Pooling for Combination of Multilevel Forecasts

In this paper, we provide a theoretical analysis of effects of applying different forecast diversification methods on the structure of the forecast error covariance matrices and decomposed forecast error components based on the bias-variance-Bayes error decomposition of James and Hastie. We express the "diversityrdquo of different forecasts in relation to different error components and propose a measure in order to quantify it. We illustrate and discuss typical inhomogeneities frequently occurring in the forecast error covariance matrices and show that previously proposed pooling based only on error variances cannot fully exploit the complementary information present in a set of diverse forecasts to be combined. If covariance values could be reliably calculated, they could be taken into account during the pooling process. We study the difficult case in which covariance information cannot be measured properly and propose a novel simplified representation of the covariance matrix, which is only based on knowledge about the forecast generation process. Finally, we propose a new pooling approach that avoids inhomogeneities in the forecast error covariance matrix by considering the information contained in the simplified covariance representation and compare it with the error-variance-based pooling approach introduced by Aiolfi and Timmermann. Applying our approach more than once leads to the generation of multistep and multilevel forecast combination structures, which have generated significantly improved forecasts in our previous extensive experimental work; the summary of which is also provided.