  We consider discrete default intensity based and logit type reduced form
models for conditional default probabilities for corporate loans where we
develop simple closed form approximations to the maximum likelihood estimator
(MLE) when the underlying covariates follow a stationary Gaussian process. In a
practically reasonable asymptotic regime where the default probabilities are
small, say 1-3% annually, the number of firms and the time period of data
available is reasonably large, we rigorously show that the proposed estimator
behaves similarly or slightly worse than the MLE when the underlying model is
correctly specified. For more realistic case of model misspecification, both
estimators are seen to be equally good, or equally bad. Further, beyond a
point, both are more-or-less insensitive to increase in data. These conclusions
are validated on empirical and simulated data. The proposed approximations
should also have applications outside finance, where logit-type models are used
and probabilities of interest are small.
