GARCH (Generalised Autoregressive Conditional Heteroskedasticity) models time-varying volatility in financial returns. The GARCH(1,1) model specifies the conditional variance as h_t = ω + α·ε_{t-1}² + β·h_{t-1}. Stationarity requires α + β < 1. GARCH captures volatility clustering, the fact that large price moves tend to follow large moves. EGARCH and TARCH extend this to capture asymmetric effects (the leverage effect). All are estimated by maximum likelihood.
The starting point: properties of financial returns
Consider the log return on a stock or index: rt = log(Pt) − log(Pt-1). Several empirical properties of financial return data are robustly documented across markets and time periods:
Near-zero autocorrelation in levels. The ACF of daily or weekly returns is close to zero at all lags. Returns are hard to forecast, roughly consistent with market efficiency. If you try to predict tomorrow's return from today's, the correlation is small (though not exactly zero, momentum effects exist, particularly at monthly frequency).
Significant autocorrelation in squared returns. The ACF of rt² is significantly positive at many lags. This is the signature of volatility clustering: periods of large returns (in either direction) tend to be followed by periods of large returns. Calm periods follow calm periods.
Fat tails (leptokurtosis). The distribution of returns has heavier tails than a normal distribution, extreme moves occur more often than normality would predict. The excess kurtosis (fourth standardised central moment minus 3) is positive for most financial return series.
Asymmetric response to shocks. Negative returns tend to increase volatility more than positive returns of the same magnitude. This is called the leverage effect, declining stock prices increase a firm's debt-to-equity ratio, increasing financial risk and hence volatility.
The basic ARMA framework captures the first property well but has nothing to say about the second, third or fourth. We need models for the conditional variance, how the variance of tomorrow's return depends on today's information.
ARCH: the first model for conditional variance
Robert Engle (Nobel Prize in Economics, 2003) introduced the ARCH (Autoregressive Conditional Heteroskedasticity) model in 1982. The key idea is to let the variance of the return at time t depend on past squared errors.
Write the return as Rt = μt + ut, where μt is the conditional mean (which could be a constant, or an ARMA model) and ut is the shock. Given information It-1 available at time t−1:
ut | It-1 ~ N(0, ht)
ht = ω + α1ut-1² + α2ut-2² + … + αqut-q²
The conditional variance ht depends on the squared past shocks ut-j². A large shock last period (|ut-1| large) increases the conditional variance this period, the model captures the fact that a large move yesterday predicts a volatile today.
The ARCH model requires non-negativity: ω > 0 and αj ≥ 0 for all j. For the unconditional variance to be finite (covariance stationarity), we also need Σαj < 1.
GARCH(1,1): the workhorse of volatility modelling
Bollerslev (1986) generalised the ARCH model by including lagged conditional variances as well as lagged squared shocks, analogous to adding MA terms to an AR model. The GARCH(1,1) is by far the most widely used specification:
ht = ω + α·ut-1² + β·ht-1
with ω > 0, α ≥ 0, β ≥ 0, and α + β < 1 for stationarity.
Reading the equation:
- ω is the long-run average variance contribution. It determines where the conditional variance settles in the long run.
- α (the ARCH coefficient) captures how much yesterday's squared shock affects today's variance. A large α means the variance responds quickly to recent shocks.
- β (the GARCH coefficient) captures persistence, how much the variance from yesterday carries over to today. A large β means volatility is persistent: a volatile period stays volatile for a long time.
Stationarity and the long-run variance
The GARCH(1,1) is covariance stationary if and only if α + β < 1. Under stationarity, the unconditional (long-run) variance is:
This is the variance the process reverts to in the long run. In practice, estimated GARCH parameters for financial returns often show α + β close to 0.95–0.99, indicating very high volatility persistence, a shock to volatility takes a long time to dissipate. When α + β = 1, we have the Integrated GARCH (IGARCH) model, which is non-stationary in variance, shocks to volatility are permanent.
Typical estimated values: For daily equity returns, GARCH(1,1) estimates are often around ω ≈ 0.00001, α ≈ 0.08, β ≈ 0.91, giving α + β ≈ 0.99. This high persistence means a volatility spike on one day will still be partially visible many days later, consistent with the empirical clustering we observe.
Mean reversion in variance
The GARCH(1,1) can be written as:
where σ̄² = ω/(1−α−β) is the long-run variance. This shows that ht mean-reverts toward σ̄² at rate (1−α−β). High current variance → next period's variance is pulled back toward the long-run level. This is the mechanism by which calm eventually returns after turbulent periods.
Asymmetric GARCH models
A well-documented asymmetry in equity markets is the leverage effect: negative returns (price falls) increase volatility more than positive returns (price rises) of the same magnitude. The symmetric GARCH(1,1) cannot capture this, it enters ut-1² which is the same whether the shock was positive or negative.
| Model | Conditional variance specification | What it captures |
|---|---|---|
| ARCH(q) | ht = ω + Σαjut-j² | Variance depends on past squared shocks only |
| GARCH(1,1) | ht = ω + αut-1² + βht-1 | Variance persistence (both past shocks and past variance) |
| TARCH / GJR-GARCH | ht = ω + αut-1² + γut-1²·𝟙[ut-1<0] + βht-1 | Asymmetry: negative shocks (γ>0) have larger impact on variance |
| EGARCH | log(ht) = ω + α|ut-1/√ht-1| + γ(ut-1/√ht-1) + β·log(ht-1) | Asymmetry through sign of standardised shock; log ensures ht>0 |
| GARCH-in-mean | Rt = μ + δ·ht + ut | Volatility in mean equation, risk-return tradeoff |
The EGARCH (Exponential GARCH, Nelson 1991) is particularly important. By modelling log(ht) rather than ht, it guarantees non-negativity of variance without needing to impose sign constraints on parameters. The term γ(ut-1/√ht-1) captures asymmetry: γ < 0 means that negative standardised shocks increase log variance more than positive ones, the leverage effect.
Estimation by maximum likelihood
GARCH models are estimated by maximising the conditional log-likelihood. Assuming conditionally normal errors:
The log-likelihood is the sum of the contributions from each period. Each term has two parts: −(1/2)log(ht) rewards a small predicted variance when the shock ut turns out to be small (not being unnecessarily alarmed), and −ut²/(2ht) penalises a large squared shock relative to the predicted variance (being surprised by a large move when you predicted calm).
Maximisation is iterative (Newton-Raphson or BFGS). The ht sequence is computed recursively: starting from an initial value h1 (typically the unconditional sample variance), each ht is computed from the preceding shock and variance. The initial value affects estimation but the effect dissipates as T grows.
Non-Gaussian alternatives: Financial returns have fat tails, the conditional normal assumption understates tail probabilities. Common alternatives are the Student-t distribution and the generalised error distribution (GED). Using the t-distribution in the likelihood adds one parameter (the degrees of freedom ν) and typically improves fit significantly for daily or weekly data.
Testing and selecting GARCH specifications
The workflow for a GARCH analysis is:
- Estimate the conditional mean. This could be a constant, an AR(p), or a GARCH-in-mean specification. Get the residuals ût.
- Test for ARCH effects. Regress ût² on a constant and q lags of ût-j²; if the R² is significant (T·R² ~ χ²q under H₀), there are ARCH effects and a GARCH model is warranted.
- Estimate GARCH(1,1). This almost always fits well as a starting point.
- Test for asymmetry. Fit the TARCH or EGARCH and test whether γ is significant. In equity markets, γ < 0 is almost always found.
- Compare specifications using AIC/BIC. GARCH(1,1) often wins on AIC/BIC even against larger models, more parameters rarely improve fit enough to justify the penalty.
- Check standardised residuals. The standardised residuals êt = ût/√ĥt should be approximately iid with unit variance. Check their ACF and ACF of their squares, both should show no significant autocorrelation if the model is adequate.
Why GARCH matters beyond modelling
GARCH estimates have direct practical uses in finance:
Option pricing. The Black-Scholes model assumes constant volatility, the famous "volatility smile" in implied volatility surfaces is evidence that markets price options using a volatility process closer to GARCH. GARCH option pricing models (Duan, 1995) incorporate time-varying volatility directly.
Value-at-Risk (VaR). Risk management requires estimating the probability of large losses. A GARCH model gives a time-varying conditional variance ĥt, from which a one-day 99% VaR can be computed as ẑ0.01·√ĥt+1 (where ẑ0.01 is the 1% quantile of the assumed distribution). GARCH-based VaR is time-varying, it rises in turbulent periods and falls in calm ones.
Portfolio optimisation. Mean-variance optimisation uses a variance-covariance matrix of returns. A multivariate GARCH model (e.g., DCC-GARCH) allows this matrix to change over time, improving portfolio allocation in volatile periods.
Free lecture notes
The full EC5609 Financial Econometrics lecture slides, including Lecture 7 on GARCH and the Lecture 1 slides on distribution theory and properties of stock returns, are available to download free at drnickygrant.com/teaching.html
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