Nonparametric GARCH: An Iterated Neural Estimator

We investigate a forecasting framework based on a simple discrete-time dynamic model with coefficients varying in time. The parameters of the model are recovered within a deep learning framework, which makes it possible to retain a transparent parametric structure while simultaneously accounting for complex and nonstationary patterns in the observed phenomenon. Our analysis covers two specifications of the noise process. Besides the standard Gaussian setting, we also consider Laplace-distributed noise, which can offer a more adequate description in the presence of heavier tails and sharper local fluctuations. For both cases, we formulate the predictive scheme of the model and analyze the associated uncertainty quantification, including the construction of prediction intervals. The results illustrate that a relatively simple model, when combined with time-dependent parameter estimation, can serve as a mathematically tractable and practically flexible tool for forecasting complex dynamics under different noise assumptions. The general model is stated for TVAR($p$), while the prediction-interval formulas and the numerical experiments are developed for the TVAR(1) case.

We consider estimating the one-step-ahead conditional distribution of a multivariate stochastic process. Many existing approaches rely on assumptions such as finite-range memory, sparsity, or additivity, which can be poorly suited to processes with long-range nonlinear interactions. However, without such structural assumptions, nonparametric estimation is challenging due to the curse of dimensionality. To address this challenge, we introduce a new estimation approach based on the predictive states of a process, possibly with infinite-range memory. We show that our estimator achieves fast convergence rates when the past history can be compressed into a low-dimensional statistic that is sufficient for predicting the future. Specifically, we show that the statistical complexity of the estimation problem is determined by the intrinsic dimension of the predictive state space. We establish guarantees for an instantiation of our method based on deep neural network estimators, and we support these theoretical results with experiments.

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.