We provide a joint treatment of two major problems that surround testing for a unit root in practice, namely uncertainty as to whether or not a linear deterministic trend is present in the data, and uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. In earlier work [Harvey, Leybourne and Taylor, 2008] we proposed methods to deal with trend uncertainty when the initial condition is assumed to be (asymptotically) negligible, together with methods to deal with uncertainty over the initial condition when the form of the trend function was taken as known. In each case we recommended a simple union of rejections-based decision rule. In the first case rejecting the unit root null whenever either of the quasi-differenced (QD) detrended or QD demeaned augmented Dickey-Fuller [ADF] unit root tests yields a rejection, and in the second case if either of the QD and OLS detrended/demeaned ADF tests rejects. Both approaches were shown to work well. In this paper we extend these procedures to allow for both trend and initial condition uncertainty, proposing a four-way union of rejections decision rule based on the QD and OLS demeaned and the QD and OLS detrended ADF tests. This is shown to work well but to lack power, relative to the best available test, in some scenarios. A modification of the basic union, based on auxiliary information including linear trend pre-test statistics, is proposed and shown to deliver significant improvements. A by-product of our analysis is that the power functions of the associated trend function pre-tests are shown to be heavily dependent on the initial condition.
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David I. Harvey, Stephen J. Leybourne and A. M. Robert Taylor
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Posted on Thursday 1st May 2008