br Methodology This section presents the methodology employe

Methodology This section presents the methodology employed to verify the impact of racial segregation on the school proficiency gap between race groups. The presentation of this section is strongly based on Card and Rothstein (2007). Primarily, we assume that students’ proficiency on the standardized test depends on their own characteristics, on the racial composition of their school and on the characteristics of the other children who live in the same city. Formally, the model can be described as:where y represents the result in the standardized test for student i who belongs to racial group r and studies in school e, which is in city c, X is a vector containing characteristics of the student, N represents the share of black students enrolled in the school, and Q is a vector containing the average characteristics of all children of race r in city c. Therefore, coefficient β captures the direct effect of the fraction of black peers in the school on students’ proficiency on the standardized test. The error component from the previous equation can be written as follows:where μ represents the unobserved “abilities” of students of race r and city c, u is a shared error component for students of race r in school e and ν is an individual-level error for each student. This model contains a problem of non-randomness in the distribution of students to schools and neighborhoods in a city. In other words, order GSK-923295 tend to live in neighborhoods which have similar characteristics to their own (like race, income, schooling, etc.), so the students from these families usually study in schools nearby their homes. This fact would cause biases in the estimation of coefficient β. They would originate from the correlation between u, which is an unobserved effect of race per school, with the explanatory variables from Eq. (1). We can eliminate the effect by averaging the achievement outcomes of each race group to the city level. Thus, we can rewrite Eq. (1) such that the average score of racial group r in city c (y) is given by:where X represents the mean characteristics of students of race group r in city c, N represent the mean characteristics of the fraction of black students to peer groups of race-r students in city c and Q is the same from Eq. (1), since it already represents the mean for race group r in city c. Averaging eliminates the effects of within-city sorting, but it does not eliminate the problem of non-randomness between cities. That is to say, there still may be differences in the average unobserved “abilities” of race group students of each city, μ, that would lead to biases in the estimation of Eq. (3) in case μ, is correlated with explanatory variables. These biases would come from the possibility of people changing cities due to their professional success, causing a non-random distribution of races through cities. For example, more successful black people could look for “whiter” cities to live in, causing bias in the model. Any differences that are common across race groups in a city can be “differenced out” by comparing blacks and whites within the same city, that is to say, calculating the difference between races with city mean variables. This strategy permits us to compare cities with different shares of blacks and whites in a more successful way. Consider, for example, that more successful blacks want to live in cities that already have an equally successful white population, so blacks who are not as successful stay in cities with an equally low-success white population. Therefore, differencing race group results eliminates any city-wide unobserved “ability” factors that affect blacks and whites equally in the same city. This way:where r=1 represents blacks and r=2 represents whites, Δy is the difference in mean test scores between blacks and whites in the same city, ΔX is the difference in mean characteristics of students and ΔQ is the difference in the mean of all the children in the same city.