An Unbiased Model Comparison Test Using Cross-Validation

Abstract

Political scientists often consider multiple empirical models of the same process. When these models are parametric and non-nested, the null hypothesis that two models fit the data equally well is commonly tested using methods introduced by Vuong (1989) and Clarke (2003, 2007). The objective of each is to compare the Kullback-Leibler Divergence (KLD) of the two models from the true model that generated the data. In this research note we show that both of these tests are based upon a biased estimator of the KLD, the individual log-likelihood contributions, and that the Clarke test is not proven to be consistent for the difference in KLDs. As a solution, we derive a test based upon cross-validated log-likelihood contributions, which represent an unbiased KLD estimate.

Keywords: Model Selection, Cross-Validation, Kullback-Leibler Divergence, Vuong Test, Clarke Test, Linear Regression, Ordinary Least Squares, Robust Regression