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The encompassing principle is used to develop a testing framework which unifies the literature on non-nested testing, allowing analysis of the relationship between alternative tests and in particular enabling asymptotic and finite sample equivalences and identities to be established easily when they exist, as well as embracing nested hypothesis testing. The concept of the implicit null hypothesis of a test is introduced to show that the effective acceptance region for some tests extends beyond the acceptance region corresponding to the null of interest, and so such tests can be inconsistent against fixed alternatives closely related to the nominal null and alternative. The analysis is illustrated by an application to two non-nested linear regression models, and it is shown that the conventional F test, as well as all the one degree of freedom non-nested tests, has an encompassing interpretation, and that the F test is a complete parametric encompassing test.
Mizon et al. (1986) studied this question.