In Pesaran [9], the test developed by Cox for comparing separate families of hypotheses was applied to the choice between two non-nested linear single-equation econometric models. In this paper, the analysis is extended to cover multivariate nonlinear models whenever full information maximum likelihood estimation is possible. This allows formal comparisons not only of competing explanatory variables but also of alternative functional forms. The largest part of the paper derives the results and shows that they are recognizable as generalizations of the single-equation case. It is also shown that the calculation of the test statistic involves very little computation beyond that necessary to estimate the models in the first place. The paper concludes with a practical application of the test to the analysis of the U.S. consumption function and it is demonstrated that formal tests can give quite different results to conventional informal selection procedures. Indeed, in the case examined, five alternative hypotheses, some of which appear to perform quite satisfactorily, can all be rejected using the test.