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Statistical inference for Markov chains with applications to credit risk

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Abstract

The focus of this paper is on the derivation of confidence and credibility intervals for Markov chains when discrete-time, continuous-time or discretely observed continuous-time data are available. Thereby, our contribution is threefold: First, we discuss and compare multinomial confidence regions for the rows of discrete-time Markov transition matrices in the light of empirical characteristics of credit rating migrations. Second, we derive an analytical expression for the expected Fisher information matrix of a continuous-time Markov chain which is used to construct credibility intervals using a non-informative Jeffreys prior distribution and a Metropolis-Hastings Algorithm. Third, we concretize profile and estimated/pseudo likelihood based confidence intervals in the continuous-time data settings, which in contrast to asymptotic normality based intervals explicitly consider non-negativity constraints for the parameters. Furthermore, we illustrate the described methods by Moody’s corporate ratings data with exact continuous-time transitions.

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Notes

  1. We use the term estimated/pseudo likelihood instead of profile likelihood since we consider here the marginal likelihood function, see e.g., Pawitan (2001).

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Acknowledgements

The authors would like to thank two anonymous referees for their helpful comments and suggestions, which significantly contributed to improving the quality of this publication. Marius Pfeuffer gratefully acknowledges funding by DZ Bank Foundation and Universitätsbund Erlangen-Nürnberg.

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  1. Institute for Statistics and Econometrics, Lange Gasse 20, 90403, Nuremberg, Germany

    Linda Möstel, Marius Pfeuffer & Matthias Fischer

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  1. Linda Möstel
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  2. Marius Pfeuffer
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  3. Matthias Fischer
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Correspondence to Marius Pfeuffer.

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Möstel, L., Pfeuffer, M. & Fischer, M. Statistical inference for Markov chains with applications to credit risk. Comput Stat 35, 1659–1684 (2020). https://doi.org/10.1007/s00180-020-00978-0

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