Part IV · Chapter 13
ARMA Models
ARMA combines persistence (AR) and shock propagation (MA). Both ACF and PACF tail off, making identification harder than pure AR or MA.
\( y_t = \phi y_{t-1} + \varepsilon_t + \theta \varepsilon_{t-1} \)
Learning objectives
Estimate ARMA(p,q) on simulated data.
Recognize ARMA fingerprint: both ACF and PACF tail off.
Use AIC/BIC to choose between candidates.
Check both stationarity (φ) AND invertibility (θ).
AR + MA Mixer
Seed:
Status
—
Both stationarity and invertibility required for a valid ARMA.
📐 ARMA(1,1)
\[ y_t = \phi y_{t-1} + \varepsilon_t + \theta \varepsilon_{t-1} \]
Stationary |φ| < 1
Invertible |θ| < 1
ACF tails off
PACF tails off
🔍 What to look for
- Identification by ACF/PACF alone is ambiguous — use AIC/BIC.
- Check residuals after fitting: should be white noise (Ljung-Box p > 0.05).
- If both ACF and PACF cut sharply, consider pure AR or MA instead.
⚠️ Pro Tip: What to Avoid
Student says
"My ARMA(2,2) has the lowest AIC — must be best."
Why this is wrong
Lowest AIC alone is insufficient; you must also check residual diagnostics (Ljung-Box, ARCH LM). Diagnostics override IC.
Correct interpretation
Pick the lowest-IC model whose residuals pass diagnostics. Prefer parsimony when ties are close.
📝 Mini-quiz
📋 Key Takeaways
| Pattern | Likely model |
|---|---|
| PACF cuts at p, ACF decays | AR(p) |
| ACF cuts at q, PACF decays | MA(q) |
| BOTH decay | ARMA(p,q) — try several |
| Both cut at lag 0 | White noise |