REPRESENTATIONS OF I(2) COINTEGRATED SYSTEMS USING THE SMITH-MCMILLANFORM

This paper presents a discussion of cointegration amongst I(2) variabl es 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 ratio nal polynomial matrix as a unifying framework to describe the null-spa ce structure of I(2)-cointegrated systems and show how different repre sentations such as the autoregressive and error correction representat ions, the common stochastic trends representation and various triangul ar array decompositions, can be derived. Hence we extend the I(1) resu lts of Hylleberg and Mizon (1989) to I(2) systems. The different repre sentations provide different insights into distinct features of multiv ariate systems that may simultaneously contain several types of equili brium behaviour that is more complex than that found with I(1) systems . We also discuss how appropriately defined state variables may ease t he interpretational difficulties that may arise in polynomially cointe grated systems. (C) 1998 Elsevier Science S.A. All rights reserved.