Part VIII · Chapter 26
GARCH Models
GARCH adds lagged variance to ARCH: volatility has memory. α + β controls persistence; α + β < 1 for stationarity, α + β near 1 ⇒ slowly decaying volatility shocks.
\( h_t = \omega + \alpha\, \varepsilon_{t-1}^2 + \beta\, h_{t-1} \)
Learning objectives
Simulate GARCH(1,1) and read α + β persistence.
Compute long-run unconditional variance.
Recognize IGARCH (α+β=1) and instability.
Forecast volatility into the future.
GARCH Volatility Memory
Seed:
Persistence
α + β: —
Long-run variance: —
Volatility half-life: — periods
📐 GARCH(1,1)
\[ h_t = \omega + \alpha\, \varepsilon_{t-1}^2 + \beta\, h_{t-1} \]
\[ \bar h = \frac{\omega}{1-\alpha-\beta} \;\;\text{ (if }\alpha+\beta<1\text{)} \]
\[ \text{Half-life: } \frac{\ln 0.5}{\ln(\alpha+\beta)} \]
ω > 0 floor variance
α shock impact (ARCH)
β volatility persistence
α + β < 1 stationarity / mean reversion
🔍 What to look for
- Conditional volatility √h_t rises during clusters of large |returns| and decays during calm periods.
- Forecast horizon: variance reverts geometrically to long-run mean.
- α+β = 0.98 ⇒ shocks ~ 34 periods to halve (slow). α+β = 0.6 ⇒ shocks decay quickly.
⚠️ Pro Tip: What to Avoid
Student says
"α + β ≈ 0.99 means volatility decays fast."
Why this is wrong
α + β CLOSE TO 1 means VERY SLOW decay (long memory). At α+β = 0.99, half-life is ~69 periods. Variance only mean-reverts when α+β < 1, and the closer to 1, the slower.
Correct interpretation
Half-life = ln(0.5)/ln(α+β). Use this to communicate volatility persistence in actual horizon units.
📝 Mini-quiz
📋 Key Takeaways
| α+β | Persistence | Half-life (periods) |
|---|---|---|
| 0.5 | Fast decay | ≈ 1 |
| 0.9 | Moderate | ≈ 6.6 |
| 0.95 | High | ≈ 13.5 |
| 0.99 | Very high | ≈ 69 |
| 1.0 | IGARCH — no mean reversion | ∞ |
| > 1 | UNSTABLE | n/a |