Overall Variance Decomposition
Past analyses of educational production functions have looked at both the level and the growth in achievement. The advantage of the growth formulation is that it eliminates a variety of confounding influences including the prior, and often unobserved, history of parental and school inputs. This formulation, frequently referred to as a value-added model, explicitly controls for variations in initial conditions when looking at how schools and other factors influence performance during, say, a given school year.4 However, standard value added models do not account for unobservable factors that affect the rate of acquisition of new knowledge, and it is the availability of test score measures for more than two grades that enables us to control for such influences through the use of fixed effects models for achievement growth.
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We use a fixed effects, value added framework to investigate mathematics and reading achievement in grades four, five and six. Equation (1) describes test score gain, AjgH, in a variance components framework: Achievement gain for student i in grade g in school s in year t is a function of a time invariant individual-specific growth component [yj, a school quality component that varies across grades (e.g. teacher quality, curriculum, or textbooks) [ugiJ, a school quality component that is constant across grades [6SJ (e.g. resources,, school leadership, or institutional structure), and an idiosyncratic random error [tigsJ The idiosyncratic error includes all measurement error in the tests. The individual-specific component could include individual ability, parental background, or neighborhoods to the extent each affects the growth in achievement; level effects are differenced out in calculating the gain in achievement.
where the subscript bgs refers to variation between grades and schools and the subscript bs refers to variation between schools. The nonrandom sorting of both students and teachers is made explicit in equations (2)-(4) by the three со variance terms and by allowing the between-school variance in у and и to differ from zero.