Density forecast evaluation in unstable environments

We propose a density forecast evaluation method in the presence of instabilities, which are defined as breaks in any conditional moment of interest and/or in the functional form of the conditional density of the process. Within the framework of the autocontour-based tests proposed by González-Rivera et al. (2011) and González-Rivera and Sun (2015), we construct Sup- and Ave-type tests, calculated over a collection of subsamples in the evaluation period. These tests have asymptotic distributions that are nuisance-parameter free and they are correctly sized and very powerful for detecting breaks in the parameters of the conditional mean and conditional variance. A power comparison with the tests of Rossi and Sekhposyan (2013) shows that our tests are more powerful across the models considered in their work. We analyze the stability of a dynamic Phillips curve and find that the best one-step-ahead density forecast of changes in inflation is generated by a Markov switching model that allows state shifts in the mean and variance of inflation changes as well as in the coefficient that links inflation and unemployment.