Matrix poyinomials and their inversion: the algebraic framework of unit-root econometrics representation theorems

In this paper the issue of the inversion of a matrix polynomial about a unit root is tackled by restoring to Laurent expansion. The principal-part matrix coefficients associated with a simple and a second order pole are properly characterized and closed-form expressions are derived by virtue of a recent result on partitioned inversion (Faliva and Zoia, 2002). This eventually sheds on the analytical foundation of unit-root econometrics which in turn paves the way to an elegant unified representation theorem for (co)integrated processes up to the second order.