The Integration Order of Vector Autoregressive Processes

Cointegration and polynomial cointegration are defined as follows

Definition 2.2. The I(d) process X_t is cointegrated if there exists β
such that β⁰Xt is I(b) with b < d. It is polynomially cointegrated if there
exists β_k for k = 0, ∙ ∙ ∙ ,d — 1, such that Pd=0 β'_k∆^kX_t is stationary.

3. Poles, order of integration and multiplicities

The characteristic polynomial of (2.1) is the p × p matrix function

k

Π(z) = Ip — X ΠiZ∖
i=1

and the characteristic equation is defined as ∣ Π(z) ∣ = 0, where ∣Π(z) ∣ =
det(Π(z)) is a polynomial of degree n ≤ kp, ∣Π(z) ∣ = ∑n=₀ d_iz^z. From
∣Π(0)∣ = 1 it follows that zero is not a root of the characteristic equa-
tion. Let nr be the number of distinct roots zi , each with multiplicity
m_i ; the determinant can thus be written as

nr

(3.1)            ∣Π(z)∣ = dn    (z — zi)^mi = (z — 1)^mg(z),

i=1

where g(1) 6= 0 and 1 ≤ m ≤ n.

Assumption 3.1. The only unstable root is at z = 1; that is ∣Π(z)∣ = 0
implies z_i = 1 or ∣z_i ∣ > 1.

Evaluating the characteristic polynomial at the roots of the char-
acteristic equation we get reduced rank matrices; at the unit root we
write Π(1) = -αβ, where α and β are p × r matrices of full rank r < p.

The inverse is defined as the adjoint matrix Πa(z) = Adj(Π(z)) di-
vided by the determinant

(3.2)              Π(z)^-1 = ,z = {1,... ,znr}.

∣Π(z)∣

Since Π(z) has reduced rank at the roots of the characteristic equation,
(3.2) is not defined at z = {1, ∙ ∙ ∙ ,z_nr}. These singularities are known
to be poles but at the moment nothing can be said about their order.

Proposition 3.2. If Assumption 3.1 holds, then Xt is I(d) if and only
if Π(z)^-1 has a pole of order d at z = 1.

Proof. By definition X_t is I(d) if ∆^dX_t = C(L)t_t with C(z) =
_i^∞₌₀ Cizⁱ convergent for ∣z∣ < 1 + δ for some δ > 0 and C(1) 6= 0; then

Π(z)^-1 = C(z)/(1 — z)^d has a pole of order d at z = 1. ■

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