Linear Plus Quadratic Approach to the Mean Square Error Optimal Combination of Forecasts

Abstract

This paper deals with linear plus quadratic approaches aiming to find a combined forecast for a scalar random variable from several individual forecasts for that variable. When combining forecasts linear approaches have been used predominantly. One reason may be the well-known fact that the linear approach with constant term is optimal with respect to the mean square prediction error loss, if the single forecasts and the target variable follow a joint normal distribution. In this paper no assumption is made on the type of the joint distribution. Its moments up to order four, however, are assumed to be given for the derivation of the optimal combination parameters. Three versions for the quadratic part of the combined forecast are discussed. As a by-product a linear plus quadratic adjustment of a single forecast is obtained. In order to apply these methods to empirical data the moments of the joint distribution have to be estimated.