where E
t
is the expectation operator at period t, A is a square matrix of size n
w
+ n
k
represents the structural coefficients matrix for future variables, B is a square matrix of the same size represent those
for contemporaneous variable, ε
t
is a n
z
× 1 white noise vector, and thus z
t
is a first-order autoregressive process whose coefficients are collected in
φ.
2.2. Solution
In this part we provide the solution of 2 based on undetermined coefficient setup, where the solution is assumed in the form of:
1
, ,
t t
t t
t t
w Mk
Nz k
Pk Qz
+
= +
= +
3 with M, N, P, Q are to be determined matrices. It is clear that the solution writes the non-predetermined
endogenous variables as a linear combination of predetermined endogenous variables and exogenous variables.
From the second equations of 2 and 3 we have
1 ,
1
.
t t t
t t
t t
E z z
E k Pk
Qz φ
+ +
= =
+ 4
By substituting into the first equation of 3 we produce
1 1
1
.
t t
t t
t t
t t
E w ME k
E z MPk
MQ N
z
φ
+ +
+
= +
= +
+
5 Hence,
1 1
1
: .
t t
t t
t t
t t
E w MP
MQ N
E k
z E k
P Q
φ
+ +
+
+ ⎡
⎤ ⎡ ⎤
⎡ ⎤
Ω =
= +
⎢ ⎥ ⎢
⎥ ⎢
⎥ ⎣
⎦ ⎣
⎦ ⎣
⎦
6 Together with the fact that
,
t t
t t
t t
t t
w M
N Bw
Cz B
Cz B
k z
Cz k
I ⎛
⎞ ⎡ ⎤
⎡ ⎤ ⎡ ⎤
+ =
+ =
+ +
⎜ ⎟
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎣ ⎦
⎣ ⎦ ⎣ ⎦
⎝ ⎠
the basic model 3 can then be written as
.
t t
t t
M MQ
N M
N A
k A
z B
k B
C z P
Q I
φ
+ ⎛
⎞ ⎡ ⎤
⎡ ⎤
⎡ ⎤ ⎡ ⎤
+ =
+ +
⎜ ⎟
⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎢ ⎥ ⎣ ⎦
⎣ ⎦
⎣ ⎦ ⎣ ⎦
⎝ ⎠
7 Next we provide the solution of the model, or in other words, we explicitly determine matrices
M, N, P, Q. From 7 it is obvious that the following equalities should be satisfied: .
MP M
MQ N
N A
B A
B C
P I
Q φ
+ ⎡
⎤ ⎡ ⎤
⎡ ⎤
⎡ ⎤ =
⇔ =
+ ⎢
⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥ ⎣
⎦ ⎣ ⎦
⎣ ⎦
⎣ ⎦ 8
Since the square matrices A and B may be singular, the first equality of 8 can be treated as a generalized eigenvalue problem. The generalized Schur decomposition or qZ-decomposition then
guarantees the existence of unitary matrices q and Z, i.e., qq = I and ZZ = I, such that ,
, qAZ
S qBZ
T =
= 9
where S and T are triangular. If S:= [s
ij
] and T:= [t
ij
] then the ratios λ
i
:= t
ii
s
ii
are the generalized eigenvalues of the pencil matrix B
−λA. It is known that we can always arranged the eigenvalues λ
i
and associated columns of Q and Z descendently in order of their moduli. Premultiply the first equality of
8 by q and consider the fact that qA = SZ
-1
and qB = TZ
-1
then we have
, MP
M SH
TH P
I ⎡
⎤ ⎡ ⎤
= ⎢
⎥ ⎢ ⎥
⎣ ⎦
⎣ ⎦
where H = Z
-1
. By partitioning S, T, dan H we can write
11 11
12 11
11 12
21 22
21 22
21 22
21 22
S H
H T
H H
MP M
S S
H H
T T
H H
P I
⎡ ⎤⎡
⎤ ⎡
⎤⎡ ⎤
⎡ ⎤ ⎡ ⎤
= ⎢
⎥⎢ ⎥
⎢ ⎥⎢
⎥ ⎢ ⎥
⎢ ⎥ ⎣ ⎦
⎣ ⎦ ⎣
⎦⎣ ⎦
⎣ ⎦⎣
⎦
10
where the first row can be written as
11 11
12 11
11 12
. S
H M H
P T
H M H
+ =
+
The above equality is satisfied only if H
11
M + H
12
= 0. Hence, we obtain
1 11
12
. M
H H
−
= − Since H = Z
-1
, i.e.,
11 12
11 12
21 22
21 22
, H
H Z
Z I
H H
Z Z
I ⎡
⎤ ⎡ ⎤ ⎡
⎤ =
⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎣ ⎦
⎣ ⎦ ⎣
⎦ 11
we have H
11
Z
12
+ H
12
Z
22
= 0 or
1 12
11 12
22 −
= − H
H Z Z
. Thus,
1 12
22
. M
Z Z
−
= 12
Now, the second row of 10 provides
21 11
12 22
21 22
21 11
12 22
21 22
. S
H M H
P S
H M H
P T
H M H
T H M
H +
+ +
= +
+ +
And further
22 21
22 22
21 22
. S
H M H
P T
H M H
+ =
+
The last equation holds since H
11
M + H
12
= 0 from the first row case. Again, from 11 we have
21 12
22 22
H Z H Z
I +
=
or
1 1
22 22
21 12
22
. H
Z H Z Z
− −
= −
Together with 12 we then obtain
1 1
22 22
22 22
S Z P T Z
− −
= or, equivalently
1 1
22 22
22 22
. P
Z S T Z
− −
= 13
So far we have already identified M and P. From now we will discover N and Q. Recall the second equation of 8. By application of the generalized Schur decomposition as before we can
process such that ,
, MQ
N N
qA qB
qC Q
MQ N
N SH
TH D
Q φ
φ +
⎡ ⎤
⎡ ⎤ =
+ ⎢
⎥ ⎢ ⎥
⎣ ⎦
⎣ ⎦ +
⎡ ⎤
⎡ ⎤ =
+ ⎢
⎥ ⎢ ⎥
⎣ ⎦
⎣ ⎦ 14
where we define D:= qC. Furthermore, equality in the first row can be expressed in term of the partition matrices as follows:
1 1
1 1
11 11
11 11
11 11
1
. N
H T S H N H T D
φ
− −
− −
− = −
By denoting
1 1
11 11
11 11
: G
H T S H
− −
= − and
1 1
11 11
1
: F
H T D
− −
= − , the last equation can be written as
, N
GN F
φ +
= which can then be transformed into standard linear equation as follows:
vec vec
,
T
I G
N F
φ +
⊗ =
where vec ⋅ denotes vectorizations by columns of a matrix and denotes the Kronecker product. And
furthermore
1
vec vec .
T
N I
G F
φ
−
= +
⊗ 15
Matrix N therefore can be obtained by inverting the vec ⋅. Next, the second row of 14
provides
22 21
22 21
11 22
21 21
11 22
21 2
. S
H M H
Q S H
S H N
T H T H
N D
φ
+ +
+ =
+ +
The solution of Q then is given by
1 1
3 2
, Q
K K
K
−
= −
16 where
1 22
21 22
2 21
11 22
21 3
21 11
22 21
2
: ,
: ,
: .
K S
H M H
K S H
S H N
K T H
T H N
D
φ
= +
= +
= +
+
3. Macroeconomic Structural Equations
