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.