A Kernel Score Perspective on Forecast Disagreement and the Linear Pool

TL;DR

This paper extends linear pooling results to all kernel scores, showing how forecast disagreement impacts performance and providing new conditions for optimal equal weighting in forecast combination.

Contribution

It generalizes linear pooling theory to kernel scores, including univariate and multivariate, discrete and continuous cases, and offers new insights into optimal weighting strategies.

Findings

Forecast disagreement influences linear pool performance.

Equal weights are optimal under certain kernel scoring rules.

Results apply to a broad family of scoring rules including CRPS and Energy Score.

Abstract

This paper generalizes several results on linear pooling from squared error loss to all kernel scores. The latter are a rich family of scoring rules that covers point and distribution forecasts for univariate and multivariate, discrete and continuous settings. Its members include the Continuous Ranked Probability Score for univariate distribution forecasting and the Energy Score for multivariate distribution forecasting. Our results indicate that forecast disagreement (measured as the average pairwise divergence of all component distributions) has important implications for the linear pool's performance. The results are useful for understanding and designing linear pools in general combination settings. In particular, they motivate using the linear pool (as opposed to other combination formulas) and yield a novel condition under which equal combination weights are optimal under a given kernel scoring rule.