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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

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.

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

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.

Paper Structure

This paper contains 61 sections, 14 theorems, 263 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Classical ARIMA and SARIMA
  4. Galerkin Projection for Function Approximation
  5. Related Work
  6. The Galerkin-ARIMA and Galerkin-SARIMA Models
  7. Problem Setup and Notation
  8. Nonseasonal Galerkin-ARIMA
  9. Seasonal Galerkin-SARIMA Extension
  10. Two-Stage Least-Squares Estimation
  11. Stage 1: autoregressive projection.
  12. Stage 2: moving-average projection.
  13. One-step-ahead forecast.
  14. Ridge-Regularized Galerkin-ARIMA
  15. Theoretical Analysis of the Unpenalized Estimator
  16. ...and 46 more sections

Key Result

Theorem 1

Suppose Assumptions (A1)–(A5) hold and that the true conditional mean is linear SARIMA. Then:

Figures (9)

  • Figure 1: Synthetic series used in the experiments. Part I.
  • Figure 2: Synthetic series used in the experiments. Part II.
  • Figure 3: Synthetic forecasting comparison under the tuned protocol. Each algorithm selects its order by BIC on the initial window and holds it fixed during rolling one-step-ahead refits. Bars report MAE, RMSE, and total wall-clock time over the rolling horizon.
  • Figure 4: Unemployment rate. Algorithm comparison under the tuned rolling protocol.
  • Figure 5: Real GDP. Algorithm comparison under the tuned rolling protocol.
  • ...and 4 more figures

Theorems & Definitions (21)

  • Theorem 1: Linear Oracle Property
  • Lemma 1: Jackson-type approximation bounds
  • Theorem 2: Asymptotic unbiasedness and consistency
  • Proposition 1: Asymptotic normality of the Galerkin coefficients
  • Proposition 2: Optimal mean squared error rates
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Lemma : Lemma \ref{['lem:jackson']}, restated
  • proof
  • ...and 11 more