Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue4.Apr2000:

Journal of Economic Dynamics & Control
24 (2000) 535}559

On the solution of the linear rational
expectations model with multiple lags
Hugo Kruiniger
University College London, Department of Economics, Gower Street, London WC1E 6BT, UK

Abstract
In this paper the symmetric linear rational expectations model from Kollintzas (1985)
is generalized by allowing for multiple lags. By using a convenient decomposition of
the matrix lag polynomial of the Euler}Lagrange equations that encompasses that for
the Kollintzas model, it is shown that the model admits a unique stable solution. The
stability condition derived by Kollintzas turns out to be valid in the more general setting.
Moreover a procedure is given to obtain a particularly useful representation of the
solution that is as close as possible to a closed form representation. ( 2000 Elsevier
Science B.V. All rights reserved.
JEL classixcation: C61; C62; D92
Keywords: Rational expectations; Closed form solution; Unique solution; Time-to-build;
Stability condition

1. Introduction
The modern macroeconomic literature includes many models which involve
the solution to a stochastic dynamic optimization problem. The objective
functions in these problems are often approximated by linear}quadratic speci"cations (e.g. see Christiano, 1990). There are now several methods available for
solving such models. Generally, they involve the factorization of the spectral
density-like matrix lag polynomial which appears in the "rst order conditions,
E-mail address: kruinige@nber.org (H. Kruiniger)

0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 0 6 - 8

536

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

the so-called Euler}Lagrange conditions (ELC). Hansen and Sargent (1981)
outlined a relatively fast and revealing solution procedure for multivariate linear
rational expectations (LRE) models. Although their procedures are applicable
to a wide class of linear}quadratic models, in particular models with dynamic
interrelation and multiple lags, their approach has some disadvantages. The

main drawback of their procedures is that the solutions are expressed in terms of
parameters from which the structural parameters cannot be uniquely identi"ed.
Several other methods have been suggested in the literature for the estimation
and solution of LRE models, that do not su!er from this drawback. Kollintzas
(1985) proposes a solution method for a class of multivariate LRE models that
satisfy a symmetry condition. When a model has this property, it is possible to
obtain an attractive representation of the forward solution of the model by
diagonalizing the matrices of the lag polynomial of the ELC simultaneously.
Epstein and Yatchew (1985) outlined a simpli"ed solution and estimation
procedure for a sub-class of symmetric LRE models that involves a reparametrization of the model and yields a closed form solution. This procedure was
extended by Madan and Prucha (1989) in order to estimate more general,
possibly nonsymmetric models. Although all these procedures allow for identi"cation of the structural parameters, they have been developed for more restrictive LRE models than the Hansen and Sargent procedures, i.e. for models that
consist of second order di!erence equations.
In this paper we derive a convenient representation of the solution to a generalized version of Kollintzas' model which allows for multiple lags in the
transition equations of the endogenous state variables. This wider framework
encompasses a model with both adjustment costs and a general time-to-build
structure as in Kydland and Prescott (1982) as well as other more complicated
dynamic models. The nonsymmetric models studied by Madan and Prucha
(1989) and Cassing and Kollintzas (1991) only allowed for time-to-build lags of
one period. This is quite restrictive since it prevents one to study variables that

have di!erent lag structures at the same time or to model the value-put-in-place
pattern of the investment process.
In order to obtain a solution of the symmetric LRE model with multiple lags,
which is as close as possible to a closed form solution (CFS) and still allows for
identi"cation of the structural parameters, we will follow Broze et al. (1995), and
restate the ELC in a reduced form in terms of realizations and revision processes. Next, we will present a convenient decomposition of the matrix lag
polynomial. This decomposition considerably facilitates the derivation of restrictions on the revision processes which restore the equivalence of both
representations of the "rst order conditions to the LRE model. Moreover, this
decomposition includes the matrix lag polynomial of the LRE model given in
Kollintzas (1985) and this allows us to show that his stability condition is also
su$cient for the uniqueness and the stability of the solution of the more general
model: after adding the restrictions on the revision processes that are implied by

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

537

the stability condition to the &equivalence' restrictions mentioned above, all
revision processes can be written as unique functions of the exogenous processes. The &stability' restrictions on the revision processes are such that after
substitution of these functions in the reduced form, the nonstable roots of the

matrix lag polynomials cancel out. A lemma provides a procedure for imposing
the stability condition on the solutions, that is, for "nding the factorization of
the matrix lag polynomial of the exogenous part of the reduced form that
includes the factor with the nonstable roots that has to cancel out.
The paper is organized as follows. In the next section the model is stated.
Section 2 also gives other examples of LRE models that feature multiple lags.
The third section contains the main results of this paper. Section 4 concludes.
2. The model
In this section the symmetric linear rational expectations model with multiple
lags is described using an example with time-to-build. The choice of this example
is motivated by the fact that it is not restrictive and empirically relevant: Rossi
(1988), Johnson (1994), Oliner et al. (1995), and Peeters (1995) all "nd evidence
favoring factor demand equations with time-to-build.
Consider a (representative) agent who maximizes the expected discounted
stream of cash#ows given technological constraints and information on the
economic environment. As far as the agent is concerned, this environment only
consists of product and factor markets. He is a price taker in both markets.
Following Kollintzas (1985), the gross cash #ows are given by
g(x , *x , k )"a@x #k@x !1x@Ex !x@H*x !1*x@ G*x , (2.1)
t t 2 t t

t
t`1 t
t
t
t`1 2 t`1
t`1
where x is a (F]1) vector of stocks of production factors, decided upon at
t
equidistant points in time. The (F]F) constant matrices E, G and H are
symmetric, E is positive (semi-)de"nite (symmetric) (PSDS) and G is PDS; in
addition [ E H ] is PSDS and G}H is nonsingular; a is a (F]1) vector of
H{ G
constants, while k is a (F]1) vector of exogenous technology shocks. Furthert
more, let I , f"1,2, F, be the gross changes of the stocks of the inputs (e.g.
f,t
gross investments), let p be the corresponding real factor prices, which are
f, t
assumed to be exogenous as well, and let c denote the real discount factor. Then
the agent chooses a contingency plan for Mx N that maximizes
t

C

D

T
F
lim E + ct g(x , *x , k )! + p I
(2.2)
0
t
t`1 t
f,t f, t
T?= t/0
f/1
subject to an initial condition, e.g. x "x6 , a terminal condition and the
0
transition equations for x. The agent has rational expectations with regard to
the exogenous processes. Furthermore these processes are assumed to be linear.

538

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

In the generalized LRE model, the transition equations allow for multiple
lags. Kydland and Prescott (1982) stressed that it takes time to build new capital
(see Montgomery (1995) for some descriptive statistics) and suggested a speci"cation for investment that takes this feature into account. Let s
be the total
f, j, t
size of an investment project of type f, j periods from completion, j"0,2, h ,
f
where h is the gestation lag. Then the building process for inputs of type f can be
f
formalized as
x
"(1!d )x #s
,
(2.3a)
f, t`1
f f, t

f,0, t`1
s
"s
, j"0,2, h !1,
(2.3b)
f, j, t`1
f, j`1, t
f
where d is the depreciation rate of inputs of type f. Furthermore, let t be the
f
f, j
share of the resources spent on investment projects that are completed in
j periods. Total investment of type f in period t equals
hf
hf
"+ t s
with + t "1.
f, t
f, j f, j, t
f, j

j/0
j/0
From combining Eqs. (2.3a)}(2.3c), it follows that

(2.3c)

I

hf`1
, f"1,2, F,
(2.4)
" + / x
f, j f, t`j~1
j/0
where the / 's depend on the t 's and d .
f,j
f,j
f
There are other examples which "t the framework described above. One can
think of a "rm which builds tunnels and bridges. Here the time-to-build lags do

not pertain to the inputs but to the outputs. Moreover, as these projects require
speci"c capital and skilled labour, the "rm tries to minimize the #uctuations in
the total size of its projects as well as in the composition of the portfolio of
projects. Another example is investment in a foreign country when pro"ts can
only be repatriated after some time.
I

f, t

3. The existence, uniqueness and representation of the solution
Maximization of Eq. (2.2) subject to Eqs. (2.1) and (2.4) implies the following
"rst order conditions (ELC) for the stocks:

G

D

C

(1#c)

E f [G !H ]x ! E #
G !2H x
t~h
f.
f. t
f.
f. t`1
f.
c

H

1
#a #E fk
# [G !H ]x
f. t~1
f
t~h f, t
c f.
hf`1
]"0,
! + / c~j`1E f[p
f, j
t~h f,t~j`1
j/0

f"1,2, F,

(3.1)

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

539

where M is generic notation for the f th row of matrix M. Because of the
f.
gestation lags, the decisions concerning x
,x
, ,x
are made at
1, t`h1 2, t`h2 2 F, t`hF
time t.
De"ning ¸ as the lag operator1 we can write system (3.1) succinctly as
E ¸~JP(¸)y #u "0
t
t
t
with y "[x
,1 x
, ,x
]@
t
1,t`h 2,t`h2 2 F,t`hF
J"h !h #1 and assuming h 5h 525h 50,
1
F
1
2
F

(3.2)

h1`1
cjE [p
]
a #E k
!c~h1 + u 1
1,h ~j`1 t 1,t`j
1
t 1,t`h1
j/0
h2`1
cjE [p
]
!c~h2 + u 2
a #E k
2,h ~j`1 t 2,t`j
2
t 2,t`h2
j/0
u"
"d #;(¸)v ,
t
t
t
F
F
hF`1
a #E k
cjE [p
]
!c~hF + u F
F
t F,t`hF
F,h ~j`1 t F,t`j
j/0
where ;(¸)"; #; ¸1#2#; ¸q, v is a white noise process and d is the
0
1
q
t
t
deterministic part of u ,2
t
J
P(j)" + A jJ~i .
(3.3)
i
i/~J
In the de"nition of the A 's we use the de"nitions
i
C"[c ], D"[d ] (F]F)
jk
jk
with
(1#c)
g !2h ,
c ,e #
jk
jk
jk
jk
c

d ,g !h , j, k"1, 2,2, F
jk
jk
jk

A "[a ] (F]F), !J4i4J have nonzero elements3
i
i,jk
aj k
"d , a j k "!c , a j k
"1d .
h ~h `1,jk
jk
h ~h ,jk
jk
h ~h ~1,jk c jk
1 The inverted lag operator ¸~1 is the forward operator.
2 For the moment we will assume that u has a "nite VMA representation of order q. A more
t
general VARIMA model for u can be handled as well, as we will see at the end of this section.
t
3 Matrices with the same subscripts should be added up.

540

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

Introducing the vectors of revision processes ej"E (y )!E (y ),
t
t t`j
t~1 t`j
j"0,2,h , which are martingale di!erences, we obtain by replacing the expec1
tations in Eq. (3.2) with realizations and revisions and after shifting the system
J periods back in time (see Appendix A)
J~1 J
(3.4)
P(¸)y " + + A ¸J~j`iei!u .
t
t~J
j
t
i/0 j/i`1
The ewective (not multiplied by zero) revision processes in Eq. (3.4) are
e0, e0,2, eh1~h2,2, e0,2, eh1~hF where the subscript now indicates a production
1 2
F
2
F
factor. Thus we have (h #1)F!+F h revision processes.4 They are subject to
i/1 i
1
constraints as we will show in the sequence. P(j) can also be represented as
P(j)"[jMijq (j)]
ij
with q (j)"d !c j#(d /c)j2, i, j"1, 2 ,2, F and
ij
ij
ij
ij
J!1
J#h !h !1 2 J#h !h !1
2
1
F
1
J#h !h !1
J!1
2 J#hF!h2!1
1
2
M"
.
F
F
F

C

J#h !h !1 J#h !h !1 2
1
F
2
F
Then it is easily seen that P(j) can be decomposed as5

J!1

D

(3.5)

P(j)"jJ~1P~1(j)Q(j)P(j),
where P(j)"diag(jh1, jh2,2, jhF)

A

B

(1#c)
1
Q(j)"[q (j)]"(G!H)! E#
G!2H j# (G!H)j2
ij
c
c
"S@f(I#(f~1)j#(I/c)j2)S,

(3.6)

where f is a diagonal matrix with the eigenvalues of !(C~1)D and S~1 is the
matrix with the corresponding eigenvectors.
Eq. (3.4) will now be rewritten in a form that is suitable for deriving constraints on the revision processes (for derivation see also Appendix A)
P(¸)y "P(¸)
t

G

H

J~1
+ ei #m !u ,
t~i
t~J
t~J
i/0

(3.7)

4 Hereafter the subscript of the revision processes will refer to a production factor or indicate time.
5 This particular decomposition emerged quite naturally. Broze et al. (1995) propose two general
decomposition methods: the adjoint operator method and the Smith form method.

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

541

where
J~1 j
J~1 J
m "! + + A ¸J`h`je j! + + A ¸J~h`je j.
t
h
t
~h
t~J
j/0 h/0
j/0 h/1
Using the fact that +J~1ei "y !E y we obtain
t
t~J t
i/0 t~i
P(¸)E y "m !u .
(3.8)
t~J t
t~J
t~J
Replacing P(¸) in the recursive equation (3.8) with its decomposition (3.5) and
premultiplying both sides by ¸~h1P(¸) yields
(3.9)
¸~hFQ(¸)P(¸)E y "¸~h1P(¸)Mm !u N.
t~J t
t~J
t~J
The premultiplication shifts the equations forward in time. The LHS of the
equation above contains only current and lagged values of y. Note that
P(¸)y "x . If h '0 the components of the second term at the RHS are based
t
t
1
on information after time t!J. This observation is at the root of the following
lemma.
Lemma 1. In order that y"My N is a solution of Eq. (3.2), the revision processes e j
t
t
have to satisfy the constraints
EM¸~h1P(¸)[m !u ] D X N
t~J
t~J
t~i
N, i"0,2, J!1.
"EM¸~h1P(¸)[m !u ] D X
t~J
t~J
t~i~1

(3.10)

Proof. Take expectations of both sides of Eq. (3.9) with respect to the information sets X , i"0,2, J, and calculate the di!erence between two adjacent
t~i
expressions.6 See also Broze et al. (1995). h
The dimension of the set of solutions that satisfy Eq. (3.1) depends on the
number of free revision processes. The number of e!ective (univariate) martingale di!erences in Eq. (3.4) equals (h #1)F!+F h . After imposing the
i/1 i
1
constraints in Eq. (3.9), there are only F free revision processes left as we will
prove now.
Lemma 2. The restrictions in Eq. (3.10) reduce the number of free revision processes in Eq. (3.4) by h F!+F h .7
i/1 i
1
6 Since m
depends only on information up to t!J, in some cases the equalities in (3.10) hold
t~J
trivially and do not imply restrictions on the revision processes. For example, subtracting the
expectation of the "rst equation in (3.7) with respect to X
from the expectation of the same
t~i~1
equation with respect to X for i"0,2, h !h does not yield a single constraint.
t~i
1
F
7 This is in fact predicted by property (37) in the paper by Broze, GourieH roux and Szafarz (1995).
However we believe that our proof, as it is con"ned to the particular model above, is more revealing
and complete than their proof.

542

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

Proof. See Appendix B.
So all the revision processes in Eq. (3.4) but F are subject to restrictions. The
space of candidate solutions to the optimal control problem that satisfy the
Euler}Lagrange conditions (3.2) will be reduced even further by imposing the
transversality condition or a slightly stronger condition like a stability condition. Such conditions guarantee uniqueness of the solution just as they do when
there is no time-to-build.8 In the latter case P(j)"Q(j).
Recall the decompositions in Eqs. (3.5) and (3.6). Then premultiplying both
sides of Eq. (3.4) by (S@f)~1¸~h1P(¸) leads to
(I#(f~1)¸#(I/c)¸2)S¸~hFP(¸)y
t

G

H

J~1 J
.
"¸~hF(S@f)~1 + + P(¸)A ¸1~j`iei!P(¸)u
t
t~1
j
i/0 j/i`1

(3.11)

By virtue of Eq. (3.10), the number of free revision processes in Eq. (3.11) can be
reduced to F, say e0.9 If we look for a solution y that belongs to the class of
t
t
VARIMA-processes, we can parameterize e0 as U0v . Likewise, we can write ei as
t
t
t
Uiv , i"1,2,J!1, where the Ui are F]F matrices. Thus the RHS of Eq. (3.11)
t
is a distributed lag of v , R(¸)v "(R #R ¸#2#R ¸g )v with g"J#q.
t
t
0
1
g
t
At this point let us introduce some notation for later. We denote the vectors
comprising the ewective revision processes by e6 i. Further, we implicitly de"ne the
t
matrices UM i by e6 i,UM iv . Now, the UM i's are subject to restrictions like Eq. (3.10).
t
t
On the other hand, we note that knowledge of the rows of the Ui corresponding
to the other revision processes in the ei is not required because they cancel out in
t
Eq. (3.11). Finally, we de"ne the matrix UM ,[(UM 0)@2(UM J~1)@ ]@ and the vector
e6 ,[(e6 0 )@2(e6 J~1)@ ]@.
A solution of the stochastic optimal control problem must also satisfy the
transversality condition. Su$cient conditions for this condition to hold are that
the exogenous stochastic process v and a solution y are of exponential order
t
t
less than c~1@2 (see Sargent, 1987, pp. 200}201). The u process has this property
t
by assumption. To verify whether y has this property, we investigate the LHS of
t
Eq. (3.11). Let us de"ne QK (j),(I#(f~1)j#(I/c)j2). Denote the roots of the fth
equation in QK (1/j)"0 by j and j . All pairs of roots Mj , j N, f"1,2,F,
1f
2f
1f 2f
satisfy j j "c~1. Let j be the smaller modulus root of each pair. Then we
1f 2f
1f
have Dj D4c~1@2 and Dj D5c~1@2. The stability condition given in Kollintzas
1f
2f

8 See Kollintzas (1985).
9 Generally, the matrix that results from leaving out the "rst F columns from the matrix at the
LHS of (A2.2) is nonsingular. See also footnote 14.

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

543

(1985) is necessary and su$cient for Dj D(c~1@2 to hold. For instance, this
1f
condition is satis"ed when the adjustment costs are strongly separable (H"0).
Notice that if Dj D(c~1@2, then j , j 3R. But in order that the solution y is
1f
1f 2f
t
of exponential order less then c~1@2, we also need that R(¸) can be factorized as
(I!K`¸)RM (¸), where K`"diag(j ,2, j ) and RM (¸) is a lag polynomial of
21
2F
order g!1, for otherwise we are not able to get rid of that part of the LHS of
Eq. (3.11) that causes violation of the condition, i.e. the factor (I!K`¸).
The lemma below provides a condition that is equivalent to the existence of
the factorization but more easily veri"able.

Lemma 3. Let N(j)"N #N j#2#N jw be a polynomial of order w. This
0
1
w
polynomial can be factorized as (I!Cj)NM (j) if and only if
w
+ Cw~iN "0.
i
i/0

(3.12)

Proof. See Appendix C.
Substituting R(¸) for N(j) and K` for C gives the condition
g
+ (K`)g~iR "0.
i
i/0

(3.13)

The matrices R ,2, R depend on F]F unknown entries of U0. But Eq. (3.13)
0
g
imposes F]F restrictions on the U0 's. In general, these restrictions uniquely
ij
identify U0. To see this consider the RHS of Eq. (3.11) apart from the last term
J~1 J
¸~hF(S@f)~1 + + P(¸)A ¸1~j`iei,(RK #RK ¸#2#RK
¸J~1)e ,
t
0
1
J~1
t
j
i/0 j/i`1
(3.14)
where e "(e02eJ~1)@, RK ,(cS@f)~1[K 2K ], i"0,2,J!1; the RK 's are
t
i
i1
iJ
i
t
t
F](J]F) matrices and the K 's are F]F matrices with K "0, j'i#1,
ij
ij
M
K "+f(i) AM
, 14j4i#1,10 where
ij
k/1 J`j~i~1,k

C

D C

0 2 0 cd
1F
0 2 0
0

AM "
J,1
F

F

F

0 2 0

0

,

0 2 0 !c6
1F
0 2 0
0

AM
"
J~1,1
F

F

F

0 2 0

0

10 AM matrices with the same indices should be added up.
ij

D

,

544

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

C

D

0 2 0 d
1F
0 2 0 0

AM
"
J~2,1
F

F

F

0 2 0

0

,

C

0 2 0 cd
0
1F~1
0 2 0
0
0

"
AM
"0,2, AM 1 F~1
J~3,1
h ~h `1,1
F

C

F

F

F

0 2 0

0

0

D

0 2 0 !c6
0
1F~1
0 2 0
0
0

AM 1 F~1 "
h ~h ,1
F

C

F

F

F

0 2 0

0

0

D

,

,2,

D

cd
0 2 0
11
0
0 2 0
AM "
,
1,1
F
F
F
0

0 2 0

C D C D
C D

0 2 0
0
0 2 0 cd
2F
AM " 0 2 0
"
0 , AM
J,2
J~1,2
F
F
F
0 2 0
0

0

0

0 2 0
0
0 2 0 !c6
0 2 0
F
F

0

0 2 0

0

2F

F

0 2 0
0 2 0

0 cd
22
0
0 2 0 ,2,
" 0
AM 1 2
h ~h `1, 2
F
F
F
F
0

C

0 2 0
F
F

0

0 2 0

0

D

F
AM "
J,F
0 2 0
0
0 2 0 cd
FF

with c6 ,cc , i, j"1, 2,2, F,
ij
ij

,2,

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

545

fM (i)"f for h
!h (i4h !h with f"1,2,F, i"0,2, J!1 and
f`1
F
f
F
h
,h !1.
F`1
F
De"ne
J~1
B , + (K`) g~j(cS@f)~1K
, i"0,2, J!1.
i
ji`1
j/1
Then condition (3.13) can be rewritten in the form

(3.15)

J~1
(3.16)
+ B ei#BM v "0,
t
it
i/0
where BM "!+g (K`)g~j(S@f)~1K
with K ";I
, K ";I
, ,
j/1
j0
10
0,F. 20
1,F. 2
K F~1 F
"(;I F~1 F #;I
),2, K ";I
where ;I
is a zero
h ~h `1,0
h ~h ,F.
0,F~1.
g0
q,1.
s,i.
matrix with the ith row replaced by ; .
s,i.
Combining the restrictions on the revision processes in (3.10) and (3.16) gives

DC D C D C D
;

C

A

0

A
~1
A
~2
F

A

1
A
0
A
~1
F

A
A

2

1
A
0
F

A

1

B
J~1

0

e0
;
t
1
e1
F
t
e2
;
t # q* v "
t
F
0

2 AJ~2 AJ~1
2 AJ~3 AJ~2
2 AJ~4 AJ~3
F
F

A
A
A
A
~J`2
~J`3
~J`4 2
0
B
B
B
B
2
0
1
2
J~2

0

eJ~2
t
eJ~1
t

0
F

0
0

F

F

0

0

BM

0

(3.17)
where q*"min (q, J!1).
We can write the system in (3.17) more compactly as
=e #Zv "0.
(3.18)
t
t
where = and Z are de"ned implicitly.
We note that, as mentioned in the proof of Lemma 2, !h F#+F h
i/1 i
F
&restrictions' in the system are false but they are included for convenience: it is
easier to show the independence of the true restrictions by looking at the
extended system as will become clear below. Furthermore in the true restrictions
the revision processes in Eq. (3.17) that are not mentioned below Eq. (3.4) are
multiplied by zeros and cancel out. For the last F rows this follows from the
observation that Eq. (3.16) is obtained from Eq. (3.14) by "rst decomposing
the A in terms, then premultiplying each term by K raised to some power, where
j
the power depends on the lag of the corresponding revision process, and "nally
collecting terms (on the basis of the horizons of the processes). Thus the
vectors of the revision processes in Eq. (3.16) (which di!er in terms of their

546

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

horizons) are multiplied by the same rows of A as in Eq. (3.14). The latter is
j
obtained by shifting the rows in Eq. (3.4) back in time and by premultiplying the
result by a constant matrix.
We write the system of the true restrictions contained in Eq. (3.18) as
=
M e6 #ZM v "0.
t
t

(3.18@)

For uniqueness of the linear solution My N to Eq. (3.1) it is su$cient that = is
t
nonsingular. We will demonstrate that in general this condition is satis"ed. In
the model without time-to-build ="A "G is nonsingular.
1
Theorem 1. The matrix W dexned in Eqs. (3.17) and (3.18) is generally nonsingular,
that is ∀c, M ,ME,G,H D det(= )"0N, the space of values of the entries of the
W,c
c
matrices E, G, and H for which det(= )"0, has Lebesgue measure zero.
c
Proof of Theorem 1. See Appendix D.
The uniqueness of the solution to the model guarantees the existence of the
closed form solution (CFS)
x "Kx #S~1RM (¸)¸hFv
t
t~1
t

(3.19)

where K"S~1K~S, and K~, diag (j ,2, j ).
11
1F
From Eq. (3.19) it is clear that the transversality condition can also be
imposed on the solutions of the model with multiple lags.
Let M"I!K. If DMDO0, then Eq. (3.19) admits the #exible accelerator form
x !x "M(xH!x ) with xH"M~1S~1RM (¸)¸hFv .
t
t
t~1
t
t~1
t

(3.20)

Notice that the autoregressive part of the closed form solution is invariant with
respect to the order of the gestation lags. Our strategy for deriving expressions of
the MA parameters RM in the CFS in terms of the structural parameters entails
i
the following steps:
1. Obtain expressions for e6 ; from Eq. (3.18@) e6 "!=
M ~1ZM v , i.e. UM "!=
M ~1ZM .
t
t
t
2. Substitute these expressions in the RHS of Eq. (3.11) and "nd formulae for
R up to R , g"J#q. Make use of Eq. (3.14).
0
g
3. Use the procedure in the proof of Lemma 3 to obtain the factorization
R(¸)"(1!K`¸)RM (¸), i.e. RM , RM ,2, RM
.
0 1
g~1
For nontrivial orders of the (gestation) lags, the MA parameters in Eq. (3.19)
depend in a complicated way on the structural parameters and estimation of the
latter is not straightforward when these restrictions are to be exploited.

H. Kruiniger / Journal of Economic Dynamics & Control 24 (2000) 535}559

547

To estimate the structural parameters in the CFS an iterative procedure can
be used which entails the three steps mentioned above but carried out in reverse
order. At each iteration one has to invert the (sparse) matrix =
M in Eq. (3.18@) to
obtain UM . We note that the dimensions of =
M (and also =) do not depend on q.
Furthermore, one can reduce the computational burden by replacing S and
K` in Eq. (3.17) (or Eq. (3.18@)) by estimates based on an initial consistent
estimate of K,11 which would result in a system that is linear in the technology
parameters and the ; . Alternatively, one can of course replace the S and K` in
i
Eq. (3.17) by new estimates at each iteration. So in practice we have found
a representation of the solution that is as close as possible to a closed form
representation.
The identi"cation of the structural parameters hinges on whether the factor
demand equations are estimated in a system together with the production
(pro"t) function and (observable) exogenous processes or separately. If the latter
are not included in the system, the MA parameters provide no information on
the structural parameters. If the production (pro"t) function is not included, one
can at best identify the parameters in C and D (up to a constant factor) but not
those in E,G, and H.
When one ignores the restrictions on the MA parameters, one still has a semi
closed form representation of the solution. Epstein and Yatchew (1985) discuss
an exhaustive set of properties of the accelerator matrix, M, that are implied by
the structure of the problem. By demanding that the estimates of M exhibit these
properties and by estimating the factor demand equations together with the
production function and the price equations, more precise estimates of the
structural parameters can be obtained.
At the outset we assumed that u &VMA(q). However, prices sometimes
t
follow more general VARIMA processes. In such cases the formula of the CFS
should be amended by using the following procedure. The autoregressive lag
polynomial of the exogenous processes Mu N can be factorized such that one
t
factor is a diagonal matrix lag polynomial with common roots on the diagonals: