Brown Bag (Lunch) seminar: Combination of Forecasts with Minimum Mean Absolute Errors
Mean absolute errors and mean squared error are among the two most widely used forecast accuracy measures in forecasting work. This paper demonstrates that the optimal solutions for mean absolute deviation loss functions and mean squared error loss functions are equivalent. This equivalence is examined in the context of forecast aggregation, that is combining forecasts into a consensus forecast when more than one forecast is available. While the literature has covered the optimal combination of forecasts extensively for mean squared errors, it is sparse with respect to mean absolute deviation. This paper starts by providing the first-order optimal condition for a mean absolute deviation loss function and then shows that the optimal solutions for both loss functions are equivalent when forecast errors are Normally distributed. More practically, it is demonstrated that the optimal solutions are also equivalent for large samples. This paper also provides a set of conditions for when the optimal solution of the mean absolute deviation loss function is an arithmetic average. These results are illustrated in forecasting the growth rate of the U.S. index of industrial production using many macroeconomic leading indicators as predictors. The results show the equivalence between the two loss functions and also that mean absolute deviation presents practical advantage when there are outliers in the data and when using many predictors.