PRACTITIONERS CORNER: Tests for Cointegration in Models with Regime and Trend Shifts

AuthorBruce E. Hansen,Allan W. Gregory
Publication Date01 Aug 1996
Tests for Cointegration in Models with Regime and
Trend Shifts
Allan W Gregoiy and Bruce E. Hansen*
Recently Gregory and Hansen (1996) developed residual-based tests for
cointegration that are valid against an alternative hypothesis that there
may be one break in the cointegrating vector. The tests are extensions of
the ADF, Z, and Z, tests for cointegration and are non-informative with
respect to the timing of the break. The null hypothesis is the same as
conventional tests (no cointegration). Gregory and Hansen (1996)
considered three alternative models: (i) level shift; (ii) level shift with
trend; and (iii) regime shift (both level shift and slope coefficients can
In this paper we introduce a more general model that permits a trend
shift as well as a regime shift and provide the critical values appropriate
for testing this alternative. This more general alternative may prove more
interesting in some applications and is a natural extension of Perron
(1989) and Zivot and Andrews (1992) who concentrated on the possibility
of a shift in the trend in the context of unit root testing. We provide no
formal proof here of the tests for this model but refer the interested
reader to an appendix in Gregory and Hansen (1996) as a guide to how
this might be accomplished. Since there are no closed-formed solutions
for the limiting distributions, critical values for the tests are calculated by
simulation methods. We follow MacKinnon (1991) and estimate response
surfaces to approximate the appropriate critical values.
The organization of this paper is as follows. In Section II we review the
standard cointegrating model and the model with regime and trend shift.
In Section III we present the tests, and report the critical values and close
in Section IV with some brief remarks.
tThe first author acknowledges financial support from the Social Sciences and Humanities
Research Council of Canada. The second author thanks the National Science Foundation for
financial support.
© Blackwell Publishers 1996. Published by Blackwell Publishers, 108 Cowley Road, Oxford 0X4 uF,
UK & 238 Main Street, Cambridge, MA 02142, USA.

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