The Integration Order of Vector Autoregressive Processes

The only polynomial cointegrating relation that involves the levels
and reduces the order of integration from two to zero is:

β ⁰Xt + α⁰ ∏(1)∆ Xt.

Note that (5.1) and Ip = ββ⁰ + β₁ β 1 + β₂β2 imply H(1) = /32β°,H(1);
thus the coefficient of the pole in (1 - z)Π(z) is actually β^X2β₂⁰ H (1). This
means that there are terms in α⁰Π(1)∆X_t which do not help eliminate
the non stationarity of / ⁰X_t and thus are not needed. Thus the minimal
choice is to take
• ---

β ⁰Xt + α⁰ Π(1) /32 β2 ∆ Xt

as in Johansen (1992).

Now consider a process integrated of order d. The Taylor expansion
of Π(z) is

and translates into

in terms of the stochastic process. Since ∆^dX_t is I(0) and Π_d(L) is a
finite order polynomial, Πd(L)∆^dXt is stationary.

Then Pn=0 Π(ⁿ)(1) ⁽ ,1/∆ⁿX_t is also stationary. The polynomial
cointegrating relation that involves the levels and reduces the order
of integration from d to zero is simply

βXt + α⁰Π(1)∆Xt - α⁰ ".,.V Xt + ∙∙∙ - (-1)^d-1 α^{0 П} _;d 1)(1,) ∆^d-1 Xt.
2                              (d - 1)!

The difficulties arise when we want to find the minimal representa-
tion (see la Cour, 1998, for the I(3) case). Further investigation is still
needed to find a tractable solution in the general case.

6. Conclusion

This paper has extended the way we understand the order of inte-
gration in the univariate case to vector autoregressive processes. It has
shown that there exists a very natural representation of the inverse of
the characteristic polynomial, in which the order of the pole at the
unit root is explicit. This result significantly simplifies the proof of the
Granger Representation Theorem in the I(1) and I(2) cases.

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