Galerkin-ARIMA: A Two-Stage Polynomial Regression Framework for Fast Rolling One-Step-Ahead Forecasting

Time-series models like ARIMA remain widely used for forecasting but limited to linear assumptions and high computational cost in large and complex datasets. We propose Galerkin-ARIMA that generalizes the AR component of ARIMA and replace it with a flexible spline-based function estimated by Galerkin projection. This enables the model to capture nonlinear dependencies in lagged values and retain the MA component and Gaussian noise assumption. We derive a closed-form OLS estimator for the Galerkin coefficients and show the model is asymptotically unbiased and consistent under standard conditions. Our method bridges classical time-series modeling and nonparametric regression, which offering improved forecasting performance and computational efficiency.