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Model will tend to stay in a particular state and then suddenly jump to a new state, remaining
in that state for some time. This is precisely the behaviour desired from such a model when
trying to apply it to market regimes. The regimes themselves are not expected to change too
quickly. Consider regulatory changes and other slow-moving macroeconomic effects, for instance.
However when they do change they are expected to persist for some time.
14.2.1 Hidden Markov Model Mathematical Specification
The corresponding joint density function for the HMM is given by (again using notation from
Murphy (2012)[71]):
p(z 1:T | x 1:T ) = p(z 1:T )p(x 1:T | z 1:T ) (14.6)
[
] [
T∏
T
]
∏
= p(z 1 ) p(z t | z t−1 ) p(x t | z t )
(14.7)
t=2
t=1
The first line states that the joint probability of seeing the full set of hidden states and
observations is equal to the probability of seeing the hidden states multiplied by the probability
of seeing the observations, conditional on the states. This makes sense as the observations cannot
affect the states, but the hidden states do indirectly affect the observations.
The second line splits these two distributions into transition functions. The transition function
for the states is given by p(z t | z t−1 ) while that for the observations (which depend upon the
states) is given by p(x t | z t ).
As with the Markov Model description above it will be assumed that both the state and
observation transition functions are time-invariant. This means that it is possible to utilise the
K × K state transition matrix A as before with the Markov Model for that component of the
model.
However for the application considered here, namely observations of asset returns, the values
are in fact continuous. This means the model choice for the observation transition function is
more complex. A common choice is to make use of a conditional multivariate Gaussian distribution
with mean µ k and covariance σ k . This is formalised below:
p(x t | z t = k, θ) = N (x t | µ k , σ k ) (14.8)
That is, if the state z t is currently equal to k, then the probability of seeing observation x t ,
given the parameters of the model θ, is distributed as a multivariate Gaussian.
In order to make this a little clearer Figure 14.2 shows the evolution of the states z t and how
they lead indirectly to the evolution of the observations, x t :
14.2.2 Filtering of Hidden Markov Models
With the joint density function specified it remains to consider the how the model will be utilised.
In general state-space modelling there are often three main tasks of interest: Filtering, smoothing
and prediction. The previous chapter on State-Space Models and the Kalman Filter described
these briefly. They will be repeated here for completeness: