on the interpretation of cointegration in the linear–quadratic inventory model

Journal of Economic Dynamics & Control26 (2002) 2037–2049

www.elsevier.com/locate/econbase

On the interpretation of cointegration in thelinear–quadratic inventory model

James D. Hamilton ∗

Department of Economics, University of California at San Diego, La Jolla,CA 92093-0508, USA

Received 1 September 1999; accepted 7 March 2001

Abstract

The traditional formulation of the linear–quadratic inventory model with unit rootspredicts cointegration between inventories and sales. That formulation implies thatmarginal production costs and the marginal bene0ts of inventories are both tending to∞, and the cointegrating coe2cient re3ects the optimal trade-o5 between these com-peting factors. This paper suggests a reformulation of the problem in which marginalproduction costs and marginal inventory bene0ts are both stationary and in which thecointegrating coe2cient is the same as the value that characterizes the target inventorylevel in the cost function. c© 2002 Elsevier Science B.V. All rights reserved.

JEL classi!cation: C32; C61

Keywords: Cointegration; Linear–quadratic; Inventories

1. Introduction

The linear–quadratic model of optimal inventory accumulation developedby Holt et al. (1960), and Hansen and Sargent (1980, 1981) has becomethe standard model for empirical analysis of inventory dynamics. Examplesinclude Blanchard (1983), West (1986), Eichenbaum (1989), Ramey (1991),Krane and Braun (1991), Kashyap and Wilcox (1993), Durlauf and Maccini

∗ Tel.: +1-858-534-3383; fax: +1-858-534-7040; webpage: http:==www.econ.ucsd.edu=∼jhamilto=.E-mail address: [email protected] (J.D. Hamilton).

0165-1889/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0165-1889(01)00021-5

2038 J.D. Hamilton / Journal of Economic Dynamics & Control 26 (2002) 2037–2049

(1995), Fuhrer et al. (1995), West and Wilcox (1994, 1996), and Humphreyset al. (2000). Excellent surveys of the method have been provided by West(1995), Anderson et al. (1996), and Ramey and West (1999).The theoretical framework is usually developed for zero-mean, stationary

variables, though the data to which the model is applied are almost alwaysnonstationary. One popular approach is to assume that the driving variablesare characterized by deterministic trends. It is then a perfectly valid procedureto apply the stationary theoretical model to the detrended data, though somee2ciency can be lost by ignoring restrictions that the model may impose onthe deterministic terms (West, 1989).More care is required in accounting for stochastic trends or unit roots.

The current dominant approach, developed by Kashyap and Wilcox (1993),is to assume that the 0rm’s sales have a unit root (follow an I(1) process)whereas the unobserved shock to the 0rm’s productivity is stationary (or I(0)).Under these assumptions, Kashyap and Wilcox show that the standard modelimplies that inventories and sales will be cointegrated. Further discussion andevidence on this point can be found in Granger and Lee (1989), West (1995),and Ramey and West (1999).This note examines an unappealing feature of this cointegration that appears

not to have been commented upon previously in this literature. The 0rmin this environment is seeing its marginal production costs tend to ∞, anddeliberately chooses to let its inventory management costs tend to ∞ so as tominimize total costs. The cointegration coe2cient relating inventories to salesis determined by the optimal balancing of these two costs and the optimalrate of divergence of inventory management costs from their minimal value.This note further suggests a simple cure for this problem. Rather than

assume that unobserved productivity shocks are I(0), we propose modelingthem as I(1) but cointegrated with the 0rm’s sales. It is perhaps surprisingthat this alternative assumption about unobserved shocks ends up implyingthe identical reduced-form dynamic behavior of sales and inventories and theidentical value for the sample likelihood function as in the Kashyap–Wilcoxframework. However, the assumptions imply a di5erent mapping from thereduced-form dynamic behavior into the structural parameters which wouldseem much more appealing theoretically. Speci0cally, in the framework sug-gested below, marginal production costs and marginal bene0ts of inventoriesare stationary along the long-run growth path, and the cointegrating relationis one and the same as the target inventory relation. The framework is sug-gested as a better way to interpret cointegration between inventories and salesusing a coherent theoretical model of the 0rm’s decision problem.It is helpful 0rst to develop this point with a simple example in which a

representation of the cointegrated system and the mapping from reduced-formto structural parameters can be examined in closed form. We then show thatthe same results hold in general.

J.D. Hamilton / Journal of Economic Dynamics & Control 26 (2002) 2037–2049 2039

2. The linear–quadratic inventory model

Consider the following decision problem, similar to that in Ramey andWest (1999): 1

max{Qt ;Ht}∞

t=0

E0

{ ∞∑t=0

�t(PtSt − Ct)

}(1)

subject to:

Ct = (12){a0(MQt)2 + a1[(Qt −Uct)2 + a2(Ht−1 − a4 − a3St)2]} (2)

Qt = St +Ht −Ht−1; (3)

where Pt is the price of the good, St the unit sales, Ct the cost of production,Qt the quantity produced, Ht the inventories, Uct the shock to marginal costof production and � the discount rate.The 0rst-order condition for cost minimization is

Et[a0(MQt − 2�MQt+1 + �2MQt+2) + a1(Qt −Uct)

−�a1(Qt+1 −Uc; t+1) + �a1a2(Ht − a4 − a3St+1)] = 0: (4)

Cost minimization must hold regardless of the relation between prices andsales. For purposes of this note, we focus exclusively on the cost-minimizationcondition and treat prices and sales as exogenous. The traditional interpreta-tion of cointegration between inventories and sales arising from such a model(e.g., Kashyap and Wilcox, 1993; West, 1995; Ramey and West, 1999) comesfrom assuming that St has a unit root while Uct is stationary. We begin byconsidering a special case for which the solution can be examined in closedform.

2.1. An example of the traditional cointegrated representation

Suppose that there are no costs of adjusting production (a0 = 0) and salesfollow a random walk with drift,

St = St−1 + a5 + vst ; (5)

1 This is slightly a di5erent notation from that used by Ramey and West. Here production costsare given by ( 12 )a1Q

2t −a1QtUct+( 12 )a1U

2ct whereas Ramey and West specify ( 12 )a1Q

2t +QtU∗

ct :The term ( 12 )U

2ct is from the point of view of the 0rm a constant which has no e5ect on

any 0rst-order conditions, and −a1Uct = U∗ct is a renormalization of the shock to marginal

production costs that simpli0es the algebra here. Also, we specify the coe2cient on inventorycosts as ( 12 )a1a2 whereas Ramey and West use ( 12 )a2: Again this is only a renormalization thatsimpli0es the algebra, though in empirical work one might want to allow a zero coe2cient onproduction costs and nonzero coe2cient on inventory costs, which our normalization does notallow.


for vst white noise and Uct=vct is also white noise. In this case, after dividingby a1; Eq. (4) becomes

Et[Qt − vct − �Qt+1 + �a2(Ht − a4 − a3St − a3a5)] = 0; (6)

since Etvs; t+1 = Etvc; t+1 = 0: Substituting the inventory identity (3) into (6)results in

Et[MHt + St − vct − �(MHt+1 + St + a5)

+�a2(Ht − a4 − a3St − a3a5)] = 0: (7)

De0ne

�0 ≡ −(a5=a2)− a4 − a3a5;

�1 ≡ 1− ��a2

− a3; (8)

wt ≡ Ht + �0 + �1St; (9)

allowing (7) to be written

Et(MHt − �MHt+1 + �a2wt − vct) = 0: (10)

Notice that Eq. (9) implies

MHt =Mwt − �1MSt: (11)

Substituting (11) into (10) results in

Et(Mwt − �1MSt − �Mwt+1 + ��1MSt+1 + �a2wt − vct) = 0

or, from (5),

Et(Mwt − �1a5 − �1vst − �Mwt+1 + ��1a5 + �a2wt − vct) = 0;

Et[(1 + �+ �a2)wt − wt−1 − �wt+1] = (1− �)�1a5 + �1vst + vct ;

Et�[(

1− 1 + �+ �a2�

L+ �−1L2)wt+1

]=−h0 − xt (12)

for L the lag operator, h0 = (1− �)�1a5 and xt = �1vst + vct :Eq. (12) is an example of the well-known di5erence equation from Sargent

(1987, p. 201). The factorization

(1− �1z)(1− �2z) =(1− 1 + �+ �a2

�z + �−1z2

)(13)


has real roots 0¡�1¡ 1 and �2 = 1=(��1)¿ 1: Recalling that xt is whitenoise, the solution to (12) is known from Sargent (1987, pp. 203 and 394)to be

wt = �1wt−1 +�1h0

1− �−12

+ �1xt : (14)

Notice that (14) is a stationary AR(1) process. It follows from (9) that Ht

and St must be cointegrated with cointegrating vector (1; �1)′:One can write the cointegrated system in traditional error-correction form

by substituting wt = MHt + (Ht−1 + �0 + �1St−1) + �1a5 + �1vst and wt−1 =Ht−1 + �0 + �1St−1 into (14):

MHt + (Ht−1 + �0 + �1St−1) + �1a5 + �1vst

= �1(Ht−1 + �0 + �1St−1) +�1h0

1− �−12

+ �1xt

or

MHt = (�1 − 1)(Ht−1 + �0 + �1St−1) +�1a5(�1 − 1)1− �1�

+ �1vct + (�1 − 1)�1vst : (15)

Eqs. (5) and (15) constitute the vector error-correction representation for thecointegrated VAR for (St; Ht)′ with cointegrating relation (9). In terms ofthe original parameters of the structural model de0ned by the optimizationproblem in (1)–(3), the discount rate � is unidenti0ed on the basis of ob-servations of (St; Ht)′ and would have to be imposed a priori. The parametera1 is also unidenti0ed and would have to be normalized (say, a1 = 1): Thevalue of �1 is inferred from the cointegrating relation between St and Ht; andthe value of �1 is then identi0ed from the response of inventory investmentto the lagged cointegrating residual wt−1 or to the current shock to sales vst .The framework is overidenti0ed in that both the contemporaneous correlationbetween the VAR residuals in (5) and (15) and the response of MHt to thelagged error-correction term wt−1 are governed by the single parameter �1:Given �1 and �, the value of a2 can then be found from (13). Knowing �,a2, and �1, one can then infer a3 from (8).A surprising feature of this solution, which has been noted by Kashyap and

Wilcox (1993), West (1995), and Ramey and West (1999), is that the cointe-grating vector (1; �1)′ is not the value (1;−a3)′ that one might have expectedon the basis of the underlying cost function. Indeed, from (8) and (9),

Ht−1 − a3St =−MHt + wt − �0 − 1− ��a2

St (16)


which, since MHt and wt are I(0), must be I(1). As such, the variance ofHt−1 − a3St is O(t), meaning the probability that |Ht−1 − a3St | exceeds any0nite bound tends to unity as t tends to ∞. In other words, the marginalcost of managing inventories necessarily goes to in0nity under the traditionalformulation.This results, in general, from the fact that if Uct is stationary and produc-

tion Qt is I(1), then the marginal cost of production a1(Qt − Uct) must goto in0nity. Cost minimization calls for continually cutting back inventoriesrelative to the target value a4 + a3St so that the in0nite marginal bene0t ofputting another unit into inventory is equated with the in0nite marginal costof producing the good. Speci0cally, the marginal bene0t of adding anotherunit to inventory (which is the negative of the marginal cost of increasingHt) is given by −�a1a2Et(Ht − a4 − a3St+1). From (16) this marginal bene0tis asymptotically dominated by the term �a1a2(1− �)=(�a2)St = a1(1− �)St:Producing another unit Qt for inventory (and using the additional inven-tory, say, to produce one less unit next period Qt+1) has a marginal cost ofEta1[Qt − Uct − �(Qt+1 − Uc; t+1)], which is asymptotically dominated by theterm a1(1−�)St . The asymptotic marginal cost equals the asymptotic marginalbene0t. Hence, the cointegration coe2cient �1 results from the 0rm’s desireto drive inventory management costs to in0nity at the optimal rate given thatmarginal production costs are also going to in0nity. Of course, this optimalplan is also causing pro0ts to tend to −∞, and if shutting down the plant isan option, that is obviously superior to the solution implied by the 0rst-orderconditions.All of this would seem to be a most unappealing theoretical framework with

which to account for the trends and comovement in inventories, production,and sales. The problem, moreover, is rather fundamental to the above frame-work. Whenever there is a unit root in sales but none in marginal productioncosts Uct , then physical production costs must go to in0nity and inventorymanagement costs will also tend to ∞ under a cost-minimizing strategy. Onthe other hand, if one tries to 0x this by assuming that Uct has a unit root,if this is unrelated to the unit root in sales, then inventories and sales wouldno longer be cointegrated.

2.2. A more appealing cointegrated representation

The obvious solution, if one wants to account for cointegration betweeninventories and sales using the model (1)–(3), is to assume that Uct andSt both have unit roots, but that they are themselves cointegrated. Such anassumption might be defended on the grounds that it may be technologicaladvance (an upward trend in Uct) that generates the upward trend in sales;the appendix provides a general equilibrium example of how this would oc-cur. Consider, for example, the consequences if we set Uct = St + vct for vct


white noise but leave the other details exactly as in Section 2.1. In this case,Eqs. (6) and (7) would become

Et[(Qt − St − vct)− �(Qt+1 − St+1 − vc; t+1)

+�a2(Ht − a4 − a3St − a3a5)] = 0;

Et[MHt − vct − �MHt+1 + �a2(Ht − a4 − a3St − a3a5)] = 0: (17)

If we now de0ne

�̃0 ≡ −a4 − a3a5;

�̃1 ≡ −a3; (18)

w̃t =Ht + �̃0 + �̃1St; (19)

expression (17) can be written

Et(MHt − �MHt+1 + �a2w̃t − vct) = 0: (20)

Eq. (20) will be recognized as identical to (10). Hence, the same analysisused in Eqs. (12)–(14) can be used to establish that w̃t is stationary, meaningthat (Ht; St)′ is again cointegrated, though the cointegrating vector is now(1; �̃1)

′ = (1;−a3)′. Note that in this solution, the deviation of inventoriesfrom target and marginal production costs are both stationary.

2.3. On the consequences of using a di6erent representation

The assumption Uct = vct was seen in Section 2.1 to imply the vectorerror-correction model

St = St−1 + a5 + vst ; (21)

MHt = (�1 − 1)(Ht−1 + �0 + �1St−1) +�1a5(�1 − 1)1− �1�

+ �1vct + (�1 − 1)�1vst ; (22)

whereas the assumption Uct = St + vct implies the system

St = St−1 + a5 + vst ; (23)

MHt = (�1 − 1)(Ht−1 + �̃0 + �̃1St−1) +�̃1a5(�1 − 1)1− �1�

+ �1vct + (�1 − 1)�̃1vst : (24)


The observable implications of the 0rst model for data on {Ht; St} are com-pletely described by (21) and (22), whereas the observable implications ofthe second model are completely described by (23) and (24). One can thinkof maximum likelihood estimation as 0rst choosing parameters (a5; �1; �0; �1;�2vc; �

2vs) on the basis of the representation in (21) and (22) and then mapping

these back into the structural parameters (a5; a2; a4; a3; �2vc; �2vs) as described

at the end of Section 2.1. It is clear that representation (21) and (22) wouldimply the numerically identical value for the likelihood function as (23) and(24) and the numerically identical observed behavior for {Ht; St}: The keydi5erence is the interpretation given to the estimated cointegrating coe2-cient �1. In the second model, the cointegrating relation is interpreted asdirectly identifying the long-run target relation between inventories and sales,Ht = a4 + a3St , whereas in the 0rst model, the cointegrating relation is in-terpreted as the outcome of balancing in0nite inventory management costsagainst in0nite production costs.The two models thus predict exactly the same behavior for {Ht; St}, but

only the 0rst model predicts pro0ts tend to −∞ and has the unpalatableinterpretation of the long-run comovement between these two variables. Itwould seem clear that the second model is the better one to use for purposesof interpreting the dynamic behavior of sales and inventories, that is, oneshould use the structural parameter values that result from interpreting thecointegrating coe2cients as �̃0 and �̃1 rather than as �0 and �1:One objection that might be raised about the framework proposed here

is that the 0rm is asymptotically operating around the point Qt = Uct + kfor some constant k implied by the other parameters. It is possible for kto be su2ciently small that the 0rm is predicted often to be operating ina region with decreasing marginal cost. This, however, is simple to changeby replacing the assumption Uct = St + vct with Uct = St + a6 + vct . Thenew parameter a6 cannot be identi0ed separately from the constant a4 in theinventory management costs. However, if in the data Qt −St is stationary (aseither the Kashyap–Wilcox model or the one here imply), then there exists avalue of a6 for which marginal production costs are virtually always positive,and given this value for a6, a value for a4 can then be found for which themodel is consistent with the data. Indeed, it is precisely because our modelhas the feature that Qt−Uct is stationary that it is possible to interpret the dataas all falling in a reasonable region of the cost function. Hence, rather thana drawback of our framework, concerns about negative or in0nite marginalproduction costs are a key reason one might prefer to base inference on theassumptions proposed here.Note that the proposed solution only works if the cointegrating vector

for (St; Uct)′ is (1;−1)′. The general equilibrium example presented in theappendix shows why this, in general, would be expected to be thecase.


3. A generalization

The issues illustrated with the two examples above are in fact quite general.Consider the behavior of (4) when (MSt; Uct)′ follows an arbitrary vector I(0)process. To reproduce the argument in Kashyap and Wilcox (1993, p. 388)and West (1995, footnote 9), from the inventory identity (3), MQt =MSt +M2Ht , and if one conjectures that both MSt and MHt are I(0), then MQt isalso I(0) and the 0rst-order condition (4) implies that the following variablemust be I(0):

(Qt −Uct)− �(Qt+1 −Uc; t+1) + �a2(Ht − a4 − a3St+1)

= (MHt + St −Uct)− �(MHt+1 + St+1 −Uc; t+1)

+�a2(Ht − a4 − a3St+1): (25)

If Uct is any I(0) process, then (25) can only be I(0) if

�a2Ht + (1− � − �a2a3)St ∼ I(0);

that is, only if (Ht; St)′ are cointegrated with cointegrating vector (1; �1)′

de0ned in (8). Hence, as noted by these authors, this vector (1; �1)′ is thecointegrating vector when Uct is any stationary process, not just for Uct whitenoise as discussed in the previous section.On the other hand, if instead Uct is I(1) but St − Uct is any I(0) process,

then (25) is stationary provided that �a2(Ht − a3St+1) ∼ I(0). Hence, the re-sult that (1;−a3)′ is the cointegrating vector holds whenever St and Uct arecointegrated with cointegrating vector (1;−1)′. Thus, the approach recom-mended in Section 2, of replacing the assumption that Uct is stationary withthe assumption that St−Uct is stationary, is a general solution to the problemof interpreting cointegration between inventories and sales with a model thatmakes theoretical sense.

Acknowledgements

I am grateful to Valerie Ramey, Scott Schuh, and an anonymous refereefor helpful comments.

Appendix A. General equilibrium example

Consider a perfectly competitive representative 0rm with productionfunction

Qt = AtN�t ; 0¡�¡ 1 (A.1)


for Nt labor input (paid wage Wt) and At an exogenous technology shock.The 0rm’s output is the economy’s sole consumption good which is takento be the numeraire. Suppose that the 0rm pays a cost Xt for inventorymanagement,

Xt =�2(Ht−1 − !St)2

St; (A.2)

where to obtain results parallel to those for (1)–(3), we treat this as a costdirectly paid to households who supply inventory management services ac-cording to a fee schedule speci0ed by (A.2). 2 The 0rm’s decision problemis

max{Qt ;Ht ; St ; Nt}∞

t=0

E0

{ ∞∑t=0

�t(St −WtNt − Xt)

}(A.3)

subject to (A.1), (A.2), and (3). Let Mt denote the marginal cost of produc-tion:

Mt =WtNt

�Qt: (A.4)

The 0rst-order conditions are

1 =Mt − �2

(H 2t−1

S2t

− !2); (A.5)

Mt = �Et

[Mt+1 − �

(Ht

St+1− !

)]; (A.6)

Qt

St= 1+

(Ht−1

St

)(Ht −Ht−1

Ht−1

): (A.7)

To close the model, suppose that at ≡ logAt follows a random walk withdrift,

at = at−1 + g+ $t ; (A.8)

while population is constant:

Nt = N: (A.9)

Households use up some of the purchased good in producing inventory man-agement services, earning zero pro0ts in equilibrium. Hence household con-sumption is given by St−Xt . With no private storage or capital, the householdbudget constraint is

St =WtNt + Xt +%t (A.10)

2 If the 0rm does not make outside payments for inventory management services, then inven-tory management costs would need to appear in either the production function or the inventoryaccumulation identity. This would complicate the mapping between the general equilibriumexample and system (1)–(3).


for %t the household’s share of the pro0ts of the representative 0rm. Note that(A.9) is the condition for labor market equilibrium and output equilibrium isthen assured by Walras’ Law.Consider 0rst the case of deterministic growth (E($2t )=0). It follows from

(A.1)–(A.10) that along the steady-state growth path, Mt is constant whilethe log of each element of Zt = (Wt;Qt; Ht; St ; Xt ;%t; At)′ grows linearly atthe rate g.With stochastic growth (E($2t )¿ 0); from the standard log linearization

around the steady-state growth path we have the familiar result that eachelement of logZt is individually I(1) and any two elements of logZt arecointegrated with cointegrating vector (1;−1)′; see for example King et al.(1991).One could also use a linear as opposed to log-linear approximation to the

stochastic growth path, for example, treating Ht −Ht−1 rather than log(Ht)−log(Ht−1) as I(0). To do so, substitute (A.1) into (A.4),

Mt =1�

(Qt

Bt

)(1=�)−1

; (A.11)

where

Bt ≡ A1=(1−�)t W−�=(1−�)

t :

Note that Qt and Bt are both Op(At) along the stationary growth path. Takinga 0rst-order Taylor approximation to (A.11) around (Q0; B0)′ and rearranging,

Mt � M0 +(1�− 1

)M0

Q0[Qt − (Q0=B0)Bt]: (A.12)

Recall that along the stationary growth path, log(Qt) and log(Bt) are bothI(1) while log(Qt)− log(Bt) is stationary with mean log(Q0=B0). In the linearapproximation to this path, Qt and Bt are regarded as I(1), and must becointegrated with cointegrating vector (1;−(Q0=B0))′ given the stationarity ofMt: De0ning a1 ≡ [(1=�)− 1]M0=Q0 and Uct ≡ (Q0=B0)Bt; expression (A.12)can be written

Mt � M0 + a1(Qt −Uct): (A.13)

We can likewise approximate

�(Ht

St+1− !

)� k0 + a1a2(Ht − a3St+1); (A.14)

where k0 ≡ �[(H0=S1) − !]; a2 ≡ �=(S1a1), and a3 ≡ (H0=S1): Substituting(A.13) and (A.14) into (A.6),

a1(Qt −Uct) = �Et[a1(Qt+1 −Uc; t+1)− a1a2(Ht − a4 − a3St+1)] (A.15)

for a4 ≡ −(�a1a2)−1[�k0 + (1 − �)M0]: Eq. (A.15) will be recognized asidentical to (4) when a0 = 0: Hence, the solution to the linear–quadratic


optimization problem (1)–(3) can be motivated as a linear approximation tothe general equilibrium stochastic growth path implied by (A.3).Recall from (A.13) that Qt − Uct must be stationary in this equilibrium.

Subtracting Uct from (3) results in

Qt −Uct = St −Uct + (Ht −Ht−1):

Since Ht−Ht−1 is also regarded as stationary for purposes of this approxima-tion, it follows that (St; Uct)′ must be cointegrated with cointegrating vector(1;−1)′.Of course, if growth is actually exponential as the model implies (and

as most economic time series appear to exhibit), a log-linear approxima-tion would be superior to the linear approximation analyzed above. Viewing(A.11) as a function of (qt; bt)′ ≡ (log(Qt); log(Bt))′; the Taylor approxima-tion would be

Mt � M0

[1−

(1�− 1

)(q0 − b0)

]+M0

(1�− 1

)(qt − bt): (A.16)

Note that (qt; bt)′ must be cointegrated with cointegrating vector (1;−1)′:Similarly, approximating the left-hand side of (A.14) with a linear functionof (ht; st+1)′ ≡ (log(Ht); log(St+1))′ gives

�(Ht

St+1− !

)� �[(H0=S1)(1− h0 + s1)− !] + �(H0=S1)(ht − st+1):

(A.17)

Substituting (A.16) and (A.17) into (A.6) and dividing both sides byM0[(1=�)− 1] yields

qt − uct = �Et[(qt+1 − uc; t+1)− a∗2(ht − a∗4 − st+1)]; (A.18)

where uct ≡ bt: Eq. (A.18) will again be recognized as an expression of theform of (4) in which the variables Qt;Ht; and St have simply been replacedby their natural logarithms. Hence, if the goal is to 0t the model using onlythe Euler equation (4), one might want to use logarithms rather than levelsof the variables in order to better capture exponential as opposed to lineargrowth. Again, obviously (st ; uct)′ are cointegrated with cointegrating vector(1;−1)′:

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