Practical considerations for optimal weights in density forecast combination

The problem of finding appropriate weights to combine several density forecasts is an important issue currently debated in the forecast combination literature. Recently, a paper by Hall and Mitchell (IJF, 2007) proposes to combine density forecasts with optimal weights obtained from solving an optimization problem. This paper studies the properties of this optimization problem when the number of forecasting periods is relatively small and finds that it often produces corner solutions by allocating all the weight to one density forecast only. This paper's practical recommendation is to have an additional training sample period for the optimal weights. While reserving a portion of the data for parameter estimation and making pseudo-out-of-sample forecasts are common practices in the empirical literature, employing a separate training sample for the optimal weights is novel, and it is suggested because it decreases the chances of corner solutions. Alternative log-score or quadratic-score weighting schemes do not have this training sample requirement.