用投影方法改进经典时间序列模型，提升经济金融数据预测精度。
A Projection-Based ARIMA Framework for Nonlinear Dynamics in Macroeconomic and Financial Time Series: Closed-Form Estimation and Rolling-Window Inference
- 用低维基函数替代线性滞后算子，保持传统AR-MA结构
- 闭式解估计支持快速滚动重估，预测准确率优于经典模型
- 适合央行预测与投资组合风险分析，可处理序列相关性
我们提出Galerkin-ARIMA和Galerkin-SARIMA，一种基于投影的经典ARIMA/SARIMA扩展方法，将刚性的线性滞后算子替换为低维Galerkin基展开，同时保留熟悉的AR-MA分解结构。在合成数据及季度GDP、日度标普500收益率上的实验表明，Galerkin-SARIMA的预测精度不低于且通常优于经典ARIMA/SARIMA。参数估计采用两阶段最小二乘法，获得闭式解，支持高效滚动窗口重估，且保持原有算子结构，便于应用于央行预测与投资组合风险管理。我们建立了弱相依条件下的逼近-估计权衡，给出未惩罚估计量的一致性与渐近分布结果，比较了预测风险与经典SARIMA，提出基于信息准则选择基函数维度的方法。进一步发展了基于自助法的外生因子块推断与块自助预测区间，能有效处理序列相关性和两阶段生成回归量结构。
原文摘要 · Abstract (English)
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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