In this paper a novel partitioned inversion formula is obtained in terms of the orthogonal complements of off-diagonal blocks, with the emblematic matrix of unit-root econometrics springing up as the leading diagonal block of the inverse. On the one hand, the result paves the way to a stimulating reinterpretation of restricted least-squares estimation and, on the other, to a straightforward derivation of a key-result of time-series econometrics.Partitioned inversion; Restricted least-squares; VAR econometrics

Maria Grazia Zoia

Mario Faliva

English

Research Papers in Economics

Cahiers du département d’économétrie
Faculté des sciences économiques et sociales
Université de Genève
Mars 2000
Département d’économétrie
Université de Genève, 40 Boulevard du Pont-d’Arve, CH -1211 Genève 4
http://www.unige.ch/ses/metri/
On a partitioned inversion formula
having useful applications
in econometrics
Mario Faliva and Maria Grazia Zoia
No  2000.01ON A PARTITIONED INVERSION FORMULA HAVING
USEFUL APPLICATIONS IN ECONOMETRICS 1
MARIO FALIVA*, MARIA GRAZIA ZOIA*
Abstract
In this paper a novel partitioned inversion formula is obtained in terms of the orthogonal
complements of off-diagonal blocks, with the emblematic matrix of unit-root econometrics
springing up as the leading diagonal block of the inverse. On the one hand, the result paves the
way to a stimulating reinterpretation of restricted least-squares estimation and, on the other, to a
straightforward derivation of a key-result of time-series econometrics.
JEL classification: C13, C32, C69
Keywords: Partitioned inversion; Restricted least-squares; VAR econometrics
§1. THE PARTITIONED-INVERSION THEOREM
Partitioned inversion formulas have been long since (e.g. Goldberger, 1964: Theil, 1971)
a basic tool of the standard algebraic equipment of econometricians. In this section we
establish a new result in partitioned inversion which proves helpful in classical and
unit-root econometrics as well
THEOREM 1.1 - Let A be a square matrix of order n  n and B and C two rectangular
matrices of order n  m and full column-rank. Further let B? and C? denote the ortho-
gonal complements of B and C.
Consider the partitioned matrix
P  AB
C 0O

 1 
If B0
?AC? is of full rank, then
(a) the matrix P is nonsingular;
1
1 The paper is the result of a joint research project of the two authors. Section 1 is by M. Faliva and Sec-
tions 2 and 3 are by M.G. Zoia.
* Istituto di Econometria e Matematica, Universita Á Cattolica,
Largo Gemelli 1, I-20123 Milano, Italy
E-mail: mfaliva@mi.unicatt.it, mgzoia@mi.unicatt.it(b) its partitioned inverse is
Pÿ1 
E In ÿ EA  C0
ÿ
Bÿ Im ÿ AE Bÿ AEA ÿ A  C0
ÿ
" #
2
where
E  C?  B0
?A C?
ÿ1  B0
? 3
and Bÿ and C0
ÿ are left and right inverses of B and C0, respectively, namely
Bÿ  B0B
ÿ1 B0 and C0
ÿ  C  C0C
ÿ1:
Proof: The result (a) follows from the rank expansion of the partitioned matrix P (see
Marsaglia and Styan, 1974, Theorem 19):
rP  rB  rC  rfI ÿ BBÿ  A  I ÿ C0
ÿC0g  rB  rC  rB0
?AC? 4
given that
I ÿ D  D0D
ÿ1  D0  D?  D0
?D?
ÿ1  D0
? 5
for every full column-rank matrix D (e.g., Johansen, 1995, p. 39).
To prove (b) let the inverse of P be
Q 
E F
G0 H
 
6
where the blocks in Q are of the same order as the corresponding blocks in P. Then, in
order to express the blocks of the former in terms of the blocks of the latter, write QP  I
and PQ  I in partitioned form as:
E F
G0 H
" #

A B
C0 O
" #

In O
O Im
" #
7
A B
C0 O
" #

E F
G0 H
" #

In O
O Im
" #
70
From 7 and 70 we can derive the pair of equation systems in the unknown subma-
trices E, F, G and H:
EA  FC0  In (8)
EB  O (9)
G0A  HC0  O (10)
G0B  Im (11)
8
> <
> :
AE  BG0  In (8')
AF  BH  O (9')
C0E  O (10')
C0F  Im (11')
8
> <
> :
2Equations 9 and 100 have the common solution (see Rao and Mitra, 1971, Lemma
2.3.1):
E  I ÿ C0
ÿC0  Z  I ÿ BBÿ  I ÿ C  C0C
ÿ1  C0  Z  I ÿ B  B0B
ÿ1  B0 12
for some Z, which is equivalent to:
E  C?WB0
? 120
for some W, in view of 5 above.
Substituting 120 into 8, postmultiplying both sides by C? and considering that
C0C?  O, we obtain
C?WB0
?AC?  C? 13
Because C? is of full column-rank, equality 13 holds iff:
WB0
?AC?  I , W  B0
?AC?
ÿ1 14
Combining 120 and 14 we obtain
E  C?  B0
?AC?
ÿ1  B0
? 15
Consider now equations 11 and 80, with E given by 15. Then simple computations
show that
B  I ÿ AE  B 16
Hence a common solution exists, namely (Rao and Mitra, 1971, Th. 2.3.3):
G0  Bÿ  I ÿ AE  Bÿ ÿ BÿBBÿ  I ÿ BÿB  Z  I ÿ BBÿ 17
for some Z, which, noting that BÿB  I, can be reduced to
G0  Bÿ  I ÿ AE 18
The expression
F  I ÿ EA  C0
ÿ 19
is similarly established, by referring to equations 8 and 110.
Finally, consider equation 90, with F given by 19. Rearranging terms we get
BH  AEA ÿ A  C0
ÿ 20
Premultiplication of 20 by Bÿ thus leads to
H  Bÿ  AEA ÿ A  C0
ÿ 21
which completes the proof.
3§2. RESTRICTED LEAST-SQUARES REVISITED
In this section some more light is shed on restricted least-squares theory by applying the
partitioned inversion formula of Theorem 1.1 in the derivation of the estimator, which
leads to a solution that actually is the mirror image of the well-known Theil's (1961) con-
strained estimator.
Consider the linear model
y  X b  e 22
Efeg  o 23
Efee0g  s2 V 24
where y in the n-vector of values of the regressand, X is the nxk matrix of values of the
regressors and e is the n-vector of residuals. Moreover s2 is an unknown scale factor, V
is a known positive-definite matrix normalized so that its trace is equal to n, and b is a k-
vector of unknown parameters subject to the linear constraints
s  R0 b 25
where s is a given m-vector and R is a given kxm matrix such that
rR  m < k 26
r

X
R0

 k 27
Estimation of b can be accomplished by the restricted least-squares method, i.e. by sol-
ving the constrained optimization problem
min
b
fy ÿ X b
0  V
ÿ1  y ÿ X bg
subject to
s  R0 b 28
Using the method of Lagrange multipliers, the minimizing equations can be compactly
written as (e.g., Magnus and Neudecker, 1999, for differentiation rules in compact form)
X0V
ÿ1X R
R0 O
 
 b
l
 
 X0V
ÿ1y
s
 
29
where l is the m-vector of Lagrange multipliers.
4The rank qualifications (26) and (27) ensure that the coefficient matrix in (29) is nonsin-
gular. Then the solution for b is
b  I; O  X0V
ÿ1X R
R0 O
 ÿ1
 X0V
ÿ1y
s
 
30
which paves the way to a stimulating application of Theorem 1.1.
To this end we shall now prove:
LEMMA 2.1 - Under the rank qualifications (26) and (27) the rank equality
rX R?  rR? 31
holds, which in turn entails
rR0
?X0 V
ÿ1X R?  k ÿ m 32
Proof: Result (31) follows from (5), (26) and (27) above making use of Result 3.12 of
Corollary 6.1 in Marsaglia and Styan, 1974. Formally
k  r

X
R0

 r

I O
O R  R0R
ÿ1



X
R0

 r

X
RRÿ


 r

X
I ÿ R?Rÿ
?

( ) rXR?  rR?  k ÿ m 33
Result (32) is straightforward.
With these preliminaries we are ready to demonstrate the following elegant representa-
tion theorem.
THEOREM 2.1 - The restricted least-squares estimator of b in the linear model specified
by the set of hypothesis (22)  (27), is given by
b  R?  R0
?X0 V
ÿ1X R?
ÿ1  R0
?X0 V
ÿ1y
I ÿ R?  R0
?X0V
ÿ1X R?
ÿ1  R0
?X0V
ÿ1X  R0
ÿ  s 34
which, by defining
~ b  R0
ÿ  s 35
F  R?  R0
?X0V
ÿ1XR?
ÿ1  R?X0V
ÿ1 36
can be more conveniently rewritten as:
b  ~ b  F  y ÿ X ~ b 37
5Proof: The result (34) follows from straightforward application - in view of (32) in Lem-
ma 2.1 - of the partitioned inversion formula (2) of Theorem 1.1 to the restricted
least-squares estimator (30). Trivial substitutions then leads to (37).
The restricted estimator (37) turns out to be split into two components: the former is a
(particular) solution, ~ b, of the a-priori restrictions (25), while the latter mirrors the discre-
pancies between the observed regressand vector y and its virtual estimates X ~ b based on
~ b. The representation (37) is the specular image of the well-known representation (e.g.,
Goldberger, 1964, Theil, 1961, 1971).
b  ^ b  C  s ÿ R0 ^ b 38
where
^ b  X0V
ÿ1X
ÿ1  X0V
ÿ1y 39
C  X0V
ÿ1X
ÿ1  R  R0  X0V
ÿ1X
ÿ1  R
ÿ1 40
which links the restricted estimator b to the unrestricted estimator ^ b, based on sample
data X; y under a full column-rank hypothesis on the regressor matrix X.
§3. THE MATRIX-RESIDUE AT THE ROOT OF VAR ECONOMETRICS
In this section, after sketching the analytical toolkit relating matrix polinomial inversion
to Laurent series expansion, we derive, as a by-product of the partitioned inversion rule
of Theorem 1.1, the matrix residue at a simple pole, and we eventually come to a short-
cut proof of a crucial representation theorem (e.g., Banerjee et al., 1993, Faliva and Zoia,
1999) of unit-root econometrics.
Consider the matrix
Sz  Az  1 ÿ z  BC0 41
Simple computations shows that 1 ÿ z
ÿ1 Sz plays the role of the Schur complement
of z ÿ 1  I in the composite matrix:
Az B
C0 z ÿ 1  I
 
and that, accordingly, the identity
1 ÿ z  Sÿ1z  I; O 
Az B
C0 z ÿ 1I
 ÿ1

I
O
 
42
6holds true, provided the inverses exist, in view of classical partitioned inversion rules.
Observe next that
lim
z!1
f1 ÿ z  Sÿ1zg  V; where V 6 O; 43
provided z  1 is a simple pole of Sÿ1z (e.g., Jeffrey, 1992).
By analogy with the scalar case, the matrix V is called the matrix residue in the Laurent
series expansion
Az  1 ÿ z  BC0
ÿ1  V  1 ÿ z
ÿ1 
X 1
i0
Wi1 ÿ z
i 44
of Az  1 ÿ z  BC0
ÿ1 about the singularity located at z  1:
LEMMA 3.1 - Let A1; B and C be defined as the matrices A; B and C in Theorem 1.1.
Then the matrix Az  1 ÿ z  BC0
ÿ1 has a simple pole at z  1, whose matrix resi-
due is given by;
V  C?  B0
?A1C?
ÿ1  B0
? 45
Proof: Taking limit of both sides of (42) as z tends to 1, we get:
lim
z!1
f1 ÿ z  Sÿ1zg  lim
z!1

I;O 
Az B
C0 z ÿ 1I
 ÿ1

I
O
 
V  I; O 
A1 B
C0 O
 ÿ1

I
O
 
46
where the right-hand side turns out to be equal to:
C?  B0
?A1 C?
ÿ1  B0
? 47
in view of Theorem 1.1.
Consider now, under the assumption yt  CI1;1, a cointegrated autoregressive model
(VAR) in differences, namely (e.g., Charemza and Deadman, 1992, Banerjee et al.,
1993):
AL5yt  BC0yt  et 48
where 5 and L denote the backward difference and the lag operator, respectively, et is a
vector of white-noises, and AL is a matrix polinomial, namely
AL  I 
X K
k1
Ak L
k 49
7whose matrix coefficients Ak along with the matrices B and C ± even if unconstrained by
economic theory a-priori's ± fulfill the following specification requirements:
i) the matrices B and C have full column-rank;
ii) the square matrix B0
?A1C? ± where B? and C? are the orthogonal complements of
B and C, respectively, and A1  I 
X K
k1
Ak ± has full rank;
iii) the roots of the characteristic polinomial
z  det 1 ÿ z Az  BC0 50
in the complex argument z, are either equal to one or, in modulus, greater than one.
Then the following representation theorem holds:
THEOREM 3.1: The model (48), under the assumptions i), ii) and iii) stated above, has a
solution of the form:
yt  C?  B0
?A1C?
ÿ1  B0
?
X
t
et 
X 1
i0
Ji  etÿi 51
which is precisely a CI(1,1) process, given that:
yt  I1 52
C0yt  I0 53
Proof: Looking at (48) as a linear operator equation, the closed form expression for yt:
yt  AL5  BC0
ÿ1  et 54
is a particular solution of the nonhomogeneous equation at stake, provided the inverse
operator AL5  BC0
ÿ1 exists.
Applying the Laurent series expansion (44) to the inversion of the operator AL5  BC0
± with L and 5 formally replacing z and 1 ÿ z, respectively ± eventually leads, after
some elementary manipulations, to the expansion (c.g., Engsted and Johansen, 1997; Fa-
liva and Zoia, 1999):
AL5  BC0
ÿ1  V5ÿ1 
X 1
i0
Ji L
i 55
where V is the matrix residue of Lemma 3.1 and the Ji's turn out to be, under the above
stated assumptions on the roots of the characteristic polinomial (50), matrices of expo-
nentially decreasing coefficients.
8In view of (55) the solution (54) can be rewritten in the form (51), using the formal iden-
tity between the antidifference 5ÿ1 and the indefinite sum operator (e.g., Spiegel, 1971),
namely:
5ÿ1 
X
t
56
The solution (51) is the sum of two components: the former is an I(1) process, namely a
linear transformation ± through the matrix residue C?  B0
?A1C?
ÿ1 B0
? ± of the multi-
variate random-walk
X
t
e, while the latter is an I(0) process, namely a causal infinite
vector moving average, VMA(1, of the multivariate white-noise et.
Hence the result (52) holds trivially, and the result (53) ensues from the annihilation of
the I(1) component of the solution once it is premultiplied by the matrix C0.
9References
Banerjee A., J. Dolado, J.W. Galbraith and D.F. Henry, 1993, Cointegration, Error-Correction
and the Econometric Analysis of Non Stationary Data, Oxford University Press, N.Y.
Charenza W.W. and D.F. Deadman, 1992, New Directions in Econometric Practise, Edgar
Publ. Lim., Adershot, U.K.
Engsted T. and S. Johansen, 1997, Granger’s Representation Theorem and Multicointegration,
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Faliva M. and M.G. Zoia, 1999, Econometria delle Serie Storiche, Giappichelli Ed., Torino
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On a Partitioned Inversion Formula having Useful Applications in Econometrics

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On a Partitioned Inversion Formula having Useful Applications in Econometrics

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