🤖 AI Summary
Traditional AR(p) models struggle to capture the time-varying dynamics and complex noise structures inherent in nonstationary time series. This work proposes a novel hybrid approach that integrates deep learning with time-varying autoregressive (TVAR) modeling, leveraging neural networks to flexibly estimate time-dependent coefficients. The method preserves model interpretability while effectively adapting to nonstationarity and enables probabilistic forecasting under both Gaussian and Laplacian noise assumptions. Empirical validation under a TVAR(1) framework demonstrates that the proposed approach accurately captures intricate temporal dynamics with a parsimonious structure and yields reliable prediction intervals, particularly excelling in scenarios involving heavy-tailed distributions or sharp volatility shifts.
📝 Abstract
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.