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Unformatted text preview: Optimal Test for Markov Switching Parameters & Marine Carrasco y Liang Hu z Werner Ploberger x May 2009 Abstract This paper proposes a class of optimal tests for the constancy of parameters in random coe cients models. Our testing procedure covers the class of Hamilton&s models, where the parameters vary according to an unobservable Markov chain, but also applies to nonlinear models where the random coe cients need not be Markov. Standard approaches do not apply. We use Bartletttype identities for the construction of the test statistics. It has several desirable properties. First, it only requires estimating the model under the null hypothesis where the parameters are constant. Second, we show that the tests of this class are asymptotically optimal in the sense that they maximize a weighted power function. Moreover, we show that the contiguous alternatives converge to the null with a nonstandard rate of decay. & An earlier version of this paper has circulated under the title "Optimal Test for Markov Switching." Carrasco gratefully acknowledges partial nancial support from the National Science Foundation under grant SES0211418. y University of Montreal, email: marine.carrasco@umontreal.ca z University of Leeds, email: lh@lubs.leeds.ac.uk x Washington University in St Louis, email: werner@ploberger.com 1. Introduction In this paper, we focus on testing the constancy of parameters in dynamic models. The parameters are constant under the null hypothesis, whereas they are random and weakly dependent under the alternative. The model of interest is very general and includes as special case the Markov switching model of Hamilton (1989) where the regime changes in the parameters are driven by an unobservable twostate Markov chain and the state space models. The random coe cients need not be Markov and the model under the null need not be linear and could be a GARCH model for instance. Two distinct features make testing the stability of coe cients particularly challenging. The &rst is that the hyperparameters that enter in the dynamics of the random coe cients are not identi&ed under the null hypothesis. As a result, the usual tests, like the likelihood ratio test, do not have a chisquare distribution. The second feature is that the information matrix is singular under the null hypothesis. This is due to the fact the underlying regimes are not observable. The &rst feature known as the problem of nuisance parameters that are not identi&ed under the null hypothesis also arise when testing for structural change or threshold e/ects. It has been investigated in many papers: Davies (1977, 1987), Andrews (1993), Andrews and Ploberger (1994), Hansen (1996) among others. However, the second feature of our testing problem implies that the right(i.e. contiguous) local alternatives are of order T & 1 = 4 , while they are of the order T & 1 = 2 in the case of structural change and threshold models. The optimality discussed below shows that there do not exist tests withthreshold models....
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This note was uploaded on 12/26/2011 for the course ECON 245b taught by Professor Staff during the Fall '08 term at UCSB.
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