Applications of state space models in finance

72 5 Conditional heteroskedasticity models
After substitution of (5.36) for p(θ) in (5.35), the likelihood of the original model can
be stated as
�
p(y|θ)g(θ|y)
L(ψ) = LG(ψ)
dθ
g(y|θ)
� �
p(y|θ)
= LG(ψ)EG , (5.37)
g(y|θ)
where EG denotes the expectation with respect to g(θ|y). The simulation smoother 15 by
Durbin and Koopman (2002) is used to obtain M independent draws θ (i) from g(θ|y).
The Monte Carlo likelihood of the basic SV model can be calculated as the product of
the Gaussian likelihood of the approximating model and a correction factor, which is
obtained by simulation:
�
M�
ˆL(ψ)
1 p
= LG(ψ)
M
� y|θ (i)�
g � y|θ (i)�
�
, (5.38)
To minimize the required number of draws, g(θ|y) has to be chosen to be a good approximation
to p(θ|y).
The importance sampling density can be obtained according to the procedure proposed
by Shephard and Pitt (1997) and Durbin and Koopman (1997), which approximates the
original model by a linear Gaussian state space model. In the approximating model, the
state equation is still provided by (5.26), while the observation equation is represented
by
yt = ht + at + btut. (5.39)
The standard normally distributed errors are assumed to be uncorrelated with the state
disturbances ηt. The location and scaling parameters at and bt are responsible for a good
match between the approximating and the original model. The conditional density g(θ|y)
of the approximating model is taken as the importance density. For more details on the
selection of the approximating model, see Durbin and Koopman (1997).
Based on the outline of Lee and Koopman (2004) the importance sampling procedure
to calculate the likelihood of y given the vector of parameters ψ can be summarized as
follows:
1. At the beginning, neither θ, at nor bt are known. The algorithm that solves at and
bt analytically for given θ and y is started by choosing a trial value for θ.
2. Taking logarithms of the conditional densities of y given θ, both for the approximating
and the original model, and equalizing the first two derivatives of the
resulting log-density functions with respect to θ, yields some first estimates of at
and bt.
15 According to Durbin and Koopman (2002, p. 603), “a simulation smoother in state space
time series analysis is a procedure for drawing samples from the conditional distribution of
state or disturbance vectors given the observations.” These samples are useful for analyzing
non-Gaussian and nonlinear state space models, both from a classical and from a Bayesian
perspective.
i=1