Lesson 3.4: GARCH Extensions

Capturing more complex volatility dynamics like the leverage effect with EGARCH and GJR-GARCH.

The Limitation of Standard GARCH

The standard GARCH(1,1) model was a huge leap forward, but it has one major limitation inherited from ARCH: it's **symmetric**. It models variance using the *squared* shock ( $\epsilon_{t-1}^2$ ), meaning that a large positive shock has the exact same effect on future volatility as a large negative shock of the same magnitude.

In real financial markets, this is not true. This asymmetry is known as the **leverage effect**.

The Leverage Effect

The "Leverage Effect"

Empirically, negative news (a negative shock) tends to increase volatility much more than positive news of the same magnitude. A -5% day is much scarier and leads to more future uncertainty than a +5% day. To capture this, we need models that can react asymmetrically to shocks.

Two Popular Asymmetric GARCH Models

1. EGARCH (Exponential GARCH)

The EGARCH model, proposed by Nelson (1991), models the **logarithm** of the variance, which has the nice property of not needing positivity constraints on the coefficients.

EGARCH(1,1) Equation

\ln(\sigma_t^2) = \omega + \beta \ln(\sigma_{t-1}^2) + \alpha \frac{\epsilon_{t-1}}{\sigma_{t-1}} + \gamma \left( \left|\frac{\epsilon_{t-1}}{\sigma_{t-1}}\right| - \sqrt{\frac{2}{\pi}} \right)

The key term is $\alpha \frac{\epsilon_{t-1}}{\sigma_{t-1}}$ . If the coefficient $\alpha$ is negative and significant, it confirms the leverage effect: a negative shock $\epsilon_{t-1}$ will have a larger impact on future volatility than a positive shock.

2. GJR-GARCH

The GJR-GARCH model, by Glosten, Jagannathan, and Runkle (1993), provides a simpler way to capture the leverage effect by adding an indicator variable.

GJR-GARCH(1,1) Equation

\sigma_t^2 = \omega + (\alpha + \gamma I_{t-1})\epsilon_{t-1}^2 + \beta \sigma_{t-1}^2

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

A positive and significant $\gamma$ indicates a leverage effect. When there is bad news ( $\epsilon_{t-1} < 0$ ), the impact of the shock on volatility is $\alpha + \gamma$ , which is larger than the impact $\alpha$ from good news.

What's Next? The Capstone Project

We now have a full suite of powerful volatility models. It's time to put them into practice.

In our final lesson for this module, we will perform an end-to-end capstone project: **Modeling and Forecasting the Volatility of the S&P 500** using a GARCH(1,1) model.

GARCH Models

Capstone: Modeling S&P 500 Volatility