Applications of state space models in finance

24 3 Linear Gaussian state space models and the Kalman filter
The matrix N t−1 can be obtained from the backwards recursion given in (3.26) or
alternatively as
N t−1 = Z ′ t DtZt + T ′ t N tT t − Z ′ t K′ t N tT t − T ′ t N tKtZt, (3.34)
which is computationally less intense as it relies directly on the sparse system matrices
T t and Zt. The system of (3.27) through (3.34), for t = T, . . . , 1, initialized with rT = 0
and N T = 0, is denoted as disturbance smoothing recursion.
3.3.3.2 Fast state smoothing
The smoothing recursion for the state disturbance term can also be used for fast state
smoothing as introduced by Koopman (1993). The idea is to calculate ˆ ξ t for t = 1, . . . , T ,
without the necessity to store at and P t. This procedure results in lower computational
costs.
Following the outline in Durbin and Koopman (2001, ¢ 4.4.2) the necessary recursion
can be derived from the state equation in (3.1) which implies
Substituting (3.28) into (3.35) yields
ˆξ t+1 = T t ˆ ξ t + Rtˆη t. (3.35)
ˆξ t+1 = T t ˆ ξ t + RtQ t R ′ t rt, t = 1, . . . , T, (3.36)
which is initialized for t = 1 by (3.23) together with (3.24). In contrast to the state
smoothing recursion presented in ¢ 3.3.2, this recursion allows for the computation of
smoothed states ˆ ξ t without getting at and P t involved.
The process of disturbance smoothing is comparable to that of state smoothing: while
the Kalman filter is employed to proceed forwards, backwards proceeding through the
data is done using the disturbance smoother. Due to its computational advantage, the
disturbance smoother will be used whenever the only objective is to form an inference
on the state vector and V t is not required.
3.3.4 Missing observations
Within the state space framework, missing observations can be easily dealt with. Durbin
¢
and Koopman (2001, 4.8) show that for a missing set of observations, denoted as
yτ , . . . , yτ ∗−1, the original filter and smoothing recursions can be used for all t. The
one-step ahead forecast error vt and the Kalman gain matrix Kt are simply set to zero
for all missing data points. With vt = 0 and Kt = 0 the filter recursions in (3.13) and
(3.14) become
at+1 = T tat, (3.37)
P t+1 = T tP tT ′ t + RtQ t R ′ t , t = τ, . . . , τ ∗ − 1. (3.38)
The smoothing recursions in (3.24) and (3.26) can be written as
rt−1 = T ′ t rt, (3.39)
N t−1 = T ′ tN tT t, t = τ ∗ − 1, . . . , τ, (3.40)
while the other relevant smoothing steps remain unaffected.