Arguably this is causal evidence that teacher expectations affect students' educational attainment

Teacher Expectations Matter. Nicholas W. Papageorge, Seth Gershenson, Kyung Min Kang. NBER Working Paper No. 25255, Nov 2018. https://www.nber.org/papers/w25255

Abstract: We develop and estimate a joint model of the education and teacher-expectation production functions that identifies both the distribution of biases in teacher expectations and the impact of those biases on student outcomes via self-fulfilling prophecies. Our approach leverages a unique feature of a nationally representative dataset: two teachers provided their educational expectations for each student. Identification of causal effects exploits teacher disagreements about the same student, an idea we formalize using lessons from the measurement error literature. We provide novel, arguably causal evidence that teacher expectations affect students' educational attainment: Estimates suggest an elasticity of college completion with respect to teachers' expectations of about 0.12. On average, teachers are overly optimistic about students' ability to complete a four-year college degree. However, the degree of over-optimism of white teachers is significantly larger for white students than for black students. This highlights a nuance that is frequently overlooked in discussions of biased beliefs: less biased (i.e., more accurate) beliefs can be counterproductive if there are positive returns to optimism or if there are socio-demographic gaps in the degree of teachers' optimism; we find evidence of both.