Abstract
We propose a novel method for Markov regime switching (MRS) model estimations by spectral clustering hidden Markov model (SC-HMM). The proposed SC-HMM exploits the Markov property of hidden states and utilizes pairwise feature similarities for latent state identifications. It can be applied to general hidden Markov models (HMMs) with continuous observations. In contrast to the maximum likelihood estimation (MLE), SC-HMM predicts latent states and yields conditional distribution statistics without knowledge of types of conditional distributions. To illustrate, SC-HMM is first applied to a simple HMM with discrete observations. We consider the MRS model estimation with continuous observations to further demonstrate SC-HMM. Specifically, based on local observations, we propose a set of features for the MRS estimation. A similarity matrix is determined from derived features and spectral clustering predicts latent states. Conditional distribution statistics and transitional probabilities are estimated based on identified latent states. By conducting simulation studies on both two-state and three-state MRS estimations, we demonstrate that, in comparison with MLE, the proposed SC-HMM is more robust. Furthermore, we demonstrate the validity of SC-HMM by estimating a two-state MRS from the S&P/TSX Composite Index daily and monthly data from 1977 to 2014.
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Notes
Transition matrix P is a Markov matrix, thus each row sum equals to one.
We find no significant performance difference using other Laplacian variants in our MRS study.
If the number of latent states is unknown a priori, it may be possible to analyze and assess structures revealed in the eigen subspace to first determine the number of latent states.
The Gaussian similarity \(W_{ij}=\exp (-\Vert x_i-x_j\Vert ^2/(2\sigma ^{2}))\) is used, where the parameter \(\sigma \) controlling the width of the neighborhoods (Shi and Malik 2000) is set to 1.
Transition probabilities \(p_{12}, p_{21}\) are typically small for a financial market.
All results are reproducible by setting the seed to one for random number generator in Matlab.
Data source: Canadian Financial Markets Research Centre summary information database(CFMRC TSX database). The closing S&P/TSX Composite Total Return Index based on prices and distributions (stocks, dividends, etc.).
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Acknowledgements
Yuying Li acknowledges funding from the National Sciences and Engineering Research Council of Canada. Kai Zheng acknowledges funding from the National Natural Science Foundation of China (No. 71671146). Weidong Xu acknowledges funding from NSFC (No. 71771197), NSFC project under Grant (No. 71821002) and Educational Commission of Zhejiang Province of China (No.Y201329832). Authors also acknowledge anonymous referees for their comments and suggestions which have led to improvement in the presentation of the paper.
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Zheng, K., Li, Y. & Xu, W. Regime switching model estimation: spectral clustering hidden Markov model. Ann Oper Res 303, 297–319 (2021). https://doi.org/10.1007/s10479-019-03140-2
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DOI: https://doi.org/10.1007/s10479-019-03140-2