The In-and-Out-of-Sample (IOS) Likelihood Ratio

A new test of model misspecification is proposed, based on the ratio of in-sample and out-of-sample likelihoods. The test is broadly applicable, and in simple problems approximates well known, intuitive methods. Using jackknife influence curve approximations, it is shown that the test statistic can be viewed asymptotically as a multiplicative contrast between two estimates of the information matrix, both of which are consistent under correct model specification. This approximation is used to show that the statistic is asymptotically normally distributed, though it is suggested that p-values be computed using the parametric bootstrap. The resulting methodology is demonstrated with a variety of examples and simulations involving both discrete and continuous data