Applications of state space models in finance

5.2 Stochastic volatility 65
or the generalized error distribution (Nelson 1991). As none of these will find consideration
in the empirical part of this thesis, the reader is referred to Bollerslev et al. (1992)
for a summary and an account of the relevant literature.
5.1.4 Parameter estimation
As a wide range of GARCH specifications can be estimated by standard econometric
software packages, parameter estimation will only be briefly summarized. Throughout
this thesis, all the computations related to GARCH models are carried out using Ox 3.30
by Doornik (2001) together with the package G@RCH 2.3 by Laurent and Peters (2002).
Following Bollerslev et al. (1994) GARCH models are usually estimated by ML,
where the assumption of an IID distribution for zt is made. It follows from (5.3) that
zt(θ) := ɛ(θ)ht(θ) (−1/2) , where the conditional mean and variance functions depend on
the finite dimensional vector θ with true value θ0. Let f(zt; ν) denote the conditional
density function for the standardized innovations with mean zero, variance unity and
nuisance parameters ν. Let ψ ′ := (θ ′ , ν ′ ) be the combined vector of the parameters
to be estimated. The conditional loglikelihood function for the t-th observation can be
expressed as
lt(yt; ψ) = log f(zt(θ); ν) − 1
2 log ht(θ), t = 1, . . . , T. (5.23)
The second term on the right hand side appears, because ht depends on the unknown
parameters in the conditional mean for yt (cf. Franses and van Dijk 2000, ¢ 4.3.1). Once
an explicit assumption for the conditional density in (5.23) has been made, the ML
estimator for the true parameters ψ 0 , denoted as ˆ ψ ML , can be calculated by maximizing
the loglikelihood for the full sample:
log L(yT , yT −1, . . . , y1; ψ) =
T�
lt(yt; ψ), (5.24)
where yT , yT −1, . . . , y1 refer to the sample realizations of the GARCH model.
As the first-order condition to be solved is nonlinear in the parameters, ˆ ψ ML is obtained
by employing iterative optimization procedures. Following Bollerslev (1986) the
most popular procedure to estimate GARCH models is the algorithm named after Berndt
et al. (1974). Although convergence may fail in specifications with many parameters,
usually no convergence problems arise in connection with univariate GARCH models
and large data sets (cf. Alexander 2001, ¢ 4.3).
5.2 Stochastic volatility
This section introduces the concept of stochastic volatility (SV). In contrast to the
class of GARCH models, the SV approach includes an unobservable shock to the return
variance, which cannot be characterized explicitly based on observable past information.
This raises the difficulty that no closed expression for the likelihood function exists.
As the parameters of the SV model cannot be estimated by a direct application of
standard maximum likelihood techniques, estimation is conducted by approximation or
t=1