Pregibit: a family of binary choice models

The pregibit binary choice model is built on a distribution that allows symmetry or asymmetry and thick tails, thin tails, or no tails. Thus, the model is much more flexible than the traditional binary choice models: pregibit nests logit, approximately nests probit, loglog, cloglog, and gosset models and incorporates the linear probability model. Greater flexibility allows a more accurate estimation of the data-generating process, including asymmetric and thick/thin tails. We prove that the maximum likelihood estimator of the pregibit model is consistent and asymptotically normally distributed. A Monte Carlo analysis and two real-world examples show that probit and logit estimates may show misleading evidence in cases where a pregibit model is statistically preferred. One example concerns enrollment in post-secondary education in Belgium: The pregibit estimate of the enrollment gap between Belgian natives and foreign students is 50 % larger, and the type of high school (general, technical, catholic) is more influential. The second example examines the outcome of mortgage applications in the USA. Here, pregibit estimates assign a stronger role to variables that measure the financial strength of mortgage applicants and a weaker role to demographic characteristics including minority status. More importantly, the distribution of the disturbances proves to be seriously skewed: Pregibit indicates that even high-risk applicants (with a probit acceptance probability of nearly 0) have a positive probability of getting their mortgage application approved. Apparently, mortgage officers are more inclined to uncover reasons to make a mortgage deal than to send clients away empty-handed.