A Projection‐Based ARIMA Framework for Nonlinear Dynamics in Macroeconomic and Financial Time Series: Closed‐Form Estimation and Fast Rolling Forecasting

We develop Galerkin–ARIMA, a projection‐based forecasting framework that expands observation‐ and innovation‐lag states with finite basis functions while nesting classical multiplicative SARIMA as a restricted linear specification. The model is estimated using preliminary residual proxies followed by a joint least‐squares or ridge regression, which supports efficient rolling refitting. For the fixed‐dimensional unpenalized estimator, we establish asymptotic normality while accounting explicitly for generated regressors. Across 10 US macroeconomic and financial series, Galerkin–ARIMA has competitive one‐step‐ahead forecast accuracy, with nonlinear gains varying across series and the linear specification providing the most stable performance. In pre‐specified runtime experiments on Real GDP, the S&P 500, and the Unemployment Rate, Ridge Galerkin–ARIMA is substantially faster than likelihood/state‐space SARIMA under fixed, initially tuned, and repeatedly retuned rolling specifications. The results suggest that the main value of the framework is flexible lag‐state modeling and computationally efficient forecasting. Nonlinear enrichment does not produce uniform accuracy gains.