Fractional Time-Varying Autoregressive Modeling: Parallel GAM and PINN Approaches to Dynamic Volatility Forecasting

Time-series modeling plays a crucial role in analyzing and forecasting financial volatility. Classical approaches, such as the Autoregressive (AR) and Fractional Autoregressive (FAR) models, capture short-term linear dependencies and long-range correlations, respectively, but their reliance on fixed structures and stationarity assumptions limits their adaptability to evolving market dynamics. To overcome these limitations, this study introduces a Fractional Time-Varying Autoregressive (FTVAR) framework that allows model parameters to evolve smoothly over time, integrating long-memory effects with nonstationary behavior. The FTVAR process is examined through two complementary methods: a Generalized Additive Model (GAM) for interpretable estimation of time-varying coefficients, and a Physics-Informed Neural Network (PINN) that embeds system dynamics to enhance forecasting under complex conditions. Simulation studies demonstrate that the FTVAR model consistently outperforms conventional AR approaches, offering superior forecasting accuracy, robustness to nonstationarity, and a more comprehensive representation of evolving volatility structures. Empirical analyses on the S&P 500 and VIX indices further confirm the effectiveness and practical relevance of the proposed framework.