Quantile Aggregation of Density Forecasts

Abstract

Quantile aggregation (or ‘Vincentization’) is a simple and intuitive way of combining probability distributions, originally proposed by S.B. Vincent in 1912. In certain cases, such as under Gaussianity, the Vincentized distribution belongs to the same family as that of the individual distributions and it can be obtained by averaging the individual parameters. This article compares the properties of quantile aggregation with those of the forecast combination schemes normally adopted in the econometric forecasting literature, based on linear or logarithmic averages of the individual densities. Analytical results and Monte Carlo experiments indicate that the properties of quantile aggregation are between those of the linear and the logarithmic pool. Larger differences among the combination schemes occur when there are biases in the individual forecasts: in that case quantile aggregation seems preferable on the whole. The practical usefulness of Vincentization is illustrated empirically in the context of linear forecasting models for Italian GDP and quantile predictions of euro area inflation.

Probability Forecast Combination via Entropy Regularized Wasserstein Distance

“…The solid line represents the performance based on the quantile aggregation, which aggregates all forecasters in the pool. As found by other research papers, e.g., [2,3] the quantile aggregation method generates a decent predictive distribution, which performs slightly better than the ex-post top 4 forecaster. The right panel in Figure 1 shows the historical performance of our proposed approach with various choices of regularization parameter, γ.…”

“…In the univariate case, an equally weighted centroid defined by a Wasserstein metric corresponds to a quantile averaging or vincentized center where quantiles of forecast densities are averaged. The resulting combined density tends to be narrower than the linear opinion rule [1][2][3], which may or not be desirable, depending on the context.…”

“…where P −1 it (•) and P −1 t (•) are the quantile function of agent i and of the combination method, respectively. This forecast aggregation rule is also known as "quantile aggregation" or "Vincentized distribution" [2,3,10]. We prefer the representation of Equation (1) because this definition can be easily extended to higher dimensional densities or mixed data types (e.g., when some inputs are continuous and others are discrete) unlike quantile aggregation.…”

Forecast combinations: an over 50-year review

“…Is it better to average quantiles (as in quantile averaging) or average probabilities (as in linear pooling)? By restricting themselves to simple averages, and Busetti (2017) theoretically and empirically compared the properties of these two combination strategies and suggested that quantile averaging seems overall a preferable and viable approach. attributed this, in part, to the fact that the average probability forecast is in general underconfident while the average quantile forecast is always sharper.…”

A Finite Mixture Modelling Perspective for Combining Experts’ Opinions with an Application to Quantile-Based Risk Measures

“…Furthermore, an important note is the meaning of the word "probabilities" in the quantitative opinion combination literature. Whilst traditionally, "probabilities" means mass or density functions for the discrete case and continuous case, respectively Clemen (1989), in recent years, there has been some evidence that combining quantiles, first suggested by Vincent (1912), might be at least as good as combining probability densities (see Lichtendahl et al (2013), Busetti (2017), Bansal and Palley (2017), Hora et al (2013), Bogner et al (2017, and Jose et al (2013)), despite some criticism from Colson and Cooke (2017). Quantiles combination was also found to be preferable when individual forecasts are biased; see Bamber et al (2016) and Lichtendahl et al (2013).…”