Separation and rare events

When separation is a problem in binary dependent variable models, many researchers use Firth's penalized maximum likelihood in order to obtain finite estimates (Firth, 1993; Zorn, 2005; Rainey, 2016). In this paper, I show that this approach can lead to inferences in the opposite direction of the separation when the number of observations are sufficiently large and both the dependent and independent variables are rare events. As large datasets with rare events are frequently used in political science, such as dyadic data measuring interstate relations, a lack of awareness of this problem may lead to inferential issues. Simulations and an empirical illustration show that the use of independent weakly-informative prior distributions centered at zero, for example, the Cauchy prior suggested by Gelman et al. (2008), can avoid this issue. More generally, the results caution researchers to be aware of how the choice of prior interacts with the structure of their data, when estimating models in the presence of separation.