Analysis of Financial Time Series, Third Edition

3.9 The Threshold GARCH Model

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

(3.34)

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

(3.35)

where the estimated parameter ...