Time Series • Statistics Background for Forecasting
Time Series / P3.04 Interpret the PACF

P3.04 Interpret the PACF

Practical 3 Statistics Background for Forecasting
Quick concept

Explain why ACF of an AR(1) tails off while PACF cuts off, using the series from P3.03.

Practical / Solution

P3.04 Interpret the PACF

Problem Statement

Explain why ACF of an AR(1) tails off while PACF cuts off, using the series from P3.03.

Learning Outcomes

  • State the tail-off vs cut-off teaching rule.
  • Use the rule cautiously.

Theory

Teaching rule: AR(p) has ACF tailing off and PACF cutting off after p. MA(q) has ACF cutting off after q and PACF tailing off. Mixed ARMA shows tails in both. Real data are mixed and samples are finite.

Dataset / Data Source

Reuse the AR(1)-like constructed series.

Analysis / Program

import numpy as np from statsmodels.tsa.stattools import acf, pacf rng = np.random.default_rng(5) e = rng.normal(size=120) y = np.zeros(120) for t in range(1, 120): y[t] = 0.7 * y[t-1] + e[t] print("ACF 1-5:", np.round(acf(y, nlags=5, fft=True)[1:], 3)) print("PACF 1-5:", np.round(pacf(y, nlags=5), 3)[1:])

Expected Output

Printed ACF values decaying and PACF large mainly at lag 1 for this simulation.

Result / Interpretation

The numbers illustrate the teaching rule. Students should still plot data and later check residuals after fitting.

Note

Identification rules are guides, not automatic model selectors.