Another volatility model commonly used to handle leverage effects is the threshold GARCH (or TGARCH) model; see Glosten, Jagannathan, and Runkle (1993) and Zakoian (1994). A TGARCH(m, s) model assumes the form
where Nt−i is an indicator for negative at−i, that is,
and αi, γi, and βj are nonnegative parameters satisfying conditions similar to those of GARCH models. From the model, it is seen that a positive at−i contributes 
to 
, whereas a negative at−i has a larger impact 
with γi > 0. The model uses zero as its threshold to separate the impacts of past shocks. Other threshold values can also be used; see Chapter 4 for the general concept of threshold models. Model (3.34) is also called the GJR model because Glosten et al. (1993) proposed essentially the same model.
For illustration, consider the monthly log returns of IBM stock from 1926 to 2003. The fitted TGARCH(1,1) model with conditional GED innovations is
where the estimated parameter of the GED is 1.51 with standard error 0.099. The standard error of the parameter for the mean equation is 0.002 and the standard errors of the parameters in the volatility equation are 1.26 × −4, 0.0314, 0.0395, and 0.049, respectively. To check the fitted model, we have Q(12) = 18.34(0.106) for the standardized residual ãt and Q(12) = 5.36 (0.95) for 
. The model is adequate in modeling the first two conditional moments of the log return series. Based on the fitted model, the leverage effect is significant at the 5% level.
S-Plus Commands Used
> ibm.tgarch = garch(ibmln∼1,∼tgarch(1,1),leverage=T,
+ cond.dist=‘ged’)
> summary(ibm.tgarch)
> plot(ibm.tgarch)
It is interesting to compare the two models in Eqs. (3.32) and (3.35) for the monthly log returns of IBM stock. Assume that at−1 = ± 2σt−1 so that ϵt−1 = ± 2. The EGARCH(1,1) model gives
On the other hand, ignoring the constant term 0.000345, the TGARCH(1,1) model gives
The two models provide similar leverage effects.