Learning with the Nash-Sutcliffe loss

Multi-step time series forecasting (MSF) is commonly evaluated using point-wise error metrics such as mean squared error (MSE), implicitly treating the conditional mean as a sufficient target. We show that this can be misleading under conditional uncertainty, where the conditional expectation becomes unrepresentative of typical realized values at longer horizons. We formalize this effect through a conditional uncertainty gap and prove that whenever this gap is nonzero, no deterministic predictor can simultaneously minimize MSE and match the marginal distribution of realized futures. This establishes a fundamental, model-agnostic trade-off between point accuracy and marginal realism in MSF evaluation. Using controlled stochastic dynamical systems and nine real-world forecasting benchmarks, we empirically characterize the resulting accuracy--realism frontier and \textbf{quantify the practical cost of MSE-only model selection}. As conditional uncertainty increases with forecast horizon, the attainable set expands into a pronounced Pareto front, separating MSE-optimal but under-dispersed predictors from methods that trade accuracy for realistic marginal variability. \textbf{Across benchmarks, we find that small relaxations in MSE ($\boldsymbol{\le 5\%}$) frequently unlock disproportionate gains in marginal realism, with median improvements of $\mathbf{17.3\%}$ and gains exceeding $\mathbf{30\%}$ in some datasets.} We further show that common forecasting strategies systematically occupy different regions of this frontier: direct multi-output predictors concentrate near the accuracy-optimal extreme, while recursive strategies and sample-based inference favors marginal realism. Together, these results expose a structural failure mode of MSE-based evaluation in long-horizon forecasting and recast strategy and inference selection as navigation of an unavoidable accuracy--realism trade-off.

We introduce a neural network-based framework for learning time series estimators through a process we term decision-theoretic pretraining. Analysts specify a generative world, a distribution over data-generating processes, and a target decision objective. A neural network trained on stratified simulations from this world approximates the corresponding optimal decision rule, yielding a neural estimator that provides forecasts, parameter estimates, predictive intervals, or model-selection for zero-shot inference on previously unseen time series. The joint specification of the generative world and objective enables the estimators to directly approximate process-level, finite-sample properties: near-optimal risk, bias control, minimax performance, and uniform calibration. Our experiments demonstrate that these neural estimators can outperform traditional baselines such as maximum likelihood estimation and model selection via AICc, for the same model structural model classes. Furthermore, even when trained purely on simulations of structural models, they achieve competitive or state-of-the-art forecasting accuracy on major real-world benchmarks, compared with statistical, neural or large pre-trained models. We illustrate the framework by addressing two longstanding challenges: finite-sample bias and miscalibration in AR(p) models, and the forecast combination puzzle. These applications highlight the approach's main advantage: its ability to approximate solutions to analytically intractable or computationally prohibitive time series problems, including complex structural equations or optimality criteria. Ultimately, by enabling explicit control over decision-theoretic trade-offs, the framework equips analysts with highly efficient estimation tools tailored to their specific analytical needs.

We study the problem of model selection among probabilistic forecasting models evaluated on datasets of multiple time series. The performance of a model on a single time series is quantified by the average value (score) of a proper scoring rule over a test set, but extending model selection to data sets of time series requires aggregating these scores. Common approaches either rely on scaling scores and averaging them (mean scaled score) or avoid scaling by using alternative statistics such as mean ranks or win rates. However, these approaches can yield conflicting conclusions. We show that such discrepancies arise from the skewness of the distribution of the scores, which is particularly pronounced when test sets are short. The skewness can cause non-mean criteria (e.g., mean rank, median, win rate) to select misspecified models. In contrast, the mean score is immune from this problem. We further show that, as the size of the test sets increases, all aggregation criteria converge to the same model selection decision, mitigating these discrepancies. Our experiments on intermittent demand time series, including data from the M5 competition, highlight the importance of sufficiently large test sets; the mean scaled score appears to be the more reliable approach, also because empirically we found its decision to remain consistent when different scaling factors are adopted.