Applications of state space models in finance

62 5 Conditional heteroskedasticity models
have been proposed to capture asymmetric effects. Below, only the two most influential
nonlinear extensions are discussed. For a more comprehensive overview of the
broad spectrum of nonlinear GARCH specifications, see, for example, Hentschel (1995)
or Franses and van Dijk (2000, ¢ 4.1.2).
5.1.2.1 Exponential GARCH
The first GARCH extension, in which the conditional volatility depends on both the
size as well as the sign of lagged shocks, was proposed Nelson (1991). In its simplest
specification with p = q = 1, the exponential GARCH (EGARCH) model can be written
as
log(ht) = ω + γ1zt−1 + ϑ1(|zt−1| − E(|zt−1|)) + δ1 log(ht−1). (5.13)
Due to the modeling of ht in logarithms, no restrictions on the coefficients have to be
imposed to ensure nonnegativity of the conditional volatility. Let the function g(zt) be
defined as g(zt) := γzt + ϑ1(|zt| − E(|zt|)) where the two summands relate to the sign
and to the magnitude effect. By rewriting it as
g(zt) = (γ1 + ϑ1)ztI(zt > 0) + (γ1 − ϑ1)ztI(zt < 0) − ϑ1(E|z1|), (5.14)
with I(·) being an indicator function, it can be seen how asymmetric effects are incorporated:
while the term (γ1 + ϑ1) is affected by positive shocks, negative shocks have an
impact on (γ1 − ϑ1). Generally, positive shocks have a smaller effect on ht than negative
shocks of equally sized positive shocks (cf. Engle and Ng 1993). This becomes clear by
having a look at the news impact curve (NIC) 13 for the EGARCH model, which is given
by
NIC � ɛt|ht = σ 2� ⎧ � �
(γ1 ⎪⎨
+ ϑ1)
A exp ɛt , for ɛt > 0,
σ
= � �
(5.15)
(γ1 ⎪⎩
− ϑ1)
A exp ɛt , for ɛt < 0,
σ
with A = σ2δ1 �
exp(ω − ϑ1 2/π), for parameter constellation γ1 < 0, 0 ≤ ϑ < 1 and
ϑ1 + δ1 < 1. As the EGARCH model is not differentiable with respect to zt−1 at zero,
its estimation is more difficult than that of alternative asymmetric models. Another
problem is related to forecasting. Usually, the researcher is interested in forecasting ht+l
and not log ht+l. This requires a transformation that depends on the complete l-step
ahead forecast distribution, f(yt+l|Ωt), which is generally not available in closed-form.
5.1.2.2 GJR-GARCH
Glosten et al. (1993) and Zakoian (1994) independently introduced an alternative nonlinear
extension to capture asymmetric effects: the GJR-GARCH or threshold GARCH
model. According to Ling and McAleer (2002), the GJR-GARCH model represents the
13 Engle and Ng (1993) introduced the NIC, defined as the functional relationship between
the conditional variance and lagged shocks, as a measure of how the arrival of new information
is reflected in volatility estimates. The NIC can be used to compare different GARCH models.