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Wiley InterScience | ||
  Journal of Time Series AnalysisVolume 21 Issue 3, Pages 249 - 260 Published Online: 4 Jan 2002 © 2009 Blackwell Publishing Ltd
Abstract | Full Text: PDF (Size: 209K) | Related Articles | Citation Tracking  The Limiting Density of Unit Root Test Statistics: A Unifying Technique Copyright Blackwell Publishers Ltd 2000 KEYWORDS unit root tests • weak convergence • partial sum process • ordinary and fractional Brownian motion • Karhunen-Loeve expansions • wavelets • normal type integrals with complex parameters ABSTRACTIn this note we introduce a simple principle to derive a constructive expression for the density of the limiting distribution, under the null hypothesis, of unit root statistics for an AR(1)-process in a variety of situations. We consider the case of unknown mean and reconsider the well-known situation where the mean is zero. For long-range dependent errors we indicate how the principle might apply again. We also show that in principle the method also works for a near unit root case. Weak convergence and subsequent Karhunen-Loeve expansion of the weak limit of the partial sum process of the errors plays an important role, along with the evaluation of a certain normal type integral with complex mean and variance. For independent and long range dependent errors this weak limit is ordinary and fractional Brownian motion respectively. AMS 1991 subject classification. Primary 62M10; secondary 62E20.
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