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Hidden Markov Switching

Hidden Markov Switching - Testing for hidden Markov...

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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 post-war 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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Hidden Markov Switching - Testing for hidden Markov...

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