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

We introduce Galerkin-ARIMA, a novel time-series forecasting framework that integrates Galerkin projection techniques with the classical ARIMA model to capture potentially nonlinear dependencies in lagged observations. By replacing the fixed linear autoregressive component with a spline-based basis expansion, Galerkin-ARIMA flexibly approximates the underlying relationship among past values via ordinary least squares, while retaining the moving-average structure and Gaussian innovation assumptions of ARIMA. We derive closed-form solutions for both the AR and MA components using two-stage Galerkin projections, establish conditions for asymptotic unbiasedness and consistency, and analyze the bias-variance trade-off under basis-size growth. Complexity analysis reveals that, for moderate basis dimensions, our approach can substantially reduce computational cost compared to maximum-likelihood ARIMA estimation. Through extensive simulations on four synthetic processes-including noisy ARMA, seasonal, trend-AR, and nonlinear recursion series-we demonstrate that Galerkin-ARIMA matches or closely approximates ARIMA's forecasting accuracy while achieving orders-of-magnitude speedups in rolling forecasting tasks. These results suggest that Galerkin-ARIMA offers a powerful, efficient alternative for modeling complex time series dynamics in high-volume or real-time applications.