Bernoulli Regression Models: Revisiting the specification of statistical models with binary dependent variables

The problem of statistical model specification was initially raised by R.A. Fisher who understood it as an informed selection whose adequacy is testable a posteriori. This, however, raised the problem of how to blend substantive subject matter with statistical information in empirical modeling, which is congruous when the adequacy of the statistical model is secured first. Statistical adequacy can be ensured by taking a purely probabilistic construal of statistical models ab initio, and then embed the structural model carrying the substantive information, in its context. Latent variable and generalized linear modeling approaches for discrete choice models may not adequately capture both sources of information. This paper is re-visits the specification of conditional statistical models with binary dependent variables using the probabilistic reduction approach. The primary contribution of the paper is a general derivation, presentation and application of the Bernoulli Regression model. A statistical modeling framework is provided for specifying statistically adequate Bernoulli Regression Models that allows the statistical and substantive information to play a crucial role.