This paper develops a classical approach to model selection using the Kullback-Leibler Information Criterion (KLIC) to measure the closeness of a model to the truth. The author proposes likelihood-ratio-based statistics for testing the null hypothesis that competing models are equally close to the true data-generating process against the alternative hypothesis that one model is closer. The tests are directional and are derived for cases where the competing models are non-nested, overlapping, or nested, and whether both, one, or neither model is misspecified. The asymptotic distribution of the likelihood ratio statistic is characterized under general conditions, showing it to be a weighted sum of chi-square or normal distributions depending on whether the distributions in the competing models closest to the truth are observationally identical. The paper also proposes a test for this condition. The approach is applied to derive new and directional likelihood ratio-based tests for model selection in all possible situations, including strictly non-nested, overlapping, and nested models. The results are compared with those of Akaike and Cox, and the paper concludes with directions for further research.This paper develops a classical approach to model selection using the Kullback-Leibler Information Criterion (KLIC) to measure the closeness of a model to the truth. The author proposes likelihood-ratio-based statistics for testing the null hypothesis that competing models are equally close to the true data-generating process against the alternative hypothesis that one model is closer. The tests are directional and are derived for cases where the competing models are non-nested, overlapping, or nested, and whether both, one, or neither model is misspecified. The asymptotic distribution of the likelihood ratio statistic is characterized under general conditions, showing it to be a weighted sum of chi-square or normal distributions depending on whether the distributions in the competing models closest to the truth are observationally identical. The paper also proposes a test for this condition. The approach is applied to derive new and directional likelihood ratio-based tests for model selection in all possible situations, including strictly non-nested, overlapping, and nested models. The results are compared with those of Akaike and Cox, and the paper concludes with directions for further research.