The Integration Order of Vector Autoregressive Processes

form of a matrix polynomial (see Engle and Yoo (1991), Ahn and Rein-
sel (1990) and Haldrup and Salomon (1998)) and on the Jordan rep-
resentation of the companion form (see Archontakis (1998) and Bauer
and Wagner (2003)). Other relevant papers in this area are Gregoire
and Laroque (1993) and Gregoire (1999), who discuss polynomial coin-
tegration in a very general setup and Neusser (2000), who points out
some interesting algebraic properties of the I(1) model. An attempt
to characterize explicitly the polynomial cointegration properties of an
I(d) process from its autoregressive representation is la Cour (1998).
See Johansen (2005a) for an exhaustive survey of the mathematical re-
sults concerning the representation theory and Johansen (2005b) for an
application of similar ideas to fractional integration and cofractionality.

In this paper we study the I(d) multivariate case and show that
one can determine the order of integration of a vector autoregressive
process as the difference between the multiplicity of the unit root in
the characteristic equation and the multiplicity of the unit root in the
adjoint matrix polynomial. This result arises from observing that the
reduced rank of the characteristic polynomial at z = 1 translates into
a zero versus non zero statement about the adjoint matrix polynomial.
This then allows to write the inverse in such a way that the order of
the pole at the unit root becomes explicit, resembling what happens in
the univariate case.

The paper is organized as follows: in section 2 we introduce the
VAR(k) model and the standard definitions of integration and cointe-
gration and in section 3 we prove the main Theorem on I(d) processes.
In section 4 we show the equivalence with the standard I(1) and I(2)
conditions (Johansen, 1996) and in section 5 we discuss polynomial
cointegration. The last section contains some concluding remarks.

2. VAR(k) model and standard definitions

Consider the p-dimensional autoregressive model

(²∙¹)         Xt = ^π1 Xt-1 + ^π2Xt-2 + ∙ ∙ ∙ + ^πkXt-k + ^ct^,

or Π(L)X_t = c_t and c_t is an i∙i∙d∙ process.

Definition 2.1. The process Xt = C(L)Ct is stationary if C(z) =
_i^∞₌₀ C_izⁱ converges for |z| < 1 + δ for some δ > 0; it is I(0) when it

is stationary and C(1) 6= 0; it is I (d), d > 0, if ∆^dX_t is I (0).

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