Applications of state space models in finance

26 3 Linear Gaussian state space models and the Kalman filter
The first is to employ a diffuse prior that fixes a1 at an arbitrary value and allows
the diagonal elements of P 1 to go to infinity. Following Durbin and Koopman (2001,
¢ 5.1) a general specification for the initial state vector is given by
ξ 1 = a + Aδ + R0η 0, δ ∼ N(0, κIq), η 0 ∼ N(0, Q 0), (3.49)
where the m × 1 vector a can be treated as a zero vector whenever none of the elements
of of ξ 1 are known constants. The m × q matrix A and the m × (m − q) matrix R0 are
fixed and known selection matrices with A ′ R0 = 0. The initial covariance matrix Q 0 is
assumed to be known and positive definite. The q × 1 vector δ is treated as a random
variable with infinite variance and is called the diffuse vector as κ → ∞. This leads to
P 1 = P ∗ + κP ∞, (3.50)
with P ∗ = R0Q 0 R ′ 0 and P ∞ = AA ′ . With some elements of ξ 1 being diffuse, the
initialization of the Kalman filter is referred to as diffuse initialization. However, in
cases where P ∞ is a nonzero matrix, the standard Kalman filter cannot be employed
as no real value can represent κ as κ → ∞. It is necessary to find an approximation
or to modify the Kalman filter in an appropriate way. The technique that will be
used throughout this thesis is based on Harvey and Phillips (1979). The two authors
propose to replace κ by a large but finite numerical value, which enables the use of the
standard Kalman filter. They showed that this will yield a good approximation, where
the remaining rounding errors are small.
As an alternative to the large κ approximation, exact initialization techniques have
been developed for κ → ∞. However, exact treatments turn out to be difficult to
implement and cannot deal with the general multivariate linear Gaussian state space
model directly. They will not find any consideration hereafter. For details on alternative
exact treatments of the initial Kalman filter, see, for example, Ansley and Kohn (1985),
de Jong (1988b, 1991) and Koopman (1997).
The assumption of an infinite variance might be regarded unnatural as all observed
time series have a finite variance. This leads to the third way of initialization: Rosenberg
(1973) considers ξ 1 to be an unknown constant that can be estimated from the first
observation y 1 by maximum likelihood. It can be shown that by this procedure, the
same initialization of the filter is obtained as by assuming that ξ 1 is a random variable
with infinite variance (cf. Durbin and Koopman 2001, ¢ 2.9).
In the following, unless genuine prior information on a1 and P 1 is available, a diffuse
prior with a = 0, P ∗ = 0 and P ∞ = I will be used such that ξ 1 ∼ N(0, κI). Following
the recommendation of Koopman et al. (1999), κ is first set to 10 6 and then multiplied
by the maximum diagonal value of the residual covariances to adjust for scale:
κ = 10 6 � �
Qt 0
× max 1,
0 Ht
��
. (3.51)
It should be noted that the initialization is not trivial for the multivariate case. It is
possible that the part of F −1
t that is linked to P ∞ is sometimes singular with different
cannot simply be expanded in powers of κ−1 for the first few
ranks. In these cases F −1
t