Part VIII · Chapter 25
ARCH Models
ARCH models conditional variance using past squared shocks: today's risk depends on yesterday's surprise. Volatility clusters in calm and turbulent regimes — exactly what ARCH captures.
\( h_t = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2,\;\; \varepsilon_t = \sqrt{h_t} z_t \)
Learning objectives
Simulate an ARCH(1) and observe volatility clustering.
Interpret α₀, α₁; stability requires α₁ < 1.
Identify ARCH effects by squared-return autocorrelation.
Understand that ARCH forecasts variance, not direction.
ARCH Clustering Simulator
Seed:
Status
Persistence α₁: —
Long-run var: — (if α₁ < 1)
Squared-return ACF should be POSITIVE — that's the ARCH signature.
📐 ARCH(1)
\[ h_t = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2,\quad \alpha_0 > 0,\;\;\alpha_1 \geq 0 \]
\[ \text{Long-run variance: }\frac{\alpha_0}{1-\alpha_1}\quad(\alpha_1<1) \]
α₀ floor variance
α₁ shock sensitivity
Stability α₁ < 1
ARCH LM test T·R² ~ χ²(q) on squared residuals
🔍 What to look for
- Returns plot shows clusters of large absolute values — calm vs turbulent periods.
- Squared-return ACF has significant positive bars (unlike return ACF which is near zero).
- α₁ near 1 ⇒ explosive variance; α₁ ≥ 1 ⇒ unstable.
⚠️ Pro Tip: What to Avoid
Student says
"GARCH/ARCH predicts whether the market will go up tomorrow."
Why this is wrong
ARCH/GARCH forecast VARIANCE (risk magnitude), not the sign of returns. Returns themselves may be ≈ white noise in the conditional mean.
Correct interpretation
"Conditional variance ĥ_{t+1} = α₀ + α₁ ε̂_t². Use for VaR, option pricing, position sizing — not direction."
📝 Mini-quiz
📋 Key Takeaways
| Quantity | Note |
|---|---|
| ARCH(1) variance | α₀ + α₁ ε²_{t-1} |
| Stability | α₁ < 1 |
| Long-run variance | α₀ / (1 − α₁) |
| ARCH LM test | T·R² on squared residuals ~ χ²(q) |
| Forecast | Variance, NOT direction |