Regularized Ensemble Forecasting for Learning Weights from Historical and Current Forecasts

Abstract

Combining forecasts from multiple experts often yields more accurate results than relying on a single expert. In this paper, we focus on forecasting a continuous outcome and introduce a novel regularized ensemble method that extends the traditional linear opinion pool by leveraging both current forecasts and historical performances to set the weights. In contrast to existing approaches that emphasize either current forecasts or past accuracy, our method accounts for both sources simultaneously. It learns weights by minimizing the variance of the combined forecast (or its transformed version) while incorporating a regularization term informed by historical performances. We also show that this approach has a Bayesian interpretation. Different distributional assumptions within this Bayesian framework yield different functional forms for the variance component and the regularization term, adapting the method to various scenarios. In empirical studies on Walmart sales and macroeconomic forecasting, our ensemble outperforms leading benchmark models both when experts' full forecasting histories are available and when experts enter and exit over time. Throughout, we also characterize how the optimal weights depend on current forecasts and historical performance, and use the empirical results to discuss where the framework's strengths lie and when each source of information plays a greater role.