Machine-Learning-Based Determination of Steinian Shrinkage Targets and Levels in Forecast Combination

While forecast combination generally improves forecast accuracy, a persisting research question is how to weight individual forecasters. One natural approach is to determine weights that minimize the mean squared error (MSE) on past error observations. Such weights can be computed from the past errors’ sample covariance matrix, which is, however, an unstable estimator of the true covariance matrix, i.e., its variance is high unless there are many error samples - often not given in forecast situations. Hence, the estimation error associated with a sample covariance matrix often leads to overfitted weights and decreased accuracy of novel forecasts derived with such weights, quickly overwhelming potential accuracy gains due to learning weights. A remedy of this overfitting problem is the (Steinian) shrinkage of the sample covariance matrix to a rather inflexible target like the Identity matrix. This decreases the matrix’s variance, albeit at the cost of introducing some bias. To apply Steinian shrinkage, two decisions must be made upfront. First, a suitable structure of the target needs to be set. Second, a shrinkage level must be determined that solves the bias—variance trade-off associated with the sample covariance and target matrix. While Steinian shrinkage exhibits promising outcomes in synthetic data experiments, we are not aware of data-driven approaches to select the target and determine the shrinkage level to be used in forecast combination. In this paper, we propose machine learning-based tuning procedures for selecting targets and tuning shrinkage levels, where experimental analyses show promising results in terms of MSE reductions on unseen data.