Bootstrap Determination of the Co-Integration Rank in Vector Autoregressive Models


  • Giuseppe Cavaliere,

    1. Dept. of Statistical Sciences, University of Bologna, via delle Belle Arti 41, I-40126 Bologna, Italy;
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  • Anders Rahbek,

    1. Dept. of Economics, University of Copenhagen, Øster Farimagsgade 5, Building 26, DK-1353 Copenhagen K, Denmark;
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  • A. M. Robert Taylor

    1. School of Economics, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom;
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    • We thank a co-editor and four anonymous referees for their very helpful and constructive comments on previous versions of this paper. We also thank Iliyan Georgiev, Niels Haldrup, Bent Nielsen, Heino Bohn Nielsen, Søren Johansen, and Anders Swensen for many useful discussions on this work. Parts of this paper were written while Cavaliere and Rahbek both visited CREATES, whose hospitality is gratefully acknowledged. Rahbek gratefully acknowledges funding from the Danish Council for Independent Research, Social Sciences (Grant 10-07977). Rahbek is also affiliated with CREATES, funded by the Danish National Research Foundation.


This paper discusses a consistent bootstrap implementation of the likelihood ratio (LR) co-integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying vector autoregressive (VAR) model that obtain under the reduced rank null hypothesis. A full asymptotic theory is provided that shows that, unlike the bootstrap procedure in Swensen (2006) where a combination of unrestricted and restricted estimates from the VAR model is used, the resulting bootstrap data are I(1) and satisfy the null co-integration rank, regardless of the true rank. This ensures that the bootstrap LR test is asymptotically correctly sized and that the probability that the bootstrap sequential procedure selects a rank smaller than the true rank converges to zero. Monte Carlo evidence suggests that our bootstrap procedures work very well in practice.