Part IV · Chapter 12
Moving Average (MA) Models
MA models depend on past SHOCKS, not past observations. Memory is finite: an MA(q) shock only lasts q periods.
\( y_t = \varepsilon_t + \theta\, \varepsilon_{t-1} \)
Learning objectives
Distinguish "MA model" from "moving-average smoother".
Trace a shock through an MA(1) process.
Recognize MA(q) by ACF cutoff at lag q.
Understand invertibility: |θ| < 1.
MA Shock Propagation
Live
Invertibility: —
ρ(1) of MA(1): —
Shock at t* shows up only at t* and t*+1, then DISAPPEARS — finite memory.
📐 MA(1) properties
\[ y_t = \varepsilon_t + \theta\varepsilon_{t-1},\quad \text{Var}(y_t)=(1+\theta^2)\sigma^2 \]
\[ \rho(1)=\frac{\theta}{1+\theta^2},\quad \rho(h)=0 \text{ for } h>1 \]
θ MA coefficient (signs propagate)
Invertibility |θ| < 1
ACF cuts off after lag q
Memory finite = q periods
🔍 What to look for
- Shock at t* creates a spike at t* and an echo at t*+1 (scaled by θ).
- For h > 1, the series returns to its baseline — finite memory.
- ACF should be near zero beyond lag 1 (MA(1) signature).
⚠️ Pro Tip: What to Avoid
Student says
"MA model means I'm modeling a moving-average smoother."
Why this is wrong
The MA in MA(q) refers to past SHOCKS in a stochastic model, not the descriptive smoothing operation from Ch 5. Different concept entirely.
Correct interpretation
MA(q) is a stochastic process: y_t = ε_t + θ_1 ε_{t-1} + … + θ_q ε_{t-q}.
📝 Mini-quiz
📋 Key Takeaways
| Quantity | MA(1) |
|---|---|
| Mean | 0 |
| Variance | (1 + θ²)σ² |
| ρ(1) | θ / (1 + θ²) |
| ρ(h) for h > 1 | 0 |
| Invertibility | |θ| < 1 |