Representation of Cointegrated Autoregressive Processes with Application to Fractional Processes

Abstract

We analyze vector autoregressive processes using the matrix valued characteristic polynomial. The purpose of this article is to give a survey of the mathematical results on inversion of a matrix polynomial in case there are unstable roots, to study integrated and cointegrated processes.

The new results are in the I(2) representation, which contains explicit formulas for the first two terms and a useful property of the third. We define a new error correction model for fractional processes and derive a representation of the solution.

Keywords:

• Error correction models
• Fractional autoregressive model
• Granger representation
• Integration of order 1 and 2

JEL Classification:

• C32

ACKNOWLEDGMENTS

I would like to thank the Danish Social Sciences Research Council for continuing support, the members of the ESF-EMM network for discussions, and Morten Ørregaard Nielsen for useful comments. The article is partly based on a lecture presented at the Joint Statistics Meeting, San Francisco 2003, in a session in honour of T. W. Anderson.

Notes

¹A counter example to Lemma 1 in the article by Engle and Granger (1987) is given by taking G(λ) = diag(1 + λ, λ², λ²). What is missing in Lemma 1 is a condition corresponding to the I(1) condition of cointegration (see Johansen, 1996, Theorem 4.5.)