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 = ʃ1, ■■■ ,znr}’
where H(1) 6= 0p 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 Xt be I(2) and β 1 = β⊥η; then β'Xt and β1Xt
are I(1), and β0Xt + α0∏(1)∆Xt is I(0).
Proof. From (5.1) we have that β0H(1) = 0r×p and β10 H(1) = 0s×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 β0Xt + α0∏(1)∆Xt being I(0). ■
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