Economic Models - Convex Optimization

210 Athanasios Athanasenas
Table 1. Determination of the value of p.
Lag length p
LR
Value of p (probability)
1 — —
2 132.91 0.0000
3 14.988 0.0047
4 14.074 0.0071
5 2.7328 0.6035
6 5.1465 0.2726
7 2.3439 0.6728
8 5.2137 0.2661
specified in Eq. (6b) has the following form:
[ Y i W i t i ]
−0.092971 0.028736 0.000320
˜ =
. (7)
0.016209 −0.024594 0.000124
4.2. Dynamic System Stability
TheVAR(4) can be transformed to an equivalent first-order dynamic system,
in the following way.
⎡ ⎤ ⎡
⎤ ⎡ ⎤
x i A 1 A 2 A 3 A 4 x i−1
⎢ Lx i
⎣ L 2 ⎥
x i
⎦ = ⎢ I 2 0 0 0
⎥ ⎢ Lx i−1
⎣ 0 I 2 0 0 ⎦ ⎣ L 2 ⎥
x i−1
⎦
L 3 x i 0 0 I 2 0 L 3 x i−1
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
δ µ w i
+ ⎢ 0
⎥
⎣ 0 ⎦ + ⎢ 0
⎥
⎣ 0 ⎦ t i + ⎢ 0
⎥
⎣ 0 ⎦ . (8)
0 0 0
L in this place, denotes the linear lag operator, such that L k z i = z i−k .
Equation (8) can be written in a compact form as:
y i = Ãy i−1 + d + zt i + v i
(8a)
where
y i ′ = [x i Lx i L 2 x i L 3 x i ], d ′ = [δ 0 0 0],
z ′ = [µ 0 0 0], v i ′ = [w i 0 0 0]
and the dimension of matrix à is (8 × 8).