High Moment Constraints for Predictive Density Combination

Financial data typically exhibit asymmetry and heavy tails, which makes forecasting the entire density of the returns critically important. We investigate the effects of aggregating, or combining, predictive densities and find that even if the individual densities are skewed and/or heavy-tailed, the combined density often has significantly reduced skewness and kurtosis. This phenomenon has important implications for measuring downside risk in financial assets. When forecasting financial risk, recently proposed combination methods have focused on specific regions of the density support. We propose an alternative approach, which modifies the popular Log-Score weighting scheme by introducing data-driven constraints on the combination weights that control the skewness and kurtosis of the resulting predictive density. An empirical application using S&P 500 daily index returns demonstrates that the corresponding skewness and kurtosis successfully track the respective sample characteristics of the returns over time. Moreover, the proposed approach outperforms its natural competitors at forecasting the 1% Value-at-Risk for a broad range of estimation-window sizes.