Halina Frydman

Credit Rating Dynamics and Markov Mixture Models

Despite mounting evidence to the contrary, credit migration matrices, used in many credit risk and pricing applications, are typically assumed to be generated by a simple Markov process. Based on empirical evidence, we propose a parsimonious model that is a mixture of (two) Markov chains, where the mixing is on the speed of movement among credit ratings. We estimate this model using credit rating histories and show that the mixture model statistically dominates the simple Markov model and that the differences between two models can be economically meaningful. The non-Markov property of our model implies that the future distribution of a firms ratings depends not only on its current rating but also on its past rating history. Indeed we find that two firms with identical current credit ratings can have substantially different transition probability vectors. We also find that conditioning on the state of the business cycle or industry group does not remove the heterogeneity with respect to the rate of movement. We go on to compare the performance of mixture and Markov chain using out-of-sample predictions.

Maximum Likelihood Estimation in the Mover-Stayer Model

Abstract The discrete time mover-stayer model, a special mixture of two independent Markov chains, has been widely used in modeling the dynamics of social processes. The problem of maximum likelihood estimation of its parameters from the data, however, which consist of a sample of independent realizations of this process, has not been considered in the literature. I present a maximum likelihood procedure for the estimation of the parameters of the mover-stayer model and develop a recursive method of computation of maximum likelihood estimators that is very simple to implement. I also verify that obtained maximum likelihood estimators are strongly consistent. I show that the two estimators of the parameters of the mover-stayer model previously proposed in the literature are special cases of the maximum likelihood estimator derived in this article, that is, they coincide with the maximum likelihood estimator under special conditions. I thus explain the interconnection between existing estimators. I also pre...

Testing the Adequacy of Markov Chain and Mover-Stayer Models as Representations of Credit Behavior

We summarize methodology for testing the compatibility of discrete time stochastic processes-stationary and nonstationary Markov chains and an extension, the mover-stayer model-with longitudinal data from an unknown empirical process. We apply this methodology to determine the suitability of these models to represent the payment behavior of a sample of retail revolving credit accounts. We are led to reject stationary and also nonstationary Markov chain models for our data and for our state space definition in favor of the mover-stayer model. The mover-stayer model, in contrast to Markov chains, incorporates a simple form of population heterogeneity. Stationary Markov chains have been used extensively in finance literature to model payment behavior of credit accounts. Our empirical study suggests, however, that stationary Markov chains may not appropriately model payment behavior. It also indicates that incorporating heterogeneity in modeling payment behavior may be more important than incorporing nonstationarity.

Malignant melanoma. Changing trends in factors influencing metastasis-free survival from 1964 to 1982

Seven hundred fourteen patients with cutaneous melanoma in clinical Stage I treated between 1964 and 1982 were included in this study. In an analysis of metastasis‐free survival, thickness of the tumor, ulceration, gender, epithelioid cells as predominant cells in the tumor, and localization of the tumor were found to be independent prognostic factors. In a time trends analysis, the distributions of three of the prognostic factors (thickness of the tumor, ulceration, and inflammatory cell infiltrate) were found to shift during the last decade in the direction of improved prognosis, indicating that tumors are detected earlier than before. The distributions of two other factors (cell type and location of the tumors) shifted in the direction of deteriorated prognosis, suggesting partly that the biologic nature of the disease may have changed and partly that other behavioral factors may have played a role.

Semiparametric estimation in a three-state duration-dependent Markov model from interval-censored observations with application to AIDS data.

The semiparametric maximum likelihood estimation is considered in a three-state duration dependent Markov process when times of the intermediate transition (e.g., onset of a disease) are interval censored and the times of transitions to an absorbing state (e.g., death) are known exactly or are right censored. It is assumed that the intensity of the transition to an absorbing state depends both on chronological time and duration in the intermediate state. De Gruttola and Lagakos (1989, Biometrics 45, 1-11) and Frydman (1992, Journal of the Royal Statistical Society, Series B 54, 853-866) discussed non-parametric estimation from the same sampling scheme under the assumption that the intensity of transition to an absorbing state depends only on the duration in the intermediate state or only on the chronological time respectively. The approach taken here generalizes, but in discrete time framework, the results from Frydman (1992). The distribution of the time to the intermediate transition is modelled nonparametrically and the intensity of onset of terminal condition semiparametrically. The algorithm is developed for the computation of the estimators. The methods are illustrated with AIDS data.

Estimation in the Mixture of Markov Chains Moving With Different Speeds

This article considers a new mixture of time-homogeneous finite Markov chains where the mixing is on the rate of movement and develops the EM algorithm for maximum likelihood estimation of the parameters of the mixture. Continuous- and discrete-time versions of the mixture are defined, and their estimation is considered separately. The developed methods are illustrated with an application to modeling bond ratings migration. The class of mixture models proposed in this article provides a framework for modeling population heterogeneity with respect to the rate of movement. The proposed mixture subsumes the mover–stayer model, which has been widely used in applications.

Total positivity and the embedding problem for Markov chains

We prove that the class of transition matrices for the finite state time-inhomogeneous birth and death processes coincides with the class of non-singular totally positive stochastic matrices. Thus we obtain a complete solution to the embedding problem for this class of Markov chains.

Cost Inefficiency, Size of Firms and Takeovers

This study, using the Cox proportional hazards model, finds that the risk of takeover rises with cost inefficiency. It also finds that a firm faces a significantly higher risk of takeover if its cost performance lags behind its industry benchmark. Moreover, these findings appear to be remarkably stable over the nearly two decades spanned by the sample. The effect of the variables used to measure the risk-size relationship, however, indicates temporal changes. Lastly, the study presents evidence from fixed-effects models of ex post cost efficiency improvements that support the hypothesis that takeover targets are selected based on the potential for improvement.

Nonparametric estimation of the cumulative intensities in an interval censored competing risks model

The nonparametric maximum likelihood estimation (NPMLE) of the distribution function from the interval censored (IC) data has been extensively studied in the extant literature. The NPMLE was also developed for the subdistribution functions in an IC competing risks model and in an illness-death model under various interval-censoring scenarios. But the important problem of estimation of the cumulative intensities (CIs) in the interval-censored models has not been considered previously. We develop the NPMLE of the CI in a simple alive/dead model and of the CIs in a competing risks model. Assuming that data are generated by a discrete and finite mixed case interval censoring mechanism we provide a discussion and the simulation study of the asymptotic properties of the NPMLEs of the CIs. In particular we show that they are asymptotically unbiased; in contrast the ad hoc estimators presented in extant literature are substantially biased. We illustrate our methods with the data from a prospective cohort study on the longevity of dental veneers.

Estimation of overall survival in an ‘illness–death’ model with application to the vertical transmission of HIV‐1

We derive a nonparametric maximum likelihood estimate of the overall survival distribution in an illness-death model from interval censored observations with unknown status of the nonfatal event. This expanded model is applied to the re-analysis of data from a randomized trial where infants, born to women infected with HIV-1 that were randomly assigned to breastfeeding or counseling for formula feeding, were followed for 24 months for HIV-1 positivity, HIV-1-free survival, and overall survival. HIV-1 positivity, assessed by postpartum venous blood tests, is the interval censored nonfatal event, and HIV-1 positivity status is unknown for a subset of infants due to periodic assessment. The analysis demonstrates that estimation of the overall and the pre- and post-nonfatal event survival distributions with the proposed methods provide novel insights into how overall survival is influenced by the occurrence of the nonfatal event. More generally, it suggests the usefulness of this expanded illness-death model when evaluating composite endpoints as potential surrogates for overall survival in a given disease setting.