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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Thursday, August 10, 2000}
\centerline{Sequoia Hall Rm. 200}

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\centerline{\sl Bent Nielsen}
\centerline{\sl Oxford University}
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\centerline{\bf The Asymptotic Distribution of Unit Root Tests of Unstable
Autoregressive Processes}
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The idea of unit root testing is to test the hypothesis that the
differences of an observed time series do not depend on its levels, or in
other words, the levels of the time series has a unit root which can be
removed by differencing. While it is in general possible to have multiple
unit roots only the hypothesis of exactly one unit root is considered here.
The available tests therefore hinge on two assumptions: (i) the levels of
the time series has exactly one unit root which can be removed by
differencing, and (ii) the remaining characteristic roots of the time
series are stationary roots. In this paper it is proved that for the
likelihood ratio test and a number of other likelihood based statistics the
assumption (ii) is redundant whereas (i) is necessary. It is also shown
that for some tests which are not likelihood based it is indeed necessary
to assume that the  differences have stationary roots.  Possible
applications will be discussed. 

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