Some Non-Nested Hypothesis Tests and the Relations Among Them

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Abstract
This paper discusses several statistical techniques which can be used to test the validity of a possibly nonlinear and multivariate regression model, using the information provided by estimating one or more alternative models on the same set of data. The techniques we propose can be regarded as alternative implementations of Cox's idea for non-nested hypothesis testing; under the null hypothesis, all of the test statistics are asymptotically the same random variable. For the univariate linear regression case, our test and Pesaran's has asymptotic relative efficiency of unity for local alternatives. Finally, we present sampling experiments for univariate linear models which show that the small-sample performance of our J test and Pesaran's test can be quite different.
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