  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).
