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Implicit Alternatives and the Local Power of Test Statistics

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Author Info
Russell Davidson
James G. MacKinnon

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Abstract

The local power of test statistics is analyzed by extending the notion of Pitman sequences to sequences of data-generating processes (DGPs) that approach the null hypothesis without necessarily satisfying the alternative hypothesis. Under quite general conditions, the three classical test statistics -- likelihood ratio, Wald, and Lagrange multiplier -- are shown to tend asymptotically to the same random variable under all sequences of local DGPs. The power of these tests depends on the null, the alternative, and the sequence of DGPs, in a simple and geometrically intuitive way. Moreover, for any test statistic that is asymptotically Chi-squared under the null, there exists an "implicit alternative hypothesis" which coincides with the explicit alternative for the classical test statistics, and against which the test statistic will have highest power.

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Publisher Info
Paper provided by Queen's University, Department of Economics in its series Working Papers with number 556.

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Length: 39 pages
Date of creation: 1984
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Publication status: Published in Econometrica, 55, 1987
Handle: RePEc:qed:wpaper:556

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Keywords: power local power local alternatives Pitman sequence classical tests LM test LR test Wald test

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  19. Russell Davidson & James G. MacKinnon, 1988. "Specification Tests Based on Artificial Regressions," Working Papers 707, Queen's University, Department of Economics. [Downloadable!]
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