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Nonparametric Estimation for Non-homogeneous semi-Markov Processes: An application to credit risk

by André Lucas of Vrije Universiteit Amsterdam,
André Monteiro of Vrije Universiteit Amsterdam, and
Georgi Smirnov of the University of Porto

March 13, 2006

Abstract: We propose procedures for estimating the time-dependent transition matrices for the general class of finite non-homogeneous continuous-time semi-Markov processes. We prove the existence, and Frechet differentiability, of a unique solution for the system of Volterra integral equations which relates the transition matrix with the subdensity functions, therefore showing that the realized transition probabilities can be consistently estimated from window censored event-history data. An implementation of the method is presented, based on nonparametric estimators of the conditional hazard rate functions in the general and separable (multiplicative) cases. We use the resulting estimators for dealing with a central issue in credit risk. We consider the problem of obtaining estimates of the historical corporate default and rating migration probabilities using a dataset on credit ratings from Standard & Poors.

JEL Classification: C13, C14, C33, C41, G11.

Keywords: Non-homogeneous semi-Markov processes, transition matrix, Volterra integral equations, separability, credit risk.

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