14.3 GARCH models and extensions

GARCH models expand on ARCH models by including past conditional variances in the current variance equation. This produces a more efficient representation of volatility processes, capturing long-term dependencies with fewer parameters and reducing overfitting risk.

Extensions like EGARCH and TGARCH address asymmetric effects in volatility, particularly the "leverage effect" in financial time series. These models allow negative and positive shocks to impact volatility differently, which improves accuracy in forecasting and risk management.

GARCH Models

GARCH model vs ARCH models

An ARCH model explains today's variance using only past squared residuals (past shocks). A GARCH model adds lagged conditional variances to that equation, so the model "remembers" its own recent variance estimates too.

The GARCH(p, q) model is written as:

$\sigma_t^2 = \omega + \sum_{i=1}^{q} \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^{p} \beta_j \sigma_{t-j}^2$

• $q$ is the number of lagged squared residuals (the ARCH terms)
• $p$ is the number of lagged conditional variances (the GARCH terms)

The most commonly used specification is GARCH(1,1), which includes just one ARCH lag and one GARCH lag. Even this simple version captures volatility clustering remarkably well in practice.

Why prefer GARCH over a pure ARCH model?

• Parsimony. A GARCH(1,1) can replicate the memory of a high-order ARCH model with only three parameters ($\omega, \alpha_1, \beta_1$) instead of dozens.
• Better long-term memory. The lagged variance term lets the model capture slow-decaying volatility patterns that a short-lag ARCH model would miss.
• Lower overfitting risk. Fewer parameters means more stable estimates, especially with limited data.

GARCH models with lag structures

Choosing lag orders (p, q):

You need to pick $p$ and $q$ that fit the data without overcomplicating the model. In practice, GARCH(1,1) is the default starting point. Move to higher orders only if diagnostics suggest the simpler model is inadequate.

Estimation methods:

• Maximum likelihood estimation (MLE) is the standard approach. You assume a distribution for the standardized residuals (typically Gaussian or Student's t) and maximize the corresponding log-likelihood.
• Quasi-maximum likelihood estimation (QMLE) is useful when the assumed distribution is wrong. QMLE still gives consistent parameter estimates under weaker assumptions, though standard errors need adjustment.

Model selection and diagnostics:

1. Fit candidate models with different $(p, q)$ combinations.

2. Compare them using information criteria (AIC, BIC). Lower values indicate a better trade-off between fit and complexity.

3. Check the standardized residuals of your chosen model:

   • Ljung-Box test on squared standardized residuals to detect remaining serial correlation in variance.
   • ARCH-LM test to check whether any ARCH effects are left unexplained.

4. If these tests reject, consider increasing the lag order or switching to an extension model.

Volatility persistence in forecasting

Volatility persistence measures how long the effect of a shock to variance lingers. For a GARCH model, it's captured by the sum:

$\sum_{i=1}^{q} \alpha_i + \sum_{j=1}^{p} \beta_j$

For a GARCH(1,1), this simplifies to $\alpha_1 + \beta_1$.

• When this sum is close to 1, shocks to volatility decay very slowly. A spike in variance today will still influence variance estimates far into the future.
• When the sum equals 1 exactly, you have an Integrated GARCH (IGARCH) process, where shocks never fully die out and the unconditional variance is undefined.
• When the sum is well below 1, volatility reverts to its long-run level relatively quickly.

Implications for forecasting:

• High persistence means current volatility has a strong influence on future volatility, so short-horizon forecasts will differ substantially from the long-run average.
• Long-term forecasts converge to the unconditional variance $\frac{\omega}{1 - \alpha_1 - \beta_1}$, but convergence is slower when persistence is high.
• Confidence intervals for multi-step-ahead forecasts widen more when persistence is high, reflecting greater uncertainty.

GARCH Model Extensions

Extensions of GARCH models

Standard GARCH treats positive and negative shocks symmetrically: a +2% return shock and a −2% return shock produce the same effect on variance. In reality, financial markets often exhibit a leverage effect, where negative shocks increase volatility more than positive shocks of equal size. The extensions below address this.

EGARCH (Exponential GARCH)

EGARCH models the log of the conditional variance, which has two benefits: it naturally prevents negative variance estimates, and it allows asymmetry through a sign term.

$\log(\sigma_t^2) = \omega + \sum_{i=1}^{q} \alpha_i \frac{|\varepsilon_{t-i}|}{\sigma_{t-i}} + \sum_{j=1}^{p} \gamma_j \frac{\varepsilon_{t-j}}{\sigma_{t-j}} + \sum_{k=1}^{p} \beta_k \log(\sigma_{t-k}^2)$

• The $\gamma_j$ coefficients capture asymmetry. If $\gamma_j < 0$, negative shocks increase volatility more than positive shocks.
• Because the equation is in logs, there's no need for non-negativity constraints on the parameters.

TGARCH / GJR-GARCH (Threshold GARCH)

These models introduce an indicator function that "switches on" an extra term when the shock is negative:

$\sigma_t^2 = \omega + \sum_{i=1}^{q} (\alpha_i + \gamma_i I_{t-i}) \varepsilon_{t-i}^2 + \sum_{j=1}^{p} \beta_j \sigma_{t-j}^2$

where $I_{t-i} = 1$ if $\varepsilon_{t-i} < 0$, and 0 otherwise.

• For a positive shock, the impact on variance is $\alpha_i$.
• For a negative shock, the impact is $\alpha_i + \gamma_i$. If $\gamma_i > 0$, negative shocks have a larger effect, confirming the leverage effect.
• The GJR-GARCH formulation is very similar to TGARCH, with minor differences in how the threshold term enters.

Why these extensions matter:

The leverage effect is well-documented in equity markets: stock price declines raise a firm's debt-to-equity ratio, increasing perceived risk and therefore volatility. Ignoring this asymmetry leads to biased volatility forecasts. These extensions are widely used in risk management applications such as portfolio optimization and option pricing, where accurate directional volatility estimates are critical.