Title: A general inversion theorem with applications to integrated processes
Authors: Paolo Paruolo - European Commission Joint Research Centre (Italy) [presenting]
Massimo Franchi - University of Rome La Sapienza (Italy)
Abstract: The aim is to present generalizations of Granger's Representation Theorem for I(1) processes and of Johansen's Representation Theorem for I(2) processes to processes integrated of any order. These generalizations exploit novel results on the inversion of matrix functions that are singular at a given point (labelled extended local rank factorization, ELRF) to provide extensions and a unifying approach to alternative representations of linear systems integrated of any order. In particular, for linear systems integrated of any order, the ELRF gives explicit expressions for the coefficients of the vector AutoRegressive and Equilibrium Correction representations in terms of their Moving Average representation. Vice-versa, for vector AutoRegressive processes, it gives explicit expressions for the coefficients of the Moving Average and the Common Trends representations. The elrf encompasses earlier inversion results for I(1) and I(2) autoregressive systems, and it is shown to be related to the Jordan pairs, Jordan chains and the local Smith form; this allows us to provide explicit links to previous representations.