Combining density forecasts

Abstract

This paper brings together two important but hitherto largely unrelated areas of the forecasting literature, density forecasting and forecast combination. It proposes a practical data-driven approach to the direct combination of density forecasts by taking a weighted linear combination of the competing density forecasts. The combination weights are chosen to minimize the ‘distance’, as measured by the Kullback–Leibler information criterion, between the forecasted and true but unknown density. We explain how this minimization both can and should be achieved but leave theoretical analysis to future research. Comparisons with the optimal combination of point forecasts are made. An application to simple time-series density forecasts and two widely used published density forecasts for U.K. inflation, namely the Bank of England and NIESR “fan” charts, illustrates that combination can but need not always help.