2. MAIN RESULT

The definition of order of integration of a p-dimensional stochastic process is as follows.

DEFINITION 2.1. The linear process X_t = C(L)ε_t, ε_t ∼ i.i.d. is called I(0) if

converges for |z| < 1 + δ for some δ > 0 and C(1) ≠ 0_p; if Δ^dX_t is I(0) then X_t is I(d).

Given the autoregressive equation

where ε_t is an independent and identically distributed (i.i.d.) sequence we want to understand when it defines an I(d) process. It is well known that the solution of (2.1) is an integrated process when the roots of the characteristic equation are either one or lie outside the unit circle. So we assume that

where

is the characteristic polynomial of (2.1), |Π(z)| = det(Π(z)), and the roots of g(z) are outside the unit circle. Note that g(1) ≠ 0 implies that m > 0 is the multiplicity of the unit root in the characteristic equation. From |Π(1)| = 0 we have that rank(Π(1)) = r < p and thus that Π(1) can be written as the product of two p × r matrices α and β of full rank r; that is, −αβ′ = Π(1). Moreover, we have the following result.

PROPOSITION 2.2. Let Π_a(z) be the adjoint of Π(z); then

with H(1) ≠ 0_p and a ≥ 0 is the multiplicity of the unit root in the adjoint matrix polynomial.

Proof. The entry j,i of the adjoint is the signed determinant of the p − 1 × p − 1 matrix derived by eliminating row i and column j from Π(z). If rank(Π(1)) = r < p − 1, any p − 1 × p − 1 submatrix of Π(1) is singular and Π_a(1) = 0_p; if instead r = p − 1, there is at least one p − 1 × p − 1 nonsingular submatrix of Π(1) and Π_a(1) ≠ 0_p. Then Π_a(z) = (z − 1)^aH(z) where H(1) ≠ 0_p, and we call a ≥ 0 the multiplicity of the unit root in the adjoint matrix polynomial. █

As a consequence of the reduced rank of Π(1) the factor z − 1 appears both in |Π(z)| and in Π_a(z); this means that Π(z)⁻¹ = Π_a(z)/|Π(z)| is then equal to

with H(1) ≠ 0_p, g(1) ≠ 0 and the pole at the unit root has order m − a > 0.

It is interesting to see that the number of unit roots in the characteristic equation does not determine the order of integration simply because the factor z − 1 appears both at the numerator and at the denominator of Π(z)⁻¹ and cancels. Exactly as it is in the univariate case the order of integration is equal to the order of the pole of the inverse at the unit root. Then, as the next theorem states, what is relevant is the difference between the two multiplicities and not the number of unit roots in the characteristic equation alone.

THEOREM 2.2. The process X_t in (2.1) is integrated of order

with m being the multiplicity of the unit root in |Π(z)| and a the multiplicity of the unit root in Π_a(z).

Proof. The roots of g(z) being outside the unit circle imply that the function C(z) = H(z)/g(z) is well defined for |z| < 1 + δ for some δ > 0. Thus

converges on the same disc, and it is such that C(1) = H(1)/g(1) ≠ 0_p; this means that Δ^m−aX_t is I(0) and thus that X_t is integrated of order d = m − a. █

We know how to calculate the roots of a polynomial; then it is clear that we can find m in |Π(z)| and the number of roots a_ij in each entry of Π_a(z). Then a = min_ij a_ij, and the process is integrated of order d = m − a.

Example 1 (Johansen, 1992)

The characteristic polynomial

has adjoint

and determinant

The basic assumption about the roots is satisfied if δ = 0 or δ ≥ 3. When δ ≥ 3, g(1) = δ and m = 1. Because Π_a(1) ≠ 0_p we have that a = 0 and d = m = 1. When δ = 0,

and m = 2; then Π_a(1) ≠ 0_p implies d = 2.

Example 2 (Modification of Paruolo, 1996)

The characteristic polynomial

has adjoint

and determinant

The basic assumption about the roots is satisfied, and m = 4. Because a = min_ij a_ij = 1, Π_a(z) = (1 − z)H(z), and the process is integrated of order d = m − a = 3.

3. EQUIVALENCE WITH THE I(1) AND I(2) CONDITIONS

Next we want to show the equivalence of m − a = 1 and m − a = 2 with the standard rank conditions in Johansen (1996). They are stated as follows:

, and it is I(2) if and only if

where ξ,η are p − r × s matrices of full rank s < p − r and |ξ_⊥′α_⊥′θβ_⊥η_⊥| ≠ 0 where

.

The identity Π(z)Π_a(z) = Π_a(z)Π(z) = |Π(z)|I_p is written as

evaluated at z = 1 this gives αβ′H = Hαβ′ = 0_p, and because H ≠ 0_p it implies

for some matrix φ_d ≠ 0_p−r.

PROPOSITION 3.1 (I(1) condition). Let m and a be the multiplicities of the unit root in |Π(z)| and in Π_a(z); then

with −αβ′ = Π.

Proof. Differentiate (3.1) at z = 1 and substitute H = β_⊥φ_dα_⊥′ and Π = −αβ′ to have

pre- and postmultiply it by α_⊥′ and α_⊥ to have that

Assume that

; then φ_d ≠ 0_p−r only if m − a = 1 and necessity is proved. Assume now m − a = 1; then

and g ≠ 0 implies |φ₁| ≠ 0 and

and the proof is complete. █

In the next proposition we consider the I(2) case.

PROPOSITION 3.2 (I(2) condition). Let m and a be the multiplicities of the unit root in |Π(z)| and in Π_a(z); then

with −αβ′ = Π,

, and

.

Proof. Let m − a > 1; then

must be reduced rank, and we write

for two p − r × s matrices ξ and η of full rank s < p − r. The first derivative of (3.1) at z = 1 is

and

.

Pre- and postmultiplying (3.3) by α_⊥′ and α_⊥ we have that

and similarly from the other version of the identity that

. Then from ξη′φ_d = φ_dξη′ = 0_p−r and φ_d ≠ 0_p−r it follows that φ_d = η_⊥ψξ_⊥′ for some matrix ψ ≠ 0_p−r−s and thus that H = β_⊥η_⊥ψξ_⊥′α_⊥′.

Note that by premultiplying (3.3) by α′ we have that

The second derivative of (3.1) at z = 1 is

where c = (m − a)(m − a − 1), and premultiplying it by α_⊥′ we have

Using I_p = β_⊥ β_⊥′ + ββ′, (3.2), and (3.4) we see that

and thus that (3.5) becomes

Premultiply by ξ_⊥′, substitute H = β_⊥η_⊥ψξ_⊥′α_⊥′, postmultiply by α_⊥ξ_⊥, and let

to see that

Assume that |ξ_⊥′α_⊥′θβ_⊥η_⊥| ≠ 0; then ψ ≠ 0_p−r−s only if m − a = 2 and necessity is proved. Assume now m − a = 2; then

and g ≠ 0 implies |ψ| ≠ 0 and |ξ_⊥′α_⊥′θβ_⊥η_⊥| ≠ 0 and the proof is complete. █

More complicated rank conditions arise for m − a > 2.