next up previous
Next: EM for HMMs and Up: Input-Output HMMs Previous: Input-Output HMMs

Markov Switching Models

 

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.


next up previous
Next: EM for HMMs and Up: Input-Output HMMs Previous: Input-Output HMMs

Yoshua Bengio
Tue Oct 7 08:34:36 EDT 1997