Simulation-based tests of forward-looking models under VAR learning dynamics

JOURNAL OF APPLIED ECONOMETRICSJ. Appl. Econ. 26: 762–782 (2011)Published online 29 January 2010 in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/jae.1138

SIMULATION-BASED TESTS OF FORWARD-LOOKINGMODELS UNDER VAR LEARNING DYNAMICS

LUCA FANELLIa* AND GIULIO PALOMBAb

a Department of Statistical Sciences, University of Bologna, Bologna, Italyb Department of Economics, Universita Politecnica, delle Marche, Ancona, Italy

SUMMARYIn this paper we propose a simulation-based technique to investigate the finite sample performance oflikelihood ratio (LR) tests for the nonlinear restrictions that arise when a class of forward-looking (FL)models typically used in monetary policy analysis is evaluated with vector autoregressive (VAR) models. Weconsider ‘one-shot’ tests to evaluate the FL model under the rational expectations hypothesis and sequencesof tests obtained under the adaptive learning hypothesis. The analysis is based on a comparison betweenthe unrestricted and restricted VAR likelihoods, and the p-values associated with the LR test statistics arecomputed by Monte Carlo simulation. We also address the case where the variables of the FL model can beapproximated as non-stationary cointegrated processes. Application to the ‘hybrid’ New Keynesian PhillipsCurve (NKPC) in the euro area shows that (i) the forward-looking component of inflation dynamics is muchlarger than the backward-looking component and (ii) the sequence of restrictions implied by the cointegratedNKPC under learning dynamics is not rejected over the monitoring period 1984–2005. Copyright 2010John Wiley & Sons, Ltd.

Received 8 August 2007; Revised 7 August 2009

Supporting information may be found in the online version of this article.

1. INTRODUCTION

The class of forward-looking (FL) models typically employed in macroeconomics and monetarypolicy analysis imposes parametric restrictions on vector autoregressive (VAR) systems (see,for example, Sargent, 1979; Campbell and Shiller, 1987; Johansen and Swensen, 1999). Theserestrictions are generally nonlinear, as shown in Pesaran (1987), Binder and Pesaran (1995),Bekaert and Hodrick (2001) and Fanelli (2008a).

Tests in VAR models are usually based on linear restrictions and large-sample approximations.However, it is well recognized that the use of asymptotic distributions can lead to misleadinginference in samples of usual sizes. Furthermore, the presence of non-stationary processes canworsen reliability problems as documented in Toda and Yamamoto (1995) and Yamada andToda (1998). The literature has mainly focused on the finite sample performance of tests forlinear restrictions (Dufour and Jouini, 2006), ignoring nonlinear restrictions and in particular theconstraints that arise with FL models. The hypothesis of rational expectations imposes nonlinearrestrictions on the statistical model for the variables, which are usually rejected, suggesting thatFL models are too simple to capture the complex probabilistic nature of the data. Focusing onthe expectations hypothesis of the term structure of interest rates and exchange rates, Bekaert and

Ł Correspondence to: Luca Fanelli, Dipartimento di Scienze Statistiche, Universita di Bologna, Via Belle Arti 41 40126Bologna, Italy. E-mail: [email protected]

Copyright 2010 John Wiley & Sons, Ltd.

SIMULATION-BASED TESTS 763

Hodrick (2001) show that the commonly employed VAR-based tests, combined with the use ofasymptotic distributions, lead to severe size and power distortion in finite samples.1 Moreover,Li (2007) has recently pointed out that the persistence in the data may further worsen the finitesample performance of commonly used tests for the rational expectations restrictions.

The contribution of this paper is twofold. First, we use simulation-based techniques to investigatethe finite sample performance of the likelihood ratio (LR) tests for the nonlinear restrictions thatarise when a class of FL models is evaluated through VAR models under the rational expectationshypothesis. We call this type of tests ‘one-shot’, as they are computed in one solution using thefull available sample. The idea is to use the consistent point estimates of the model parameters toimplement the local Monte Carlo (LMC) LR-type tests described in Dufour (2006) and Dufour andJouini (2006). In practice, LMC amounts to a ‘parametric bootstrap’ procedure through which, fora given sample, the p-value associated with the LR test statistic can be calculated by simulation.Although LMC tests are reliable under restrictive regularity conditions (Dufour, 2006), our MonteCarlo results show that they provide a good control of the size of the test when the restrictionsimplied by FL models are imposed; this holds when the maximization of the likelihood functionunder the null hypothesis is based on the combination of Newton’s method with a grid search forthe structural parameters and the ‘true’ values are contained in the chosen grid. In these situations,the rejection of a large class of FL models can be ascribed to the poor finite sample performancesof the tests based on asymptotic critical values.

Secondly, we develop a method to analyze sequences of LR tests that arise when FL modelsare tested under the adaptive learning hypothesis.2 In this context VARs are used as the agents’forecast model (perceived law of motion); as the information set increases over time and coefficientestimates are recursively updated, a new set of cross-equation restrictions arises and become partof the empirical assessment of the FL model.

Recently, Mavroeidis et al. (2008) have emphasized that the distribution of estimators and teststatistics is generally non-standard under adaptive learning. Moreover, in the sequential testingsetup, conventional critical values are not suited since by the law of iterated logarithms theprobability of rejecting the null is asymptotically one and a new theory is needed (see Inoueand Rossi, 2005; Anatolyev, 2009). In the learning framework, the empirical assessment of the FLmodel involves relatively small sample sizes, especially at the beginning of the agents’ learningprocess, suggesting that the use of simulation (LMC) techniques might deliver reliable inference.

Mavroeidis et al. (2008) propose a test of FL models under learning dynamics based on a‘limited information’ approach. They provide a measure related to the goodness of fit of the actuallaw of motion, namely the model obtained by substituting expectations in the FL model with theforecasts implied by the agents’ perceived law of motion. Unlike Mavroeidis et al. (2008), ourmethod relies on a likelihood-based sequential approach and the test of the model is performedby simulation techniques.3

1 The paper by Mankiw and Shapiro (1986) is a seminal contribution in this field.2 The traditional approach to modelling ‘boundedly rational’ expectations assumes agents form expectations by usingadaptive updating rules (see, for example, Branch and Evans, 2006). We refer, for example, to Pesaran (1987, ch. 3),Sargent (1999) and Evans and Honkapohja (1999, 2001) for a comprehensive discussion about ‘bounded rationality’ andlearning behaviour.3 Li (2008) has recently proposed a method for estimating and testing Euler equations when the parameters of the impliedreduced form are time-varying and drift in a small but persistent manner, similarly to what happens with the constant-gainrecursive least squares algorithm (see Section 2) in the adaptive learning literature.

Copyright 2010 John Wiley & Sons, Ltd. J. Appl. Econ. 26: 762–782 (2011)DOI: 10.1002/jae

764 L. FANELLI AND G. PALOMBA

For both contributions, we consider two possible approaches to the empirical evaluation ofthe FL model. In the former, we address the situation in which the objective of the analysis istesting the VAR coefficient restrictions, ignoring the possible non-stationarity of the variables. Inthe latter, we extend the analysis to the case in which unit roots and cointegration are explicitlyaccounted for.

The paper provides an empirical illustration based on the ‘hybrid’ version of the New KeynesianPhillips Curve (NKPC) for the euro area (Galı et al., 2001). Our results show that the cross-equation restrictions implied by the NKPC with rational expectations are rejected when the LRtest statistics are compared with standard asymptotic critical values. Conversely, the cross-equationrestrictions are not rejected when the non-stationarity of the variables is properly taken into accountand an LMC LR test for the NKPC is computed. This finding supports the idea that, in finitesamples, tests based on asymptotic critical values may falsely lead one to reject FL models.Moreover, when the variables are highly persistent, it is convenient to impose the underlying (ifany) cointegration restrictions.

As regards the evaluation of the NKPC under learning dynamics, we show that the magnitudeof the recursively estimated forward-looking parameter is significantly larger than the magnitudeof the recursively estimated backward-looking parameter; the model is not rejected when LMCLR-type tests are computed by using recursive least squares (RLS) learning over the monitoringperiod (1984–2005), provided the cointegration restrictions are accounted for.

The paper is organized as follows. Section 2 introduces the FL model and derives the impliedcross-equation restrictions on VAR coefficients. Section 3 describes the LMC procedure and thecorresponding LR tests for the FL model. Section 4 documents some Monte Carlo experimentsin which the finite sample performance of the LR tests for the FL model is investigated. Theempirical application, based on the NKPC for the euro area, is the subject of Section 5. TheAppendix extends the analysis to the case of cointegrated variables and describes the data. Proofsand technical details are reported in a technical supplement.4

2. FORWARD-LOOKING MODEL AND VAR RESTRICTIONS

We focus on the following class of FL models:

Qyt D �Et QytC1 C υ Qyt�1 C � Qwt C �t �1�

Qwt D g� Qyt�1, Qwt�1, . . .�C et �2�

in which Qyt is a scalar endogenous variable, Et QytC1 D E� QytC1jFt� is the expected value of QytC1

conditional on the sigma-field Ft, Ft � FtC1, Qwt is an explanatory (driving) variable representedin (2) in reduced form, g�Ð� is a linear function whose arguments are a finite number of lags of Qytand Qwt respectively, �t D ��t, et�0 a 2 ð 1 martingale difference sequence (MDS) conditional onFt (Et�tC1 D 02ð1) and � D ��, υ, ��0, with 0 < � < 1, 0 < υ < 1, � > 0, is the vector of structuralparameters.

The specification (1)–(2) covers many of the FL models typically used in monetary policyanalysis (see Section 5) and can be generalized to the case in which Qyt and Qwt are vectors.In this context, the tilde symbol indicates that the variables do not embody any deterministic

4 The technical supplement is available online.



components. We have deliberately left the reduced form equation for Qwt not fully specified in (2);the structure of the process generating Qwt is fundamental for computing the rational expectationssolution of the model and for understanding whether there exists a unique non-explosive solutionor not, with consequences on the identifiability of the structural parameters (see Pesaran, 1987;Mavroeidis, 2005). The approach based on the econometric investigation of the unique reducedform solution associated with (1)–(2) maintains that the data-generating process belongs to therational expectations solution of the FL model.

Our method, based on ‘VAR expectations’,5 takes a slightly different perspective and is basedon the idea that, if the FL model holds and has a unique solution, its reduced form must be nestedwithin the agents’ statistical model. The implied cross-equation restrictions can be retrieved bythe method of undetermined coefficients and can be used to discuss the identifiability of the FLmodel, to estimate its structural parameters and to test its empirical validity.

Let Xt D �yt, wt, u0t�

0 be the pð 1 vector of observable variables, where ut is a qð 1 �p D2 C q� sub-vector of ‘additional’ variables that might enter the information set. The vectorXt collects the variables whose dynamics is potentially affected by deterministic components(constant, trend, dummies, etc.); QXt D � Qyt, Qwt, Qu0

t�0 is the corresponding vector of variables net of

determinist components, hereafter called the ‘detrended’ process.Given the sigma field Ht D �X1, . . . , Xt� � Ft, it is assumed that Xt is generated by the

finite-order VAR processes of the form

Xt D QXt C dt, A�L� QXt D εt, A�L�dt D �CDt �3�

whereA�L� D Ip � A1L � Ð Ð Ð � AkL

k

L is the lag operator, Ai, i D 1, 2, . . . , k are pð p matrices of parameters, k ½ 2, � is a pð 1constant, Dt is a fð 1 vector containing other deterministic terms with associated pð f matrixof parameters and εt is a Gaussian distributed MDS conditional on Ht, with pð p covariancematrix ε. The quantities X0, X�1, . . . , X�kC1 are fixed. It is further assumed that the roots ofdet[A�s�] D 0 are jsj ½ 1, hence explosive solutions are ruled out. We also impose the necessarycondition for the identifiability of the structural parameters, pk ½ 4; see below.

Henceforth, the VAR in equation (3) will read as the agents’ expectations generating system, orPerceived Law of Motion (PLM). Since QXt D Xt � dt D � Qyt, Qwt, Qu0

t�0 is expressed as A�L� QXt D εt,

the first-order companion form representation of the detrended process is given by

XŁt D AXŁ

t�1 C εŁt

where the state vector XŁt D � QX0

t, QX0t�1, Ð Ð Ð , QXt�kC1�0 and εŁ

t D �ε0t, 01ðp�k�1��0 are pk ð 1, and A is

the nð n companion matrix

A D

A1 Ð Ð Ð AkIp Ð Ð Ð 0pðp

. . ....

0pðp Ð Ð Ð Ip 0pðp

�4�

5 The use of ‘VAR expectations’, which does not necessarily coincide with rational expectations (Brayton et al., 1997), iswidespread in the literature, and goes back to Sargent (1979) and Campbell and Shiller (1987).



At time t � 1, agents’ j-step-ahead forecast of yt and wt, net of deterministic components, aregiven by

E� QytCj j Ht�1� D s0yE�XŁtCj j Ht�1� D s0yA

jC1XŁt�1 �5�

E� QwtCj j Ht�1� D s0wE�XŁtCj j Ht�1� D s0wA

jC1XŁt�1 �6�

where sy and sw are respectively pk ð 1 selection vectors such that s0yXŁt D yt and s0wX

Łt D wt.

Using the law of iterated expectations (Ht�1 � Ht � Ft) and the MDS property of �t, the FLequation (1) can be rewritten as

E� Qyt j Ht�1� D �E� QytC1 jHt�1�C υ Qyt�1 C � E� Qwt j Ht�1� �7�

so that using the forecasts (5)–(6) in (7) yields

s0yAXŁt�1 D �s0yA

2XŁt�1 C υs0yX

Łt�1 C �s0wAX

Łt�1

As XŁt 6D 0 a.s. for each t, rearranging terms above gives rise to the following set of cross-

equation restrictions:s0yA�Ipk � �A�� υs0y � �s0wA D 01ðpk �8�

which involve both structural parameters and the VAR coefficients.The technical supplement associated with this paper reports two propositions. The former

discusses the conditions which ensure that the structural parameters are locally identifiable underthe restrictions (8). The latter provides the explicit form representation of the cross-equationrestrictions. We show that the number of over-identifying constraints is c D �2 C q�pk � [�1 Cq�pk C 3] D pk � 3, where 3 is the dimension of the vector of structural parameters �. It turns outthat pk ½ 4 is a necessary condition for c ½ 1, i.e. for having binding cross-equation restrictions.‘One-shot’ LR tests of the FL model can be computed by estimating the VAR unrestrictedly andsubject to the nonlinear over-identifying restrictions respectively, using the whole available sample.We argue that simulation-based inference, as discussed in Section 3, can be used to overcome thepoor finite sample performance of the LR tests based on asymptotic critical values.

When the system (3) is treated as the agents’ perceived law of motion in the adaptive learningframework, there are likely to be feedbacks from expectations to outcomes and the VAR coefficientsshould be allowed to change over time. In this case, expectations at time t are obtained by replacingthe expressions in (5)–(6) by

E� QytCj j Ht�1� D s0yAjC1t XŁ

t�1 �9�

E� QwtCj j Ht�1� D s0wAjC1t XŁ

t�1 �10�

where At denotes the companion matrix in (4) with the coefficients replaced with their time-varyingcounterparts.6 Using (9)–(10) in (7) yields a sequence of restrictions of the form

s0yAt�Ipk � �At�� υs0y � �s0wAt D 01ðpk, t D T0, T0 C 1, . . . �11�

6 The expressions in (9)–(10) can be thought as the result of an anticipated-utility approximation; see Cogley and Sbordone(2008) for details.



where T0 is the ‘first monitoring’ time. Using the terminology in Anatolyev (2009), in this setupone can consider ‘retrospective’ tests and ‘monitoring’ tests: in the former, a researcher tracksthe validity of the restrictions over time in a historical sample, whereas in the latter a researcherconducts testing ‘in real time’ as new observations arrive.

In the following sections, we use likelihood-based methods and simulation techniques to test‘retrospectively’ the collection of restrictions that arise from (11). We focus on the case where theagents follow the two most popular learning algorithms, namely RLS and constant-gain recursiveleast squares (CG-RLS); see Sargent (1999), Evans and Honkapohja (2001) and Branch andEvans (2006) for details. The technical supplement shows that, since with the RLS algorithmthe underlying VAR coefficients are time invariant, i.e. At D A for each t and (11) collapses to(8), it is the estimator of A that is recursively updated as Ht increases, generating a collection oftest statistics. With the CG-RLS algorithm, the underlying VAR coefficients are time-varying andtheir evolution is governed by the so-called constant-gain parameters (Sargent, 1999; Branch andEvans, 2006; Milani, 2007).

Irrespective of whether the VAR coefficients are treated as time-varying or not, we address theproblem of recursively evaluating the sequence of restrictions (11) by simulation techniques.

3. SIMULATION-BASED TESTS

In this section we consider the problem of testing the cross-equation restrictions implied by theFL model (1) by using LR tests and simulation-based techniques. For simplicity, we refer to thesystem (3) and the restrictions in (8). This framework covers the case in which simulation-basedinference is used to test FL models under rational expectations and RLS learning. However, thesuggested technique can be generalized, with qualifications, to more complex learning algorithms.The test will be presented assuming a stationary VAR but the analysis can be extended to thecointegrated version of the model, as shown in the appendix.

In principle, for a given sample of length t, one can estimate the unrestricted VAR and the VARsubject to the cross-equation restrictions and compute LR statistics of the form

LRt D 2�log L��t�� log L��t�� 12�

where L�Ð� is the likelihood function, �t is the vector containing the ML estimates of the unrestrictedVAR parameters conditional on information Ht, and �t is the vector containing the constrainedML (CML) estimates conditional on information Ht. The unrestricted estimates are given by�t D �a0

t, �0t, �

0t�

0, where at D �a0yt, a

0wt, a

0ut�

0, �t D ��0t, vec�t�0�0 and �t D vech�εt�, whereas the

constrained estimates are given by �t D �v0t, �

0t, �

0t�, vt D �a0

yt, a0ut, �

0t�

0, �t D ��0t, vec�t�0�0, and

�t D vech�εt�. The CML estimate �t can be obtained by combining Newton-like methods witha grid search for �; see the technical supplement for details. With ‘one-shot’ tests, the time indexin (12) is set to t D Tmax, where Tmax is the full sample size. In the learning framework, t rangesfrom the first monitoring time, T0, until Tmax, hence we focus on a ‘retrospective’ sequence oftests.

In this context, caution is needed in carrying out the LR tests using the asymptotic quantile�2

1��c�, where c D pk � dim�� D pk � 3 is the number of over-identifying restrictions (see thetechnical supplement) and � is the nominal level of the test. Indeed, the comparison of each LRt



with �21��c� may lead to severe size distortions for t D T0, T0 C 1, . . . , Tmax because

limt!1P[LRt ½ �2

1��c�] D 1

according to the law of iterated logarithms. Size distortions may arise also for ‘one-shot’ testscomputed under rational expectations when Tmax is not large enough. Thus for both ‘one-shot’and recursive tests the adaptation of finite-sample simulation-based inference along the lines ofDufour (2006) and Dufour and Jouini (2006) seems a reasonable solution.

The algorithm for computing the LMC p-values associated with each LRt statistics in (12) isdescribed in the following steps:7

1. For fixed k and for given t, the test statistic based on the observed data is computed, LRt D LRst .This requires both the unconstrained ML (O�t) and CML (�t) estimation of the VAR.

2. Given the CML estimate �t D �v0t, �

0t, �

0t� obtained in step 1, a sequence of M i.i.d. vectors

ε11, ε

22, . . . , ε

Mt , drawn from the Gaussian distribution N�0, εt�, is generated and M samples of

length t are constructed according to

A�L� Xmj D �t CtDj C εmj , m D 1, 2, . . . ,M, j D 1, 2, . . . , t �13�

where the elements of the matrices in A�L� depend on �t �vt�, and Xm0 D X0, Xm�1 D X�1, . . .,Xm�kC1 D X�kC1 are fixed at the initial sample values.3. For each sample of length t generated in step 2, the unrestricted and restricted VAR models are

estimated and M independent LR statistics LRmt , m D 1, 2, . . . ,M are calculated.4. The LMC p-value associated with LRst is computed as

p�LRst , �t� D M Ð GM�LRst �C 1

MC 1

where

GM�LRst � D 1

M

M∑mD1

fLRmt > LRstg

fÐg is the indicator function and the notation p�LRst , �t� points out that the simulated p-valuedepends on the consistent point estimates of the restricted VAR. If p�LRst , �t� > �, the LMC testis not significant at level �.8

5. Under learning, steps 1–4 are repeated sequentially for t D T0, T0 C 1, . . . , Tmax.

7 The LMC procedure can be viewed as a degenerate version of the maximized Monte Carlo (MMC) tests proposed inDufour (2006) and Dufour and Jouini (2006); MMC tests are based on a ‘consistent set estimator’ of the parameters andrequire the maximization of a p-value function over the chosen consistent set estimator. MMC tests have better asymptoticproperties, compared to LMC tests, but they are computer intensive, especially in the presence of nonlinear restrictions.For a detailed comparison between MMC and LMC tests see Dufour and Jouini (2006).8 Strictly speaking, p�LRst , �t� � � is not sufficient to reject the null hypothesis at level �. Indeed, as explained in Dufourand Jouini (2006), the rejection of the null should be based on the maximized p-value: pMMC D supfpM�LRst , ��, � 2 �0g,where �0 is the parameter space restricted by the null hypothesis, or a consistent set restricted estimator.



The LMC power associated with each LRt statistic can be obtained similarly by generatingthe samples in step 2 (equation (13)) from the unconstrained ML estimates �t D �a0

t, �0t, �

0t�

0. Non-parametric versions of the LMC tests can be obtained by replacing the pð 1 vectors εmj instep 2 (equation (13)) with Oεmj , where Oεmj is resampled from the matrix [Oεs1, Oεs2, . . . , Oεst ] and thevectors Oεs1, Oεs2, . . . , Oεst are the VAR residuals obtained in step 1 from the estimation of the restricted(unrestricted) model on the observed data.

4. SIMULATION EXPERIMENT

In this section we present some Monte Carlo evidence on the performance of ‘one-shot’ LMCLR-type tests for the nonlinear restrictions in (8). We consider a simple DGP, represented by abivariate VAR process (p D 2) with k D 2 lags. Results are reported in Table I.9

To evaluate the size of the test, we generate the data from the restricted VAR using equation (8)and the propositions reported in the technical supplement. The vector of structural parameters is�Ł D �0.70, 0.20, 0.15�0 and VAR disturbances have Gaussian distribution with covariance matrixε having 1 as diagonal elements and 0.5 as off-diagonal terms.

In order to mimic the observed time-series persistence, the largest eigenvalue of the (constrained)companion matrix A is set to 0.9888. The nominal level of the test is � D 0.05.

After generating N D 1000 samples of size T D 50, T D 100, T D 150 and T D 200 respec-tively, we computed the LR test on each sample (see below for details). The first column in Table I(ASYLR) reports the number of times in which the LR test rejects the null hypothesis according tothe asymptotic critical value �2

1��1�.10 The LMC p-value associated with each LR test is obtained

following steps 1–4 of the algorithm in Section 3 using M D 100 replications; the second columnof Table I (LMCLR) summarizes the corresponding percentages of rejection.

Table I. Rejection frequencies (percentages) of asymptotic and LMC LR-type tests for the cross-equationrestrictions (8) implied by the FL model (1)

DGP: bivariate VAR(2)(p D 2, k D 2)Largest eigenvalue of (restricted) companion matrix A: 0.9888

Size Powerjjdjj2 D 0.3

Powerjjdjj2 D 0.8

ASYLR LMCLR ASYLR LMCLR ASYLR LMCLRT D 50 27.78 4.3 63.7 26.0 98.4 87.8T D 100 21.28 4.3 86.1 48.5 100 99.7T D 150 19.98 4.5 97.7 84.0 100 100T D 200 19.28 5.3 99.9 98.2 100 100

Notes: The structural parameters are fixed at � D ��, υ, �� D �0.70, 0.20, 0.15�; the nominal level of the tests is � D 0.05;ASYLR stays for the asymptotic (chi-squared distributed) test based on the LR statistic, while LMCLR is the correspondingLMC test; power is obtained under the alternative of variables generated by nonstationary cointegrated processes and dmeasures the distance from the null (see Section 4 for details); rejection frequencies are based on N D 1000 samples andthe simulated p-values and powers are computed using M D 100 replications.

9 The simulation experiment has been performed in Ox 4.0. All codes, including those relative to the estimates in Section5, are available upon request.10 In this case, the number of over-identifying restrictions is c D pk � dim�� D 1.



The CML estimation is carried out by combining the BFGS method with a grid search for �.The grid has been constructed around the point �Ł D �0.70, 0.20, 0.15�0, using š0.02 as incrementfor all parameters. The grid includes 180 points.

To evaluate the power of the test, any unrestricted VAR model whose coefficients do not fulfilthe restrictions (8) can potentially be used to generate the data. We have arbitrarily consideredtwo experiments, based on two different specifications of the alternative hypothesis. Define thepk ð 1 vector

d D �Ipk � �A0�A0sy � υsy � �A0sw

which measures the distance between the unrestricted and restricted VAR coefficients; it turnsout that d D 0pkð1 under the null and d 6D 0pkð1 under the alternative for suitable choices of thecoefficients in A. We selected the coefficients in A such that jjdjj2 D 0.3 in the former experiment,and jjdjj2 D 0.8 in the latter, jj Ð jj2 being the Euclidean norm. In both cases, the largest eigenvalueof the constrained companion matrix has been set to one, assuming a cointegrated VAR as DGP.Intuitively, one should expect higher power in the second experiment in which a greater distancefrom the null is allowed.

Under these two alternatives, we generated N D 1000 samples of size T D 50, T D 100,T D 150 and T D 200 respectively and computed the LR tests. Columns ‘ASYLR’ of Table I reportthe rejection frequencies obtained with the �2

1��1� quantile. Also in this case the LMC p-valuesassociated with each LR test are computed according to steps 1–4 of Section 3 with M D 100.The rejection frequencies based on the LMC p-values are summarized by the ‘LMCLR’ columns.

Overall, our results confirm that the rejection of FL models may partly be ascribed to the useof asymptotic critical values in finite samples (for instance, we obtain an empirical size of 19.28%instead of 5% with T D 200).11 On the other hand, the LMC procedure seems to control thelevel of the test satisfactorily. This evidence supports the idea that simulation-based methods candeliver reliable inference about FL models. Even with samples of length T D 50 and T D 100,the empirical size implied by the LMC procedure is slightly inferior to the nominal size of thetest. The power of the LMC LR-type tests against backward-looking cointegrated specificationsbehaves in a rather normal way, increasing as the model moves away from the null hypothesis.

5. APPLICATION TO THE NEW KEYNESIAN PHILLIPS CURVE

The so called ‘hybrid’ version of the NKPC (Galı and Gertler, 1999) represents a special case ofequation (1) and can be regarded as one of the most prominent FL models of inflation dynamics.In this framework, Qyt is the inflation rate and Qwt a measure of firms’ real marginal costs. In theCalvo (1983) setup, the structural parameters in � D ��, υ, ��0 are related to other structural ‘deep’parameters by the mapping

� D �ϑ

ϑ C �[1 � ϑ�1 � ��]�14�

11 The empirical sizes in Table I may depend on the grid chosen for � , υ and �. Interestingly, the rejection frequenciesobtained with the asymptotic critical values (ASYLR) are more sensitive than the LMC p-values (LMCLR) to variationsin the grid. As an example, for T D 150, our Monte Carlo experiments suggest that, if the grid constructed around thepoint �Ł D �0.70, 0.20, 0.15�0 comprises 990 points instead of 180 points (using š0.02 as increment for all parameters),then ASYLR D 10.79% and LMCLR D 4.5%.



υ D �

ϑ C �[1 � ϑ�1 � ��]�15�

� D �1 � ��1 � ϑ��1 � ϑ��

ϑ C �[1 � ϑ�1 � ��]�16�

where � is firms’ discount factor, � the fraction of forward-looking firms and �1 � ϑ��1 is theaverage time upon which prices are kept fixed (see Galı and Gertler, 1999; Galı et al., 2001).

Using a generalized method of moments approach, Galı et al. (2001) find support for the NKPCin the euro area over the period 1971–1998. Other existing studies based on model consistentexpectations (Bardsen et al., 2004) and ‘VAR expectations’ (Fanelli, 2008a) reject the model.Milani (2005, 2007), Lansing (2008), Mavroeidis at al. (2008) (for US data) and Fanelli (2008b)(for euro area data) investigate versions of model (1) under learning dynamics. Mavroeidis et al.(2008) and Fanelli (2008b) are the only examples in which the data adequacy of the NKPC withlearning is tested. The following subsections provide an LMC approach to the NKPC under rationalexpectations and learning dynamics, respectively.

5.1. Results

Our analysis is based on quarterly data for the inflation rate �t, the wage share wst and theshort-term interest rate it, taken from the Area-Wide Model (AWM) dataset (Fagan et al., 2001);see Appendix for details. We focus on two types of VAR-based LR tests: ‘one-shot’ tests andsequences of recursively computed tests, respectively. ‘One-shot’ tests are computed using thefull available sample (1970 : 2–2005 : 4) and investigate the validity of the NKPC under ‘VARexpectations’. As explained in Section 2, ‘VAR expectations’ amount to a particular formulationof the rational expectations hypothesis; hence the two concepts will be used interchangeably.Sequences of recursive LR tests investigate the validity of the NKPC under the adaptive learninghypothesis. These tests are computed by splitting the available sample into two distinct parts: thesubsample 1970 : 2–1983 : 4 (including initial values) is used to produce initial estimates of VARcoefficients; the subsample from T0 D 1984:1 until Tmax D 2005:4 is used to evaluate the NKPCrecursively.12

We estimate a VAR model for Xt D ��t, wst, it�0 with k D 3 lags and a constant (i.e. A�L�dt D �, � 0, Dt � 0). The constant is restricted to lie in the cointegration space (� D ˛�0, where�0 is the intercept entering the cointegrating relation) when the analysis is performed with thecointegrated model, since the variables do not show any linear trend. Table II reports the estimatedroots of the VAR companion matrix, the LR cointegration trace test and the cointegrating relationsestimated over the sub-period 1971 : 1–1983 : 4 and the whole estimation sample 1971 : 1–2005 : 4.The largest estimated root of the companion matrix has modulus close to unity. The LR trace testsuggests that the hypothesis of unit roots is highly supported by the data; moreover, inflation andthe wage share cointegrate with cointegrating vector (1, �0.5, 0)0, roughly confirming the findingin Fanelli (2008a) based on a bivariate system for �t and wst and a previous release of the AWMdataset.

12 Other solutions are potentially available to set initial conditions in recursive algorithms (see Carceles-Poveda andGiannitsarou, 2007). The analysis of the effects that the initial conditions produce on the outcomes of the recursive testsis beyond the scopes of this paper.



Table II. LR trace test for cointegration rank and estimated cointegrating relationship and adjustmentcoefficients

VAR: Xt D ��t, wst, it�0 lags k D 3 1971 : 1–1983 : 4Ł 1971 : 1–2005 : 4Ł

Largest eigenvalues of companion matrix: 0.92 š 0.066iŁ 0.977 š 0.027iŁ

Cointegration rank test

H0 : r � j Trace p-value

j D 0 43.74Ł53.18Ł

0.004Ł0.000Ł

j D 1 13.66Ł14.29Ł

0.321Ł0.276Ł

j D 2 4.79Ł1.64Ł

0.318Ł0.839Ł

Estimated cointegrating relationship and adjustment coefficients0Xt D ot D �t � 0.491�0.089�

wst C 1.88�0.357�

, ˛ D ��0.54�0.20�

, 0.22�0.05�

, 0�0

*0Xt D ot D �t � 0.515�0.043�

wst C 1.98�0.170�

*, *˛ D ��0.25�0.08�

, 0.15�0.03�

, 0�0*

LR D 2.37[0.30]

, *LR D 4.99Ł[0.08]

Notes: Test statistics and estimates are computed on the sub-period 1971 : 1–1983 : 4 (not including initial values) exceptthose between asterisks, **, which are computed on the full-sample 1971 : 1–2005 : 4 (not including initial values).Standard errors in parentheses, p-values in brackets.

Given these results, we consider two possible specifications of the agents’ forecast model: inSection 5.1.1, the NKPC is analyzed using the VAR in levels disregarding the non-stationarity ofthe variables; in Section 5.1.2, the NKPC is analyzed by incorporating the cointegration restrictions,as outlined in the Appendix. In both approaches, the estimation of the constrained VAR is carriedout by combining the BFGS method with a grid search for the structural parameters (see thetechnical supplement for details).

5.1.1 Model in LevelsFor t D T0, T0 C 1, . . . , Tmax, the VAR model is estimated recursively under the cross-equationrestrictions using CML, and unrestrictedly using ML. The CML estimates show that, for all t,the restricted log-likelihood is maximized for �t D �Ł D ��Ł, υŁ, �Ł� D �0.93, 0.05, 0.05�0. It turnsout that the magnitude of the estimated forward-looking parameter (�) is much larger than themagnitude of the estimated backward-looking parameter (υ); in the literature (in which attention isgenerally devoted to purely forward-looking versions of model (1), i.e. with υ D 0); this evidence istypically interpreted as an indication that learning is a source of inflation persistence (Milani, 2005,2007). The issue is investigated in details in Section 5.1.2. In terms of the Calvo parameterization(14)–(16), these estimates correspond to �Ł D 0.95 (discount factor), �Ł D 0.043 (fraction ofbackward-looking firms) and �1 � ϑŁ��1 ³ 5.3 (average number of quarters over which pricesare kept fixed).

The upper panel of Figure 1 plots the entire sequence of LR test statistics and the (constant)critical value �2

0.95(6), according to which the NKPC is sharply rejected for all t D T0, T0 C1, . . . , Tmax. The same happens for the ‘one-shot’ LR test.



1985 1990 1995 2000 2005

14

16

18

20

22

24

26

γ∗=0.93 , δ∗=0.05 , κ∗=0.05

LR

χ20.95(6)

1985 1990 1995 2000 2005

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η =0.05η =0.01

LMC sizeLMC power

Figure 1. Upper panel: sequence of LR statistics for the restrictions implied by the NKPC under learningdynamics (VAR in levels) with the corresponding optimal estimates of the structural parameters �Ł D��Ł, υŁ, �Ł�0 and �2

0.95�6� quantile. Lower panel: LMC simulated p-values and powers associated with eachtest statistic in the sequence. This figure is available in color online at wileyonlinelibrary.com/journal/jae

The lower panel of Figure 1 plots the sequence of LMC p-values associated with each LRstatistic, computed following the procedure described in steps 1–5 of Section 3. More precisely,for each t D T0, . . . , Tmax, we generatedM D 499 samples from the restricted VAR with coefficient�t D �a0

yt, a0ut, �

0t, �

0t, �

0t�, �

0t D vech�εt� and then computed the frequency of rejections of the null

hypothesis. The lower panel of Figure 1 also plots the LMC power associated with each LR teststatistics; in this case the frequency of rejections of the null hypothesis has been computed fromM D 499 samples drawn from the unrestricted VAR with coefficients �t D �a0

yt, a0wt, a

0ut, �

0t, �

0t�

0,�t D vech�εt�.13

13 We have not reported in Figure 1 and Figure 2 the nonparametric version of the LMC powers. The idea is thatthe parametric bootstrap is usually appropriate when one is using maximum likelihood estimation, whereas with othernonparametric estimation methods, a nonparametric bootstrap involving re-sampling is more suited. Moreover, the



The sequence of simulated p-values suggests that the NKPC is rejected at the 5% significancelevel. Accordingly, also the ‘one-shot’ test, corrected for the finite sample bias, rejects therestrictions implied by the NKPC.

The results of this subsection, as well as the results of Section 5.1.2 below, have been obtainedby assuming that the agents’ learning algorithm is the RLS which allows us to exploit the directlink with ‘standard’ ML estimation. The use of RLS implies that the underlying VAR coefficientsare time-invariant. In the technical supplement we argue that with our data similar results are likelyto be achieved via the CG-RLS algorithm, which is consistent with a time-varying specificationof the VAR coefficients.14 The equivalence between RLS and CG-RLS in the chosen monitoringsample can be explained by observing that the analysis covers a relatively ‘stable’ inflationaryregime, i.e. in which inflation expectations in the major European countries have been anchoredto the need for converging towards the European Monetary Union before 1998 and to the pricestability commitment of the European Central Bank from 1999 onwards.15

5.1.2 Cointegrated ModelTable II shows that the system Xt D ��t, wst, it�0 can be approximated as a non-stationary VARin which �t and wst cointegrate and the short-term interest rate is weakly exogenous. Hereafter,0 D �1,��, 0, �0� with � D 0.5 and �0 D 1.88 (�0 is the restricted constant) will be taken asthe estimated cointegration vector and will be treated as time-invariant.

The empirical analysis of the ‘hybrid’ NKPC with cointegrated variables is based on theformulation (17) of the model, discussed in the Appendix. The agents’ forecast model isspecified as a stationary VAR for Wt D �ot,�t,it�0 with three lags, in which ot D 0Xt DQ�t � 0.5wst C 1.88 is the estimated disequilibrium relationship taken from the original VAR and�t and it are selected from Xt by the selection matrix � (see equation (25)).

As for the model in levels, for t D T0, T0 C 1, . . . , Tmax, the system is estimated recursivelyin restricted and unrestricted form. The resulting CML estimates are given by � t, ω0

t� D� Ł, ωŁ�0 D �18.6, 0.4�0 for all t, and the resulting structural parameters correspond to �t D�ŁŁ D ��ŁŁ, υŁŁ, �ŁŁ� D �0.93, 0.05, 0.01�0. In particular, note that �ŁŁ D �1 � �ŁŁ � υŁŁ�� D 0.01is obtained by inverting the relationship (20) discussed in the Appendix. As for the VAR inlevels, the magnitude of the recursively estimated forward-looking parameter is significantly largerthan the magnitude of the recursively estimated backward-looking parameter, suggesting, as inMilani (2005, 2007), that typical persistence inflation mechanisms such as indexation, contractsand rules of thumbs are no longer crucial ingredients of inflation dynamics under adaptive learningschemes.16 In terms of the Calvo parameterization, these estimates correspond to �ŁŁ D 0.98, �ŁŁ D

techniques proposed by Davidson and MacKinnon (1999) to reduce computational costs when bootstrap tests are computedin models with nonlinear restrictions are not easily adaptable in our setup.14 In particular, the constant-gain parameters which maximize the likelihood function under the CG-RLS algorithm tend tobe very small over the monitoring sample, possibly suggesting small changes in the underlying VAR coefficients. It turnsout that the values of the likelihood function obtained with the CG-RLS algorithm in correspondence of the ‘optimal’constant-gain parameters are almost indistinguishable from the values obtained with the RLS (standard ML) algorithm.See the technical supplement for details.15 The data availability does not allow a thorough investigation of learning behavior in the 1970s and in the first part ofthe 1980s.16 Preston (2005) has recently shown that the dynamic structure of the NKPC does not involve only forecasts of a singleperiod in the future when the rational expectations hypothesis is replaced with a more general specification of the agents’beliefs. In this paper we investigate the ‘standard’ formulation of the NKPC, i.e. the specification based on forecasts ofa single period in the future as in Milani (2005, 2007) and Mavroeidis et al. (2008).



0.048 and �1 � ϑŁŁ��1 ³ 11, respectively. Compared to the estimates obtained in Subsection 5.1.1,the probability that firms are able to change their prices across periods falls from �1 � ϑŁ� ³ 0.19to �1 � ϑŁŁ� ³ 0.09. This discrepancy is due to the magnitude of the estimated � in the twospecifications; in the cointegrated model, the estimated cointegrating coefficient � affects themagnitude of � with consequences on the implied estimate of the degree of price stickiness.

The upper panel of Figure 2 plots the entire sequence of LR statistics, along with the constantcritical value �2

0.95�5�, according to which the model is rejected. For t D Tmax, the ‘one-shot’ testbased on standard asymptotic critical values rejects the cointegrated version of the hybrid NKPCunder the rational expectations paradigm, as in Fanelli (2008a).

The lower panel of Figure 2 plots the sequence of LMC p-values and power associated witheach LR test statistic comprising the sequence. Also in this case, for each t D T0, . . . , Tmax we

1985 1990 1995 2000 2005

9

10

11

12

13

14

γ∗=0.93 , δ∗=0.05 , κ∗=0.01

LR

χ20.95(5)

1985 1990 1995 2000 2005

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η =0.05

LMC sizeLMC power

Figure 2. Upper panel: sequence of LR statistics for the restrictions implied by the cointegrated versionof the hybrid NKPC under learning dynamics with the corresponding estimates of structural parameters�ŁŁ D ��ŁŁ, υŁŁ, �ŁŁ�0 and �2

0.95�5� quantile. Lower panel: LMC simulated p-values and powers associated witheach test statistic in the sequence. This figure is available in color online at wileyonlinelibrary.com/journal/jae



generated M D 499 samples from the restricted VAR for Wt to compute the sequence of LMCp-values; similarly, we generated M D 499 samples from the unrestricted VAR for Wt to computethe sequence of LMC powers. The simulated p-values do not reject the cointegrated version of the‘hybrid’ NKPC over the entire sequence, giving empirical room to the RLS version of the adaptivelearning hypothesis. Moreover, as the model is not rejected for t D Tmax (though marginally), itturns out that, once corrected for the finite sample bias, also the ‘one-shot’ test is consistent withthe restrictions implied by the rational expectations hypothesis. Overall, the empirical evidenceseems to be consistent with a situation in which, if the agents replace the expectations in theNKPC with the forecast implied by a cointegrated VAR, they eventually converge to equilibrium.

Compared to Milani (2007), who considers US data and uses a fully specified small-scaledynamic stochastic general equilibrium model, our result has been obtained by recursively testingthe validity of the restrictions implied by the NKPC with learning. Compared to Mavroeidis et al.(2008), in which results are also based on US data, we use a sequential testing approach andcapture the persistence in the data by imposing the cointegration restrictions.

The difference between the tests obtained through the VAR in levels (Section 5.1.1) and the testsobtained through the cointegrated VAR can be explained by using the results in Li (2007), whodemonstrates that, in finite samples, the persistence in the data leads to large size distortion andpower loss in tests for the cross-equation restrictions. Accordingly, even if bootstrap (parametricor nonparametric) tests potentially correct the tendency of the asymptotic test to over-reject thenonlinear restrictions implied by the NKPC, the correction is more reliable in the model based onthe cointegrated VAR, since in this case the unit roots are removed from the system. Furthermore,the results provided in Table I remark that LMC LR-type tests for the nonlinear restrictions impliedby FL models have reasonable power against cointegrated systems. These considerations suggestthat the results reported in Section 5.1.1 can be reasonably affected by the non-stationarity of thevariables.

6. CONCLUSIONS

In this paper we have investigated the finite sample performance of LR tests for the nonlinear cross-equation restrictions that arise when FL models are tested against VARs. We consider two typesof tests: (i) ‘one-shot’ tests calculated under rational expectations by using the whole availablesample based on the comparison of the unrestricted and the constrained VAR likelihoods; (ii)sequences of tests obtained under the assumption that ‘boundedly rational’ agents use VARs torecursively update their beliefs, also based on the comparison of the unrestricted and constrainedlikelihoods.

As regards the first type of tests, our Monte Carlo experiments remark that the use of asymptoticcritical values may induce one to falsely reject FL models in finite samples. This result generalizesthe class of models and restrictions investigated in Bekaert and Hodrick (2001) and helps to explainwhy FL models receive a marginal empirical support in the literature. We show that the LMCLR-type tests discussed in Dufour and Jouini (2006), provide a substantial control of the levelof the test if the maximization of the likelihood function under the cross-equation restrictions isproperly designed. This result is valid also with highly persistent variables. In addition, these testsexhibit reasonable power against backward-looking cointegrated VARs. The empirical analysisinvestigates the ‘hybrid’ NKPC on euro area data over the period 1971–2005 and is based on aVAR including the inflation rate, the wage share and a short-term interest rate. The main result



is that the implied cross-equation restrictions are sharply rejected when the LR test is comparedwith the asymptotic critical values, but they are not rejected, albeit marginally, when the LMCp-value is computed and the non-stationarity of the variables is taken into account. Moreover,the imposition of the cointegration restrictions entails a change in the estimated degree of pricestickiness.

As regards the second group of tests, the paper shows that simulation-based inference and VARmodels can be opportunely combined for recursively testing FL models under learning dynamics.We argue that, especially in the first stages of the learning process characterized by relatively smallsamples, LMC techniques are particularly advantageous. Indeed, in this setup it is not clear whichare the correct critical values that should be used to test the model. The empirical applicationshows that the ‘hybrid’ NKPC is not rejected when LMC LR-type tests are recursively computedover the chosen monitoring period, 1984–2005, if the agents follow the typical learning algorithmsand the cointegration restrictions are imposed. The magnitude of the forward-looking parameterdominates the magnitude of the backward-looking parameter, suggesting that learning dynamicscan potentially replace ‘traditional’ persistence inflation mechanisms such as indexation, contractsand rules of thumb.

In this paper we have focused on the VAR restrictions that stem from a single-equation structuralFL model, keeping the other equations of the system in reduced form. However, the suggestedapproach can be potentially extended to cover the class of nonlinear restrictions that arise in thepresence of simultaneous systems of FL equations and small-scale dynamic stochastic generalequilibrium models.

ACKNOWLEDGEMENTS

We acknowledge helpful comments from Hashem Pesaran, Riccardo ‘Jack’ Lucchetti, PaoloParuolo, Massimo Franchi and two anonymous referees. We are responsible for all remainingerrors. We thank seminar participants at the following conferences: 63rd European Meeting of theEconometric Society, Milan, 27–31 August 2008; the First International Conference in memory ofCarlo Giannini, Bergamo, 25–26 January 2008; 20 Years of Cointegration: Theory and Practice inProspect and Retrospect, 2nd Tinbergen Institute Conference, Rotterdam, 23–24 March 2007. Thefirst author acknowledges partial financial support from Italian MIUR and University of BolognaRFO grants.

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APPENDIX

Cointegrated Variables

In this appendix we extend the analysis of Section 3 to the case of non-stationary cointegratedvariables.

Two parameterizations of the FL model (1) are discussed. The former is given by

Qyt D Et QytC1 � ω� Qyt � � Qwt�C �Łt �17�

where

D �/υ, �18�

ω D �1 � υ� ��/υ, �19�

� D �/�1 � υ� �� 20�

and �Łt D �t/υ. It is assumed that υC � < 1. According to equation (20), one of the structural

parameters is automatically anchored to the cointegration parameter �. For instance, given �, �and υ, � is automatically determined. As in the FL model the parameter � is related to the ‘pass-through’ from Qwt to Qyt, in turns out that cointegration plays an effective role in determining oneof the three structural parameters of the NKPC.

The latter parameterization is based on the restriction υC � D 1 and is obtained by manipulatingequation (1) in the form

Qyt D 1 � υ

υEt QytC1 C �

υQwt C �Ł

t �21�

Here Qwt behaves as the driving variable of Qyt. Whereas equation (17) is based on the hypothesisthat Qyt and Qwt cointegrate, equation (21) requires that Qwt is individually stationary.

Consider the vector equilibrium correction (VEqC) representation of the non-stationary‘detrended’ process QXt D Xt � dt D � Qyt, Qwt, Qu0

t�0

QXt D ˛ˇ0 QXt�1 Ck�1∑jD1

i QXt�j C εt �22�

where ˛ˇ0 D �(Ip � ∑k

iD1 Ai)

, ˛ and ˇ are pð r full rank matrices, and i D � ∑kjDiC1 Ai, i D

1, . . . , k � 1; see Johansen (1996) for details. If Qyt and Qwt are cointegrated and conform to (17)and there are no additional cointegration relations in the system, then r D 1 and

ˇ0 D �1,��, 01ðq� �23�

so that ˇ is subject to q zero over-identifying restrictions. On the other hand, if (21) is the ‘correct’FL model, then r D 1 and

ˇ0 D �0, 1, 01ðq� �24�



and there are q C 1 zero over-identifying restrictions. If the cointegration matrix of the VEqCdoes not match either the structure (23) or the structure (24), the agents’ forecast model does notcomply with the long-run empirical implications of the FL model.

Suppose that ˇ takes the form (23) so that the term ot D ˇ0 QXt D Qyt � � Qwt measures deviationsfrom equilibrium. Define the pð 1 process Wt D � QX0

tˇ, QX0t��

0, where � is a pð �p� r� selectionmatrix such that det�� 0ˇ?� 6D 0; Paruolo (2003, Theorem 2) shows that the VEqC (22) can bewritten in terms of a VAR process for Wt without loss of information, i.e.

B�L�Wt D ε0t �25�

where ε0t D �ε0

tˇ, ε0t��

0 is a MDS with respect to Ht, and

B�L� D Ip � B1L � Ð Ð Ð � BckLk, Bck D �B1k, 0pð�p�r��

with the coefficients in Bi, i D 1, . . . k which depend on the VEqC coefficients ˛, 1, . . . , k�1

(see also Mellader et al., 1992). Because of the super-consistency result, one can replace ˇ in Wt

with the maximum likelihood estimate D �1,��, 01ðq�0 without affecting the asymptotics. TheVAR in (25) can be cast in companion form, and the j-step-ahead conditional forecast of yt andot can be computed as

E�ytCj j Ht�1� D s0yBjC1WŁ

t�1

E�otCj j Ht�1� D s0oBjWŁ

t�1

where sy and so are selection vectors, B is the pk ð pk companion matrix and WŁt D

�W0t,W

0t�1, . . . ,W

0t�kC1�

0 is the nð 1 state vector.As in Section 2, the cross-equation restrictions between the VAR in equation (25) and the FL

model (17) can be written as

s0yB�Ipk � B�C ωs0oB D 01ðpk �26�

and Propositions 1 and 2 in the technical supplement can be applied. In this case, the number ofoveridentifying restrictions is c D pk � �p� r�� 2, where 2 D dim� , ω� and r is the cointegra-tion rank of the system, see Fanelli (2008a). Moreover, the analysis can be generalized to the caseof adaptive learning along the lines of sections 2 and 3 either using recursive estimates of ˇ, orkeeping ˇ fixed at the super-consistent estimate based on information HT0�1. In both cases, theanalysis requires the absence of structural breaks from T0 to Tmax.

Data

We consider quarterly data for the euro area, using a release of the AWM dataset (Fagan et al.,2001) available before 2007. Variables, including initial values, cover the period 1970 : 2–2005 : 4.To measure inflation we use the GDP deflator, i.e. yt D 4 ð �pt � pt�1� D �t, where pt is the logof the GDP deflator. The GDP deflator is YED in the AWM dataset. Firms’ average marginalcosts are proxied by the wage share (log of real unit labour costs), wt D wst. The wage share iscomputed as wst D log�100 ð WINt/YERt�� pt, where WIN is ‘Compensation to Employees’



(in nominal terms) and YER is real GDP in the AWM dataset. A short-term interest rate, ut � it(qu D 1), is included in the system. We have used it D 1

100STNt as proxy of the short-term interestrate, where STN is the short-term interest rate in the AWM dataset.


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Simulation-based tests of forward-looking models under VAR learning dynamics