The MSPE paradox is not only about MSPE!

Recent literature demonstrates that traditional comparisons of Mean Squared Prediction Error (MSPE) between two competing forecasts may be controversial. This is so because, under certain inefficiency conditions, the forecast with the lowest MSPE coincides with the forecast displaying the lowest correlation with the target variable: this is known as ‘The MSPE Paradox’. In this paper, we show that this paradoxical result may also appear using another popular loss function: Mean Absolute Error (MAE). We show examples of this result for unbiased errors distributed as either Gaussian, t-Student, Uniform, and Logistic. We also present results for biased errors across a range of different distributions. We offer a characterization of ‘paradox zones’ and we further illustrate the relevance of the MAE Paradox with an empirical application when forecasting commodity returns.