Scalar mean square error optimal linear combination of multivariate forecasts

When a forecaster predicts the future value of a certain random variable it is very likely that he will not only forecast that certain variable but he will also forecast other variables from the same field. In the literature on the combination of several individual forecasts univariate approaches have been used almost exclusively. They deal with each forecasted variable at a time. In doing so all the information stemming from the interaction of the variables is neglected. The aim of this report is to show how a set of such multivariate forecasts can be combined efficiently. We will focus on various linear combinations and determine how the combination weights should be chosen optimally with respect to the scalar mean square prediction error (SMSPE) criterion. For this purpose we will assume that the first and second order moments of the joint distribution of target variable and individual forecasts are given. As a by-product linear adjustments of single forecasts are obtained. An example illustrating the potential inherent in the multivariate approaches compared to the classical univariate methods is presented. The performance of these methods has to be reassessed if the moments of the joint distribution are unknown and have to be estimated. Further investigations have to be carried out.