Alternative Procedures to Discriminate Non Nested Multivariate Linear Regression Models

ABSTRACT

This article builds classical and Bayesian testing procedures for choosing between non nested multivariate regression models. Although there are several classical tests for discriminating univariate regressions, only the Cox test is able to consistently handle the multivariate case. We then derive the limiting distribution of the Cox statistic in such a context, correcting an earlier derivation in the literature. Further, we show how to build alternative Bayes factors for the testing of nonnested multivariate linear regression models. In particular, we compute expressions for the posterior Bayes factor, the fractional Bayes factor, and the intrinsic Bayes factor.

Keywords:

• Bayes factors: Intrinsic
• Posterior
• Fractional
• Non nested hypothesis
• Separate families of hypothesis

Mathematics Subject Classification:

• 62F15
• 62F03

Acknowledgments

The authors are grateful to Aziz Lazraq and Robert Cléroux for valuable comments. Fernandes and Pereira, respectively, thank research fellowships from CNPq and CAPES, Brazil. The usual disclaimer applies.

Notes

¹Atkinson (1970) develops a general theory for testing H ₀:λ = 0 against H ₁:λ = 1. To estimate α₁ under the null hypothesis H ₀, Atkinson uses α_1|0 rather than αˆ₁. However, Atkinson's procedure does not always entail consistent tests (see Pereira, 1977a).

²One could also use other descriptive statistic, e.g., the geometric mean and the median, to summarize the information given by the R partial Bayes factors. Moreover, if R is too large, one may randomly select a sample from the collection of possible training samples.

³We thank M. H. Pesaran for supplying a proof that rests on the properties of the duplication matrix.