Amadeo Grob, Maurizio Daniele, Johanna Ziegel
arXiv 24 Aug 2026 · Statistics — Methodology
arXiv:2608.23064 · PDF · Extracted main text
Traditional statistical tests are poorly suited for the sequential evaluation of probabilistic forecast calibration. We address this limitation in macroeconomic forecasting by applying a new sequential testing method based on e-values. The e-value-based methodology enables anytime-valid inference. It allows practitioners to test against calibration continuously without invalidating statistical guarantees. To illustrate the framework's practical value, we apply it to probabilistic inflation forecasts for the United States, the Euro Area, and Switzerland. Our analysis shows that the sequential approach gives detailed insights into the timing and nature of forecast misspecification. We find these diagnostics are particularly insightful during major structural breaks. During these events, we find evidence against calibration that static, full-sample tests often miss. Therefore, this work shows that e-value-based tests are a practical method for the evaluation of forecast calibration in empirical macroeconomics.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Arnold, Sebastian and Henzi, Alexander and Ziegel, Johanna F (2023) Sequentially valid tests for forecast calibration self | 1.000 | 25 | 3 | 100% |
| 2 | Gneiting, Tilmann and Balabdaoui, Fadoua and Raftery, Adrian E (2007) Probabilistic Forecasts, Calibration and Sharpness | 0.874 | 5 | 2 | 100% |
| 3 | Shafer, Glenn (2021) Testing by Betting: A Strategy for Statistical and Scientific Communication | 0.737 | 3 | 2 | 100% |
| 4 | McCracken, Michael W and Ng, Serena (2016) FRED-MD: A monthly database for macroeconomic research | 0.644 | 4 | 2 | 50% |
| 5 | Grünwald, Peter and De Heide, Rianne and Koolen, Wouter (2024) Safe testing | 0.644 | 2 | 2 | 100% |
| 6 | Ramdas, Aaditya and Grünwald, Peter and Vovk, Vladimir and Shafer, G… (2023) Game-Theoretic Statistics and Safe Anytime-Valid Inference | 0.585 | 3 | 1 | 100% |
| 7 | Vovk, Vladimir and Wang, Ruodu (2021) E-values: Calibration, combination and applications | 0.585 | 3 | 1 | 100% |
| 8 | Cevid, Domagoj and Michel, Loris and Näf, Jeffrey and Bühlmann, Pete… (2022) Distributional Random Forests: Heterogeneity Adjustment and Multivariate Distributional Regression | 0.511 | 3 | 2 | 33% |
| 9 | Diebold, Francis X. and Gunther, Todd A. and Tay, Anthony S (1998) Evaluating Density Forecasts with Applications to Financial Risk Management | 0.511 | 2 | 1 | 100% |
| 10 | Gneiting, Tilmann (2011) Making and Evaluating Point Forecasts | 0.511 | 2 | 1 | 100% |
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