Empirically-transformed Linear Opinion Pools

Many studies have found that combining density forecasts improves predictive accuracy
for macroeconomic variables. A prevalent approach known as the Linear Opinion Pool
(LOP) combines forecast densities from “experts”; see, among others, Stone (1961),
Geweke and Amisano (2011), Kascha and Ravazzolo (2011), Ranjan and Gneiting
(2010) and Gneiting and Ranjan (2013). Since the LOP approach averages the experts’
probabilistic assessments, the distribution of the combination generally differs from the
marginal distributions of the experts. As a result, the LOP combination forecasts
sometimes fail to match salient features of the sample data, including asymmetries in
risk. In this paper, we propose a computationally convenient transformation for a target
macroeconomic variable with an asymmetric marginal distribution. Our methodology
involves a Smirnov transform to reshape the LOP combination forecasts using a
nonparametric kernel-smoothed empirical cumulative distribution function. We illustrate
our methodology with an application examining quarterly real-time forecasts for US
inflation based on multiple output gap measures over an evaluation sample from 1990:1
to 2017:2. Our proposed methodology improves combination forecast performance by
approximately 10% in terms of both the root mean squared forecast error and the
continuous ranked probability score. We find that our methodology delivers a similar
performance gain for the Logarithmic Opinion Pool (LogOP), a commonly-used
alternative to the LOP.