Abstract
In this paper we consider the problem of testing for the co-integration rank of a vector autoregressive process in the case where a trend break may potentially be present in the data. It is known that un-modelled trend breaks can result in tests which are incorrectly sized under the null hypothesis and inconsistent under the alternative hypothesis. Extant procedures in this literature have attempted to solve this inference problem but require the practitioner to either assume that the trend break date is known or to assume that any trend break cannot occur under the co-integration rank null hypothesis being tested. These procedures also assume the autoregressive lag length is known to the practitioner. All of these assumptions would seem unreasonable in practice. Moreover in each of these strands of the literature there is also a presumption in calculating the tests that a trend break is known to have happened. This can lead to a substantial loss in finite sample power in the case where a trend break does not in fact occur. Using information criteria based methods to select both the autoregressive lag order and to choose between the trend break and no trend break models, using a consistent estimate of the break fraction in the context of the former, we develop a number of procedures which deliver asymptotically correctly sized and consistent tests of the co-integration rank regardless of whether a trend break is present in the data or not. By selecting the no break model when no trend break is present, these procedures also avoid the potentially large power losses associated with the extant procedures in such cases.
Highlights
Macroeconomic series are typically characterized by piecewise linear trend functions; see, inter alia, Stock and Watson (1996, 1999, 2005) and Perron and Zhu (2005)
As we will show in the simulation results we report in this paper, an un-modelled trend break causes substantial over-sizing in the standard rank tests, consistent with the findings for standard unit root tests in Perron (1989)
Consistent with the theoretical arguments provided by these authors, we found the choice of 2 for the breakpoint parameter in the penalty function gave better finite sample results than a penalty of 1, in that the latter did not appear to penalise the inclusion of the break sufficiently strongly, such that the trend break was retained too often when no break was present, resulting in correspondingly lower power in that case; see the accompanying supplement, Harris et al (2015)
Summary
Macroeconomic series are typically characterized by piecewise linear (or broken) trend functions; see, inter alia, Stock and Watson (1996, 1999, 2005) and Perron and Zhu (2005). In the context of the component DGP of Trenkler et al (2007), a multivariate generalisation of the first difference trend break estimator used in Harris et al (2009) is proposed, along with a corresponding estimator obtained from the levels of the data, while for the Johansen et al (2000) setup a maximum likelihood estimator of the break date is used Based on these break date estimators, for each of the two approaches an information-based method using a Schwarz-type criterion is employed to select between the version of the model which includes a trend break (included at the relevant estimated break date) and that which does not.
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