Weighted likelihood ratio scores for evaluating forecast densities in tails

We propose and evaluate several new scoring rules based on likelihood ratios, for comparing forecast densities in the context of VaR modelling and expected loss estimation. Our approach is motivated by the observation that existing scoring rules tend to favour fat-tailed models when compared with thin-tailed models. 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 extreme case. After generalizing these scoring rules to smooth weight functions, their properties are investigated numerically and illustrated by an empirical application.