مقاله سال ۲۰۱۶

The question of finding weights for combining density forecasts is non-trivial, and is currently being debated in the forecast combination literature. The latest work in this area is by Kapetanios, Mitchell, Price, and Fawcett (2015), and examples of early contributions are provided by Tay and Wallis (2000)and Corradi and Swanson (2006). The reader is also invited to peruse the review by Timmermann (2006) for a thorough review of the forecast combination literature.

In a recent paper, Hall and Mitchell (2007) propose a practical way of obtaining weights in a linear combination of density forecasts. The weights are found by maximizing the average logarithmic score of the combined density forecast. Hall and Mitchell (2007) call these weights ‘‘optimal’’ because they minimize the ‘‘distance’’ between the forecast and true (but unknown) densities, as measured by the Kullback–Leibler Information Criterion (KLIC).

Although Hall and Mitchell(2007) show how these weights can be used, the paper does not detail the theoretical properties of the estimators of the weights. The motivation for the study relies on asymptotic theory, namely that the number of time periods grows to infinity (T → ∞). Geweke and Amisano (2011) propose an approach that is similar to that of Hall and Mitchell (2007) using Bayesian methods, and provide a theoretical justification for the use of optimal linear combinations.