Testing for hidden Markov switching
Jin Seo Cho
Victoria University of Wellington
PO Box 600, Wellington, 6001, New Zealand
http://www.victoria.ac.nz/sef
Robert B Davies
Statistics Research Associates Limited
PO Box 12649, Wellington, 6144, New Zealand
http://www.statsresearch.co.nz
5 March 2008
Abstract
1
Introduction
1.1
Motivation and paper plan
We have a stationary sequence of observations. We consider the problem of distinguishing
between their having independent normal distributions and their being generated by a
hidden Markov switching process. We also consider the situation where the hypothesis is
that they are Markov and the alternative is that they follow a Hamilton (1989) process.
Hidden Markov models (HMMs) are frequently used for modelling processes with
autocorrelation.
This is particularly the case where the autocorrelation is thought to
result from a hidden Markov process which switches between a number of levels.
For
example, Hamilton (1989) applies them to explain the cyclic behaviour of the postwar
US GNP. [Include additional references here] For the general theory of Hidden Markov
models, see, for example, Cappé, Moulines & Rydén (2005) and MacDonald & Zucchini
(2000).
1
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The problem of selecting the correct number of levels of the Markov hidden process
involves the kind of testing situation investigated by Davies (1977), Davies (1987) where
there are parameters present only under the alternative. In this situation the standard
asymptotic theory does not apply.
In the present paper, we consider just the two level situation under the alternative. We
consider the situation where the expected values of the process switch and the situation
where the variance switches. Then we adapt the mean switiching case to the Hamilton
model where the process is a
fi
rst order autoregressive process under the hypothesis and
includes Markov switching under the alternative.
Our approach is to
fi
nd an analog of Neyman’s C(
α
) test, assuming that those para
meters present only under the alternative are known. Then we follow Davies (1977) to
fi
nd a test.
Cho & White (2007) follow a somewhat similar approach to that considered here for
investigating the
fi
nite mixture model with independent observations.
Carrasco, Hu &
Ploberger (2005) derive a test similar to ours for the hidden Markov problem, but using
a somewhat di
ff
erent approach.
The analysis of the switching model with the expected value being switched is in
section 2.
The corresponding analysis where the variance is switched is in section 3.
Section 4 considers the Hamilton model. A series of lemmas used throughout the paper
are stated in section 1.4.
Finally, we gather together the mathematical proofs of the
theorems in the Appendix.
1.2
Notation
We use
X
n
=
{
X
1
, . . . , X
n
}
to denote the hidden Markov process and
Y
n
=
{
Y
1
, . . . , Y
n
}
to denote the observed data. We sometimes use lower case letters
x
n
=
{
x
1
, . . . , x
n
}
and
y
n
=
{
y
1
, . . . , y
n
}
to denote normalised or centred versions of these variables.
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 Fall '08
 Staff
 Normal Distribution, probability density function, yn, Tn, Markov chain, Hidden Markov model

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