Markov Switching Models have been introduced in the econometrics literature [&make_named_href('', "node31.html#Goldfeld73","[97]"), &make_named_href('', "node31.html#Shumway82","[98]"), &make_named_href('', "node31.html#Cosslett85","[99]"), &make_named_href('', "node31.html#Hamilton89","[12]"), &make_named_href('', "node31.html#Shumway91","[13]")] for modeling non-stationarities due to abrupt changes of regime in the economy [&make_named_href('', "node31.html#Hamilton90","[100]"), &make_named_href('', "node31.html#Bonomo94","[101]"), &make_named_href('', "node31.html#Sola94","[102]"), &make_named_href('', "node31.html#Hamilton94","[103]"), &make_named_href('', "node31.html#Cai94","[104]"), &make_named_href('', "node31.html#Garcia-Perron96","[37]")].
The point of view taken by most econometricians is to extend time-series regression models by the addition of a discrete hidden state variable, which allows changing the parameters of the regression models when the state variable changes its value.
Consider for example the time-series regression model
where the vector
is the observed (or output) variable at time t,
is a random vector variable with a zero-mean Gaussian distribution
representing noise, and
is a given vector of input variables (e.g., past values of y, as
in [&make_named_href('',
"node31.html#Hamilton89","[12]")], or present and past values of other observed variables).
is a matrix, and there are different sets of parameters
for the different (discrete) values
of the hidden state variable
. This basically specifies a particular form for the emission
distribution
of a IOHMM: a Gaussian distribution whose
mean is a linear function of
, with different parameters for the
different values of
.
To obtain a complete picture of the joint distribution
of
and
(given the input
)
one then needs to specify the distribution of the
state variable. In most of the cases described in
the econometrics literature, this distribution
is assumed to be time-invariant, and it is
specified by a matrix of transition probabilities
(as in ordinary HMMs), although more complicated
specifications have been
suggested [&make_named_href('',
"node31.html#Sichel91","[105]"), &make_named_href('',
"node31.html#Diebold93b","[106]"), &make_named_href('',
"node31.html#Ghysel93","[107]"), &make_named_href('',
"node31.html#Garcia-Schaller95","[108]"), &make_named_href('',
"node31.html#Diebold93","[109]")].
The representation of the variance of
in equation 12 can
be made more complex than a single constant parameter: variance can also be a
function of the state variable as well as of the input variables. See for
example [&make_named_href('',
"node31.html#Hamilton94","[103]"), &make_named_href('',
"node31.html#Cai94","[104]")] for Markov-switching ARCH models applied to
analyzing respectively the changes in variance of stock returns and interest
rates.
The parameters of Markov switching models
can generally be estimated using the EM
algorithm [&make_named_href('',
"node31.html#Kiefer80","[110]"), &make_named_href('',
"node31.html#Shumway82","[98]"), &make_named_href('',
"node31.html#Hamilton90","[100]"), &make_named_href('',
"node31.html#Kim94","[111]")]
to maximize the likelihood
(see next section).
Other inference algorithms are used in
econometrics applications [&make_named_href('',
"node31.html#Hamilton93","[112]")], for
filtering, smoothing, and prediction.
A filtering algorithm is used to compute an
estimate of the current distribution
for the state given past inputs and outputs.
Note that after
the forward recursion has been performed
(computation of
, equation 11),
filtering can be done easily:
A smoothing
algorithm is used to compute an a-posteriori
estimate of the distribution
for the state path, given the whole sequence of
inputs and outputs (see equation 18).
Finally, using the results of the filtering,
a prediction algorithm
allows one to compute the distribution of
future states given past
input and output observations:
In section 7, we consider state-space models (in which the hidden state variable is continuous) and hybrids with both discrete and continuous state variables, which have been used in similar time-series modeling applications.