Estimation in the Mixture of Markov Chains

This paper 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 the maximum likelihood estimation of the parameters of the mixture. A continuous and discrete time versions of the mixture are defined and their estimation is considered separately. The simulation study is carried out for the continuous time mixture. To simplify the exposition the results are derived for a mixture of two Markov chains, but can be easily extended to a mixture of any finite number of Markov chains. The class of mixture models proposed in this paper provides a framework for modeling population heterogeneity with respect to the rate of movement. The proposed mixture generalizes the mover-stayer model, which has been widely employed in applications.Statistics Working Papers Serie

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. These findings, moreover, appear to be remarkably stable over the nearly two decades spanned by the sample. The effect of the variables measuring the risk-size relationship, however, indicate 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

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.Statistics Working Papers Serie

Estimation in the continuous time mover-stayer model with an application to bond ratings migration

The usual tool for modeling bond ratings migration is a discrete, timehomogeneuous Markov chain. Such model assumes that all bonds are homogeneous with respect to their movement behavior among rating categories and that the movement behavior does not change over time. However, among recognized sources of heterogeneity in ratings migration is age of a bond (time elapsed since issuance). It has been observed that young bonds have a lower propensity to change ratings, and thus to default, than more seasoned bonds. The aimof this paper is to introduce a continuous, time-nonhomogeneuous model for bond ratings migration, which also incorporates a simple form of population heterogeneity. The specific form of heterogeneity postulated by the proposed model appears to be suitable for modeling the effect of age of a bond on its propensity to change ratings. This model, called a mover-stayer model, is an extension of a time-nonhomogeneuous Markov chain. This paper derives the maximum likelihood estimators for the parameters of a continuous time mover-stayer model based on a sample of independent continuously monitored histories of the process, and develops the likelihood ratio test for discriminating between the Markov chain and the mover-stayer model. The methods are illustrated using a sample of rating histories of young corporate issuers. For this sample, the likelihood ratio test rejects a Markov chain in favor of a mover-stayer model. For young bonds with lowest rating the default probabilities predicted by the mover-stayer model are substantially lower than those predicted by the Markov chain.Statistics Working Papers Serie

Discrete Quantile Estimation

We consider estimation of a quantile from a discrete distribution. This gives rise to three new ideas, the confidence set for such a quantile, the notion that the associated confidence level can be increased after the data are collected, and that it is legitimate to strive to obtain a singleton confidence set. We develop properties of the sample quantile noting that the behavior for discrete populations is very different from the behavior for continuous populations. We illustrate the results with simulations and examples.Statistics Working Papers Serie

Credit Rating Dynamics and Markov Mixture Models

Despite overwhelming 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. In this paper we propose a parsimonious model that is a mixture of (two) Markov chains. 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 firm's ratings depends not only on its current rating but also on its past rating history. Indeed we find that two firms with identical credit ratings can have substantially different transition probability vectors

Maximum likelihood estimation for a general mixture of Markov jump processes

We estimate a general mixture of Markov jump processes. The key novel feature of the proposed mixture is that the transition intensity matrices of the Markov processes comprising the mixture are entirely unconstrained. The Markov processes are mixed with distributions that depend on the initial state of the mixture process. The new mixture is estimated from its continuously observed realizations using the EM algorithm, which provides the maximum likelihood (ML) estimates of the mixture's parameters. To obtain the standard errors of the estimates of the mixture's parameters, we employ Louis' (1982) general formula for the observed Fisher information matrix. We also derive the asymptotic properties of the ML estimators. Simulation study verifies the estimates' accuracy and confirms the consistency and asymptotic normality of the estimators. The developed methods are applied to a medical dataset, for which the likelihood ratio test rejects the constrained mixture in favor of the proposed unconstrained one. This application exemplifies the usefulness of a new unconstrained mixture for identification and characterization of homogeneous subpopulations in a heterogeneous population.Comment: 21 pages, 1 figur

PARAMETRIC ESTIMATION OF HAZARD FUNCTIONS WITH STOCHASTIC COVARIATE PROCESSES

Let X(t), t âÂÂ¥ 0, be a real or vector valued stochastic process and T a random killing-time of the process which generally depends on the sample function. In the context of survival analysis, T represents the time to a prescribed event (e.g. system failure, time of disease symptom, etc.) and X(t) is a stochastic covariate process, observed up to time T. The conditional distribution of T, given X(t), t âÂÂ¥ 0, is assumed to be of a known functional form with an unknown vector parameter ø; however, the distributions of X(â¢) are not specified. For an arbitrary fixed ñ > 0 the observable data from a single realization of T and X(â¢) is min(T, ñ), X(t), 0 ⤠t ⤠min(T, ñ). For n âÂÂ¥ 1 the maximum likelihood estimator of ø is based on n independent copies of the observable data. It is shown that solutions of the likelihood equation are consistent and asymptotically normal and efficient under specified regularity conditions on the hazard function associated with the conditional distribution of T. The Fisher information matrix is represented in terms of the hazard function. The form of the hazard function is very general, and is not restricted to the commonly considered cases where it depends on X(â¢) only through the present point X(t). Furthermore, the process X(â¢) is a general, not necessarily Markovian process.Statistics Working Papers Serie

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. These findings, moreover, appear to be remarkably stable over the nearly two decades spanned by the sample. The effect of the variables measuring the risk-size relationship, however, indicate 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