# A Projection-Based ARIMA Framework for Nonlinear Dynamics in Macroeconomic and Financial Time Series: Closed-Form Estimation and Rolling-Window Inference

arXiv: https://arxiv.org/abs/2507.07469 · HTML: https://mlsanity.com/paper/2507.07469/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 23 references that are in the map.
- First posted on arXiv 10 Jul 2025.

## Builds on (most cited first, 3 of 3)

- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [N-BEATS: Neural basis expansion analysis for interpretable time series forecasting](https://mlsanity.com/paper/1905.10437/) (2019)
- [Manifold Markov chain Monte Carlo methods for Bayesian inference in diffusion models](https://mlsanity.com/paper/1912.02982/) (2019)

## Abstract (arXiv)

We introduce Galerkin-ARIMA and Galerkin-SARIMA, a projection-based extension of classical ARIMA/SARIMA that replaces rigid linear lag operators with low-dimensional Galerkin basis expansions while preserving the familiar AR-MA decomposition. Experiments on synthetic series and on quarterly GDP and daily S&P 500 returns show that Galerkin-SARIMA matches or improves forecast accuracy relative to classical ARIMA/SARIMA. Estimation is closed-form via a two-stage least-squares procedure, and the closed-form two-stage estimator enables efficient rolling-window re-estimation while preserving the familiar AR-MA operator structure, facilitating applications in central bank forecasting and portfolio risk management. We establish approximation-estimation trade-offs under weak dependence, provide consistency and asymptotic distributional results for the unpenalized estimator, compare prediction risk to classical SARIMA, and propose information-criterion selection of basis size. We further develop bootstrap-based inference for exogenous factor blocks and block-bootstrap prediction intervals that account for serial dependence and the two-stage generated-regressor structure.