Empirical prediction intervals applied to short term mortality forecasts and excess deaths

In the winter of 2022/2023, excess death estimates for Germany indicated a 10% elevation, which has led to questions about the significance of this increase in mortality. Given the inherent errors in demographic forecasting, the reliability of estimating a 10% deviation is questionable. This research addresses this issue by analyzing the error distribution in forecasts of weekly deaths. By deriving empirical prediction intervals, we provide a more accurate probabilistic study of weekly expected and excess deaths compared to the use of conventional parametric intervals.

Using weekly death data from the Short-term Mortality Database (STMF) for 23 countries, we propose empirical prediction intervals based on the distribution of past out-of-sample forecasting errors for the study of weekly expected and excess deaths. Instead of relying on the suitability of parametric assumptions or the magnitude of errors over the fitting period, empirical prediction intervals reflect the intuitive notion that a forecast is only as precise as similar forecasts in the past turned out to be. We compare the probabilistic calibration of empirical skew-normal prediction intervals with conventional parametric prediction intervals from a negative-binomial GAM in an out-of-sample setting. Further, we use the empirical prediction intervals to quantify the probability of detecting 10% excess deaths in a given week, given pre-pandemic mortality trends.

The cross-country analysis shows that the empirical skew-normal prediction intervals are overall better calibrated than the conventional parametric prediction intervals. Further, the choice of prediction interval significantly affects the severity of an excess death estimate. The empirical prediction intervals reveal that the likelihood of exceeding a 10% threshold of excess deaths varies by season. Across the 23 countries studied, finding at least 10% weekly excess deaths in a single week during summer or winter is not very unusual under non-pandemic conditions. These results contrast sharply with those derived using a standard negative-binomial GAM.

Our results highlight the importance of well-calibrated prediction intervals that account for the naturally occurring seasonal uncertainty in mortality forecasting. Empirical prediction intervals provide a better performing solution for estimating forecast uncertainty in the analyses of excess deaths compared to conventional parametric intervals.