Optimal forecast combinations under general loss functions and forecast error distributions

Existing results on the properties and performance of forecast combinations have been derived in the context of mean squared error loss. Under this loss function empirical studies have generally found that equally-weighted combined forecasts lead to better performance than estimates of optimal forecast combination weights which in turn outperform the best individual predictions. We show that this and other results can be overturned when asymmetries are introduced in the loss function and the forecast error distribution is skewed. We characterize the optimal combination weights for the most commonly used alternatives to mean squared error loss and demonstrate how the degree of asymmetry in the loss function and skews in the underlying forecast error distribution can significantly change the optimal combination weights. We also propose estimation methods and investigate their small sample properties in simulations and in an inflation forecasting exercise.