VAR-based Estimation of Euler Equations with an Application to New Keynesian Pricing∗ Andr´e Kurmann† Universit´e du Qu´ebec `a Montr´eal ´ and CIRPEE January 18, 2006

Abstract VAR-based estimation of Euler equations exploits cross-equation restrictions that the theory imposes on a vector-autoregressive (VAR) process for market expectations. This paper shows that Sargent’s (1979) original approach of imposing these restrictions on the endogenous variable of the Euler equation (i.e. computing the rational expectations solution) implies multiple solutions and is therefore impracticable. The paper shows that this identification problem can be circumvented by reverse-engineering the cross-equation restrictions on the forcing variable of the Euler equation. The proposed reverse-engineering approach contrasts with the conventional approach in the literature that avoids the identification problem by constraining the system formed by the Euler equation and the VAR process to yield a unique stable rational expectations equilibrium. This uniqueness condition makes little economic sense because the VAR process is just a reduced-form approximation of true market expectations, and a simulation experiment shows that imposing uniqueness can lead to severe misspecification bias. The severity of this misspecification bias is illustrated in practice with an application to Gali and Gertler’s (1999) hybrid New Keynesian Phillips Curve. JEL classification: C13, E31, E32 Keywords: Maximum Likelihood, Rational Expectations, New Keynesian Phillips Curve, Inflation, Real marginal cost. ∗

The paper has greatly benefited from comments by Jeff Fuhrer, Alain Guay, Peter Ireland, Jinill Kim, Bob King, Jesper Lind´e, Chris Otrok, Frank Schorfheide, an anonymous referee and seminar participants at the Sverige Riksbank and the Econometric Society World Congress 2005. The author gratefully acknowledges financial support from the Bankard Fund of the University of Virginia, the Swiss National Science Foundation (bourse de jeune chercheur), and the PAFARC research fund of UQAM. † Contact address: Andr´e Kurmann, Universit´e du Qu´ebec ` a Montr´eal, D´epartement des sciences ´economiques, P.O. Box 8888, Downtown station, Montr´eal (QC) H3C 3P8, Canada. Email : [email protected]. Phone: (514) 987-3000, ext. 3503. Fax: (514) 987-8494.

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1

Introduction

A wide variety of dynamic stochastic theories give rise to linear(ized) Euler equations of the form yt = aE[yt+1 |Zt ] + bxt + cyt−1 + ut ,

(1)

where yt is an endogenous choice variable; xt a forcing variable that agents take as given; E[·|Zt ] expectations conditional on information Zt ; and ut an error term that is uncorrelated with past information. Most frequently in the literature, this class of models is estimated via instrumental variable techniques such as General Method of Moments (GMM). On the one hand, these estimators have the advantage that they depend on very few auxiliary assumptions about expectations and can be implemented at relatively little cost. On the other hand, studies by Fuhrer, Moore and Schuh (1995) or Stock, Wright and Yogo (2002) document that GMM has poor small sample properties when instruments are weakly identified; is relatively inefficient; and suffers from normalization problems. An alternative approach consists of estimating the Euler equation in (1) conditional on a fully structural general equilibrium model. Compared to instrumental variable techniques, this full-information approach has the advantage that expectations are based on optimal projections implied by the model, and that estimation takes into account the entire likelihood. As Monte-Carlo simulations by Fuhrer, Moore and Schuh (1995) or Lind´e (2005) show, these estimates often have better small sample properties than GMM; are more efficient; and do not suffer from normalization problems. The flipside of the full-information approach is that the estimates of the Euler equation of interest are conditional on strong assumptions about the rest of the economy and may suffer from important biases when (some of) the remaining structural equations are incorrectly specified.1 This concern is particularly acute when the general equilibrium model as a whole fails to replicate key features of the data. This paper considers a third estimation approach that consists of proxying market expectations with forecasts from a vector-autoregressive (VAR) process in x and y (and possibly other variables), and deriving the cross-equation restrictions that the theory imposes on the VAR under rational expectations. The parameters a, b and c are then estimated from the restricted VAR via classical Maximum Likelihood (ML) or Bayesian inference. Originally proposed by Sargent (1979), this VAR-based estimator has been applied by Fuhrer and Moore (1995) and Jondeau and Le Bihan (2001) to New Keynesian pricing models; Fuhrer and Rudebusch (2004) to an Euler equation for output; and King and Kurmann (2002) and Cogley (2005) to the expectations theory of interest rates. The advantage of the VAR-based approach is that it relies on relatively weak assumptions about market expectations while 1

Ruge-Murcia (2003), for example, reports that even misspecification in the number of structural shocks can lead to subtantial bias in full-information estimates.

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simultaneously imposing the dynamic constraints of the theory and exploiting information from the entire likelihood. As such, it provides a valuable alternative for estimating Euler equations. The objective of this paper is to expose pitfalls of commonly used techniques to implement the VAR-based estimator that can greatly hamper its accuracy. The paper then proposes an alternative implementation that circumvents these pitfalls, and illustrates the different contributions with an application to a popular New Keynesian pricing equation. In his original application, Sargent (1979) implements the cross-equation restrictions by expressing the coefficients of the VAR equation for y (i.e. the rational expectations solution for y given VAR expectations) as a function of the Euler equation parameters and the coefficients of the remaining VAR equations. The present paper uncovers that the thus restricted coefficients are higher-order polynomials in the unconstrained coefficients. Hence, multiple rational expectations solutions for y satisfy the cross-equation restrictions, with the number of solutions rapidly increasing in the dimensions of the VAR. This identification problem — thus far ignored by the literature — is a direct consequence of relaxing the restrictive assumption of strict exogeneity of x with respect to y that underlies Hansen and Sargent’s (1980) classical likelihood estimator of linear rational expectations models. The highly non-linear nature of the different solutions means that computing the likelihood subject to each solution and then selecting the one with the highest value is impracticable. Instead, this paper shows that the identification problem can be circumvented altogether by ”reverse-engineering” the cross-equation restrictions as constraints on the VAR coefficients of the forcing variable x. There exists a single mapping from these coefficients to the parameters of the Euler equation and the coefficients of the remaining VAR equations. Imposing the cross-equation restrictions in this way boils the estimation down to a simple constrained optimization problem.2 The paper then contrasts the proposed reverse-engineering approach with what has emerged as the conventional approach to implementing VAR-based estimation. This approach expresses the Euler equation and the VAR process for the forcing variable as a full-information system, and restricts the estimation to yield a unique stable rational expectations equilibrium.3 In doing so, the conventional approach implicitly avoids the identification problem because the estimates are such that all but one solution to the cross-equation restrictions can be discarded on grounds of non-stationarity. At the same time, imposing uniqueness implies additional constraints on the estimation — constraints that make little 2

In deriving their Wald tests, Campbell and Shiller (1987) and Campbell (1987) have previously expressed the cross-equation restrictions on the coefficients of the equation for the forcing variable. However, they did so only under the restrictive assumption that the Euler equation holds exactly; i.e. ut = 0. As the paper shows, the identification problem does not occur in this case. 3 Studies that employ this alternative method are Fuhrer and Moore (1995), Jondeau and Le Bihan (2001) or Fuhrer and Rudebusch (2004).

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economic sense because the VAR component of the underlying general equilibrium model is only an approximation of market expectations rather than the structural description of the economy. As the paper illustrates by means of a simple experiment, this uniqueness condition lead to severe misspecification bias when the likelihood of the true structural model is maximized for a combination of parameters that implies multiple stable rational expectations equilibria for the system formed by the Euler equation and the VAR process. The dangers of imposing uniqueness in practice are underlined with an application to the hybrid New Keynesian Phillips Curve (NKPC) by Gali and Gertler (1999). The theory implies a log-linear Euler equation as in (1) that links current inflation to expected future inflation, past inflation and current real marginal cost. Conditional on real marginal cost being measured with labor income share, ML estimation with the proposed reverseengineering approach indicates that the hybrid NKPC cannot be rejected by a conventional likelihood ratio test. In addition, labor income share enters significantly into the model and forward-looking behavior is the predominant determinant of inflation. Interestingly, these results coincide by and large with Gali and Gertler’s (1999) GMM or Ireland’s (2001) fullinformation ML estimates, thus confirming that conditional on marginal cost being correctly measured by labor income share, forward-looking behavior is an important feature of price setting.4 By contrast, if the same ML estimator is implemented with the conventional approach, the results change dramatically: the backward-looking inflation component of the hybrid NKPC becomes equally important than the forward-looking component; real marginal cost enters insignificantly; and the model as a whole is rejected. This difference is due to the additional constraints imposed by the condition that the system formed by the NKPC and the VAR equation for labor income share must have a unique stable rational expectations equilibrium. The application to the hybrid NKPC therefore provides a telling illustration that the conventional approach to VAR-based estimation can lead to severe misspecification bias. The rest of the paper is organized as follows. Section 2 considers a stylized example to show the identification problem with Sargent’s (1979) original implementation and to derive the reverse-engineering approach. Section 3 illustrates the pitfalls of the conventional approach that imposes additional constraints. Section 4 applies the proposed reverse-engineering approach to the hybrid NKPC and compares the results with the existing literature. Section 5 concludes. 4

Sbordone (2002, 2004) finds results similar to Gali and Gertler (1999) based on alternative minimum distance estimators.

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2

Cross-equation restrictions and the identification problem

To illustrate the identification problem, consider a purely forward-looking version of the Euler equation in (1) yt = aE[yt+1 |Zt ] + bxt + ut .

(2)

Both yt and xt can be measured by the econometrician while ut is unobserved. Traditionally, Hansen and Sargent (1980) interpret ut as a deviation from exact rational expectations models due to structural shocks and/or latent variables that are only observed by agents in the model. But from a more applied point of view, ut can also be interpreted as an error due to non-linearities in the theory and/or a data mismeasurement.5 Throughout the paper, we assume that the econometrician observes only a subset zt of the agent’s information set Zt and that the econometrician forecasts agents’ expectations for √ x and y with a VAR process of mean exponentional order less than 1/ a. We express this VAR process in companion form as zt = Mzt−1 +et , where xt , yt ⊂ zt and [et e0t+k ] v (0, Σ) with Σ = 0 for all k 6= 0. Furthermore, we assume ut to be uncorrelated with zt−1 . This environment represents a generalization of Hansen and Sargent’s (1980) likelihoodbased estimator of linear rational expectations models in the sense that it relaxes the assumption of strict exogeneity of x with respect to y. At the same time, the assumption that ut is uncorrelated with past information is more restrictive than Hansen and Sargent’s setup. We return to discussing these differences in more detail at the end of this section. For the sake of exposition, the following derivations focus on the purely forward-looking Euler equation in (2) and a stylized VAR(1) process in the two variables of interest only # " #" # " # " mxx mxy xt−1 ex,t xt = + . (3) yt myx myy yt−1 ey,t But all of the results that we derive also hold for the Euler equation in (1) and for general VAR processes, with the appendix providing the formal proofs.

2.1

Rational expectations cross-equation restrictions

Under rational expectations, the Euler equation in (2) implies cross-equation restrictions between the structural parameters a, b and the VAR coefficients. Following Sargent (1979), 5

Following Sargent (1989), several studies introduce measurement errors into structural general equilibrium models by first computing the rational expectations equilibrium without measurement errors and then estimating the resulting state-space solution with error terms tacked onto it. See Ireland (2004) for a comprehensive review. This strategy is appealing because it allows the econometrician to estimate a fullinformation system of equations using more variables than structural shocks without facing a stochastic singularity problem. When the object of interest is a single Euler equation, however, it seems preferable to explicitly allow for the possibility of measurement errors in the derivation of the cross-equation restrictions.

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these restrictions are derived in three steps.6 First, we rewrite the Euler equation in (2) as yt = aE[yt+1 |zt ] + bxt + ut + ζ t , where the term ζ t ≡ a(E[yt+1 |Zt ] − E[yt+1 |zt ]) appears because the econometrician uses only a subset of the agents’ full information set; i.e. zt ⊆ Zt . Second, we project this equation onto zt−1 E[yt |zt−1 ] = aE[yt+1 |zt ]|zt−1 + bE[xt |zt−1 ] + E[ut |zt−1 ] + E[ζ t |zt−1 ]. By the law of iterated expectations we have E[yt+1 |zt ]|zt−1 = E[yt+1 |zt−1 ] and E[ζ t |zt−1 ] = a(E[yt+1 |zt−1 ] − E[yt+1 |zt−1 ]) = 0. Furthermore, our assumption that ut is uncorrelated with zt−1 implies E[ut |zt−1 ] = 0 under rational expectations. Third, we use the econometrician’s VAR process to form multi-period forecasts for x and y E[yt+i |zt−1 ] = hy Mi+1 zt−1

E[xt+i |zt−1 ] = hx Mi+1 zt−1 , with hy = [0 1] and hx = [1 0]. Substituting these forecasts into the above expression and realizing that the resulting relationship needs to hold for general zt−1 , we obtain hy M = ahy M2 + bhx M,

(4)

or written more explicitly myx (1 − a(mxx + myy )) = bmxx myy (1 − amyy ) − amxy myx = bmxy . The 2 equations characterize the cross-equation restrictions between the structural parameters a, b of the theory and the VAR parameters mxx , mxy , myx , myy . Intuitively, these restrictions arise because the VAR forecasts of xt and yt must be consistent with the dynamic relationship between the two variables as predicted by the theory.7 6

Sargent (1979) derives these cross-equation restrictions for a finite-horizon, first-differenced version of the expectations theory of interest rates that does not allow for an error term; i.e. ut = 0. The steps used to derive the cross-equation restrictions for the Euler equations are exactly the same. Note that the different form of the Euler equation in Sargent’s paper implies cross-equation restrictions that are more complicated than the ones we derive here for the special case of ut = 0, and that his implementation of the restrictions is subject to exactly the identification problem that is the focus of the present paper. 7 The same cross-equation restrictions could have been derived by (i) iterating the Euler equation forward; (ii) applying the transversality condition limT →∞ aT +1 E[yt+T +1 |zt ] = 0; and (iii) using the VAR process to project both sides of the resulting equation on information zt−1 .

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2.2

Maximum Likelihood estimation

As noted in the introduction, the VAR-based approach can be estimated via classical ML or Bayesian inference. To keep with most of the literature, we describe here only the ML estimator. However, all the issues on which the paper focuses would also arise with Bayesian inference. Under the assumption that the VAR error terms et are multivariate normal, the loglikelihood for sample {zt }Tt=1 can be expressed as8 T nT log(2π) − [log[det(Σ)] + n] (5) 2 2 with n = 2. VAR-based estimation of the Euler equation in (2) based on information zt P ˆ where Σ ˆ = 1 T ˆ ˆ t−1 , therefore involves minimizing det(Σ), et 0 with ˆ et = zt − Mz t=1 etˆ T subject to the cross-equation restrictions in (4). As Sargent (1979) proposed, the natural way to implement this constrained optimization problem is to express the coefficients of the VAR equation for yt as a function of the parameters of the Euler equation and the coefficients of the remaining VAR equations. Specifically, this means transforming (4) into two explicit functions of the form £=−

my· = f (a, b, mx· ),

(6)

where my· ≡ {myx , myy } and mx· ≡ {mxx , mxy }. This is commonly referred to as computing the rational expectations solution of the Euler equation. The log-likelihood in (5) is then maximized with respect to a, b and mx· .9 Sargent’s approach would be straightforward if transforming (4) into an explicit function of the form given by (6) was easy. Unfortunately, this is not the case. Even for low-dimensional VAR processes, multiple combinations of my· satisfy the cross-equation restrictions. For the stylized example here, there are three different solutions. The first solution is myx = −b/a, myy = 1/a.

(7)

The second and the third solution take the form

p (1 − amxx )2 − 4abmxy 2a p (1 − amxx ) + (1 − amxx )2 − 4abmxy . = 2a

myx =

(1 − amxx ) − mxx myy , myy = mxy

(8)

myx =

mxx myy , myy mxy

(9)

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See Hamilton (1994) for a derivation. For the stylized example here, the cross-equation restrictions could also be imposed directly on a and b after having estimated an unrestricted VAR. This is just an artifact of the simple VAR(1) specification, however. For VAR specifications with more variables and/or more lags, the number of cross-equation restrictions is larger than two, in which case a and b are over-identified and the cross-equation restrictions need to be imposed on VAR coefficients. 9

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The multiplicity of solutions is not an artifact of the simple VAR specification or the purely forward-looking Euler equation used here. To the contrary, the number of mappings my· = f (a, b, m−y· ) that satisfy (4) increases with the dimension of the VAR process. For example, if the bivariate VAR is extended to three lags instead of one (as in the application of Section 4), the number of solutions increases to 7. This raises two important questions. First, do the solutions differ in their economic interpretation? Second, since each of the solutions implies different constraints on the ML estimation (and thus a different likelihood), which one should be selected? Consider solution (7) first. It could have been obtained by simply lagging the variables of the Euler equation by one period and rewriting it as b 1 1 yt = − xt−1 + yt−1 − ut−1 + εt , a a a where εt ≡ yt − E[yt |Zt−1 ] is defined as the agents’ rational expectations forecast error. In this case, the rational expectations solution for yt is simply a function of the structural parameters a, b but does not depend on the econometrician’s specification of the VAR forecasting process. The cross-equation restrictions therefore reduce to single-equation restrictions, which could have also been obtained by postmultiplying each term of the cross-equation restrictions in (4) by M −1 . But as is obvious from the derivation of these restrictions worked out above, this operation is admissible only if zt−1 is uncorrelated with ut−1 ; i.e. E[ut−1 |zt−1 ] = 0. Since yt ∈ zt and ut is one of the determinants of yt , however, this condition holds true only for the special case of ut = 0. In other words, imposing the rational expectations solution in (4) amounts to estimating an exact linear rational expectations model that is reminiscent of efficient markets regressions in the finance literature.10 Now, consider the second and third solution to the cross-equation restrictions, (8) and (9). These solutions are fundamentally different from the first one for two reasons. First, the my· terms are non-linear functions of the structural parameters a, b as well as the VAR coefficients mx· ≡ {mxx , mxy }. Hence, both of these solutions depend on the econometrician’s VAR process to forecast market expectations. Second, both solutions imply that the VAR matrix M is singular. To see this, rewrite M with the restrictions (8) and (9) imposed " # " # mxx mxy mxx mxy M≡ = mxx . myx myy mxy myy myy The eigenvalue polynomial of this matrix equals det(M − λI) = λ[λ − (myy + mxx )] = 0. Hence, the eigenvalues of M are λ1 = 0 and λ2 = myy + mxx ; or rewritten more explicitly p (1 + amxx ) − (1 − amxx )2 − 4abmxy (10) λ1 = 0, λ2 = p 2a (1 + amxx ) + (1 − amxx )2 − 4abmxy . (11) λ1 = 0, λ2 = 2a 10

See for example Roll (1969). Another application is Campbell and Shiller (1987) who derive their Wald tests and F-tests of present value models under the assumption that ut = 0.

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Since λ1 = 0, M is singular and the cross-equation restrictions in (4) cannot be postmultiplied by M−1 as is possible for the solution under ut = 0 above. Hence, VAR-based estimation subject to either the second or the third solution of the cross-equation restrictions amounts to testing an inexact rational expectations model in the sense that it allows for an unpredictable stochastic error term ut . All of these results also hold for the more general case and are summarized by the following proposition.11 Proposition 1 The rational expectations solutions to the Euler equation yt = aE[yt+1 |Zt ]+ bxt +cyt−1 +ut conditional on a VAR process can be classified according to two types. Under the assumption ut = 0, there is a single solution that is independent of assumptions about market expectations. Under the assumption ut 6= 0, multiple solutions exists with each one of them implying that the VAR companion matrix M is singular. Proof: see appendix.

2.3

Circumventing the identification problem

Focusing on the case where ut 6= 0 allows the econometrician to eliminate one of the solutions to the cross-equation restrictions. Yet, this still leaves open the question of how to select among the remaining solutions (two in the present stylized example; many more if the VAR specification is of higher dimensions). Theoretically, one could think of estimating the VAR subject to each of the different solutions and then selecting the one with the highest likelihood value. In practice, however, this approach is unworkable because the number of solutions increases rapidly in the dimensions of the VAR and because each of the solutions is a highly non-linear function of the unrestricted parameters.12 There is a simple fix to this problem — thus far ignored by the literature — that consists of expressing the VAR coefficients for the forcing variable xt as a function of the Euler equation parameters and the coefficients of the remaining VAR equations. For the stylized example of this section, this yields the following solution for the cross-equation restrictions in (4) 11

Campbell (1987) anticipates the singularity result for the case ut 6= 0. However, he does not formally derive it, nor does he notice the identification problem. Campbell’s Wald test also allows for uncorrelated ut terms. However, his approach does not allow estimation of the structural parameters of the Euler equation (i.e. in Campbell’s application to the permanent income hypothesis, the Wald test is conditional on fixing a value for the steady state real interest rate, which is the structural parameter of the theory). 12 On a standard computer, MATLAB’s symbolic mathematics toolbox even has trouble solving for the 7 analytical solutions that arise when the VAR process has 3 lags.

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mxx = mxy =

myx (1 − amyy ) amyx + b myy (1 − amyy ) , amyx + b

(12) (13)

or more compactly mx· = f (a, b, my· ). As is immediately apparent, this mapping from mx· to the structural parameters a, b and the VAR coefficients of the endogenous variable my· is unique and therefore circumvents the multiplicity problem of Sargent’s original approach. The following proposition states the generality of this reverse-engineering approach to imposing the cross-equation restrictions, as it will be called henceforth. Proposition 2 Cross-equation restrictions implied by Euler equations of the form yt = aE[yt+1 |Zt ] + bxt + cyt−1 + ut have a single solution in the VAR coefficients of the forcing variable xt . Proof: see appendix.

2.4

A comparison with Hansen and Sargent (1980)

Before continuing, it is instructive to contrast the present VAR-based estimator with Hansen and Sargent’s (1980) original likelihood estimator of linear rational expectations models. The key difference between the two is that Hansen and Sargent’s estimator assumes xt to / Zt . Under this assumption, the coefficients be strictly exogenous with respect to yt ; i.e. yt ∈ of the companion matrix M are unrestricted by the rational expectations solution for the endogenous variable of interest yt because the latter is not part of the VAR system. Under this assumption, the rational expectations solution for yt can be computed and estimated without running into the identification problem that is the main focus of this paper. Yet, there are many instances where Hansen and Sargent’s assumption of strict exogeneity is inappropriate. First, by iterating the Euler equation in (2) forward and imposing the transversality condition limT →∞ aT +1 E[yt+T +1 |Zt ] = 0, we obtain its present value form P j yt = b ∞ j=0 a E[xt+j |Zt ] + ut . As long as movements in yt are not entirely driven by ut , yt should embody information about future x and thus, yt should be part of the econometrician’s information set.13 Second, there may be structural reasons why yt has predictive power for Zt and thus for xt . One example is a policy function that affects elements of Zt as a function of current or past y. Third, and as Hansen and Sargent note themselves, even when Zt is strictly exogenous with respect to yt , yt may Granger cause zt for ”omitted 13

Campbell and Shiller (1987) make this argument for the case where the rational expectations model holds exactly.

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variable” reasons.14 In sum, the present paper contributes to the literature because it uncovers the identification problem that arises in the more general case where y is part of the econometrician‘s information set and proposes a reverse-engineering approach that circumvents the problem. At the same time, it is important to emphasize that all of the results here are conditional on the assumption that ut is uncorrelated with zt−1 , which is more restrictive than Hansen and Sargent’s environment and implies important constraints on the nature of the deviations that underlie this error term.15 Relaxing this assumption would imply more complicated cross-equation restrictions. Whether and how we can circumvent the identification problem in such a more general setting is the topic of future research.

3

Pitfalls of imposing a unique stable equilibrium

The proposed reverse-engineering approach to circumvent the identification problem contrasts with what has emerged as the conventional approach to implement VAR-based estimation of Euler equations. This section shows that by contrast to the proposed reverseengineering approach, the conventional approach — employed for example by Fuhrer and Moore (1995), Jondeau and Le Bihan (2001) or Fuhrer and Rudebusch (2004) — avoids the identification problem by imposing additional constraints on the estimated parameters. These constraints make little economic sense but can lead to serious misspecification bias.

3.1

The stylized example reconsidered

To understand the conventional approach, reconsider the stylized example from above. The Euler equation in (2) and the VAR equation for the forcing variable xt in (3) can be expressed as a full-information system of the form yt = aEt yt+1 + bxt + ut

(14)

xt = mxx xt−1 + mxy yt−1 + ext , or equivalently AEt Yt+1 = BYt + CXt , where Yt+1 = [xt+1 yt+1 xt yt ]0 , Yt = [xt yt xt−1 yt−1 ]0 , Xt = [ext ut ]0 and A, B and C are matrices filled with the appropriate parameters. The conventional approach solves this 14

Instead of working out the cross-equation restrictions for this omitted variable case, Hansen and Sargent (1980) resort to an instrumental variable estimator that can be seen as a predecessor of Hansen’s (1982) GMM estimator. 15 For example, ut may not follow an autoregressive process as in Hansen and Sargent because this would automatically imply that yt−1 is correlated with ut .

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system by requiring that there exists a unique stable rational expectations equilibrium. This solution necessarily has the same VAR form than the forecasting process used in the reverseengineering approach, with the coefficients satisfying the same cross-equation restrictions than in (4). From there, the likelihood is easily computed as in (5) and estimation becomes straightforward. How does the conventional approach avoid the identification problem uncovered in the previous section? First, nothing restricts ut to be constant, which by definition rules out the exact solution (ut = 0) in (7). Second, it is well known from Blanchard and Kahn (1980) that a unique stable rational expectations equilibrium exists if and only if the number of non-predetermined variables of the system equals the number of generalized eigenvalues with modulus larger than one. For the stylized example in (14), the generalized eigenvalues are16 p (1 + amxx ) ∓ (1 − amxx )2 − 4abmxy , λ4 = ∞. λ1 = 0, λ2,3 = 2a By definition, |λ1 | < 1 and |λ4 | > 1. Since there are two non-predetermined variables in Yt (xt and yt ), the system has a unique stable solution if the combination of parameters a, b, mxx , mxy is such that |λ2 | < 1 while at the same time |λ3 | > 1. Forward iteration then allows to eliminate the two unstable eigenvalues. What remains are the two stable eigenvalues λ1 and λ2 , which are exactly the eigenvalues of M in (10). Hence, the conventional approach avoids the identification problem highlighted in Section 2 by restricting a, b, mxx , mxy in (14) to a region of the parameter space where only the solution in (8) implies a stationary (stable) rational expectations equilibrium and where the other solution in (9) can be discarded on grounds of non-stationarity.17 Under this condition, the map my· = f (a, b, m−y· ) necessarily becomes unique. The same logic applies, of course, for larger VAR specifications of the forcing variable. In this case, the uniqueness requirement constrains the parameters such that all but one of the solutions to the cross-equation restrictions imply non-stationarity.

3.2

Does imposing uniqueness matter and does it make sense?

The inequality constraints imposed by the conventional approach restrict a, b, mxx , mxy to a region of the parameter space where the system in (14) has a unique stable solution. Hence, for any case where the likelihood is maximized for a combination of a, b, mxx , mxy that lies outside this region, uniqueness prevents estimation with the conventional 16

The generalized eigenvalues are the solutions to det(B − λA) = 0. When A is singular, one or more of the eigenvalues are infinite. See King and Watson (1998) for details. 17 This logic applies for the case 0 < λ2 < 1 and λ3 > 1. Likewise, uniqueness restricts a, b, mxx , mxy such that λ2 < −1 and 0 < λ3 < 1, in which case the solution in (8) can be discarded on grounds of non-stationarity.

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approach from converging to the optimum. The proposed reverse-engineering approach, by contrast, is not subject to inequality constraints. Estimation therefore converges to the optimum. The following simulation experiment illustrates this crucial difference between the two approaches. Suppose that the dynamics of x and y are generated from the following system yt = aEt yt+1 + bxt + ut

(15)

xt = c1 yt + c2 xt−1 + c3 yt−1 + vt , where ut and vt are possibly correlated contemporaneously. The first equation is the stylized Euler equation from above. The second equation could be interpreted, for example, as a policy rule that directly affects xt given yt , xt−1 and yt−1 . Let the different parameters take the values a = 1, b = 0.5, c1 = −0.5, c2 = 1 and c3 = 0.4. For this combination, the system (15) has a unique rational expectations equilibrium that takes the form of a VAR(1) xt = 0.34xt−1 + 0.14yt−1 + ext yt = 1.31xt−1 + 0.53yt−1 + eyt . Now, consider an econometrician who wants to estimate the stylized Euler equation in (15), but who does not know the true structure of the rest of the economy. We assume, however, that the econometrician correctly describes the reduced-form dynamics of x and y with a VAR(1). Hence, the experiment we are about to consider by definition rules out any estimation bias that could arise from an incorrectly specified likelihood. Using the proposed reverse-engineering approach, the econometrician estimates a and b by maximizing the likelihood of a VAR(1) in x and y subject to the cross-equation restrictions (12) and (13) that are implied by the Euler equation. The resulting estimates asymptotically converge to the true maximum a ˆ = 1, ˆb = 0.5. Figure 1 depicts the contour plot of the log-likelihood surface as a function of a and b (conditional on the parameters c1 , c2 , and c3 being kept at their true values), with point A depicting this maximum.18 By contrast, if the econometrician follows the conventional estimation approach, she expresses the stylized Euler equation and the VAR(1) equation for x as a general equilibrium system as in (14) and estimates its unique stable rational expectations solution. Here is where the problem occurs. For the true values a = 1, b = 0.5, mxx = 0.34 and mxy = 0.14, the two non-trivial generalized eigenvalues of this system are both smaller than one (λ2 = 0.48 and λ3 = 0.87). Hence, the system in (14) has multiple stable solutions even though the solution to the true data-generating process in (15) is itself unique. The conventional estimation approach will therefore not reach the true maximum because the parameters a, b, mxx , mxy are constrained to remain in the region with a unique rational expectations 18

The contour plot was generated from a sample of 1000 observations under the assumption that the structural error terms ut and vt are i.i.d.

13

equlibrium. In Figure 1, this region is shown by the shaded triangle in the lower-left part, limited by the boundary {a, b} = {0.94, 0.30} to {a, b} = {0.80, 0.95}. Any combination above this boundary is not admissible for estimation because it violates uniqueness. Instead, the conventional approach moves along the boundary to the highest possible point, which is located at a ˆ = 0.82, ˆb = 0.92 (point B) — far away from the true maximum at a = 1 and b = 0.5. This experiment illustrates that the uniqueness restrictions imposed by the conventional approach can lead to serious misspecification bias even if the likelihood behind the estimation is correctly specified. Intuitively, this bias arises because the true dynamics of x are described by xt = c1 yt + c2 xt−1 + c3 yt−1 + vt and not by a VAR(1) as the conventional approach imposes. The restrictions that the resulting system in (14) implies for uniqueness are very different from the ones implied by the true structural model in (15).19 Of course, one may ask whether imposing uniqueness matters in practice and if so, whether it could be justified on economic grounds. The next section shows that at least for one prominent application, the NKPC, imposing uniqueness matters greatly and leads to an important misspecification bias. As for economic justification, modern macroeconomic theory emphasizes intertemporal decisions under uncertainty, which typically results in optimality conditions that contain expectational and/or contemporaneous terms as explanatory variables. As such, treating the VAR equation as a true structural description of the forcing variable and imposing uniqueness on the resulting general equilibrium system makes little economic sense. Furthermore, even if the true dynamics of x were described by a VAR process, imposing uniqueness would by definition rule out rational bubbles and sunspot equilibria; i.e. equilibria where non-fundamental shocks lead agents to revise their forecasts of endogenous variables. As Meese (1986) or Benhabib and Farmer (1999) argue, such equilibria may be important to explain business cycle dynamics. Hence, it is not clear why one would want to exclude such equilibria, especially when the main interest is in estimating a single Euler equation.20 19

Note that in this simple experiment, the econometrician could not separately estimate the parameters c1 , c2 , c3 because they are not identified from the reduced-form solution that defines the likelihood. However, the point of VAR-based estimation is exactly that we are interested in estimating only part of the model (here the Euler equation for y) rather than the entire set of structural equations. All that matters is that the parameters of interest a, b are identified, which is true by the cross-equation restrictions in (12) and (13). 20 This may remind the reader of recent work by Lubik and Schorfheide (2004) who develop a method to solve and estimate dynamic general equilibrium models in the presence of multiple stable equilibria. In principle, their method could also be applied to VAR-based estimation of Euler equations. Aside from being considerably more costly to implement, it would require auxiliary assumptions about the relationship between structural shocks and sunspot shocks. The approach proposed here does not need to take a stand on these issues because it is not necessary to specify whether the VAR forecasting process approximates an economy with a unique or multiple rational expectations equilibria.

14

This discussion raises the question of how well a finite-order VAR process approximates market expectations. The above simulation experiment was designed so that the VAR(1) process underlying the estimation coincided with the rational expectations solution of the true model. Hence, the proposed reverse-engineering approach converged towards the true parameter values. In general, however, the dynamics of the variables involved are likely to follow more complicated processes. In particular, most DSGE models in use today imply rational expectations solutions that follow a VARMA process. As Fern´ andez-Villaverde, Rubio-Ramirez and Sargent (2005) show, these VARMA processes have an infinite-order VAR represention only under certain conditions. And even if these conditions are fulfilled, it remains an open question how the estimation results are affected by the truncation necessary to obtain a finite-order VAR process. Quantifiying the bias induced by VAR approximation is an open question and the topic of ongoing research.21 Ultimately, the issue of misspecification bias and the econometric properties of the VAR-based estimator need to be contrasted to misspecification issues and econometric properties of alternative estimation methods, as discussed in the introduction of the paper.

4

Application to New Keynesian pricing

This section applies the VAR-based estimator to a popular New Keynesian pricing equation. The application has received great attention in the recent literature and thus, the results in this section are interesting because they add to the current debate. More importantly, however, the application provides an example where imposing uniqueness greatly affects ML estimates.

4.1

The hybrid New Keynesian Phillips curve

The hybrid NK pricing model has been proposed by Gali and Gertler (1999) as an extension of the sticky price model by Calvo (1983). Since the primary focus of this section is about estimation, we only discuss key aspects of the model that are important for the interpretation of the results. Calvo’s sticky price model consists of a large number of imperfectly competitive but otherwise identical firms as proposed by Blanchard and Kiyotaki (1987). It is assumed that in every period, a random fraction 1 − θ of these firms adjusts their price. The remaining fraction θ of firms must keep their price fixed, disregarding of the number of periods they 21

Recent work by Chari, Kehoe and McGrattan (2004) calls into question the accuracy of of approximating the solution to a full-blown dynamic general equilibrium model with a finite-order VAR in a small subset of the model’s variables. However, their critique is directed mostly towards the structural VAR literature exploiting long-run restrictions. As Erceg, Guerreri and Gust (2004) and Christiano, Eichenbaum and Vigfusson (2005) show, the bias introduced by truncation seem to be of relatively small for short-run restrictions similar to the ones imposed in VAR-based estimation.

15

have kept their price unchanged in the past. Hence, a firm keeps its price fixed for an average of 1/(1 − θ) periods. An adjusting firm will set its new price rationally such as to maximize current and expected future profits (taking into account that it may not readjust for several periods). Gali and Gertler’s extension consists of imposing that among the price adjusting firms, only a fraction 1 − ω is forward-looking and set its price optimally. The remaining fraction ω of adjusting firms is assumed to use a rule-of-thumb instead that consists of setting the new price equal to the average of last period’s new price, updated with last period’s inflation rate. Aggregating over the different firms and log-linearizing, these assumptions result in an Euler equation that links current inflation π t to expected future inflation Et π t+1 , real marginal cost ψ t and lagged inflation π t−1 π t = γ f Et π t+1 + ϕψ t + γ b π t−1 + ut ,

(16)

where the error term ut can take on different interpretations.22 All variables denote percentage deviations from their respective steady states. The slope coefficients γ f , γ b and ϕ are functions of the underlying structural model parameters and are defined as ϕ = γf

=

γb =

(1 − ω)(1 − θ)(1 − βθ) φ βθ φ ω , φ

(17)

where φ = θ + ω[1 − θ(1 − β)] and β is the discount factor. Gali and Gertler call (16) the ’hybrid’ New Keynesian Phillips Curve (NKPC) because inflation is determined both by (forward-looking) expectations about future inflation and a (backward-looking) lagged inflation term. For the particular case where the fraction of rule-of-thumbers is zero, i.e. ω = 0, the hybrid NKPC reduces to π t = βEt π t+1 + ϕψ t + ut , which is the log-linearized inflation equation of the original Calvo model.

4.2

Maximum Likelihood estimator

The cross-equation restrictions that the hybrid NKPC imposes on a general VAR process in n variables and p lags are derived analogous to the ones in the proof of Proposition 1 (see appendix) and take the form hπ [M − γ f M2 − γ b I] = ϕhψ M, 22

(18)

On the theoretical side, ut may capture exogenous shocks to firms’ desired markups (see Steinsson, 2004). On the empirical side, ut may take into account errors due to log-linearization (see Rotemberg and Woodford, 1997) or mismeasurement of either real marginal cost or inflation.

16

where M is the (np × np) companion matrix of a VAR; I is a np × np identity matrix; and hπ and hψ are (np × 1) selection vectors for π t and ψ t , respectively. As discussed in Section 2, these restrictions are obtained under the assumption that ut is uncorrelated with t − 1 information contained in the VAR. By Proposition 2, the system of np equations in (18) has a single solution in the np coefficients of the VAR equation for real marginal cost (i.e. the forcing variable of the hybrid NKPC) mψ· = f (γ f , γ b , ψ, m−ψ· ),

(19)

where mψ· is the np × 1 coefficient vector of the VAR equation for real marginal cost; and m−ψ· is the vector containing the n(n − 1)p coefficients of the remaining VAR equations. Analogous to Section 2, the log-likelihood of the VAR is

£=−

T nT log(2π) − [log[det(Σ)] + n] . 2 2

Hence, ML estimation using the proposed reverse-engineering approach boils down to minˆ subject to the restrictions in (19). imizing det(Σ), The resulting ML estimates are contrasted to the estimates obtained with the conventional estimation aproach. As discussed in the previous section, this approach is implemented by expressing the hybrid NKPC together with all the VAR equations except the one for inflation as a full-information general equilibrium system AEt Yt+1 = BYt + CXt . The rational expectations equilibrium to this system implies the same cross-equation restrictions than the ones imposed with the reverse-engineering approach. In addition, for there to be a unique equilibrium, we need to impose inequality restrictions such that the number of non-predetermined variables in Yt is equal the number of generalized eigenvalues with modulus larger than one.23 For the sake of completeness, we also estimate the hybrid NKPC under the strong assumption that it holds exactly; i.e. ut = 0. For this case, the rational expectations solution is obtained by rearranging the NKPC as πt = −

ϕ 1 γ ψ t−1 + π t−1 − b π t−2 + εt . γf γf γf

ML estimation reduces to independent Ordinary Least Squares (OLS) estimations of the np different VAR equations, with the coefficients of the inflation equation restricted to one lag of real marginal cost and two lags of inflation. 23

These uniqueness restrictions are imposed by estimating directly the eigenvalues of the full-information system instead of the parameters. In each estimation step, the unique stable solution is then computed numerically using the algorithm by King and Watson (1998).

17

4.3

Data

Real marginal cost is not directly observed in the data. Hence, additional theory is needed to tie it to observables. Following Gali and Gertler (1999) and Sbordone (2002), we assume that firm production is log-linear in its inputs. This implies that real marginal cost can be measured by labor income share (i.e. real unit labor cost). The dynamics of labor income share and inflation are assumed to be well approximated by a bivariate VAR process in the two variables. For the sake of comparison, all estimates are based on exactly the same U.S. data series that Gali and Gertler (1999) use in their study. The sample covers the period 1960:1-1997:4.24 The rate of inflation π t is represented by the first difference of the overall GDP deflator. Labor income share st is measured by nominal non-farm business unit labor cost deflated with the non-farm business GDP deflator. The Aikake Information Criterion selects an optimal lag number of three for the candidate bivariate VAR. For reference, Table 1 presents the unrestricted OLS coefficient estimates.

4.4

Results

Table 2 reports the results for the different estimation methods. All estimates are obtained using the simulated annealing algorithm by Goffe (1996).25 The parameter β is fixed to unity, and the other two structural parameters are constrained to lie within the range of theoretically admissible values; i.e. 0 ≤ θ ≤ 1 and 0 ≤ ω ≤ 1.26 The table also reports standard errors (in parenthesis), which are computed as the square roots of the diagonal elements of minus one times the inverted matrix of second derivatives of the maximized log-likelihood. These standard errors need to be interpreted with caution, however, because their computation is based on a numerical approximation of the second-order derivatives of the log-likelihood and inversion of the resulting matrix.27 24

I thank Jordi Gali for providing me with the data. When appropriately specified, simulated annealing identifies the global maximum with probability one as the number of grid searches goes to infinity. This property represents an important advantage over traditional numerical gradient algorithms because it is well-known that constrained log-likelihood surfaces often have multiple local maxima (as in the simple experiment in Section 3.2). See Goffe (1996) and Judd (1999) for further discussion. A MATLAB version of the simulated annealing code (adapted from Goffe’s FORTRAN code) as well as all other programs used in this paper are available on the author’s website at http://www.er.uqam.ca/nobel/r16374. 26 If β is estimated within the admissible bounds of 0 ≤ β ≤ 1, the ML estimator pushes β towards 1. The other estimates are not sensible to setting β slightly smaller than 1. Also note that for β = 1, we obtain the special case of γ f + γ b = 1. See Woodford (2003, page 217) for a discussion. 27 We also computed the standard errors from the cross-product of the gradients of the log-likelihood (outer-product or Berndt-Hall-Hall-Hansen method). The difference in results is in general small and does not affect any of the conclusions. As an extension, one could apply White’s (1982) test of the significance of this difference to obtain a measure of misspecification of the model. 25

18

4.4.1

Estimates using the reverse-engineering approach

The first row of Table 2 reports the ML estimates using the proposed reverse-engineering approach. Overall, the results are very encouraging for the theory. First, the likelihood ratio of 4.844 implies a p-value of 56% and thus, the hybrid NKPC (up to an uncorrelated error term) cannot be rejected at conventional significance levels. Second, labor income share enters positively and significantly. Third, the estimated fraction of backward-looking ruleof-thumbers of ω ˆ = 0.040 indicates that most firms behave rationally and set new prices so as to maximize the discounted present value of current and expected future profits. Fourth and finally, the frequency of price adjustment is estimated at 1− ˆθ = 0.152, which implies an average price duration of 1/(1 − ˆθ) = 6.67 quarters. This is too high compared to evidence from micro studies where firms are reported to change their price on average every 2.5 to 4 quarters.28 However, Sbordone (2002), Woodford (2003) as well as Eichenbaum and Fischer (2004) show that the same NKPC slope coefficients imply a considerably lower degree of average price duration under the assumption of fixed firm-specific capital stock. 4.4.2

Estimates using the conventional approach

The second row of Table 2 reports the ML estimates using the conventional approach, which constrains the parameters to yield a unique stable equilibrium for inflation. The estimates are dramatically different. In particular, the fraction of backward-looking price setters jumps to ω ˆ = 0.740, which results in a hybrid NKPC with roughly equal coefficients on the forward- and the backward-looking part. Furthermore, labor income share is insignificantly different from zero, and the likelihood ratio test strongly rejects the NKPC as a whole. This stark difference in results is explained by the fact that the general equilibrium system formed by the hybrid NKPC and the VAR(3) for real marginal cost has two nonpredetermined variables but, when evaluated at the ML estimates obtained with the reverseengineering approach, only one generalized eigenvalue with modulus larger than one. Hence, there are multiple stable solutions for inflation at the maximum, which is why the impact of imposing uniqueness is so important. The striking difference in results illustrates that the conventional approach can severely constrain the ML estimation of Euler equations and lead to strong misspecification bias. 4.4.3

Estimates for ut = 0

The third row of Table 2 reports the ML estimates under the assumption that the hybrid NKPC holds exactly; i.e. ut = 0. The estimated fraction of firms that keep their prices fixed in any given period equals ˆθ = 0.92, while the estimated fraction of backward-looking price setters ω is virtually zero. Together with the value of β = 1, these estimates imply a NKPC 28

See Taylor (1998) and Wolman (2000) or Bils and Klenow (2004).

19

that is purely forward-looking. However, the likelihood ratio test versus the unrestricted VAR(3) strongly rejects the model and labor income share enters insignificantly into the NKPC. These results strongly contrast with the ones obtained for ut 6= 0, indicating that the assumption of no error term is too restrictive for the NKPC to fit the data.

4.5

Comparison to existing literature

In recent years, a large number of studies have estimated the hybrid NKPC or variants thereof. The following discussion is by no means intended to provide an exhaustive review but rather puts the results of this paper into perspective relative to some of the more prominent studies on the topic. The above ML estimates for the solution under ut 6= 0 confirm results by Gali and Gertler (1999) who use GMM, and Sbordone (2002) who uses an alternative minimum distance estimator.29 In particular, Gali and Gertler report a price fixity θ between 0.803 and 0.838, and a fraction of backward-looking price-setters ω between 0.244 and 0.522 (depending on the moment specification). These estimates imply a lagged inflation coefficient γ b that ranges from 0.233 to 0.383, which is somewhat larger than in the present case. However, the relatively large standard error of 0.122 for the present estimate of γ b indicates that we can reject neither that the NKPC is purely forward-looking nor that γ b falls in the range of Gali and Gertler’s GMM estimate. Likewise, the ML estimate of the slope coefficient on labor income share of ϕ ˆ = 0.024 lies in the range of estimates reported by Gali and Gertler (between 0.009 and 0.027). The present estimates are also consistent with full-information likelihood-based estimates of the NKPC by Ireland (2001) whose results indicate overwhelming forward-looking behavior by firms. The general equilibrium model underlying his estimation implies that real marginal cost is proportional to labor income share and thus, his empirical specification of the NKPC is comparable to the one here. In sum, the present ML estimates corroborate the conclusions from both method-of-moments and full-information likelihood estimation techniques: conditional on real marginal cost being appropriately measured by labor income share, the NKPC with predominantly forward-looking behavior provides a good approximation of U.S. inflation dynamics.30 29

Sbordone’s (2002) estimation approach consists of minimizing the distance between observed inflation and a theoretical inflation series implied by the model. This theoretical inflation series is conditional on an unrestricted VAR forecasting process and hence, Sbordone does not directly impose the cross-equation restrictions. By contrast, it can be shown that GMM asymptotically imposes the cross-equation restrictions under the assumption that the instruments follow a VAR process. The following discussion therefore concentrates on Gali and Gertler’s GMM results. 30 The present estimates strongly contrast, however, with Jondeau and Le Bihan (2001) who estimate the hybrid NKPC using the conventional approach to VAR-based estimation. They report a much more important degree of backwardness similar to the results reported here when using the conventional approach, thus suggesting that their estimation suffers from the same misspecification bias.

20

This conclusion contradicts earlier studies by Fuhrer and Moore (1995) among others, who proxy real marginal cost with the output gap and overwhelmingly reject various forward-looking versions of the NKPC. As Gali and Gertler (1999) document, however, the output gap has very different cyclical properties than labor income share, thus providing a rationale for their difference in results.31 At the same time, Fuhrer and Moore’s (1995) results are based on a VAR-based estimator that is implemented via the conventional approach; i.e. by imposing uniqueness. It is thus interesting to ask whether the difference in results is due to their use of the output gap instead of labor income share or their more restrictive estimation technique. Table 3 reports the ML estimates for the case where real marginal cost is measured by the output gap.32 As the first row of this table shows, the likelihood ratio test now rejects the hybrid NKPC at a high significance level. In addition, the degree of backwardness is much more important than in the labor income share case, resulting in a hybrid NKPC with about equally important forward- and backward-looking inflation components. These results are very similar to Fuhrer and Moore (1995) and underline that the fit of the hybrid NKPC, and more particularly the importance of the forward-looking component, depends crucially on how real marginal cost is measured. Hence, the main empirical difficulty is not in explaining inflation with a forward-looking pricing model, but in reconciling the behavior of output with the behavior of real marginal cost. Interestingly, as the second row of Table 3 shows, none of the estimates conditional on the output gap are affected by imposing uniqueness on the estimation. This is because the likelihood surface conditional on the output gap is maximized in a region of the parameter space that already implies a unique stable equilibrium. This explains why the estimates conditional on the output gap are very similar to Fuhrer and Moore’s who impose uniqueness in their estimation.33 In sum, whether the uniqueness condition unnecessarily constrains the estimates depends on the likelihood surface implied by the data. If the likelihood is maximized for a 31

Gali and Gertler (1999) and Sbordone (2002) prefer labor income share as a proxy for real marginal cost since using the output gap is problematic for two reasons. Empirically, the output gap is itself difficult to measure because the natural level of output, from which the output gap is constructed, is unobserved and ad-hoc proxies are likely to be ridden with considerable error. Theoretically, the output gap is equivalent to real marginal cost only under the restrictive assumption that capital stocks are fixed; an assumption that is not necessary when measuring real marginal cost with labor income share. 32 The output gap is defined as xt ≡ yt − yt∗ , where yt and yt∗ denote the observed level of output and the natural level of output, respectively. As in Fuhrer and Moore (1995), I measure output with non-farm business GDP and proxy the natural level yt∗ with a fitted linear trend. The citibase codes to construct the different variables are ’lbgdpu’ for the overall GDP deflator, ’lblcpu’ for nominal non-farm business unit labor cost and ’gbpuq’ for real non-farm business GDP. All variables are transformed to logarithms and demeaned prior to estimation. 33 For the record, Table 3 also reports the results for the case ut = 0 (third row). The estimates are at the boundary of admissible values and therefore, standard errors are not reported.

21

combination of parameters that imply by themselves a unique stable solution to the system formed by the Euler equation and the VAR process for the exogenous variable, the uniqueness condition of the conventional approach has no effect. Conversely, if the likelihood is maximized at a point with multiple stable solutions, imposing uniqueness can lead to an important misspecification bias.

5

Conclusion

The paper improves on the existing literature along three dimensions. First, it uncovers that there are multiple rational expectations solutions to the cross-equation restrictions of Sargent’s (1979) original VAR-based likelihood estimator and that these solutions can be classified according to whether they allow for uncorrelated deviations from the theory or not. Second, for the presumably more realistic case that admits uncorrelated deviations, the paper proposes a reverse-engineering approach to implement the cross-equation restrictions that circumvents the identification problem of Sargent’s (1979) original work. Third, the paper contrasts the proposed strategy to an alternative, conventionally employed method that consists of restricting the ML estimates to yield a unique stable rational expectations equilibrium for the full-information system formed by the Euler equation and the VAR equation(s) for the forcing variable. Imposing such a uniqueness condition makes little economic sense since the VAR equation(s) for the forcing variable is unlikely to be its true structural description. The application to Gali and Gertler’s hybrid NKPC illustrates that the restrictions thus imposed can severely constrain the ML estimates. An important avenue to investigate concerns the implementation of cross-equation restrictions that arise for more complicated models. As mentioned in the paper, the proposed estimation strategy applies to cross-equation restrictions derived under the assumption that deviations from the theory ut are uncorrelated with past information. But such an assumption could be overly restrictive. Likewise, we may want to estimate Euler equations that involve multiple leads and lags of the different variables, or finite horizon present value models. The cross-equation restrictions that result from such specifications are more complicated. Hence, there may no longer be a single solution in the coefficients of the forcing variable. Whether and how we can circumvent the identification problem in these more general settings is the topic of future research.

22

A A.1

Appendix Cross-equation restrictions and singularity of M

Consider the general Euler equation (1) yt = aEt yt+1 + bxt + cyt−1 + ut , and suppose that the dynamics of the forcing variable xt are described by a VAR process in n variables and p lags thereof zt = M1 zt−1 + M2 zt−2 + ... + Mp zt−p + ez,t , where zt is the n-variable vector of date t information and contains as a minimum the two variables of the Euler equation, xt and yt . For the sake of concreteness, let xt and yt take the first and second position in zt , respectively. Analogous to the illustrative example in Section 2, this VAR process can be rewritten in companion form as zt = Mzt−1 + et , where zt = [zt zt−1 ...zt−p+1 ]0 is the np × 1 vector of relevant information; et = [ez,t 0...0]0 is the (np × 1) vector of rational expectations errors with E[et |zt−1 ] = 0; and M is the (np × np) companion matrix given by ⎤ ⎡ M1 M2 . . . Mp−1 Mp ⎥ ⎢ ⎢ In 0n . . . 0n 0n ⎥ ⎥ ⎢ ⎢ 0n In . . . 0n 0n ⎥ ⎥ ⎢ ⎥ ⎢ M=⎢ . . . . . . . ⎥. ⎥ ⎢ ⎥ ⎢ . . . . . . . ⎥ ⎢ ⎥ ⎢ . . . . . . ⎦ ⎣ . 0n 0n 0n . . . In The (n × n) blocks Mi , i = 1...p, contain the projection coefficients of zt−i on zt while In and 0n are (n × n) identity and null matrices, respectively. Hence, M contains n2 p non-trivial coefficients that we can stack in a column vector m = [m1 m2 ...mn ]0 = vec([M1 M2 ...Mp ]0 ), where mj , j = 1, 2...n holds the np coefficients of the VAR equation for the j-th variable in zt . Under the assumption that the econometrician’s information set is a subset of the agents’ full information set (i.e. zt ⊆ Zt ) and under the assumption that E[ut |zt−1 ] = 0, the rational expectations cross-equation can be derived analogous to Section 2 as hy [M − aM2 − cInp ] = bhx M, 23

(20)

where hx and hy are (1 × np) selection vectors. Given the supposed placement of xt and yt as the first and second element of zt , these selection vectors are defined as hx = [1 0 0...0] and hy = [0 1 0 0...0]. To prove the singularity of M under the approximate theory solution, note that a scalar λ and a vector t are said to be a (left) eigenvalue / eigenvector pair of M if tM = λt, which can be rewritten as (Inp − λM)t = 0. Thus, as long as a nonzero vector t exists, an eigenvalue λ of M is a number that satisfies the characteristic equation det(Inp − λM) = 0. Equivalently, this condition can be expressed as a np-th order polynomial in λ; i.e. det(Inp − λM) = λnp + c1 λnp−1 + c2 λnp−2 + ...cnp−1 λ +cnp = 0. For M to be singular, there needs to be at least one zero eigenvalue.34 Hence, one needs to show that cnp = 0 under the approximate theory solution. To do so, I first show that the particular structure of the companion matrix implies cnp = det(Mp ). Second, I derive that det(Mp ) = 0 under the approximate theory solution. For the first part, rewrite the argument of the characteristic equation as ⎡ ⎤ M1 − λIn M2 . . Mp−2 Mp−1 Mp ⎢ ⎥ ⎢ In ⎥ −λIn . . . 0n 0n ⎢ ⎥ ⎢ 0n ⎥ I . . . 0 0 n n n ⎢ ⎥ ⎢ ⎥ [M − λInp ] ≡ C = ⎢ . ⎥. . . . . . . ⎢ ⎥ ⎢ . ⎥ 0n . . . −λIn 0n ⎢ ⎥ ⎢ ⎥ . . . In −λIn 0n ⎣ . ⎦ 0n 0n . . 0n In −λIn Now, note that determinant for any (r × r) matrix A can be expressed as r X det(A) = (−1)j+1 ar,j det(Ar,j ), j=1

where ar,j denotes the r-th row / j-th column element of A, and where Ar,j denotes the (r −1 × r −1) matrix formed by deleting row r and column j from A. Applying this formula to the present case, I obtain ¡ ¢ det ([M − λInp ]) ≡ det(C) = − det Cnp,n(p−1) + λ det (Cnp,np ) ,

This follows directly from the Jordan decomposition V MV −1 = J since a zero eigenvalue implies det(J) = 0 and det(J) = det(M) by definition. 34

24

as all the other elements in the last row of [M − λInp ] ≡ C are zero. To derive the part cnp that does not contain any λ, I only need to further consider the first term of det(C), which ¢ ¡ I redefine as − det Cnp,n(p−1) ≡ − det(C 1 ) for notational convenience. Applying the same determinant formula again, I can write − det(C 1 ) as ´ ³ ´ ³ 1 1 − λ det Cn(p−1),n(p−1) . − det(C 1 ) = det Cn(p−1),n(p−2) ³ ´ 1 Again, one only needs to further consider the first term det Cn(p−1),n(p−2) , which is rede2 fined in similar fashion as det(C ). Its determinant is ´ ³ ´ ³ 2 2 + λ det Cn(p−2),n(p−2) . det(C 2 ) = − det Cn(p−2),n(p−3)

This reduction can be done for total of n(p − 2) + 1 times until the only element of det ([M − λI]) = det(C) left to consider is Ã" #! M1 − λIn Mp n(p−2)+1 − det(C ) = − det . In 0n Finally, it can be shown that for any matrix with the (n x n) blocks D, E, F 35 #! Ã" D E = − det(EF 0 ). det F 0n Applying this result to the present case, one obtains − det(C n(p−2)+1 ) = det(Mp ), and thus, it must be the case that cnp = det(Mp ). Turning to the second part of the proof, write out the rows of the cross-equation restrictions in (20) as

[hy,n

⎜⎢ ⎜⎢ ⎢ 0n ...0n ] ⎜ ⎜⎢ ⎝⎣

= b[hx,n

35

⎛⎡

⎡

⎢ ⎢ 0n ...0n ] ⎢ ⎢ ⎣

M1 In ... 0n M1 In ... 0n

M2 0n ... ... M2 0n ... ...

... ... ... In ... ... ... In

Mp 0n ... 0n Mp 0n ... 0n

This result is taken from The (www.ee.ic.ac.uk/hp/staff/dmb/matrix/).

⎡

⎤

⎢ ⎥ ⎢ ⎥ ⎥− a⎢ ⎢ ⎥ ⎣ ⎦

⎤

⎥ ⎥ ⎥, ⎥ ⎦

Matrix

25

M1 In ... 0n

M2 0n ... ...

Reference

... ... ... In

Mp 0n ... 0n

Manual

⎤2

⎡

⎢ ⎥ ⎢ ⎥ ⎥ − c⎢ ⎢ ⎥ ⎣ ⎦

by

In 0n ... 0n

Mike

0n In ... ...

... ... ... 0n

Brookes

0n 0n ... In

⎤⎞ ⎥⎟ ⎥⎟ ⎥⎟ ⎥⎟ ⎦⎠

where hy,n and hx,n are (1 x n) selection vectors. Rewriting hy and hx in this form highlights that only the first n rows of the cross-equation restrictions matter and thus, (20) can be reduced without loss of generality to h i hy,n M1 − a(M12 + M2 ) − cIn M2 − a(M1 M2 + M3 ) ... Mp − aM1 Mp h i = bhx,n M1 M2 ... Mp . The last n equations of this expression are

hy,n [In − aM1 ] Mp = bhx,n Mp , or equivalently (hy,n [In − aM1 ] − chx,n ) Mp = 0. There are two possible solutions to this condition. Either υ = (hy,n [In − aM1 ] − chx,n ) = 0 (i.e. υ is a zero-vector), in which case we cannot say anything about the characteristics of Mp . Or, υ is not a zero-vector, in which case υ represents an eigenvector associated with a zero eigenvalue of Mp ; i.e. det(Mp ) = 0. Hence, the cross-equation restrictions imply a singularity on the VAR companion matrix M as long as υ 6= 0. To only part left in the proof is to examine the conditions for which υ = 0; i.e. the conditions for which the cross-equation restrictions do not imply a singularity. Write out the rows of υ = (hy,n [In − aM1 ] − chx,n ) as ⎤ ⎡ ⎤ ⎡ −amxy,1 − c υ1 ⎥ ⎢ ⎥ ⎢ ⎢ υ2 ⎥ ⎢ 1 − amyy,1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ... ⎥ ⎢ ... ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥, ⎢ ⎥ ⎢ υj ⎥ ⎢ −amyj,1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ... ⎥ ⎢ ... ⎦ ⎣ ⎦ ⎣ υn −amyn,1

which makes clear that υ = 0 if and only if myx,1 = −c/a, mxx,1 = 1/a and myj,1 = 0 for all other j. Furthermore, the condition υ = 0 can be used to derive restrictions for the other coefficients my· of the VAR equation for the endogenous variable. Consider the first n terms of the cross-equation restrictions above ¡ ¢ hy,n M1 − a(M12 + M2 ) − cIn = bhx,n M1 ,

which can be rearranged as

(hy,n [In − aM1 ] − bhx,n ) M1 = hy,n (aM2 + cIn ) ,

26

or equivalently υM1 = hy,n (aM2 + cIn ) . The condition υ = 0 therefore implies hy,n (M2 + bIn ) = 0, or written more explicitly myx,2 = 0, myy,2 = −b/a and myj,2 = 0 for all other j. Analogously, all the other k = 2, 3...p − 1 sets of cross-equation restrictions can be expressed as (hy,n [In − aM1 ] − bhx,n ) Mk = −hy,n Mk+1 , or equivalently υMk = −hy,n Mk+1 . For υ = 0, one obtains myj,k = 0 for all j = 1, 2, ...n. To summarize, the cross-equation restrictions result in a singular companion matrix M for all cases but υ = 0. This latter case implies coefficient restrictions of the form myx,1 = −b/a, myy,1 = 1/a, myx,2 = 0, myy,2 = −c/a and myj,k = 0 for all other j and k. But these are exactly the restrictions of the pure theory solution. Hence, it is exactly the pure theory solution that does not imply a singular M, which means that one can postmultiply each term of (20) by M−1 such as to recover restrictions that hold under the condition ut = 0 and that do not depend on the specification of the VAR process. This proves the first proposition.

A.2

Circumventing the identification problem

Consider the cross-equation restrictions (20) hy [M − aM2 − cInp ] = bhx M,

27

and use the same definition of the companion matrix M as before. Writing out these np restrictions equation by equation, we obtain myx,1 − a(myx,1 mxx,1 + myy,1 myx,1 + myy,1 − a(myx,1 mxy,1 + myy,1 myy,1 +

n X s=3

my3,1 − a(myx,1 mx3,1 + myy,1 my3,1 +

n X

myj,1 mjx,1 + myx,2 ) = bmxx,1

j=3

mys,1 msy,1 + myy,2 ) − c = bmxy,1 n X

mys,1 ms3,1 + my3,2 ) = bmx3,1

s=3

...

myj,k − a(myx,1 mxj,k + myy,1 myj,k + myn,p − a(myx,1 mxn,p + myy,1 myn,p +

n X s=3

n X

mys,k msj,k ) = bmxj,k ... mys,p msn,p ) = bmxn,p

s=3

where j = 3, 4, ...n denotes the j − th variable in the vector of date t information zt ; and k = 2, 3, ...p denotes the k − th lag. According to Sargent’s (1979) approach, the crossequation restrictions are imposed by solving this system for the VAR coefficients of the equation for the endogenous variable y; i.e. my· = f (a, b, c, m−y· ). Since several terms of my· appear in each equation non-linearly, these solutions turn out to be complicated higher-order polynomials. Therefore, multiple my· satisfy the solution to the cross-equation restrictions under ut 6= 0. By contrast, it is straightforward to see that each equation only contains one VAR coefficient of the equation for the exogenous variable; i.e. mxx,1 is the only one of these coefficients that enters the first equation; mxy,1 is the only coefficient that enters the second equation; and so forth. Hence, there is a single mapping of the form mx· = f (a, b, c, m−x· ). This proves the second proposition.

28

References [1] Benhabib, J., Farmer R.E.A., 1998. Indeterminacy and Sunspots in Macroeconomics, in: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics, Vol. 1A. Elsevier Science B.V., North-Holland. [2] Bils, M., Klenow P.J., 2004. Some Evidence on the Importance of Sticky Prices. Journal of Political Economy (forthcoming). [3] Blanchard, O. J., Kahn C. M., 1980. The Solution of Linear Difference Models under Rational Expectations. Econometrica 48, 1305-1311. [4] Blanchard, O.J., Kiyotaki N., 1987. Monopolistic Competition and the Effect of Aggregate Demand,” American Economic Review 77, 647-666. [5] Calvo, G. A., 1983. Staggered Prices in an Utility-Maximizing Framework. Journal of Monetary Economics 12, 383-398. [6] Campbell, J. Y., 1987. Does Saving Anticipate Declining Labor Income? An Alternative Test of the Permanent Income Hypothesis. Econometrica 55, 1249-1273. [7] Campbell, J. Y., Shiller R.J., 1987. Cointegration and Tests of Present-Value Models. Journal of Political Economy 95, 1062-1088. [8] Chari, V. V., Kehoe P. J., McGrattan, E. R., 2005. A Critique of Structural VARs Using Real Business Cycle Theory. Federal Reserve Bank of Minneapolis Research Paper 631. [9] Christiano, L. J., Eichenbaum M., Vigfusson R., 2005. Are Structural VARs Useful? Northwestern University. [10] Cogley, T., 2005. Changing Beliefs and the Term Structure of Interest Rates: CrossEquation Restrictions with Drifting Parameters. Review of Economic Dynamics 8. [11] Eichenbaum, M., Fisher J. D. M. , 2004. Evaluating the Calvo Model of Sticky Prices. Northwestern University and Federal Reserve Bank of Chicago. [12] Erceg, C. J., Guerrieri L., Gust C., 2004. Can Long-Run Restrictions Identify Technology Shocks? Federal Reserve Boad of Governors International Finance Discussion Paper No. 792. [13] Fern´ andez-Villaverde, J., Rubio-Ramirez, J. F., Sargent T. J., 2005. A, B, C’s (and D’s) of Understanding VARS. Federal Reserve Bank of Altanta Working Paper No. 2005-9.

29

[14] Fuhrer, J.C., Moore G., 1995. Inflation Persistence. Quarterly Journal of Economics 110, 127-153. [15] Fuhrer, J.C., Moore G.R., Schuh, S.D., 1995. Estimating the Linear Quadratic Inventory Model. Maximum Likelihood versus Generalized Method of Moments. Journal of Monetary Economics 110, 115-157. [16] Fuhrer, J.C., Rudebusch, G. D., 2004. Estimating the Euler Equation for Output,” Journal of Monetary Economics 51, 1133-1153. [17] Gali, J., Gertler M., 1999. Inflation Dynamics: A Structural Econometric Analysis. Journal of Monetary Economics 44, 195-222. [18] Goffe, W.L., 1996. SIMANN: A Global Optimization Algorithm using Simulated Annealing. Studies in Nonlinear Dynamics & Econometrics 1, 169-176. [19] Hamilton, J. D., 1994. Time Series Analysis. Princeton University Press, Princeton. [20] Hansen, L. P., 1982. Large Sample Properties of Generalized Method of Moments Estimators. Econometrica 50, 1029-1054. [21] Hansen, L. P., Sargent T. J., 1980. Formulating and Estimating Dynamic Linear Rational Expectations Models. Journal of Economic Dynamics and Control 2. [22] Ireland, P. N., 2001. Sticky Price Models of the Business Cycle. Journal of Monetary Economics 47, 3-18. [23] Ireland, P. N., 2004. A Method for Taking Models to the Data. Journal of Economic Dynamics and Control 28, 1205-1226. [24] Jondeau, E., Le Bihan H., 2001. Testing for a Forward-Looking Phillips Curve. Additional Evidence from European and US Data. Universit´e Paris XII. [25] Judd, K.L., 1998. Numerical Methods in Economics. MIT Press, Cambridge. [26] King, R.G., Kurmann, A, 2002. Expectations and the Term Structure of Interest Rates: Evidence and Implications. Federal Reserve Bank of Richmond Economic Quarterly 88, 49-95. [27] King, R.G., Watson, M.W., 1998. The Solution of Singular Linear Difference Systems Under Rational Expectations. International Economic Review 39, 1015-1026. [28] Lind´e, J., 2005. Estimating New-Keynesian Phillips Curves: A Full-Information Maximum Likelihood Approach. Journal of Monetary Economics 52, 1135-1149.

30

[29] Lubik, T., Schorfheide, F., 2004. Testing for Indeterminacy: An Application to U.S. Monetary Policy,” American Economic Review 94, 190-217. [30] Meese, R. A., 1986. Testing for Bubbles in Exchange Markets: A Case of Sparkling Rates? Journal of Political Economy 94, 345-373. [31] Roll, R., 1969. The Behavior of Interest Rates: An Application of the Efficient Market Model to U.S. Treasury Bills. Basic Books, New York. [32] Rotemberg, J.J., Woodford, M., 1997. An Optimization-Based Econometric Framework for the Evaluation of Monetary Policy. NBER Macroeconomic Annual 12, 297-346. [33] Ruge-Murcia, F., 2003. Methods to Estimate Dynamic Stochastic General Equilibrium Models. Universit´e de Montr´eal. [34] Sargent, T. J., 1979. A Note on Maximum Likelihood Estimation of the Rational Expectations Model of the Term Structure. Journal of Monetary Economics 5, 133143. [35] Sargent, T. J., 1989. Two Models of Measurement and the Investment Accelerator. Journal of Political Economy 97, 251-287. [36] Sbordone, A.M., 2002. Prices and Unit Labor Costs: Testing Models of Pricing. Journal of Monetary Economics 49, 265-292. [37] Sbordone, A.M., 2004. Do Expected Future Marginal Costs Drive Inflation Dynamics? Journal of Monetary Economics (forthcoming). [38] Steinsson, J., 2004. Erratum to: Optimal Monetary Policy in an Economy with Inflation Persistence. Harvard University, manuscript. [39] Stock, J.H., Wright, J., Yogo, M. 2002. A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments. Journal of Business and Economic Statistics 20, 518-529. [40] Taylor, J.B., 1998. Staggered Wage and Prices in Macroeconomics, in: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics, Vol. 1A. Elsevier Science B.V., North-Holland. [41] White, H., 1982. Maximum Likelihood Estimation of Misspecified Models. Econometrica 50, 1-25. [42] Woodford, M., 2003. Interest and Prices. Princeton University Press, Princeton. [43] Wolman, A.L., 2000. The Frequency and Costs of Individual Price Adjustment. Federal Reserve of Richmond Economic Quarterly 86, 1-22. 31

-7900

1.15 1.1

-7950 -7850 -7800 -7750

-7900

-7700 -7650 -7600

-7700

a

1.05 1

-8000 -7950

A -75 50

-7550

-7850 -7800 -7750

-7650

-7600

True maximum

0.95 -7 7 0.9 -78 50 -7 00 85 0 0.85 - 7 - 79 95 0 0 0 0.8 0.3 0.4

-76 -7 65 00 0 -7 70 0

-7500

-75 50

0.5

0.6

0.7

0.8

Maximum with uniqueness imposed B 0.9

1

b

Figure 1: Loglikelihood plot and bounds imposed by uniqueness condition

32

Table 1 Unrestricted VAR estimates (a) Labor income share and inflation s t-1 π t-1 st

πt

s t-2

π t-2

s t-3

π t-3

R2 0.780

0.884

0.036

-0.004

-0.038

-0.078

0.271

(0.083)

(0.244)

(0.111)

(0.289)

(0.084)

(0.240)

0.074

0.642

-0.060

0.047

-0.014

0.240

(0.028)

(0.083)

(0.038)

(0.099)

(0.029)

(0.082)

0.824

Notes: This table reports the coefficient estimates for the OLS regressions of labor income share and inflation on lags thereof. The sample period is 1961:1-1997:4. Standard errors are shown in brackets.

Table 2 Maximum Likelihood estimates for labor income share Structural parameters β

θ

Reverse-engineering approach

1.000

Conventional approach (imposing uniqueness)

1.000

Exact model (ut = 0)

1.000

NKPC slope coefficients ω

γf

γb

ϕ

0.848

0.050

0.944

0.056

0.025

(0.031)

(0.116)

(0.122)

(0.122)

(0.011)

0.874

0.740

0.542

0.458

0.003

(0.010)

(0.053)

(0.020)

(0.020)

(0.001)

0.923

0.000

1.000

0.000

0.006

(0.084)

(0.082)

(0.481)

(0.087)

(0.017)

Likelihood ratio

p-value

4.844

0.564

22.417

0.001

29.097

0.000

Notes: This table reports the ML estimates of the NKPC for the different solutions to the cross-equation restrictions. Estimates are reported for both the case where real marginal cost is measured by labor income share. Standard errors are reported in parenthesis.

Table 3 Maximum Likelihood estimates for output gap NKPC slope coefficients

Structural parameters β Reverse-engineering approach

1.000

Conventional approach (imposing uniqueness)

1.000

Exact model (ut = 0)

1.000

θ

ω

γf

γb

ϕ

0.530

0.439

0.547

0.453

0.128

(0.057)

(0.054)

(0.018)

(0.018)

(0.055)

0.530

0.439

0.547

0.453

0.128

(0.057)

(0.054)

(0.018)

(0.018)

(0.055)

1.000

0.000

1.000

0.000

0.000

Likelihood ratio

p-value

17.718

0.007

17.718

0.007

42.487

0.000

Notes: This table reports the ML estimates of the NKPC for the different solutions to the cross-equation restrictions. Estimates are reported for both the case where real marginal cost is measured by the linearly detrended output gap. Standard errors are reported in parenthesis.