GJR-GARCH and the Leverage Effect

Standard GARCH models future volatility as depending on the magnitude of yesterday's return shock but not its sign — a -3% day and a +3% day feed into tomorrow's variance forecast identically. Equity markets don't actually behave that way: volatility empirically jumps more after a sharp decline than after an equally sized rally, a pattern long called the leverage effect (originally attributed to falling equity value mechanically raising a firm's debt-to-equity ratio, though the effect is now understood to be driven more broadly by risk aversion and volatility feedback than leverage itself).

GJR-GARCH (Glosten-Jagannathan-Runkle) captures this asymmetry by adding one indicator term to the standard GARCH variance equation:

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

where $I_{t-1}$ equals 1 if yesterday's shock $\epsilon_{t-1}$ was negative and 0 otherwise. In plain English: a negative shock gets an extra dose of variance impact, $\gamma \epsilon_{t-1}^2$, on top of the ordinary $\alpha \epsilon_{t-1}^2$ term that both positive and negative shocks receive — so a positive $\gamma$ directly measures how much harder down-days hit future volatility than up-days of the same size.

Fitting GJR-GARCH to an equity index almost always finds $\gamma > 0$ and statistically significant, confirming the asymmetry is real and not just noise — which matters directly for options pricing and risk models, since ignoring it understates the volatility spike that tends to follow a market drop.