The Integration Order of Vector Autoregressive Processes



4. Equivalence with the standard I(1) and I(2) conditions

We want to prove the equivalence with the standard conditions in
Johansen (1996) and derive the explicit expression of
H(1). We intro-
duce the following notation: let
A be the orthogonal complement of
an
m × n matrix A of rank n < m ,let A = A ( AA )- 1 and write the
Taylor expansion of Π(
z) at z = 1 as

Π(г) = Π(1) + Π(1)(г - 1) + Π21)(г - 1)2 + (г - l)3Π3(г).

The order of integration is established (Johansen, 1996) by some re-
duced and full rank conditions on specific matrices:
Xt is I(1) if and
only if
IΠ(1) I = 0 and '/ Π(1)(l = 0; similarly, the I(2) condition
states that the order of integration is two if and only if
Π(1) = 0,
(0lΠ(1)(l = 0 and α2θβ21 = 0 where θ = -ɑ + ∏(1)βa0∏(1),
α2 = (l^l, β2 = (LnL, (0lΠ(1)(l = ξη' and ξ, η are p — r × s matrices
of full rank
s < p - r. Using (3.1) and Theorem 3.3 we rewrite the
identity Π(
гa(г) = Πa(г)Π(г) = Π(г)Ip as

(4.1)                  Π( г ) H ( г ) = ( г — 1) dg ( г ) Ip

and H(г)Π(г) = (г — 1)dg(г)Ip. At г = 1 they read αβ0H(1) =
H (1)0 = 0p and mean that

H(1) =βlζd(0l,

for some p — r × p — r matrix ζd of rank 0 < rd ≤ p — r.

Proposition 4.1 (I (1) case). A necessary and sufficient condition for
(0lΠ(1)(l = 0 is that

a=m—1

in Πa(г) = (г — 1)aH(г) and H(1) = 0p. The explicit expression for
H
(1) is

H (1) = g (1) (l ( (0l Π(1) (l ) -1 αL.

Proof. Assume a = m — 1 so that d = 1; differentiate (4.1) at г = 1
to get ∏(1)
H(1) — αβ0H(1) = g(1)Ip and thus

αL ∏(1) β±ζ 1 = g (1) Ip-r.

Then g(1) = 0 implies ζ 11 = 0, αL∏(1)(l = 0 and ζ 1 = g(1)((0lΠ(1)(l)-1,
and thus the
I(1) condition is satisfied.

Assume now the I(1) condition holds and suppose d = m — a > 1;
differentiating (4.1) at
г = 1 we get αL∏(1)( ,zd = 0p-r ; since ζd = 0p-r
this contradicts (0l∏(1)(l = 0 and implies m — a = 1. ■



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