This paper uses detailed administrative data from one of the largest community colleges in the United States to quantify the extent to which academic performance depends on students being of similar race or ethnicity to their instructors. To address the concern of endogenous sorting, we use both student and classroom fixed effects and focus on those with limited course enrolment options. We also compare sensitivity in the results from using within versus across section instructor type variation. Given the computational complexity of the 2-way fixed effects model with a large set of fixed effects we rely on numerical algorithms that exploit the particular structure of the model's normal equations. We find that the performance gap in terms of class dropout and pass rates between white and minority students falls by roughly half when taught by a minority instructor. In models that allow for a full set of ethnic and racial interactions between students and instructors, we find African-American students perform particularly better when taught by African-American instructors.