THE INTEGRATION ORDER OF VECTOR
AUTOREGRESSIVE PROCESSES
MASSIMO FRANCHI
UNIVERSITY OF COPENHAGEN
AND
UNIVERSITY OF ROME “LA SAPIENZA”
Abstract. We show that the order of integration of a vector au-
toregressive process is equal to the difference between the mul-
tiplicity of the unit root in the characteristic equation and the
multiplicity of the unit root in the adjoint matrix polynomial. The
equivalence with the standard I(1) and I(2) conditions (Johansen,
1996) is proved and polynomial cointegration discussed in the gen-
eral setup.
1. Introduction
An autoregressive process is integrated of order d, if its characteristic
equation has d roots at z = 1 and the remaining lie outside the unit
circle. This is not true in the multivariate case, because the order of
integration of a vector autoregressive processes is not established by
the multiplicity of the unit root in the characteristic equation. For this
reason, some extra conditions are needed in order to write down the
moving average representation. Johansen (1988, 1992) imposes neces-
sary and sufficient conditions on the parameters of the autoregressive
process and derives the corresponding moving average representation
for I(1) and I(2) processes. His work is related to Engle and Granger
(1987), who start from the moving average representation of an I(1)
process which exhibits cointegration and derive the corresponding error
correction model; unfortunately the proof of the Granger Representa-
tion Theorem is not correct (see Johansen (2005a) for a counterexample
to Lemma 1). Other proofs of the same theorem are based on the Smith
Date : February 17, 2006. I am very grateful to S0ren Johansen for his precious
insights and his continuous help throughout the development of the paper. E-mail
address: [email protected].