A corrected Clarke test for model selection and beyond

We introduce a large family of model selection tests based on the expectation of an arbitrary, possibly non-smooth, parametric criterion function of the data. The considered methodology is illustrated for several econometric problems, including linear and quantile regression. It covers the case of strictly locally non-nested models and some overlapping models. The asymptotic theory of the proposed test statistic is stated. A general exchangeable bootstrap scheme allows the evaluation of its limiting law as well as its asymptotic variance. Our framework includes the tests for non-nested model selection of Vuong (1989) and Clarke (2007) as particular cases. We show that the statistic of the latter test is not Binomial distributed as originally stated and we provide its corrected limiting law. In a simulation study, we empirically verify the distributional approximation of our test statistic in a finite sample and examine the empirical level and power of the corresponding model selection tests in various settings. Finally, an analysis of a financial dataset illustrates the proposed model selection procedure at work.