Improving density forecasts and value-at-risk estimates by combining densities

We investigate the added value of combining density forecasts for asset return prediction in a specific region of support. We develop a new technique that takes into account model uncertainty by assigning weights to individual predictive densities using a scoring rule based on the censored likelihood. We apply this approach in the context of recently developed univariate volatility models (including HEAVY and Realized GARCH models), using daily returns from the S&P 500, DJIA, FTSE and Nikkei stock market indexes from 2000 until 2013. The results show that combined density forecasts based on the censored likelihood scoring rule significantly outperform pooling based on the log scoring rule and individual density forecasts. The same result, albeit less strong, holds when compared to combined density forecasts based on equal weights. In addition, VaR estimates improve a t the short horizon, in particular when compared to estimates based on equal weights or to the VaR estimates of the individual models.