Chapter 06

Hard

Volatility Clustering and GARCH

00 · Symbol Glossary

$\sigma_t^2$sigma sub t squared — conditional variance

$\sigma_t^2=\mathrm{Var}(r_t\mid \mathcal{F}_{t-1})$ , the variance of the return given the past. In GARCH this is a parametric function of past shocks and past $\sigma^2$ .

$\varepsilon_t$epsilon sub t — return innovation

$r_t=\mu+\varepsilon_t$ with $\varepsilon_t=\sigma_t z_t$ , $z_t$ iid mean $0$ variance $1$ (often $N(0,1)$ or Student- $t$ ). Uncorrelated levels; dependent squares.

$\omega,\alpha,\beta$GARCH(1,1) parameters

$\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\beta\sigma_{t-1}^2$ . Require $\omega>0$ , $\alpha\geq 0$ , $\beta\geq 0$ for a positive variance recursion (sufficient conditions).

$\alpha+\beta$persistence

In GARCH(1,1), $\alpha+\beta$ is the AR coefficient of $\varepsilon_t^2$ in its ARMA representation. Covariance stationarity of $\varepsilon_t$ needs $\alpha+\beta<1$ .

01 · Stylized Facts

Daily asset returns are close to uncorrelated (ACF of $r_t$ near zero after lag 0) but not iid: large $\lvert r_t\rvert$ is followed by large $\lvert r_{t+1}\rvert$ more often than a constant-variance model allows. Squared (or absolute) returns have a slowly decaying ACF. That is volatility clustering. Unconditional histograms are heavy-tailed relative to $N(\hat\mu,\hat\sigma^2)$ .

A constant- $\sigma$ ARMA on $r_t$ can pass Ljung–Box on levels and still be a bad risk model. The missing equation is for $\sigma_t^2$ .

02 · ARCH and GARCH

Definition — ARCH$(q)$ (Engle)

\varepsilon_t=\sigma_t z_t, \qquad \sigma_t^2 = \omega + \sum_{i=1}^{q}\alpha_i\varepsilon_{t-i}^2

with $z_t$ iid $(0,1)$ independent of the past. Conditional mean of $\varepsilon_t$ is $0$ ; conditional variance tracks recent squared shocks. For $\sigma_t^2>0$ you need $\omega>0$ and $\alpha_i\geq 0$ (and usually $\sum_i\alpha_i<1$ for a finite unconditional variance in ARCH $(q)$ alone). High-order ARCH is flexible but needs many parameters; GARCH $(1,1)$ below replaces a long ARCH lag sum with one shock term and one “memory” term $\beta\sigma_{t-1}^2$ .

Definition — GARCH$(1,1)$ (Bollerslev)

\sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2

Write $v_t=\varepsilon_t^2-\sigma_t^2$ and substitute $\sigma_t^2=\varepsilon_t^2-v_t$ into the recursion. The result is an ARMA $(1,1)$ in $\varepsilon_t^2$ :

\varepsilon_t^2 = \omega + (\alpha+\beta)\varepsilon_{t-1}^2 + v_t - \beta v_{t-1}

AR coefficient $\alpha+\beta$ , MA coefficient $-\beta$ . The error $v_t$ is a martingale difference (mean $0$ given the past), but its own variance moves with $\sigma_t^2$ , so this is an ARMA in form only — do not feed it to the Gaussian ARMA likelihood of Chapter 05. GARCH $(1,1)$ is the default: one shock lag, one “lagged variance” that parsimoniously mimics a long ARCH.

Definition — Covariance Stationarity of GARCH(1,1)

If $0<\alpha+\beta<1$ and $\omega>0$ , then $\varepsilon_t$ is weakly stationary with

\mathbb{E}[\varepsilon_t^2]=\frac{\omega}{1-\alpha-\beta}

If $\alpha+\beta=1$ (IGARCH) the unconditional variance is infinite (or undefined) while $\sigma_t^2$ still evolves; used sometimes for very persistent vol, at the cost of second-order stationarity.

Strict stationarity is the weaker, and more relevant, condition

Covariance stationarity ( $\alpha+\beta<1$ ) is about the second moment. Strict stationarity of GARCH(1,1) with $\omega>0$ holds under $\mathbb{E}[\log(\alpha z_t^2+\beta)]<0$ , which is weaker. Jensen’s inequality makes the gap concrete: $\log$ is strictly concave, so $\mathbb{E}[\log(\alpha z_t^2+\beta)]<\log(\alpha\mathbb{E}[z_t^2]+\beta)=\log(\alpha+\beta)$ . At $\alpha+\beta=1$ the right side is $0$ and the left is strictly negative — IGARCH is strictly stationary even though its unconditional variance does not exist. That is why software happily fits IGARCH and why $\hat\alpha+\hat\beta\approx 1$ is not, by itself, a broken model.

Half-life of a variance shock

For GARCH(1,1), a shock to $\sigma_t^2$ decays like $(\alpha+\beta)^h$ . The half-life is $\log(1/2)/\log(\alpha+\beta)$ periods (when $0<\alpha+\beta<1$ ). Equity daily GARCH often has $\alpha+\beta\approx 0.98$ , so persistence is measured in months, not days.

What Gaussian QMLE does and does not buy you

Fitting GARCH with a Gaussian likelihood when $z_t$ is not Gaussian is quasi-maximum likelihood. Two separate conditions:

Consistency needs the variance equation to be correctly specified. The innovation distribution may be wrong; $\hat\omega,\hat\alpha,\hat\beta$ still converge to the truth.
Asymptotic normality at the usual $\sqrt{T}$ rate additionally needs a finite fourth moment of the standardized innovation, $\mathbb{E}[z_t^4]<\infty$ , plus interior parameters.

So misspecifying the shape of $z_t$ is survivable; misspecifying the variance recursion is not. And if $z_t$ is heavy enough that $\mathbb{E}[z_t^4]=\infty$ — a Student- $t$ with $\nu\leq 4$ — the standard asymptotics behind every printed $t$ -statistic simply do not apply.

❌ Interpreting a GARCH $t$-stat on $\beta$ with Gaussian MLE as ‘proof’ after a crash

Quasi-MLE tolerates a misspecified innovation distribution, but the reported standard errors do not come free.

Why it breaks: the Gaussian information-matrix equality fails once $z_t$ is not normal, so naive SEs are wrong in an unknown direction. Separately, $\varepsilon_t^2$ is highly leveraged by outliers — one 1987-style day dominates $\hat\alpha$ .

Consequence: report Bollerslev–Wooldridge sandwich SEs, or estimate with $t$ /GED innovations directly. Do not treat $\hat\alpha+\hat\beta=0.999$ as a precisely estimated unit root in variance without a dedicated IGARCH test.

Testing at the boundary

The parameter space imposes $\alpha\geq 0$ and $\beta\geq 0$ . A test of “no ARCH” ( $\alpha=0$ ) therefore sits on the boundary, where the usual $t$ and likelihood-ratio asymptotics fail — the LR limit is a mixture of $\chi^2$ distributions, not a $\chi^2_1$ , and the naive $p$ -value is conservative. Use Engle’s LM test on squared residuals for “is there ARCH at all,” and save $t$ -statistics for parameters you already believe are interior.

03 · Leverage and Asymmetry (What to Know)

Equity vol rises more after negative returns than after positive ones of the same size (leverage / news-impact). Symmetric GARCH cannot capture that.

Standard extensions, not derived here in full:

GJR-GARCH / TARCH: extra term $\gamma\varepsilon_{t-1}^2\mathbf{1}_{\{\varepsilon_{t-1}<0\}}$ . Negative returns raise tomorrow’s variance more than positive returns of the same size — the leverage / news-impact asymmetry equity indexes show.
EGARCH: models $\log\sigma_t^2$ , so positivity is automatic, and a signed $z_{t-1}$ term encodes the same asymmetry without forcing $\sigma_t^2>0$ by parameter bounds alone.

If the application is equity index risk, at least test for leverage (sign bias) before locking GARCH(1,1). FX often looks closer to symmetric.

04 · Use in Risk: One-Day VaR

Step-by-step — Gaussian GARCH(1,1) 1-day 99% VaR

Fit GARCH(1,1) to

r_t

(demean first if

\mu\neq 0

). Obtain

\hat\sigma_{T+1}^2=\hat\omega+\hat\alpha\hat\varepsilon_T^2+\hat\beta\hat\sigma_T^2

z_t\sim N(0,1)

, the 1% quantile of

r_{T+1}

given the past is

\hat\mu+\hat\sigma_{T+1}\Phi^{-1}(0.01)

with

\Phi^{-1}(0.01)\approx -2.33

For a long position, VaR is the positive loss at that left-tail quantile: scale

-\text{quantile}

by notional. Step 2 gives the return threshold; step 3 turns it into a currency loss number.

Gaussian

z_t

understates 99% VaR for real returns — the tail misspecification in step 4 is exactly why step 2’s

\Phi^{-1}

is too small. Replace

\Phi^{-1}

by a fitted

t

quantile or by historical

z

(filtered historical simulation).

05 · Exercises

EXERCISE 6.1

Take $\mathbb{E}[\,\cdot\,]$ in the GARCH(1,1) recursion and use $\mathbb{E}[\varepsilon_{t-1}^2]=\mathbb{E}[\sigma_{t-1}^2]$ under stationarity.

$\mathbb{E}[\sigma_t^2]=\omega+\alpha\mathbb{E}[\varepsilon_{t-1}^2]+\beta\mathbb{E}[\sigma_{t-1}^2]$ . Stationarity equates the three expectations of squares, so $\mathbb{E}[\varepsilon_t^2](1-\alpha-\beta)=\omega$ , hence $\omega/(1-\alpha-\beta)$ provided $\alpha+\beta<1$ .

Derive $\mathbb{E}[\varepsilon_t^2]=\omega/(1-\alpha-\beta)$ for covariance-stationary GARCH(1,1).

EXERCISE 6.2

Levels can be uncorrelated while squares are not — that is still WN in the Chapter 01 sense if second moments exist and $\mathrm{Cov}(\varepsilon_t,\varepsilon_{t-h})=0$ .

For GARCH with $\mathbb{E}[\varepsilon_t\mid\mathcal{F}_{t-1}]=0$ , $\mathrm{Cov}(\varepsilon_t,\varepsilon_{t-h})=0$ for $h\geq 1$ . So $\{\varepsilon_t\}$ is WN (and a martingale difference) when unconditional variance is finite. It is not iid, because $\mathrm{Var}(\varepsilon_t\mid\mathcal{F}_{t-1})$ varies. ACF of $\varepsilon_t$ is uninformative; ACF of $\varepsilon_t^2$ is the GARCH signature.

Is a covariance-stationary GARCH(1,1) process white noise? Is it iid? Which ACF should you plot?

EXERCISE 6.3

Half-life $h$ solves $(\alpha+\beta)^h=1/2$ .

$h=\log(1/2)/\log(0.98)\approx 34.3$ trading days — on the order of seven weeks. A one-off shock to variance is still about half present a month and a half later. That is why GARCH forecasts for next week are not “the unconditional $\sigma$ .”

If $\alpha+\beta=0.98$ , what is the half-life of a variance shock in trading days? Interpret.

06 · Chapter Summary

Item	Statement
Clustering	$\lvert r_t\rvert$ persists; $r_t$ itself may not
ARCH	$\sigma_t^2$ linear in lagged $\varepsilon^2$
GARCH(1,1)	$\omega+\alpha\varepsilon_{t-1}^2+\beta\sigma_{t-1}^2$
Covariance stationarity	$\alpha+\beta<1$ ; $\mathbb{E}[\varepsilon^2]=\omega/(1-\alpha-\beta)$
Strict stationarity	$\mathbb{E}[\log(\alpha z^2+\beta)]<0$ ; weaker, holds for IGARCH
Gaussian QMLE	consistent if the variance equation is right; normality needs $\mathbb{E}[z^4]<\infty$
Persistence	$(\alpha+\beta)^h$ decay; half-life $\log 2/\lvert\log(\alpha+\beta)\rvert$
Leverage	equities: bigger vol response to negative $r_t$
VaR	use $\sigma_{T+1}$ and a tail for $z$ , not only $N(0,1)$

Next: Chapter 07 — Vector Autoregressions and Granger Causality, when more than one series must move together in the mean.