Across grades within individual schools, the influence of nonrandom student sorting is likely to be minor and aggregation by grade will reduce the problem of measurement error. Equation (5) does not completely isolate the within school variation in teacher quality per se, because и includes all systematic aspects of between grade differences. Nonetheless, the within-school-between-grade variations of other school inputs and average student characteristics are likely to be small in comparison to the between school variations. More importantly, comparisons across cohorts at different grades and the removal of individual fixed effects can effectively isolate the contribution of within school differences in teacher quality to the total variation in achievement.
Table 1 reports the variance decomposition of math and reading test score gains for all 4th, 5th and 6th graders in Texas in the 1995 school year.1 The between school variance as a percentage of the total is 5.5 percent for math and 3.3 percent for reading, while the contribution of the between school and grade variance rises to 15.3 percent foi math and 8.9 percent for reading. Given that the within school variation in cohort average characteristics is likely to be small, this pattern of results makes a prima facie case that there are substantial within school differences in schools, likely arising from differences in teacher quality as well as other school inputs. Consequently, studies that use the between school variance component as an upper bound for the potential contribution of schooling may seriously underestimate the importance of schools.
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Even here, the potential contribution of school differences is likely understated. First, any measurement error in the tests enlarges the overall variance in test performance and is implicitly attributed to the within school variance. Second, the aggregation across teachers within a grade also implicitly attributes within grade variation in teachers to the within school variance. While neither of these factors has substantial effects on standard regression analyses, each contributes to understating the estimated importance of systematic school and teacher factors.