In testing for the cointegrating rank of a vector autoregressive
process it is important to take into account level shifts that have
occurred in the sample period. Therefore the properties of estimators of
the time period where a shift has taken place are investigated. The
possible structural break is modeled as a simple shift in the level of the
process. Two alternative estimators for the break date are considered, and
their asymptotic properties are derived under various assumptions
regarding the size of the shift. In particular, properties of the shift
date estimators are obtained under the assumption of an increasing or
decreasing size of the shift when the sample size grows. These results are
used to explore the implications for testing the cointegrating rank of the
process. A previously proposed likelihood ratio type test for the
cointegrating rank and a modified version are considered, and their
asymptotic properties are derived. It is shown that their asymptotic null
distributions are unaffected by the level shift under the assumptions made
for the shift size. The performance of the shift date estimators and the
cointegrating rank tests in small samples is investigated by
simulations.We thank two referees for
helpful comments, and we are grateful to the Deutsche
Forschungsgemeinschaft, SFB 373, and the European Commission under the
Training and Mobility of Researchers Programme (contract ERBFMRXCT980213)
for financial support. The first author also acknowledges financial
support by the Yrjö Jahnsson Foundation, the Academy of Finland, and
the Alexander von Humboldt Foundation under a Humboldt research award.
Part of this research was done while he was visiting the Humboldt
University in Berlin, and part of the research was carried out while he
and the third author were visiting the European University Institute,
Florence. An extended version of this paper is available as an EUI
discussion paper under the title “Break Date Estimation and
Cointegration Testing in VAR Processes with Level Shift,” ECO
2004/21.