AI 中文总结
本研究构建了统一的基于回归的教学框架，将ACF、PACF、Durbin-Levinson递推及一步预测与回归知识关联，降低了回归与时间序列课程的概念障碍。
AI 中文摘要
自相关函数（ACF）和偏自相关函数（PACF）是识别自回归移动平均（ARMA）模型的基础工具，但它们常以与学生已掌握的回归框架脱节的计算方法形式被介绍给学生。本笔记针对平稳时间序列的ACF、PACF、Durbin-Levinson递推及一步预测，构建了一个统一的、基于回归的教学框架。研究表明，ACF恰好是平稳过程对其自身某一阶滞后项进行简单线性回归的系数，PACF是扩展多元回归模型中最新滞后项的系数，Durbin-Levinson递推本质上是一系列偏回归的序列，每一个都可通过回归入门课程中的普通最小二乘恒等式表达。针对MA(1)和AR(1)过程的实例，展示了ACF和PACF的拖尾与截尾行为如何直接从该视角推导得出。研究认为，这种框架降低了回归课程与时间序列课程之间的概念障碍，并提供了面向课堂教学的建议。
英文摘要
The autocorrelation function (ACF) and partial autocorrelation function (PACF) are foundational tools for identifying autoregressive moving-average (ARMA) models, yet they are often introduced in ways that appear disconnected from the regression concepts students already know. This paper develops a unified, regression-based instructional framework for the ACF, PACF, Durbin--Levinson recursion, and one-step-ahead prediction for weakly stationary time series. We show that the ACF is the coefficient from a simple linear regression of a mean-zero stationary process on one of its lagged values, while the PACF is both the coefficient of the newest predictor in an expanding multiple regression and the corresponding partial correlation. Using partial regression, we derive the Durbin-Levinson updates for the newly added coefficient, the existing regression coefficients, and the prediction-error variance from familiar ordinary least-squares principles. Worked MA(1) and AR(1) examples show how the characteristic cutoff and tailing-off patterns of the ACF and PACF emerge naturally from this regression perspective. The same recursive regression coefficients also determine the optimal linear one-step-ahead predictor. The resulting framework provides a coherent instructional pathway from regression to model identification, recursive estimation, and prediction and suggests practical ways to connect introductory regression and time series courses.
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