Statistical Inference for Score Decompositions

We introduce inference methods for score decompositions, which partition scoring functions for predictive assessment into three interpretable components: miscalibration, discrimination, and uncertainty. Our estimation and inference relies on a linear recalibration of the forecasts and is applicable to general point forecasts such as means and quantiles due to its validity for non-smooth scoring functions. This approach ensures non-negative decomposition terms in finite samples, enables asymptotic inference under model misspecification, and establishes a direct connection to the classical Mincer-Zarnowitz regression. The resulting inference framework facilitates novel tests for equal linearized forecast calibration or discrimination, which yield three key advantages. They enhance the information content of predictive ability tests by decomposing scores, can improve detection power in scenarios where predictive differences are attributable to specific score components, and formally connect scoring-function-based evaluation to traditional calibration tests, such as financial backtests. Applications demonstrate the method's utility. We find that for survey inflation forecasts, discrimination abilities can differ significantly even when overall predictive ability does not. In an application to financial risk models, our tests provide deeper insights into the calibration and information content of volatility and Value-at-Risk forecasts. By disentangling forecast accuracy from backtest performance, the method exposes critical shortcomings in current banking regulation.