MATH7970 Chap.10 ARCH and GARCH Volatility Models
ARCH and GARCH Volatility Models
Predictable variance
Returns can be uncorrelated while their magnitudes cluster. ARCH connects current conditional variance to recent squared innovations; GARCH adds lagged conditional variance for persistent memory.
Mean before variance
Clean the conditional mean before testing squared residuals.
Positivity, persistence and standardised-residual diagnostics control interpretation of the variance recursion.
Separate mean and variance equations
Begin with a mean model that removes predictable level and direction. Then inspect its residual magnitudes. Serial correlation in raw residuals points back to the mean equation, while dependence in squared residuals motivates conditional variance modelling.
If mean structure remains, ARCH parameters can absorb the wrong pattern and their interpretation becomes misleading.
Read news and memory together
The squared-innovation coefficient controls response to recent news; the lagged-variance coefficient carries earlier conditional scale forward. A model can react sharply and decay quickly or respond modestly and remain persistent.
Their sum summarises persistence in the basic model, while the separate values describe how that persistence is created. Interpret both before comparing long-run variance or shock decay.
Check positivity and stable scale
Variance coefficients must keep every conditional variance nonnegative.
A persistence sum below one supports a finite long-run variance in the standard specification, while a value near one implies slow reversion. State these conditions before dividing by one minus persistence.
A numerical long-run variance has no stable interpretation when the required denominator condition fails.
Diagnose standardised residuals
Divide residuals by fitted conditional standard deviations, then inspect raw and squared ACFs. Remaining squared dependence means volatility structure survives.
Heavy tails may call for a different innovation distribution, while asymmetric news effects require a model that explicitly distinguishes signs.
A variance forecast describes scale, not whether the next return is positive or negative; direction belongs to the separate mean equation.
Compare volatility forecasts honestly
Evaluate conditional variance with measures designed for scale forecasts and with common origins. A large realised return is not by itself proof that a high variance forecast was accurate, because direction and magnitude are random.
Examine whether standardised residuals behave consistently across calm and turbulent periods, whether interval coverage improves and whether the persistence estimate is stable after an extreme episode is removed. Report sensitivity when one crisis dominates the fitted parameters instead of treating unusually slow decay as a permanent property of all periods.
What this chapter covers
- 01
Volatility clustering
- 02
ARCH news term
- 03
GARCH persistence
- 04
Long-run variance
- 05
Conditional standard deviation
- 06
Squared-residual ACF
Worked example · free
Update conditional variance
- 1Square the innovation.
- 4Compute variance.
- 2Convert to standard deviation.
Key terms
- Conditional Variance
- Variance predicted using information available before the next innovation.
- ARCH Effect
- Dependence of variance on recent squared innovations.
- GARCH Model
- A variance model using squared innovations and lagged conditional variance.
- Volatility Persistence
- The speed at which a variance shock decays through time.
- Standardised Residual
- A residual divided by its fitted conditional standard deviation.
ARCH and GARCH Volatility Models FAQ
Can returns be uncorrelated but volatile in clusters?
Yes. Linear mean dependence can be absent while squared or absolute returns retain strong serial structure. That distinction is why squared residuals are inspected after the conditional mean is fitted.
What does alpha plus beta describe?
In a basic GARCH model it summarises volatility persistence, with values near one implying slow decay. The long-run variance interpretation additionally requires the persistence sum to remain below one.
Does a variance forecast predict direction?
No. It predicts conditional magnitude or scale, not whether the next return will be positive or negative. Direction belongs to the conditional mean model, while GARCH supplies the changing scale.
Exam move
Write every variance update with units. Diagnose the mean using residuals and the variance using squared standardised residuals, then check positivity and persistence before interpretation. Work variance recursions in a table containing squared innovation, previous variance, news contribution, memory contribution and updated variance. Predict how increasing each coefficient changes response and decay.
Compare residual and squared-residual ACFs so mean and variance targets remain separate. Check positivity and persistence before deriving long-run variance. Convert to standard deviation only when required, state that the result forecasts scale rather than direction, and repeat with one extreme shock to watch volatility fade. Draw the variance response after one large innovation and label its decay.
The path should agree with persistence and catches a recursion using standard deviation where squared units belong. Check standardised residuals again after any change to the conditional mean specification.
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