The Cointegrated VAR without Unit Roots: Representation Theory and Asymptotics
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It has been known since Elliott (1998) that efficient methods of inference on cointegrating relationships break down when autoregressive roots are near but not exactly equal to unity. This paper addresses this problem within the framework of a VAR with non-unit roots. We develop a characterisation of cointegration, based on the impulse response function implied by the VAR, that remains meaningful even when roots are not exactly unity. Under this characterisation, the long-run equilibrium relationships between the series are identified with a subspace associated to the largest characteristic roots of the VAR. We analyse the asymptotics of maximum likelihood estimators of this subspace, thereby generalising Johansen's (1995) treatment of the cointegrated VAR with exactly unit roots. Inference is complicated by nuisance parameter problems similar to those encountered in the context of predictive regressions, and can be dealt with by approaches familiar from that setting.
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