Comparing density forecasts via weighted likelihood ratio tests

We propose a test for comparing the out-of-sample accuracy of competing density forecasts of a variable. The test is valid under general conditions: The data can be heterogeneous and the forecasts can be based on (nested or nonnested) parametric models or produced by semiparametric, nonparametric, or Bayesian estimation techniques. The evaluation is based on scoring rules, which are loss functions defined over the density forecast and the realizations of the variable. We restrict attention to the logarithmic scoring rule and propose an out-of-sample "weighted likelihood ratio" test that compares weighted averages of the scores for the competing forecasts. The user-defined weights are a way to focus attention on different regions of the distribution of the variable. For a uniform weight function, the test can be interpreted as an extension of Vuong's likelihood ratio test to time series data and to an out-of-sample testing framework. We apply the tests to evaluate density forecasts of U.S. inflation produced by linear and Markov-switching Phillips curve models estimated by either maximum likelihood or Bayesian methods. We conclude that a Markov-switching Phillips curve estimated by maximum likelihood produces the best density forecasts of inflation.