European Central Bank publishes research proposing parametric tilting as a more stable alternative to entropic tilting for density forecasts

The European Central Bank published a Working Paper presenting “parametric tilting”, a methodology for incorporating external information into econometric model-based density forecasts. The paper argues that the approach provides more reliable and numerically stable adjusted distributions than traditional entropic tilting, which can generate unrealistic outcomes when the imposed information is far from the model’s baseline distribution. Parametric tilting reframes the adjustment problem by fitting a parametric skew‑t distribution to the original forecast distribution by minimising the Kullback-Leibler divergence subject to user-specified moment constraints (including the mean, variance, skewness and mode). By controlling the final distribution’s shape, the approach is designed to avoid multimodal or degenerate outcomes that can arise when entropic tilting reweights empirical draws with limited overlap between the baseline and target distributions. The paper also sets out practical implementation steps when the baseline density is only available through draws, including histogram-based approximations and resampling via importance weights, and illustrates the method on a COVID-19 episode example that tilts a model-based GDP growth density towards Survey of Professional Forecasters moments. The authors conclude with an outlook for further research, including potential extensions of the methodology beyond the single-variable setting.