Abstract
This paper discusses the notion of cointegrating space for linear processes integrated of any order. It first shows that the notions of (polynomial) cointegrating vectors and of root functions coincide. Second, it discusses how the cointegrating space can be defined (i) as a vector space of polynomial vectors over complex scalars, (ii) as a free module of polynomial vectors over scalar polynomials, or finally (iii) as a vector space of rational vectors over rational scalars. Third, it shows that a canonical set of root functions can be used as a basis of the various notions of cointegrating space. Fourth, it reviews results on how to reduce polynomial bases to minimal order—i.e., minimal bases. The application of these results to Vector AutoRegressive processes integrated of order 2 is found to imply the separation of polynomial cointegrating vectors from non-polynomial ones.
Highlights
In their seminal paper, Engle and Granger (1987) introduced the notion of cointegration and of cointegrating (CI) rank for processes integrated of order 1, or I(1)
This section motivates the study of the represention of cointegrating vectors in terms of bases of suitable spaces, for systems integrated of order two, which are more formally introduced in Section 3 below
Results in Gohberg et al (1993) shows that the order of a root functions is finite, because it is bounded by the order of zω as a zero of det C (z), a result that is reported in the proposition
Summary
In their seminal paper, Engle and Granger (1987) introduced the notion of cointegration and of cointegrating (CI) rank for processes integrated of order 1, or I(1).
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