Weighted Likelihood Ratio Scores for Evaluating Density Forecasts in Tails∗

We propose and evaluate several new scoring rules based on likelihood ratios, for comparing density forecasts in the context of VaR modelling and expected loss estimation. Our approach is motivated by the observation that some existing weighted scoring rules tend to favour fat-tailed predictive densities over thin-tailed predictive densities. Rather than restricting the weight functions, we impose some restrictions on the score functions. Our benchmark case has fixed weights, equal to one in the left tail and zero elsewhere. Two different scoring rules based on partial likelihood are proposed for this zero-one case. After generalizing the new scoring rules to smooth weight functions, their properties are investigated numerically and illustrated by an empirical application.