Some Theoretical Results on Forecast Combinations

This paper proposes a unified framework to analyse the theoretical properties of forecast combination. The proposed framework not only is useful for deriving all existing results with ease but also provides important insights into two unresolved puzzles of forecast combination. Specifically, this paper aims to explain why a simple average of forecasts often outperforms forecasts from single models in the sense of mean squared forecast errors (MSFE) and to determine why a more complicated weighting scheme does not always perform better than a simple average. While this paper obtains several new theoretical results, two of them are particularly important in practice. First, the MSFE of forecast combination decreases as the number of models increases. Second, the conventional approach to selecting optimal models based on a simple comparison of MSFEs without further statistical testing will lead to biased results