1.9.2 ARCH & GARCH

Basic Prerequisites: Econometrics & Time Series (ARMA, stationarity, forecasting) and Statistics & Inference (maximum likelihood).

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1. Intuition & Practical Objective

ARCH and GARCH answer a single question: how should yesterday’s shock enter today’s variance forecast? The ARCH answer (Engle 1982) is “through yesterday’s squared shock”. The GARCH answer (Bollerslev 1986) adds “…and through yesterday’s variance forecast”, which is the crucial generalisation: it lets a long history of shocks feed today’s risk through a short memory.

The economic reading of GARCH(1,1),

$${\sigma }_t^2=\underset{\text{floor}}{\underbrace{{\alpha }_0}}+\underset{\text{news from yesterday}}{\underbrace{{\alpha }_1{a}_{t-1}^2}}+\underset{\text{memory of the past}}{\underbrace{{\beta }_1{\sigma }_{t-1}^2}},$$

is a weighted, exponentially-decaying average of past squared shocks with a floor. It is exactly the same object as RiskMetrics’ EWMA, but with two differences that make it a proper statistical model: an intercept ${\alpha }_0$ restoring mean reversion, and estimated (not fixed) coefficients. Because the recursion feeds back, a single large shock propagates forward for many days - the model generates clustering rather than being told about it.

The practical payoff: a one-line, closed-form, one-step-ahead variance forecast that a risk system can consume directly. Everything downstream - VaR, position sizing, option-hedging vol input - starts here.

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2. Mathematical Ground Truth & Derivations

ARCH($q$). With $a_t={\sigma }_t{\epsilon }_t$, ${\epsilon }_t\stackrel{iid}{\sim }N(0,1)$,

$${\sigma }_t^2={\alpha }_0+\sum _{i=1}^q{\alpha }_i{a}_{t-i}^2,{\alpha }_0>0,{\alpha }_i\ge 0.$$

Taking unconditional expectations, $\mathbb{E}[{\sigma }_t^2]={\alpha }_0+(\sum {\alpha }_i)\mathbb{E}[a_t^2]$, and $\mathbb{E}[{\sigma }_t^2]=\mathrm{Var}(a_t)$, so

$$\mathrm{Var}(a_t)=\frac{{\alpha }_0}{1-{\sum }_{i=1}^q{\alpha }_i},\text{finite iff }\sum {\alpha }_i<1.$$

ARCH($1$) is already fat-tailed: its unconditional excess kurtosis is $3(1-{\alpha }_1^2)/(1-3{\alpha }_1^2)-3>0$ whenever $3{\alpha }_1^2<1$ (Tsay §3.4.1). Fat tails emerge without fat-tailed shocks - the clustering does it.

GARCH($p,q$).

$${\sigma }_t^2={\alpha }_0+\sum _{i=1}^q{\alpha }_i{a}_{t-i}^2+\sum _{j=1}^p{\beta }_j{\sigma }_{t-j}^2.$$

Stationarity requires ${\sum }_{i=1}^{\max (p,q)}({\alpha }_i+{\beta }_i)<1$ (set ${\alpha }_i=0$ for $i>q$, ${\beta }_j=0$ for $j>p$). Then the unconditional variance is

$$\mathrm{Var}(a_t)=\frac{{\alpha }_0}{1-{\sum }_i{\alpha }_i-{\sum }_j{\beta }_j}.$$

Derivation: take unconditional expectations of the recursion: $\mathbb{E}[{\sigma }_t^2]={\alpha }_0+\sum {\alpha }_i\mathbb{E}[a_{t-i}^2]+\sum {\beta }_j\mathbb{E}[{\sigma }_{t-j}^2]$; stationarity gives both $\mathbb{E}[a_t^2]$ and $\mathbb{E}[{\sigma }_t^2]$ equal to a common ${\bar{\sigma }}^2$, so ${\bar{\sigma }}^2={\alpha }_0+(\sum {\alpha }_i+\sum {\beta }_j){\bar{\sigma }}^2$.

GARCH(1,1) - the workhorse.

$$\boxed{{\sigma }_t^2={\alpha }_0+{\alpha }_1{a}_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2},\pi \equiv {\alpha }_1+{\beta }_1<1,\mathrm{Var}(a_t)=\frac{{\alpha }_0}{1-{\alpha }_1-{\beta }_1}.$$

ARMA($1,1$) representation. Substituting ${\sigma }_{t-1}^2=a_{t-1}^2-{\eta }_{t-1}$ (with ${\eta }_t\equiv a_t^2-{\sigma }_t^2$ a martingale difference) gives

$$a_t^2={\alpha }_0+({\alpha }_1+{\beta }_1)a_{t-1}^2+{\eta }_t-{\beta }_1{\eta }_{t-1}.$$

So $a_t^2$ is an ARMA(1,1), and the persistence $\pi ={\alpha }_1+{\beta }_1$ is its autoregressive root. This is the bridge that lets ARMA theory (stationarity, ACF decay, forecasting) transfer to the variance world - and it is why GARCH(1,1), not high-order GARCH, is usually enough: a low-order ARMA captures most of the dynamics.

Forecasting. ${\mathbb{E}}_t[a_{t+1}^2]={\sigma }_{t+1}^2$ is known, and for $\ell \ge 2$,

$${\sigma }_h^2(\ell )={\alpha }_0+\pi {\sigma }_h^2(\ell -1)⟶\frac{{\alpha }_0}{1-\pi }(\ell \to \infty ).$$

The forecast decays geometrically from the current level to the unconditional variance - the “volatility term structure”. The half-life of a shock is $\ln \frac{1}{2}/\ln \pi$; for $\pi =0.98$, $\approx 34$ days.

Estimation. With ${\epsilon }_t\sim N(0,1)$, the Gaussian log-likelihood is

$$\ln L=-\frac{1}{2}\sum _{t=1}^T[\ln 2\pi +\ln {\sigma }_t^2+\frac{a_t^2}{{\sigma }_t^2}],$$

maximised numerically (the recursion is not linear in the parameters). Method-of-moments / ARMA estimation of the $a_t^2$ series gives a starting point; MLE is the standard.

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3. Computational Implementation - simulate, fit by MLE, forecast

Standard library only. Simulates a GARCH(1,1), estimates $({\alpha }_0,{\alpha }_1,{\beta }_1)$ by Gaussian MLE with a compact Nelder–Mead optimiser (unconstrained reparametrisation: ${\alpha }_0=e^{q_0}$, persistence via a logistic so $\pi <0.999$, and a share so ${\alpha }_1,{\beta }_1\ge 0$), then produces the term-structure forecast.

Two things to read off. First, MLE recovers the true process closely: the estimated unconditional variance $4.02\text{e-}05$ is within 0.5% of the truth $4.00\text{e-}05$, and persistence is $0.917$ vs $0.950$ - the small downward bias is expected at $n=1500$ for a persistent series (a symptom of the flat likelihood surface discussed in §4). Second, the forecast path decays monotonically to the unconditional variance, exactly as ${\sigma }_h^2(\ell )={\alpha }_0+\pi {\sigma }_h^2(\ell -1)$ predicts.

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4. Failure Modes & First-Principles Breakdowns

1. Flat, non-quadratic likelihood $\Rightarrow$ unstable estimates. ${\alpha }_1$ and ${\beta }_1$ are weakly identified separately (only $\pi$ is well-determined), so estimates trade off along a ridge. This is why ”${\hat{\beta }}_1$ moved 0.05 between samples” is usually noise, not news.
2. High-order ARCH/GARCH over-parameterisation. Adding lags multiplies the positivity constraints and widens the identification ridge; the likelihood barely improves. GARCH(1,1) is empirically near-optimal - extra order buys complexity, not accuracy (Hansen–Lunde 2005’s forecast comparison reaches the same conclusion).
3. Symmetry. The model cannot tell $+3\sigma$ from $-3\sigma$; the single largest empirical mis-specification for equity risk. Fixed in 03.
4. Persistence $\to 1$ degeneracy. When $\hat{\pi }\approx 1$, the unconditional variance estimate ${\alpha }_0/(1-\pi )$ blows up (denominator near zero) and the model becomes IGARCH-in-practice - often a structural-break artefact, not true persistence (see 05).
5. Non-negativity constraints. ${\alpha }_i,{\beta }_j\ge 0$ are needed only for the “all shocks raise variance” interpretation; if the optimiser wanders into a negative coefficient the recursion can go negative and the likelihood is undefined - a common coding failure.

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5. References

• Engle, Robert F. (1982): Autoregressive Conditional Heteroscedasticity…, Econometrica 50(4), 987–1007
• Bollerslev, Tim (1986): Generalized Autoregressive Conditional Heteroskedasticity, J. Econometrics 31(3), 307–327
• Tsay, Ruey S.: Analysis of Financial Time Series (3rd ed., 2010)
• Hansen, Peter R. & Lunde, Asger (2005): A Forecast Comparison of Volatility Models: Does Anything Beat a GARCH(1,1)?, J. Applied Econometrics 20(7)

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6. Connected Graph Bridges

• Base: Econometrics & Time Series (ARMA ↔ $a_t^2$ rep.) · Statistics & Inference (MLE)
• Prior: 01 · From Zero · Hub: Index
• Continue: 03 · Asymmetric Models · 05 · Failure Modes
• Applied: VaR & Expected Shortfall (consumes ${\sigma }_{t+1}^2$)