Instabilities in Cox proportional hazards models in credit risk

Joseph L. Breeden, Anthony Bellotti, Yevgeniya Leonova

Research output: Journal PublicationArticlepeer-review


When the underlying system or process being observed is based upon observations versus age, vintage (origination time) and calendar time, Cox proportional hazards (PH) models can exhibit instabilities because of embedded assumptions. The liter-ature on age–period–cohort (APC) models provides clues as to why this is, which we use to explore possible instabilities in applying Cox PH models. This model structure occurs frequently in applications such as loan credit risk, website traffic, customer churn and employee retention. Our numerical studies, designed to capture the dynamics of credit risk modeling, demonstrate that the same linear specification error from APC models occurs in Cox PH estimation when the covariates contain basis functions with linear terms, or if those covariates are correlated to the baseline hazard function. This demonstrates that the linear trend specification error in APC models is equivalent to multicollinearity instabilities in Cox PH models or in any regression context when applied to comparable problems. As part of our study of multicollinearity, we show equivalences between Cox PH estimation methods and other regression techniques, including an original proof that Breslow’s method for Cox PH estimation is equivalent to Poisson regression in the case of ties.

Original languageEnglish
Pages (from-to)29-55
Number of pages27
JournalJournal of Credit Risk
Issue number2
Publication statusPublished - Jun 2023


  • Cox proportional hazards (PH) models
  • Poisson regression
  • age–period–cohort (APC) models
  • conditional logistic regression
  • data science

ASJC Scopus subject areas

  • Finance
  • Economics and Econometrics


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