Abstract
Correlated ordinal data typically arises from multiple measurements on a collection of subjects. Motivated by an application in credit risk, where multiple credit rating agencies assess the creditworthiness of a firm on an ordinal scale, we consider multivariate ordinal regression models with a latent variable specification and correlated error terms. Two different link functions are employed, by assuming a multivariate normal and a multivariate logistic distribution for the latent variables underlying the ordinal outcomes. Composite likelihood methods, more specifically the pairwise and tripletwise likelihood approach, are applied for estimating the model parameters. Using simulated data sets with varying number of subjects, we investigate the performance of the pairwise likelihood estimates and find them to be robust for both link functions and reasonable sample size. The empirical application consists of an analysis of corporate credit ratings from the big three credit rating agencies (Standard & Poor’s, Moody’s and Fitch). Firm-level and stock price data for publicly traded US firms as well as an unbalanced panel of issuer credit ratings are collected and analyzed to illustrate the proposed framework.
Highlights
The analysis of univariate or multivariate ordinal outcomes is an important task in various fields of research from social sciences to medical and clinical research
In the context of credit risk one may think of the underlying latent variable as the latent creditworthiness of a firm, which is measured on a continuous scale
Low values of the latent creditworthiness will translate to the worst rating classes, while the right tail of the distribution of the latent variables will correspond to the best rating classes
Summary
The analysis of univariate or multivariate ordinal outcomes is an important task in various fields of research from social sciences to medical and clinical research. To motivate this study we focus on a data set of US corporates over the period 1999– 2013 for which at least one corporate credit rating from the big three CRAs is available For this purpose we propose the use of multivariate ordered probit and logit regression models. Difficulties in Bayesian inference arise due to the fact that absolute scale is not identifiable in ordinal models In this case, the covariance matrix of the multiple outcomes is often restricted to be a correlation matrix which makes the sampling of the correlation parameters non-standard. The main advantages of using the multivariate logistic distribution in Eq (3) are (i) it allows for a flexible dependence structure between the underlying latent variables Y through the unconstrained correlation matrix of the t-copula and (ii) the regression coefficients can be interpreted in terms of log odds ratios.
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