Vuong (1989) develops a classical approach to model selection using likelihood ratio tests for non-nested hypotheses. The paper introduces a method to compare competing models based on the Kullback-Leibler Information Criterion (KLIC), which measures the distance between a model and the true data-generating process. The likelihood ratio (LR) statistic is used to test whether one model is closer to the true distribution than another. The tests are directional and account for whether the models are nested, overlapping, or non-nested, and whether they are correctly specified. The paper characterizes the asymptotic distribution of the LR statistic under general conditions. It shows that the distribution depends on whether the models closest to the true distribution are observationally identical. If they are, the LR statistic follows a weighted sum of chi-square distributions; otherwise, it follows a normal distribution. The paper also proposes a test to determine whether the models are observationally identical, based on the variance of the LR statistic. The paper considers three cases: strictly non-nested models, overlapping models, and nested models. For strictly non-nested models, the LR statistic is asymptotically normal under the alternative hypothesis and follows a weighted sum of chi-square distributions under the null hypothesis. For overlapping models, the paper proposes a sequential procedure to test whether the models are observationally identical before conducting the model selection test. For nested models, the paper discusses the implications of the asymptotic distribution of the LR statistic. The paper also compares its approach to those of Akaike and Cox. It shows that the LR statistic can be used to test non-nested hypotheses and that the asymptotic distribution of the LR statistic depends on whether the models are nested or correctly specified. The paper concludes that the LR statistic is a useful tool for model selection and non-nested hypothesis testing, and that its asymptotic distribution can be characterized under general conditions.Vuong (1989) develops a classical approach to model selection using likelihood ratio tests for non-nested hypotheses. The paper introduces a method to compare competing models based on the Kullback-Leibler Information Criterion (KLIC), which measures the distance between a model and the true data-generating process. The likelihood ratio (LR) statistic is used to test whether one model is closer to the true distribution than another. The tests are directional and account for whether the models are nested, overlapping, or non-nested, and whether they are correctly specified. The paper characterizes the asymptotic distribution of the LR statistic under general conditions. It shows that the distribution depends on whether the models closest to the true distribution are observationally identical. If they are, the LR statistic follows a weighted sum of chi-square distributions; otherwise, it follows a normal distribution. The paper also proposes a test to determine whether the models are observationally identical, based on the variance of the LR statistic. The paper considers three cases: strictly non-nested models, overlapping models, and nested models. For strictly non-nested models, the LR statistic is asymptotically normal under the alternative hypothesis and follows a weighted sum of chi-square distributions under the null hypothesis. For overlapping models, the paper proposes a sequential procedure to test whether the models are observationally identical before conducting the model selection test. For nested models, the paper discusses the implications of the asymptotic distribution of the LR statistic. The paper also compares its approach to those of Akaike and Cox. It shows that the LR statistic can be used to test non-nested hypotheses and that the asymptotic distribution of the LR statistic depends on whether the models are nested or correctly specified. The paper concludes that the LR statistic is a useful tool for model selection and non-nested hypothesis testing, and that its asymptotic distribution can be characterized under general conditions.