Cointegration without Unit Roots

TL;DR

This paper develops a new framework for identifying and testing cointegration in structural VAR models without requiring exact unit roots, addressing issues of inference failure near unit roots.

Contribution

It introduces a novel impulse response based characterization of cointegration that remains valid without exact unit roots and adapts robust testing methods for this setting.

Findings

New impulse response approach to cointegration without unit roots

Tests that control size effectively despite nuisance parameters

Framework maintains identification of long-run relationships

Abstract

It has been known since Elliott (1998) that standard methods of inference on cointegrating relationships break down entirely when autoregressive roots are near but not exactly equal to unity. We consider this problem within the framework of a structural VAR, arguing this it is as much a problem of identification failure as it is of inference. We develop a characterisation of cointegration based on the impulse response function, which allows long-run equilibrium relationships to remain identified even in the absence of exact unit roots. Our approach also provides a framework in which the structural shocks driving the common persistent components continue to be identified via long-run restrictions, just as in an SVAR with exact unit roots. We show that inference on the cointegrating relationships is affected by nuisance parameters, in a manner familiar from predictive regression; indeed the two problems are asymptotically equivalent. By adapting the approach of Elliott, M\"uller and Watson (2015) to our setting, we develop tests that robustly control size while sacrificing little power (relative to tests that are efficient in the presence of exact unit roots).