Applications of state space models in finance

52 4 Markov regime switching
Taking the logarithm of LT gives the loglikelihood function as
log LT =
T�
t=1
� �
wt
log . (4.24)
wt−1
It can be shown that the ratio of the scaling factors can be obtained as wt/wt−1 =
φ t−1Bt1 ′ . An efficient algorithm for recursively evaluating the loglikelihood based on
the scaled forward probabilities can be derived with starting equations
and updating equations
log L0 = 0, (4.25)
φ 0 = πs, (4.26)
vt = φ t−1Bt, (4.27)
ut = vt1 ′ , (4.28)
log Lt = log Lt−1 + log ut, (4.29)
φ t = vtu −1
t . (4.30)
To obtain the loglikelihood function, the loop is repeated for t = 1, . . . , T (cf. Zucchini
et al. 2006, ¢ 4.3).
Together with a statistical software package that offers functions for numerical maximization
(or minimization), this algorithm can be employed to estimate the unknown
parameters of a hidden Markov model by means of direct numerical maximization. In
this thesis, the ML parameters of HMMs are estimated using the R-functions nlm() and
optim(), which can be employed to minimize a negative loglikelihood.
4.3.3 Standard errors of ML estimates
In the context of HMMs it is not trivial to determine the accuracy of the ML parameters
estimated by one of the methods described above. As the elements of the vector
of estimated parameters can be shown to be correlated, it is not possible to compute
the standard error of the overall model directly. One possibility to obtain distributional
properties of the parameter estimates is to apply parametric bootstrap methods. As
these procedures are beyond the scope of this thesis, in the following the quality of a
hidden Markov model will be evaluated based on its forecast performance rather than
using classical statistical fit statistics. For details on how parametric bootstrap procedures
can be employed to analyze the distributional properties of parameter estimates,
the reader is referred to Zucchini et al. (2006, ¢ 4.4) and the references given therein.
4.4 Forecasting and decoding
Based on Zucchini et al. (2006, ¢ 5) this section briefly summarizes how HMMs can be
used to produce forecasts ( ¢ 4.4.1), and how information with regard to the unobservable
states of the Markov chain can be obtained ( ¢ 4.4.2). The inference regarding the hidden
states is commonly referred to as decoding.