Equivalence of optimal forecast combinations under affine constraints

Forecasts are usually produced from models and expert judgements. The reconciliation of different forecasts presents an interesting challenge for managerial decisions. Mean absolute deviations and mean squared errors scoring rules are commonly employed as the criteria of optimality to aggregate or combine multiple forecasts into a consensus forecast. While much is known about mean squared errors in the context of forecast combination, little attention has been given to the mean absolute deviation. This paper establishes the first-order condition and the optimal solutions from minimizing mean absolute deviation. With this result, the paper derives the conditions in which the optimal solutions for minimizing mean absolute deviation and mean squared error loss functions are equivalent. More generally, this paper derives a sufficient condition which ensures the equivalence of optimal solutions of minimizing different loss functions under the same affine constraint that each feasible solution must sum to one. A simulation study and an illustration using expert forecasts data corroborate the theoretical findings. Interestingly, the numerical analysis shows that even with skewness in the data, the equivalence is unaffected. However, when outliers are presented in the data, mean absolute deviation is more robust than the mean squared error in small samples, which is consistent with the conventional belief relating the two loss functions.