Applications of state space models in finance

28 3 Linear Gaussian state space models and the Kalman filter
observations, y 1, . . . , y T , which are assumed to be IID, the joint density is given by the
product of the individual densities, denoted by p(·):
L(y, ψ) = p(y 1, . . . , y T ) =
T�
p(yt), (3.52)
where L(y, ψ) is the joint probability density function of the t-th set of observations.
When the joint density is evaluated at a given data set, L(y, ψ) is referred to as a
likelihood function of the model. To avoid computational difficulties caused by extreme
numbers that may result from the product above, it is generally simpler to work with
the natural logarithm of the likelihood function:
log L(y, ψ) =
t=1
T�
log p(yt ). (3.53)
The likelihood function and its logarithm are often simply denoted as L(y) and log L(y),
respectively. If the vector of parameters is identifiable, the ML estimate of the parameters
( ˆ ψ) is found by maximizing the likelihood with respect to ψ. 7 For a general introduction
into the methodology of ML, see, for example, Greene (2003, ¢ 17) or Davidson
and MacKinnon (2004, ¢ 10).
3.4.1.1 Prediction error decomposition
As the observations for time series models are not generally independent, Equation (3.53)
is replaced by a probability density function. The distribution of y t is conditioned on
Y t−1, the information set at time t − 1:
log L(y) =
T�
t=1
t=1
log p(y t |Y t−1), (3.54)
with Y t = {y 1, . . . , y t} and p(y 1|Y 0) := p(y 1).
If the observation disturbances and the initial state vector in the general state space
model (3.1)–(3.7) have a multivariate normal distribution, it can be shown that the
conditional distribution of y t itself is normal with conditional mean
and conditional covariance
E(y t|Y t−1) = Ztat, (3.55)
V ar(y t |Y t−1) = F t. (3.56)
The variance of the one-step ahead forecast error, F t, is defined as in (3.16). For Gaussian
models, y t is conditionally distributed as N(Ztat, F t) with conditional probability
density function
p(yt|Y t−1) = 1
2π |F t| 1
�
2 exp − 1
2 v′ tF −1
�
t vt . (3.57)
7 Identifiability means that the estimation yields a unique determination of the parameter
estimates for a given set of data. For more details on the identifiability of structural time series
models, see Harvey (1989, 4.4).