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: X_t 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,
∣(⁰_lΠ(1)(_l∣ = 0 and ∣α2θβ21 = 0 where θ = -ɑ + ∏(1)βa0∏(1),
α2 = (_l^_l, β2 = (_Ln_L, (⁰_lΠ(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(г)Π_a(г) = Π_a(г)Π(г) = ∣Π(г)∣I_p as

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

and H(г)Π(г) = (г — 1)^dg(г)I_p. At г = 1 they read αβ⁰H(1) =
H (1)(β ⁰ = 0_p 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
∣(⁰_lΠ(1)(l∣ = 0 is that

a=m—1

in Π_a(г) = (г — 1)^aH(г) and H(1) = 0_p. 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)I_p 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)((⁰_lΠ(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)( ,z_d = 0_p-r ; since ζ_d = 0_p-r
this contradicts ∣(⁰_l∏(1)(_l∣ = 0 and implies m — a = 1. ■

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