discussion is not meant to be a comprehensive guide to the inversion of block-tridiagonal matrices, but rather simply intends to convey how nite-
horizon MLRE models may be eciently solved using numerical schemes from the literature on the inversion of block-tridiagonal matrices.
Case
1: B possibly singular. Consider rst the case where B is possibly singular. The standard textbook
approach to solving linear equation systems of the form Cutht, dened by 4.2, is by means of Gaussian elimination, or equivalently, LU-factorization;
namely factorization of C into the product of a block lower triangular matrix L
H and a block upper triangular matrix UH, and then solving the resultant
block-triangular equation systems
L Hltht,
5.1 by forward substitution, and
U
Hutlt, 5.2
by backward substitution. Such an algorithm does not exploit the sparse nature of C, however, and in general can be quite inecient. An alternative
numerical procedure that utilizes the block-tridiagonal structure of C is the LDU
-factorization, discussed, for example, in Axelsson 1994. Decompose C as CD
c
L
c
U
c
, 5.3
where D
c
, L
c
, and U
c
are the block-diagonal, block-subdiagonal, and block-
superdiagonal entries in C, and consider the factorization CLD
~1U,
5.4 where
LDL
c
, 5.5
and
UDU
c
. 5.6
Noting that
C DL
c
D ~1DU
c
DU
c
L
c
L
c
D ~1U
c
, 5.7
it is easily seen that DD
c
L
c
D ~1U
c
, and thus the matrix D satises the
recursions
D1C11, DiCiiCi,i~1D~1
i~1
Ci~1,i, i2, 3,2,¹t1, 5.8
where Cii denotes the i,ith block of C, and Di the ith diagonal block of D.
M. Binder, M.H. Pesaran Journal of Economic Dynamics Control 24 2000 325}346 333
As discussed in Axelsson 1994, sucient conditions for the recursions in 5.8
to be well dened are: i C is symmetric positive denite, or ii C is a block H
-matrix. 12
Having factorized C as in 5.4, it is then a simple step to solve for ut by splitting Cutht into
L gtht or gt,1D~1
1
ht,1, gt,iD~1
i
ht,iCi,i~1gt,i~1,
i 2, 3,2, ¹t1,
5.9 and
D ~1Uutgt or ut,T~t`1gt,T~t`1, ut,igt,iD~1
i
Ci,i`1ut,i`1,
i¹t , ¹t1,2,1.
5.10
The solution xt to the MLRE model 2.2 is given by the last m entries in ut.
We therefore have the following proposition: Proposition 5.1
Solution based on LDU-factorization. Consider the xnite-hor- izon MLRE model
2.2. Let H
i
ImB
H ~1
i~1
A ,
i 2, 3,2, ¹t1,
5.11 C
i H
~1 i
BCi~1wT`1~i, i2,3,2, ¹t1, 5.12
with the initial conditions H1Im, and C1BExT`1DXTwT. Suppose the matrices Hi are nonsingular for i2,3,2, ¹t1. Then the solution for xt to
2.2 is given by
xt H
~1 T~t`1
Axt~1E C
T~t`1 DXt.
5.13 It is easily veried that the recursions in 5.11 and 5.12 match the
recursions in Proposition 3.1. The ith recursion matrices Hi and Ci in Proposi- tion 5.1 are related to the ith recursion matrices UT~t`1~i and WT`1~i
in Proposition 3.1 as follows: HiUT~t`1~i, i1, 2,2,¹t1, and Ci U
~1 T~t`1~i
W T`1~i
, i1,2,2,¹t1. Notice that this equivalence also im-
plies that inversion of C does not only yield the solution for xt, but also yields the
solution for
Mxt`qNT~t
q1 . We have
xt`q H
~1 T~t`1~q
Axt`q~1E C
T~t`1~q DXt`q, q1,2,2, ¹t. 5.14
Proposition 5.1 provides a link between the recursions of Proposition 3.1 and the LDU-factorization, and establishes conditions under which these recursions
12 See Axelsson 1994, Chapters 6 and 7 for a denition of block H-matrices. The class of block H-matrices encompasses, but is not restricted to, matrices that are generalized diagonally
dominant. 334 M. Binder, M.H. Pesaran Journal of Economic Dynamics Control 24 2000 325}346
are well dened. Notice that for these recursions to be well dened, it is by no means necessary that the coecient matrices A and B are nonsingular.
13 While the recursions in 5.8}5.10 exploit the block-tridiagonal structure of
C , the Di matrices in general are full, even if the superdiagonal, diagonal and
subdiagonal blocks of C are sparse, as the inverse of a sparse matrix is, in general, full. A fully ecient solution scheme to Cutht in the case where
A andor B are sparse will therefore also incorporate sparse approximations of
the inverses D
~1 i
, i1, 2,2, ¹t1, and sparse approximations of the
matrix product terms in the recursions in 5.8, Ci,i~1D~1
i~1
Ci~1,i. A variety of
numerical schemes accomplishing this by allowing the user to control the sparse blocks are discussed in Axelsson 1994, Chapter 8.
Case 2: B nonsingular.
In the case where the block-subdiagonal matrix B is nonsingular, one may
also solve the nite-horizon MLRE model 2.2 by adapting the recursions suggested in Bowden 1989 for the inversion of block-tridiagonal matrices:
Proposition 5.2 Solution based on Bowdens procedure. Consider the xnite-
horizon MLRE model 2.2 with the coezcient matrix B nonsingular. Then the
solution for xt to 2.2 is given by xtBT~t`1Axt~1
T~t +
i0
BT~t`1~iEwt`iDXtB1BExT`1DXt, 5.15
where
BiF~1
T~t`2
Fi, i1, 2,2, ¹t1,
5.16
F1Im, F2B~1, Fi`1FiFi~1Ai~1B~1, i2, 3,2,¹t,
5.17 and
FT~t`2FT~t`1FT~tA.
5.18 A proof of Proposition 5.2 can be constructed following the arguments in
Bowden 1989. Note that Bowdens procedure merely requires the inversion of two m]m
dimensional matrices. It is clearly an eective and straightforward method for the solution of the MLRE model, 2.2. To compute
Mxt`qNT~t
q1 , however, the
13 While we have not encountered in any application that we have considered near-singularity or singularity of any of the Hi matrices, if such near- singularities did arise, the recursions in
Proposition 5.1 could be stabilized using techniques developed in the recent numerical analysis literature on near- rank-decient problems. See, for example, Hansen 1998, for an up-to-date
survey. M. Binder, M.H. Pesaran
Journal of Economic Dynamics Control 24 2000 325}346 335
analog of 4.1 needs to be constructed for periods t1, t2,2,¹1, before the solution technique of Proposition 5.2 can be applied. Therefore, Bowdens
procedure will typically be less ecient for the computation of
Mxt`qNT~t
q1 than the LDU-factorization based procedure of Proposition 5.1. However,
in the case of MLRE models with a nite and xed planning horizon at all current and future dates, the structure of the matrix C in 4.2
remains unchanged in all periods, and Bowdens procedure is likely to be computationally more ecient than the LDU-factorization based
procedure.
6. An illustration : A consumers optimal expenditure shares
In this section, we illustrate Propositions 5.1 and 5.2 by applying them to the solution of a model of a consumers optimal expenditure shares if share adjust-
ment to the target level is costly both in terms of the level of adjustment and in terms of the speed of adjustment.
14 Consider the following nite-horizon adjustment cost problem:
min
Mst`qNT~t
q0
E
G
T~t +
q0 bq
C
st`qsHt`qHst`qsHt`qst`qGst`q 2st`qK2st`q
D
DXt
H
6.1
for given initial and terminal conditions st~1, st~2, EsT`1DXT~1, EsT`1DXT, and EsT`2DXT, and subject to
i st`q1, q0,1,2,¹t.
6.2
In 6.1 and 6.2, st`q is an r]1 dimensional vector of the consumers expendi- ture shares, H, G, and K are r]r dimensional symmetric matrices of xed
coecients, b30, 1 is a constant discount factor, i is an r]1 dimensional vector
of ones, and sHt`q is a vector of desired target expenditure shares, derived, for
example, from the Almost Ideal Demand System of Deaton and Muellbauer 1980,
s
H t`q
aC ln pt`qdln
A
yt`q pt`q
B
, q0, 1,2, ¹t,
6.3
14 See Pesaran 1991 for a detailed discussion of adjustment costs both for the level and the speed of adjustment. See also Price 1992 and Binder and Pesaran 1995.
336 M. Binder, M.H. Pesaran Journal of Economic Dynamics Control 24 2000 325}346
where
a
A
a1 a2
F ar
B
, C
A
c11 c12 2 c1r c21 c22 2 c2r
F F
F cr1 cr2 2 crr
B
,
d
A
d 1
d2 F
dr
B
, and
ln pt`q
A
lnp1,t`q lnp2,t`q
F lnpr,t`q
B
.
In 6.3, pi,t`q is the price deator of commodity group i, yt`q is the consumers expenditure on all the commodities, and pt`q is the general price index, approxi-
mated using the Stone formula
ln pt`qx0lnpt`q , 6.4
where x0u10, u20,2,ur0, and ui0 is the budget share of the ith commodity group in the base year. Consumer theory imposes the following restrictions on
the parameters of the target share equations: a adding-up restrictions:
r +
i1 ai1,
r +
i1 cij0, and
r +
i1 di0,
6.5 b homogeneity restrictions:
r +
j1 cij0,
6.6 and
c symmetry restrictions: cijcji, jOi.
6.7
We also assume that for given observations on pt`q and yt`q, the parameters in
a , C, and d are such that the desired shares sHi,t`q in 6.3 lie in the range [0,1] for
i 1, 2,2, r, and q0, 1,2, ¹t. To compute EsHt`qDXt, we also assume that
mtln p1t, ln p2t,2,ln prt, lnyt follows the vector autoregressive process of
order s:
mta
s +
i1
Rimt~it, ti.i.d. N0rC1, R
, 6.8
for all t.
M. Binder, M.H. Pesaran Journal of Economic Dynamics Control 24 2000 325}346 337
Forming the Lagrangian for 6.1}6.8, the Euler equations at time t q can
after some algebraic manipulation be written as 15
M00st`qN
C
M10st`q~1M20st`q~2M01Est`q`1D X
t`q
M02Est`q`2D X
t`q
HsHt`q
D
hi, 6.9
q0, 1,2, ¹t, where M00H1bG14bb2K, M10G
21
bK, M20K, M01bM10, M02b2M20, NIrhiiM~1
00 , and
h1iM ~1
00
i 0.16 Noting that iM~1
00
N0r, it is easily seen that ist`q1, for
all q, as it should. The Euler equations 6.9 constitute a special case of the
nite-horizon MLRE model 2.1, and can be eciently solved using Proposi-
tion 5.1, or, in the case where the adjustment costs are of rst-order only, K0r,
using Proposition 5.2. Case
1: KO0r.
