A nondegenerate Vuong test and post selection confidence intervals for semi/nonparametric models

This paper proposes a new model selection test for the statistical comparison of semi/non‐parametric models based on a general quasi‐likelihood ratio criterion. An important feature of the new test is its uniformly exact asymptotic size in the overlapping nonnested case, as well as in the easier nested and strictly nonnested cases. The uniform size control is achieved without using pretesting, sample‐splitting, or simulated critical values. We also show that the test has nontrivial power against all<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msqrt><mi>n</mi></msqrt></math>‐local alternatives and against some local alternatives that converge to the null faster than<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msqrt><mi>n</mi></msqrt></math>. Finally, we provide a framework for conducting uniformly valid post model selection inference for model parameters. The finite sample performance of the nondegenerate test and that of the post model selection inference procedure are illustrated in a mean‐regression example by Monte Carlo.