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This paper examines whether instructor demographic match affects student persistence and graduation, extending a large literature that has primarily focused on intensive-margin outcomes such as course performance, major choice, and field switching. While prior studies document that demographic alignment between students and instructors can influence academic trajectories within institutions, little causal evidence exists on whether these effects translate into the extensive margin of remaining enrolled or completing a degree. This distinction is central for policy, as institutions increasingly consider workforce diversification as a potential lever for improving retention. We leverage administrative data from a broad-access public university where first-semester course assignments are largely determined by the registrar, generating quasi-random exposure to instructor characteristics. Using this institutional feature, we implement an instrumental variables strategy based on leave-one-out, course-by-cohort variation in instructor gender and national origin. This design isolates exogenous variation in instructor demographics while addressing potential selection arising from student course choice or add/drop behavior. Across all specifications, we find no evidence that instructor demographic match affects persistence or graduation. Instrumental variables estimates are consistently close to zero, with all effects bounded within two percentage points under a formal equivalence testing framework. These results hold across multiple persistence measures, alternative specifications, and student subgroups. Small positive associations observed in OLS estimates disappear under instrumentation, consistent with selection bias rather than causal effects. We also explore mechanisms by examining course grades. While OLS results suggest modest positive effects of female instructors on grades, these patterns are fully explained by instructor-level grading differences rather than improvements in student learning. Instrumental variables estimates show no meaningful grade effects overall. However, we document an important secondary finding: in STEM courses, female instructors assign lower grades on average, with the effect concentrated among male students. Despite these shifts in grades, we find no corresponding changes in persistence outcomes, providing direct evidence against an academic-momentum channel linking instructor characteristics to retention. Taken together, the findings suggest that while instructor demographics may influence within-institution sorting, they do not affect whether students remain enrolled or graduate in this setting. This has direct implications for higher education policy: efforts to improve retention may be more effective when targeted toward interventions with established causal impacts, such as financial aid, advising, or early-alert systems, rather than relying on demographic composition alone.