Modeling Volatility and GARCH Family › Asymmetric GARCH Models

In our study of stylized facts, we noted that negative returns tend to increase volatility more than positive returns of the same magnitude. This asymmetry, called the leverage effect, has a financial explanation. When stock prices fall, firms become more leveraged (debt-to-equity ratio rises), increasing the riskiness of equity. A firm with fixed debt obligations becomes more precarious as its equity value declines, making future returns more uncertain.

Beyond the mechanical leverage explanation, psychological and behavioral factors may contribute. Negative returns may trigger fear and uncertainty among investors, leading to more volatile trading behavior. Margin calls during market declines force liquidations that amplify price movements. Regardless of the cause, the asymmetry remains: bad news moves volatility more than good news.

Standard GARCH treats positive and negative shocks symmetrically through the $\epsilon_{t-1}^2$ term, so a positive two percent return and a negative two percent return contribute equally to tomorrow's variance. Several extensions address this limitation by allowing the model to distinguish between good and bad news.

Nelson's 1991 EGARCH model specifies the log of conditional variance:

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

where:

• $\ln(\sigma_t^2)$: natural log of conditional variance
• $\omega$: constant parameter
• $\frac{\epsilon_{t-1}}{\sigma_{t-1}}$: standardized residual $z_{t-1}$
• $\alpha$: coefficient on the magnitude of the shock (symmetric effect)
• $\gamma$: coefficient on the sign of the shock (asymmetric/leverage effect)
• $E\left| \frac{\epsilon_{t-1}}{\sigma_{t-1}} \right|$: expected absolute value of standardized residual
• $\beta$: persistence coefficient

The key features of EGARCH distinguish it from other asymmetric specifications:

• Log specification: Modeling the log of variance rather than variance itself ensures $\sigma_t^2 > 0$ without requiring parameter constraints. The exponential function always produces positive variance.
• Standardized residuals: Uses $z_{t-1} = \epsilon_{t-1}/\sigma_{t-1}$ for stability. Dividing by the conditional standard deviation normalizes the shock, making coefficients more interpretable and comparable across different volatility levels.
• Asymmetry parameter $\gamma$: Negative $\gamma$ captures the leverage effect, where negative shocks increase volatility more than positive shocks. negative standardized residual adds to log variance both through the magnitude term (which responds to the absolute value) and directly through the sign term (which is now positive when $z_{t-1}$ is negative).

Glosten, Jagannathan, and Runkle (1993) proposed a simpler asymmetric specification that modifies standard GARCH in a more intuitive way:

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

where $I_{t-1}$ is an indicator function:

$$I_{t-1} = \begin{cases} 1 & \text{if } \epsilon_{t-1} < 0 \\ 0 & \text{if } \epsilon_{t-1} \geq 0 \end{cases}$$

where:

• $\sigma_t^2$: conditional variance
• $\omega$: constant parameter
• $\alpha$: coefficient on symmetric response
• $\gamma$: coefficient on asymmetric leverage effect
• $I_{t-1}$: binary indicator for negative shocks
• $\epsilon_{t-1}$: past return shock
• $\beta$: coefficient on past variance
• $\sigma_{t-1}^2$: lagged conditional variance

The interpretation is straightforward, requiring no logarithms or standardization.

• Positive shocks contribute $\alpha \epsilon_{t-1}^2$ to variance
• Negative shocks contribute $(\alpha + \gamma) \epsilon_{t-1}^2$ to variance
• $\gamma > 0$ implies negative shocks have a larger impact (leverage effect)

The indicator function acts as a switch. It turns on the additional $\gamma \epsilon_{t-1}^2$ term only when the previous shock was negative. This creates a piecewise linear response to shocks; it has a steeper slope for negative shocks than for positive ones. The simplicity of this specification makes it easy to interpret and estimate, which is why it appears often in applied work.