A generalized variance bounds test with an application to the Holt et al. inventory model

JOURNALOF

ELSEVIER

Economic Dynamics & Contd

Journal of Economic Dynamics and Control 19 (1995) 59-89

A generalized variance bounds test with an application to the Holt et al. inventory model Tryphon Kollintzas Department

qf Economics,

Athens

University

qf Economies

and Business,

Athens

104 34, Greece

(Received February 1993; final version received August 1993)

Abstract This paper derives a variance bounds test for a broad class of linear rational expectations equilibrium models. According to this test, if observed data accord with the model, then a weighted sum of autocovariances of the covariance-stationary components of the endogenous state variables should be nonnegative. The new test reinterprets West’s (1986) variance bounds test and extends its applicability by not excluding unobservable exogenous state variables, nonstationary exogenous and endogenous state variables, and nonzero initial values for the endogenous state variables. The new test is computed together with West’s test for the Holt et al. inventory model using nondurable industry data. While West’s test almost always rejects the model, the latter almost always passes the new test. This and other evidence provide strong support for the importance of supply shocks in explaining aggregate inventory behavior via the Holt et al. model. Key ~;ordst Variance bounds test; Linear rational expectations tory behavior; Production smoothing; Supply shocks JEL class$cation: C12; C22; C52; C61; E22

model; Aggregate

inven-

Most of the research reported in this paper was done while I was a visiting scholar at the Research Department of the Federal Reserve Bank of Minneapolis. 1 would like to thank that institution and its staff for their support and hospitality. I have benefited from comments by Lawrence Christiano, Sophia Dimelis, Zvi Eckstein, Edward Green, Lars Hansen, Hugo Hopenhayn, Patrick Kehoe, Michael Magdalinos, Thomas Sargent, Kenneth West, and other conference participants at the Business Cycle, Inventory Fluctuations, and Monetary Policy at the Certosa di Pontignano (Siena), June 1992. In addition, 1 am grateful to Kenneth West for making available to me the data used in this paper and answering several clarifying questions on his computer work in West (1986). Finally, the financial support of the National Science Foundation is gratefully acknowledged.

0165-1889/95/$07.00 0 1995 Elsevier Science B.V. All rights reserved SSDI 0165 18899300775 Y

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1. Introduction

In a seminal paper West (1986) derived a variance bounds test for the Holt et al. inventory model. His idea was essentially the following: If production, sales, and inventories are consistent with the optimal policy of the underlying dynamic maximization problem, then the unconditional expectation of the difference between the value of the objective function under the optimal policy and its value under a policy with inventories identically zero should be nonnegative. Then he showed that if production, sales, and inventories are covariance-stationary processes, this difference is a weighted sum of variances and covariances of production, sales, and inventories. Thus, he tested whether the nonnegativity of this sum is satisfied by the covariance-stationary components of the data in the case of several nondurable industries at the two-digit SIC code level. He found that the test ‘almost always rejects the model, even though the model does well by traditional criteria’. Several other authors have used ‘West’s variance bounds test’ (WVBT) to obtain similar rejections of the Holt et al. inventory model on aggregate data sets.’ The purpose of this paper is to reinterpret and generalize West’s variance bounds test for a broad class of linear rational expectations equilibrium (LREE) models.2 It is shown that the unconditional expectation of the difference between the values of the objective function of the underlying dynamic problem under an optimal policy and under any feasible policy, such that the difference between the corresponding state paths is a covariance-stationary process, is a weighted sum of autocovariances of that process. Hence, this sum must be nonnegative for any covariance-stationary difference between an optimal state path and a feasible state path. Two implications of this result stand out. One is that, if observed data accord with the optimal policy (that is, the theory), then their covariance-stationary components should satisfy the above condition. For, as shown below, the nonstationary components of the endogenous state variables qualify as the state path associated with a feasible policy, this is the reinterpretation result. The other noteworthy implication is that, unlike West’s condition, the one derived here does not require observable exogenous state variables, covariance-stationary exogenous or endogenous state variables, or a zero initial value for the endogenous state variable. The generalization result is

‘See, e.g., Kahn

(1989), Krane

and Braun

(1991), and Dimelis (1991).

‘The LREE models are stochastic dynamic equilibrium models where objective functions are quadratic, transition equations of exogenous variables are linear, subjective discount factors are constant, expectations of exogenous state variables are replaced by their MMSE counterparts, and subjective expectations of endogenous state variables are generated by the equilibrium of the model. Examples of LREE models can be found in Aoki (1989), Lucas and Sargent (1981), Sargent (1989), and Hansen and Sargent (1991). LREE models are a special class of what is commonly referred to as Linear-Quadratic Stochastic Control models. The results of this paper should be applicable to this more general class of models.

T. Kollintzas /Journal of Economic Dynamics and Conrrol 19 (1995) 59-89

61

a consequence of the fact that the condition derived here exploits some of the other necessary conditions, while West’s does not. A dramatic illustration of this result is obtained when the test corresponding to the new condition is computed alongside the WVBT for the Holt et al. inventory model. That is, while the WVBT almost always rejects the model, confirming West (1986) this model almost always passes the new test. This provides for further strong evidence for the importance of unobserved supply shocks in explaining aggregate inventory behavior within the Holt et al. framework.3 Four sections follow. Section 2 sets up a general LREE model, reviews the standard necessary conditions for its solution, derives the new condition, and interprets it. Section 3 illustrates the economic importance of the new condition in the context of the Holt et al. inventory model. Section 4 contains the empirical illustration. And, the last section includes some concluding comments. All proofs are in the Appendix.

2. A general linear rational expectations

model

Let {t(t): t E N), N = (0, IfI 1, . . . }, be a stochastic process on a probability space (0, s2, P), where t(t) is an (n; x 1)-dimensional vector of exogenous state variables at the beginning of period t. Also, let 52, be the a-algebra generated by the sequence of random variables ( . . . , ((t - l), t(t)), t E N. The term R, represents the information available to the system at the beginning of period t. c Q, Vt E N. The term E(a) denotes the unconditional Clearly, Q, c Q,, 1 expectations operator with respect to P. That is, for any integrable function ( -) with respect to P, E( .) = JO( .)P(dw). The term E,( .) denotes the conditional expectations operator with respect to P, given Q. That is, for any integrable function ( a) with respect to P, such that E( e) < co, JA( -)P(dw) = iA E,( -)P(dw), V’A E R,. The {5,(t): t,~ IV} process takes values in W,. That is, the space of sequences 5 = (t(r), ((r + l), . . ), < = (t(r), ((T + l), .I. ), and so on, such that (El)

i /jl-’ , = .,

E{(t)‘&)

< x,

VTE N,

“The basic feature of this model is: (1) The production smoothing motive for holding inventories which results from the combination of convex production costs in the face of a stochastic demand. However, some of the major stylized facts of aggregate inventory behavior are (Blinder, 1986): (i) the variance of (detrended) production exceeds that of sales, (ii) the covariance between sales and inventory, investment is positive, (iii) the speed of adjustment in stock-adjustment inventory equations are ‘implausibly slow’. are not consistent with this motive. And, in the absence of: (II) The stockout-avoidance motive (convex stockout-avoidance costs that are proportional to expected sales) or nonnegativity-type constraints on inventories. (111) Sufficiently large cost shocks, these stylized facts, are not consistent with the model. See. Blinder and Maccini (1991) for a survey of the pertinent literature.

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where /I E (0, 1) is the discount factor in all periods. Also, let x(t) be an (n, x l)dimensional vector of endogenous state variables at the beginning of period t. Then, a variety of LREE models can be stated as a problem of the form:

subject to x(t + 1) be Q-measurable,

(2)

X(T) = x,

(3)

{x(t)),m_TE w,,

(4)

where h[5(r), x(r),x(t + 1) -

x(t)1=

(5) (Al)

N’ = N,

Q’ = Q,

s’ = s,

and ZX is the space of sequences ?(T + l), ,.. ), and so on, such that tzr /I-’ Ex(t’)’ g(t) < cc,

x = (x(r),

x(r+

l), . . . ), X = (x”(r),

tJT E N.

It should be noted that x(t + 1) Q-measurable, Vt E {T, T+ 1,...}. means that decisions at time t depend only on the history of the {t(t): t E N} process and X. Also, note that no a priori curvature restrictions have been imposed on h[<(t), x(t), x(t + 1) - x(t)]. Moreover, no explicit law of motion for the {t(t): t E IV} process has been postulated. Finally, condition (4), that {x(t)),“O=2E W,, can be relaxed. All that is necessary here is that /?(T-T)tZI/E,x(r)ll + 0 as T + co, VX E R”, where II-11 denotes the standard Euclidean norm. Clearly then, if {x(t + l)}gr is an optimal policy for this problem, x(t + 1) must be: arg max{h[5(t),x(t),x(t - x(t +

l)l),

+ 1) - x(t)] + BE,h[r(t

x(t + 1) &&-measurable.

+ l), x(t + l),x(t + 2) (6)

T. Kollintzas /Journal qf Economic Dynamics and Control 19 (1995 j 59

Therefore, if {x(t + l)}Er hold, Vt E(T, r+ 1, . ..}.”

is an optimal

policy, the following

89

conditions

63

must

Euler condition (S - R)E,x(t

+ 2) - [Q - (R + R’) + (1 + P-‘)S]x(t

+ 1)

+ p-‘(S - R)‘x(t) = 1 + (p-1 - 1)~ + (U - P)‘E,t(t Legendre

+ 1) + fi-‘P’S(~),

(7)

condition

Q - (R + R’) + (1 + pm ’ )S negative

semidefinite.

(8)

Condition (7) is the discrete time analogue of the Euler-Lagrange condition of continuous time deterministic models. Likewise, (8) is the discrete time analogue of the Legendre condition. Loosely speaking, (7) implies that at the beginning of any period t within the planning horizon the planner [i.e., whoever solves problem (lH4)] adjusts the state vector so as the associated discounted expected net benefits from this adjustment should be zero at the margin. Then, (8) requires that these net benefits be nonincreasing at the margin.

3. The new condition First, we need the following: Lemma 1. Ifs(t) = x(t)+ - x-(t), where {x’(t + l)}zl is an optimalpolicy and IS an y feasible policy, that is, satisfies (2H4), then the ,following (x-(1 + l)& condition must hold.

lim E i T+-C

p’-‘[

- 26(t + l)‘(S - R)6(t

+ 2)

r=T

+ d(t + l)‘(Q - R - R’ + (1 + J-‘)S)h(t

+ l)] I 0.

(9)

4For the general theory see Gihman and Skohorod (1979, Ch. l), Aoki (1981, Ch. 3), and Stockey and Lucas with Prescott (1989, Ch. 9). This formulation follows the last study. In particular, for the Euler Condition, see Stokey and Lucas with Prescott (1989, pp. 280-283).

T. KoNintzaslJournal of Economic Dynamics and Control 19 (1995) 59- 89

64

(The proof is in the Appendix.) Then, it is obvious that: Theorem. stationary

If s(t) is deBned process, then.

in Lemma

I and {s(t):

t E N} is a covariance-

(1 - /3-l E{ - 2&r + l)‘(S - R)&(r + 2) +C?(T+ l)‘[Q-@+K)+(l

+p-‘)S]G(r+

l)}lO,

VTEN.

(10)

Several comments are in order. First, condition (10) is indeed new. In particular, it is not the other second-order (Legendre) condition or implied by that condition. Actually, (1) has an interesting economic interpretation. From the proof of Lemma 1 (in the Appendix) it can be shown that

fB l+l-TET{

- 2&t + l)‘(S - R)6(t + 2)

+ i?(t + l)‘[Q - (R - R’) + (1 + B-‘)S]&t

+ l,}

is the conditional expectation of the net benefit associated with any deviation {d(t)}y”=. from the optimal plan in period t. Thus, (10) simply states that the expectation of the net benefit associated with any covariance-stationary deviation {d(t)}P)= T from the optimal plan, in any period, should be nonpositive. Second, what makes this result useful is that if observed data accord with the optimal policy, then their deviations from their nonstationary components should satisfy (10). For as it shown below these nonstationary components trivially satisfy all the requirements for a feasible solution. Or, if the maintained hypothesis is that the covariance-stationary components of the endogenous variables are generated by the optimal solution of the model and a zero feasible solution is meaningful (as in West, 1986) then again (10) should be satisfied by the covariance-stationary component of the endogenous state variables. Thus, (10) provides a natural and easily implementable test for the optimal solution, a test that does not require strong curvature restrictions or a specification of the law of motion of the exogenous state variables.5

5This is important for estimation purposes, because it implies that (10) can be tested by estimating the ‘structural’ parameters of the model without having to estimate the ‘expectations’ parameters of the model. Thus, (10) can be tested by estimating the Euler condition, (7), by GMM estimation procedures. This is the tack we follow in the empirical section.

T. Kollintzas /Journal of’ Economic

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It remains to show how to construct observable covariance-stationary {s(t): t E fW> processes and thereby check for (10). Here, I plan to illustrate this by comparing ‘open-loop’ and ‘closed-loop’ optimal policies. To do this, it is necessary to be more specific about the law of motion of the {r(t): t E fV} process. Let $0) = 4(r) - EC(r). And suppose

03)

that

$(t + 1) = A@(t) - E(C+ l),

where: (E3)

the eigenvalues

of A are of modulus

(E4)

E(r) - M(O a,

vt E N,

(E5)

Es(t)a(t’)

Vt # t’.

= 0,

less than one,

{e(t): t E N} is a covariance-stationary

Note that, by construction, Further, assume that

642)

Q - (R + R’) + (1 + b-‘)S

(A3)

det(S - R) # 0,

and let J, ( J2) be a Jordan

matrix

negative

process.

semidefinite,

with the eigenvalues

of

with modulus less (greater) than p- 1’2. Also, let Hi (H2) be the matrix with the eigenvectors and the generalized eigenvectors of 4(z) corresponding to Jr (J2), and add these regularity restrictions: 644)

{z E @: detl4(z)l

(A5)

rank (Hi) = n,,

= 0} n {z E C: i=

IzI = am”‘)

= 0,

1,2.

Restriction (A2) is a necessary condition. Restriction (A3) is a regularity condition. Restriction (A4) implies that the eigenvalues of 4(n) are not on the circle of radius p- ‘12. This, in turn, implies that the eigenvalues of 4(n) can be divided into two sets with the same number of elements. The modulus of the elements of the first set is less than p-“2 and the modulus of the second set is greater than

T. KoNintzas/.Joumal

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/I-“’ (i.e., the saddle-point property). This is a stability restriction. Restriction (A5) requires that the matrices that are composed with the eigenvectors and the generalized eigenvectors of 4(L) corresponding to the above partition of the eigenvalues of +(I.) are nonsingular. This is again a regularity restriction. These restrictions are weaker than the standard conditions imposed in stochastic linear-quadratic control problems: (i) the negative definiteness of S, (ii) the negative semidefiniteness of Q, (iii) stabilizability, and (iv) detectability of the underlying system dynamics.6 Then the unique solution to (7) subject to (3) and (4) is given by

x+(t + 1) = Klx+(t) + Ml&) + N(t),

X+(T) =

x,

(11)

where Ki = HiJiH;‘) -

M

=

K;‘(S

+ f

i = 1,2, - R)-‘p-‘p’

K;‘[(S

- R)-‘(U

- P)’ + K;‘(S

-

R)-'/?-'PIA',

i=1

+ K;‘(S

- R)-‘fl-‘P’E&)

+ f

+ K;‘(S

- R)-‘B-‘P’]E&

+j).

j= 1

K;j[(S

- R)-‘(U

- P)

Consider, now, the deterministic counterpart of problem (l)-(S). Under the restrictions mentioned earlier, any optimal policy for this problem should satisfy x-(t

+ 1) = K,x-(t)

+ N(t),

X_(T) = x.

(12)

6The solution method employed here is developed in Kollintzas (1986). It exploits the fact that the eigenvalues and eigenvectors of +(,I) form what is called a ‘complete pair of right divisors’ for +(A). The claims about the restrictions imposed are discussed in Kollintzas (1985) and Cassing and Kollintzas (1991). Also, they are shown to be true in the case of the Holt et al. inventory model in Kollintzas (1989).

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Obviously, {x- (t + l)},“=., is a feasible policy for problem (lH5). This policy, which is sometimes referred to as the open-loop policy, and (11) imply that x+(t + 1) = K’imTP1X + 1 K; [Mij(t - i) + N(t - i)],

(13)

i=O

c K;N(t-i).

l)=K;-T+lX+

x-(t+

(14)

i=O

Thus, s(t) = x+(t) - x-(t) = c K; MI& - i).

(15)

i=O

Hence, s(t) can be obtained as the finite sum of covariance-stationary processes and therefore it is itself covariance-stationary. This result can be easily extended to account for deterministic and moving average components in the law of motion of the {$(t): t E N} process. Finally, it should be noted that there are other classes of alternative policies that may be of interest for comparison purposes. The following section illustrates this for the ‘zero’ state policies.

4. Comparison with West’s variance bounds test In order to illustrate the economic importance of the new condition it is appropriate to compare it with the WVBT in the context of West’s version of the Holt et al. inventory model. One way to look at this model is to consider a firm that takes as given its sales of a single homogeneous good and seeks a production schedule that will minimize its expected discounted future stream of real costs:

WQ@))2=o~W - WI = E. f B’{aoCQ@)- QO - Ul’ + dQW12 t=o + Ui

E

[w,

azCff(t)

-

a,S(r

+

1)1’),

(16)

i = 0,1,2,3,

subject to Q(t) = S(t) + H(t) - H(t - l),

(17)

H( - 1) = 0,

(18)

68

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where E,, and B are as in Section 2, Q(t) is production in period t, H(t) is inventories at the end of period t, and S(t + 1) is the covariance-stationary component of sales in period t + 1. The term ao[Q(t) - Q(t - l)]’ represents adjustment costs brought about by changing production levels. The term al[Q(t)]’ represents production costs, and the term a2[H(t) - a3S(t + 1)12 represents inventory holding and backlog costs. In this model, firms hold inventories for two reasons: to smooth production in the face of randomly fluctuating sales and to avoid stockouts and/or sales backlogs. Neither the cost minimization hypothesis nor any particular market structure hypothesis is crucial here.7 4.1. Deriving West’s test To derive West’s variance bounds test, we hypothesize that the optimal production plan (Q*(t)}c,, and its associated inventory plan (H*(t)},“=, are zero-mean covariance-stationary processes. The production plan {Q”(t)}~O, where production is set equal to sales, is

Q'(t) = SW,

= {OJ, . ..}

VtEN+

(19)

so that no inventories are held: HO(t) = 0,

VtE N,.

Since that production WVBT=

plan is feasible, it must be true that

E(C[Q’(t),“=,,

= E

E, f

t=o

H( - l),O] - C[(Q*(t)}$o,

P’(uo[S(t) - S(t - l)]’

H( - l),Ol)

+ u1 [S(t)12

+ u2[ - usS(t + 1)12} - Eo f P’{ao[Q*(t) r=o + a, [Q*(t)]’

‘See, e.g., Blinder

and Mac&i

+ uz[H*(t)

(1991).

- asS(t +

1)12} >

- Q*(t - l)]’

T. Kollintzas /Journal

= fO

of Economic Dynamics and Control I9 i 199s) 59~ 89

P’{Q,{E(IW

-

W

-

69

1)l’ - ECQ*H- Q*O- N’s

+ ~I{ECW)I* - ECQ(Ol*)- uzECH*Wl* + 2a2a3E[H*(t)S(t = f.

8’{a0CWN -

- a,var(H*) = (1 - /?-’

var(AQ*)]

+ a, [var(S) - var(Q*)]

+ 2u2u3cov(H*,S+

{uO[var(dS)

- u2var(H*)

+ l)]}

- var(dQ*)]

,)} + a, [var(S) - var(Q*)]

+ 2~~u~cov(H*,S+~))

2 0,

(21)

where

VfEN+,

= ECQ*Wl’,

var(Q*) var(dQ*)

= E[Q*(t)

WQ*,Q*+l)

- Q*(t -

l)]*,

V’t E fV+,

= ECQ*@)Q*(t + 111, b”tc N+,

and so on. The first equality in (21) follows from (17H19). The second equality follows from the fact the EEo( s) = E(s). The third and fourth equalities in (21) follow from the assumed covariance-stationarity of {S(t)}tm,,, {Q*(t)}Zo, and {H*(t)}zo. Finally, the inequality in (21) follows, simply, from the fact that {Q”(t)}l%O is not an optimal plan for (16H18), but just a feasible plan for this problem. Thus, based on the hypothesis that the covariance-stationary component of observed production is part of the optimal plan {Q”(t)}130_0, West developed a simple statistical test for (21), which we will consider later. To compare the new condition to (21), we must map West’s model in the format of problem (l)-(5). Let:

4(t) = (S(t + l), S(t),w

N/2 =

0 I

-

I))‘,

x(t) = (H(t -

l),H(t

- 2))‘,

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-a2a3 P/2 = a0 + a,

-ao+az

a0

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and Control 19 (1995) 59- 89

aO+az

0

! I ! I -a0

R/2 =

qf Economic

0 ,

-a0

Q/2 =

0

0

s/2 =

0’

a0 + a, +a2

0

0

0

I

1.

Now the new condition (10) gives

or GVBT= (1 - /3-‘E([(5

+ P-‘)ao + (1 + p-‘)al

- 2(4ao + al)H(t)H(t

- 1) + 2aoH(t)H(t

+ p-‘a2] -2))

2 0.

[H(t)]’ (22)

Lemma 2 clarifies the relationship between (21) and (22). Lemma 2. Given H( - 2), H( - l), H- (0), H-(l), . . . = 0 and given West’s (1986) cost function (IO), (21) holds if and only if (22) holds. (The proof is in the Appendix).

Thus, (21) is equivalent to (22) when it is applied to West’s version of the Holt et al. inventory model.

4.2. Relaxing West’s assumptions

We turn now to juxtaposing (21) to (22), when some of West’s assumptions are relaxed.

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Case I. Unobserved exogenous state variables Suppose that production technology is subject production costs are given by

to a random

71

shock such that

r(t)Q(t) + alQ(t)', where {y( . )} is a covariance-stationary the econometrician. It is straightforward changed to (1 - b)-’ {a,[var(dS)

- var(dQ)]

process observed by the firm but not by to show that in this case (21) should be

- cov(y, AH) + a, [var(S) - var(Q)]

+ 2aza3cov(H*,Stl))

-a,var(H)

2 0.

However, since cov(y, AH) is not estimable, such problem arises with (22).

(23)

(23) cannot

Case 2. Nonstationary exogenous state variables Suppose that sales follow a nonstationary process s E (l,p-‘I”) such that E,S(t +j)

=

f?s’,

Vt,je

be evaluated.

Clearly, no

and there exist $ > 0 and

N,.

Clearly then E[S(t)]’ is not independent is violated. Nevertheless, (22) continues

oft, and hence the third equality to remain valid.

in (21)

Case 3. Nonstationary endogenous state variables Suppose that a, = a3 = 0, aI > 0, and - (B-i” - 1)ai < a2 I 0. Further suppose that sales are governed by S(t) -pJ(t - 1) = s,(t) for lpsl < 1 and {c,( .)> is a white noise process. Now, the Euler condition (7) reduces to E,H(t

+ 2) - (1 + b-’

+ a;‘a,)E,H(t)

+ BP’E,H(t

- 1)

= p-l E,S(t) - E,S(t + 1). The characteristic

polynomial

associated

(24) with (24) is

lb2- (1 + fi-’ + a11a2)12 + 8-l. The smallest modulus root of this polynomial is 2 E [l, fi-“‘]. are satisfied and the unique solution to the Euler condition is H(t) = ilH(t - 1) - (1 -&-‘L(l

-pp,)S(t).

But (A2)-(A5)

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Clearly, {H(t)},“=, is not covariance-stationary. violated. But again (22) remains valid. Case 4.

Thus, the third equality in (21) is

A zero initial value for the endogenous variable

Clearly, if H( - 1) i 0, then (20) should be replaced by

HO(t)= H( - l),

V’rEN+,

which violates the first equality in (21). These four cases are, of course, possible theoretical extensions of West’s inventory model in which (21) is no longer valid but (22) continues to be. In the next section we show that these extensions are not without empirical content.

5.

An empirical application

Partly to facilitate comparisons with West’s variance bounds test and since the ‘production smoothing’ debate is important in its own right, the Holt et al. inventory model was employed to implement the new test in practice. The data used for this purpose is an extension of the quarterly seasonally adjusted data for the six nondurable US manufacturing industries - apparel, chemicals, food, petroleum, rubber, and tobacco - studied by West (1986). The period covered is 1958.QI to 1984.QI. The procedure for computing the new test is very similar to that used for computing the WVBT. This is because both the WVBT and the GVBT are functions of two kinds of parameters. First, the deep structural parameters ao, al. a2, and as. And second, the parameters in an appropriate ordered vector autoregression for inventories and sales that can be used to compute the autocovariances of the covariance-stationary components of these two processes as well as the production process. Therefore, if the vector of all these parameters is 8 E 8, we have WVBT

= f(e),

GVBT

= g(O),

j?e-

R,

g: e -+ R,

f E c(e), g E c’(e),

wherefand g are defined by (21) and (22), respectively. Now,if 8 is a consistent and aysmptotically normal estimator of 8 with variance C, the WVBT and GVBT are asymptotically normally distributed with meansf(@ and g(b) and variances [V’(@] C [V’(J)]’ and [Vg(#)] C [Vg(e)]’ , respectively. Given 4, these partial derivatives can be estimated numerically.

73

T. Kollinlzas JJournal of Economic Dynamics and Control 19 (1995) 59- 89

5.1.

Estimation of ‘deep’ structural parameters

It is straightforward to show that the Euler and Legendre Conditions associated with the Holt et al. inventory model with unobservable cost shocks (i.e., the model of Section 4.2) are given by

E,laoh(r + 2) - [al + aoU + B)l& + 1) + aodt) +a2H(t)

- a,a3S(t + 1) + v(t)} + ‘linear time trend’ = 0

(25)

and a, + ao(l + 8) 2 0, respectively,

(26)

where

q(t) = PQW -QO - 11.’ There are several ways to estimate (25). In what follows we have estimated (25) by a Two-Step Two-Stage Squares (TSTSLS) procedure, similar to West (1986), and by a Generalized Method of Moments (GMM) procedure, similar to Eichenbaum (1989). The first procedure allows for nonstationary data but only for i.i.d. cost shocks, which enable us to compare the WVBT to the GVBT using the same estimation procedure. Note that there is no contradiction in the fact that the WVBT cannot be computed with any kind of unobservable cost shocks and the fact that West’s estimation procedure allows for i.i.d. cost shocks. It is precisely this happy coincidence that allows for the direct comparison between the WVBT and the GVBT.9 On the contrary, the second procedure allows for AR(l) cost shocks but does not allow for nonstationary data. The TSTSLS estimation procedure Given the arbitrary normalization q(t + 1) = ao[&(t

a, + a,,(1 + /3) = 1, (25) can be written aslo

+ 2) + q(t)] + a,H(t)

-aa,a3S(t

+ 1)

+ Us, + ‘linear time trend’

“Observant readers may have realized that these linear time terms are not consistent with West’s model and the derivation of his test. However, as West points out these terms can easily be taken into account in both the derivation of the Euler condition and the WVBT. ‘This

realization

has also been made in West and Wilcox (1992) and Dimelis and Ghali

“‘As noted by Krane parameters a2 and a,.

and

Braun

(1991) this normalization

may

be crucial

(1992).

for the sign of the

14

T. Kollinfzas/Journal

of Economic Dynamics and Control 19 (1995) 59-89

or Yl, =

XA + Ull?

(27)

where yl, = q(t + l), X, = (1, t,pq(t + 2) + q(t),H(t),S(t + l)), 6r is a vector of coefficients corresponding to X,, and u or = (YI, -X&I) -K(YI~ -Xtb) + y(t) is a disturbance term that follows an MA(2) law of motion. According to the TSTSLS estimation procedure br is estimated as

6, = (X’Z~_‘z’X)_‘X’Z~_‘Z’y,, where

T = sample size, ult = estimate of urt from a two-stage least squares estimation of (27), X’ = (Xi,

. . . )X’,),

Yl = (Yll,

‘..

z’ = (z;,

9

YlTY,

. . . zl,), )

z, = (1, t, H(t - l), H(t -2) H(t - 3) S(t - l), S(t - 2), s(t - 3)) is the vector of instruments in period t, and h- ’ is a consistent estimator of Hansen’s (1982) optimal weighting matrix that minimizes the asymptotic covariance matrix of bl . This matrix is not guaranteed to be positive definite in finite samples. For that matter, whenever 6 failed to be positive definite, we estimated it, following Cumby et al. (1983), as follows. First, note that the eight-dimensional vector process Ztult has an MA(2) representation of the form

where [, is an eight-dimensional vector white noise process with zero mean and variancecovariance matrix C,. Second, note that B(z) can be obtained by estimating a truncated B(z)- ‘. Here we followed Cumby et al. (1983) and

T. KollintzasiJournal

B(z)- l to be a second-order

we restricted as !5 = &l)

75

qf Economic D~namirs and Control 19 (1993-j S9- 89

polynomial.

Then, 6 can be estimated

c^
where b(z) = I + B^,z + @,z’ and c^[ stand for the estimated B(z) and C,, respectively. In only two out of the six industries d was estimated by this method. The varianceecovariance matrix of the data was estimated in two steps. First we estimated the vector autoregression of order three: (28)

Y2t = Ztb2 + u2t> y2t = W(t), s(t)), b2 is an appropriately cients, and u2t is an appropriately dimensioned with variance-covariance matrix where

c2

dimensioned vector of coeffivector of white noise processes

O22O.23.

=

[

023 c33 1

Estimates of b2 and C2 where obtained via a FIML estimation procedure. Second, the Yule-Walker equations were used to obtain estimates of the covariance and cross-covariances needed in (21) and (22). Hansen’s (1982) asymptotic variance-covariance matrix of 0, CO, was estimated as

c,= [W,(6)] - l i=i

h,(6)h,(e,l[v&(e)- ‘1, -2

where

is the vector of all the orthogonality conditions respect to the Z, vector of instruments.

imposed

by the theory

with

The GMM estimation procedure In order to estimate (25) allowing for persistence in the unobservable, to the econometrician, cost shocks, one must take a specific stand on the law of motion

76

T. Kollintzas/Journal

of Economic Dynamics and Control 19 (1995) 59- 89

of these shocks. Assuming that {y(s)} follows an AR(l) law of motion of the form

I4 + 1)= PY@)+

Et+19

IPI < 17

where {E,: t E N} is a white noise process, and abstracting from linear trend, (25) yields

WY,, -

g,b;) -

PLI(YI~-I

- %I&, = ~1,

(29)

where x”, = (l,fiq(t + 2) + q(t),H(t),S(t + 1)) and b; is a vector of coefficients corresponding to 2,. It follows by taking expectations conditional on 52, 1 that (29) gives the testable form Yl, -

PYl,-1

-

Ml

+ &lb;

= u”l,,

(30)

where tii, is orthogonal to any random variable contained in Q,_ 1. Relation (30) induces the set of unconditional orthogonality condition E(&o

filt) = 0,

(31)

where ft = (l,H(t

- l), H(t - 2),H(r

- 3),S(t

-

l),S(t - 2),S(t - 3)).

In this case (p, b;) can be consistently estimated by minimizing the expression (32)

where fi is again a truncated estimator of Hansen’s (1982) optimal weighting matrix that minimizes the asymptotic covariance matrix of (p, b;).’ ’ The basic difference between (30) and (27) is, of course, that (30) is no longer exactly identified so that we can no longer use TSTSLS. For this reason the general GMM method was employed to minimize (32). Three alternative detrending procedures were used: 1) first differences, 2) the Hodrick-Prescott filter, 3) second-order trend polynomial.

11See Eichenbaum

(1989) for details.

II

T. Kollintzas JJournal of Economic Dynamics and Control 19 (199.5) 59% 89

In all cases the detrended sales and inventory data were used to construct the output, QCti,and the weighted output difference, q(t) = /3Q(t) - Q(t - I), in a manner consistent with the inventory transition constraint (17). The variances and cross-covariances needed for the WVBT and the GVBT were computed directly from the detrended data. Finally, the varianceecovariance of the estimated parameters was obtained directly from the GMM estimation procedure. 5.2.

Estimation

results

Tables 1-6 present the estimates of the structural parameters and some of the variances and covariances used in the WVBT and the GVBT. The second column of these tables reports the estimates from the TSTSLS. The next three columns report the estimates from the GMM. In particular, columns 3, 4, and 5 correspond to procedure first differencing, HP detrending, and quadratic

Table 1 Apparel (standard

errors in parentheses)

Estimation method and shocks

TSTSLS i.i.d.shocks

GMM AR(l) shocks

Data

Levels

First differences

HP filter

UC1

0.173 (0.127)

- 0.131 (0.915)

~ 0.0228 (0.2 15)

- 0.176 (0.258)

01

0.655 (0.253)

1.262 (1.825)

1.455 (0.430)

I.351 10.516)

u2

0.186 (0.022)

0.544 (2.300)

~ 0.444 (1.114)

- 0.048 (0.148)

u3

3.022 (3.099)

I.760 (2.875)

- 2.490 (4.134)

~ 5.674 (19.213)

- 0.104 (0.858)

0.0444 (0.413)

0.863 (0.520)

/’

Quadratic

w(Q)

0.023

I.031

I .OY4

1.708

var(S)

0.025

0.532

0.82Y

1.387

cov(S,dH)

0.000

0.038

0.019

0.045

Jr

1.95

4.44

0.55

0.17

0.51 (0.28)

0.06 (0.05)

WVBT

- 2.45 (1.11)

GVBT

2.91 (0.80)

0.10 (0.15)

trend

78

Table 2 Chemicals

T. KollintzaslJournal

(standard

of Economic Dynamics and Control 19 (1995) 59- 89

errors in parentheses)

Estimation method and shocks

TSTSLS i.i.d.shocks

GMM AR(I) shocks

Data

Levels

First differences

HP filter

0.06 1 (0.098)

- 0.510 (0.105)

0.381 (0.167)

- 0.425 (0.100)

0.878 (0.196)

2.017 (0.209)

1.750 (0.334)

1.847 (0.200)

0.017 (0.013)

0.672 (0.549)

- 0.249 (0.394)

1.176 (0.264)

2.420 (1.965)

1.091 (0.673)

- 2.552 (4.253)

1.659 (0.95 I)

0.114 (0.157)

0.599 (0.489)

0.753 (0.244) 7.580

01

03

P

WQ)

0.096

1.144

2.226

var(S)

0.095

0.848

1.946

cov(S, dff)

- 0.001

JT

11.76

WVBT

- 0.02 (2.84)

GVBT

6.45 (2.43)

Table 3 Food (standard

- 0.176 7.03

- 0.06 (0.18)

3.60

- 0.06 (0.53)

TSTSLS i.i.d.shocks

GMM AR(l) shocks

Data

Levels

First differences

7.421 - 0.114 1.03

1.53 (1.64)

HP filter

Quadratic

0.209 (0.160)

- 0.310 (0.098)

1.108 (0.153)

- 0.307 (0.025)

1.416 (0.320)

1.618 (0.077)

- 1.210 (0.305)

1.612 (0.049)

0.069 (0.320)

- 0.368 (0.279)

- 0.305 (0.071)

- 0.107 (0.129)

4.79 1 (1.947)

3.520 (2.534)

4.775 (0.354)

9.112 (8.976)

- 0.048 (0.102)

- 0.151 (0.064)

0.852 (0.012)

P

MQ)

0.053

3.985

3.386

4.995

var(S)

0.046

2.683

2.605

4.286

cov(S, dff)

- 0.001

- 0.213

- 0.063

- 0.118

34.16

23.55

6.53

- 0.84 (0.26)

- 0.20 (0.06)

0.21 (1.03)

JT

trend

errors in parentheses)

Estimation method and shocks

a,

- 0.038

Quadratic

3.45

WVBT

8.47 (10.21)

GVBT

13.97 (7.69)

trend

T. Kollintzas/Journal

Table 4 Petroleum

(standard

of Economic Dynamics and Control 19 (199s)

TSTSLS i.i.d.shocks

GMM AR( 1) shocks

Data

Levels

First differences

- 0.084

- 0.230

HP filter - 0.338

Quadratic

(0.349) 1.460 (0.697)

(0.364) 1.675 (0.725)

(0.440) 0.568 (0.878)

a2

0.039 (0.041)

- 3.692 (6.135)

- 0.798 (3.611)

0.010 (0.0541

a.?

0.096 (1.259)

0.373 (0.441)

0.072 (1.007)

3.124 (1 X.532)

~ 0.230 (0.260)

0.938 (0.080)

0.924 (0.035)

P var(Q)

0.035

0.508

0.575

2.550

var(S)

0.034

0.332

0.478

2.446

- 0.000

- 0.044

- 0.016

~ 0.014

cov(S,dH)

Jl WVBT GVBT

Table 5 Rubber (standard

6.85

trend

0.216

(0.050) 1.167 (0.101)

a,

79

errors in parentheses)

Estimation method and shocks

a0

59- 89

0.50

1.85

0.40

0.29 (0.73)

0.15 (1.27)

0.14 (0.04)

- 1.12 (0.45) 1.67 (0.47)

errors in parentheses)

Estimation method and shocks

TSTSLS i.i.d.shocks

GMM AR(l) shocks

Data

Levels

First differences

HP filter

Quadratic

a0

0.068 (0.127)

0.171 (0.547)

0.361 (0.325)

- 0.300 (0.375)

01

1.135 (0.253)

0.656 (1.091)

0.279 (0.648)

1.599 (0.749)

a2

0.019 (0.03 1)

- 0.846 (1.519)

~ 0.014 (0.134)

0.05 I (0.281)

a3

1.934 (2.995)

1.067 (1.262)

- 5.802 (44.499)

3.048 (11.671)

0.369 (0.246)

- 0.321 (0.096)

P

-

0.569

(1.042)

var(Q)

0.100

0.369

0.553

var(S)

0.099

0.236

0.457

2.021

cov(S, AH)

O.ooO

0.002

0.012

0.032

JT

3.70

0.09

0.07

0.02

WVBT

3.13 (7.84)

GVBT

2.91 (5.19)

-

- 0.23 (0.15)

- 0.07 (0.03)

2.160

0.16 (0.39)

trend

80

T. KollinrzaslJournal

Table 6 Tobacco (standard

of Economic Dynamics and Control I9 (1995) 59-89

errors in !xwentheses)

Estimation method and shocks

TSTSLS i.i.d.shocks

GMM AR(l) shocks

Data

LW&

First differences

HP filter

Quadratic

a0

-0.176 (0.115)

- 0.386 (0.503)

~ 0.205 (1.198)

- 0.193 (2.560)

aI

0.648 (0.349)

1.771 (1.00)

1.408 (2.391)

1.385 (5.107)

a2

- 0.072 (0.145)

- 1.104 (9.024)

0.526 (18.105)

0.040 (1.128)

a3

2.597 (3.754)

1.187 (10.936)

- 2.860 (93.292)

2.534 (49.780)

- 0.201 (3.971)

0.390 (4.412)

0.120 (3.391)

P

var(Q)

0.0018

0.22

0.11

0.16

var(S)

0.0012

0.10

0.05

0.10

cov(S, AH)

O.OQCKl

- 0.001

- 0.006

JT

28.79

WVBT

- 0.65 (0.22)

GVBT

0.55 (0.26)

- 0.008 0.19

-0.01 (0.60)

0.07

0.15

0.11 (1.70)

0.07 (0.14)

trend

detrending of the data, respectively. In the last three rows we report the Jr, WVBT, and GVBT statistics, respectively. The TSTSLS results confirm basically those obtained by West (1986). Thus, most structural parameters have the ‘correct’ signs and several of them are significant at the 0.05 level. Adjustment costs seem to be concave (i.e., a0 < 0) rather than convex in three out of six industries, giving rise to an incentive for rapid production changes. Production costs seem to be convex (i.e., al > 0) in all industries, giving rise to an incentive to smooth production. Combined inventory holding and backlog costs seem to be convex (i.e., a2 > 0) in five out of six industries and the target level of inventories is a positive function of backlog sales (i.e., a3 > 0) in all industries, giving rise to an incentive for stockout avoidance. The second-order necessary condition is automatically satisfied and via the normalization, a, + a,(1 - b) = 1, implies that a0 and/or al must be positive. The sign distribution of these parameters, with the unique exception of the negative a2 in the tobacco industry, is exactly the same as in West. The a,

T. Kollintzas/Journal

of Economic Dynamics and Control 19 (1995) 59- 89

81

and especially a3 are rarely significant at the 0.05 level. l2 But these parameters are of the same absolute order and of the same order to each other as in West and other studies. The variancecovariance estimates of the inventories, sales, and production series are summarized in rows 8, 9, and 10. These estimates are also similar to those obtained by West. Thus, in all six industries the variance of production and production changes is greater than the variance of sales and sales changes, respectively. Finally, it should be noted that inventory changes covary negatively with sales in four industries. We will return to this finding later. The Hansen (1982) test of the overidentifying restrictions, Jr, is distributed as x2 with three degrees of freedom. This corresponds to critical levels 7.81 at 0.05, 11.34 at 0.01, and 12.84 at 0.005. It follows that the model is rejected only for the case of tobacco at the 0.005 level. The model is rejected only for the case of chemicals at the 0.05 level, but is accepted at the 0.005 level. The model is accepted at the 0.01 level in all remaining cases, With the unique exception of the tobacco industry, these results parallel West’s Thus, we follow West and interpret these results as ‘supportive of the model’. In general, before we turn to the tests of the model it seems reasonable to conclude that the results of this study, so far, confirm those of West (1986). It is, therefore, a striking result of this analysis the contrast between the last two rows of Tables l-6, where the WVBT is computed alongside the WVBT. The WVBT is negative in four cases and significantly so at the 0.05 level in three of them (apparel, petroleum, and tobacco). In the two cases that the WVBT is positive it is insignificantly so at the 0.05 level. These results also confirm West?. The GVBT, on the other hand, is positive in all six cases and significantly so at the 0.05 level in four of them (apparel, chemicals, petroleum, and tobacco). The GVBT is also significantly positive at the 0.10 level in the case of the food industry. These results not only reverse West’s inference that the Holt et al. inventory model is inconsistent with the data, but they also point to the reason why he obtained his result. That is, because the WVBT excludes stochastic supply shocks. A direct indication that in the presence of cost shocks the GVBT will tend to be a greater real number than the WVBT is the fact that in (23) cov(y, AH) will tend to be negative, if inventory changes move in the opposite direction of marginal production costs. This is to say that if one could have measured cov(y, AH), the WVBT in (23) would have been a greater real number than the one computed by West in (21). It should be pointed out again, that, the TSTSLS estimation procedure is consistent with i.i.d. cost shocks. This happy coincidence enables us to compare the WVBT to the GVBT using the same estimation procedure. The other reasons that restrict the application of the WVBT ~ growing endogenous and exogenous variables and a ‘zero’ feasible alternative - do not “This has been noted in West and Wilcox (1992) who provide Monte estimators of several of these parameters have even wider dispersion asymptotic theory.

Carlo evidence that the than that predicted by

82

T. KoNintzaslJournal of Economic Dynamics and Control I9 (1995) 59- 89

seem to be a factor in these data sets and estimation methodologies. Further, the production cost or supply shocks can also account for the variancecovariance estimates of the tables. Thus, we may conclude that the application of the GVBT to the Holt et al. inventory model suggests that earlier rejections of this model based on variance bounds tests were casting this model in restrictive environments which are prejudiced against its acceptance. Further evidence for the last conclusion is provided from the model that allows for persistence in these shocks. There are few cases where the model is rejected under the GMM estimation procedure. The Jr statistic is, now, asymptotically distributed as x2 with two degrees of freedom. This corresponds to critical levels of 5.99,9.21, and 10.60 at the 0.05,0.01, and 0.005 levels of significance, respectively. Confirming Eichenbaum (1989), we find that marginal adjustment costs are declining in most cases (a0 is negative in 13 out of 18 cases) and marginal production costs are increasing in most cases (ai is positive in 17 out of 18 cases). Inventory holding/stockout avoidance cost parameters are mostly insignificant (a2 and a3 are insignificant in 30 out of 36 cases). p is insignificant in the case of first differencing and HP detrending, indicating overdifferencing. On the other hand, confirming Eichenbaum (1989), this parameter is positive and highly significant in all cases with quadratic detrending.13 The WVBT cannot, of course, be computed for this class of models. The GVBT turns out to be positive two, three, and six times out of six times for first differencing, HP detrending, and quadratic detrending, respectively. This is consistent both with the fact that p seems to be positive and statistically significant much more for quadratic detrending than in the other two cases, as one thinks of cost shocks as been rather permanent in nature. I4 Moreover, this result is also consistent with the relative ratios of the variances of sales and output. That is, these ratios are much higher for first differencing and HP detrending than for quadratic detrending. In the spirit of Eichenbaum (1984, 1989) and West (1990), we interpret the GVBT results as providing further evidence for the importance of cost smoothing as a force behind aggregate inventory behavior. 6. Concluding remarks This paper derives a new necessary condition for a broad class of linear rational expectations equilibrium models. According to this condition the weighted average of autocovariances of any covariance-stationary difference ‘sEichenbaum (1989) estimates the parameters of the full solution rather than parameters of the Euler condition. This is due to the fact that he is interested in the coefficient in the stock adjustment law of motion of inventories, which is a function of the smallest root of the characteristic polynomial associated with the Euler equation, estimated here. “See,

e.g., Prescott

(1986).

T. KollintzaslJournal

of Economic Dynamics and Control I9 (1995) 59% 89

83

between an optimal state path and any feasible state path should be nonnegative. This induces a variance bounds - Generalized Variance Bounds Test (GVBT) - test for the covariance-stationary components of the data that generalizes and reinterprets West’s (1986) Variance Bounds Test (WVBT). Most importantly, the GVBT allows for nonstationary endogenous and exogenous state variables and unobserved exogenous state variables. Illustration of the GVBT in the cast of the Holt et al. inventory model, using nondurable data aggregated to the two-digit SIC code level, gave a striking result. That is, while the WVBT almost always rejects this model, the GVBT almost always does not reject this model. This turns out to provide strong evidence for the importance of cost shocks in explaining aggregate inventory behavior. Further evidence for this finding is provided when the GVBT is applied to environments with AR(l) cost shocks where the WVBT cannot be used. In future research it seems natural to see whether the GVBT can be extended to nonlinear rational expectations equilibrium models. Although one can easily determine the necessary conditions for this extension - the constancy of the Hessian of the objective function evaluated at some steady state - these conditions seem to be difficult to satisfy. Important such cases are growth models with a steady state. Finally, direct evidence of the power of the GVBT was not provided. This could, in principle, be done by Monte Carlo methods using a misspecified alternative version of the Holt et al. inventory model.

Appendix Proqf qf Lemma

I

Since hi’ is quadratic,

Taylor’s

theorem

implies

that

T =

E,tzrB”

-

I’x:h+(‘[x+(t)

f[xf(t) - x_(t)]’

- +[u+(t) - u-(t)]’ -

- x-(t)]

+ IQ++I’[v+(t)

- v-(t)]

vx”h+I’[u+(t) - u_(t)] lQl+I’[x+(t)

- x_(t)]

j[u+(t) - u_(t)]’ ~,.“h+I’[u+(t) - o-(t)]

I )

(A.11

84

T. KoNintza.7 /Journal

qf Economic

Dynamics

and Control 19 (1995) 59- 89

where V’;h+ 1’ stands for the gradient of h with respect to x evaluated at h[C(t), x’(t), u’(t)], VX:,h+1’ stands for the Hessian of h with respect to x evaluated at h[<(t), x!(t), u’(t)], and so on. Since u(t) = x(t + 1) - x(t) and d(t) = x’(t) - x-(t), (A.l) gives

[V’h’l’

A; = E, ; fl’-’ f=l

-

Vuh+ I’]cS(t) + K>h+l’6(t + 1)

- )d(t)‘[

V,xh+ If -

V,,h+ If -

- fd(t)‘[

Vx:,h+ 1’ -

VL,‘,,h+ l’]b(t + 1)

- @(t - l)‘[ VL.,h+ 1’ -

- f&t

Vp:.,h+1’ + VL,‘,,h+ 1*-j&t)

VL,;.,h+ [‘l&t + 1)

+ 1)’ [ VL.>,h+ 1’]6(t + 1) .

(A.3

Since both {x+(t + l)},“=, and (x-(t + l)};“=, are assumed 6(r) = 0. Then, a change of time indexes produces this:

A;=

E,. i

bf+l-’

[Vxh+l’+’

-

Vph+l’“]d(t

to be feasible,

+ 1)

,=I

-i&t

-

+ l)‘[Vxxh+~‘+l

-

Vxuh+I’+’

Vuxh+l’+’ + V~‘,,h+~“‘]i5(t + 1)

- s(t + l)‘[ Vx;,h+ l’+l -

K.h+ I’+‘lW

+ 2)

+ i B’-’ [ K>h+I’b(t + 1) - :d(t + 1)’ Vu.,h+ I’s(t + l)] *=,

T. KoNinrzaslJournal

of Economic Dynamics and Control 19 (1995) 59- 89

85

T-l =

E,

j3 1 /Y’{[fl-’ f=T

VL,,h+l’+ VX;h+If+’ -

+pT{vJl+lTG(T+

1)-f&7-+

V0.h+l’]8(t + 1))

l)‘K,.,hlT&T+

1))

T-l

- @

+

c /I-‘{ *=T

rqt+ l)‘[

+ (1 + B-l)

or, in explicit

- 2&t + l)‘[vX;,h(‘+’

vXXhl’+’ -

vXLJll’+’ -

-

v”Jl1’“]s(t

+ 2)

vJIl’+’

(A.3)

+ 1,;. j ,

vJll’“]s(t

notation, T- 1 /I 1 /TT[(S-R)x+(t+2) f =T

dT=ET

-(Q - R - R’ + S + ,K’S)x+(t + fi-‘(S

- R)‘x+(t) - I-

- p- ‘p’S(t)]‘s(t

+ 1)

(fi-’ - 1)m - (U - P)‘[(t + 1)

+ 1) 1

+

j?T-‘{[Sx+(T+

- fi3(7-+ l)‘S6(T+

1) + (R - S)‘x+(T)

+ m + P’E(T)]‘G(T+

1)

1))

T-l -

&p

,;T

at-‘{

-

26(t + l)‘(S - R)d(t + 2)

+ d(t + l)‘(Q - R - R’ + S + P-‘S)&t

+ 1)).

(A.4)

VtE{T,T+ 1, . . }. [See, Now, 0, = fit+ I, V~EN, so E,(s)= E,[$(.)], for example, Billingsley (1976, Theorem 34.4).] Since {x+(t + l)}Er and {x-(t + l)}ZT are assumed to be S2,-measurable, (12) implies that s(t + 1) is

T. Kollintzas/Journal

86

of Economic Dynamics and Control 19 (1995) 59- 89

!&measurable. Therefore, E&t + 1) = s(t + 1) and E,[( .)s(t + l)] = [E,( .)]s(t + 1). [See, for example, Billing&y (1986, Theorem 34.3).] These facts imply that - R - R’ + S + /I-‘S)x+(t

ET[(S - R)x+(t + 2) -(Q

+ 1)

+ B-‘(S - R)‘x+(t) - 1 - (p-r - 1)m - (U - P)‘@ + 1) -

p-‘P’((t)]‘s(t

+ 1)

= ET{ [(S - R)E,x+(t

+ 2) -(Q

- R - R’ + S + fi-‘S)E,x+(t

+ 1)

+ /I-‘(S - R)E,x(t) - 1 - (p-’ - 1)~ - (U - P)‘E,t(t + 1) - /I-‘P’E,<(t)]‘6(t

+ l)}.

(A.5)

Therefore, since {x+(t + l)},oo’Tis assumed to be an optimal policy, (17) implies that the first term on the right side of (A.4) is zero. Furthermore, since ({t(t)}: T, {x+(t + l)}Zr), ({t(t)}:,, {x-(t + l)},“=.) E WCx U-U,,each of the bilinear and quadratic forms of the second term on the right side of (19) goes to zero as T + CO. Therefore, T-l

lim A: = - 38 Ji:

E, c

+

T-W

- 26(t + l)‘(S - R)h(t + 2)

p’-‘[

f=T

+ d(t + l)‘(Q - R - R’ + S + fi-‘S)d(t

+ l)].

(A.6)

Now, since E( .) = E [E,( .)I, Vt E N, it follows immediately that if {x’(t + l)}zT is an optimal policy and {x-(t + l)}zT is any feasible policy, then (20) must hold. Q.E.D. Proof of Lemma 2

Since

{Q(t), S(t), H(t):

t E N>

is a covariance-stationary

process

EL-E,(-)1= V-1, @ = (1 - B)-’ ECU,,{[S(t) - S(t - 1)12 - [Q(t) - Q(t - 1)12}

+

a~

([s(t)]’

-

[QW12}

-

a2

[H(t)12 + %%H(t)S(t + 111

and

T. Kollintzas

=

EEo

5

t=o

JJournal

qf Economic Dynamics

BfCao(L-W

-

W

-

and Control

19 (1995)

59- 89

87

1)12- [Q(t) - Q@ - U12}

+ a,{CW12 - CQ(W) - a2CffW12 + 2a2eH(f)W

+ 111

S(t + 1)

S(t) =EEo

fp t=o

S(t -

1)

H(t -

1)

H(t - 2) H(t) - H(t _ H(t -

a24

1)

1) - H(t - 2)

1

0

0

- a24

0

- a2a3

0

a, + a,

- uo

- a0

a0

a0 + a1

0

a0

a0

- a0

- a0

0

a0 + cl2

- a0

- a0 + u2

0

a0

a0

0

uo+u,+u2

0 0

88

T. KoNintzaslJournal

of Economic Dynamics and Control 19 (1995) 59- 89

64.7)

On the other hand, it follows, as in the proof of Lemma X = - $(l - fl)-‘E{[(5 - 2(4a,, + al)H(t)H(t

+ fi-‘)a0

+ (1 + j?-‘)a,

- 1) + 2a,,H(t)H(t

1, that + fl-‘aJCH(t)‘]

- 2))

Thus, if H( - 2) = H( - 1) = H-(t) = H- (1) = 0, then B E (0, l), (21) holds if and only if (22) holds. Q.E.D.

C$= X. Hence,

since

References Aoki, Masanao, 1989, Optimization of stochastic systems (Academic Press, San Diego, CA). Billingsley, Patrick, 1986, Probability and measure (Wiley, New York, NY). Blinder, Alan S., 1986, Can the production smoothing model of inventory be saved?, Quarterly Journal of Economics 101, 431454. Blinder, Alan S. and Louis J. Maccini, Taking stocks: A critical assessment of research on inventories, Journal of Economic Perspectives 5, 73-96. Cassing, Shirley and Tryphon Kollintzas, 1991, Recursive dynamic interrelations and endogenous cycling, Internal Economic Review 32, 417440. Dimelis, Sophia P., 1992, A test of the linear rational expectations equilibrium inventory model in the EEC, International Journal of Production Economics 26, 25532. Dimelis, Sophia P. and Moheb A. Ghali, 1992, Classical and variance bounds tests of production and inventory behavior, Paper presented at the Business Cycle, Inventory Fluctuations and Monetary Policy Meeting, University of Siena, June 15-18. Dimelis, Sophia P. and Tryphon Kollintzas, 1989, A linear rational expectations equilibrium model of the American petroleum industry, in: Tryphon Kolhntzas, ed., Rational expectations equilibrium inventory model: Theory and applications (Springer-Verlag, New York, NY).

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89

Eichenbaum, Martin S., 1984, Rational expectations and the smoothing properties of inventories of finished goods, Journal of Monetary Economics 14, 71-96. Eichenbaum, Martin S., 1989, Some empirical evidence on the production level and production cost smoothing models of inventory investment, American Economic Review 79, 853-864. Eichenbaum, Martin S., Lars P. Hansen, and Kenneth J. Singleton, 1988, A time series analysis of representative agent models of consumption and leisure choice under uncertainty, Quarterly Journal of Economics 103, 51-58. Gihman, losif 1. and Anatoli V. Skorohod, 1979, Controlled stochastic processes (Springer-Verlag. New York, NY). Hansen, Lars P., 1982, Large sample properties of generalized methods of moments estimators, Econometrica 50, 1029-1053. Hansen, Lars P. and Thomas J. Sargent, 1991, Rational expectations econometrics (Westview Press. Boulder, CA). Holt, Charles C., Franc0 Modigliani, John F. Muth, and Herbert A. Simon, 1960, Planning production, inventories and work force (Prentice-Hall, Englewood Cliffs, NJ). Kahn, James A., 1989, The seasonal and cyclical behavior of inventories. Mimeo. (Department of Economics, University of Rochester, Rochester, NY). Kollintzas, Tryphon, 1985, The symmetric linear rational expectations model, Econometrica 53. 963-976. Kollintzas, Tryphon, 1986, A nonrecursive solution for the linear rational expectations model, Journal of Economic Dynamics and Control IO, 327-332. Kollintzas, Tryphon, 1989, The linear rational expectations equilibrium inventory model: An introduction, in: Tryphon Kollintzas, ed., Rational expectations equilibrium inventory model: Theory and applications (Springer-Verlag, New York, NY). Krane, Spencer and S. Braun, 1991, Production smoothing evidence from physical product data. Journal of Political Economy 99, 558-581. Lucas, Robert E. Jr. and Thomas J. Sargent, eds., 1981, Rational expectations and econometric practice (University of Minnesota Press, Minneapolis, MN). Stockey, Nancy L. and Robert E. Jr. Lucas, with Edward C. Prescott, 1989, Recursive methods in economics dynamics (Harvard University Press, Cambridge, MA). West, Kenneth D., 1986, A variance bounds test of the linear quadratic inventory model, Journal of Political Economy 94, 374401. West, Kenneth D., 1990, The sources of fluctuations in aggregate inventories and GNP, Quarterly Journal of Economics 103,939-971. West, Kenneth D. and David W. Wilcox, 1992, Some evidence on the finite sample distribution of an instrumental variables estimator of the linear quadratic inventory model, Paper presented at the Business Cycle, Inventory Fluctuations and Monetary Policy Meeting, University of Siena, June 15-m18.