A Regression-Based Framework for the ACF, PACF, Durbin-Levinson Recursion, and One-Step-Ahead Prediction

Abstract

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.