Applications of state space models in finance

46 4 Markov regime switching
based on X can be computed by a linear combination of the single components:
p(x) =
f(x) =
m�
πipi(x) (discrete case), (4.2)
i=1
m�
πifi(x) (continuous case), (4.3)
i=1
with the k-th moment of the mixture defined as a linear combination of the respective
components moments
E(X k ) =
m�
i=1
πiE(X k i ), k = 1, 2, . . . . (4.4)
As it can be shown that the variance of a mixture model cannot be simply computed as
a linear combination of the respective components variances, the standard equality
V ar(X) = E(X 2 ) − (E(X)) 2 , (4.5)
can be employed together with (4.4) to estimate the variance of a mixture model (cf.
Zucchini et al. 2006, ¢ 2).
The parameters of a mixture distribution are usually estimated by ML. For the example
of the continuous case, the likelihood of an m-components mixture model can
generally be stated as
L(ψ 1, . . . , ψ m, x1, . . . , xT ) =
T�
j=1 i=1
m�
πifi(xj, ψi), (4.6)
with observations x1, . . . , xT . The mixture weights π1, . . . , πm and the parameter vectors
of the component distributions are included in ψ 1, . . . , ψ m. As the ML estimates
ˆψ 1, . . . , ˆ ψ m represent a solution to a system of nonlinear equations, the likelihood can
be maximized analytically only for rather trivial models. In most cases, the unknown
parameters have to be estimated by employing direct numerical maximization procedures
or by using the EM algorithm (cf. ¢ 3.4.3). Figure 4.2 shows the results of fitting
mixtures of two (a) and three normals (b) to the weekly log-returns of the Technology
sector. The components’ weights have been obtained by a hidden Markov model for
which more details will be provided in the section below. Compared to the case where
the returns are fitted by a single normal distribution, the fit is clearly improved by both
mixture models.
4.1.2 Markov chains
Let {St : t = 1, . . . , T } be a stochastic process, i.e. a sequence of random variables that
can assume an integer value in S = {1, . . . , m}, the state space. If for each date t,
the probability that St+1 is equal to a particular value st ∈ S depends only on St, the
current state of the process, such a process is called an m-state Markov process:
P (St+1 = st+1|St = st, St−1 = st−1, . . . , S1 = s1) = P (St+1 = st+1|St = st). (4.7)