Applications of state space models in finance

64 5 Conditional heteroskedasticity models
The SB test checks whether the squared current residual depends on the sign of lagged
residuals. It is defined as the t-statistic for the coefficient β1 in the following regression:
ɛ ∗2
t = β0 + β1S − t−1 + ηt, (5.20)
with ɛt := yt − µ and ɛ∗ t := ɛt/σ, where µ and σ refer to the unconditional first two
moments of yt, the series to be considered. S − t−1 represents a dummy variable that
takes the value of unity for negative ɛt and zero otherwise. Replacing S − t−1 in (5.20) by
S − t−1ɛ∗t−1 , or by S+ t−1ɛ∗t−1 with S+ t−1 = 1 − S− t−1 , leads to the NSB test and PSB test,
respectively. They test whether the conditional volatility depends on the size of past
negative or positive shocks. As the corresponding test statistics are all t-ratios, they
asymptotically follow the standard normal distribution.
A general test for nonlinear GARCH can be derived by conducting these tests jointly:
ɛ ∗2
t = β0 + β1S − t−1 + β2S − t−1 ɛ∗ t−1 + β3S + t−1 ɛ∗ t−1 + ηt, (5.21)
where the null hypothesis is defined as H0 : β1 = β2 = β3 = 0. The test-statistic is equal
to T times the R 2 from this regression. It asymptotically follows a χ 2 distribution with
three degrees of freedom.
5.1.3 Non-Gaussian conditional densities
Even though GARCH models have thicker than normal tails, it is still possible that the
kurtosis induced by a GARCH model with conditional normal errors does not capture
the leptokurtosis present in high-frequency financial time series completely (cf. Bollerslev
et al. 1994). If not taken explicitly into account, this leads to a formally misspecified
likelihood function. However, as long as E(zt|Ωt−1) = 0 and V ar(zt|Ωt−1) = 1, the
future conditional variance does not depend on the distribution of zt. Under these circumstances,
QML estimation still yields asymptotically valid volatility forecasts without
having the distribution of zt fully specified (cf. Andersen et al. 2005).
Alternatively, instead of relying on QML based procedures, fully efficient ML estimates
can be obtained by considering alternative distributions with fatter than normal tails
for zt. The most common choice is to employ a standardized Student-t distribution with
ν degrees of freedom as proposed by Bollerslev (1987). The t-distribution is symmetric
around zero; the number of existing moments is restricted by ν. For the standard
GARCH(1,1) model, the fourth moment of zt exists for ν > 4. It is given by
Kz =
3(ν − 2)
. (5.22)
ν − 4
The value of Kz is greater than the normal value of three. This results in an unconditional
kurtosis of ɛt that is also greater than in the normal case. The number of degrees
of freedom is treated as an extra parameter and can be estimated together with the other
model parameters. In case of the nonlinear GJR-GARCH(1,1) model with zt ∼ t(ν) and
ν ≥ 5, and Kɛ and Kz being defined as in (5.18) and (5.22), respectively, the fourth
moment exists if (5.17) holds (cf. Ling and McAleer 2002).
To account for the observed leptokurtosis, several other parametric distributions have
been suggested. Examples include the normal-Poisson mixture distribution (Jorion 1988)