The school-by-year fixed effects remove, in a very general way, not only school-specific performance trends but also idiosyncratic variation over time in school administration and in neighborhood and local eco- nomic conditions that likely affect mobility patterns, including such things as the introduction of new race-related school policies or the myriad changes documented to occur in “transitional neighborhoods.” For ex- ample, an economic shock that reduces neighborhood employment and income is absorbed and will not bias the estimates, nor will a shock to local school finances or the quality of the local school board, because each of these would affect all grades in a school. The seventh term, t , is the full three-way interaction between school, grade, and year; it cannot be included in our estimation because there
356 Hanushek et al. would be no variation left in racial composition across time or grades. 11 Ignoring this three-way interaction means that grade-specific variation over time in school average teacher quality or other achievement deter- minants could potentially bias the estimates if also correlated with racial composition. Yet, because of the nontrivial costs of switching schools and the fact that teacher assignments and other relevant aspects of school decisions are typically not known until immediately prior to the beginning of a school year, we do not expect changes over time in school and teacher quality for specific grades to be systematically linked with yearly changes in racial composition through parental behavioral responses. 12 We do in- clude information on teacher experience and class size, because these var- iables (which have been shown to be significant determinants of achieve- ment) might be incidentally linked to racial composition. The sensitivity of the racial composition estimates to these controls provides information about the likely effects of both observed and unobserved changes in school and teacher quality not accounted for by the included fixed effects. The variation used to identify the parameter estimates for racial com- position can be illustrated by considering a single school. (In a more general case with multiple schools, the coefficients would reflect the av- erage of these within school relationships across the sample). With mul- tiple years of data for one grade, we could use cohort differences in achievement and racial composition to identify the racial composition effect entirely within a grade of the school. However, unobserved changes over time could bias the estimates produced by this “school-by-grade fixed effects” model that removes any variation across schools and grades within the school. Alternatively, with multiple grades of data for a single year, we could use grade differences in achievement and racial composition 11 We restrict attention to variations in racial composition at the grade rather than classroom level. We believe there are conceptual reasons for doing this, but we also have no alternative because our data do not support classroom-specific analysis. The complication of any classroom analysis comes from selective place-
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