Auto-Correlation (ACF & PACF)

Definition

Core Statement

ACF (Auto-Correlation Function) measures the correlation between a time series and its lagged values (e.g., $Y_{t}$ vs $Y_{t - 1}$ ).
PACF (Partial Auto-Correlation Function) measures the correlation between $Y_{t}$ and $Y_{t - k}$ after removing the effects of the intermediate lags ( $Y_{t - 1} \dots Y_{t - k + 1}$ ).

These are the primary tools for identifying the order (p, q) of ARIMA Models.

Purpose

Identify Seasonality: Spikes at regular intervals (e.g., every 12 lags for monthly data).
Determine Model Order: Use the shape of ACF/PACF plots to choose $p$ (AR) and $q$ (MA).
Residual Checking: Are the errors "White Noise"? (Ideally, ACF should be zero for all lags > 0).

The Rules of Thumb (Box-Jenkins)

Plot	AR Process ( $p$ )	MA Process ( $q$ )	ARMA ( $p, q$ )
ACF	Decays gradually (Geometric/Sinusoidal)	Cuts off after lag $q$	Decays gradually
PACF	Cuts off after lag $p$	Decays gradually	Decays gradually

Mnemonic

AR(p): Look at PACF. Significant spike at $p$ , then zero.
MA(q): Look at ACF. Significant spike at $q$ , then zero.

Worked Example: Identifying a Model

Problem

You plot ACF and PACF for a stationary series.

Observation 1 (PACF):

Huge spike at Lag 1 ( $r = 0.8$ ).
Spike at Lag 2 is small/insignificant.
Conclusion: This suggests AR(1).

Observation 2 (ACF):

ACF starts high (0.8) and slowly decays (0.64, 0.51...).
This confirms it is an AR process (gradual decay).

Model Proposal: ARIMA(1, 0, 0).

Assumptions

Stationarity: Both diagnostics assume mean and variance are constant. If ACF decays very slowly (linear), the data is non-stationary. Differencing ( $d = 1$ ) is required.

Python Implementation

import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

# Load Data
# data = pd.read_csv(...)['Sales']

# Plot
fig, ax = plt.subplots(1, 2, figsize=(12, 4))
plot_acf(data, lags=20, ax=ax[0])
plot_pacf(data, lags=20, ax=ax[1])
plt.show()

# Interpretation:
# Blue shaded area is the 95% Confidence Interval. 
# Anything outside the blue zone is statistically significant.

Common Pitfall

The "Intermediate" Trap

Why do we need PACF?

If $Y_{t - 1}$ causes $Y_{t}$ , and $Y_{t - 2}$ causes $Y_{t - 1}$ ...
Then $Y_{t - 2}$ will correlate with $Y_{t}$ purely because of the chain reaction.
ACF shows this "echo" (Lag 2 is correlated).
PACF removes the middleman ( $Y_{t - 1}$ ) and shows the pure correlation of Lag 2. (Result: Zero).
Mistake: Using ACF to set AR order usually leads to picking a $p$ that is way too high.

ARIMA Models - The model built using these tools.
Stationarity (ADF & KPSS) - Prerequisite.
Ljung-Box Test - Statistical test for "White Noise" (checking group of lags).