Applications of state space models in finance

60 5 Conditional heteroskedasticity models
lagged squared shocks to the function of the conditional variance. However, allowing q to
become large results in many parameters to be estimated. This would be infeasible given
the positiveness and stationarity conditions that have to be imposed. In many cases, an
ARCH(q) process is not capable of capturing both the height and shape dimensions of
the autocorrelation function; further generalization is required (cf. Pagan 1996).
By including a lagged conditional variance term to the conditional variance function,
Bollerslev (1986) generalized the ARCH(q) model. The GARCH(p, q) class of models
allows for a more flexible lag structure and for longer memory effects:
ht = ω +
q�
i=1
γiɛ 2 t−i +
p�
j=1
δjht−j
= ω + γ(L)ɛ 2 t + δ(L)ht, (5.5)
with ω > 0 and L denoting the lag operator, where γ(L) = γ1L + · · · + γqL q and
δ(L) = δ1L + · · · + δpL p . For δj = 0 and j = 1, . . . , p, the GARCH(p, q) process reduces
to an ARCH(q) model. Following Bollerslev et al. (1994) the process is well-defined if
all parameters in the infinite-order autoregressive representation
ht = (1 − δ(L)) −1 γ(L)ɛ 2 t
(5.6)
are nonnegative. It is assumed that all the roots of 1−δ(L) lie outside the unit circle. The
condition for covariance-stationary of the GARCH(p, q) process is �q i=1 γi+ �p j=1 δj < 1.
5.1.1.1 Statistical properties
A specification with p = q = 1 represents the simplest and most common GARCH
model. As it is found to be sufficient for most empirical applications (cf. Bollerslev
et al. 1992), only the GARCH(1,1) process will be considered in the following. The
GARCH(1,1) model corresponds to an ARCH(∞) process, in which the conditional
variance is guaranteed to be nonnegative with ω > 0, γ1 > 0 and δ1 ≥ 0. Under the
assumption of covariance-stationarity, which is assured for γ1 +δ1 < 1, the unconditional
variance of ɛt is defined. It can be calculated as
σ 2 ω
=
. (5.7)
1 − γ1 − δ1
If (γ1 + δ1) 2 + 2γ 2 1 < 1, the unconditional fourth moment of ɛt exists. For normally
distributed zt, the kurtosis of ɛt is defined as
Kɛ = 3 � 1 − (γ1 + δ1) 2�
1 − (γ1 + δ1) 2 − 2γ2 . (5.8)
1
The kurtosis of ɛt is always greater than three. It captures some of the excess kurtosis
usually inherent to financial time series.
When the GARCH(1,1) model is applied to high-frequency data, a common finding is
that current information has an impact on conditional variance forecasts for any horizon.
This leads to a sum of γ1 and δ1 that is close or equal to unity. Engle and Bollerslev