(9)

Proof. See Appendix A.  There is a continuum of non-fundamental components in Eq. (8) because ηt is a vector of arbitrary i.i.d processes, possibly including the structural shocks et . However, there are finite number of  and it has a simple structure: non-zero eigenvalues of  are the inverses of some or all of the non-zero eigenvalues of F.7 This implies that we have only to study F in order to deduce stability of wt , and determinacy of the model without solving (8) directly. For a compact exposition regarding stability and determinacy, we define the spectral radius operator: Definition 1 (Spectral Radius). r(X ) = max{1≤i≤n} |ξi | : Rn×n → R+ where ξ i is the i-th eigenvalue of an n × n matrix X. Now we can state that a fundamental solution (3) is dynamically stable if and only if r() < 1. In addition, there exist stationary processes wt if r(F) > 1 because  in Eq. (8) can always be constructed such that it contains an eigenvalue equal to 1/r(F) < 1. Therefore, r(F) ≤ 1 is the condition under which there is no stationary stochastic process of wt . 3. Determinacy A model is said to be determinate if the model has a unique stable rational expectations solution. Different researchers use different representations of the underlying model, but many use essentially the same matrix decomposition theorem to derive the conditions for determinacy, i.e., the main theorem of Blanchard and Kahn (1980). But this method is available only when A is non-singular. Thus, the procedure of researchers often is to reformulate a model into a canonical form of Blanchard and Kahn (1980) if it has a singular A, following the steps proposed by King and Watson (1998), for instance. Instead, however, one can use a simpler way to identify determinacy, without transformation of (1), following (Klein, 2000) or McCallum(1998); 2007),  In −B ˜ and who utilize the generalized Schur decomposition theorem. To do so, define 2n × 2n dimensional matrices B= In 0n×n



A



0n×n . Solving for real-valued  amounts to choosing n roots out of the 2n generalized eigenvalues of the matrix 0n×n In ˜ namely, λ(B, ˜ A˜ ) = {λ1 , λ2 , . . . , λ2n }, where |λ1 | ≤ |λ2 |, … , ≤|λ2n |.8 Thus the number of fundamental solutions, pencil [B˜ − λA],

A˜ =

denoted by 2n Cn , is finite. Following (McCallum, 2007), determinacy can be stated in terms of the matrices  and F in Eqs. (4a) and (5), which govern the stability of fundamental and non-fundamental components of the rational expectations solutions, respectively. His representation then enables us to relate determinacy and the property of forward convergence. To proceed, we introduce an important property regarding the eigenvalues of  and F: for any  associated with n eigenvalues ˜ A˜ ), the eigenvalues of the corresponding F are the inverses of the remaining eigenvalues in λ(B, ˜ A˜ ). Let MOD denote in λ(B, the  associated with n smallest eigenvalues, (λ1 , λ2 , … , λn ) where MOD stands for minimum of modulus. The fundamental solution xt = MOD xt−1 +  MOD zt is known as the MOD solution where  MOD is uniquely defined in (4b) when  = MOD . Then the eigenvalues of FMOD are the set (1/λn+1 , 1/λn+2 , . . . , 1/λ2n ). Determinacy is the case in which the n smallest generalized eigenvalues are inside the unit circle and the remaining eigenvalues are greater than or equal to unity in absolute value. This is the case in which r(MOD ) < 1 and r(FMOD ) ≤ 1 . Therefore, if a model is determinate, the determinate solution must be the MOD solution. Following this idea of McCallum (2007), determinacy (indeterminacy) conditions can be stated in the following way. Proposition 2. Linear rational expectations models of the form (1) can be classified as follows. 1. The model (1) is determinate if and only if r(MOD ) < 1 and r(FMOD ) ≤ 1. 2. The model (1) is indeterminate if and only if r(MOD ) < 1 and r(FMOD ) > 1. 3. The model (1) has no stable rational expectations solutions if and only if r(MOD ) ≥ 1. Proof. See (McCallum, 2007).9



7 (Lubik and Schorfheide, 2004) also derive an alternative representation of an indeterminate solution based on the result of Sims (2002) in which the forecast errors of the forward-looking variables are partitioned into a vector of structural shocks and a vector of sunspot shocks. For instance, the arbitrary matrices ˜ and Mς in their equation (25) govern the propagation of the structural shocks and sunspot shocks, respectively. In our case, ηt can be decomposed into M ηt = A1 et + A2 ςt where ς t is a sunspot shock that is unrelated to et , and A1 and A2 correspond to M˜ and Mς of Lubik and Schorfheide (2004). Their set of indeterminate solutions are equivalent to ours (9) as long as the underlying model is the same. 8 If a complex root is included, its conjugate member must also be included for  to be real-valued. Modulus equality holds when two eigenvalues form a complex conjugate. 9 A subtle issue may arise in the case of r(F ) = 1. For instance, consider a univariate model. When r(F ) = 1, wt = a solves (7) for any arbitrary constant a. There seems to be no general agreement as to whether this case is considered as indeterminate or determinate. However, since such a solution is non-stochastic, we include r(F ) = 1 as determinacy. If one alternatively treats this case as indeterminate, then one may define determinacy that excludes r(F ) = 1 in Assertion 1 and includes it in Assertion 2 of Proposition 2.

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Assertions 2 and 3 of Proposition 2 can also be easily understood. In the case of indeterminacy, r(MOD ) < 1 still holds but at least one of the generalized eigenvalues of (λn+1 , λn+2 , . . . , λ2n ) is less than one in absolute value, implying that r(FMOD ) > 1. When the maximum eigenvalue of (λ1 , λ2 , … , λn ) is greater than 1, the condition r(MOD ) ≥ 1 must be true. Recall that any non-fundamental solution is the sum of a fundamental solution and a sunspot component from Eq. (8). Therefore, there is no stable rational expectations solution in this case. 4. The forward convergence and no bubble conditions Nothing in the foregoing determinacy/indeterminacy conditions serves to establish economic relevance for the determinate or indeterminate solutions on any basis. Multiple stable solutions or even an unstable solution can be accepted as plausible predictions of a given model. The forward method of Cho and Moreno (2011), however, provides a model refinement scheme, which generalize the process consistency of Flood and Garber (1980) in the sense of ruling out explosive bubbles and a solution selection criterion within the class of fundamental solutions. It should also be stressed that the refinement schemes of the forward method are completely independent of determinacy and indeterminacy based on the number of stable solutions. Following their method, we solve the model (1) into the future so as to derive its forward representation as follows:

xt = Mk Et xt+k + k xt−1 + k zt ,

(10)

where (Mk , k ,  k ) is given by: M1 = A, 1 = B, 1 = C, and for k > 1,

Mk = Fk−1 Mk−1 ,

(11)

k = (I − Ak−1 )−1 B,

(12a)

k = (I − Ak−1 )−1C + Fk−1 k−1 R,

(12b)

where F1 = A and for k > 1, Fk is given by:

Fk = (I − Ak−1 )−1 A,

(13)

provided that the regularity condition det (I − Ak ) = 0 is satisfied for all k = 1, 2, 3, . . .. Definition 2 (Forward Convergence Condition, FCC). The model (1) is said to satisfy the forward convergence condition if (k ,

 k ) defined in (12) converge as k → ∞.

Note that if (k ,  k ) converges to (∗ ,  ∗ ), Eq. (12) fulfill the conditions (4). Note also that Fk in (13) converges to F∗ if and only if k converges. Hence, under the FCC, the matrix F∗ defined by Eq. (13) fulfills the condition (5) as well. Therefore, the following forward solution,

xt = ∗ xt−1 +  ∗ zt

(14)

is a fundamental solution and it exists if and only if the model satisfies the FCC. Hence the FCC and the existence of the forward solution are equivalent. Therefore, the matrices (k ,  k ) are unique and implied by the model. Accordingly, the forward solution is a model-implied relation and, consequently, it is economically relevant by itself, i.e., inherently. Evidently, forward convergence is exactly the same concept for general linear rational expectations models as the process consistency of Flood and Garber (1980), a paper that seems accordingly to have been undervalued in the literature. A key implication of the forward method is that the expectational term Mk Et xt+k depends on the particular solution with which expectations are formed. In principle, the limiting behavior of this “bubble” term should be verified for each rational expectations solution, instead of assuming its behavior. Formally we define the no-bubble condition: Definition 3 (No-bubble condition, NBC). A solution to the model (1) is said to satisfy NBC if limk→∞ Mk Et xt+k = 0n×1 in (10) when expectations are formed with that solution. The following proposition provides a central result of the forward method. Proposition 3. The forward solution is the unique rational expectations solution that satisfies the NBC. Proof. See Appendix B.



Thus the NBC is the unique feature that differentiates the forward solution from all other fundamental solutions. The NBC then removes two kinds of equilibria. First, it refines away all the solutions for those models that fail to satisfy the FCC. Suppose that a stable fundamental solution xt = xt−1 +  zt exists for a model where one or both of (k ,  k ) explodes as k tends to infinity. Since this solution must satisfy the forward representation (10) for all k, this implies that the expectational term evaluated with this solution must explode as well. This would be the major consideration underlying the restriction in Blanchard and Kahn (1980). In this sense, the FCC can be interpreted as a model refinement scheme and the NBC refines away all those fundamental solutions. The role of the NBC is, however, not confined to models that are not forward-convergent. In addition, the NBC also refines away all the stable fundamental solutions, different from the forward solution for the models that are convergent, that would arise in the case of indeterminacy. Fundamental solutions are often conceived of and referred to as bubble-free solutions.

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Thus, the notion of bubble-free solutions that violate our no-bubble condition would be basically incoherent. This refinement is of important when, unlike (Flood and Garber, 1980), we consider models with predetermined variables. In both cases, since any rational expectations solution can be written as sum of a fundamental solution and the associated bubble term, then if any fundamental solutions violate the NBC, any member of the set of rational expectations solutions associated with those solutions can hardly be justifiable economically – i.e., is not analytically coherent. 5. Complete characterization of the rational expectations solutions under FCC Now we bridge the relation between the FCC and determinacy, which is absent from Cho and Moreno (2011). Under the FCC, F ∗ = limk→∞ Fk exists from Eq. (13). The following proposition shows that under FCC, determinacy corresponds to r(F∗ ) ≤ 1 and indeterminacy to r(F∗ ) > 1, and thus characterizes the full set of rational expectations solutions. Proposition 4. Suppose that the rational expectations model (1) satisfies the forward convergence condition. 1. If r(∗ ) < 1 and r(F∗ ) ≤ 1, then model (1) is determinate and the unique stationary solution is given by the forward solution:

xt = ∗ xt−1 +  ∗ zt .

(15)

2. If r(∗ ) < 1 and r(F∗ ) > 1, then (1) is indeterminate and the set of stable rational expectations solutions for which the fundamental solutions satisfy the NBC is given by

xt = ∗ xt−1 +  ∗ zt + wt ,

(16)

where wt is an arbitrary stationary process such that wt = F ∗ Et wt+1 and the whole set of wt associated with F∗ can be constructed from Proposition 1. Proof. See Appendix C.  Assertion 1 indicates why a determinate solution selected by a statistical criterion of stability is usually an economically relevant equilibrium of the underlying model: it is because ∗ = MOD for a very large fraction of all rational expectations models. The conditions of Assertion 1 are straightforward to check and these conditions ensure that the determinate solution is the unique stable equilibrium that is independent of the bubble term, i.e., the expectational effect of the future endogenous variables. The conditions in Assertion 1 of Proposition 4 are, however, sufficient for determinacy, but not necessary. This implies that there can be cases in which a model is determinate but the determinate solution is not the forward solution from Proposition 2; and therefore the determinate solution violates the NBC. It seems that the determinate solution may differ from the forward solution only when a model has a block-recursive structure such that a self-contained autonomous block of the model is behaviorally independent of the remaining block. The implicit assumption is that agents in the autonomous block use only the information embedded in that block, when forming expectations. For example, consider a real business cycle model featuring neutrality of money. The real sector is by itself a well-defined sub-model insulated from the monetary sector. The real variables then become exogenous, determining the money stock in equilibrium. If the real sector is explosive and the remaining block is indeterminate, the model can have a determinate solution in which the equilibrium path of the real variables depend on the stock of money, which is contradictory to the presumption of neutrality of money. In contrast, the forward solution in this case, however is consistent with neutrality of money, although is not stable.10 Now we consider the indeterminate case in which the model satisfies the FCC, but r(F∗ ) > 1. In this case, there exists a continuum of non-fundamental rational expectations solutions associated with the forward solution. But there may well exist ˜ ) < 1 and r(F˜ ) > 1.11 Such a ˜ different from ∗ and the corresponding F˜ as long as r( fundamental solutions associated with  solution must violate the NBC from Proposition 3. Therefore, the set of rational expectations solutions that are consistent with the FCC and NBC are the ones associated with the forward solution. ˜ ) ≥ 1, then Proposition 4 excludes the following cases as they have no relevant rational expectations equilibria. First, if r( the model has no stationary solution of which the fundamental component satisfies the NBC. Second, those models that are not forward convergent have no fundamental solutions satisfying the NBC. This latter case is the one for which the NBC is the most important as a solution refinement: without verifying the FCC, one may find solutions to a model that fails to satisfy the FCC. We present an example of this kind in the following section. Our results show the importance of examining the FCC of rational expectations models. Moreover, our methodology is sufficient to identify determinate and indeterminate cases and it provides a complete set of solutions to any linear rational expectations models satisfying the FCC. Another important feature of our methodology is that the solution method and the solution refinements are obtained using only the rationality assumption and the recursive structure of the underlying macroeconomic model, without solving for all the mathematical solutions using matrix decomposition techniques.

10 Another economic example is an open economy model of Gali and Monacelli (2005) in which the foreign country is a large closed economy and the home country is a small open economy. The equilibrium of the foreign country is self-determined and serves as an exogenous process for the equilibrium of the home country. If the foreign country has no stable solution and the block of the home country is independent, the foreign variables does depend on the home country in the determinate solution. An atheoretical numerical example of this kind can also be found in Cho and McCallum (2009). 11 ˜ ) < 1 and r(F˜ ) ≤ 1, because this would imply that the model is determinate. ˜ different from ∗ , such that r( There cannot be a fundamental solution with 

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ρ = 0.8 1 φy

Indeterminacy FCC

Determinacy FCC

0.5 Indeterminacy Non−FCC 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ρ = 0.9 1 φy

Indeterminacy FCC

Indeterminacy Non−FCC

φπ

2

Determinacy FCC

0.5 Case 2 Case 1

Case 3 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

φπ

2

Fig. 1. Regions of Determinacy, Indeterminacy and Forward Convergence. This figure plots the determinacy and indeterminacy regions of the New-Keynesian model (17). The indeterminacy region is decomposed into the one where the FCC holds and the other one where the FCC fails to hold.

6. Example In this section, we present a New-Keynesian model similar to the one considered by Cho and Moreno (2011), but detect determinacy and indeterminacy using Proposition 4 and further investigate the reasons why the solutions to the model are difficult to serve as relevant equilibria when the model is not forward convergent. The model is given by the three equations.

πt = β Et πt+1 + κ yt ,

(17a)

yt =

μEt yt+1 + (1 − μ)yt−1 − θ (it − Et πt+1 ) + zt ,

(17b)

it =

φπ πt + φy yt ,

(17c)

where π t , yt , it are respectively inflation, the output gap and the nominal interest rate. zt is an aggregate demand shock that follows an AR(1) process: zt = ρzt−1 + t where 0 ≤ ρ < 1 and  t is an i.i.d. shock. β is the time discount factor, κ is the Phillips curve parameter and θ is the elasticity of the output gap with respect to the real interest rate. We allow the output gap depends on both the expected future with μ and lagged output gaps with 1 − μ. Eq. (17c) is a Taylor rule reacting to inflation and the output gap. By substituting out it , the model (17) can be cast into a bivariate system as xt = AEt xt+1 + Bxt−1 + Czt where xt = [πt yt ] and A1 , B = B−1 B2 , C = B−1 C1 with A = B−1 1 1 1



B1 =

1

θ φπ



−κ , 1 + θ φy



A1 =

β 0

0

μ





,

B2 =

 



0 0 , 0 1−μ

C1 =

0 . 1

We set the parameter values as β = 0.99, κ = 0.3, μ = 0.7, θ = 1, following the literature, for example, Lubik and Schorfheide (2004). Now, we partition the parameter space in terms of φ π and φ y in Taylor rule in two dimensions: determinacy and forward convergence condition. First, determinacy and indeterminacy regions can be identified by the combinations of φ π and φ y such that r(∗ ) < 1 and r(F ∗ ) = 1 from Proposition 4. The parameter space where the FCC holds can be identified by examining the existence of ∗ and  ∗ defined in (12). Fig. 1 plots these partitions of the parameter space when ρ = 0.8 and 0.9. In the absence of the lagged output gap in (17b), (Bullard and Mitra, 2002) show analytically that the model can be determinate when the coefficient of inflation in Taylor rule φ π is slightly smaller than 1 if φ y is large enough. In our example, although the determinacy condition cannot be obtained in closed form due to the presence of the lagged variable, Figure 1 shows a result similar to that of Bullard and Mitra (2002). Recall that k = (I − Ak−1 )−1C + Fk−1 k−1 R from (12b). Since R = ρ in this example, the matrix governing convergence of  k is ρ F∗ . That is,  ∗ exists if and only if ∗ exists and ρ r(F∗ ) < 1. Therefore, if the model is determinate, the FCC is always satisfied. When the model is indeterminate, i.e., r(F∗ ) > 1, the FCC still holds if r(F∗ ) < 1/ρ . However, it fails to hold as long as r(F∗ ) ≥ 1/ρ . The regions denoted by FCC and Non-FCC in Fig. 1 represent these two cases respectively. Comparing the top and bottom panels, it is easy to see that the Non-FCC region becomes larger, the higher is ρ .

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To assess economic relevance of the model and the solutions corresponding to the three cases displayed in Fig. 1, we provide numerical examples when φy = 0.1 and ρ = 0.9 as follows. Case 1. When φπ = 1.5, the FCC holds and r(∗ ) = 0.208 with r(F ∗ ) = 0.624. Therefore, the model is determinate and the determinate forward solution is given by:









0 0.079 1.490 xt = x + z. 0 0.208 t−1 0.430 t This solution implies that when there is a rise in aggregate demand shock of size 1, both output and inflation increase, which is the case that would be expected from economic theory. Case 2. When φπ = 0.9, the FCC holds and r(∗ ) = 0.254 with r(F ∗ ) = 1.073. Hence, the model is indeterminate. The forward solution exists and it is qualitatively similar to the one under determinacy. A set of indeterminate solutions can be constructed using Eq. (16). There exists, however, another stable fundamental solution:









0 3.629 −9.369 xt = x + z. 0 0.932 t−1 −0.262 t In contrast to the forward solution, both coefficients in  are negative. This is clearly counterintuitive because it implies that an exogenous increase in aggregate demand decreases both inflation and the output gap. We rule out this solution on the ground of the NBC. In this case, we find that this solution is also rejected by other solution refinement schemes such as the MSV criterion of McCallum (1983) and expectational stability of Evans and Honkapohja (2001).12 However, this is not the case in the following model, which fails to satisfy the FCC. Case 3. When φπ = 0.8, the FCC does not hold: whereas ∗ (and F∗ ) still exists,  k explodes as k → ∞. In this example, r(∗ ) = 0.265 and r(F ∗ ) = 1.151. Therefore, the model is again indeterminate, and there are two stable fundamental solutions. Unlike Case 2, however, the model fails to satisfy the FCC because r(ρ F ∗ ) = 1.038 > 1, implying that  k grows without bound. The MOD solution is given by:13









0 0.108 −18.999 xt = x + z. 0 0.265 t−1 −5.092 t Just like the fundamental solution in Case 2 other than the forward solution, a rise in aggregate demand decreases both inflation and the output gap. In fact, in this case of indeterminacy and Non-FCC displayed at the bottom panel of Fig. 1, the values of  in the MOD solution above are all negative, and approach to negative infinity as φ π increases to 0.850 from below. In contrast, expectational stability, one of the most popular solution selection criteria, does not distinguish Case 2 and Case 3, and fails to refine away the MOD solution in the latter case as it turns out to be expectationally stable.14 This example illustrates the importance of the forward convergence requirement as a model refinement scheme. 7. Conclusion The forward convergence and no-bubble conditions generalize and modify the process consistency criterion of Flood and Garber (1980) and the restrictions on the expectations of the variables in the future in Blanchard and Kahn (1980), respectively, both of which have often been assumed to hold but not examined for general rational expectations models in the literature. We demonstrate the importance of forward convergence as a powerful model refinement scheme for linear rational expectations models. Within the class of forward convergent models, we completely characterize the set of economically relevant equilibria to a given model based on the solution refinement scheme. Our analysis indicates that the forward solution is the unique stable solution independent of the bubble term for almost all determinate economic models and, consequently, provides some economic justification for emphasis on a determinate equilibrium. In the case of indeterminacy, moreover, the forward convergence and no-bubble condition detect whether a given model is by itself economically reasonable, and if so, the forward method provides the set of relevant equilibria. We show, through a standard New-Keynesian example, that some indeterminate models may not be forward-convergent and, the equilibrium path predicted by any solution to such models is completely opposite – thus counterintuitive - to what is predicted by the forward solution to the forward-convergent models. Therefore, the forward convergence condition must be verified, not assumed, and should be both useful and important in practice as a model refinement in such cases.

12 Of course, this does not imply that all of these solution selection criteria are equivalent. An obvious distinction is that whereas it is possible for more than one solution to be E-stable, only the forward solution can pass the NBC. 13 The other fundamental solution is also qualitatively similar to the MOD solution. 14 To compare the expectational stability criterion with the FCC (and the NBC as well) on the same basis, we assume that a vector of constants is not included in the perceived law of motion (PLM).

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Acknowledgments Seonghoon Cho acknowledges that this work was partly supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2010-327-B00089) and by the Yonsei University Research Fund of 2011. Appendix A. Proof of Proposition 1 Consider an arbitrary stochastic process wt and let V be a n × k matrix with 0 ≤ k ≤ n where its columns are orthonormal, spanning the support of wt for all t. This implies that Et wt+1 ∈ Col (V ) almost surely as well. Since wt is a solution to wt = F Et wt+1 , wt must be in the column space of F. Hence, the columns of V can be interpreted as the members of an orthonormal basis for Col(F) without loss of generality. Note that the number of columns of V cannot be greater than that of F, i.e., 0 ≤ k ≤ m ≤ n. Next, we apply the real Schur decomposition theorem to the matrix F to find an n × m orthonormal matrix U and an m × m block diagonal matrix  of full rank such that F = U  U  . This can be written as

FU  −1U  = UU  .

(A1)

 −1 .

Note that since  is block-diagonal, so is Since the columns of V are some or all of the columns of U, we can partition U into two parts such that U = [V V0 ] where V0 has the remaining columns of U. We also partition  −1 into two parts such that   O  −1 = where  is the k × k block-diagonal matrix associated with V and 0 is the (m − k) × (m − k) block-diagonal O 0 matrix associated with V0 . Since we consider a real-valued stochastic process wt only, if  contains eigenvalues,  complex-valued    I  k =[V V0 ] =V . Now their conjugate members must also be the eigenvalues of . Then, for any V, U  −1U V =[V V0 ] −1 Om−k Om−k post-multiply V to both sides of Eq. (A1) to yield FU  −1U V = UU V = V . Therefore, we have:

FV V  = F  = VV  ,

(A2)

where  = V V  . Post-multiplying V −1V  to both sides of (A2) yields FVV  = V −1V  .15 Note that Et wt+1 = VV  Et wt+1 almost surely. Therefore, the model (7) is almost surely identical to wt = FVV  Et wt+1 = V −1V  Et wt+1 . Pre-multiplying  to both sides of (7), we have16

wt = VV  Et wt+1 .

(A3)

Finally, we define the vector of rational expectations errors ut+1 ≡ wt+1 − Et wt+1 . Since wt+1 and Et wt+1 are in the column space of V, ut+1 must also be in Col(V), implying that ut+1 can be written as V ηt+1 . Therefore, from (A3), VV  Et wt+1 = wt+1 − V ηt+1 = wt , which is Eq. (8). Finally, we subtract xt−1 from xt in Eq. (6) to eliminate wt , and write the indeterminate solution in the following way.

xt = ( + )xt−1 − xt−2 +  zt −  zt−1 + V ηt , which is Eq. (9). Q.E.D. Appendix B. Proof of Proposition 3 We basically repeat the formal proof given in Cho and Moreno (2011, p. 266) as it is a crucial step for establishing our main result. Consider the model (1) and suppose that the FCC holds. Then the forward solution exists and is given by Eq. (14) xt = ∗ xt−1 +  ∗ zt . Since the matrices (k ,  k ) in equation (12) are unique and real-valued, so are the limiting values (∗ ,  ∗ ). Since the forward solution is a fundamental solution, it must solve the forward representation of the model (10) as k goes to infinity:

xt = lim Mk Et xt+k + ∗ xt−1 +  ∗ zt . k→∞

(B1)

Therefore, it must be true that limk→∞ Mk Et xt+k = On×1 when expectations are formed with the forward solution, implying that the forward solution satisfies the NBC. Suppose that the NBC holds for another (fundamental or non-fundamental) solution. Since the solution must solve (B1), (B1) becomes the forward solution, which contradicts the supposition that this solution differs from the forward solution. When the FCC does not hold, (k ,  k ) does not converge or is not well-defined if the regularity condition is violated. Consequently, for any other solution, limk→∞ Mk Et xt+k is not well-defined, implying the violation of the NBC. Q.E.D. Appendix C. Proof of Proposition 4 Assertion 1. Note that the eigenvalues of ∗ and the inverses of the eigenvalues of F∗ constitute the generalized eigenvalues of the model. Therefore, if r(∗ ) < 1 and r(F∗ ) ≤ 1, then there are exactly generalized eigenvalues inside the unit circle. Therefore, ∗ is MOD and the model is determinate from Proposition 2. 15 16

This operation does not lose any information as we can recover (A2) by post-multiplying VV to FVV  = V −1V  . This operation does not lose any information as we can recover (7) by multiplying F to (A3).

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Assertion 2. Suppose that r(∗ ) < 1 and r(F∗ ) > 1. From Proposition 3, the forward solution component of (16) satisfies the NBC and all other fundamental solutions violate the NBC from Proposition 3. Note that F∗ must contain at least one root outside the unit circle. Let k be the number of some or all of such unstable roots (1 ≤ k ≤ m ≤ n). Following Proposition 1, one can construct ∗ a stationary process wt∗ = Et wt+1 + V ηt where the non-zero eigenvalues of  are the inverses of the chosen unstable roots of F∗ . Q.E.D. References Blanchard, O.J., 1979. Backward and forward solutions for economies with rational expectations. Am. Econ. Rev., Papers Proc. 69 (2), 114–118. Blanchard, O.J., Kahn, C.M., 1980. The solution of linear difference models under rational expectations. Econometrica 48 (5), 1305–1311. Bullard, J., Mitra, K., 2002. Learning about monetary policy rules. J. Monet. Econ. 49 (6), 1105–1129. Campbell, J.Y., Pflueger, C., Viceira, L.M., 2014. Monetary policy drivers of bond and equity risks. NBER Working Paper No. 20070. Cho, S., McCallum, B.T., 2009. Another weakness of “determinacy” as a selection criterion for rational expectations models. Econ. Lett. 104 (1), 17–19. Cho, S., Moreno, A., 2011. The forward method as a solution refinement in rational expectations models. J. Econ. Dyn. Control 35 (3), 257–272. Driskill, R., 2006. Multiple equilibria in dynamic rational expectations models: a critical review. Eur. Econ. Rev. 50 (1), 171–210. Evans, G.W., Honkapohja, S., 2001. Learning and Expectations in Macroeconomics. Princeton University Press. Flood, R.P., Garber, P.M., 1980. An economic theory of monetary reform. J. Polit. Econ. 88 (1), 24–58. Gali, J., Monacelli, T., 2005. Monetary policy and exchange rate volatility in a small open economy. Rev. Econ. Stud. 72 (3), 707–734. Honkapohja, S., Mitra, K., 2004. Are non-fundamental equilibria learnable in models of monetary policy? J. Monet. Econ. 51 (8), 1743–1770. King, R.G., Watson, M.W., 1998. The solution of singular linear difference systems under rational expectations. Int. Econ. Rev. 39 (4), 1015–1026. Klein, P., 2000. Using the generalized schur form to solve a multivariate linear rational expectations model. J. Econ. Dyn. Control 24 (10), 1405–1423. Lubik, T.A., Schorfheide, F., 2004. Testing for indeterminacy: an application to US monetary policy. Am. Econ. Rev. 94 (1), 190–217. McCallum, B.T., 1983. On non-uniqueness in rational expectations models: an attempt at perspective. J. Monet. Econ. 11 (2), 139–168. McCallum, B.T., 1998. Solutions to linear rational expectations models: A compact exposition. Economics Letters 61 (2), 143–147. McCallum, B.T., 2007. E-stability vis-a-vis determinacy results for a broad class of linear rational expectations models. J. Econ. Dyn. Control 31 (4), 1376–1391. McCallum, B.T., 2012. Determinacy, learnability, plausibility, and the role of money in new keynesian models. NBER Working Paper No 18215. Sims, C.A., 2002. Solving linear rational expectations models. Comput. Econ. 20 (1), 1–20.