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Consistent estimation for discretely observed Markov jump processes with an absorbing state

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  • Alexander Kremer
  • Rafael Weißbach

Abstract

For a continuous-time Markov process, commonly, only discrete-time observations are available. We analyze multiple observations of a homogeneous Markov jump process with an absorbing state. We establish consistency of the maximum likelihood estimator, as the number of Markov processes increases. To accomplish uniform convergence in the continuous mapping theorem, we use the continuity of the transition probability in the parameters, the compactness of the parameter space and the boundedness of probabilities. We allow for a stochastic time-grid of observation points with different intensities for each observation process. Furthermore, we account for right censoring. The estimate is obtained via the EM algorithm with an E-step given in closed form. In our empirical application of credit rating histories, we fit the model of Weißbach and Mollenhauer (J Korean Stat Soc 40:469–485, 2011 ) and find marked differences, compared to the continuous-time analysis. Copyright Springer-Verlag Berlin Heidelberg 2013

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  • Alexander Kremer & Rafael Weißbach, 2013. "Consistent estimation for discretely observed Markov jump processes with an absorbing state," Statistical Papers, Springer, vol. 54(4), pages 993-1007, November.
    • Handle: RePEc:spr:stpapr:v:54:y:2013:i:4:p:993-1007
    DOI: 10.1007/s00362-013-0515-0
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    References listed on IDEAS

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    1. Robert A. Jarrow & David Lando & Stuart M. Turnbull, 2008. "A Markov Model for the Term Structure of Credit Risk Spreads," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 18, pages 411-453, World Scientific Publishing Co. Pte. Ltd..
    2. Weißbach, Rafael & Mollenhauer, Thomas, 2011. "Modelling Rating Transitions," VfS Annual Conference 2011 (Frankfurt, Main): The Order of the World Economy - Lessons from the Crisis 48698, Verein für Socialpolitik / German Economic Association.
    3. Mogens Bladt & Michael Sørensen, 2005. "Statistical inference for discretely observed Markov jump processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 395-410, June.
    4. Koopman, Siem Jan & Lucas, André & Schwaab, Bernd, 2011. "Modeling frailty-correlated defaults using many macroeconomic covariates," Journal of Econometrics, Elsevier, vol. 162(2), pages 312-325, June.
    5. Weißbach, Rafael & Walter, Ronja, 2010. "A likelihood ratio test for stationarity of rating transitions," Journal of Econometrics, Elsevier, vol. 155(2), pages 188-194, April.
    6. Hobolth Asger & Jensen Jens Ledet, 2005. "Statistical Inference in Evolutionary Models of DNA Sequences via the EM Algorithm," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-22, August.
    7. Mogens Bladt & Michael SØrensen, 2009. "Efficient estimation of transition rates between credit ratings from observations at discrete time points," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 147-160.
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    Cited by:

    1. Benjamin Strohner & Rafael Weißbach, 2016. "Altersspezifische Querschnittsanalyse der Fertilität in Mecklenburg-Vorpommern mit dem EM-Algorithmus [Age-Specific Cross-Sectional Analysis of the Fertility in Mecklenburg-West Pomerania with the ," AStA Wirtschafts- und Sozialstatistisches Archiv, Springer;Deutsche Statistische Gesellschaft - German Statistical Society, vol. 10(4), pages 269-288, December.
    2. Camilla Ferretti & Giampaolo Gabbi & Piero Ganugi & Federica Sist & Pietro Vozzella, 2019. "Credit Risk Migration and Economic Cycles," Risks, MDPI, vol. 7(4), pages 1-18, October.
    3. Voß, Sebastian & Weißbach, Rafael, 2014. "A score-test on measurement errors in rating transition times," Journal of Econometrics, Elsevier, vol. 180(1), pages 16-29.
    4. Greig Smith & Goncalo dos Reis, 2017. "Robust and Consistent Estimation of Generators in Credit Risk," Papers 1702.08867, arXiv.org, revised Oct 2017.

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