Authorship attribution techniques are increasingly being used in online contexts such as sock puppet detection, malicious account linking, and cross-platform account linking. Yet, it is unknown whether these models perform equitably across different demographic groups. Bias in such techniques could lead to false accusations, account banning, and privacy violations disproportionately impacting users from certain demographics. In this paper, we systematically audit authorship attribution for bias with respect to gender, native language, and age. We evaluate fairness in 3 ways. First, we evaluate how the proportion of users with a certain demographic characteristic impacts the overall classifier performance. Second, we evaluate if a user's demographic characteristics influence the probability that their texts are misclassified. Our analysis indicates that authorship attribution does not demonstrate bias across demographic groups in the closed-world setting. Third, we evaluate the types of errors that occur when the true author is removed from the suspect set, thereby forcing the classifier to choose an incorrect author. Unlike the first two settings, this analysis demonstrates a tendency to attribute authorship to users who share the same demographic characteristic as the true author. Crucially, these errors do not only include texts that deviate from a user's usual style, but also those that are very close to the author's average. Our results highlight that though a model may appear fair in the closed-world setting for a performant classifier, this does not guarantee fairness when errors are inevitable.