Title: Expectations vs. Realities: The Cost of MSE-Optimal Forecasting Under Conditional Uncertainty

Abstract:

Standard multi-step time series forecasting (MSF) protocols typically rely on point-wise error metrics like mean squared error (MSE), operating under the assumption that the conditional mean serves as a sufficient prediction target. However, we demonstrate that this approach can be deceptive in scenarios characterized by conditional uncertainty. In such contexts, particularly at extended horizons, the conditional expectation often fails to reflect typical observed outcomes. We define this phenomenon via a "conditional uncertainty gap" and provide a proof showing that if this gap is non-zero, no deterministic model can simultaneously achieve minimum MSE and replicate the marginal distribution of actual future values. This finding reveals a fundamental, model-independent trade-off between point-wise accuracy and marginal realism in MSF assessments.

Through experiments involving controlled stochastic dynamical systems and nine real-world forecasting benchmarks, we empirically map the resulting accuracy–realism frontier and measure the tangible expense of selecting models based solely on MSE. As forecast horizons lengthen and conditional uncertainty rises, the set of achievable models forms a distinct Pareto front. This front distinguishes MSE-optimal predictors, which tend to be under-dispersed, from methods that sacrifice some accuracy to achieve more realistic marginal variability. Our analysis across benchmarks indicates that minor concessions in MSE performance (≤ 5%) often yield significant improvements in marginal realism, delivering median gains of 17.3% and increases surpassing 30% in certain datasets. Furthermore, we observe that standard forecasting techniques cluster in specific areas of this frontier: direct multi-output predictors gravitate toward the accuracy-optimal pole, whereas recursive strategies and sample-based inference lean toward marginal realism. Collectively, these insights highlight a structural limitation of MSE-centric evaluation in long-horizon forecasting and frame the choice of forecasting strategies and inference methods as a necessary navigation of the inherent accuracy–realism compromise.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC