Determinacy, indeterminacy and dynamic misspecification in linear rational expectations models

Determinacy, indeterminacy and dynamic misspecification in linear rational expectations models

Journal of Econometrics 170 (2012) 153–163 Contents lists available at SciVerse ScienceDirect Journal of Econometrics journal homepage: www.elsevier...

304KB Sizes 3 Downloads 67 Views

Journal of Econometrics 170 (2012) 153–163

Contents lists available at SciVerse ScienceDirect

Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom

Determinacy, indeterminacy and dynamic misspecification in linear rational expectations models Luca Fanelli ∗ Department of Statistical Sciences, University of Bologna, via Belle Arti 41, I-40126, Italy

article

info

Article history: Received 3 February 2011 Received in revised form 4 April 2012 Accepted 21 April 2012 Available online 2 May 2012 JEL classification: C12 C32 C52 E31 Keywords: Determinacy GMM Indeterminacy LRE model Identification Maximum likelihood VAR VARMA

abstract This paper proposes a testing strategy for the null hypothesis that a multivariate linear rational expectations (LRE) model may have a unique stable solution (determinacy) against the alternative of multiple stable solutions (indeterminacy). The testing problem is addressed by a misspecification-type approach in which the overidentifying restrictions test obtained from the estimation of the system of Euler equations of the LRE model through the generalized method of moments is combined with a likelihood-based test for the cross-equation restrictions that the model places on its reduced form solution under determinacy. The resulting test has no power against a particular class of indeterminate equilibria, hence the non rejection of the null hypothesis can not be interpreted conclusively as evidence of determinacy. On the other hand, this test (i) circumvents the nonstandard inferential problem generated by the presence of the auxiliary parameters that appear under indeterminacy and that are not identifiable under determinacy, (ii) does not involve inequality parametric restrictions and hence the use of nonstandard inference, (iii) is consistent against the dynamic misspecification of the LRE model, and (iv) is computationally simple. Monte Carlo simulations show that the suggested testing strategy delivers reasonable size coverage and power against dynamic misspecification in finite samples. An empirical illustration focuses on the determinacy/indeterminacy of a New Keynesian monetary business cycle model of the US economy. © 2012 Elsevier B.V. All rights reserved.

1. Introduction It is well known that linear rational expectations (LRE) models can have multiple equilibria, a situation referred to as indeterminacy. Determinacy refers to a LRE model that has a unique stable (asymptotically stationary) solution. The time series representation of a determinate LRE model differs substantially from that of an indeterminate one because the latter is characterized by a set of arbitrary parameters not related to the structural parameters and, in addition, by a set of stochastic disturbances independent from the structural disturbances that reflect the agents’ self-fulfilling beliefs, see e.g. Benhabib and Farmer (1999). Recently, macroeconomists have become increasingly interested in evaluating the determinacy/indeterminacy of a class of linear(ized) business cycle monetary LRE models of the New Keynesian tradition, often termed small-scale dynamic stochastic general equilibrium (DSGE) models, with the objective of investigating



Tel.: +39 0541 434303; fax: +39 051 232153. E-mail address: [email protected]

0304-4076/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jeconom.2012.04.002

possible sources of the ‘US Great Moderation’, see, inter alia, Lubik and Schorfheide (2004) and Benati and Surico (2009). Indeed, the equilibrium implied by these models may not be unique if the central bank does not raise aggressively enough the nominal interest rate in response to inflation. According to a recent interpretation, the ‘passive’ monetary policy conduct and the resulting indeterminacy of the system would explain the ‘US Great Inflation’ observed before the appointment of Paul Volcker as Chairman of the Federal Reserve at the end of the 1970s and, instead, the more aggressive monetary policy undertaken from the eighties onwards would explain why the system moved from indeterminacy to a unique equilibrium. On the econometric side, abstracting from the ‘explosive indeterminacy’ associated with rational bubbles, only a few studies have dealt with the problem of assessing determinacy/ indeterminacy in stable dynamic macroeconomic models featuring rational expectations. Salemi (1986), Pesaran (1987), Broze and Szafarz (1991) and Salemi and Song (1992) show that the nuisance parameters that index the equilibria generated by LRE models under indeterminacy can be estimated consistently by likelihood methods but do not address the testing problem. Imrohoroğlu (1993) presents a frequentist approach for testing the existence of

154

L. Fanelli / Journal of Econometrics 170 (2012) 153–163

a unique stable solution by confining attention to a very special case and his setup is also consistent with the occurrence of non-stationary explosive equilibria. In the current literature, the only formalized test (model comparison) of the hypotheses of determinacy against indeterminacy in stable LRE models has been put forth by Lubik and Schorfheide (2004) through a Bayesian approach. This paper addresses the problem in a family of multivariate LRE models that covers many of the models currently used in finance and macroeconomics. In this class of models, if a unique stable solution exists, it can be represented as a finite order vector autoregression (VAR) whose coefficients are subject to a set of nonlinear cross-equation restrictions (CER), see Hansen and Sargent (1980, 1981) and Binder and Pesaran (1995). Conversely, if multiple stable solutions occur, system equilibria can be represented as vector autoregressive moving average (VARMA) processes with coefficients subject to nonlinear CER. These VARMA-type solutions are characterized by two independent sources of indeterminacy: a set of auxiliary parameters that are not related to the structural parameters and enter the moving average part of the solution, and a set of ‘sunspot shocks’ which do not depend on the structural disturbances. For a particular configuration of the auxiliary parameters, the autoregressive and moving average polynomials of the indeterminate VARMAtype solutions share ‘common’ roots. The cancellation of these common roots generates, if sunspot shocks are absent, a minimum state variable (MSV) solution (McCallum, 1983, 2003) which is observationally equivalent to the VAR representation of the LRE model under determinacy. It turns out that the statistical assessment of determinacy/indeterminacy in multivariate LRE models involves nuisance parameters. The direct comparison of the likelihood of the solution obtained under determinacy with the likelihood of the solutions obtained under indeterminacy gives rise to a nonstandard inferential problem because the auxiliary parameters that index the multiplicity of solutions are not identifiable under determinacy. Unfortunately, the general solution for these type of problems suggested by Andrews and Ploberger (1994) is difficult to implement in a multivariate setup. In some cases the determinacy (indeterminacy) region of the parameter space of the LRE model can be uniquely associated with a set of inequality restrictions, see e.g. Lubik and Schorfheide (2004) and Farmer and Guo (1995). In these situations, one can potentially rely on a consistent estimator of the structural parameters robust to determinacy/indeterminacy and check the validity of these restrictions.1 However, hypothesis testing in this case requires nonstandard inference, see e.g. Kodde and Palm (1986). Another complication in testing determinacy against indeterminacy arises from the possible dynamic misspecification of the LRE model, where by the term ‘dynamic misspecification’ is meant the omission of relevant lags/leads and/or variables from the specified equations. Since the VARMA-type solutions obtained under indeterminacy present a richer dynamic structure compared to the VAR solution obtained under determinacy, the investigation of the time series properties of the data might erroneously lead one to mix up the omission of propagation mechanisms in the structural

1 For instance, Farmer and Guo (1995) suggest using instrumental variable techniques to estimate the structural parameters of their small-scale business cycle model of the US economy: they argue that the parametric inequality restrictions that are sufficient for indeterminacy are fulfilled in their estimated model but do not provide any test in support of their claim. Binder and Pesaran (1995) follow a similar route in the estimation of a real business cycle of the US economy. They check ex-post that the largest eigenvalue of the matrix that governs solution uniqueness/multiplicity in their solution method (see our S (θ) matrix in Section 3) lies within the unit disk, but do not provide any test.

model with the hypothesis of indeterminacy. According to Lubik and Schorfheide (2004), all system-based approaches to the evaluation of indeterminacy are affected by this weakness. The testing strategy proposed in this paper: (i) is not based on prior distributions and maximizes the role attached to the data in testing determinacy against indeterminacy; (ii) circumvents the nonstandard inferential problem implied by the direct comparison of the likelihoods of the LRE models obtained under determinacy and indeterminacy; (iii) is not based on inequality restrictions; (iv) controls by construction for the possible dynamic misspecification of the LRE model; (v) is computationally straightforward and can be implemented with any existing econometric package. The analysis is based on a simple intuition: in a correctly specified LRE model, a test for determinacy/indeterminacy can be indirectly formulated, in the spirit of Hansen and Sargent (1980, 1981), as a likelihood-based test for the validity of the CER that the LRE model places on its finite order VAR representation under determinacy. Under the maintained assumption of correct specification, the rejection of the CER obtained under determinacy can automatically be associated with the hypothesis of indeterminacy. This argument breaks down, however, when the correct specification of the LRE model can not be taken for granted, i.e. when the rejection of the CER can also be ascribed to dynamic misspecification. The idea, therefore, is to test for the correct specification of the LRE model by a method robust to determinacy/indeterminacy, prior to focusing on the CER. To achieve this task, the overidentifying restrictions test (Hansen’s J-test) resulting from the direct (joint) estimation of the Euler equations of the LRE model by a suitable version of the generalized method of moments (GMM) is applied. If the LRE model is not rejected by the overidentification restrictions test, a Lagrange multiplier (LM) test for the CER obtained from the estimation of the determinate VAR solution by maximum likelihood (ML) is applied. If the J-test rejects the LRE model, it does not make sense to test whether the model equilibrium is in a determinate or indeterminate state. This procedure, hereafter denoted as the ‘J → LM’ test, gives rise to a multiple hypothesis testing issue but is based on standard inference. However, when the ‘J → LM’ test does not reject, i.e. when the J-test does not reject the LRE model and the LM test does not reject the CER at their chosen levels of significance, it is not possible to conclusively rule out the hypothesis of indeterminacy. Indeed, as noticed above, the restricted VAR equilibrium obtained under determinacy is observationally equivalent to the MSV reduced form that can be obtained as a special case from the class of indeterminate VARMA-type solutions. Thus the testing strategy can only find evidence against determinacy, not in favour of it. It is proved that the ‘J → LM’ is consistent against a well defined set of indeterminate alternatives and, notably, that is consistent against the hypothesis of dynamic misspecification of the LRE model. As regards finite sample properties, the Monte Carlo simulations presented below show that the ‘J → LM’ test has a reasonable size coverage using as data generating process (DGP) a linearized three equation sticky price LRE model featuring an interest rate rule as in Benati and Surico (2009). The finite sample power of the ‘J → LM’ test against the omission of relevant propagation mechanisms suggests that the risk of confounding the dynamic misspecification of the LRE model with indeterminacy is under control. The implementation of the ‘J → LM’ testing procedure is straightforward and any existing econometric package that features GMM estimation and ML estimation under nonlinear constraints can be used. To show the usefulness of this test we present an empirical illustration based on a New Keynesian monetary business cycle model of the US economy, using the same data set as Lubik and Schorfheide (2004). The paper is organized as follows. Section 2 introduces the reference multivariate LRE model. Section 3 summarizes the

L. Fanelli / Journal of Econometrics 170 (2012) 153–163

joint test and Section 4 focuses on its asymptotic properties. Section 5 reports some Monte Carlo evidence on the finite sample performance of the joint test, and Section 6 provides an empirical illustration based on US data. Section 7 contains some concluding remarks. All proofs are in the Appendix. 2. Model Let Xt be the n × 1 vector of observable variables at time t. The structural system is given by

Γ0 Xt = Γf Et Xt +1 + Γb Xt −1 + ωt

(1)

where Γi , i = 0, f , b are n × n matrices whose elements depend on the structural parameters, Et · := E (· | It ) is the conditional expectations operator, It is a nondecreasing information set containing the sigma field σ (Xt , Xt −1 , . . . , X1 ), and ωt is a n × 1 structural disturbance. X0 , X−1 and X−2 are conventionally treated as non-stochastic at time t = 1. The structural disturbances ωt obey the asymptotically stable first-order VAR process

ωt = Rωt −1 + ut

(2)

in which ω0 is fixed, R is a n × n stable matrix (i.e. with eigenvalues lying inside the unit disk) and ut is a fundamental disturbance modelled as a martingale difference sequence (MDS) with respect to It , with covariance matrix Σu . The structural parameters are collected in the m × 1 vector θ , and θ0 denotes the ‘true’ value of θ . The space of theoretically admissible values of θ , P ⊂ Rm , is assumed to be compact and θ0 is an interior point of P . It is considered the partition θ := (γ ′ , σu+′ )′ , where the m1 × 1 vector γ collects the non-zero elements of Γi , i = 0, f , b and R, and the g × 1 vector σu+ collects the elements of vech(Σu ) (m := m1 + g). A particular family of LRE models nested in system (1)–(2) is obtained under the restriction σu+ := 0g ×1 (Σu := 0n×n ), which implies ut := ωt := 0n×1 , ∀t, and γ = θ . Borrowing the terminology from Hansen and Sargent (1991), LRE models that do not feature structural (fundamental) disturbances are ‘exact’, while LRE models with σu+ ̸= 0g ×1 are ‘inexact’. The importance of ‘exact’ specifications in the analysis of the determinacy/indeterminacy of LRE models has been pointed out by Lubik and Schorfheide (2004, 2007) and Beyer and Farmer (2007), who emphasize the observational equivalence that may occur between determinacy in ‘inexact’ LRE models and indeterminacy in their ‘exact’ counterparts. This observational equivalence limits the scope of any test for the hypothesis of determinacy against indeterminacy. The LRE model in Eqs. (1)–(2) may be rewritten in the form

Γ0R Xt = Γf Et Xt +1 + ΓbR,1 Xt −1 + ΓbR,2 Xt −2 + uRt Γ0R

:= (Γ0 + RΓf ),

ΓbR,2

ΓbR,1

(3)

:= (Γb + RΓ0 ),

:= −RΓb

so that the ‘composite’ structural disturbance uRt := ut + RΓf ηt , ηt := (Xt − Et −1 Xt ) is a MDS with respect to It . When R := 0n×n in Eq. (2), system (3) coincides with system (1) and ωt ≡ ut ≡ uRt . A solution to the LRE system is any stochastic process {Xt }∞ t =0 such that E (Xt | It ) exists and the model is verified at any t for given initial conditions. As in Broze and Szafarz (1991), define a reduced form solution as a system of equations derived from the class of solutions that explains Xt in terms of ut , lags of Xt , ut and, possibly, other ‘external’ stochastic variables unrelated to ut . As is known, the solution properties depend on the location of θ in P . In this paper attention is devoted, as in Evans and Honkapohja (1986), to the class of stationary VARMA-type reduced

155

form solutions associated with the multivariate LRE model in Eq. (3), because these solutions are of interest from the econometric standpoint. The assumptions that follow ensure that the time series representation of the equilibria implied by system (3) belongs to the class of asymptotically stationary VARMA-type processes with constant parameters. Assumption 1 (Existence of Stable Reduced Form Solutions). ∀θ ∈ P and for fixed initial conditions, there exists a stable (asymptotically stationary) reduced form solution of system (3). Assumption 2 (Non-Singularity). The matrices Γ0 , Γ0R and Θ := (Γ0R − Γf Φc ,1 ) in system (3) are non-singular, where Φc ,1 := Φc ,1 (θ ) is an n × n matrix, defined below, whose elements depend nonlinearly on θ . Assumption 3 (Parameter Constancy (Within Regimes)). θ0 does not change over the sample X1 , . . . , XT . Assumption 1 ensures that all points that lie in the theoretically admissible parameter space lead to stationary reduced form solutions. Let P D be the determinacy region of the parameter space (P D ⊂ P ); parameter values that lead to non-stationary or to non-existing solutions are automatically ruled out, hence the set P \ P D identifies the indeterminacy region of the parameter space. Assumption 2 is introduced to simplify the derivation of the reduced form solutions of system (3) but can be relaxed as in e.g. Sims (2002) and Lubik and Schorfheide (2003, 2004). Finally, Assumption 3 clarifies that the objective of the analysis is not detecting potential breakpoints that induce a switch from one regime to the other but assessing whether the observed sample X1 , . . . , XT is consistent with a determinate or an indeterminate equilibrium. The class of reduced form solutions consistent with the LRE system (3) and Assumptions 1–3 will be introduced in the next section in the presentation of the testing strategy. 3. Testing strategy Any test of determinacy against indeterminacy should preferably be set out under the assumption that all components of θ are locally (identifiable) both under determinacy and under indeterminacy. This condition, however, is not fulfilled in all LRE models nested in system (3), see e.g. Lubik and Schorfheide (2004). A comprehensive treatment of identification issues in DSGE models may be found in Iskrev (2010) and in Komunjer and Ng (2011). It is known that under a set of regularity conditions, including a proper choice of instruments, GMM estimation of system (3) provides consistent, asymptotically Gaussian, estimates of γ irrespective of whether the LRE model has a unique stable solution or multiple stable solutions (σu+ is not directly recoverable from the system of Euler equations), see e.g. Wickens (1982), West (1986) and Pesaran (1987). The robustness of the GMM estimation of γ to determinacy/indeterminacy suggests that ‘partial identification’ issues can be circumvented by adding the assumption that follows. Assumption 4 (Local Identifiability of γ Under Determinacy/ Indeterminacy). Given Assumptions 1–3, the GMM estimation of system (3) provides, under the assumption of correct specification and for a suitable choice of instruments, consistent estimates of γ . The technical conditions embodied in Assumption 4 are relatively standard and may be found in e.g. Hall (2005, Chapter 3). The testing strategy is based on the idea that conditional on the correct specification of the LRE system (3), uniqueness can be rejected in favour of multiplicity of solutions when the restrictions that the model entails under determinacy are not supported by

156

L. Fanelli / Journal of Econometrics 170 (2012) 153–163

the data. To fully understand the mechanics of this approach, it is useful to introduce the time series representations of the LRE system under determinacy and indeterminacy, and discuss the cases in which observational equivalence between the two type of solutions may occur. Let N (θ0 ) be a neighbourhood of θ0 in P D . Using Binder and Pesaran’s (1995) method and Assumptions 1–4, the unique stable solution to the LRE model (3) can be represented as the VAR system

(In − Φc ,1 L − Φc ,2 L2 )Xt = Υ −1 ut

(4)

where L is the lag/lead operator (Lh Xt := Xt −h for h := ±1, ±2, . . .), Υ := (Θ − RΓf ) and the n × n matrices Φc ,1 := Φc ,1 (θ ), Φc ,2 := Φc ,2 (θ ) are blocks of the companion matrix   Φc ,1 Φc ,2 Φ˚ c := Φ˚ c (θ ) := In

0n×n

which is the (locally) unique stable solution to the quadratic matrix equation

 2 Γ˚ f Φ˚ c − Γ˚ 0 Φ˚ c + Γ˚ b = 02n×2n

(5)

where

Γ˚ 0 := Γ˚ b :=



Γ0R 0n×n

 R Γb,1 In



0n×n , In

ΓbR,2



0n×n

Γ˚ f :=



Γf



0n×n , 0n×n

0n×n

.

The reduced form coefficients of the VAR in Eq. (4) are subject to the set of nonlinear CER in Eq. (5) and the stability of the matrix S (θ) := Θ −1 Γf := (Γ0R − Γf Φc ,1 )−1 Γf =: S (γ ) is sufficient for the existence of the time series representation in Eqs. (4)–(5), see Binder and Pesaran (1995) and Fanelli (2011). By extending the argument in Beyer and Farmer (2007) to the multivariate framework, a VAR such as that in Eq. (4) might be regarded as the indeterminate solution of a parametrically restricted ‘exact’ counterpart of system (3). This type of observational equivalence is ruled out by focusing on ‘inexact’ specifications of system (3). Let P I bethe sub-set of (P \ P D ) defined  by P I := (P \ P D ) \ P E , P E := θ ∈ P \ P D , σu+ := 0g ×1 ; each θ in P I is such that the LRE system (3) is in ‘inexact’ form and has multiple solutions. The occurrence of multiple solutions can be associated with the situation in which for θ ∈ P I , the matrix S (θ ) has at least one eigenvalue lying outside the unit disk. Let L(θ0 ) be a neighbourhood of θ0 in P I . It can be shown that for θ ∈ L(θ0 ), the matrix S (θ ) has 1 ≤ n2 ≤ n eigenvalues that lie outside the unit disk and can be decomposed as S (θ) := Q (θ )



Λ1

0n1 ×n2



Λ2

0n2 ×n1

Q −1 (θ )

(6)

where Q := Q (θ ) is n × n non-singular, Λ1 is the normal Jordan block that collects the n1 := n − n2 eigenvalues that lie inside the unit disk, and Λ2 is the n2 × n2 normal Jordan block that collects the eigenvalues that lie outside the unit disk. Using the decomposition in Eq. (6), Binder and Pesaran’s (1995) solution method and Assumptions 1–4, the reduced form solutions to the LRE model in Eq. (3) can be represented as the class of VARMA-type processes (see Fanelli (2011)):

(In − Π L)(In − Φc ,1 L − Φc ,2 L2 )Xt = (Ξ − Π L)Ψ −1 ut + τt τt := (Ξ − Π L)Ψ

−1

Q ζt + Q ζt .

Π := Π (θ ) := Q (θ )

Λ1

0n2 ×n1

0n1 ×n2 1 Λ− 2

 Q

−1

(θ ),

Ξ := Ξ (θ , κ) := Q (θ )



In 1 0n2 ×n1

0n1 ×n2



κ

Q −1 (θ ),

Ψ := Ψ (θ , κ) := (Θ − RΓf Ξ ); finally, κ is an n2 × n2 matrix containing arbitrary elements not related to the structural parameters, and ζt := (0′n1 ×1 , s′t )′ :=

(0, . . . , 0, s1,t , . . . , sn2 ,t )′ is a n × 1 MDS with respect to It with (singular) covariance matrix Σζ , called ‘sunspot shock’, which may be independent on ut . The vector of arbitrary parameters, vec(κ), 2 is assumed to lie in the open space K ⊂ R(n2 ) and the q × 1 vector + σζ , which collects the elements of vech(Σζ ), is assumed to lie in q the space Z ⊂ R0 , where R0 := R+ ∪ {0}, and R+ is the set of real positive numbers. There are two sources of indeterminacies characterizing system (7)–(8): the auxiliary parameters in vec(κ) that are not identifiable from system (4); the vector of sunspot shocks ζt which generates the additional stochastic term τt when σζ+ ̸= 0q×1 (Σζ ̸= 0n×n ). Interestingly, in correspondence of the restrictions vec(κ) := vec(In2 ) and σζ+ := 0q×1 , system (7)–(8) collapses to a MSV solution observationally equivalent to the reduced form in Eqs. (4)–(5). This phenomenon has consequences on the properties of any test of determinacy/indeterminacy which exploits the estimation of the VAR system (4)–(5) like the one presented here. Let ν := (θ ′ , ψ ′ )′ be the ‘extended’ vector of parameters associated with the indeterminate equilibria in Eqs. (7)–(8), where ψ := (vec(κ)′ , ′ σζ+′ ) and ν ∈ V , V := P I × K × Z . By construction, the sub-set

V ∗ := V \ M, where M := {ν ∈ V , vec(κ) := In2 and σζ := 0g ×1 }, is such that each ν ∈ V ∗ is associated with a reduced form solution which is not observationally equivalent to the unique solution in Eqs. (4)–(5). Let X1 , . . . , XT be a sample of observations that is thought of as being generated, under the assumption of correct specification, by a solution of the ‘inexact’ version of system (3). Consider the null hypothesis +

H0 : X1 , . . . , XT is generated from the VAR system (4)–(5) (θ0 ∈ P D ),

(9)

against the alternative H1 : X1 , . . . , XT is generated from the VARMA-type system (7)–(8)

subject to ν ∈ V ∗ (θ0 ∈ P I ).

(10)

A candidate test for H0 against H1 can be obtained from the combination of two standard tests which are reviewed below. Test 1 The first test is a test of misspecification of the LRE model that, under Assumption 4, does not require the explicit knowledge of its reduced form solutions. Let JT be the overidentifying restrictions test statistic resulting from the direct estimation of γ from system (3) by GMM. Estimation is based on the orthogonality conditions E [et (γ ) ⊗ Zt −1 ] = 0nr ×1 ,

t = 1, . . . , T˜ , T˜ := T − 1

(11)

− ) := uRt − where et (γ ) := ( − Γ f X t +1 − Γf ηt +1 is the n × 1 disturbance, Zt −1 is the r × 1 vector of overidentifying instruments (nr > m1 ), and the criterion function is Γ0R Xt

(7)



(8)

min 

In this system, the matrices Φc ,1 and Φc ,2 are the same as in system (4) and are subject to the CER of Eq. (5); the matrix Π is defined by



with Q (θ ) and Λ2 given in Eq. (6), and

γ

T˜ 1

T˜ t =1

′

ΓbR,1 Xt −1



bt (γ ) WT˜ 

T˜ 1

T˜ t =1

ΓbR,2 Xt −2

 bt (γ )

(12)

where bt (γ ) := (et (γ )⊗ Zt −1 ), WT˜ is an nr × nr symmetric positive semidefinite weighting matrix that converges in probability to a symmetric positive definite matrix of constants W .

L. Fanelli / Journal of Econometrics 170 (2012) 153–163

The time series structure of the reduced forms in Eq. (4) and Eqs. (7)–(8) suggests that the vector of instruments Zt −1 := G(Xt′−1 , Xt′−2 )′ , where G is an r × 2n selection matrix, is relevant (other than valid) under determinacy as well as indeterminacy. Since both et (γ ) and bt := bt (γ ) have, under general regularity conditions, a vector moving average structure, the ‘optimal’ choice of weighting matrix corresponds to the inverse of W := Ω0 +(Ω1 + Ω1′ ), Ωi := E (bt b′t −i ), i = 0, 1. Recent research shows that by using relatively few instruments and parametric efficient estimates of W as in e.g. Cumby et al. (1983) and West (1997), the JT test has finite sample power against the hypothesis of dynamic misspecification of the LRE model, see Mavroeidis (2005) and Jondeau and Le Bihan (2008). The JT test assesses the validity of the LRE model. Under Assumptions 1–4 and the hypothesis of correct specification, the GMM estimator of γ is consistent and asymptotically Gaussian, and JT is asymptotically χ 2 (c1 ), c1 := nr − m1 , see e.g. West (1986) and Hall (2005, Chapter 3). Test 2 The second test assesses the validity of the CER that the LRE system (3) places on its reduced form solution under determinacy. Let LMTCER be the LM test statistic for the CER in Eq. (5) obtained by estimating θ by ML from the VAR in Eqs. (4)–(5) assuming ut Gaussian. Under determinacy and Assumptions 1–4, the ML estimator of θ (or quasi-ML estimator if the actual distribution of ut is not Gaussian) is consistent and asymptotically Gaussian, and LMTCER is χ 2 (c2 ), c2 := 2n2 + 12 n(n + 1)− m, see e.g. Godfrey (1988).2 Joint test The joint test for H0 in Eq. (9) against H1 in Eq. (10) is based on the following sequence: α

α

Step 1 Compute the JT test. If JT ≥ cr1 1 , where cr1 1 is the 100(1 − α1 ) percentile of the χ 2 (c1 ) distribution, the LRE model is rejected and it does not make sense to investigate its α determinacy/indeterminacy; if JT < cr1 1 , consider the next step. Step 2 Compute the LMTCER test for the CER obtained under α α determinacy. If LMTCER ≥ cr2 2 , where cr2 2 is the 100(1 − α2 ) percentile of the χ 2 (c2 ) distribution, accept indeterminacy; α if LMTCER < cr2 2 accept the CER implied by the VAR in Eqs. (4)–(5). Henceforth, the testing procedure obtained by combining the JT test in Step 1 with the LMTCER test in Step 2 will be denoted as the ‘J → LM’ test. It is worth emphasizing that the non rejection in Step 2 is not sufficient to rule out indeterminacy (θ0 ∈ P \ P D ) conclusively because of the above mentioned observational equivalence that may occur between the determinate equilibrium and particular indeterminate solutions. It follows that the ‘J → LM’ test can only find evidence against determinacy, but not in favour of it. Noteworthy, the procedure is explicitly designed to circumvent the likelihood-based estimation of system (7)–(8) and is computationally straightforward: the JT test has become a standard diagnostic for models estimated by GMM and is routinely calculated in most computer packages; likewise, the LMTCER test can be implemented with any econometric package that features numerical optimization algorithms.3

2 LM tests are preferred to likelihood ratio (LR) tests because the latter can be poorly sized compared to the former in LRE models involving highly nonlinear parametric restrictions, see e.g. Bekaert and Hodrick (2001). Fanelli (2010) presents an alternative version of the testing strategy in which the LMTCER test is replaced with a LM test for the null of absence of residual autocorrelation in the VAR system (4), against the alternative of residual autocorrelation up to order l ≥ 1. 3 A family of LRE models widely used in monetary policy and business cycle analysis is the class of ‘purely forward-looking’ LRE models, which is obtained from

157

4. Asymptotic properties The sequential nature of the ‘J → LM’ test presented in Section 3 suggests that the null hypothesis H0 in Eq. (9) can be rejected either because the JT test rejects the LRE model (Step 1), or because the LMTCER test rejects the CER when the JT test accepts the LRE model (Step 2). Under Assumptions 1–4, the asymptotic properties of the JT and LMTCER tests are standard, hence no new result is needed. The scope of this section is to ascertain whether the joint test resulting from the sequential combination of JT and LMTCER retains desirable properties: Section 4.1 deals with the size, Section 4.2 with the power against the hypothesis of indeterminacy, and Section 4.3 with the power against the hypothesis of dynamic misspecification of the LRE model. 4.1. Size The asymptotic size of the ‘J → LM’ test for H0 in Eq. (9) is given by

α∞ := lim sup αT , T →∞

α

αT := sup [PθJ ,T (JT ≥ cr1,1T )

(13)

θ∈P D

α

α

CER 1 + PθJ ,,LM ≥ cr2,2T )] T (JT < cr1,T ; LMT

where αT is the exact size of the test based on a sample of length J J T , Pθ ,T (·) = Pγ ,T (·) is the probability measure associated with the J ,LM

distribution of JT , Pθ ,T (·; ·) is the probability measure associated α with the joint distribution of JT and LMTCER , and cri,Ti , i = 1, 2 are the exact critical values of the two tests at nominal levels αi , i = 1, 2. In turn, the asymptotic sizes of the JT and LMTCER tests are given by α

α1,∞ := lim supT →∞ α1,T , α1,T := supθ ∈P D PθJ ,T (JT ≥ cr1,1T ), and α CER α2,∞ := lim supT →∞ α2,T , α2,T := supθ ∈P D PθLM ≥ cr2,2T ), ,T (LMT respectively; moreover, αi,∞ = αi , i = 1, 2, because both tests have ‘correct’ asymptotic size under Assumptions 1–4 and correct specification. Proposition 1 ensures that the asymptotic size of the ‘J → LM’ test does not exceed the sum of the nominal levels fixed for the JT and LMTCER tests. Proposition 1 (Asymptotic Size). Consider the class of LRE models which satisfy system (3), Assumptions 1–4, and the ‘J → LM’ test for H0 in Eq. (9) against H1 in Eq. (10). Let α∞ be the asymptotic size defined in Eq. (13), where α1 is the nominal level of the JT test and α2 is the nominal level of the LMTCER . Then α∞ ≤ α1 + α2 . The main consequence of Proposition 1 is that the practitioner can test determinacy/indeterminacy by relying on standard inference. If the overall nominal level of the procedure is set at the 5% level, one can fix α1 and α2 at e.g. the 2.5% level (α1 := α2 := 0.025). If the p-value associated with JT is greater than α1 and the

system (3) under the restrictions R := Γb := 0n×n . In this case, Assumption 4 does not generally hold and the ‘J → LM’ test can not be applied in the form presented above. In particular, the determinate equilibrium in Eq. (4) reduces to Xt = Γ0−1 ut , while the indeterminate equilibria in Eqs. (7)–(8) collapse to the class of VARMA(1, 1)-type models (Lubik and Schorfheide, 2004); accordingly, under correct specification, the presence of serial correlation in Xt , t = 1, . . . , T , is evidence of indeterminacy (while the absence of serial correlation is not sufficient to accept determinacy conclusively). A testing strategy inspired to the logic of the ‘J → LM’ procedure might work as follows: the data adequacy of the ‘purely forwardlooking’ model can be investigated in a preliminary step by an identification-robust test along the lines of e.g. Dufour et al. (2009) and, successively, conditional on the non rejection of the structural model, any statistical test for the absence/presence of serial correlation can be applied to the data Xt , t = 1, . . . , T .

158

L. Fanelli / Journal of Econometrics 170 (2012) 153–163

p-value associated with LMTCER is greater (lower) than α2 , the null H0 in Eq. (9) (the alternative H1 in Eq. (10)) is selected by the data; conversely, if the p-value associated with JT is lower or equal than α1 , the LRE model is rejected and it does not make sense to infer whether the system is in a determinate or indeterminate state.

The asymptotic power of the ‘J → LM’ test against all alternatives consistent with DM is given by mis pmis ∞ := lim sup pT , T →∞

pmis T

α

:= [PθJ ,T (JT ≥ cr1,1T | DM)

(16) α

α

CER 1 ≥ cr2,2T | DM)] + PθJ ,,LM T (JT < cr1,T ; LMT

4.2. Power against indeterminacy The asymptotic power of the ‘J → LM’ test against H1 in Eq. (10) is given by p∞ := lim sup pT ,

J

J

J ,LM

where Pθ ,T (· | DM) = Pγ ,T (· | DM) and Pθ ,T (·; · | DM) are the probability measures associated with the distributions of the tests under DM. The next proposition establishes the consistency of the ‘J → LM’ test against dynamic misspecification.

T →∞



α

J

pT := sup Pθ,T (JT ≥ cr1,1T )

(14)

θ ∈P I

α

J ,LM CER 1 + Pθ, T (JT < cr1,T ; LMT

 α ≥ cr2,2T )

where pT is the exact power based on a sample of length T , and α J J J ,LM Pθ ,T (·) = Pγ ,T (·), Pθ,T (·; ·), cri,Ti and αi i = 1, 2, are defined as in Section 4.1. Under H1 and Assumptions, the LMTCER test (Step 2) is Op (T ) because of the misspecification (Godfrey, 1988) of system (4)–(5), hence p2,∞ := lim supT →∞ p2,T = 1, where p2,T := supθ∈P I α CER ≥ cr2,2T ); conversely, the JT test (Step 1) is asympPθLM ,T (LMT totically χ 2 (c1 )-distributed because of its robustness to determinacy/indeterminacy, hence p1,∞ := lim supT →∞ p1,T ≤ α1 , where α

J

p1,T := supθ∈P I Pθ,T (JT ≥ cr1,1T ). Proposition 2 establishes the consistency of the ‘J → LM’ test against H1 .

Proposition 2 (Consistency Against a Class of Indeterminate Equilibria). Consider the class of LRE models which satisfy system (3), Assumptions 1–4, and the ‘J → LM’ test for H0 in Eq. (9) against H1 in Eq. (10). Let p∞ be the asymptotic power defined in Eq. (14). Then p∞ = 1.

Γf ,h Et Xt +h + ΓbR,1 Xt −1

h=2

+ ΓbR,2 Xt −2 +

k1 

Γb,j Xt −j + uRt ,

5. Monte Carlo study The finite sample properties of the ‘J → LM’ test are investigated by some Monte Carlo simulations based on a linearized three equations sticky price LRE model widely used in the literature, given by (see Benati and Surico (2009)) yt = ϖf Et yt +1 + (1 − ϖf )yt −1 − δ(it − Et πt +1 ) + ωy,t

(17)

πt = γf Et πt +1 + γb πt −1 + ϱyt + ωπ ,t

(18)

it = λr it −1 + (1 − λr )[λπ πt + λy yt ] + ωi,t

(19)

−1 < ρa < 1, ua,t ∼ N (0, σa2 ), a := y, π , i.

To analyze the power of the ‘J → LM’ test against the hypothesis of dynamic misspecification of the LRE model, we introduce the system of Euler equations: k2 

According to Proposition 3, if H0 in Eq. (9) is tested against indeterminacy by estimating a LRE model that does not capture the dynamics of the data, the ‘J → LM’ test will incorrectly select the hypothesis of indeterminacy with probability one in the limit.

ωa,t = ρa ωa,t −1 + ua,t ,

4.3. Power against dynamic misspecification

Γ0R Xt = Γf Et Xt +1 +

Proposition 3 (Consistency Against Dynamic Misspecification). Assume that X1 , . . . , XT is generated under DM and that the ‘J → LM’ test for H0 in Eq. (9) against H1 in Eq. (10) is applied under Assumptions 1–4. Let pmis ∞ be the asymptotic power of the test against DM defined in Eq. (16). Then pmis ∞ = 1.

(15)

j =3

which, compared to system (3), includes (k1 − 2) additional lags of Xt associated with the matrices of parameters Γb,j ̸= 0n×n , j = 3, . . . , k1 , (k1 ≥ 3), and (k2 − 1) additional expectations terms associated with the matrices of parameters Γf ,h ̸= 0n×n , h = 2, . . . , k2 , (k2 ≥ 2). All reduced form models discussed in Section 3 are (non-locally) misspecified if the solutions generated by system (15) are treated as the DGP and at least one among Γb,j , j = 3, . . . , k1 and Γf ,h , h = 2, . . . , k2 is different from zero. All DGPs consistent with Eq. (15) will be indicated with the acronym ‘DM’, standing for ‘dynamic misspecification’. Under DM and Assumptions 1–4, both JT (Step 1) and LMTCER (Step 2) are Op (T ). Indeed, the GMM estimator of γ obtained from system (3) does not fulfill the orthogonality conditions in Eq. (11), while the ML estimator of θ is derived from the VAR system (4)–(5) which omits important propagation mechanisms.

(20)

yt , πt and it are the output gap, inflation, and the nominal interest rate, respectively; Eq. (17) is an intertemporal IS curve, Eq. (18) is a Phillips curve, and Eq. (19) is a policy rule; the structural disturbances in Eq. (20) obey (Gaussian) autoregressive processes. By a suitable specification of the matrices Γ0 , Γf and Γb , system (17)–(20) can be cast in the notation in Eqs. (1)–(2), with R := dg (ρy , ρπ , ρi ) and ut ∼ N (0, Σu ), where σy2 , σπ2 and σi2 are the diagonal elements of Σu . The reduced form solutions of system (17)–(20) are used as the DGP in the investigation of the empirical size of the ‘J → LM’ test and the empirical power against the hypothesis of indeterminacy. Instead, the investigation of the empirical power of the test against the dynamic misspecification of the LRE system is performed by maintaining that system (17)–(20) is nested within the model in Eq. (15). The vector θ := (γ ′ , σu+ )′ is chosen from Benati and Surico (2009), where γ := (ϱ, δ, ϖf , γ˘f , γ˘b , λr , λy , λπ , ρy , ρπ , ρi )′ and γf := γ˘f := β/(1 + ~β) and γb := γ˘b := ~/(1 + ~β), with the discount factor β fixed at 0.99. In all experiments, M = 1000 samples of length T = 75 have been considered, except where indicated, to mimic situations often encountered in practice. The results are summarized in the next subsections: Section 5.1 deals with the empirical size of the ‘J → LM’ test, Section 5.1.1 with the finite sample power against the hypothesis of indeterminacy and Section 5.2 with the finite sample power against the dynamic misspecification of the LRE system.

L. Fanelli / Journal of Econometrics 170 (2012) 153–163

159

Table 1 Size of the ‘J → LM’ test when the data are generated from the New–Keynesian business cycle monetary model in Eqs. (17)–(20) under determinacy (H0 in Eq. (9)). Determinacy: ‘artificial’ DGP

λmax [S (γ0 )] := 0.91 γ0 :

ML estimates (det.)

GMM estimates

ϱ

0.05

0.056 (0.020)

0.063 (0.026)

δ

0.50

0.520 (0.091)

0.467 (0.087)

T = 75

ϖf

0.25

0.238 (0.060)

0.269 (0.049)

Rej(LM CER T ) = 0.074

~

0.75

0.720 (0.146)

0.699 (0.357)

Rej(JT ) = 0.032

λi

0.75

0.747 (0.043)

0.741 (0.048)

Rej(JT → LM CER T ) = 0.068

λy

0.15

0.232 (0.153)

0.210 (0.220)

λπ

1.5

1.356 (0.406)

1.441 (0.506)

ρy := ρπ := ρi

0.25

0.233 (0.095)

0.290 (0.171)

γ˜f :=

0.99 1+0.99~

:= 0.568

0.578 –

0.585 –

γ˜b :=

1+0.99~

~

:= 0.430

0.420 –

0.414 –

Rej0 (JT → LM CER T ) = 0.106

Notes: Results are obtained using M = 1000 replications. Each simulated sample is initiated with 100 additional observations to get a stochastic initial state and these are then discarded. λmax [·] denotes the largest eigenvalue in absolute value of the matrix in the argument and S (γ ) is defined in Section 3. γ0 and the diagonal elements of Σu are taken from the fourth column of Table 1 of Benati and Surico (2009), except for the value of λπ (long run response to inflation) that has been changed to ensure determinacy. Σu also includes the off-diagonal terms σy,π := −0.05, σy,i := −0.07 and σπ,i := 0.08. GMM estimates have been computed using Zt −1 := (Xt′−1 , Xt′−2 )′ as vector of instruments and a parametric estimate of the weighting matrix to account for VMA(1) disturbances; VMA(1) disturbances have been approximated by a finite order VAR model with truncation order TR := int(T /1.3). ML estimates are obtained from the VAR in Eq. (4) subject to the CER in Eq. (5), using GMM estimates as initial values. Estimates and Monte Carlo standard errors, reported in parentheses, are averages across simulations. JT is the overidentification restrictions test resulting from GMM estimation, LM CER is the LM test for the T CER that the LRE model entails under determinacy. Rej(·) stands for ‘rejection frequency’. Rej0 (·) denotes the rejection frequency obtained with ρy := ρπ := ρi := 0 in the DGP and estimating the system by erroneously assuming correlated disturbances. In all cases the nominal levels of the JT and LM CER tests are fixed at 0.05 and the overall nominal level of the ‘J → LM’ test is fixed at 0.05 and α1 := α2 := 0.025. T

5.1. Empirical size The empirical size of the ‘J → LM’ test is analysed by considering two distinct DGPs. In the first (Table 1), the data are generated under determinacy by fixing γ := γ0 (henceforth γ0 denotes the calibrated value of γ in the DGP) and the diagonal elements of Σu to the modes of the prior distributions reported in the fourth column of Table 1 of Benati and Surico (2009), except for the policy parameter λπ (long run response to inflation, see Eq. (19)), which is set to 1.5 to ensure determinacy. This design characterizes an ‘artificial’ DGP. In the second (Table 2), the Monte Carlo design is calibrated to match an estimated model and in particular the data are generated under determinacy such that all chosen components of γ0 and the diagonal elements of Σu lie within the 90% coverage percentile of the posterior distribution reported in the last column of Table 1 (‘After the Volcker stabilization’) of Benati and Surico (2009). This design characterizes a situation in which the GMM minimand and the likelihood function associated with the constrained VAR obtained under determinacy provide ‘poor information’ about some components of γ (including the policy parameters λy and λπ ), mirroring the ‘weak identification’ phenomenon documented in this class of models by e.g. Dufour et al. (2009) and Mavroeidis (2010). Tables 1 and 2 report the averages of the GMM and ML estimates of γ across simulations with their associated Monte Carlo standard errors. The last column of Tables 1 and 2 reports the rejection

frequencies of the JT and LMTCER tests obtained with nominal level 0.05, and the rejection frequency of the ‘J → LM’ test obtained by setting the overall nominal level at 0.05 and the nominal levels of the individual tests at α1 := α2 := 0.025 (see Proposition 1). As expected, the estimates in Table 1 are considerably more precise than the estimates in Table 2. In general, with T = 75 observations it is difficult to estimate precisely the parameters λy and λπ of the policy rule, other than the inverse of the elasticity of intertemporal substitution in consumption, δ , and the slope of the Phillips curve, ϱ. Overall, the empirical size of the ‘J → LM’ test appears satisfactory in both Monte Carlo experiments and seems to guarantee a substantial control of the null hypothesis H0 in Eq. (9).4 5.1.1. Finite sample power against indeterminacy To analyse the power of the ‘J → LM’ test against indeterminacy, the data are generated by considering the ‘indeterminacy without sunspots’ case (Lubik and Schorfheide, 2004), which is

4 Since GMM estimation features a parametric estimate of the weighting matrix (Cumby et al., 1983; West, 1997), the finite sample performance of the JT test depends on the truncation order used to approximate the vector moving average disturbances of the Euler equations by a finite order VAR. As is known, an appropriate rule for picking the optimal truncation order in finite samples is an open question: the results in Table 2 suggest that the method used to estimate the weighting matrix may have remarkable consequences on the finite sample performance of the JT test and, accordingly, of the ‘J → LM’ procedure.

160

L. Fanelli / Journal of Econometrics 170 (2012) 153–163 Table 2 Size of the ‘J → LM’ test when the data are generated from the New-Keynesian business cycle monetary model in Eqs. (17)–(20) under determinacy (H0 in Eq. (9)). Determinacy: DGP calibrated to estimated model

λmax [S (γ0 )] := 0.92 γ0 :

ML estimates (det.)

GMM estimates

ϱ

0.05

0.184 (0.547)

0.048 (0.347)

δ

0.125

0.112 (0.274)

0.190 (0.267)

T = 75

ϖf

0.75

0.717 (0.099)

0.705 (0.123)

Rej(LM CER T ) = 0.021

~

0.05

0.07 (0.260)

0.029 (0.273)

Rej(JT ) = 0.026

λi

0.85

0.751 (0.145)

0.758 (0.135)

Rej(JT∗ ) = 0.056

λy

0.75

0.542 (0.758)

0.428 (0.597)

Rej(JT → LM CER T ) = 0.036

λπ

2.5

2.232 (1.836)

1.986 (1.487)

Rej(JT∗ → LM CER T ) = 0.063

ρy

0.75

0.610 (0.236)

0.671 (0.311)

ρπ

0.50

0.469 (0.246)

0.597 (0.447)

ρi

0.40

0.405 (0.168)

0.379 (0.171)

γ˜f :=

0.99 1+0.99~

:= 0.943

0.929 –

0.962 –

γ˜b :=

1+0.99~

~

:= 0.048

0.065 –

0.028 –

Rej0 (JT → LM CER T ) = 0.097

Notes: Results are obtained using M = 1000 replications. Each simulated sample is initiated with 100 additional observations to get a stochastic initial state and these are then discarded. See caption of Table 1 for λmax [·] and S (γ ). The components of γ0 and the diagonal elements of Σu belong to the 90% coverage percentile of the posterior distributions reported in the last column of Table 1 (‘After the Volcker stabilization’) of Benati and Surico (2009). Σu includes also the off-diagonal terms σy,π := −0.05, σy,i := −0.07 and σπ,i := 0.08. GMM and ML estimation and the JT , LM CER and ‘J → LM’ tests have been carried out as described in Table 1. In the parametric procedure for T the weighting matrix the JT test has been calculated with truncation order TR := int(T /1.3) while the JT∗ test has been calculated with TR := int(T /1.2). Estimates and Monte Carlo standard errors, reported in parentheses, are averages across simulations. Rej(·) stands for ‘rejection frequency’. Rej0 (·) denotes the rejection frequency obtained with ρy := ρπ := ρi := 0 in the DGP and estimating the system by erroneously assuming correlated disturbances. In all cases the nominal levels of the JT and LM CER tests are fixed at 0.05 and the overall T nominal level of the ‘J → LM’ test is fixed at 0.05 and α1 := α2 := 0.025.

obtained by the restriction σζ+ := 0q×1 (τt := 0n×1 a.s. ∀t) in system (7)–(8) (maintaining the condition vec(κ) ̸= vec(In2 )). Furthermore, it is assumed that the ‘true’ value of θ in P I leads to a configuration of the S (θ0 ) = S (γ0 ) matrix in Eq. (6) such that only one eigenvalue lies outside the unit circle (n2 := 1); with this design, vec(κ) collapses to a scalar. Two DGPs are considered. In the former (DGP1), γ := γ0 and the diagonal elements of Σu are fixed as in Table 1, except for the parameter λπ of the policy rule, which is set to 0.80 to ensure indeterminacy. In the latter (DGP2), γ := γ0 and the diagonal elements of Σu are calibrated to values taken within the 90% coverage percentile of the posterior distribution reported in Table 1 of Benati and Surico (2009), last column (‘Before October 1979’). The results are sketched in Table 3. Table 3 investigates the sensitivity of the rejection frequency of the ‘J → LM’ test to different values of the nuisance parameter κ (keeping γ := γ0 and Σu fixed). The reported rejection frequencies show that the finite sample power is satisfactory for values of the nuisance parameters that are relatively far from the point κ := 1 that generates a MSV solution and tends to decline, as expected, as long as κ approaches 1. Moreover, the results obtained under DGP2 show that the capacity of the ‘J → LM’ test to correctly reject the null in Eq. (9) deteriorates significantly in samples of length T = 75 if some of the structural parameters are ‘weakly identified’. In general, it is reasonable to expect a substantial increase of the

finite sample power of the ‘J → LM’ test when sunspot shocks characterize the indeterminate equilibria (σζ+ ̸= 0q×1 ). 5.2. Finite sample power against dynamic misspecification To investigate the power of the ‘J → LM’ test against the hypothesis of dynamic misspecification, we consider a version of system (15) in which the matrices Γ0 , Γf , Γb and R have the structure implied by system (17)–(20), Γf ,h := 03×3 for h ≥ 2 and k1 := 3 with Γb,3 := µI3 ; with this design, the (scalar) parameter µ captures the extent of the (non-local) misspecification of the theoretical model. Hence, the vector of structural parameters associated with the LRE system that generates the data is θ ∗ := (γ ′ , µ, σu+′ )′ . The data are generated under determinacy by fixing γ := γ0 as in Table 2 and setting the extra parameter µ to the values given in the first column of Table 4. The ‘J → LM’ procedure is applied to each simulated sample by estimating the misspecified LRE system (17)–(20). Table 4 shows that the finite sample power of the ‘J → LM’ test depends, as expected, on the magnitude of µ. The JT test, which is based on a parametric estimate of the weighting matrix (see Section 3), appears well designed to capture the omission of propagation mechanisms in the LRE model, supporting the findings in Mavroeidis (2005) and Jondeau and Le Bihan (2008). Also the

L. Fanelli / Journal of Econometrics 170 (2012) 153–163

161

Table 3 Power of the ‘J → LM’ test when the data are generated from the New–Keynesian business cycle monetary model in Eqs. (17)–(20) under indeterminacy (H1 in Eq. (10) with no sunspot shocks). Indeterminacy γ0 : DGP1, λmax [S (γ0 )] := 1.05 DGP2, λmax [S (γ0 )] := 1.02

ϱ

δ

ϖf

~

γ˜f

γ˜b

λi

λy

λπ

ρy := ρπ := ρi

0.05 0.05

0.50 0.125

0.25 0.75

0.50 0.05

0.568 0.943

0.430 0.048

0.75 0.60

0.10 0.50

0.80 0.80

0.25 0.75

Rej(JT → LM CER T ) T = 75

κ: −0.5

T = 150

DGP1 0.728 0.595 0.799 0.373 0.910 0.909

0.5

−0.8 0.8

−15 15

DGP2 0.250 0.038 0.297 0.033 0.218 0.157

DGP1 0.920 0.817 0.944 0.804 0.971 0.969

DGP2 0.725 0.043 0.794 0.054 0.613 0.447

Notes: Results are obtained using Monte Carlo simulations of M = 1000 samples. Each simulated sample is initiated with 100 additional observations to get a stochastic initial state and these are then discarded. λmax [·] denotes the largest eigenvalue in absolute value of the matrix in the argument and S (γ ) is decomposed as in Eq. (6). In DGP1 the structural parameters γ0 and the diagonal elements of Σu are taken from the fourth column of Table 1 of Benati and Surico (2009), except for the value of λπ (long run response to inflation), which has been changed to ensure indeterminacy. In DGP2 the components of γ0 (except ρy , ρπ and ρi ) and the diagonal elements of Σu belong to the 90% coverage percentile of the posterior distributions reported in the last column of Table 1 (‘Before October 1979’) of Benati and Surico (2009). Rej(·) stands for ‘rejection frequency’; the overall nominal level of the ‘J → LM’ test is fixed at 0.05 and α1 := α2 := 0.025. Table 4 Power of the ‘J → LM’ test against dynamic misspecification. DGP: determinate solution of LRE system in Eq. (15) with Γb,3 := µI3 , Γb,k1 := Γf ,k2 := 03×3 , k1 > 3, k2 ≥ 2γ0 and Σu are fixed as in Table 2 except for ρy := ρπ := ρi := 0.70

µ:

Rej(JT )

Rej(LM CER T )

Rej(JT → LM CER T )

−0.5 −0.4 −0.3 −0.20 −0.10

0.986 0.958 0.846 0.644 0.587

0.998 0.989 0.862 0.584 0.277

0.984 0.962 0.874 0.697 0.511

T = 75

Notes: Results are obtained using Monte Carlo simulations of M=1000 samples. Each simulated sample is initiated with 100 additional observations to get a stochastic initial state and these are then discarded. The econometrician is assumed to estimate system (17)–(20). All values of the misspecification parameter µ in the first column, combined with γ0 , lead to a determinate equilibrium. GMM and ML estimates have been obtained as detailed in Table 1. JT is the overidentification restrictions test resulting from GMM estimation, LM CER is the LM test for the T CER that the LRE model entails under determinacy. Rej(·) stands for ‘rejection frequency’; the nominal levels of the JT and LM CER tests are fixed at 0.05; the overall T nominal level of the ‘J → LM’ test is fixed at 0.05 and α1 := α2 := 0.025.

finite power of the LMTCER test against this type of misspecification appears reasonable. The rejection frequencies in Table 4 show that even in samples of length T = 75, the risk of confounding the dynamic misspecification of the LRE model with the hypothesis of indeterminacy is under control. 6. Empirical illustration: a New Keynesian monetary business cycle model A vast empirical literature has studied New Keynesian monetary business cycle models similar to that specified in Eqs. (17)– (20) to investigate possible sources of the ‘US Great Moderation’ and attempt to disentangle the relative contributions of two main explanations: ‘good policy’ and ‘good luck’, see, inter alia, Clarida et al. (2000), Lubik and Schorfheide (2004), Boivin and Giannoni (2006), Benati and Surico (2009) and Mavroeidis (2010). A detailed investigation of these hypotheses goes well beyond the scope of the present section, which shows the empirical usefulness of the ‘J → LM’ procedure. The analysis is based on the same data as in Lubik and Schorfheide (2004). Data are quarterly and cover the period 1960.q1–1997.q4 and refer to the log real per capita GDP detrended with the Hodrik Prescott filter (yt ), the inflation rate computed as the annualized percentage change of the CPI-U (πt ), and the (annualized) Federal Funds rate in percent (it ).

The sample is split into two sub-samples: the 1960.q1–1979.q2 ‘Pre-Volcker’ period (hereafter Period 1), and the 1979.q3–1997.q4 ‘Volcker–Greenspan’ period (hereafter Period 2). The estimated version of the New Keynesian monetary business cycle model is based on a specification of system (17)–(20) in which γf := γ˘f := β/(1 + ~β), γb := γ˘b := ~/(1 + ~β), the discount factor β is fixed at 0.99 and δ and κ are fixed at the values 0.5 and 0.05, respectively.5 The vector γ is in this case given by

γ := (ϖf , ~, λi , λy , λπ , ρi , ρy , ρπ )′ .

The upper panel of Table 5 summarizes the GMM and ML estimates of γ obtained over Periods 1 and 2 without imposing any bound restriction. The lower panel of Table 5 reports the results of the ‘J → LM’ test; the (overall) nominal level is fixed at 5% (α := 0.05) and the nominal levels of the JT and LMTCER tests are fixed at 2.5% (α1 := α2 := 0.025). In Period 1, the ‘J → LM’ test selects the hypothesis of indeterminacy (the p-value associated with the JT test is well above 0.025 while the p- value associated with the LMTCER test is well below 0.025). In Period 2, the ‘J → LM’ test rejects the estimated New Keynesian model (the p-value associated with the JT test is well below 0.025), hence it does not make sense to inquire whether a determinate or indeterminate equilibrium prevailed. Overall, the results obtained for Period 1 support the conclusions of Lubik and Schorfheide (2004) obtained through a Bayesian approach, and confirm the thesis of Clarida et al. (2000) and Boivin and Giannoni (2006), achieved without any formal testing strategy, that monetary policy prior to the Volcker–Greenspan period had been passive and had opened up the possibility of sunspot fluctuations induced by self-fulfilling expectations. On the other hand, the results obtained for Period 2 suggest that the structural model does not fully capture the properties of the data, hence it is not possible to support the conclusions of other authors unless an extended dataset is investigated and/or a different structural system is estimated.6

5 The results reported in this empirical illustration are robust to different choices of the inverse intertemporal elasticity of substitution δ . Moreover, ML estimation is based on a non-diagonal covariance matrix Σu . 6 In a companion paper (Castelnuovo and Fanelli, 2011), the analysis has been extended until 2008, using a different definition of the output gap, proxied by the Congressional Budget Office measure. In this case, the ‘J → LM’ test does not provide any evidence against the hypothesis H0 in Eq. (9), in line with the results in Lubik and Schorfheide (2004), Clarida et al. (2000) and Boivin and Giannoni (2006).

162

L. Fanelli / Journal of Econometrics 170 (2012) 153–163

Table 5 GMM and ML estimates of the New–Keynesian business cycle monetary model in Eqs. (17)–(20) on US quarterly data, and ‘J → LM’ test. 1960q1–1979q2 (T = 76)

1979q3–1997q4 (T = 74)

GMM

ML

GMM

ML

ϖf

0.761 (0.138)

0.629 (0.264)

0.631 (0.153)

0.805 (0.147)

~

0.954 (0.374)

−0.270 (0.177)

1.145 (0.498)

−0.092

λi

0.795 (0.077)

0.831 (0.082)

0.899 (0.029)

0.931 (0.059)

λy

1.504 (0.473)

1.838 (0.807)

1.545 (0.565)

1.388 (1.01)

λπ

0.619 (0.329)

0.498 (0.288)

0.960 (0.315)

0.491 (0.497)

ρy

0.888 (0.105)

0.892 (0.146)

0.997 (0.061)

0.974 (0.083)

ρπ

0.494 (0.216)

0.931 (0.171)

0.520 (0.334)

0.585 (0.148)

ρi

0.178 (0.136)

0.102 (0.139)

−0.047

−0.115

(0.067)

(0.077)

0.509 0.490

– –

0.463 0.536

– –

γ˜f := γ˜b :=

0.99 1+0.99~

~

1+0.99~

(0.271)

JT = 12.59

JT = 60.58

LM T = 30.60

LM T = 39.70

[0.182]

[0.000]

[0.000]

[0.000]

Notes: Data are the same as in Lubik and Schorfheide (2004) and refer to the US economy. GMM estimates have been obtained as detailed in Table 1, using TR := int(T /6) as truncation lag order in the parametric procedure for the weighting matrix. The outcome of the JT test is robust to alternative choices of TR. ML estimates have been obtained as detailed in Table 1, assuming a non-diagonal covariance matrix Σu . Standard errors in parentheses; p-values in brackets.

7. Conclusions This paper presents a ‘frequentist’ approach for testing determinacy against indeterminacy in stable LRE systems. The ‘J → LM’ test combines the overidentification restrictions test obtained from the estimation of the Euler equations of the LRE system by a suitable version of GMM, with a LM test for the CER obtained from the estimation of the determinate reduced form solution by ML. This test involves standard asymptotics, is explicitly designed to control for the possible omission of relevant propagation mechanisms from the specified LRE model, but can only find evidence against determinacy, not in favour of it. Some Monte Carlo experiments, based on a prototype New Keynesian monetary business cycle model show that the ‘J → LM’ test displays reasonable empirical size and finite sample power against a well defined class of indeterminate equilibria. Interestingly, the risk of confounding the omission of relevant lags/leads with the hypothesis of indeterminacy is under control. An empirical illustration shows the usefulness of the ‘J → LM’ approach by testing the hypothesis of determinacy in a widely investigated small-scale monetary business cycle model of the US economy. As it stands, the ‘J → LM’ test can be applied when all components of Xt are observed. To implement this test when the LRE system features unobserved state variables, it is necessary to address one major problem: the presence of unobserved components may lead to a VARMA-type representation for the observed variables even when a unique solution occurs (see e.g. Ravenna (2007)), hence it may be difficult to identify precisely the set of testable restrictions that arise under determinacy. The investigation of this issue is left for future work.

Acknowledgments I thank two anonymous referees, an Associate Editor and the following people for useful comments on previous versions of this paper: Gunnar Bårdsen, Efrem Castelnuovo, Davide Delle Monache, Massimo Franchi, Alain Hecq, Marco Lippi, Franz Palm, Franco Peracchi, Tommaso Proietti, Anders Rahbek, Jean-Pierre Urbain, seminar participants at the Einaudi Institute for Economics and Finance, Maastricht University and Norvegian University of Science and Technology, conference participants at the ‘Fourth Italian Congress of Econometrics and Empirical Economics’, Pisa, January 19–21, 2001; ‘Macro and Financial Econometrics Conference’, Heidelberg, September 29–30, 2011. I am solely responsible for any remaining errors. Partial financial support is gratefully acknowledged from RFO grants University of Bologna. Appendix. Proofs α

Proof of Proposition 1. Define AT 1 α1 

α

:= α2 



α α JT ≥ cr1,1T , A¯ T 1



:=

JT < cr1,T and BT 2 := LMTCER ≥ cr2,T . Given the definition of αT in the text, we used the inequality αT ≤ α1,T + α2,T , which α α J J holds because (i) α1,T := supθ∈P Pθ ,T (AT 1 ) = supθ∈P D Pθ ,T (AT 1 ) due to the robustness of the JT test   to determinacy/indeterminacy,





J ,LM

α

α

α

2 and (ii) supθ∈P D Pθ ,T (A¯ T 1 ; BT 2 ) ≤ supθ∈P D PθLM ,T (BT ) =: α2,T .





By applying the ‘limsup’ to this inequality and using αi,∞ = αi , i = 1, 2,

  α∞ := lim sup αT ≤ lim sup α1,T + lim sup α2,T T →∞

T →∞

T →∞

= α1,∞ + α2,∞ = α1 + α2 .  α

α

Proof of Proposition 2. Given B¯ T 2 := LMTCER < cr2,2T and the definition of the set V ∗ in Section 3, the following inequality holds ∀ν ∈ V ∗ : α

J

J ,LM

α





α

pT ≥ Pθ ,T (AT 1 ) + Pθ ,T (A¯ T 1 ; BT 2 ) α α ¯ α2 ≥ PθJ ,T (AT 1 ) + [1 − PθJ ,T (AT 1 ) − PθLM ,T (BT )] α

2 = PθLM ,T (BT ).

α

2 Accordingly, pT ≥ supθ∈P I PθLM ,T (BT ), and taking the ‘limsup’ it follows that p∞ ≥ p2,∞ = 1; since p∞ ≤ 1, the result is obtained. 

J ,LM

α

α

Proof of Proposition 3. Consider the inequality Pθ ,T (A¯ T 1 BT 2   J

α1

DM) ≥ 1 − Pθ ,T (AT | DM) − J

PθLM ,T

|

α (B¯ T 2 | DM) , which implies

α

pmis ≥ Pθ ,T (AT 1 | DM ) T

  α ¯ α2 + 1 − PθJ ,T (AT 1 | DM) + PθLM ,T (BT | DM) α

2 = PθLM ,T (BT | DM).

α

2 By applying the ‘limsup’ and using lim supT →∞ PθLM ,T (BT | DM) =

mis 1, it follows that pmis ∞ ≥ 1; since p∞ ≤ 1, the result is obtained. 

References Andrews, D., Ploberger, W., 1994. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414. Bekaert, G., Hodrick, R., 2001. Expectations hypotheses tests. Journal of Finance 56, 1357–1394. Benati, L., Surico, P., 2009. VAR analysis and the great moderation. American Economic Review 99, 1636–1652.

L. Fanelli / Journal of Econometrics 170 (2012) 153–163 Benhabib, J., Farmer, R.E.A., 1999. Indeterminacy and sunspots in macroeconomics. In: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics, Vol. 1A. North-Holland, Amsterdam, pp. 387–448. Beyer, A., Farmer, R.E.A., 2007. Testing for indeterminacy: an application to US monetary policy: comment. American Economic Review 97, 524–529. Binder, M., Pesaran, M.H., 1995. Multivariate rational expectations models and macroeconomic modelling: a review and some new results. In: Pesaran, M.H., Wickens, M. (Eds.), Handbook of Applied Econometrics. Blackwell, Oxford, pp. 139–187 (Chapter 3). Boivin, J., Giannoni, M.P., 2006. Has monetary policy become more effective? Review of Economics and Statistics 88, 445–462. Broze, L., Szafarz, A., 1991. The econometric analysis of non-uniqueness in rational expectations models. In: Contribution to Economic Analysis. North Holland, Amsterdam. Castelnuovo, E., Fanelli, L., 2011. Monetary policy indeterminacy in the US: results from a classical test, Quaderni di Dipartimento, Serie Ricerche 2011 No. 8, University of Bologna. Clarida, R.J., Galì, J., Gertler, M., 2000. Monetary policy rules and macroeconomic stability: evidence and some theory. Quarterly Journal of Economics 115, 147–180. Cumby, R.E., Huizinga, J., Obstfeld, M., 1983. Two-step two-stage least squares estimation in models with rational expectations. Journal of Econometrics 21, 333–355. Dufour, J.-M., Khalaf, L., Kichian, M., 2009. Structural multi-equation macroeconomic models: identification-robust estimation and fit, Bank of Canada, Working Paper 2009–19. Evans, G., Honkapohja, S., 1986. A complete characterization of ARMA solutions to linear rational expectations models. Review of Economic Studies 53, 227–239. Fanelli, L., 2011. Robust identification conditions for determinate and indeterminate linear rational expectations models, Quaderni di Dipartimento, Serie Ricerche 2011 No.1, University of Bologna. Updated version available at http://www.rimini.unibo.it/fanelli/robust_identification.pdf. Fanelli, L., 2010. Determinacy, indeterminacy and dynamic misspecification in linear rational expectations models, Quaderni di Dipartimento, Serie Ricerche 2010 No. 4, University of Bologna. Farmer, R., Guo, J., 1995. The econometrics of indeterminacy: an applied study. In: Carnegie–Rochester Conferences on Public Policy, Vol. 43, pp. 225–271. Godfrey, L.G., 1988. Misspecification Tests in Econometrics. Cambridge University Press, Cambridge. Hall, A.R., 2005. Generalized Method of Moments. Oxford University Press, Oxford. Hansen, L.P., Sargent, T.J., 1980. Formulating and estimating dynamic linear rational expectations models. Journal of Economic Dynamics and Control 2, 7–46. Hansen, L.P., Sargent Jr., T.J., 1981. Linear rational expectations models for dynamically interrelated variables. In: Lucas, R.E., Sargent, T.J. (Eds.), Rational Expectations and Econometric Practise. University of Minnesota Press, Minneapolis, pp. 127–156.

163

Hansen, L.P., Sargent, T.J., 1991. Exact linear rational expectations models: specification and estimation. In: Hansen, L.P., Sargent, T.J. (Eds.), Rational Expectations Econometrics. Westview Press, Boulder. Imrohoroğlu, S., 1993. Testing for sunspot equilibria in the German hyperinflation. Journal of Economic Dynamics and Control 17, 289–317. Iskrev, N., 2010. Local identification in DSGE models. Journal of Monetary Economics 57, 189–202. Jondeau, E., Le Bihan, H., 2008. Examining bias in estimators of linear rational expectations models under misspecification. Journal of Econometrics 143, 375–395. Kodde, D.A., Palm, F.C., 1986. Wald criteria for jointly testing equality and ineqality restrictions. Econometrica 54, 1243–1248. Komunjer, I., Ng, S., 2011. Dynamic identification in DSGE models. Econometrica 79, 1995–2032. Lubik, T.A., Schorfheide, F., 2004. Testing for indeterminacy: an application to US monetary policy. American Economic Review 94, 190–217. Lubik, T.A., Schorfheide, F., 2003. Computing sunspot equilibria in linear rational expectations models. Journal of Economic Dynamics and Control 28, 273–285. Lubik, T.A., Schorfheide, F., 2007. Testing for indeterminacy: an application to US monetary policy: reply. American Economic Review 97, 530–533. Mavroeidis, S., 2005. Identification issues in forward-looking models estimated by GMM, with an application to the Phillips curve. Journal of Money Credit and Banking 37, 421–448. Mavroeidis, S., 2010. Monetary policy rules and macroeconomic stability: some new evidence. American Economic Review 100, 491–503. McCallum, B.T., 1983. On non-uniqueness in rational expectations models: an attempt at perspective. Journal of Monetary Economics 11, 139–168. McCallum, B.T., 2003. Multiple-solution indeterminacies in monetary policy analysis. Journal of Monetary Economics 50, 1153–1175. Pesaran, H.M., 1987. The Limits to Rational Expectations. Basil Blackwell, Oxford. Ravenna, F., 2007. Vector autoregressions and reduced form representations of DSGE models. Journal of Monetary Economics 54, 2048–2064. Salemi, M.K., 1986. Solution and estimation of linear rational expectations models. Journal of Econometrics 31, 41–66. Salemi, M.K., Song, J., 1992. Saddlepath solutions for multivariate linear rational expectations models. Journal of Econometrics 53, 245–269. Sims, C.A., 2002. Solving linear rational expectations models. Computational Economics 20, 1–20. West, K.D., 1986. Full-versus limited-information estimation of a rationalexpectations model. Journal of Econometrics 33, 367–385. West, K.D., 1997. Another heteroskedasticity- and autocorrelation-consistent covariance matrix estimator. Journal of Econometrics 76, 171–191. Wickens, M.R., 1982. The efficient estimation of econometric models with rational expectations. Review of Economic Studies 49, 55–67.