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Measuring Corporate Default Risk$

Darrell Duffie

Print publication date: 2011

Print ISBN-13: 9780199279234

Published to Oxford Scholarship Online: September 2011

DOI: 10.1093/acprof:oso/9780199279234.001.0001

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(p.89) Appendix E: Testing for Frailty

(p.89) Appendix E: Testing for Frailty

Source:
Measuring Corporate Default Risk
Publisher:
Oxford University Press

(p.89) Appendix E:

Testing for Frailty

In order to judge the relative fit of the models of Chapters 3 and 6, that is, without frailty and with frailty, we do not use standard tests, such as the χ 2 test. Instead, we compare the marginal likelihoods of the models. This approach does not rely on large‐sample distribution theory and has the intuitive interpretation of attaching prior probabilities to the competing models.

Specifically, we consider a Bayesian approach to comparing the quality of fit of competing models and assume positive prior probabilities for the two models “noF” (the model without frailty) and “F”(the model with a common frailty variable). The posterior odds ratio is

( F | W , D ) ( noF | W , D ) = F ( γ ^ F , β ^ F | W , D ) n o F ( γ ^ n o F , β ^ n o F | W , D ) ( F ) ( noF ) ,
(E.1)

where β̂M and ℒM denote the maximum likelihood estimator and the likelihood function for a given model M, respectively. Substituting (6.5) into (E.1) leaves

( F | W , D ) ( noF | W , D ) = ( γ ^ F | W ) F ( β ^ F | W , D ) ( γ ^ n o F | W ) n o F ( β ^ n o F | W , D ) ( F ) ( noF ) = F ( β ^ F | W , D ) n o F ( β ^ n o F | W , D ) ( F ) ( noF ) ,
(E.2)

using the fact that the time‐series model for the covariate process W is the same in both models. The first factor on the right‐hand side of (E.2) is sometimes known as the “Bayes factor.”

Following Kass and Raftery (1995) and Eraker et al. (2003), we focus on the size of the statistic Φ given by twice the natural logarithm of the Bayes factor, which is on the same scale as the likelihood ratio test statistic. An outcome of Φ between 2 and 6 provides positive evidence, an outcome between 6 and 10 provides strong evidence, and an outcome larger than 10 is very strong evidence for the alternative model. This criterion does not necessarily favor more complex models due to the marginal nature of the likelihood functions in (E.2). See Smith and Spiegelhalter (1980) for a discussion of the penalizing nature of the Bayes factor, sometimes referred to as the “fully automatic Occam's razor.” In our case, the outcome of the test statistic is 22.6. In the sense of this approach to model comparison, we see relatively strong evidence in favor of including a frailty variable. Unfortunately, the Bayes factor cannot be used for comparing the model with frailty to the model with frailty and unobserved heterogeneity, since for the latter model evaluating the likelihood function is computationally prohibitively expensive.