Comparing Density Forecasts via Weighted Likelihood Ratio Tests: Asymptotic and Bootstrap Methods

This paper proposes tests for comparing the accuracy of density forecasts. The evaluation makes use of scoring rules, which are loss functions defined over the density forecast and the realizations of the variable. In particular, a logarithmic scoring rule leads to the development of asymptotic and bootstrap 'weighted likelihood ratio' tests. I conclude with an application to S&P500 daily returns, comparing the performance of density forecasts obtained from GARCH models with different distributional assumptions.