The Integration Order of Vector Autoregressive Processes

These two equivalences allow us to understand the standard I(1) and
I(2) conditions as imposing a particular shape to the adjoint matrix
polynomial, which in turn ensures that the pole at the unit root has
order one or two; in these cases the principal part of the Laurent expan-
sion of (3.2) around z = 1 consists of one or two terms and translates
into a moving average representation which involves the cumulation
(or the double cumulation) of the stationary process that generates the
stochastic trends.

5. Polynomial cointegration

The results of the previous section can thus be interpreted as al-
ternative proofs of the Granger Representation Theorem in the I(1)
and I(2) cases: the order of integration is established by Theorem
3.3, the explicit expression for H(1) indicates the directions in which
cointegration is to be found, and the restrictions implied by (4.1) and
its derivatives define the (polynomial) cointegration properties of the
process. In the I(2) case, for example, we write the inverse of Π(z) as
^π(^z ) 1 =( z -1)22g ( z ) ^,z = ʃ^1, ■■■ ^,zn^r}’
where H(1) 6= 0_p and g(1) 6= 0, and H(z) as

H ( z ) = H (1) + TH(1)( z — 1) + ( z — 1)2 H2( z ).

From the calculations in the proof of Proposition 4.2, we have that

(5.1)               H(1) = β⊥η⊥ψξ⊥α⊥, ∣ψ∣=0,

(5.2)             β 0H(1) = α 0 Π(1) H (1).

Thus the polynomial cointegration properties of the process are the
following:

Proposition 5.1. Let X_t be I(2) and β 1 = β⊥η; then β'X_t and β1X_t
are I(1), and β0X_t + α⁰∏(1)∆X_t is I(0).

Proof. From (5.1) we have that β⁰H(1) = 0_r×p and β₁⁰ H(1) = 0_s×p;
thus the functions β⁰Π(z)^-1 and β₁⁰ Π(z)^-1 have a pole of order one at
z = 1 and correspond to β⁰Xt and β₁⁰ Xt being I (1). Using (5.2) it is
easy to see that the function ∣β⁰ + (1 — z)ex⁰∏(1) ∣ ∏(z) ¹ has no pole
at z = 1 and corresponds to β⁰X_t + α⁰∏(1)∆X_t being I(0). ■

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