This paper studies the determinants of children's scores on tests of cognitive achievement in math and reading. Using rich longitudinal data on test scores, home environments, and schools, we implement alternative specifications for the production function for achievement and test their assumptions. We do not find support for commonly used restrictive models that assume test scores depend only on contemporaneous inputs or that assume conditioning in a lagged score captures the effects of all past inputs. Instead, the results show that both contemporaneous and lagged inputs matter in the production of current achievement and that it is important to allow for unobserved child-specific endowment effects and endogeneity of inputs. Using a specification that incorporates these features, we analyze sources of test score gaps between black, white and Hispanic children. The estimated model captures key patterns in the data, such as the widening of minority-white test score gaps with age, which is most pronounced for black children. The parameter estimates indicate that home inputs are significant determinants of achievement, while the effects of school inputs (as measured by pupil-teacher ratios and teacher salaries) are imprecisely measured in specifications that allow for unobserved child endowments. We find that equalizing home inputs at the average levels of white children would close the black-white test score gap by about 25% and close the Hispanic-white gap by about 30%.