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Tests for Unit Roots and the Initial Observation

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  • Ulrich K. Müller

    ()

  • Graham Elliott

    ()

Abstract

The paper analyzes the impact of the initial observation on the problem of testing for unit roots. To this end, we derive a family of optimal tests that maximize a weighted average power criterion with respect to the initial observation. We then investigate the relationship of this optimal family to unit root tests in an asymptotic framework. We find that many popular unit root tests are closely related to specific members of the optimal family, but the corresponding members employ very different weightings for the initial observation. The popular Dickey-Fuller tests, for instance, are closely related to optimal tests which put a large weight on extreme deviations of the initial observation from the deterministic component, whereas other popular tests put more weight on moderate deviations. At the same time, the power of the various unit root tests varies dramatically with the initial observation. This paper therefore helps to explain the results of comparative power studies of unit root tests, and allows a much deeper understanding of the merits of particular tests in specific circumstances.

Suggested Citation

  • Ulrich K. Müller & Graham Elliott, 2001. "Tests for Unit Roots and the Initial Observation," University of St. Gallen Department of Economics working paper series 2002 2002-02, Department of Economics, University of St. Gallen.
  • Handle: RePEc:usg:dp2002:2002-02
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    File URL: http://ux-tauri.unisg.ch/RePEc/usg/dp2002/dp0202mueller_ganz.pdf
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    Cited by:

    1. Ulrich K. Müller, 2002. "Size and Power of Tests for Stationarity in Highly Autocorrelated Time Series," University of St. Gallen Department of Economics working paper series 2002 2002-26, Department of Economics, University of St. Gallen.
    2. Muller, Ulrich K., 2005. "Size and power of tests of stationarity in highly autocorrelated time series," Journal of Econometrics, Elsevier, vol. 128(2), pages 195-213, October.

    More about this item

    Keywords

    Unit root tests; point optimal tests; weighted average power; asymptotic distributions;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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