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The optimal combination of forecasts, detailed in Bates and Granger (1969), has empirically often been overshadowed in practice by using the simple average instead. Explanations of why averaging might in practice work better than constructing the optimal combination have centered on estimation error and the effects variations of the data generating process have on this error. The flip side of this explanation is that the size of the gains must be small enough to be outweighed by the estimation error. This paper examines the sizes of the theoretical gains to optimal combination, providing bounds for the gains for restricted parameter spaces and also conditions under which averaging and optimal combination are equivalent. The paper also suggests a new method for selecting between models that appears to work well with SPF data.
We show that when outcomes are difficult to forecast in the sense that forecast errors have a large common component that (a) optimal weights are not affected by this common component, and may well be far from equal to each other but (b) the relative MSE loss from averaging over optimal combination is small. Hence researchers could well estimate combining weights that indicate that correlations could be exploited for better forecasts only to find that the difference in terms of loss is negligible. The results then provide another explanation for the commonly encountered practical situation of the averaging of forecasts being difficult to improve upon.