On Optimal Covariance Matrix Shrinkage Levels in Forecast Combination

Forecast combination is an established technique to improve forecast accuracy and enterprise planning, where a key research question is (still) how to weight individual forecasts. One common but largely unsuccessful approach is to learn weights that minimize the mean squared error (MSE) on known observations, usually from (instable) sample covariance matrices of past errors. These weights are then shrunk to mitigate over-fitting and avoid high errors when using the weights in novel forecasts. This can be done by shrinking the sample covariance matrix to a less flexible matrix, e.g. the unit diagonal matrix, where even formulas for the shrinkage level minimizing the expected deviation between the shrunk and the true covariance matrix exist. We provide analyses with synthetic error data showing that such shrink-levels generally do not lead to MSE-minimizing weights and argue that adjusted shrinkage criteria or machine-learning-based shrinkage tuning is advised to successfully apply such approaches in forecast combination.