Skip to content

Extensions of the GARCH Model

Standard GARCH models assume that positive and negative shocks of the same magnitude have the same effect on volatility. However, real-world financial data often exhibits asymmetric effects. These extensions are tailored to capture specific volatility dynamics like the leverage effect, long memory, and multivariate co-movements.

GJR-GARCH

Developed by Glosten, Jagannathan, and Runkle, this model specifically targets asymmetric effects in volatility.

\[ \sigma_{t} ^{2} = \omega + \alpha \epsilon_{t-1}^{2} + \gamma\epsilon_{t-1}^{2}I(\epsilon_{t-1}\lt 0) + \beta \sigma_{t-1}^{2} \]
  • \(I(\epsilon_{t-1} < 0)\) is an indicator function that equals 1 if the shock is negative (bad news) and 0 otherwise.
  • \(\gamma > 0\) captures the additional impact of negative shocks.
  • Application: Financial markets, where "bad news" typically increases volatility much more than "good news."

EGARCH (Exponential GARCH)

The EGARCH model handles leverage effects without requiring non-negativity constraints on the parameters, as it models the logarithm of variance. This ensures that conditional variance is always positive.

\[ \log(\sigma_{t}^{2}) = \omega + \beta \log(\sigma_{t-1}^{2}) + \alpha \dfrac{\epsilon_{t-1}}{\sigma_{t-1}} + \gamma\left(\lvert \dfrac{\epsilon_{t-1}}{\sigma_{t-1}} \rvert - E\left[\lvert \dfrac{\epsilon_{t-1}}{\sigma_{t-1}} \rvert \right] \right) \]
  • The log transformation guarantees \(\sigma_{t}^{2} > 0\).
  • \(\gamma > 0\) captures the impact of negative shocks.
  • Application: Financial assets with pronounced leverage effects.

TGARCH (Threshold GARCH)

Similar to GJR-GARCH, but it models the standard deviation (\(\sigma_t\)) instead of the variance (\(\sigma_t^2\)).

\[ \sigma_{t} = \omega + \alpha |\epsilon_{t-1}| + \gamma |\epsilon_{t-1}| I(\epsilon_{t-1}\lt 0) + \beta \sigma_{t-1} \]

APARCH (Asymmetric Power ARCH)

A highly flexible model that allows the data to determine the power transformation (\(\delta\)) of the volatility.

\[\sigma_{t}^\delta = \omega + \alpha(|\epsilon_{t-1}| - \gamma \epsilon_{t-1})^{\delta} + \beta \sigma_{t-1}^{\delta}\]
  • \(\delta\): Power parameter for the transformation.
  • \(\gamma\): Asymmetry parameter.
  • Application: Markets with non-linear and asymmetric patterns, such as energy and commodity markets.

FIGARCH (Fractionally Integrated GARCH)

This model accounts for the long memory property of volatility, where the impact of shocks decays at a slow, hyperbolic rate rather than exponentially.

\[\phi(L)(1-L)^{d} \sigma_{t}^{2} = \omega + (1- \beta(L))\epsilon_{t}^{2}\]
  • \((1-L)^{d}\) is the fractional differencing operator (\(0 < d < 1\)).
  • Application: Capturing persistent volatility in bond yields and exchange rates.

MGARCH (Multivariate GARCH)

Used when analyzing multiple time series simultaneously to capture co-movements in volatilities.

  • VEC Model: Directly models the variance-covariance matrix (suffers from overparameterization).
  • BEKK Model: Imposes a structure that ensures the covariance matrix is always positive definite.
  • DCC-GARCH (Dynamic Conditional Correlation): A two-step approach that models individual volatilities first, then time-dependent correlations.

Specialized GARCH Variants

  • NGARCH (Nonlinear GARCH): Captures complex non-linear interactions via a term multiplying the shock and the standard deviation (\(\lambda\epsilon_{t-1}\sigma_{t-1}\)).
  • HARCH (Heterogenous ARCH): Based on the Heterogenous Market Hypothesis, it aggregates returns over multiple time horizons (e.g., daily, weekly, monthly).
    • Logic: Market participants react to news over different time horizons.
  • GARCH-X: Incorporates an exogenous covariate \(X_t\) (like a news sentiment index or macroeconomic indicator) directly into the variance equation.

ARMA + GARCH Framework

In practice, these are combined. We use an ARMA model to capture serial correlation in the mean, and a GARCH model to capture the conditional variance.

  1. Mean Equation: \(y_t = \text{ARMA}(p,q) + \epsilon_t\)
  2. Variance Equation: \(\sigma_t^2 = \text{GARCH}(r,s)\)

Applications:
- Stock returns and Value-at-Risk (VaR) calculations.
- Modeling the relationship between GDP growth and interest rate volatility.
- Pricing energy commodities (Oil/Electricity).