find m, the number of roots at z = 1 in (3.1) and mij, the number
of roots at z = 1 in entry i, j of Πa(z). Then a = minij mij and the
process is integrated of order d = m - a.
Example 1 (Johansen, 1992): Consider the model
- ( 12 ! Xt + ( 0 1 ! ∆Xt + ( 0 -1 ! ∆2Xt = et,
[24J t U 2 + δ J t U -1 J t
with characteristic polynomial
Π(z) =
-1 - 2 + 2 (1 - - ) !
— 2 + 2 (1 - z ) - 3 + δ (1 - z ) - z - j
and characteristic equation
z2
IΠ(z)I = -(1 - z)(δ +1 - z + 4(1 - z)).
Assumption 3.1 is satisfied if δ = 0 or δ ≥ 3. When δ ≥ 3, m = 1 and
g(z) = δ +1 - z + z42 (1 - z) is such that g(1) = δ. Since Πa(1) = 0 we
have that a = 0 and d = m = 1. When δ = 0, m = 2 and g(z) = 1 +1 z2
is such that g(1) = 54; then Πa (1) = 0 implies d = 2.
Example 2 (Paruolo, 1996): Consider the model
00
Xt = 0 1
U 0
n
0 Xt-1 +
2
0 - 2 ʌ
0 0 Xt-2 + et,
0 -1
with characteristic polynomial
1
Π( z ) = 0
∖ - 2(1 - z )
0 - 2 (1 - z )ʌ
1-z
0 (1 - z)2
and characteristic equation
z2
IΠ(z)I = (1 - z)3(1 - ɪ).
2
Then Assumption 3.1 is satisfied, m = 3 and g(z) = 1 - -4 is such that
from which it is easily seen that a = 1 and thus that d = m - a = 2;
thus the process is integrated of order 2.
g(1) = 4 ; the adjoint matrix polynomial is
Πa(z) =
(1 - z)3
0
U(1 - z)2
0
(1 - z)2(1 - ,)
0
2(1 - z)2 ∖
0
1-z