Applications of state space models in finance

30 3 Linear Gaussian state space models and the Kalman filter
Substituting ˜σ 2 ∗ back into (3.60) gives the concentrated or profile loglikelihood function
that has to be maximized with respect to ψ∗:
T − d
log Lc(y) = − log(2π) −
2
1
2
T�
t=d+1
log ft −
T − d
(log ˜σ
2
2 ∗ (ψ∗) + 1). (3.62)
When (3.62) is maximized instead of (3.59), the dimension of the vector of parameters
to be estimated is reduced by one. In addition to the obtained gains in computational
efficiency, the results are likely to be more reliable. As no exact gradients are available
for the concentrated loglikelihood, it has to be maximized numerically.
If the Kalman filter is initialized by employing the large κ approximation, rounding
errors can lead to numerical problems. In this thesis, a feasible possibility to overcome
this problem in the univariate case is to calculate starting values from the first observations,
as explained by Harvey (1989, ¢ 3.3.4). For a more general algorithm, see Ansley
and Kohn (1985) who propose an analytical but complex and difficult to implement
solution to the exact initialization problem. An alternative exact approach is discussed
by de Jong (1991) who uses an augmentation technique to the exact Kalman filter. In
a more recent work, Koopman (1997) proposes an exact analytical approach based on
a trivial initialization and develops a diffuse loglikelihood, log Ld(y|ψ). However, as
it can be shown that the estimate ˆ ψ, obtained by maximizing log L(y|ψ) for fixed κ,
converges to the estimate obtained by maximizing the diffuse loglikelihood as κ → ∞
(cf. Durbin and Koopman 2001, ¢ 7.3.1), the approach taken here can be considered a
valid procedure for univariate models.
3.4.2 Numerical maximization
Given the sample observations, the loglikelihood can be maximized by means of direct
numerical maximization. The basic idea behind this method is to find the value ˆ ψ for
which the loglikelihood is maximized. An algorithm is used to make different guesses for
ψ and to compare the corresponding numerical values of the loglikelihood. To compute
the ML estimates, the algorithm performs a series of steps, each time starting with a first
guess for the unknown parameters. The algorithm then chooses the direction where to
search, determines how far to move in the chosen direction, and computes and compares
the value of the loglikelihood for the chosen values of ψ. If ψ leads sufficiently close
to a maximum of the loglikelihood, the algorithm stops, otherwise the search continues.
Generally, numerical maximization methods differ with respect to the direction to search,
the step size and the stopping rule (cf. Davidson and MacKinnon 2004, ¢ 6.4).
Many numerical maximization techniques are based on Newton’s method: for a given
starting value for ψ, the direction of search is determined by the gradient or score vector,
denoted as g(ψ); the step size is calculated from the Hessian matrix, denoted as H(ψ),
which has a unique maximum only if it is negative definite for all ψ. For a more detailed
description of the different available algorithms, see, for example, Hamilton (1994b, ¢ 5.7)
or Judge et al. (1985, ¢ B.2). In practical applications, it is often impossible or computationally
expensive to calculate the gradient and the Hessian analytically. However, it
is usually feasible to compute g(ψ) numerically. For details on the calculation of the