Representations of I(2) cointegrated systems using the Smith-McMillan form

This paper presents a discussion of cointegration amongst I(2) variables and provides a synthesis of various ways I(2) cointegrated systems may be characterized and represented. Following Yoo (1986), Engle and Yoo (1991) and Salmon (1988) we use the Smith-McMillan form of a rational polynomial matrix as a unifying framework to describe the null-space structure of I(2)-cointegrated systems and show how different representations such as the autoregressive and error correction representations, the common stochastic trends representation and various triangular array decompositions, can be derived. Hence we extend the I(1) results of Hylleberg and Mizon (1989) to I(2) systems. The different representations provide different insights into distinct features of multivariate systems that may simultaneously contain several types of equilibrium behaviour that is more complex than that found with I(1) systems. We also discuss how appropriately defined state variables may ease the interpretational difficulties that may arise in polynomially cointegrated systems.