Optimal Forecast Combination with Mean Absolute Error Loss

The optimal aggregation of forecasts produced either from models or expert judgements presents an interesting challenge for managerial decisions. Mean absolute error (MAE) and mean squared error (MSE) losses are commonly employed as criteria of optimality to obtain the weights that combine multiple forecasts. While much is known about MSE in the context of forecast combination, less attention has been given to MAE. This paper shows that the optimal solutions from minimizing either MAE or MSE loss functions, i.e., the optimal weights, are equivalent provided that the weights sum to one. The equivalence holds under mild assumptions and includes a wide class of symmetric and asymmetric error distributions. The theoretical results are supported by a numerical study that features skewed and fat-tailed distributions. The practical implications of combining forecasts with MAE and MSE optimal weights are investigated empirically with a small sample of data on expert forecasts on inflation, growth, and unemployment rates for the European Union. The results show that MAE weights are less sensitive to outliers, and MSE and MAE weights can be close to equivalent even when the sample is small.