Tests for Unit Roots and the Initial Observation

U of St. Gallen, Econ. Discussion Paper No. 2002-02

48 Pages Posted: 30 Jan 2002

See all articles by Graham Elliott

Graham Elliott

University of California, San Diego (UCSD) - Department of Economics

Ulrich K. Müller

Princeton University - Department of Economics

Date Written: December 2001

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.

Keywords: Unit root tests, point optimal tests, weighted average power, asymptotic distributions

JEL Classification: C12, C22

Suggested Citation

Elliott, Graham and Müller, Ulrich K., Tests for Unit Roots and the Initial Observation (December 2001). U of St. Gallen, Econ. Discussion Paper No. 2002-02, Available at SSRN: https://ssrn.com/abstract=298155 or http://dx.doi.org/10.2139/ssrn.298155

Graham Elliott

University of California, San Diego (UCSD) - Department of Economics ( email )

9500 Gilman Drive
La Jolla, CA 92093-0508
United States
858-534-4481 (Phone)
858-534-7040 (Fax)

Ulrich K. Müller (Contact Author)

Princeton University - Department of Economics ( email )

Princeton, NJ 08544-1021
United States
609-258-3216 (Phone)
609-258-4026 (Fax)

HOME PAGE: http://www.princeton.edu/~umueller

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