Higher Moment Constraints for Predictive Density Combinations

Abstract

The majority of financial data exhibit asymmetry and heavy tails, which makes forecasting the entire density critically important. Recently, a forecast combina- tion methodology has been developed to combine predictive densities. We show that combining individual predictive densities that are skewed and/or heavy-tailed results in significantly reduced skewness and kurtosis. We propose a solution to over- come this problem by deriving optimal log score weights under Higher-order Moment Constraints (HMC). The statistical properties of these weights are investigated the- oretically and through a simulation study. Consistency and asymptotic distribution results for the optimal log score weights with and without high moment constraints are derived. An empirical application that uses the S&P 500 daily index returns illustrates that the proposed HMC weight density combinations perform very well relative to other combination methods.