Multi-equational linear quadratic adjustment cost models with rational expectations and cointegration

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Journal of Economic Dynamics & Control 30 (2006) 445–456

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Multi-equational linear quadratic adjustmentcost models with rational expectations

and cointegration

Luca Fanelli�

Department of Statistics, University of Bologna, via Belle Arti, 41, I-40126 Bologna, Italy

Accepted 5 March 2005

Available online 4 May 2005

Abstract

In this paper the econometric analysis of linear quadratic adjustment cost models with

rational expectations and cointegrated variables is extended to the multi-equational SET-UP

and the case of second-order adjustment costs. The proposed method is based on the idea of

nesting the system of interrelated Euler equations stemming from the intertemporal

optimization problem within a cointegrated Vector Equilibrium Correction Model represent-

ing the agent forecast tool. Contrary to previous practise a likelihood-based procedure can be

set out without appealing to numerical optimization algorithms. Cointegration and general-

ized least squares techniques can be used to estimate and test the model.

r 2005 Elsevier B.V. All rights reserved.

JEL classification: C32; C52; C61

Keywords: Euler equations; MLQAC model; Rational expectations

1. Introduction

The linear quadratic adjustment cost (LQAC) model under rational expectations(RE) is widely applied for investigating intertemporal behavior of economic agents

see front matter r 2005 Elsevier B.V. All rights reserved.

.jedc.2005.03.002

0541434303; fax: +39051232153.

dress: [email protected].

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L. Fanelli / Journal of Economic Dynamics & Control 30 (2006) 445–456446

subject to convex adjustment costs. In the standard specification a representativeforward-looking agent chooses a single decision variable (the level of employment,the stock of money, the price level) subject to quadratic adjustment costs. Kennan(1979), Hansen and Sargent (1980), Dolado et al. (1991), Gregory et al. (1993),Engsted and Haldrup (1994, 1999), Huang and Shen (2002) and Fanelli (2002) areexamples where the LQAC model with first-order adjustment costs is estimated inthe presence of both stationary and non-stationary variables. Pesaran (1991), Price(1992) and Tinsley (2002) deal with the case of higher-order adjustment costs.

Economic decisions usually imply the simultaneous choice of a set of variablesrather than a single scalar. The extension of the LQAC model with RE to the case ofmultiple decision variables leads to the class of systems that we denote with theacronym MLQAC, with ‘M’ standing for ‘multi-equational’. Since Hansen andSargent (1981) the MLQAC model has been applied to different fields of research,e.g., the demand for factors and for consumption goods, see, inter alia, Nickell(1984), Eichenbaum (1984) and Weissenberger (1986).

This paper focuses on the MLQAC model. It extends the maximum likelihoodapproach proposed in Fanelli (2002) for estimating and testing the LQAC modelwith cointegrated variables to the case of multiple decision variables and second-order adjustment costs. The main assumption is that conditional forecasts on theobservable (possibly non-stationary) variables of the system are computed through avector equilibrium correction model (VEqCM); adapting the original idea of Sargent(1979), the method is based on the restrictions that the system of Euler equationsimplied by the MLQAC model imposes on the parameters of the VEqCM under theRE hypothesis.

The proposed method differs from the existing likelihood-based approaches to theMLQAC model in the following aspects. First, differently from Kozicki and Tinsley(1999), attention is here devoted to the system of interrelated Euler equationsstemming from the intertemporal optimization problem rather than the forwarddecision rule. The idea is that the parameters of interest are those associated with theEuler equations;1 the advantage is that under RE and for fixed values of the discountfactor, the model can be nested within the VEqCM without involving non-linearparametric constraints. The restricted model can be estimated and tested through asimple procedure requiring the implementation of cointegration and generalizedleast squares (GLS) techniques.2

Second, the focus is on the ‘exact’ version of the MLQAC model. FollowingHansen and Sargent (1991) this means that expectations do not involve processeswhich are unobservable to the econometrician. Examples where ‘exact’ RE modelsare estimated and tested through likelihood techniques include e.g. Sargent (1979),

1Observe, however, that these parameters are opportunely related to that of the forward solution, see

e.g. Binder and Pesaran (1995).2In order to apply likelihood methods to estimate forward decision rules implied by dynamic adjustment

cost models it is necessary to solve the equations for the unknown future expectations involving the forcing

variables. Vector Autoregressive models are usually employed to this purpose; the cross-equation

restrictions between the parameters of the two models are typically non-linear and numerical optimization

algorithms must be applied, see e.g. Binder and Pesaran (1995) and Kozicki and Tinsley (1999).

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L. Fanelli / Journal of Economic Dynamics & Control 30 (2006) 445–456 447

Campbell and Shiller (1987) and Johansen and Swensen (1999); we refer toBinder and Pesaran (1995) for the analysis of the ‘non-exact’ version of the MLQACmodel, see Pesaran (1991) for the case of a single decision variable. If from theperspective of model specification and solution the analysis of a ‘non-exact’ modelis identical with that of an ‘exact’ one, from the point of view of identification andtestable restrictions the analysis of the latter can be quite different (Hansen andSargent, 1991).3

Third, while in Kozicki and Tinsley (1999) possible equilibrium pathscharacterizing first-order non-stationary ðIð1ÞÞ variables are identified by ‘prior’cointegration analysis, here no auxiliary model is required to capture thecointegration implications of the MLQAC model. The analysis can be exclusivelybased on the VEqCM subject to the forward-looking constraints; moreover,differently from the technique outlined in Johansen and Swensen (1999), numericaloptimization methods are not required. Any existing econometric software can beused to implement the method.

The plan of the paper is the following. Section 2 introduces the MLQAC model withsecond-order adjustment costs and Section 3 deals with the estimation. Subsection 3.1derives the restrictions that the system of interrelated Euler equations imposes on theVEqCM and Subsection 3.2 deals with estimation and testing. Section 4 contains someconcluding remarks. Technical details are summarized in the appendix.

2. The model

We consider an economic agent acting intertemporally at discrete time with thetask of choosing a vector of decision variables grouped in the py � 1 vector yt. Let yn

t

be the py � 1 vector containing the long-run targets pursued by the agent; ynt depends

linearly on a pz � 1 vector of observable variables zt, as detailed below. Thestochastic processes fytþjg

1j¼0 and fztþjg

1j¼0 are defined on the same probability

space and the agent’s information set is given by the increasing sigma-fields, Aat ,

Aat �Aa

tþ1. Et� ¼ Eð� jAat Þ is the conditional expectation upon the information

available at time t.The optimal sequence fytþjg

1j¼0 is chosen to satisfy the problem:

minfytþjg

Et

X1j¼0

rj½ðytþj � yn

tþjÞ0Y0ðytþj � yn

tþjÞ

þ Dy0tþjY1Dytþj þ D2y0tþjY2D2ytþj�, ð1Þ

3As remarked by a referee, the terminology ‘exact’ vs. ‘non-exact’ RE models we adopt throughout the

paper, albeit largely used in the literature, can be misleading. It would suggest, indeed, that researchers

working with ‘exact’ RE models deal with theories that must hold exactly in the data, while researcher

working with ‘non-exact’ ones incur in some unspecified error of approximation. Probably for this reason

Campbell and Shiller (1987) suggest informal (other than statistic) methods to evaluate ‘exact’ present

value models. We turn in the Sections 2 and 3 on the precise characterization of the RE model we devote

attention in the paper.

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where Dytþj ¼ ytþj � ytþj�1, D2ytþj ¼ Dytþj � Dytþj�1, r (0oro1) is a time-invariant

discount factor and Y0, Y1 and Y2 are py � py matrices which are assumed to besymmetric and positive definite. The first quadratic form in (1) measures the cost ofnot attaining the target yn

t (disequilibrium costs); the second term measures the costof changing yt at a different level (first-order adjustment cost); the third termcaptures the cost of adjusting the speed with which changes in yt are put into effect(second-order adjustment costs). The matrices Y0, Y1 and Y2 are in general notdiagonal, thus cross adjustment and cross disequilibria costs are allowed.

Pesaran (1991) and Price (1992) deal with the problem (1) in the presence of asingle decision variable (py ¼ 1). We shall refer to the system stemming from (1) asthe MLQAC model.

The first-order necessary conditions for (1) are given by the system of (second-order) interrelated Euler equations4

r2Y2EtD2ytþ2 � rðY1 þ 2Y2ÞEtDytþ1 þ ðY1 þ 2rY2ÞDyt

þY2D2yt þY0ðyt � yn

t Þ ¼ 0py�1

ð2Þ

and the long run target ynt is assumed to be generated by

yn

t ¼ Fzt þ cþ vt, (3)

where F is a py � pz matrix of parameters, c is a py � 1 constant and vt is a py � 1disturbance term.

Unless there are specific economic reasons for including an exogenous disturbanceterm on the right-hand side of (3) (and hence in the model (2)), the presence of vt

within this class of RE models is usually motivated by the idea that in forecasting thefuture private agents use larger information sets than the econometrician canconsider because of data limitation, see e.g. Hansen and Sargent (1991) and Engstedand Haldrup (1994).5 The vt term in (3) can be thus regarded as a quantity known tothe agent but unobservable to the econometrician at time t; as argued in Hansen andSargent (1991), this is the only interpretation consistent with the class of ‘exact’ REmodels we consider in this paper.

By substituting (3) into (2), using D2ytþj ¼ Dytþj � Dytþj�1 and rearranging termsthe system can be expressed in the form

EtDytþ2 ¼ r�1½Cþ ð2þ rÞIpy�EtDytþ1

� r�2½Cþ 2rIpy�Dyt � r�2Dyt�1 � r�2Uðyt � Fzt � cÞ þ vnt , ð4Þ

where vnt ¼ r�2Uvt and the matrices C ¼ Y�12 Y1 and U ¼ Y�12 Y0 need not to besymmetric. The elements of C measure the relative importance of second-orderand first-order adjustment (and cross-adjustment) costs, whereas the elements in Umeasure the relative importance of second-order and disequilibria adjustment (andcross-adjustment) costs.

4See Binder and Pesaran (1995), Section 4.2 and Kozicki and Tinsley (1999), Section 3.5For instance, in the model of Hansen and Sargent (1980) disturbance terms are interpreted explicitly as

technology or preference shocks.

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It is assumed that yt ðDytÞ does not Granger-cause zt ðDztÞ; if indeed there werefeedbacks from yt ðDytÞ to zt ðDztÞ the model (4) might not admit a unique stablesolution.6

We refer to Hansen and Sargent (1981), Binder and Pesaran (1995) and Kozickiand Tinsley (1999) for the derivation of the forward-solution associated with thesystem (4); see Pesaran (1991) for the case of a single decision variable (py ¼ 1). Thehypothesis of this paper is that the parameters of interest are embodied in the systemof Euler equations. A special case is obtained when the Y2 matrix is null in (1), i.e.when agents face first-order adjustment costs only; with Y2 ¼ 0 the dynamicstructure of the model simplifies and the derivation of the corresponding forwarddecision rule can be obtained as in e.g. Nickell (1984). It can be recognized that withY2a0 a more flexible dynamic structure occurs.

3. Econometric issues

The conventional approach for estimating the structural parameters ðF, c, r, C, UÞof the system of Euler equations (4) is based on generalized method of moments(GMM). In this paper we propose a likelihood-based method which providesefficient estimates of the structural parameters and a direct test of the model.

Given the p� 1 vector X t ¼ ðy0t, z0tÞ

0 of observable variables (p ¼ py þ pz), it isassumed that in forecasting the future at time t, the econometrician uses theinformation set Ae

t ¼ ðX0t, X 0t�1; . . . ;X

01Þ which is a subset of Aa

t , Aet �Aa

t . Thevariables in (4) can be conditioned with respect to Ae

t generating the model

EðDytþ2 jAet Þ ¼ r�1½Cþ ð2þ rÞIpy

�EðDytþ1 jAet Þ

� r�2½Cþ 2rIpy�Dyt � r�2Dyt�1 � r�2Uðyt � Fzt � cÞ, ð5Þ

where consistently with the ‘exact’ interpretation of the RE model specified inSection 2, it has been assumed that

Eðvt jAet Þ ¼ 0

py�1

in order to rule out processes which are unobservable for the econometrician.Following Sargent (1979) and Johansen and Swensen (1999), and generalizing the

approach set out in Fanelli (2002) for the LQAC model with variables integrated oforder one ðIð1ÞÞ, the idea of the paper is to compute conditional expectations in thereference model (5) through a Vector Autoregressive model for X t reparameterizedin VEqCM form. It is then possible to nest the parameters of the MLQAC within theVEqCM (Section 3.1) and the resulting model can be estimated through maximumlikelihood (ML) (Section 3.2).

6See Timmermann (1994) for a discussion on the role of feedbacks in present value models.

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3.1. Restrictions

Assume that for t ¼ 1; . . . ;T the forecast model for X t ¼ ðy0t, z0tÞ

0 belongs to theclass of Ið1Þ cointegrated systems

DX t ¼ ab0X t�2 þ G1DX t�1 þ � � � þ Gk�1DX t�kþ1 þ mþ et, (6)

where k is the lag length, X 0;X�1; . . . ;X ð1�kÞ are fixed, et is a Gaussian White Noiseprocess with (non-singular) covariance matrix O, Gi, i ¼ 1; 2; . . . ; k � 1 are ðp� pÞ

matrices of parameters, m is a p� 1 constant and b and a are p� r matrices of rank r,0orop, see Johansen (1996) for details.

The VEqCM (6) is the model used to compute conditional expectations in (5).Given X t ¼ ðy

0t, z0tÞ

0 and et ¼ ðe0yt; e0ztÞ0, consider the following partitions (with

dimension of matrices alongside blocks):

b ¼byy bzy

byz bzz

" #py � ry py � rz;

pz � ry pz � rz;(7)

a ¼ayy ayz

azy azz

" #py � ry py � rz;

pz � ry pz � rz;(8)

Gi ¼Gyyi Gyzi

Gzyi Gzzi

" #py � py py � pz;

pz � py pz � pz;m ¼

my

mz

" #py � 1;

pz � 1:(9)

Here the cointegration rank, r, is given by r ¼ ry þ rz, where ry is thenumber of columns of byy and rz is the number of columns of bzz. The hypothesisof absence of feedbacks from yt (Dyt) to zt (Dzt) corresponds to the set of zeroconstraints

bzy ¼ 0py�rz

; azy ¼ 0pz�ry

; Gzyi ¼ 0pz�py

; i ¼ 1; 2; . . . ; k � 1. (10)

By evaluating the VEqCM (6) at time tþ 2 one obtains the expression

EðDX tþ2 jAet Þ ¼ ab0X t þ G1EðDX tþ1 jA

et Þ þ G2DX t þ G3DX t�1

þ � � � þ Gk�1DX t�kþ3 þ m,

so that using the partitions (7)–(9) and rearranging terms, the forecast EðDytþ2 jAet Þ

reads as

EðDytþ2 jAet Þ ¼ Gyy1EðDytþ1 jA

et Þ þ Gyy2Dyt þ Gyy3Dyt�1 þ � � �

þ Gyyk�1Dyt�kþ1 þ ayyðb0yyytþGyyk�1Dy þ b0yzztÞ

þ ayzðb0zyyt þ b0zzztÞ þ Gyz1EðDztþ1 jA

et Þ þ � � �

þ Gyzk�1Dzt�kþ3 þ my. ð11Þ

It can be now recognized that by equating the expressions in (5) and (11) andincluding the no-feedback constraints (10), the following set of restrictions between

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ðb; a;G1; . . . ;Gk�1;mÞ and ðF, c, r, C, UÞ holds:7

rXpy; b ¼byy bzy

byz bzz

" #¼ b0 ¼

Ipy0

�F0 bzz

" #, (12)

Gyy1 ¼ r�1½Cþ ð2þ rÞIpy�, (13)

Gyy2 ¼ �r�2½Cþ 2rIpy�, (14)

Gyy3 ¼ �r�2Ipy, (15)

Gyyj ¼ 0py�py

; j ¼ 4; . . . ; k � 1; Gyzj ¼ 0py�pz

; j ¼ 1; . . . ; k � 1, (16)

ayy ¼ �r�2U; ayz ¼ 0py�rz

, (17)

my ¼ �r�2Uc, (18)

where Gyy2 ¼ ð2�rr2 ÞIpy

� r�1Gyy1.Observe that in (12)–(17) the parameters of the F matrix are embedded in the

cointegration space, whereas the adjustment parameters ðr, C, UÞ are nested in(a;G1; . . . ;Gk�1Þ.

8Moreover, non-linear restrictions are ruled out in (13)–(17) if r istreated as a known quantity. It is precisely the linear nature of the restrictions in(13)–(17)(for fixed values of r) that makes it possible the estimation and testingprocedure we outline in the next section.

3.2. Estimation and testing

According to (12)–(18) the structural parameters ðF, c, r, C, UÞ of the system ofEuler equations (4) are embedded in the VEqCM (6).9 The maximization of the log-likelihood of the restricted VEqCM can be obtained through a procedure wesummarize in the five steps that follow (see Appendix A for details).

7It is assumed that the restrictions between the parameters of the two models do not involve those

associated with deterministic seasonal and/or intervention dummies which could characterize the right-

hand side of (5) and thus the deterministic part of the VEqCM (6).8From (12) it turns out that as the (sub)matrices azz and bzz are pz � rz, in addition to the ry ¼ py long

run relations ðyt�1 � Fzt�1 � cÞ�Ið0Þ implied by the MLQAC model, a number rz of cointegrating

relations involving the variables in zt are allowed, see Fanelli (2002) for details. Clearly, when r ¼ py (i.e.

rz ¼ 0, azz ¼ 0), the variables in zt are strongly exogenous with respect to b (see e.g. Hendry, 1995).9The analysis of identification of the parameters of the MLQAC model under the restrictions (12)–(18)

can be carried out as in Fanelli (2002). Intuitively, from (13)–(18) it turns out that if in the VEqCM the lag

length is kX4, the structural parameters in ðr;C; U; cÞ are locally identifiable; if k ¼ 2 and k ¼ 3 at least

one parameter in ðr;C; U; cÞ is not identifiable and must be pre-fixed; finally, if k ¼ 1 the parameters in

ðr;CÞ are not identifiable. The ‘natural’ choice to achieve identification when k ¼ 2 or k ¼ 3 is to fix the

intertemporal discount factor, see below.

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1.

1

En

pro

the

par

Fit a VAR model for X t ¼ ðy0t, z0tÞ

0 and determine the lag length k.
2. Test through Johansen’s trace whether the cointegration rank in the VEqCM (6)
is greater or equal to the number of decision variables, consistently with (12). Ifropy reject the MLQAC model, otherwise proceed to the step 3.

3.
Test whether the cointegrating relations conform to b ¼ b0 as in (12), leavingthe remaining parameters (a;G1; . . . ;Gk�1;mÞ of the VEqCM unrestricted. If thestructure of cointegrating relations is not consistent with (12) reject the MLQACmodel, otherwise recover the ML estimate of F from bb0.
4.
For fixed b to the ML estimate bb0 obtained in the step 3, and given a plausibleeconomic value for the discount factor r,10 estimate the parameters (a,G1; . . . ;Gk�1Þ of the VEqCM (6) subject to the exclusion restrictions implied bythe Eqs. (15)–(17), including the constraints setting the diagonal elements of Gyy3
equal to r�2 (Eq. (15)) and the constraints characterizing the elements of Gyy2

(i.e. Gyy2 ¼ ðð2� rÞ=r2ÞIpy� r�1Gyy1). The ML estimator corresponds to GLS.

Test the validity of these restrictions through (w2-distributed) LR or Wald-typestatistics; if the model passes the test go to the step 5, otherwise reject theMLQAC model.

5.
Use the relations (13), (17) and (18) to recover indirect estimates of C, U and c
from the ML estimates of ayy, Gyy1 and my obtained in the step 4.

Notice that the steps from 1 to 4 can be carried out by means of any existingeconometric package, while the algebra involved in the step 5 can be found in theAppendix A.

The proposed method is similar, in spirit, to that proposed by Kozicki and Tinsley(1999) for dynamic adjustment cost models and to the technique introduced byJohansen and Swensen (1999) for testing a more general class of RE models. Themain difference with respect to Kozicki and Tinsley (1999) is that these authorsdevote attention to the parameters of the forward decision rule rather than thoseassociated with the system of Euler equations; furthermore, they account for higher-order (‘polynomial’) adjustment costs. As in the forward decision rule expectationsinvolve future values of explanatory variables, a forecast model for zt (Dzt) isrequired to solve the model and highly non-linear cross-equation restrictions arise. Inour set-up the restrictions between the parameters of the two models are linear if thediscount factor is treated as a known quantity hence Newton-type numericaloptimization methods are not required. Moreover, the econometric analysis of theMLQAC model summarized in the steps 1–5 above can be exclusively based on theVEqCM for X t ¼ ðy

0t, z0tÞ

0; no auxiliary model or ‘prior’ cointegration analysis isrequired when the variables are Ið1Þ.

0It is generally difficult to estimate the intertemporal discount factor r within this class of models;

gsted and Haldrup (1994) suggest to prefix r within the range 0.95–0.99. Johansen and Swensen (1999)

pose grid search techniques. Observe that a grid search for r may not necessarily be advantageous in

MLQAC model, especially when one thinks about measures of uncertainty for the other model

ameters interacting with the uncertainty surrounding estimates of r.

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Differently from Johansen and Swensen (1999) the structural parameters of theRE model are assumed to be unknown and subject to estimation. AlthoughJohansen and Swensen’s (1999) approach can be extended to the case where thestructural parameters are known and have to be estimated, the implementation oftheir likelihood-based procedure requires numerical optimization algorithms.

4. Conclusions

In this paper we have extended the approach proposed in Fanelli (2002) forestimating and testing the LQAC model with RE and Ið1Þ cointegrated processes tothe case of multiple decision variables and second-order adjustment costs. This classof models arises in all fields of economic research where intertemporal behavior ofagents facing adjustment costs is involved. The focus is on the system of interrelatedEuler equations stemming from the intertemporal optimization problem. Theeconometric analysis is carried out under the hypothesis of VEqCM-based forecastsand estimation through ML hinges on the restrictions that the MLQAC modelimposes on the parameters of the VEqCM under the RE hypothesis. Cointegrationtechniques and GLS methods can be used to estimate and test the model withoutappealing to numerical optimization algorithms.

Acknowledgements

I thank an anonymous referee for helpful comments and suggestions on earlierversions of the paper. Any remaining errors, of course, are my own responsibility.Partial financial support from Italian M.I.U.R. grants is gratefully acknowledged.

Appendix A

In this appendix we discuss in detail the estimation and testing procedure sketchedin Section 3.2.

Consider the VEqCM (6) where the cointegration rank r and cointegrationparameters in b are restricted as in (12) (step 3). It is then possible to write the system(6) as

DX t ¼ aðb00X t�2Þ þ GW t þ mþ et, (19)

where G ¼ ½G1; . . . ;Gk�1� and W t ¼ ðDX t�1; . . . ;DX t�kþ1Þ0. The log-likelihood

function of (19) can be concentrated with respect to ða;G;m;OÞ and the MLestimation of b0 can be obtained as in e.g. Johansen (1995) or Pesaran and Shin(2002). The estimator bb0 is super-consistent (i.e. p limðbb0Þ ¼ b0 at a rate OpðT

�1Þ) andmixed-normal so that (conditionally on the estimate of the mixing covariance matrixof bb0) LR tests for the over-identifying restrictions have w2-distribution (Johansen,1996).

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If the (identifying) constraints characterizing b in (12) are not rejected, b0 can bereplaced in (19) by its ML estimate bb0 (step 4). The resulting system involvesstationary variables and the substitution of b0 with bb0 does not affect the asymptoticinference on ða;GÞ due to the super-consistency result. Write the VEqCM (19)compactly as

DX ¼ PM þ E,

where DX ¼ ½DX 1; . . . ;DX T �, P ¼ ½a;G; m�; M ¼Mðbb0Þ ¼ ½M1ðbb0Þ; . . . ;MT ð

bb0Þ�0,Mtð

bb0Þ ¼ ðX 0t�2bb0, W 0t, 1Þ0; t ¼ 1; . . . ;T , E ¼ ½e1; . . . ; eT �

0, and apply the ‘vec’operator obtaining

vecðDX Þ ¼ ðM 0 � IpÞvecðPÞ þ vecðEÞ. (20)

Suppose that the discount factor r is known and consider the partitions in Eqs. (8)and (9); the zero restrictions in (15)–(17) along with the constraints setting thediagonal elements of Gyy3 equal to r�2 and those characterizing the (sub)matrix Gyy2,Gyy2 ¼ ðð2� rÞ=r2ÞIpy

� r�1Gyy1, can be summarized in the expression

vecðPÞ ¼ H � Zþ h, (21)

where H and h are respectively a known full-rank matrix and a known vector ofsuitable dimensions and the vector Z summarizes the short run parameters of theVEqCM left unrestricted, i.e.

Z ¼ ðvecðayyÞ0; vecðazzÞ

0; vecðGyy1Þ0; vecðGzz1Þ

0; . . . ; vecðGzzk�1Þ0; m0Þ0

¼ ðeZ0; m0Þ0.We deliberately leave the constant m free from the constraint of Eq. (18), see

below. The ML estimator of Z in the model (20)–(21) corresponds to the (estimated)GLS (see Lutkepohl, 1993, Chapter 5)

bZ ¼ fH 0

½ðMM 0Þ � bO�1�Hg�1H 0½M � bO�1�vecðDX hÞ, (22)

where vecðDX hÞ ¼ vecðDX Þ � ðM 0 � IpÞh, bO is a consistent estimator of O and ‘�’ is

the Kronecker product. The quantity T�1=2ðbeZ� eZÞ is asymptotically normally

distributed with covariance matrix ½H 0ðVM � bO�1ÞH��1, p limðT�1MM 0Þ ¼ VM , andthe restrictions in (21) can be tested through LR or Wald statistics with standardasymptotic distribution.11

If the restrictions (21) are not rejected (step 5), indirect ML estimates of thestructural parameters ðC, U, cÞ can be obtained by inverting the relationships Gyy1 ¼

r�1½Cþ ð2þ rÞIpy� (Eq. (13)), ayy ¼ �r�2U (Eq. (17)) and my ¼ �r

�2Uc (Eq. (18)),yieldingbC ¼ rbGyy1 � ð2þ rÞIpy

,

11The asymptotic distribution of the ML estimator of m is non-standard in cointegrated systems, see e.g.

Johansen (1991).

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bU ¼ �r2bayy,

bc ¼ �r2bU�1bmy,

where bGyy1, bayy and bmy are the ML estimates obtained by (22).

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Multi-equational linear quadratic adjustment cost models with rational expectations and cointegration