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

Haojie Liu; Zihan Lin

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arXiv · 2507.07469

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

Haojie Liu·Zihan Lin

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Abstract

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.

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Haojie Liu, Zihan Lin. 2025-07-10. A Projection-Based ARIMA Framework for Nonlinear Dynamics in Macroeconomic and Financial Time Series: Closed-Form Estimation and Rolling-Window Inference. https://arxiv.org/abs/2507.07469

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