The goal of this paper is to develop formal techniques for analyzing the relative in-sample performance of two competing, misspecified models in the presence of possible data instability. The central idea of our methodology is to propose a measure of the models' local relative performance: the "local Kullback-Leibler Information Criterion" (KLIC), which measures the relative distance or the two (misspecified) likelihoods from the true likelihood at a particular point in time. We then investigate estimation and inference about the local relative KLIC; in particular, we also propose to investigate its stability over time by means of statistical tests. Compared to previous approaches to model selection, which are based on measures of "global performance", our focus is on the entire time path of the models' relative performance, which may contain useful information that is lost when looking for a globally best model. The empirical application provides insights into the time variation in the performance of a representative DSGE model of the European economy relative to that of VARs.
