Part III · Chapter 9
ACF and PACF
ACF tails off + PACF cuts off ⇒ AR. ACF cuts off + PACF tails off ⇒ MA. Both tail off ⇒ ARMA. This pattern recognition is half of model identification.
Learning objectives
Compute and read sample ACF and PACF.
Apply the Box-Jenkins identification rules.
Identify AR, MA, ARMA from their fingerprints.
Distinguish significant lags from sampling noise.
Pattern Matcher
Seed:
Pattern reading
—
ρ(h) bars beyond ±1.96/√n are statistically significant.
📐 Box-Jenkins rules
| Process | ACF | PACF |
|---|---|---|
| AR(p) | tails off (decay) | CUTS off after lag p |
| MA(q) | CUTS off after lag q | tails off (decay) |
| ARMA(p,q) | tails off | tails off |
🔍 How to read
- Count significant bars in PACF: gives candidate p.
- Count significant bars in ACF: gives candidate q.
- If both keep going, use ARMA — pick by AIC/BIC.
- Confidence band ±1.96/√n; spikes inside it are noise.
⚠️ Pro Tip: What to Avoid
Student says
"ACF spike at lag 2 → it's AR(2)."
Why this is wrong
ACF spike alone doesn't identify AR. AR(p) is characterized by PACF cutoff at lag p, not by an ACF spike. Could be MA(2) instead.
Correct interpretation
Use ACF and PACF TOGETHER. AR ↔ PACF cutoff. MA ↔ ACF cutoff.
📝 Mini-quiz
📋 Key Takeaways
| Pattern | Diagnosis |
|---|---|
| ACF decays, PACF spikes at lag 1 | AR(1) |
| ACF spikes at lag 1, PACF decays | MA(1) |
| Both decay | ARMA — use AIC |
| ACF decays VERY slowly | Suspect unit root — difference |