We present a method to estimate and predict fixed effects in a panel probit model when N is large and T is small, and when there is a high proportion of individual units without variation in the binary response. Our approach builds on a bias-reduction method originally developed by Kosmidis and Firth (2009) for cross-section data. In contrast to other estimators, our approach ensures that predicted fixed effects are finite in all cases. Results from a simulation study document favorable properties in terms of bias and mean squared error. The estimator is applied to predict period-specific fixed effects for the extensive margin of health care utilization (any visit to a doctor during the previous three months), using German data for 2000-2014. We find a negative correlation between fixed effects and observed characteristics. Although there is some within-individual variation in fixed effects over sub-periods, the between-variation is four times as large.