The Integration Order of Vector Autoregressive Processes



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


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




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