Granger's representation theorem: A closed-form expression for I(1) processes

The Granger representation theorem states that a cointegrated vector autoregressive process can be decomposed into four components: a random walk, a stationary process, a deterministic part, and a term that depends on the initial values. In this paper, we present a new proof of the theorem. This proof enables us to derive closed-form expressions of all terms of the representation and allows a unified treatment of models with different deterministic specifications. The applicability of our results is illustrated by examples. For example, the closed-form expressions are useful for impulse response analyses and facilitate the analysis of cointegration models with structural changes. Copyright 2005 Royal Economic Society